Mechanics for a New Millennium
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Mechanics for a New Millennium

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Mechanics for a New Millennium Proceedings of the 20th International Congress of Theoretical and Applied Mechanics Chicago, Illinois, USA 27 August – 2 September 2000

Edited by

Hassan Aref and James W. Phillips Department of Theoretical and Applied Mechanics University of Illinois at Urbana-Champaign

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

0-306-46956-1 0-792-37156-9

©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2000 Kluwer Academic / Plenum Publishers New York All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:

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PREFACE The 20th International Congress of Theoretical and Applied Mechanics, ICTAM 2000, took place August 27 – September 2, 2000, in the conference facilities of the Chicago Marriott Downtown on Michigan Avenue in Chicago, Illinois, USA. The Congress was invited by the US National Academy of Sciences on the recommendation of its National Committee on Theoretical and Applied Mechanics. A consortium of university departments and programs acted as the local hosts. The undersigned served, respectively, as President and Secretary-General of the Congress. In this Proceedings volume we have attempted to capture the excitement of that memorable week. We have laid out the book so that it, roughly, tracks the events during the week of the Congress in much the order in which they occurred. We have given addresses and reports in context, although sometimes with embellishments to provide more data than could reasonably be presented orally. We have tried to reproduce essentials of the Opening and Closing ceremonies for the benefit of those who could not attend one or the other. We have tried to assemble an attractive volume that will have lasting value by adding an extensive name index and a briefer keyword index, and by exercising considerable care in the consistency of the layout of the full manuscripts. The international congresses of mechanics are major events of the field—we compare them to the Olympic Games in sports. It is essential that future generations can assess exactly where our field stood in the year 2000. We hope this volume—pre-ordered at the Congress in record numbers—will be found useful for many years to come. Hassan Aref

James W. Phillips

President of ICTAM 2000

Secretary-General of ICTAM 2000

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TABLE OF CONTENTS Committees, Bureau of IUTAM (1996–2000), Congress Committee of IUTAM, Local Organizing Committee, Chairs of mini-symposia and pre-nominated sessions, vii Opening ceremony, xi Congress poster, xxvii Program at a glance, xxxi Scientific program, xxxix Invited papers, 1 JAMES R. RICE: New perspectives on crack and fault dynamics, 1 RICHARD M. CHRISTENSEN: A survey of and evaluation methodology for fiber composite material failure theories , 25 JEAN-LOUIS CHABOCHE: On constitutive and damage modeling in metal matrix composites, 41 LORNA J. GIBSON: Metallic foams: Structure, properties, and applications, 57 MARTIN NEMER, JERZY BLAWZDZIEWICZ, and MICHAEL LOEWENBERG: Linear viscoelasticity of concentrated emulsions, 75 PETER A. DAVIDSON: Aluminum: Approaching the new millennium, 85 TAKEHIKO TOH and EIICHI TAKEUCHI: Electromagnetic phenomena in steel continuous casting, 99 RONALD J. ANDERSON, JOHN A. ELKINS, and BARRIE V. BRICKLE: Rail vehicle dynamics for the 21st century, 113 ROBIN S. SHARP: Fundamentals of the lateral dynamics of road vehicles, 127 MASATO ABE and J. KARL HEDRICK: A mechatronics approach to advanced vehicle control system design, 147 E M M A N U E L V I L L E R M A U X : Mixing: Kinetics and geometry, 165 S TÉPHANE R O U X and F ARHANG R ADJAÏ: Statistical approach to the mechanical behavior of granular media, 181 GEORGE J. DVORAK: Damage analysis and prevention in composite materials, 197 JOHN S. WALKER: Electromagnetic phenomena in crystal growth, 211 WILLI KORTÜM, WERNER O. SCHIEHLEN, and MARTIN ARNOLD: Software tools: From multibody system analysis to vehicle system dynamics, 225

iv MICHEL Y. LOUGE, JAMES T. JENKINS, HAITAO XU, AND BIRGIR Ö. ARNARSON: Granular segregation in collisional shearing flows, 239

PEDRO PONTE CASTAÑEDA and PIERRE SUQUET: Nonlinear composites and microstructure evolution, 253 DENIS L. WEAIRE and STEFAN HUTZLER: Hard problems with soft materials: The mechanics of foams, 275 HAMDA BENHADID: Magnetohydrodynamic damped buoyancy-driven convection, 289

TIMOTHY J. GORDON: Adaptive, nonlinear, and learning techniques for the control of vehicle ride dynamics, 307

P AUL E. D IMOTAKIS : Recent advances in turbulent mixing, 327 ISAAC G OLDHIRSCH : Kinetic and continuum descriptions of granular flows, 345

VLADIMIR A. PALMOV: Stationary waves in elasto–plastic and visco–elasto–plastic bodies, 359

ERWIN STEIN, STEPHAN OHNIMUS, and MARCUS RÜTER: Hierarchical model- and discretization-error estimation of elasto–plastic structures, 373 STEPHEN J. COWLEY: Laminar boundary-layer theory: A 20th century paradox?, 389 ERIK VAN DER GIESSEN: Plasticity in the 21st century, 413

DAVID M. MCQUEEN AND CHARLES S. PESKIN: Heart simulation by an immersed boundary method with formal second-order accuracy and reduced numerical viscosity, 429 S HIGEO K IDA : Vortical structure of turbulence, 445

R. NARAYANA IYENGAR: Probabilistic methods in earthquake engineering, 457 GEDEON DAGAN: Effective, equivalent, and apparent properties of heterogeneous media, 473 A MABLE L IÑÁN : Diffusion-controlled combustion, 487

SUBRA SURESH: Nanomechanics and micromechanics of thin films, graded coatings, and mechanical/nonmechanical systems, 503 OLE SIGMUND: Optimum design of microelectromechanical systems, 505

H. KEITH MOFFATT: Local and global perspectives in fluid dynamics, 521

Closing ceremony, 541 List of attendees, 553 List of exhibitors, 561 Index to authors, co-authors, and session chairs, 563 Subject index, 581

COMMITTEES Bureau of IUTAM Jüri Engelbrecht (Estonia) L. Ben Freund (USA), Treasurer Michael A. Hayes (Ireland), Secretary-General H. Keith Moffatt (UK)

Werner O. Schiehlen (Germany), President Tomomasa Tatsumi (Japan) Ren Wang (China) Leen van Wijngaarden (The Netherlands), Vice-President

Congress Committee of IUTAM Andreas Acrivos* (USA) 2002† Hassan Aref (USA) 2000 Sol R. Bodner* (Israel) 2000 David B. Bogy (USA) 2000 Jüri Engelbrecht (Estonia) 2000 Norman A. Fleck (UK) 2002 L. Ben Freund (USA) 2000 Graham M. L. Gladwell (Canada) 2000 Michael A. Hayes (Ireland) 2002, Rep. of ISIMM Tatsuo Inoue (Japan) 2002, Rep. of ICM Javier Jiménez (Spain) 2002 Alfred Kluwick (Austria) 2002 Anthony N. Kounadis (Greece) 2002 Y. H. Ku (USA) Peter Lugner (Austria) 2000, Rep. of IAVSD Bengt Lundberg (Sweden) 2000 Gerd E. A. Meier (Germany) 2000 H. Keith Moffatt (UK) 2000 René Moreau* (France) 2000

Bahadur C. Nakra (India) 2002 Robert I. Nigmatulin (Russia) 2000 J. Tinsley Oden (USA) 2002, Rep. of IACM Neils Olhoff* (Denmark) 2000, Secretary J. R. Anthony Pearson (UK) 2000, Rep. of ICR Timothy J. Pedley* (UK) 2000 Mahir B. Sayir (Switzerland) 2000 Werner O. Schiehlen* (Germany) 2000, Chairman Bernhard A. Schrefler (Italy) 2002 Kazimierz Sobczyk (Poland) 2002 Pierre Suquet (France) 2000 Roger I. Tanner (Australia) 2002 Tomomasa Tatsumi (Japan) 2000 Eiichi Watanabe (Japan) 2002 Leen van Wijngaarden (The Netherlands) 2000 Fen-Gan Zhuang (China) 2000

*Members of the Executive Committee (1996–2000) †2000, 2002—Year, where stated, indicates end of term (applies to members elected after 1972)

Local Organizing Committee‡ Carol J. Porter, Administrative Secretary Hassan Aref, President of ICTAM 2000 James W. Phillips, Secretary-General of On-site support: Patricia J. Franzen ICTAM 2000 Chicago Marriott Downtown: Willie Clay, Robin Enke James C. Onderdonk, Head, Conferences and Institutes Science press contact: Susan K. Mumm ‡Except as noted, all members of the Local Organizing Committee are affiliated with the University of Illinois at Urbana-Champaign, Urbana, Ill.

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CHAIRS OF MINI-SYMPOSIA AND PRE-NOMINATED SESSIONS Mini-symposium chairs Mini-symposium

Chair

Co-chair

Electromagnetic René Moreau (Prance) processing of materials (jointly with HYDROMAG)

Nagy El-Kaddah (USA)

Granular flows

Stuart B. Savage (Canada)

Robert P. Behringer (USA)

Turbulent mixing

Emil J. Hopfinger (Prance)

Paul E. Dimotakis (USA)

Damage and failure of composites

Zvi Hashin (Israel)

George J. Dvorak (USA)

Mechanics of foams and Andrew M. Kraynik (USA) Norman A. Fleck (UK) cellular materials Vehicle systems Peter Lugner (Austria) J. Karl Hedrick (USA) dynamics (jointly with IAVSD)

Pre-nominated session chairs Topics in fluid mechanics Pre-nominated session

Chair

Co-chair

Biological fluid dynamics Kazuo Tanishita (Japan)

Timothy J. Pedley (UK)

Boundary layers

Hans H. Fernholz (Germany)

Anatoly I. Ruban (UK/Russia)

Combustion and flames

John D. Buckmaster (USA) Moshe Matalon (USA)

Complex and smart fluids Compressible flow

Eric S. G. Shaqfeh (USA)

Roger I. Tanner (Australia)

John-Paul Bonnet (France)

Alexander J. Smits (USA)

Computational fluid John Kim (USA) dynamics (jointly with IACM)

Alien J. Baker (USA)

Convective phenomena

Friedrich H. Busse (Germany)

Nigel O. Weiss (UK)

Drops and bubbles

John R. Blake (UK)

Andrea Prosperetti (USA)

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Environmental fluid

Paul F. Linden (USA)

Julian C. R. Hunt (UK)

mechanics Experimental methods in Henrik Alfredsson (Sweden) Helmut Eckelmann fluid mechanics (Germany) Flow control

Mohamed Gad-el-Hak

Roddam Narasimha (India)

(USA) Flow in porous media

George M. Homsy (USA)

Joel Koplik (USA)

Flow instability and

Sidney Leibovich (USA)

Michio Nishioka (Japan)

Yves Couder (France)

Leonard W. Schwartz

transition

Flows in thin films

(USA) Fluid mechanics of materials processing

Wilhelm Schneider (Austria)

J. Iwan D. Alexander

Geophysical fluid

Kolumban Hutter (Germany)

Peter Lynch (Ireland)

Low-Reynolds-number flow

Howard A. Stone (USA)

Antonio Delgado (Germany)

Microfluid dynamics

John F. Brady (USA)

Leen van Wijngaarden (Netherlands)

Multiphase flows

Grétar Tryggvason (USA)

L. Gary Leal (USA)

Topological fluid

Arkady B. Tsinober (Israel) Richard B. Pelz (USA)

mechanics Turbulence

Olivier Métais (France)

dynamics

(USA)

Katepalli R. Sreenivasan

(USA) Vortex dynamics

Dale I. Pullin (USA)

Viatcheslav V. Meleshko (Ukraine)

Waves

Roger H. J. Grimshaw (Australia)

Chiang C. Mei (USA)

Topics in solid mechanics Pre-nominated session

Chair(s)

Co-chair(s)

Biological solid mechanics Computational solid mechanics (jointly with IACM)

Stephen C. Cowin (USA)

Ivars Knets (Latvia)

Pierre Ladevèze (France), Walter Wunderlich (Germany)

Computational strategies J. Tinsley Oden (USA) for multiscale phenomena in mechanics (jointly with IACM)

Erwin Stein (Germany)

Chairs of mini-symposia and pre-nominated sessions

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—

Contact and friction problems (jointly with IAVSD)

Anders Klarbring

Control of structures

Dick H. van Campen (Netherlands)

Felix L. Chernousko (Russia)

Damage mechanics

Dusan Krajcinovic (USA)

Jean Lemaitre (France)

Dynamic plasticity of structures

Stephen R. Reid (UK)

Victor P. W. Shim (Singapore)

Elasticity

Raymond W. Ogden (UK)

James K. Knowles (USA)

(Sweden), James R. Barber (USA)

Experimental methods in Isaac M. Daniel (USA) solid mechanics Fatigue Robert O. Ritchie (USA)

Jörg F. Kalthoff (Germany) Keisuke Tanaka (Japan) —

Fracture and crack mechanics (jointly with ICF)

Bhushan L. Karihaloo

Functionally graded materials

Marek-Jerzy Pindera

Impact and wave propagation

Rodney J. Clifton (USA)

Anders Boström (Sweden)

Material instabilities

Viggo Tvergaard

—

(UK), Jean-Baptiste Leblond (France)

Glaucio H. Paulino (USA)

(USA)

(Denmark), Alan

Needleman (USA) Alain Molinari (France)

Franz-Josef Ulm (USA)

Mechanics of porous materials

Alberto Carpinteri (Italy)

Wolfgang Ehlers (Germany)

Mechanics of thin films and nanostructures

L. Ben Freund (USA)

Jürg Dual (Switzerland)

Multibody dynamics

Werner O. Schiehlen (Germany)

John J. McPhee (Canada)

Plasticity and

Zenon Mróz (Poland),

—

Mechanics of phase transformations (jointly with IACM)

viscoplasticity

Bertil Storåkers (Sweden) Herbert A. Mang (Austria)

Arthur W. Leissa (USA)

Rock mechanics and geomechanics

Bernhard A. Schrefler (Italy)

Ioannis Vardoulakis (Greece)

Smart materials and

Richard D. James (USA)

Yuji Matsuzaki (Japan)

Friedrich G. Pfeiffer (Germany)

David B. Bogy (USA), Tatsuo Inoue (Japan)

Plates and shells (jointly

with IACM)

structures

Solid mechanics in manufacturing

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Stability of structures

Anthony N. Kounadis

George J. Simitses (USA)

(Greece)

Raphael T. Haftka (USA)

Structural optimization (jointly with ISSMO)

Martin P. Bendsøe (Denmark)

Structural vibrations

Peter Hagedorn (Germany), — James F. Doyle (USA)

Viscoelasticity and creep

John L. Bassani (USA)

Nobutada Ohno (Japan)

Topics involving both fluid mechanics and solid mechanics Pre-nominated session

Chair(s)

Co-chair(s)

Acoustics

Ann P. Dowling (UK)

Allan D. Pierce (USA), Leif Bjørnø (Denmark)

Cellular and molecular mechanics

Christopher R. Calladine

Chaos in fluid and solid

Tom Mullin (UK)

Sheldon Weinbaum (USA)

(UK)

Francis C. Moon (USA)

mechanics

Continuum mechanics

Gérard A. Maugiu (France) James Casey (USA)

Fluid–structure interaction

Michael P. Païdoussis (Canada), P. Terndrup

John Grue (Norway)

Pedersen (Denmark) Microgravity mechanics

Simon S. Ostrach (USA)

Hans J. Rath (Germany), Vadim I. Polezhaev

(Russia)

OPENING CEREMONY Monday, August 28, 2000, 10:00 Prof. Hassan Aref, President of ICTAM 2000 “Esteemed colleagues, distinguished guests, ladies and gentlemen, “Welcome to the 20th International Congress of Theoretical and Applied Mechanics, or ICTAM 2000 as we call it. The city of Chicago has an advertising slogan. It says: ‘We are glad you are here!’ I have to concur: We are really, really glad that you are all finally here! Unless you have recently organized a conference of this general magnitude and complexity yourself, you can’t begin to fathom my pleasure and relief that you are really, finally all here. You are registered, we have managed to put all the many papers into a program, your lodging is taken care of, even the audio-visuals seem to be working. This is a truly great moment, believe me!

“To get things off to a proper start, I will first call upon Prof. Werner Schiehlen of the University of Stuttgart, in Germany, the current president of the International Union of Theoretical and Applied Mechanics, to formally open the Congress. Please welcome Professor Schiehlen to the podium.” Prof. Werner O. Schiehlen, President of IUTAM “Dear Mr. President Stukel, dear members of the Host Consortium, dear colleagues from all over the world, ladies and gentlemen, “Mathematics and physics, as well as civil engineering and mechanical engineering, are the roots of mechanics. Theoretical mechanics uses mathematical methods and physical principles, today enriched with computational tools. Applied mechanics deals with constructions and machines subject to new designs and materials. “While theoretical mechanics was established as a branch of science more than 300 years ago by Isaac Newton, applied mechanics dates back to antiquity with famous names like Archimedes. However, the ancient Greeks built not only grandiose temples, mechanically well designed, but they invented the Olympic Games, too. Let us have a short look at Webster’s dictionary. The entry on Olympics reads as follows: ‘An ancient Panhellenic festival held every fourth year and made up of con-

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Prof. Hassan Aref, President of ICTAM 2000, welcomes participants to the Congress

in the Grand Ballroom of the Chicago Marriott Downtown.

tests of sports, music, and literature with the victor’s prize, a crown of wild olive.’ “Thus, the ancient games were not restricted to sports. They involved

literature as well. Literature, in general, comprises belles-lettres and textbooks, which may be also scientific. And there must be a contest. And there must be a prize. All these are essential features of the 20th International Congress of Theoretical and Applied Mechanics, and enable us to call it the Olympics of Mechanics. “But how is the contest performed? What about the prize? Well, it is known in the mechanics community that the International Papers Committee of the mechanics congresses is very selective, and nearly half of the submitted papers are rejected. Nevertheless, the number of submissions is increasing from congress to congress. This means a real contest of scientists. “On the other hand, since 1988 the Bureau of IUTAM has awarded prizes for outstanding presentations by young scientists. In particular, at this millennium congress, the three prizes are especially attractive by the ceramic plaque designed and manufactured by Illinois artist SitiMariah Jackson. The Hellenic prize of a crown of wild olive is finished as an artistic plaque in a unique way to be awarded to young victors of mechanics. “Ladies and gentlemen, it is my great honor to call the 20th International Congress on Theoretical and Applied Mechanics to order and to welcome all of you here in Chicago on behalf of IUTAM. The Marriott on Chicago’s Magnificent Mile is an excellent location for a world congress and we are looking forward to learn about the latest developments in mechanics in a most pleasant environment. After three years of extensive preparation under the guidance of Professor Hassan Aref, your active participation in the Congress will crown it with the success we all expect.

Opening Ceremony

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“I have to admit that the Congress is also the central scientific event of IUTAM, opening the door to the new millennium for mechanics. Despite its long history, mechanics is a very lively subject today. The Mechanics Calender maintained for IUTAM by the University of Illinois at UrbanaChampaign lists for the year 2000 altogether 99 scientific meetings in 24 different countries. “In addition, to look back and ahead IUTAM is publishing a report entitled Mechanics at the Turn of the Century. This report is the result of an initiative of the Bureau of IUTAM to provide some landmarks on the development in mechanics during the 20th century, to report on the 50 years of impulse to mechanics by IUTAM, and to look ahead on a very personal basis to future areas of research, and also to show the broad international involvement of scientists in IUTAM in recent years. “Ladies and gentlemen, enjoy a world congress in a fascinating city on a most challenging subject: Theoretical and Applied Mechanics.”

Professor Aref “Thank you President Schiehlen. ICTAM 2000 came about through a lengthy, competitive proposal process. In fact, last night during the first meeting of the Congress Committee we began the similar process for the next congress in 2004. The formal invitation to host the 20th congress in Chicago was issued by the U.S. National Academy of Sciences on the recommendation of the U.S. National Committee for Theoretical and Applied Mechanics. While the National Academy, then, is the official sponsor, it was a consortium of universities, most of them in the Midwest, that really stepped up to the plate and provided the base of financial backing necessary to embark on this project. The host consortium includes 13 university groups with strong interests in mechanics, including, of course, the two remaining departments in the U.S. that have the name ‘Theoretical and Applied Mechanics’. The university sponsors are listed in the accompanying table. “Several professional societies, all but one represented on the National Committee for Theoretical and Applied Mechanics, also provided support to ICTAM 2000. The financial contributions of the professional societies were used to subsidize our banquet at the Museum of Science and Industry on Thursday so that we could induce as many of you as possible to attend. I am pleased to report that this incentive worked wonderfully and about 1000 delegates and accompanying persons will attend the banquet. I would like to acknowledge them, again in alphabetical order. [See accompanying table.] “Our stalwart supporters within the Federal government provided much appreciated funding for participant support to younger attendees

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and to attendees from countries with weak financial support structures for science. I should like to thank them. [See accompanying table.]

Opening Ceremony

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“As a special feature at this Congress we are conducting a Science Teachers Day on Thursday, August 31. Some 65 teachers from the Northern half of Illinois will come to the Congress and be treated to a special program in the morning before mixing with the rest of us in the afternoon. Illinois artist Billy Morrow Jackson and I will tell you and the teachers a bit about the creation of the Congress poster during lunch on that day in Ballroom I. Many of the teachers will be joining us in the evening for the banquet. This special event was generously underwritten by the Chicago Engineers Foundation of the Union League

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Prof. Werner O. Schiehlen, President of IUTAM (left), and Prof. James J. Stukel, President of the University of Illinois, welcome participants to the Congress.

Club of Chicago. Jerry Moyar, an officer of the Foundation and an alum of my department, was responsible for securing this funding for us. “Finally, I should like to acknowledge a few individuals who contributed personal financial support directly or indirectly to the Congress. They are Professors Andy Acrivos, Stanley Berger, Tom Lundgren, and Franz Ziegler. Thank you all for your generous financial support. “My university, the University of Illinois, is represented by two campuses in the host consortium—my home campus in Urbana and the Chicago campus not too far from here. The president of the University of Illinois system, Professor James J. Stukel, whose faculty background is in mechanical engineering, has taken a personal interest in ICTAM 2000 and has generously underwritten the Welcome Reception that you will enjoy this evening. President Stukel is with us this morning, and he will now welcome you on behalf of the academic institutions hosting ICTAM 2000. Please welcome President James Stukel to the podium.”

Prof. James J. Stukel, President of the University of Illinois “Let me add my welcome to everyone here attending this conference on behalf of the host consortium universities and the academic community. As I looked at this final program, I realized that this really is going to be a very exciting conference. As a mechanical engineer, I fully appreciate the challenges of basic research in the mechanical sciences dealing with elasticity, fracture, plasticity, fatigue, turbulence, and combustion. As president of a large, public research university, I appreciate the contributions of the mechanical sciences to other academic

Opening Ceremony

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disciplines. Examples include research on the mechanics of emulsions, porous materials, bio-materials, thin films, composites, and micro- and nano-technologies. As a citizen of the world, I benefit from the technology transfer from universities, which result in a better understanding of things like oceanic mixing, a better management of the world’s water resources, making available the possibility of space exploration, biotechnology, and recent advances in micro- and nano-technologies. We as a society and as a country would not be experiencing our robust economy were it not for the transfer of technology from the universities to the industrial sector. The same is true around the world. “Another wonderful event that I noticed on the agenda, which I think is really first rate, is bringing world class scientists in contact with high school science teachers. I can’t think of a more inspirational thing for a science teacher than to come in contact with the people who are here today. They need that inspiration in this country, and I suspect it is the case worldwide. So, I would like to thank all the scientists for taking part in this very important program. “I am also very, very proud to welcome you to one of the great cities in the US, in my view one of the great cities of the world. It is a city with superb architecture, wonderful museums, a beautiful arid accessible lake-front, and world class theater and music establishments. I certainly hope you have the chance to take advantage of these wonderful opportunities here in Chicago. And, speaking of art, as we have just indicated, I hope you have enjoyed, as much as I did, the watercolor by Billy Morrow Jackson, entitled ‘Meters of Motion’, depicting the history of your discipline. I thought it was a wonderful piece of art. Don’t you agree? “The Governor has also recognized this wonderful event and how important it is to the State of Illinois. Let me read the Proclamation as it is unveiled by Professor Aref: ‘Whereas the 20th International Congress of Theoretical and Applied Mechanics, ICTAM2000, is being held in Chicago, Illinois, during the week of August 28, 2000, and ‘Whereas, the international congresses of the International Union of Theoretical and Applied Mechanics have been ongoing for more than 75 years, and have visited major cities of the world including several sister cities of Chicago, and ‘Whereas, prior congresses have been held in the United States of America on only two prior occasions in 1938 and in 1968, and ‘Whereas, ICTAM 2000 is invited by the U.S. National Academy of Sciences, the pre-eminent scientific body of this nation, and ‘Whereas, the host university consortium includes several illustrious institutions of higher learning in the State of Illinois, to wit University of Illinois at UrbanaChampaign, University of Illinois at Chicago, University of Chicago, and Illinois Institute of Technology, and

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Left: Proclamation of Theoretical and Applied Mechanics Day in Illinois, 28 August 2001; right: Illinois artist Billy Morrow Jackson (center) with the ICTAM 2000 poster based on his watercolor Meters of Motion, accompanied by his wife, Siti-Mariah Jackson (right), creator of the IUTAM Bureau Prize plaques, and Dr. Gerald J. Moyar, representing the Chicago Engineers Foundation of the Urban League Club of Chicago, sponsors of the ICTAM 2000 Science Teachers Day on

31 August 2000.

‘Whereas, the attendees at ICTAM 2000 represent a gathering of leading researchers and scholars in the mechanical sciences from the international scientific community encompassing more than 50 nations, ‘Therefore, I, George H. Ryan, Governor of the State of Illinois, proclaim August 28, 2000, Theoretical and Applied Mechanics Day in Illinois. In witness whereof, I have hereunto set my hand and caused the Great Seal of the State of Illinois to be affixed. Done at the Capitol, in the city of Springfield, this twenty-first day of August, in the Year of Our Lord two thousand, and of the State of Illinois the one hundred and eighty second. ‘Signed George H. Ryan, Governor.’

“And lastly, I’d like to invite you to the reception this evening to eat on our dollar! So, enjoy it. I know you will have a wonderful conference.” Professor Aref

“Thank you President Stukel, and thanks to Governor Ryan for this nice show of support for our congress and our field. Today, Monday, August 28, 2000, is officially Theoretical and Applied Mechanics Day in the State of Illinois! The declaration is here for your inspection. We will move it to the foyer area following the Opening Ceremony. “From the consortium of universities and from the State of Illinois let me turn next to the city of Chicago. What links are there between our science of mechanics and the city of Chicago? “Well, Chicago has always been a kind of junction within the United States, a hub through which goods passed, by rail and by truck and,

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of course, these days by airplane. Chicago is called the city of the big shoulders, the tool maker, and the nation’s freight handler in the famous 1916 poem by Pulitzer Prize winning poet and biographer Carl Sandburg. This is certainly a most appropriate location for a mini-symposium on vehicle dynamics! “As you approach the city, you immediately see, of course, those huge manifestations of mechanics-in-action called skyscrapers. The skyscrapers in Chicago are among the tallest in the world. The Sears Tower, completed in 1974, with 110 stories, is 1450 ft, or 443 meters, tall to the

so-called structural or architectural top with only the Petronas Towers in Kuala Lumpur, built some 25 years later, being taller. However, the highest occupied floor in the Sears Tower is at 1431 ft, some 200 feet higher than the corresponding point in the Petronas Towers. The top of the roof is also higher, as is the tip of the spiral antenna on the roof. Even taller buildings were at various times planned for Chicago but never built. Between the structural engineers and the architects the rebuilding of Chicago after the Great Fire of 1871, when wooden buildings were banned and land became ever more expensive, produced the first city of skyscrapers. Design pioneers, such as Daniel Burnham and Louis Sullivan, who led what became known worldwide as the Chicago School of Architecture, gave Chicago its unique skyline. As the famous architect Frank Lloyd Wright said, ‘Beautiful buildings are more than scientific. They are true organisms, spiritually conceived; works of art, using the best technology by inspiration rather than the idiosyncrasies of mere taste or any averaging by the committee mind.’ “Since bio-mechanics is prominently featured at ICTAM 2000, I might mention that Chicago’s firm of William Wrigley Jr. is the world’s largest chewing gum manufacturer. “The World’s Columbian Exposition of 1893 in Chicago introduced a couple of mechanical firsts associated with the City. On a small scale the idea of a slide fastener or what we today would call a zipper was exhibited by Whitcomb L. Judson. This invention, later refined by others, was to become a mainstay of clothing for at least the next century. On a large scale, the Ferris Wheel was the engineering highlight of the Exposition and one of the most pervasive, lasting influences of the 1893 fair. The Ferris Wheel was Chicago’s answer to the Eiffel Tower, the landmark of the 1889 Paris exhibition. The wheel was created by bridge builder George W. Ferris. Supported by two 140 foot steel towers, its 45 foot axle was the largest single piece of forged steel at the time in the world. The wheel itself had a diameter of 250 feet and was powered by two 1000 horsepower reversible engines . . .

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Ray Wild as Archimedes (left) and Craig Spidle as Galileo Galilei.

Isaac Newton has entered on the far side of the room and is negotiating a seat with members of the audience. Newton: “Oh! That’s a nice seat! Do you mind if I sit right up there?” Aref: “So, these are some of the historical links between mechanics and Chicago. Now, continuing this grand tradition, we celebrate ICTAM 2000 in the windy city ... Excuse me, sir . . . ” Newton ignores him; continues wandering towards the front of the room. Aref: Newton:

“Sir! Are you registered for the Congress? Sir?” “Who are you? And what, may I inquire, is this gathering?” Galileo has entered from a door on the opposite side of the Ballroom. Galileo: “Isaac! My dearest friend! What are you doing here?” Newton: “Finally a familiar face! Where are we? Who are all these people?” Archimedes enters in a rush, half-running, waving the Congress program. Archimedes: “Eureka! Eureka! We are at an international congress of mechanics!” Galileo: “Slow down my friend!” Newton: “Of what?” Archimedes: “We’re at the 20th International Congress of Theoretical and Applied Mechanics, ICTAM 2000 . . . ”

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Genevieve VenJohnson as ‘Chicago’ (left) and Robert Ayres as Isaac Newton.

Galileo:

“Icky-what?”

Archimedes:

“ . . . a worldwide conference to map out advances in the mechanical sciences for the new millennium. These congresses are held every four years, just like the Olympic

Games.” Galileo: Newton:

“Theoretical and Applied?” “I have always preferred the terms ‘Rational’ and ‘Practical’.”

Archimedes:

“International!”

Newton:

“But what are all these people doing here? In my day it took only a few of us to discover the universal law of gravitation, determine the figure of the Earth and the orbits of the planets, write down my first, second and third laws, of course...”

Galileo:

“Your first law was really mine, Isaac!”

Newton: Galileo:

“Your one and only, I might add!” “Wait a minute, per favore! I started dialogues on two new sciences, and I spent a lot of time paving the way for you, arguing the Earth actually moves...”

Archimedes: “Gentlemen, gentlemen, please! My dear friends, this is all irrelevant. Look at what we have wrought: Hundreds of natural philosophers improving on our discoveries, making new ones, debating all kinds of issues. Look at this massive program of presentations.”

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Newton:

“What? There is a paper here on non-Newtonian fluid mechanics. Preposterous!” Galileo: “I see a paper here that uses Galilean invariance—and one that mentions your principle, Archimedes.” Newton: “Non-Newtonian fluid mechanics! Hypothesis non fingo!” Chicago enters and stands behind the podium in relative darkness. Newton: “Where are we anyway?” Galileo: “Not Roma!” Archimedes: “Certainly not Syracusa. Though, if you gave me a fixed fulcrum and a place to stand I can move you there!” Newton: “Well, it is definitely not grand old London!” Archimedes: “It’s a new world! A city called Chick-a-goo!” Galileo: “Chick-a-goo?” Newton: “Chick-a-goo?” Chicago is now at the podium with a spotlight directed at her. Chicago Chicago! HOG Butcher for the World, Tool Maker, Stacker of Wheat, Player with Railroads and the Nation’s Freight Handler; Stormy, husky, brawling, City of the Big Shoulders:” Archimedes: Chicago: Galileo: Chicago:

Newton: Chicago:

“They say you are wicked!” “And I believe them. For I have seen the painted women under the gas lamps luring the farm boys.” “They tell me you are crooked!” “And my answer is: Yes, for I have seen the gunman kill and go free to kill again.”

“And they tell me you are brutal!” “And my reply is: On the faces of women and children I have seen the marks of wanton hunger. But having answered so I turn once more to those who sneer at this my city, and I give them back their sneer and say to them: Come and show me another city with lifted head singing so proud to be alive and coarse and strong and cunning. Flinging magnetic curses amid the toil of piling job on job, here is a tall bold slugger set vivid against the little soft cities;”

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Newton, Galileo, Archimedes protest. Chicago: “Fierce as a dog, tongue lapping for action, cunning as a savage pitted against the wilderness,” Newton, Galileo, Archimedes (take turns):

“Bareheaded, Shoveling, Wrecking, Planning, Building, breaking, rebuilding,”

Chicago:

“Under the smoke, dust all over his face, laughing with white teeth, Under the terrible burden of destiny laughing as a young man laughs, Laughing even as an ignorant fighter laughs who has never lost a battle, Bragging and laughing that under his wrist is the pulse, and under his ribs the heart of the people,”

Galileo: Archimedes: Newton: Chicago:

“Laughing!” “Laughing the stormy, husky, brawling laughter of Youth...” “ . . . half-naked, sweating,” “Proud to be Hog Butcher, Tool Maker, Stacker of Wheat, Player with Railroads and Freight Handler to the Nation.”

Chicago steps away from the podium; exits at the front of the room. Galileo:

“Eppur si muove! But still it moves! I hope it is not too late to register! I would like to discuss with these people the latest developments in the Galilean invariance.”

Newton: Archimedes: Newton:

“Non-Newtonian fluid mechanics!” “I will give my poster at the session on Thursday.” “I think I shall have to look up the President and address the General Assembly about what needs to happen in natural philosophy. And it will be strictly Newtonian!”

Newton, Galileo and Archimedes are now together at the front of the room. Galileo: “Look at these social events! Have you ever been to a baseball game?” Newton: Archimedes:

“It just doesn’t match the excitement of cricket.” “I think the Art Institute is more my style.”

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Newton:

“It might be interesting to throw an apple from the Sears Tower.”

Galileo:

“Or drop it!”

Archimedes:

“I wonder how they get water to the top floors of these tall buildings.”

Galileo: Newton:

“Is there an open staircase where one can hang a pendulum?” “I am famished. What shall we have for lunch?”

Galileo: Archimedes:

“I hear the Italian food here is excellent.” “The Greek food is very good as well.”

Newton:

“I must immediately find a fruit vendor to secure an

apple.” Archimedes picks up an apple from a member of the audience. Archimedes:

“That’s alright. Wait, wait a minute. Here is an apple! Right here. Thank you!”

Throws it to Newton who catches it. Galileo: “Then it’s off to the tower!” Galileo sets off towards the entry doors at the back of the room. The others follow. Archimedes: “You’re going to drop it from that height?” Newton: “Throw it! Throw it! Dropping is a trivial, special case, hardly warranting the consideration of philosophers.”

Galileo protests; advances towards Newton. Archimedes intervenes.

Galileo:

“How do we get there?”

Archimedes:

“By hydraulic lift, of course! And the view must be magnificent!” “With my telescope you can see forever.”

Galileo: Newton:

“If I have seen farther than others it is from the shoulders of giants!”

Archimedes pats his own shoulder.

Archimedes:

“That is my shoulder.”

Galileo:

“Would it be a good idea for me to stand on your shoulders?” “Not so far as I can see!”

Archimedes:

Galileo: “Okey-dokey.” They have reached the doors and exit the Ballrooms.

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Professor Aref “So this is another, very important aspect of Chicago—culture! Those of you who go to the Art Institute on Wednesday, with Archimedes, will see one important aspect of Chicago’s cultural life. And, surely, the Chicago Symphony and the Lyric Opera of Chicago will be household names to many of you. Of course, when it comes to music, Chicago is particularly well known for blues and jazz. We’ll have a small sampling tonight at the Reception. But there is also a vibrant theater life in the city, practitioners from which you have just met. Please give a warm round of applause to our four actors,

Robert Ayres as Sir Isaac Newton and the director of our group, Craig Spidle as Galileo Galilei, Ray Wild as Archimedes, and Genevieve VenJohnson as ‘Chicago’. “This cultural interlude, as it was called in the program—we had to call it something that didn’t give away what it was—was kindly and generously underwritten by our friends at the Illinois Institute of Technology through the Office of the Chief Scientist of the IIT Research Institute—among friends better known as Hassan Nagib.

I thank the

actors for taking this on and for doing such a great job with it. I suspect the organizers of the next congress in 2004 will be in touch, and you will be off to the bright lights of Manchester, Brussels, Dresden or Warsaw. “Well, following this recital of Carl Sandburg’s poem, albeit in a somewhat unorthodox setting, it is time for a welcome from the City of Chicago. We had originally tried to get Mayor Daley himself, but as you can imagine with his involvement in the presidential race as a major force behind the Democratic ticket, he was not available. In fact, there is so much going on in Chicago that a number of business executives who are enthusiastic about the city volunteer to come to groups like ours and provide a welcome on behalf of the City. With us this morning is Mr. Martin Chasen, President of SMC Photo Promotions, Inc., who will say a few words of welcome on behalf of the City of Chicago.”

Martin Chasen welcomes the attendees on behalf of the City, introduces them to a printed Chicago guide that was made available, and invites them to explore the many attractions of Chicago. “Last, but certainly not least, we would like to extend a welcome to you all from the Chicago Marriott Downtown, one of the great hotel properties in this city, a full service conference hotel that hosts events like ours year in and year out and that will be our main stomping grounds

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Left: Grigory Isaakovich Barenblatt (left), closing lecturer at ICTAM 1992, jokes with H. Keith Moffatt (center), closing lecturer at ICTAM 2000, and Frithiof I. Niordson, IUTAM President 1976–1980, before Niordson introduces the ICTAM 2000 opening

lecturer; right: James R. Rice delivers the ICTAM 2000 opening lecture on “New perspectives in crack and fault dynamics”.

for the next five days. Mr. John Adams, the general manager, was regrettably out of town today, but he has asked Mr. Bob Pierce, director of marketing, to say a few words of welcome on behalf of the Chicago Marriott Downtown. Please welcome to the podium Mr. Bob Pierce.” Bob Pierce welcomes the attendees on behalf of the Chicago Marriott Downtown and promises that the hotel and its staff will do everything

to make the week successful and enjoyable. “Thank you Mr. Pierce. Thank you to all our welcoming speakers, thanks to our actors, and thanks to you for your attention. Have a great time and please let us know if you need assistance with any practical matters. We are here to keep the Congress running smoothly and to assure that you have a good week. This concludes the Opening Ceremony. The Opening Lecture will begin in this room in just a few moments at 11:00 a.m.”

Note: QuickTime™ movies of all parts of the Opening Ceremony are

available on the ICTAM 2000 web site at http://www.tam.uiuc.edu/ICTAM2000/ Program/ Opening-Ceremony/

CONGRESS POSTER Meters of Motion Billy Morrow Jackson, 1998 Meters of Motion, a watercolor by Illinois artist Billy Morrow Jackson, was the basis for the poster announcing the 20th International Congress of Theoretical and Applied Mechanics. Jackson’s image depicts several of the giants of mechanics. In the center we have Isaac Newton (1642–1727) holding the device known as Newton’s cradle, illustrating the laws of impact and momentum, today frequently used as an executive toy because of its pleasing periodic motion. Above Newton the phases of the Moon and the falling apple—one of his key insights was the unification of the orbital motion of the Moon about Earth and the accelerated fall of an apple in Earth’s gravitational field. Newton’s results on his universal law of gravitation were published in 1687 in the Principia, probably the most influential scientific treatise of all time. Newton stands on the word ‘PRINCIPIA’ and his famous ‘second law’, force equals mass times acceleration or appears at top right in the image. To the left of Newton we find his intellectual predecessor Galileo (1564–1642) watching a swinging pendulum, timing its oscillations using his pulse. Galileo made seminal contributions not only to kinematics, dynamics and observational astronomy, but also to the subject of strength of materials. His work on kinematics and strength of materials, entitled Dialogues on Two New Sciences, is represented by the word ‘DIALOGUES’, itself a cantilever beam anchored in a brick wall and carrying a suspended weight. Behind Galileo a ball and feather fall with the same acceleration. Above Galileo, in the uppermost left-hand corner, we see Archimedes (287–212 B.C.), one of the great geniuses of antiquity who gave us the laws of hydrostatics and buoyancy, holding forth the symbol the ratio of the circumference to the diameter of a circle. Archimedes was the first to realize the universality of this ratio and to determine approximations to its numerical value. On the right we find Leonhard Euler (1707– 1783), the great mathematician and mechanician, who cast Newton’s laws in their enduring mathematical form using the calculus, and who

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solved legions of problems in both fluid and solid mechanics. He is shown with the famous elastica curve emanating from his forefinger. The elastica is the curve that gives the shape of a thin, perfectly elastic rod when it buckles under load. This curve also arises in other contexts, for example in the shape of a concentrated vortex, such as the tornado seen in the background between Newton and Euler. Below and between Newton and Euler we see another legend of mechanics and mathematics, Joseph Louis Lagrange (1736–1813), whose famous work Mecanique Analytique, published a century after Newton’s Principia, set out the equations of dynamics in a novel variational formulation that gave a teleological interpretation to all motion in the Universe. Lagrange’s work has served as a paradigm of all physical theories since then, from optics to electromagnetism to modern quantum mechanical field theories. Lagrange’s formulation of dynamics also lends itself to the theory of vibrations. Thus patterns of nodal lines on vibrating plates, so-called Chladni figures, flow like sheets from the manuscript he is holding. The three remaining large figures in the picture are of three of the greats of mechanics in the 20th century, individuals whose initiative

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was important also for the inception of the international congresses of mechanics. In the lower left-hand corner Theodore von Kármán (1881–1963) gazes out at sequences of vortices produced in the wakes of bluff bodies. These organized patterns, today known as Kármán vortex streets, are ubiquitous in the wakes of everything from huge tankers to power lines ‘singing in the wind’. Between Newton and Galileo we see Ludwig Prandtl (1875–1953), von Kármán’s teacher, and a seminal figure in both solid arid fluid mechanics. Prandtl is known for his work on the dynamics of the boundary layer, the thin layer of fluid adjacent to a moving rigid body, be it an airplane wing or a ship hull, a layer of fluid very much responsible for the resistance to motion experienced by the body. Golf balls have dimples precisely so that they can ‘fool’ the boundary layer into offering less resistance to their motion than to that of a smooth sphere. Prandtl is also known for his work on aerodynamics (hence the propeller churning to the right of him), on the strength of materials, and for conducting comprehensive experiments on a variety of mechanical phenomena. In the lower right hand corner is G. I. Taylor (1886–1975), whose many contributions included a theory of dislocations in solids, early studies of turbulence in the atmosphere (hence the weather balloon), a theory of the fireball from a nuclear explosion (hence the series of fireballs depicted—a film record was released of the first nuclear explosion and G. I. Taylor developed a theory of the blast phenomenon that predicted the total, classified yield of the bomb). Taylor was an avid sailor and he was very proud of his invention of a drop anchor (depicted). The mathematician and novelist Sofia Kovalevskaya (1850–1891) discovered a very important solution to the problem of the ‘heavy top’. She made her lasting contributions to mechanics during a time when women were not allowed to attend lecture courses in universities. She is standing between Newton and Galileo, below Prandtl, writing an equation (the famous Korteweg–deVries equation that describes water waves) on a blackboard. Off to the right we have a bridge structure, where the members spell out ‘IUTAM’, the acronym for the International Union of Theoretical and Applied Mechanics, the professional body responsible for the international congresses. On the bridge we find current and past officers of IUTAM: past presidents Sir James Lighthill, Paul Germain, Leen van Wijngaarden and Daniel C. Drucker, and current president Werner Schiehlen. We also see the current and past secretaries of lUTAM’s Congress Committee, Niels Olhoff and H. Keith Moffatt. The president of the 19th Congress in Kyoto, Tomomasa Tatsumi, is

in the middle of this group. Holding up the bridge are the president and

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secretary-general of the 20th Congress, Hassan Aref and James W. Phillips of the University of Illinois at Urbana-Champaign. Billy Morrow Jackson has a website at http://www.soltec.net/ jacksonstudios/.

PROGRAM AT A GLANCE Monday, 28 August 2000 10:00—Opening Ceremony Hassan Aref, President of ICTAM 2000, presiding 11:00—Opening Lecture, plenary session AZ—James R. Rice, Harvard University, USA Frithiof I. Niordson, IUTAM President 1976–80, presiding 12:00—Lunch 14:00—Mini-Symposia introductory lectures, parallel sessions BD—Damage and failure of composites (Christensen, Chaboche) BE—Mechanics of foams and cellular materials (Gibson, Loewenberg) BP—Electromagnetic processing of materials (Davidson, Toh) BQ—Vehicle system dynamics (Anderson, Sharp, Abe/Hedrick) BX—Turbulent mixing (Ottino, Villermaux) BY—Granular flows (Roux, Nagel/Jaeger) 16:00—Break 16:30—Contributed lectures, parallel sessions CA—Large-eddy simulation CB—Biological fluid dynamics—Lighthill Memorial Session CC—Convective phenomena CD—Fracture and crack mechanics (jointly with ICF) CE—Structural optimization (jointly with ISSMO) CF—Multibody dynamics CG—Material instabilities CH—Acoustics CK—Flow control CO—Stability of structures CS—Mechanics of phase transformations (jointly with IACM) CV—Environmental fluid mechanics CW—Elasticity CX—Flow instability and transition

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19:00—Welcome reception Host: James J. Stukel, President, University of Illinois

Tuesday, 29 August 2000 08:00—Mini-Symposia introductory lectures, parallel sessions DD—Damage and failure of composites (Dvorak) DE—Mechanics of foams and cellular materials (Kyriakides) DP—Electromagnetic processing of materials (Walker) DQ—Vehicle system dynamics (Schiehlen) DW—Turbulent mixing (Wunsch) DX—Granular flows (Louge) 09:00—Sectional lectures, parallel sessions ED—Pedro Ponte Castañeda, USA EE—Denis L. Weaire, Ireland EP—Hamda BenHadid, France

EQ—Timothy J. Gordon, UK

EW—Paul E. Dimotakis, USA EX—Isaac Goldhirsch, Israel 10:00—Break

10:30—Contributed lectures, parallel sessions FA—Turbulence FB—Fluid mixing FC—Boundary layers FD—Mechanics of foams and cellular materials FE—Damage and failure of composites FF—Vehicle systems dynamics (jointly with IAVSD) FG—Plasticity and viscoplasticity FH—Impact and wave propagation FK—Electromagnetic processing of materials (jointly with HYDROMAG) FL—Granular flows FO—Computational strategies for multiscale phenomena in mechanics (jointly with IACM) FR—Experimental methods in solid mechanics FS—Mechanics of porous materials FV—Combustion and flames 12:30—Lunch

Program at a glance

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14:00—Seminar presentation session, parallel sessions—three-minute overviews GA—Mechanics of foams and cellular materials

GB—Boundary layers and waves GC—Geomechanics and geophysical fluid dynamics GD—Fracture and crack mechanics (jointly with ICF) GE—Structural optimization (jointly with ISSMO) GF—Structural vibrations GG—Multibody dynamics GH—Computational solid mechanics (jointly with IACM) GK—Mixing and multiphase flow GL—Fluid–structure interaction GM—Plasticity, viscoplasticity, and dynamic plasticity of

structures GN—Continuum mechanics and nonlinear dynamics GR—Elasticity GS—Plates, shells, and stability of structures GV—Low-Reynolds-number flow, microfluid mechanics, and fluid mechanics of materials processing 15:00—Seminar presentation session, plenary session—poster display and discussion 16:00—Break

16:30—Contributed lectures, parallel sessions HA—Convective phenomena HB—Flow control HC—Flow instability and transition HD—Elasticity HE—Structural optimization (jointly with ISSMO) HF—Fatigue HG—Computational solid mechanics (jointly with IACM) HH—Contact and friction problems (jointly with IAVSD) HK—Waves HL—Granular flows HO—Dynamic plasticity of structures HR—Control of structures HS—Rock mechanics and geomechanics HV—Complex and smart fluids 18:30—General Assembly of IUTAM Werner O. Schiehlen, President of IUTAM, presiding

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Wednesday, 30 August 2000 08:00—Contributed lectures, parallel sessions IA—Boundary layers IB—Biological fluid dynamics IC—Turbulence ID—Fracture and crack mechanics (jointly with ICF) IE—Structural optimization (jointly with ISSMO) IF—Plates and shells (jointly with IACM) IG—Mechanics of thin films and nanostructures IH—Impact and wave propagation IK—Flows in thin films IL—Multiphase flows IO—Stability of structures IR—Willis Symposium IS—Viscoelasticity and creep IV—Vortex dynamics 10:00—Break 10:30—Contributed lectures, parallel sessions JA—Flow instability and transition JB—Convective phenomena JC—Fluid mechanics of materials processing JD—Elasticity JE—Material instabilities JF—Multibody dynamics JG—Computational solid mechanics (jointly with IACM) JH—Impact and wave propagation JK—Waves JL—Drops and bubbles JO—Cellular and molecular mechanics JR—Willis Symposium JS—Microgravity mechanics JV—Topological fluid mechanics 12:30—Lunch and Excursions 14:00—General Assembly of IUTAM Werner O. Schiehlen, President of IUTAM, presiding

Program at a glance

Thursday, 31 August 2001 08:00—Sectional lectures, parallel sessions KD—Peter A. Monkewitz, Switzerland KE—Vladimir A. Palmov, Russia KP—Erwin Stein, Germany KQ—Stephen J. Cowley, UK KW—Erik van der Giessen, The Netherlands KX—Charles S. Peskiri, USA 09:00—Sectional lectures, parallel sessions LD—Shigeo Kida, Japan

LE—R. Narayana Iyengar, India LP—Gedeon Dagan, Israel LQ—Amable Liñán, Spain LW—Subra Suresh, USA LX—Ole Sigmund, Denmark 10:00—Break 10:30—Contributed lectures, parallel sessions MA—Flow instability arid transition MB—Geophysical fluid dynamics MC—Low-Reynolds-number flow MD—Mechanics of foams and cellular materials ME—Damage and failure of composites MF—Structural vibrations MG—Plasticity and viscoplasticity MH—Contact and friction problems (jointly with IAVSD)

MK—Waves ML—Drops and bubbles MO—Biological solid mechanics MS—Chaos in fluid and solid mechanics MV—Vortex dynamics

12:30—Lunch 14:00—Seminar presentation session, parallel sessions—three-minute overviews NA—Granular flows and complex and smart fluids NB—Biomechanics NC—Transition and turbulence ND—Fracture and crack mechanics (jointly with ICF) NE—Contact and friction problems (jointly with IAVSD) NF—Damage and failure of composites

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NG—Impact and wave propagation NH—Smart materials and structures NK—Convective phenomena and fluid mechanics of materials processing NL—Computational solid mechanics (jointly with IACM) NM—Functionally graded materials, porous materials, phase transformations, and thin films NN—Vehicle dynamics and control of structures NS—Viscoelasticity, creep, and fatigue NV—Combustion, compressible flow, and computational fluid dynamics 15:00—Seminar presentation session, plenary session—poster display and discussion 16:00—Break

16:30—Contributed lectures, parallel sessions OA—Microfluid dynamics OB—Biological fluid dynamics OC—Turbulent mixing OD—Boundary layers OE—Elasticity OF—Damage mechanics OG—Mechanics of thin films and nanostructures OH—Fatigue

OK—Fluid–structure interaction OL—Drops and bubbles OO—Electromagnetic processing of materials (jointly with HYDROMAG) OS—Mechanics of phase transformations (jointly with IACM) OW—Turbulence 17:30—Contributed lectures, parallel sessions PA—Microfluid dynamics PB—Biological fluid dynamics PC—Turbulent mixing PD—Granular flows PE—Fracture and crack mechanics (jointly with ICF) PF—Material instabilities PG—Plasticity and viscoplasticity PH—Fatigue PK—Fluid–structure interaction PL—Multiphase flows

Program at a glance

PO—Electromagnetic processing of materials (jointly with

HYDROMAG) PS—Mechanics of porous materials PW—Flow instability and transition 19:00—Banquet at the Museum of Science and Industry

Friday, 1 September 2000 08:00—Contributed lectures, parallel sessions QA—Turbulence QB—Biological fluid dynamics QC—Electromagnetic processing of materials (jointly with

HYDROMAG) QD—Fracture and crack mechanics (jointly with ICF) QE—Functionally graded materials QF—Control of structures QG—Continuum mechanics QH—Smart materials and structures QK—Drops and bubbles QL—Computational solid mechanics (jointly with IACM)

QO—Fluid–structure interaction QR—Compressible flow QS—Solid mechanics in manufacturing QV—Complex and smart fluids 09:00—Contributed lectures, parallel sessions RA—Flow control RB—Biological fluid dynamics RC—Fluid mechanics of materials processing RD—Fracture and crack mechanics (jointly with ICF) RE—Functionally graded materials RF—Vehicle systems dynamics (jointly with IAVSD) RG—Continuum mechanics RH—Smart materials and structures RK—Waves RL—Computational solid mechanics (jointly with IACM) RO—Fluid–structure interaction RR—Compressible flow RS—Solid mechanics in manufacturing RV—Low-Reynolds-number flow 10:00—Break

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10:30—Contributed lectures, parallel sessions SA—Flow instability and transition SB—Biological solid mechanics SC—Geophysical fluid dynamics SD—Continuum mechanics

SE—Damage mechanics SF—Structural vibrations SG—Mechanics of thin films and nanostructures SH—Smart materials and structures SK—Chaos in fluid and solid mechanics SL—Plates and shells (jointly with IACM) SO—Plasticity and viscoplasticity SR—Acoustics SS—Solid mechanics in manufacturing SV—Flows in thin films 11:30—Contributed lectures, parallel sessions

TA—Flow instability and transition TB—Multibody dynamics TC—Rock mechanics and geomechanics TD—Lighthill Memorial Session TE—Damage mechanics TF—Control of structures TG—Damage and failure of composites TH—Smart materials and structures TL—Experimental methods in solid mechanics TV—Flow in porous media 12:30—Lunch

14:00—Closing Lecture, plenary session UJ—H. Keith Moffatt, Cambridge University, UK G. I. Barenblatt, Closing Lecturer, ICTAM XVIII, presiding 15:00—Closing Ceremony Werner O. Schiehlen, President of IUTAM, presiding

SCIENTIFIC PROGRAM Listed below are the technical sessions and session chairs, along with the authors and titles of papers presented within each session. For each paper, the country of the first-named author, who was the designated presenter, appears in parentheses following the list of authors. Please refer to the Index to Authors, Coauthors, and Session Chairs at the end of this volume to locate a particular author or chair. AZ—Opening Lecture by James R. Rice (Frithiof I. Niordson, Denmark, chair) AZ1—James R. Rice (USA): New perspectives on crack and fault dynamics BD—Damage and failure of composites—Introductory Lectures (George J. Dvorak, USA, chair) BD1—Richard M. Christensen (USA): A survey of and evaluation methodology for fiber composite material failure theories BD2—Jean-Louis Chaboche (France): On constitutive and damage modeling in metal matrix composites BE—Mechanics of foams and cellular materials—Introductory Lectures (Andrew M. Kraynik, USA, chair) BE1—Lorna J. Gibson (USA): Metallic foams: Structure, properties and applications BE2—Michael Loewenberg (USA): Numerical simulation of dense emulsion flows BP—Electromagnetic processing of materials—Introductory Lectures (René J. Moreau, France, chair) BP1—Peter A. Davidson (UK): Aluminum: Approaching the new millennium BP2—Takehiko Toh and Eiichi Takeuchi (Japan): Electromagnetic phenomena in steel continuous casting BQ—Vehicle system dynamics—Introductory Lectures (Peter Lugner, Austria, chair) BQl—Ronald J. Anderson; John A. Elkins; Barrie V. Brickie (Canada): Rail dynamics for the 21st century BQ2—Robin S. Sharp (UK): Fundamentals of the lateral dynamics of road vehicles BQ3—Masato Abe; J. Karl Hedrick (Japan): A mechatronics approach to advanced vehicle control system design BX—Turbulent mixing—Introductory Lectures (Paul E. Dimotakis, USA, chair) BX1—Julio M. Ottino (USA): The kinematics of mixing flows BX2 —Emmanuel Villermaux (France): Mixing: Kinetics and geometry BY—Granular flows—Introductory Lectures (Stuart B. Savage, Canada, chair) BY1—Stéphane Roux; Farhang Radjai (France): Statistical approach to the mechanical behavior of granular media BY2—Sidney R. Nagel and Heinrich M. Jaeger (USA): Signatures of microstructure in granular shear flows: The use of non-invasive probes CA—Large-eddy simulation (Ronald J. Adrian, USA, chair) CA1—Carlo Cercignani, Antonella Abbà, and Lorenzo Valdettaro (Italy): Analysis of subgrid scale models CA2—Marcel R. Lesieur (France): Large-eddy simulations of turbulence in shear flows CA3—Robert D. Moser, Stefan Volker, Prem Venugopal, and Jacob A. Langford (USA): Optimal models for large eddy simulation of turbulence

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CA4—Daniele Carati; Grégoire S. Winckelmans and Hervé Jeanmart (Belgium): Filtering and discretization in large eddy simulation of turbulent flows CA5—Andrew Pollard and Stuart McIlwain (Canada): Large eddy simulations of a coannular jet with low swirl CA6—James G. Brasseur, Anurag Juneja, and Yong Zhou (USA): Weakest links in practical application of large-eddy simulation: The surface layer

CB—Biological fluid dynamics—Lighthill Memorial Session (Andreas Acrivos, USA, chair) CB1—John R. Blake ( U K ) : Microbiological fluid dynamics: A tribute to Sir James Lighthill CB2—Timothy J. Pedley and Vanesa Magar ( U K ) : Model of nutrient uptake by a spherical squirming micro-organism CB3—Stanley A. Berger and Brian E. Carlson (USA): Flow of sickle cell blood in a microcirculatory bed CB4—Z. Jane Wang (USA): Unsteady aerodynamics of insect flight CB5—John O. Kessler (USA): Turbulence, mixing, and dynamics of bacterial suspensions CC—Convective phenomena (Friedrich H. Busse, Germany, chair) CC1—Alexander V. Getling (Russia): Magnetoconvection: Structuring of the amplified magnetic field and freezing of the flow pattern CC2—Keith Julien; Edgar Knobloch; Steve M. Tobias (USA): Highly supercritical magnetoconvection CC3—Wilhelm Schneider (Austria): Lift force induced by buoyancy at a finite horizontal plate CC4—Sergey A. Suslov; Samuel Paolucci (Australia): Spatially developing instabilities in non-Boussinesq convection flows CC5—Alexander A. Golovin; Alexander A. Nepomnyashchy; Leonid M. Pismen (Israel): Non-potential effects in nonlinear dynamics of Marangoni convection patterns CC6—Valentina M. Shevtsova, Denis E. Melnikov, and Jean-Claude Legros (Belgium): Influence of the temperature dependent viscosity on the onset of instability in liquid bridge

CD—Fracture and crack mechanics (jointly with ICF) (Bhushan L. Karihaloo, UK, chair) GD1—Daniel Rittel (Israel): Transient thermomechamcal effects in dynamic fracture CD2—Yoichi Sumi and Yang Mu (Japan): Thermally induced quasi-static wavy crack propagation in a brittle solid CD3—Asher A. Rubinstein ( U S A ) : Crack path development in materials with nonlinear process zone CD4—Krishnaswamy Ravi-Chandar (USA): Dynamic shear crack propagation

CD5—Giulio Maier, Gabriella Bolzon, and Roberto Fedele (Italy): Parameter identification of cohesive crack models by Kalman filter CD6—Jean-Baptiste Leblond and Joel Frelat (France): Crack kinking from an initially closed crack in the presence of friction

CE—Structural optimization (jointly with ISSMO) (Martin P. Bends0e, Denmark, chair) CE1—Helder C. Rodrigues, Paulo S. Miranda, and José M. Guedes; Christopher Jacobs

(Portugal): Optimization study of the interface stiffness in non-cemented hip prostheses CE2—Andreas Rottler, Hans A. Eschenauer, and Robert Dragos; Guido Bieker and Matthias Flach (Germany): Dynamic behavior and durability of rail vehicle components— modeling, simulation, optimization CE3—Andrzej Osyczka and Krenich Stanislaw (Poland): Structural optimization of robot grippers using genetic algorithms CE4—Niels Olhoff and Jacob P. Foldager; Jorn S. Hansen (Denmark): Optimization of the buckling load for composite shells taking thermal effects into account CE5—Guillaume Renaud and Jorn S. Hansen (Canada): Shape optimization of a composite patch repair CE6—John E. Taylor; Martin P. Bendsøe (USA): A mutual energy formulation for the design of continuum structures for generalized objective CF—Multibody dynamics (John J. McPhee, Canada, chair) CF1—Ahmed A. Shabana (USA): Use of the absolute nodal coordinate formulation in modeling flexible multibody systems

CF2—Wojciech Blajer (Poland): An improved formulation for multibody systems with possible singularities and redundancy

CF3—Peter Eberhard (Germany): Transitions in mixed multibody systems/finite element contact simulation CF4—Jiazhen Hong, Anping Guo, and Yanzhu Liu (China, PRC): A dynamic model with substructures for contact-impact analysis of flexible multibody systems

CF5—Florian Wegmann and Friedrich Pfeiffer (Germany): Couplings with neglectable bending stiffness in multibody systems

Scientific program

xli

CF6—Olivier A. Bauchau and Yuri G. Nikishkov (USA): An implicit transition matrix approach to stability analysis of flexible multibody systems CG—Material instabilities (Viggo Tvergaard, Denmark, chair) CG1—Ahmed Benallal; Claudia Comi (France): Bifurcation and perturbation of elasticplastic saturated porous media CG2—John W. Rudnicki and Dmitry I. Garagash (USA): Stability of undrained deformation

of fluid-saturated dilating/compacting solids CG3—Matthew R. Kuhn (USA): The response of dense granular materials to large strain gradients CG4—Lambertus J. Sluys, Harm Askes, Alena Pozivilova, and Wei M. Wang (The Nether-

lands): Large strain viscoplasticity for the modelling of shear banding and necking CG5—Yoshihiro Tomita (Japan): Computational prediction of mechanical characteristics of blended polymer under combined loading CG6—Klaus Thermann; Henryk Petryk (Germany): Post-critical plastic deformation pattern in incrementally nonlinear materials CH—Acoustics (Allan D. Pierce, USA, chair) CH1—Fernando Lund (Chile): Interaction of acoustic waves with singularities in solid and fluid mechanics CH2—Hafiz M. Atassi (USA): Propagation of acoustic and vortical disturbances in a duct with swirling mean flow CH3—Mikael A. Langthjem and Niels Olhoff (Denmark): On analysis and minimization of flow noise in a centrifugal pump CH4—William R. Graham (UK): The effect of modal coupling on low frequency vibration and radiation of fluid-loaded plates CH5—Tim Colonius, Clarence W. Rowley, and Richard M. Murray (USA): Simulation, modeling, and control of self-sustained oscillations in the flow past an open cavity CH6—Ted A. Manning and Sanjiva K. Lele (USA): A mechanism for the generation of shock-associated noise

CK—Flow Control (Mohamed Gad-el-Hak, USA, chair) CK1—Peruvemba R. Viswanath and Kinikkara T. Madhavan (India): Separation control by energization of bubble flow: A novel concept CK2—Frank Urzynicok and Hans H. Fernholz (Germany): Flow-induced acoustic resonators for separation control CK3—Peter W. Carpenter and Duncan A. Lockerby; Christopher Davies (UK): Is

Helmholtz resonance important for boundary-layer control by micro-jet actuators? CK4—J. David A. Walker, Chi-Young Kim, and Hediye Atik (USA): Control of boundarylayer separation with localized suction CK5—Sedat Tardu (France): Regularization of the near wall turbulence activity by unsteady local blowing/suction CK6—Junwoo Lim, Sung M. Kang, John Kim, and Jason L. Speyer (USA): Application of a linear controller to turbulent flows CO—Stability of Structures (Anthony N. Kounadis, Greece, chair) CO1—Simon D. Guest and Sergio Pellegrino (UK): Bi-stable shell structures CO2—James G. A. Croll; Seishi Yamada (UK): Numerical validation of analytical lower bounds for buckling of axially loaded cylinders CO3—James H. Starnes; Mark W. Hilburger (USA): Effects of imperfections on the buckling response of compression-loaded composite shells

CO4—Rivka Gilat; Jacob Aboudi (Israel): Parametric stability of nonlinearly elastic composite plates by Lyapunov exponents CO5—Christos C. Chamis (USA): Probabilistic stability of composite stiffened shells CS—Mechanics Of phase transformations (jointly with IACM) (Alain Molinari, France, chair) CS1—Franz D. Fischer; Narendra Simha (Austria): Phase transformations in elastic-plastic solids: Experiments and simulations CS2—Edgar N. Mamiya, Fabio A. Ferreira, and Dianne M. Viana (Brazil): Description of

internal hysteresis loops in pseudoelastic materials CS3—Mohammed Cherkaoui and Marcel Berveiller (France): Nucleation and growth of martensitic domains in TRIP steels CS4—Xiujie Gao and L. Catherine Brinson (USA): Simplified multivariant model and SEM/ EBSD verification of variant formation and switching

CS5—Yves M. Leroy and Joumana Ghoussoub (France): Solid-fluid phase transformation within grain boundaries

xlii

ICTAM 2000

A waiter presents 1988–1992 IUTAM president Paul Germain (center) with an 80th birthday cake during the ICTAM 2000 Welcome Reception at the Chicago Marriott Downtown.

Joining Germain are (clockwise) John R. Willis, René J. Moreau,

Hassau Aref, and Werner O. Schiehlen (back to camera).

Enjoying the Welcome Reception are (from left) Jette Olhoff, 1976–1980 IUTAM president Frithiof I. Niordson, ICTAM 2000 president Hassan Aref, 1992–2000 IUTAM Congress Committee chair Niels Olhoff, and Susanne Aref.

CV—Environmental fluid mechanics (Peter Lynch, Ireland, chair)

CVl—Colm-cille P. Caulfield; Andrew W. Woods; Adrian McBurnie and Jeremy C. Phillips ( U S A ) : Mixing induced by a finite mass flux plume in a ventilated room CV2— Carlos Haertel; Fredrik Carlsson and Mattias Thunblom (Switzerland): Analysis and simulation of the lobe formation at a gravity-current head CV3 —Bruce R. Sutherland and Kathleen Dohan (Canada): Excitation of large amplitude internal waves from a turbulent mixing region CV4—Julian C. R. Hunt; James W. Rottman (UK): Singular Coriolis effects in stable mountain flows

Scientific program

xliii

CV5—Pierre Carlotti (UK): Spectra of turbulence near the ground CV6—Paul F. Fischer and Gary K. Leaf; Juan M. Restrepo (USA): Forces on spherical particles in oscillatory boundary layers CW—Elasticity (Raymond W. Ogden, UK, chair)

CW1—Michael A. Hayes; Philippe S. Boulanger (Ireland): Unsheared pairs and limiting directions in finite strain CW2—Thomas C. T. Ting (USA): Materials that uncouple antiplane and inplane displacements but not the stresses, and vice versa CW3—Paolo Podio-Guidugli; Paola Nardinocchi (Italy): Angled plates CW4—Pierre Seppecher and Mohamed Camar-Eddine (France): Non-local interactions resulting from the homogenization of an heterogeneous linear elastic medium CW5—Alan S. Wineman (USA): Network scission and healing effects in an elastomeric body due to deformation and high temperatures CW6—Peter Wriggers and Tarek I. Zohdi (Germany): Some aspects of the computational testing of microheterogeneous solids CX—Flow instability and transition (Alexander Solan, Israel, chair) CX1—Bart S. Ng and William H. Reid (USA): Simple asymptotics for the spectra of viscous shear flows CX2—Alison P. Hooper (UK): Energy methods and convergence problems relating to two-fluid flows CX3—Eli Reshotko; Anatoli Tumin (USA): Optimal spatial disturbances in a circular pipe flow CX4—Norman J. Zabusky, Alexei D. Kotelnikov, and Yuriy F. Gulak (USA): Nonlinear evolution and vortex localization in Richtmyer-Meshkov accelerated interfaces CX5—Jason G. Oakley, Bhalchandra P. Puranik, Mark H. Anderson, and Riccardo Bonazza (USA): An investigation of shock-induced interfacial instabilities for strong incident shocks CX6—Katherine Prestridge, Paul M. Rightley, and Robert F. Benjamin; Peter Vorobieff (USA): Validation of an instability growth model using particle image velocimetry measurements DD—Damage and failure of composites—Introductory Lectures (Zvi Hashin, Israel, chair)

DDl —George J. Dvorak (USA): Damage analysis and prevention in composite materials DE—Mechanics of foams and cellular materials—Introductory Lectures (Norman A. Fleck, UK, chair) DE1—Stelios Kyriakides (USA): In-plane crushing of honeycomb DP—Electromagnetic processing of materials—Introductory Lectures (Nagy El-Kaddah, USA, chair) DP1—John S. Walker (USA): Electromagnetic phenomena in crystal growth

DQ—Vehicle system dynamics—Introductory Lectures (J. Karl Hedrick, USA, chair) DQ1—Werner O. Schiehlen; Willi Kortüm and Martin Arnold (Germany): Software tools: From multibody system analysis to vehicle system dynamics DW—Turbulent mixing—Introductory Lectures (Emil J. Hopfinger, France, chair) DW1—Carl I. Wunsch (USA): Overview of oceanographic mixing problems from the microscale to the general circulation DX—Granular flows—Introductory Lectures (Robert P. Behringer, USA, chair) DX1—Michel Y. Louge and James T. Jenkins (USA): Particle segregation in collisional shearing flows ED—Sectional Lecture by Pedro Ponte Castañeda (Zvi Hashin, Israel, chair) ED1—Pedro Ponte Castañeda (USA): Nonlinearity and microstructure evolution in composite

materials

EE—Sectional Lecture by Denis L. Weaire (Norman A. Fleck, UK, chair) EE1—Denis L. Weaire (Ireland): Hard problems with soft materials: The mechanics of foams EP—Sectional Lecture by Hamda Ben Hadid (Nagy El-Kaddah, USA, chair) EPl—Hamda Ben Hadid (France): MHD damped buoyancy-driven flows

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ICTAM 2000

EQ—Sectional Lecture by Timothy J. Gordon (J. Kari Hedrick, USA, chair) EQl—Timothy J. Gordon ( U K ) : Adaptive, nonlinear and learning techniques for the control of vehicle ride dynamics EW—Sectional Lecture by Paul E. Dimotakis (Emil J. Hopfinger, France, chair) EW1—Paul E. Dimotakis (USA): Recent advances in turbulent mixing EX—Sectional Lecture by Isaac Goldhirsch (Robert P. Behringer, USA, chair) EX1 —Isaac Goldhirsch (Israel): Kinetic and continuum descriptions of granular flow FA—Turbulence (Olivier J. Métais, France, chair) FA1—Markus Klein, Amsini Sadiki, and Johannes Janicka (Germany): Direct numerical simulations of plane turbulent jets at moderate Reynolds numbers FA2—Venkatesa I. Vasanta Ram and Maren Schmidts; Gerta Rocklage-Marliani (Germany): Swirling turbulent flow in a pipe: 3-D laser-Doppler velocimetry measurements and comparison with theories FA3—Joël Delville, Fabienne Ricaud, Christophe Picard, Laurent Neau, and Laurent E. Brizzi (France): Large scale structures identification in an axisymmetric shear layer from the near field pressure FA4—Stanislav V. Gordeyev and Flint O. Thomas (USA): Extraction and dynamical modeling of large-scale coherent structure in the planar turbulent jet FA5—Pierre Comte; Emmanuel Briand (France): Formation of coherent structures in spatially-growing boundary layers in large-eddy simulation FB—Fluid mixing (Robert S. Brodkey, USA, chair) FB1—Igor Mezic (USA): Chaotic advection in bounded Navier-Stokes flows FB2—Vered Rom-Kedar; Andrew C. Poje (USA): Universal properties of chaotic transport in the presence of diffusion FB3—Yiannis Ventikos (Switzerland): Spatio-temporal correlations for chaotic advection in aperiodic flows FB4—Jerry P. Gollub, David Rothstein, Emeric Henry, and Wolfgang Losert (USA):

Comparison of chaotic and turbulent transient mixing in two dimensions FB5—Alain Pumir; Misha Chertkov; Boris I. Shraiman (France): Geometry and statistics in Lagrangian dispersion FB6—Haris J. Catrakis, Joshua W. Hearn, Brenda A. McDonald, and Robert D. Thayne (USA): Structure and evolution of fluid interfaces in turbulence

FC—Boundary layers (Yury S. Kachanov, Russia, chair) FC1—Peter W. Duck (UK): On a class of unsteady, non-parallel, three-dimensional disturbances to boundary-layer flows FC2—Stefan Braun and Alfred Kluwick (Austria): Three-dimensional marginal separation of laminar viscous wall jets and boundary layer flows FC3 —Aleksandr V. Obabko and Kevin W. Cassel (USA): Navier–Stokes solutions of thickcore vortex induced unsteady separation FC4—Christopher D. Tomkins and Ronald J. Adrian (USA): Spanwise perturbation of energetic modes in a turbulent boundary layer

FD—Mechanics of foams and cellular materials (Howard A. Stone, USA, chair) FDl—Michael Thies, Carolin Körner, and Robert F. Singer (Germany): Foam formation

and evolution: Numerical simulation with a lattice Boltzmann model FD2—Andrew M. Kraynik; Douglas A. Reinelt (USA): Microrheology of dry soap foams with random structure FD3—Douglas J. Durian, Anthony D. Gopal, Arnaud Saint-Jalmes, and Loic Vanel

(USA): The melting of aqueous foams FD4—Stephan A. Koehler, Sascha Hilgenfeldt, and Howard A. Stone (USA): Foam drainage: Polydispersity and two-dimensional flow FD5—Dmitri L. Vainchtein and Hassan Aref (USA): A phase transition in compressible foam FD6—Bernd Markert and Wolfgang Ehlers (Germany): Finite viscoelastic deformations in fluid-saturated porous solids FE—Damage and f a i l u r e of composites (William A. C u r t i n , USA, chair) FE1—Y. Jack Weitsman (USA): Damage, deformation and failure in randomly reinforced polymeric composites FE2—Peter Gudmundson and Jan-Erik Lundgren (Sweden): Moisture absorption and desorption in composite laminates containing transverse matrix cracks

Scientific program

xlv

FE3—Linda S. Schadler and Chaohui Zhou; S. Leigh Phoenix; Irene J. Beyerlein; Maher S. Amer (USA): Time dependent damage development in graphite/epoxy composites under constant load FE4—Peter Matic, Andrew B. Geltmacher, Kirth E. Simmonds, and Richard K. Everett; Carl T. Dyka (USA): Microtomography and finite element simulations of TiB2 reinforced 1100 aluminum microstructures FE5—David H. Alien (USA): A rate dependent fracture model for predicting multiple damage

during impact in laminated composites FE6—Gyaneshwar P. Tandon; Nicholas J. Pagano (USA): Effect of process-induced damage on the micromechanical response of a multiphase composite

FF—Vehicle systems dynamics (jointly with IAVSD) (Robin S. Sharp, UK, chair) FFl—Hiroshi Yabuno, Takuto Okamoto, and Nobuharu Aoshima (Japan): Stabilization control for the hunting motion of railway wheelset FF2—Oldrich Polach (Switzerland): Influence of locomotive tractive effort on the forces between wheel and rail FF3—Thomas B. Meinders (Germany): The influence of unbalances on the development of polygonalized railway wheels FF4—Hans True; Rolf Asmund (Denmark): The dynamics of a railway freight wagon wheelset with dry friction damping FF5—Yoshihiro Suda, Takefumi Miyamoto, and Norihiko Katoh (Japan): Active controlled rail vehicles for improved curving performance and response to track irregularity FF6—Steven A. Velinsky (USA): Mechanics of wheeled-vehicle based mobile manipulators FG—Plasticity and viscoplasticity (Zenon Mróz, Poland, chair)

FG1—Morton E. Gurtin; Paolo Cermelli (USA): The characterization of geometrically necessary dislocations in finite plasticity FG2—Otto T. Bruhns (Germany): A consistent description of finite elastoplasticity FG3—Keh-Chih Hwang and Han Q. Jiang; Yonggang Huang; Huajian Gao (China, PRC): A finite deformation theory of strain gradient plasticity

FG4—David L. McDowell, George C. Butler, and Robert D. McGinty (USA): An internal state variable model for dislocation substructure formation in polycrystal plasticity FG5—Laurent Langlois and Marcel Berveiller (France): Softening induced by change of dislocation micro structure in metals FG6—Henryk Zorski (Poland): Statistical theory of dislocations in 2D elastic body

FH—Impact and wave propagation (Rodney J. Clifton, USA, chair) FH1—Tsolo Ivanov; Radijanka Savova (Bulgaria): Surface wave propagation FH2—John G. Harris (USA): Elastic surface waves in curved structures FH3—Xiaodong Wang (Canada): High-frequency elastic wave propagation induced by piezoelectric actuators FH4—Lester W. Schmerr Jr. and Matthias Rudolph; Alexander Sedov (USA): Ultrasonic transducer wave field modeling for complex geometries and anisotropic materials

FH5—Paul Fromme and Mahir B. Sayir (Switzerland): On the scattering of flexural waves at a notched hole in a plate FH6—R. Bruce Thompson (USA): Ultrasonic models for the influence of micro structure on flaw detection in aircraft engines

FK—Electromagnetic processing of materials (jointly with HYDROMAG) (John S. Walker, USA, chair) FK1—Duane Johnson; Ranga Narayanan (USA): The effect of magnetic suppression on the convective flow patterns in cylindrical containers FK2—Amnon J. Meir and Paul G. Schmidt (USA): Liquid-metal flow in a cylindrical crucible rotating in a transverse magnetic field FK3—Claude B. Reed; Sergei Molokov (USA): The effect of a uniform magnetic field on a liquid-metal flow in a rivulet

FK4—Koichi Kakimoto (Japan): Heat and mass transfer in Czochralski silicon crystal growth under magnetic fields FK5—Thierry Alboussière; Jean-Paul Garandet; René J. Moreau ( U K ) : Asymptotic magnetohydrodynamic convection and symmetries FK6—René J. Moreau, Alban Potherat, and Karim Messadek; Joel Sommeria (France): Quasi-two-dimensional turbulent shear flows: Modelling versus experiment FL—Granular flows (Stéphane Roux, France, chair)

FL1—John M. N. T. Gray (UK): An exact solution for the mixing of a monodisperse granular

material in a rotating drum

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ICTAM 2000

FL2—Prabhu R. Nott (India): Kinematics of mixing and segregation of granular flows in rotating drums FL3—Kimberly M. Hill; Nitin Jain and Julio M. Ottino (USA): Axial segregation patterns in non-circular tumbling mixers FL4—Daniel M. Mueth, Georges F. Debregeas, Gregory S. Karczmar, Peter J. Eng, Heinrich M. Jaeger, and Sidney R. Nagel (USA): Non-invasive studies of shear bands in dense, three-dimensional granular flow FL5 —Daniel W. Howell and Robert P. Behringer; Christian Veje (USA): Stress fluctuations in slow shear flow FL6—Wolfgang Losert and Jerry P. Gollub (USA): Particle dynamics and shear forces in sheared, air-fluidized granular matter

FO—Computational strategies for multiscale phenomena in mechanics (jointly with IACM) (Erwin Stein, Germany, chair) FO1—J. Tinsley Oden, Gregory Rodin, Kumar Vemaganti, and Yuhong Fu (USA): Adaptive modeling of heterogeneous materials: Theory, fast summation methods and applications FO2—Jacob Fish and Qing Yu (USA): Multiscale models for problems in heterogeneous media FO3 —William A. Goddard, Tahir Cagin, Alejandro Strachan, and Richard P. Muller (USA): First principles multiscale approaches to prediction of mechanical properties FO4—Nicholas J. Pagano and Valeri A. Buryachenko (USA): Multiscale analysis and edge effects for multiply interacting inclusion problems FO5—Christian H. Huet (Switzerland): Integrated micromechanics strategies in random composites bodies with distributed granulometry FO6—Krishna Garikipati; Thomas J. R. Hughes (USA): Embedding micromechanical laws in continuum macromechanics: A variational multiscale approach

FR—Experimental methods in solid mechanics (Isaac M. Daniel, USA, chair) FRl—Michel M. Géradin, Georges Magonette, and Artur V. Pinto (Italy): Large dynamic and pseudo-dynamic testing at the European Laboratory for Structural Assessment FR2—Eiichi Watanabe, Tomoaki Utsunomiya, Kazutoshi Nagata, and Yousuke Tsumura; Yukihide Kajita (Japan): A shaking table test for pounding between girders in elevated bridges and the numerical simulation FR3—Eric Clément, Guillaume Reydellet, and Loic Vanel; Daniel W. Howell, Junfei Geng, and Robert P. Behringer (France): Response functions in granular pilings FR4—Lily D. Poulikakos and Mahir B. Sayir; Manfred N. Parti (Switzerland): In-line nondestructive evaluation of asphalt plug joints FR5—Junji Yoshida, Masato Abe, and Yozo Fujino (Japan): Constitutive law for high damping rubber material and its experimental verification

FS—Mechanics

of porous materials (Alberto Carpinteri, Italy, chair)

FSl—Zhuping Huang; Yi Liu (China, PRC): The macroscopic strain potentials in nonlinear porous materials FS2—Alain Molinari and Sebastien Mercier (France): Micromechanics of porous materials at high strain rates FS3—Pia Redanz; Norman A. Fleck (Denmark): On affine motion of metal powder particles undergoing compaction

FS4—Sankara J. Subramanian and Petros Sofronis (USA): Micromechanical modeling of powder compaction FS5—Ragnar Larsson and Jonas Larsson (Sweden): Nonlinear analysis of porous medium with compressible fluid FS6—Stefan Diebels (Germany): Compressible porous media and the evolution of the volume fractions

FV—Combustion

and flames (Amable Liñán, Spain, chair)

FV1—Francisco J. Higuera (Spain): Thermocapillary flow due to a flame spreading over a liquid fuel FV2—Portonovo S. Ayyaswamy, Ira M. Cohen, and Srinivas S. Sripada (USA): Electric field induced convection effects on flames FV3—John Brindley, John F. Griffiths, and Andy C. Mclntosh (UK): Initiation of combustion by spherical hotspots in reactive solids FV4—Thomas L. Jackson; John D. Buckmaster and Abdelkarim Hegab; Gregory M. Knott ( U S A ) : Random packing of heterogeneous propellants and the flames they support

FV5—Bernard J. Matkowsky; Anatoly P. Aldushin (USA): Steady state combustion wave characteristics

FV6—John D. Buckmaster (USA): The dynamics of edge-flames

Scientific program

xlvii

GA—Mechanics of foams and cellular materials (Stelios Kyriakides, USA, chair) GA0l—William E. Warren; Esben Byskov (USA): Three-fold symmetry restrictions on twodimensional micropolar materials GA02—Xiaolin Wang; William J. Stronge (Singapore): Micropolar theory for elastodynamics

of honeycomb GA03—Jörg Hohe and Wilfried Becker (Germany): A refined analytical model for the effective

elasticity tensor of general cellular sandwich cores GA04—Changqing Chen, Tian Jian Lu, and Norman A. Fleck ( U K ) : Stiffness and strength of two dimensional cellular solids containing holes and inclusions GA05—Ashraf Bastawros (USA): Notch effects and associated length scales in the mechanics of cellular metals GA06—Karam Sab; Myriam Laroussi and Amina Alaoui (France): Scale effects in high strain compression of periodic open cell foams GA07—Yu Wang and Alberto M. Cuitiño; Gustavo Gioia (USA): Inhomogeneous deformation in compressed open-cell solid foams GA08—Xavier Chateau; Luc Dormieux (France): The behavior of unsaturated porous media:

A micromechanical approach GA09—Chris G. Poulton, Ross C. McPhedran, and Nicolae A. Nicorovici; Alexander B. Movchan; Yuri A. Antipov (Australia): Spectral problems and phononic band gaps for periodic structures GA10—Rafael J. Mora and Anthony M. Waas; Jacob Pawlicki (USA): Strength scaling of brittle graphitic foams GA11—Tian Jian Lu and Changqing Chen; Guangrui Zhu ( U K ) : The crushing of corrugated

board panels GA12—Martin Schanz; Alex H.-D. Cheng (Germany): Compressional waves in a onedimensional poroelastic column GA13—Hanxing Zhu and Alan H. Windle ( U K ) : The high strain compression of irregular open-cell foams

GB—Boundary layers and waves (Roger H. J. Grimshaw, UK, chair) GB01—Georgiy Korolev ( U K ) : Flow separation and non-uniqueness of the solution of the boundary-layer equations

GB03—David H. Wood and Yuhua Guo (Australia): Measurements in the vicinity of a stagnation point GB04—Ronald Abstiens, Stefan Fühling, and Wolfgang Schröder (Germany): Boundarylayer measurements on the ELAC configuration at Re = 20 × 106 GB05—Rudolph A. King; Kenneth S. Breuer (USA): Oblique transition in a Blasius boundary layer GB06—Xuesong Wu ( U K ) : Receptivity of boundary layers with distributed roughness to vortical disturbances GB07—Christian A. Maresca, Eric C. Berton, Daniel P. Favier, and Christine M. Allain (France): Investigation of boundary layers of moving walls by use of embedded laser-Doppler velocimetry methods GB08—Stefan Herr, Werner Wuerz, Anke Woerner, Ulrich Rist, and Siegfried Wagner; Yury S. Kachanov (Germany): Experimental and numerical investigation of 3D wall roughness acoustic receptivity on an airfoil

GB09—Alexander D. Kosinov (Russia): Flow characteristics measurements in attachment line

boundary layer on a swept cylinder at Mach 2 GB10—Alric P. Rothmayer (USA): A modified Kuramoto-Sivashinsky equation for air driven films GB11—Tim David and Peter G. Walker ( U K ) : Platelet activation and adhesion in general 2D flows GB12—Oleg G. Derzho and Roger H. J. Grimshaw (Australia): Axisymmetric waves and eddies in swirling flows in a tube GB13—Pearu Peterson; Brenny (E.) van Groesen (Estonia): The direct and inverse problem of wave crests

GB14—Janet M. Becker and David Bercovici (USA): Pattern formation on the interface of a two-layer fluid with a bi-viscous lower layer

GC—Geomechanics and geophysical fluid dynamics (Julian C. R. Hunt, UK, chair) GC01—María-Arántzazu Alarcón, John W. Rudnicki, Richard J. Finno, and Amy Rechenmacher (USA): Constitutive modeling of direct measures of strain in simulated fault gouge GC02—Michio Kurashige, I. Taguchi, and Kazuwo Imai (Japan): Integral equations for an arbitrarily-loaded crack in oil reservoirs of poroelastic solid

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ICTAM 2000

GC03—Jean-François Thovert and Farchid Yousefian; Pierre Adler and Samir Bekri (France): Macroscopic properties of heterogeneous porous media GC04—Krzysztof Wilmanski (Germany): Propagation of weak and strong discontinuities in poroelastic materials GC05—Marie-Angele Abellan and Jean-Michel Bergheau; Paul Jouanna (France): Impact of initial conditions on environmental previsions: A case study in unsaturated soils GC06—Xiang Song Li; Yannis F. Dafalias (Hong Kong): Unified modeling of stress-strain behavior of loose and dense sand GC07—Thomas Zwinger and Alfred Kluwick; Peter Sampl (Austria): Dynamics of dry snow avalanches on arbitrarily shaped slopes GC08—Fridtjov Irgens (Norway): Simplified simulation model of snow avalanches and landslides GC09—Chiu-On Ng (Hong Kong): Mass transport following release of contaminated particles into a steady channel flow GC10—Michihisa Tsutahara, Takeshi Kataoka, and Masaya Tanaka (Japan): Point sink flow in a rotating stratified fluid GC11—Takeshi Miyazaki and Tomoyuki Shimonishi (Japan): Stability of a quasigeostrophic ellipsoidal vortex GC12—Rajesh Bhaskaran and Sidney Leibovich (USA): Eulerian and Lagrangian Langmuir circulation patterns GC13—Hideshi Hanazaki; Julian C. R. Hunt (Japan): Linear processes in unsteady stably stratified turbulence with mean shear GC14—Patrick F. Hodnett and Raymond McNamara (Ireland): Zonal influences in a modified Stommel–Arons model of the abyssal ocean circulation

GD—Fracture and crack mechanics (jointly with ICF) (L. Ben Freund, USA, chair) GD01—Mark E. Mear (USA): A weakly-singular, weak-form traction integral equation for analysis of cracks in a half-space GD02—Christian F. Niordson (Denmark): Analysis of steady-state ductile crack growth along a laser weld GD03—Mikhail Perelmuter (Russia): Bridge model for an interface crack GD04—Chunyu Li; George J. Weng; Zhuping Duan (China, PRC): A cylindrical crack in a functionally graded material interlayer under torsional impact GD05—Nadim I. Shbeeb and Wieslaw K. Binienda (USA): Analysis of the driving forces for multiple cracks in an infinite non-homogeneous plate GD06—Masaaki Watanabe (Japan): The angled crack problem revisited GD07—Igor A. Guz and Alexander N. Guz’ (Ukraine): Stability of two different half-planes in compression along interfacial cracks: Analytical solutions GD08—Chien H. Wu and Ming L. Wang (USA): The effect of crack-tip point loads on fracture GD09—Kavi Bhalla and Alan T. Zehnder; Robert Thomas and Lawrence Favro (USA):

Thermal analysis of crack tearing GD10—Adélaïde Feraille and Alain Ehrlacher (France): Behavior of a wet crack submitted to heating up to high temperature GD11—Nicolas Moes and Ted Belytschko (USA): An extended finite element method to model crack growth without remeshing GD12—Alina A. Zalounina and Jens H. Andreasen (Denmark): An edge crack in a coated solid subjected to contact loading GD13—Jens H. Andreasen (Denmark): Straight sided buckling of a coating on a stiff substrate GD14—Liviu Marsavina; Matthew J. Ekman and Andrew D. Nurse (Romania): On the near-tip stress intensity factor mode mixed at bimaterial interface cracks under small-scale

yielding GD15—Vera E. Petrova; Klaus Herrmann (Russia): Thermal cracking of a bimaterial containing an interface crack and microcracks GE—Structural optimization (jointly with ISSMO) (Niels Olhoff, Denmark, chair) GE01—Valeri L. Markine, Amy de Man, and Coenraad Esveld (The Netherlands): Optimal design of ballastless railway track structure GE02—Sergio R. Turteltaub (USA): Optimal distribution of material properties for transient phenomena

GE03—Seil Song, Jong S. Im, and Yung M. Yoo; Jung K. Shin, Kwon H. Lee, and Gyung J. Park (Korea): Automotive door design with the ultra light steel auto body concept using structural optimization GE04—Julian A. Norato, Tyler E. Bruns, and Daniel A. Tortorelli; Kevin J. Hinders; I. Dennis Parsons (USA): Computational tools for architectural design GE05—Alejandro R. Diaz and Brian Feeny (USA): Optimization of two-phase material distributions generated by iterated affine maps

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GE06—Ani P. Velo; Robert P. Lipton (USA): Optimal design of gradient fields with application to problems of heat conduction GE07—Yuanxian Gu, Biaosong Chen, and Hongwu Zhang (China, PRC): Thermalstructural optimization and sensitivity analysis with precise time integration GE08—Ciro A. Soto; John E. Taylor and Jianbin Du; Helder C. Rodrigues (USA): Treatments for simultaneous structural topology and material properties design optimization GE09—Yoon Y. Kim and Gil H. Yoon (Korea): Multi-resolution multi-scale topology optimization in general domains GE10—Thomas Borrvall and Joakim Petersson (Sweden): Topology optimization using regularized intermediate density control GE11—Sándor Kaliszky and János Lógó (Hungary): Layout optimization of elastoplastic structures with plastic deformation and displacement constraints GE12—Niels L. Pedersen (Denmark): On topology optimization of plates with prestress GE13—Thomas Buhl (Denmark): Topology optimization with cost of supports GE14—Chien Jong Shih and Ting Jhang Tseng (Taiwan): Reliability based optimum structural design for robust performance using fuzzy formulation strategy GE15—Gih Keong Lau, Hejun Du, and Mong King Lim (Singapore): Topology optimization of compliant mechanism subjected to harmonic loads GF—Structural vibrations (Irina Goryacheva, Russia, chair) GF01—Shanshin Chen and Daniel A. Tortorelli (USA): Domain decomposition of energy conserving flexible multibody dynamic systems GF02—Peter H. Ruge and Nils Wagner (Germany): Continuous versus discontinuous timefinite-elements for vibration systems with fading memory GF03—Ye Ping Xiong; Jing Tang Xing and W. Geraint Price (China, PRC): A generalized mobility progressive method of power flow analysis for complex coupled dynamic systems GF04—Manfred W. Zehn and Alexander Saitov (Germany): Improvement of finite element models for the dynamics of active composite materials GF05—Carlos E. N. Mazzilli, Mário E. S. Soares, and Odulpho G. P. Baracho Neto (Brazil): Perturbation techniques applied to the evaluation of nonlinear modes GF06—Matti Martikainen, Erno K. Keskinen, and Veli-Matti Jarvenpää; Michel Cotsaftis (Finland): Transient oscillation dynamics of elastic structure due to rolling contact source GF07—Robert G. Parker and Yehong Lin (USA): Parametric instability of axially moving media under multi-frequency tension and speed fluctuations GF08—Vladimir I. Babitsky and Alexander M. Veprik (UK): Synchronization of vibration by collisions in structures and periodic Green’s functions GF09—Igor E. Poloskov (Russia): Analysis of random regimes and computer algebra GF10—Thomas P. Mitchell (USA): The response of an axially translating one-dimensional continuum to stochastic excitation GF12—M. Alaeddin Arpaci and S. Ergun Bozdag (Turkey): Triply coupled vibrations of thin-walled open cross-section beams including rotary inertia effect GF13—Noël Challamel (France): Some limit cycles in drilling mechanics GF14—Xia Lu and Sathya V. Hanagud (USA): Irreversible thermodynamic models for damping GG—Multibody dynamics (Timothy J. Gordon, UK, chair)

GG01—Michel Fayet (France): Dynamics of semi-free systems GG02—Bodo Heimann and Martin Grotjahn (Germany): Newton–Euler equations for multibody systems in minimal dimensional parameter-linear form GG03—Martin Arnold (Germany): Coupled dynamical simulation of coupled mechanical systems GG04—Jens Wittenburg; Ljubomir Lilov (Germany): Decomposition of a screw displacement into three consecutive screw displacements with fixed axes GG05—Brian Moore; Jean-Claude Piedboeuf; Pierre Sicard (Canada): Dynamics identification of mechanisms using a symbolic approach GG06—Hiroaki Yoshimura and Takehiko Kawase (Japan): A method of tearing and interconnecting for multibody dynamics GG07—Yu-Hung Hsu and Kurt S. Anderson (USA): Analytical full-recursive sensitivity analysis for multibody system designs GG08—Viktor Berbyuk; Anders Boström (Ukraine): Optimization problems of controlled multibody systems having spring-damper actuators GG09—Andreas Voile (Germany): The influence of wobbling masses on biomechanical systems GG10—Anatoly A. Khentov (Russia): The principle of strong interaction and the orbital motion of some Saturn’s and Jupiter’s satellites

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ICTAM 2000

GG11—Vigen Arakelian and Marc Dahan; Mike Smith (France): Shaking force and shaking moment balancing of slider-crank mechanisms GG12—Johannes Gerstmayr and Hans Irschik (Austria):

Computational strategies for

vibrating elastoplastic structures with rigid-body degrees of freedom

GH—Computational solid mechanics (jointly with IACM) (J. Tinsley Oden, USA, chair) GH01—Muneo Hori and Tsuyoshi Ichimura (Japan): Macro–micro analysis for computation of earthquake wave propagation in metropolis

GH02—Grzegorz Zboinski and Wieslaw Ostachowicz (Poland): A unified approach to adaptive modeling and analysis of complex structures GH03—Basudeb Bhattacharyya; Subrata Chakraborty (India): Probabilistic sensitivity analysis of structures involving parametric uncertainty GH04—Per Kettil and Nils-Erik Wiberg (Sweden): Adaptive solid modeling and analysis applied to design of bridges GH05—Stanislaw A. Lukasiewicz; Marcin Kaja and Marek Stanuszek (Canada): Filtering and enhancement of the numerical and experimental data at arbitrarily located points GH06—Joaquim B. Cavalcante Neto; Luiz F. Martha (Brazil): A prototype for a system for adaptive analysis in three dimensions GH07—Michael L. Dambach; Alexander Tessler; Donald W. Oplinger (USA): Adaptive mesh refinement strategies for modeling adhesively bonded joints GH09—Gakuji Nagai and Akira Wada; Takahiro Yamada (Japan): 3-D mesoscopic failure simulation of concrete materials by using image-based finite element method GH10—Martin Ammann and Wolfgang Ehlers (Germany): H-adaptive finite element strategies applied to multi-phase problems GH11—Bruno Cochelin; Noureddine Damil; Michel Potier-Ferry (France): Advances in the asymptotic numerical method

GH12—Gang Li and Narayana R. Aluru (USA): Finite cloud meshless method for geometrically nonlinear analysis of micro-electro-mechanical systems GH13—Kent T. Danielson, Shaofan Li, and Wing Kam Liu; R. Aziz Uras (USA): Parallel computational methods for meshfree analysis GH14—Noriko Katsube and Dan Zeng; Wole Soboyejo (USA): Hybrid finite element method for heterogeneous materials and cracks GH15—Zhenhan Yao, Yongqiang Chen, and Xiaoping Zheng (China, PRC): Numerical simulation of failure processes in 3-D heterogeneous brittle material GK—Mixing and multiphase flow (Grétar Tryggvason, USA, chair)

GK01—Fotis Sotiropoulos, Tahirih C. Lackey, and Donald R. Webster (USA): Chaotic and quasi-periodic dynamics in 3D steady vortex breakdown bubbles GK02—Clotilde Chagny-Regardin, Cathy Castelain, and Hassan Peerhossaini (France): The limits of chaotic advection flow regime on convective heat transfer GK03—Patrick D. Anderson, Peter G. M. Kruijt, Oleksey S. Galaktionov, Gerrit W. M. Peters, and Han E. H. Meijer (The Netherlands): A mapping approach for 3D distributive mixing flows

GK04—Gregory P. King; Murray Rudman; Athanasios N. Yannacopoulos; George Rowlands; Igor Mezic (UK): Can effective diffusion coefficients be extracted from symmetry measures? GK05—Keisuke Araki; Katsuhiro Suzuki (Japan): A Riemannian geometrical analysis of relative diffusion of Lagrangian particle pair in turbulence GK06—Andrew W. Cook; Paul E. Dimotakis (USA): Investigation of mixing effects in incompressible Rayleigh–Taylor instability GK07—Xiaolin Li, James Glimm, Andrea Marchese, and Zhiliang Xu ( U S A ) : Computation of 3-d Rayleigh–Taylor mixing rate with randomly perturbed interface

GK08—Yuan-nan Young, Anshu Dubey, and Robert Rosner; Henry M. Tufo (USA): Miscible Rayleigh–Taylor instability: 2D and 3D GK09—Jerzy Baldyga and Wioletta Podgorska (Poland):

Application of the multifractal

model of turbulence to chemical engineering mixing problems GK10—Sami A. Bayyuk, De-Ming Wang, and Samuel A. Lowry (USA): A self-consistent higher-order accurate scheme for the volume-of-fluid methodology for multi-phase flows GK11—Christopher M. Varga and Juan C. Lasheras; Emil J. Hopfinger (USA): On the steady and unsteady breakup of a liquid jet by a coaxial gas stream GK12—Frédéric Risso and Kjetil Ellingsen (France): Dynamics of the path oscillations of a rising ellipsoidal bubble

GK13—Nelya K. Vakhitova, Robert I. Nigmatulin, Iskander Sh. Akhatov, Raisa Kh. Bolotnova, Andrei S. Topolnikov, and Elvira Sh. Nasibullayeva (Russia): Resonant supercompression of gas bubbles (sonoluminescence, sonochemistry and bubble fusion)

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GL—Fluid-structure interaction (Earl H. Dowell, USA, chair) GL01—Hong-Hui Shi and Motoyuki Itoh; Takuya Takami (Japan): The motion of an underwater blunt body after water entry GL02—Jean-Francis Ravoux; Ali Nadim; Hossein Haj-Hariri (USA): An embedding method for bluff body flows GL04—Alexander I. Zubkov and Anatoly I. Glagolev (Russia): Reduction of aerodynamic drag of bodies of revolution using heat and mass injection in near wake GL05—Morten H. Hansen and Joergen T. Petersen (Denmark): Flutter phenomena in wind turbines GL06—Gregory Hagen and Igor Mezic (USA): Fluid modeling and control of axial compressors GL07—Emmanuel de Langre (France): Absolutely unstable waves in hydroelastic systems GL08—Henrik Møller and Erik Lund (Denmark): Shape sensitivity analysis of strongly coupled fluid–structure interaction problems GL09—Lawrence N. Virgin, Steve T. Trickey, and Earl H. Dowell (USA): Sudden jumps in the oscillatory behavior of a nonlinear aeroelastic system GL10—Alexander Galper and Touvia Miloh (Israel): Rotational flow hydrodynamics GL13—Viorel I. Anghel (Romania): Resonances analysis of hingeless rotor helicopter using the equivalent spring restrained blade model

GM—Plasticity, viscoplasticity, and dynamic plasticity of structures (Stephen R. Reid, UK, chair) GM02—Elhem Ghorbel and Didier Baptiste (France): A generalized yield criterion for unoriented polymers GM03—Douglas J. Bammann (USA): Incorporation of a natural length scale into a crystal plasticity model GM04—Hyszard B. Pecherski; Katarzyna Korbel; Andrzej Korbel (Poland): Finite plastic deformation due to the sequence of slips: Experimental evidence and theoretical model GM05—Franck Ferrer, Thierry Bretheau, and Jérorne Crépin; Alain Barbu (France): Plasticity of zirconium at intermediate temperatures: Effect of a small quantity of sulfur GM06—Wenhui Zhu, Shinji Tanimura, Koji Mimura, and Tsutomu Umeda (Japan): Dynamic response of new sensing block systems and its application to dynamic buckling tests GM07—Michael L. Falk; James S. Langer (USA): A microscopically motivated theory of dynamic plastic deformation in noncrystalline solids GM08—Jilin Yu; Marcilio Alves (China, PRC): On the choice of material model and its parameters in structural impact GM09—Miroslav D. Nestorovic and Nicolas Triantafyllidis; Yves M. Leroy (USA): On the stability of rate-dependent solids with application to the uniaxial plane strain test GM10—Mark R. Liberzon (Russia): Impulse treatment by high-energetic electromagnetic field on materials GM11—K. Lawrence DeVries and Paul R. Borgmeier (USA): Stress distribution in and

failure of standard adhesive joints GM12—Jesus Chapa-Cabrera and Ivar E. Reimanis; Eric D. Steffler (USA): Fracture behavior for cracks perpendicular to the gradient in functionally graded materials GM13—Mark E. Walter (USA): Analysis of the deformation of an aluminide coating system under thermal loading

GN—Continuum mechanics and nonlinear dynamics (Gerard A. Maugin, France, chair) GN01—Peter A. Dashner (USA): Elastic shadow flow and its theoretical implications for inelastic solids GN02—Alain Merlen and Saloua Ben Khélil; Vladimir Preobazhenski; Philippe Pernod (France): Numerical simulation of acoustic waves phase conjugation in active media GN03—Judith E. Skolnik and Mark H. Wähling (Germany): Thermo-mechanical processes in saturated porous solids GN04—Miles B. Rubin; Oleg Yu. Vorobiev, Ilya Lomov, Lewis A. Glenn, and Tarabay Antoun; Alexander Chudnovsky (Israel): Fracture surface energy in a thermomechanical model for bulking of porous viscoplastic material GN05—Valery I. Levitas (USA): Unified approach to phase transitions, fracture and chemical reactions in inelastic materials GN06—Dimitris C. Lagoudas and David H. Allen; Chongho Yoon (USA): A model for predicting fracture toughness and damage evolution in viscoelastic solids GN07—Klaus Hackl (Germany): On induced microstructures occurring in models of finitestrain elastoplasticity

GN08—Kam Tim Chau and Xiao Yang (Hong Kong): Resistance of nonlinear soil to a horizontally vibrating pile

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ICTAM 2000

GN09—Pawel Dluzewski and Grzegorz Maciejewski; Slawomir Kret (Poland): Finite element simulation of residual stress in semiconductor epilayers GN10—Markus Zahn (USA): Nonlinear ferrofluid duct flow profiles in alternating and travelling wave magnetic fields GN11—Philip J. Morrison (USA): An integral transform for the continuous spectrum in fluid and plasma dynamics GN12—Jacob P. Meijaard (The Netherlands): Non-classical bifurcations and chaos in a positioning mechanism GN13—S. John Hogan, Mario Di Bernardo, and Martin E. Homer; Mark I. Feigin (UK): Local analysis of c-bifurcations in n-dimensional piecewise-smooth dynamical systems GN14—Haider N. Arafat and Ali H. Nayfeh (USA): Energy transfer between widely-spaced in-plane bending modes of cantilever beams GR—Elasticity (Michael A. Hayes, Ireland, chair) GR01—Xanthippi Markenscoff (USA): An inverse problem in elasticity: The Eshelby inclusion GR02—George V. Jaiani (Georgia): On a mathematical model of bars with variable rectangular cross-sections GR03—George Mejak (Slovenia): Shape optimization of composite bars in torsion GR05—David P. Mason and Nicolette Roussos (South Africa): Nonlinear superposition principle for radial oscillations of a thin Mooney–Rivlin cylindrical tube GR06—Yi-chao Chen (USA): Dynamic solutions of thermoelastic rods under impact loads GR07—Anthony J. Paris and Sarah E. Zeller (USA): On the mechanics of cord-composite materials GR08—Jian Cao and Nan Song (USA): Effect of tooling design on the variation of final product geometry GR09—Allan X. Zhong (USA): Effects of incompressibility in numerical submodeling of hyperelastic materials GR10—Bing-Zheng Gai and Qing-Cai Chen (China, PRC): BEM analysis of interaction of elastic waves with unilateral interface crack GR11—Youn-Sha Chan; Glaucio H. Paulino; Albert D. Fannjiang (USA): Gradient elasticity theory for a mode I crack in homogeneous and nonhomogeneous materials GR12—Natasha V. Movchan (UK): Effect of interfacial bonding on stress singularity at the vertex of a conical inclusion GR13—Jurgen Jäger (Germany): A generalized Coulomb law for elastic bodies

GS—Plates, shells, and stability of structures (Arthur w. Leissa, USA, chair) GS01—Eric M Mockensturm (USA): On the finite twisting of thick elastic plates GS02—Yu-Hsuan Su and S. Mark Spearing (USA): Large deflection analysis of an annular plate with a rigid boss under axisymmetric loading GS03—Sachiko Nakagoshi and Hirohisa Noguchi (Japan): A new wavelet Galerkin method for analysis of Mindlin plates GS04—Manfred Bischoff; Kai-Uwe Bletzinger (USA): Stabilized discrete shear gap plate and shell elements GS05—Natacha M. Buannic and Patrice M. Cartraud (France): Higher-order asymptotic modeling for heterogeneous periodic plates GS06—Eliza M. Haseganu and Ying Liu (Canada): Equilibrium analysis of cylindrical elastic membranes subjected to conservative pressure loading GS07—Dobroslav D. Ruzic and Ljubisa S. Markovic (Yugoslavia): New formula for the critical external lateral pressure around the cylindrical shell GS08—Guo-Hua Nie (China, PRC): An exact analysis of buckling of imperfect shallow spherical shells on an elastic foundation GS09—Tyler E. Bruns and Daniel A. Tortorelli (USA): Topology optimization of nonlinear elastic structures and compliant mechanisms GS10—Bogdan Bochenek and Jacek Kruzelecki (Poland): Selected problems of optimization for postbuckling behavior GS11—Makoto Ohsaki (Japan): Sensitivity analysis of critical load factor of symmetric systems for minor imperfection GS12—Donald W. Raboud, A. William Lipsett, and M. Gary Faulkner (Canada): An evaluation of the stability of flexible end loaded cantilever beams GS13—Sylvain Drapier, Lionel Léotoing, and Alain Vautrin (France): Influence of scale effects on the buckling of sandwich structures GS14—Rüdiger Schmidt; Liviu Librescu (Germany): Geometrically nonlinear theory of laminated shells weakened by interlaminar bonding imperfections

Scientific program

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GV—Low-Reynolds-number flow, microfluid mechanics, and fluid mechanics of materials processing (Patrick Huerre, France, chair) GV01—Evgeny S. Asmolov (Russia): Unsteady flow past a sphere undergoing unidirectional motion at small Reynolds number GV02—Avinoam Nir and Yaron Rosenstein (Israel): Hindered rotation in a dilute suspension GV03—Alexander Prokunin (Russia): Towards a paradox in the rough-particle motion along a wall in a liquid

GV04—Nadav Liron and Efrath Barta (Israel): Motion of a particle towards and within an orifice of finite length at zero Reynolds number GV05—Xueyan Guo and Antonio Delgado (Germany): Numerical simulation of pure elongational flow around particle-aggregates GV06—Asimina Sierou and John F. Brady (USA): Accelerated Stokesian dynamics simulations of concentrated suspensions GV07—Slavtcho G. Slavtchev and Svetla P. Miladinova; Georgy Lebon; Jean-Claude Legros (Bulgaria): Long wave instabilities of non-uniformly heated falling films GV08—Sahraoui Chaïeb; Gareth H. McKinley (USA): Mixing immiscible fluids: Drainage induced cusp formation GV09—Antonio Castellanos, Heliodoro Gonzalez, Antonio Ramos, and Francisco J. Garcia (Spain): Dynamics of liquid columns subjected to magnetic or electric fields GV10—Andrey V. Kuznetsov (USA): Investigation of diffusion effects in free surface flows

with solidification GV11—Emily S. Nelson; Phillip Colella (USA): Infiltration dynamics of reactive melt infiltration GV12—John B. Szczech and Constantine M. Megaridis; Dan Gamota and Jie Zhang (USA): Manufacture of microelectronic circuitry by drop-on-demand dispensing of nanoparticle suspensions GV13—Carl D. Meinhart and Shannon W. Stone; Steve Wereley (USA): A microfluidicbased nanoscope HA—Convective phenomena (Detlef Lohse, The Netherlands, chair) HA2—Alexander Yu. Gelfgat, Pinhas Z. Bar-Yoseph, and Alexander Solan; Eliezer Kit (Israel): Axisymmetry breaking of natural convection in a cylinder with a parabolic sidewall

temperature HA3—Blas Echebarria and Hermann Riecke (USA): Defect chaos in non-Boussinesq rotating convection HA4—Friedrich H. Busse, Oliver Brausch, Markus Jaletzy, and Werner Pesch (Germany): Hexarolls, knot convection and phase turbulence in the rotating cylindrical annulus HA5—Karen E. Daniels, Brendan B. Plapp, and Eberhard Bodenschatz (USA): Inclined layer convection HA6—Rafael Delgado-Buscalioni and Emilia Crespo del Arco (Spain): Dynamical coupling between shear rolls and a longitudinal thermal wave in a 3D inclined cavity HB—Flow control (Peter W. Carpenter, U K , chair) HB1—Jeffrey D. Crouch, Gregory D. Miller, and Philippe R. Spalart (USA): Active control for breakup of airplane trailing vortices HB2—Markus Schatz, Holger Lübcke, and Ulf Bunge (Germany): Flow simulation around movable flaps under high-lift conditions HB3—Nadine Aubry and Zhihua Chen (USA): Electro-magnetic feedback control of wake flows HB4—Dietmar Rempfer and John L. Lumley (USA): Low-dimensional models for turbulent shear-flow control HB5—Owen R. Tutty; Lubomir Baramov and Eric Rogers ( U K ) : Low dimensional robust control of channel flow HB6—Rama Govindarajan and Balaji T. Ranganathan (India): Control of flow stability in a channel by location of viscosity stratified fluid layer HC—Flow instability and transition (Sidney Leibovich, USA, chair)

HC1—Paul Manneville (France): Modeling of transitional plane couette flow HC2—Bruno Eckhardt, Armin Schmiegel, and Holger Faisst (Germany): Transition to turbulence in plane Couette flow HC3—Alp Akonur and Richard M. Lueptow (USA): Two-plane velocity field in wavy Taylor– Couette flow HC4—Gerd Pflster, Jan Abshagen, and Arne Schulz (Germany): Higher instabilities in Taylor–Couette flow HC5—Hendrik C. Kuhlmann, Stefan Albensoeder, and Hans J. Rath (Germany): 3D-flow instability in generalized lid-driven cavities

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ICTAM 2000

HC6—Alexander Solan, Alexander Yu. Gelfgat, and Pinhas Z. Bar-Yoseph (Israel): Threedimensional instability of flow in a rotating lid-cylinder enclosure HD—Elasticity (Alan S. Wineman, USA, chair) HD1—Christian Licht; Oana Iosifescu and Gérard Michaille (France): Variational limit of a one-dimensional discrete system of material points HD2—Lev Truskinovsky (USA): Is a chain with bi-stable springs elastic? HD3—David M. Haughton (UK): Nonlinear elastic materials which behave differently in tension and compression HD4—Roderic Lakes (USA): Deformation of elastic and viscoelastic chiral materials HD5—Raymond W. Ogden (UK): A pseudo-elastic model for damage in elastic solids under large strain HD6—Millard F. Beatty (USA): The Mullins effect in combined triaxial stretch and simple shear of a block HE—Structural Optimization (jointly with ISSMO) (Sándor Kaliszky, Hungary, chair) HE1—George I. N. Rozvany (Hungary): Latest developments in the exact layout theory of grid-like continua HE2—Michael Ryvkin, Moshe B. Fuchs, and Eyal Moses (Israel): Topologtcal optimization of structures with 2D translational symmetry HE3—Richard J. Balling (USA): A blueprint for simultaneous analysis and optimization of structural topology via multigrid HE4—Tadeusz S. Burczynski, Witold Beluch, Marek Nowakowski, and Piotr Orantek (Poland): Hybrid evolutionary methods in structural optimization HE5—Martin P. Bendsøe; Ole Sigmund (Denmark): Grey-scales in topology optimization: Hashin–Shtrikman bounds and realization by composites HE6—Bing Chung Chen and Noboru Kikuchi (USA): Topology optimization of structures and compliant mechanism with design dependent loads HF—Fatigue (Bruno A. Boley, USA, chair) HF1—Masahiro Jono and Atsushi Sugeta (Japan): Atomic force microscope observation and mechanism of fatigue crack growth HF2—Tung Hua Lin; Naigang G. Liang; Kevin F. F. Wong; Ning J. Teng (USA): Micromechanic analysis of crack initiation and hysteresis loops of aluminum single crystals HF3—Yoshikazu Nakai (Japan): Micro-mechanisms of slip-band growth and crack initiation in fatigue of 70-30 brass HF4—Jian-Ku Shang (USA): Micromechanisms of interfacial fatigue in solder interconnects HF5—Keisuke Tanaka and Yoshiaki Akiniwa (Japan): Microstructural effect on crack closure and propagation thresholds of small fatigue cracks HF6—Robert O. Ritchie and Da Chen (USA): Fatigue of ceramics at elevated temperatures HG—Computational Solid mechanics (jointly with IACM) (Pierre Ladevèze, France, chair)

HG1—Walter Wunderlich and Rudolf Findeiss; Harald Cramer (Germany): Space and time adaptive computation of localized failure in geomechanics HG2—Scott G. Bardenhagen; Jeremiah U. Brackbill; Deborah L. Sulsky (USA): A numerical study of sheared granular material HG3—Osamu Kuwazuru; Nobuhiro Yoshikawa (Japan): Nonlinear finite element for plain woven fabrics HG4—Sohichi Hirose; Can-Yun Wang; Jan D. Achenbach (Japan): Boundary element method for elastic wave scattering by a crack in an anisotropic solid HG5—Tod A. Laursen (USA): New conservative finite element algorithms for dynamic inelastic impact HG6—Erik Lund; Jorn S. Hansen (Denmark): Two-dimensional shape sensitivity analysis: A fixed basis function finite element approach

HH—Contact and friction problems (jointly with IAVSD) (James R. Barber, USA, chair) HH1—Nadia Lapusta; James R. Rice (USA): Instability of dynamic frictional sliding HH2—Michel Raous and Iulian Rosu; João A. C. Martins (France): Instabilities and nonsmooth solutions related to frictional contact HH3—Bernardino M. Chiaia, Mauro Borri-Brunetto, and Alberto Carpinteri (Italy): Contact force transmission at rough interfaces: Numerical and experimental investigation HH4—Pakalapati T. Rajeev and Thomas N. Farris; Irina G. Goryacheva (USA): Wear in partial slip contact HH5—George G. Adams and Mikhail Nosonovsky (USA): Elastic waves induced by the frictional sliding of two elastic half-spaces HH6—Marius Cocu, Elaine Pratt, and Jean-Marc Ricaud (France): Analysis of a class of dynamic viscoelastic contact problems with friction

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HK—Waves (Chiang C. Mei, USA, chair) HK1—Roger H. J. Grimshaw (UK): Models for instability in inviscid fluid flows HK2—Lev Shemer, Haiying Jiao, and Eliezer Kit (Israel): Nonlinear wave groups: Experi-

ments versus simulations based on the Zakharov and Dysthe models HK3—Brenny (E.) van Groesen and Jaap-Harm Westhuis; Renee H. M. Huijsmans (The Netherlands): Unstable evolution of bichromatic wave trains HK5—Alfred Kluwick; Edward A. Cox; Stefan Scheichl (Austria): Negative and nonclassical hydraulic jumps in two layer fluid flow HK6—Gregory P. Chini; Sidney Leibovich (USA): Nonlinear Langmuir circulation–internal wave interactions HL—Granular flows (James T. Jenkins, USA, chair) HL1—Stefan Luding, Mark Laetzel, and Hans J. Herrmann (Germany): Macroscopic material properties from microscopic simulations of a 2D shear-cell HL2—Olivier Pouliquen and Yoel Forterre (France): Friction law for dense granular flows down rough planes HL3—Stephen C. Keast and Michel Y. Louge (USA): On dense granular flows down flat frictional inclines HL4—E. Bruce Pitman (USA): Modeling and simulation of debris flows

masses across 3-D terrain HL6—Howard A. Stone and Johann M. Schleier-Smith (USA): Puddling, heaping, and cracking in vertically vibrated liquid-saturated granular material HO—Dynamic plasticity Of Structures (Victor P. W. Shim, Singapore, chair)

HO1—Li-Lih Wang (China, PRC): Dynamic unloading response of structures at high strain rates HO2—Michael J. Forrestal; Andrew J. Piekutowski (USA): Penetration of aluminum targets by ogive-nose steel projectiles at striking velocities to 3.0 km/s HO3—Stephen R. Reid; Jia Ling Yang; Tongxi Yu (UK): Large deflection elastic-plastic, hardening-softening model for pipe whip HO4—William J. Stronge (UK): Generalized impulse and generalized momentum applied to

multibody impact with an energetic coefficient of restitution HO5—Victor P. W. Shim and Liming Yang (Singapore): Relationship between microstructure and the dynamic plastic response of crushable foam under impact HO6—Fan Song, Xiaolei Wu, and Yilong Bai (China, PRC): Some distinctive microstructures and mechanical behaviors of abalone nacre HR—Control of Structures (Alexander F. Vakakis, USA, chair) HR1—Vekatesh Deshmukh and S. C. Sinha (USA): Control of a class of nonlinear systems with time periodic coefficients HR2—Gabor Stepan and Laszlo Kovacs (Hungary): Dynamics of structures subjected to digital force control HR3—Kurt Schlacher and Andreas Kugi (Austria): Active control of smart structures, a Lagrangian approach HR4—Fabrizio Vestroni; Vincenzo Gattulli (Italy): Non-collocated longitudinal control for oscillating suspended cables

HR5—Alois Steindl (Austria): Stabilizing a flexible dumbbell satellite by tension control HR6—Uwe Jungnickel and Peter Maißer (Germany): Dynamic control of under-actuated systems HS—Rock mechanics and geomechanics (Bernhard A. Schrefler, Italy, chair) HS1—Joseph F. Labuz and Li-Hsien Chen; Luigi Biolzi (USA): Fracture of rock from wedge indentation HS2—Alexei A. Savitski and Emmanuel Detournay; Dmitry I. Garagash (USA): Asymptotic solutions for a penny-shaped fluid-driven fracture HS3—Ioannis Vardoulakis; Maria Stavropoulou; George Exadaktylos (Greece): Nonlinearity of hard crystalline rocks in uniaxial tension–compression loading HS4—Roberto Nova and Riccardo Castellanza (Italy): Modelling weathering effects on the mechanical behavior of soft rocks HS5—Felix A. Darve and Farid Laouafa (France): Instability mechanisms in granular materials

HS6—Sol R. Bodner; Moshe Merzer (Israel): An explanation of a possible magnetic field precursor to the 1989 Loma Prieta earthquake

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ICTAM 2000

HV—Complex and Smart fluids (Eric S. G. Shaqfeh, USA, chair) HV1—Lei Li and Ronald G. Larson (USA): Brownian dynamic simulations of dilute polystyrene and DNA solutions in extension and shear flows HV2—Bamin Khomami and Madan Somasi (USA): Modeling of viscoelastic flows in complex

geometries by Brownian dynamics/finite element techniques HV3—Peter Wapperom and Roland Keunings (Belgium): Numerical simulation of transient

flow of polymer melts with integral models HV4—Roger I. Tanner, Yurun Fan, and Nhan Phan-Thien (Australia): Computations of

chaotic rheological mixing HV5—Daniel J. Klingenberg and Yuri M. Shkel (USA): Magnetorheology and magnetostriction of anisotropic composites of ferromagnetic particles HV6—Andrejs Cebers and Ivars Drikis (Latvia): Magnetic fluid free interface dynamics in Hele-Shaw cells IA—Boundary layers (Hans H. Fernholz, Germany, chair) IA1—Richard E. Hewitt and Peter W. Duck ( U K ) : Non-axisymmetric rotating-disk flows IA2—Jan L. Van Ingen and Dirk M. Passchier (The Netherlands): Experimental investigations of transitional and turbulent flows for airfoil analysis and design IA3—Vladimir B. Zametaev and Marina A. Kravtsova (Russia): 3D interacting flow theory and lateral edge stall IA4—Nickolay V. Semionov, Alexander D. Kosinov, and Viktor V. Levchenko (Russia): Experimental study of disturbances development in a supersonic boundary layer on a swept wing IA5—Anatoly I. Ruban and Imrahim Turkyilmaz (UK): On laminar separation from a corner point of a rigid body contour in transonic flow

IA6—Thierry Maeder, Nikolaus A. Adams, and Leonhard Kleiser (Switzerland): Direct

numerical simulation of supersonic turbulent boundary layers IB—Biological fluid dynamics (Timothy J. Pedley, UK, chair) IB1—Bruce G. Bukiet, Hans R. Chaudhry, and Thomas Findley; Arthur B. Ritter (USA): Residual stresses in oscillating thoracic arteries increase blood flow IB2—Shi-Gui Wu, Li-Min Hu, Ai-Ke Qiao, Yu-Hua Peng, Jia-Quan Wang, and You-Jun

Liu (China, PRC): A numerical approach to nonlinear blood flow in arterial stenosis IB3—Marie Oshima and Ryo Torii; Toshio Kobayashi; Kiyoshi Takagi (Japan): The hemodynamic study of the cerebral artery using numerical simulations based on the CT angiog-

raphy IB4—Kazuo Tanishita, Takaaki Deguchi, and Tetsuya Tsuji; Takako Nishiya and Yasuo Ikeda (Japan): Motion of artificial platelet in the model small artery IB5—Joe S. Hur and Eric S. G. Shaqfeh; Hazen P. Babcock, Douglas E. Smith, and

Steven Chu (USA): Dynamics of dilute and semi-dilute dna solutions in shear flow IB6—Toshiro Ohashi, Shinji Seo, Yasuaki Ishii, and Masaaki Sato (Japan): Intracellular stress distribution and cytoskeletal structure in sheared endothelial cells IC—Turbulence (Marcel R. Lesieur, France, chair) IC1—Elias Balaras, Ugo Piomelli, and Andrea Pascarelli (USA): Turbulent structures in accelerating boundary layers IC2—Alexander P. Kozlov and Nikolay I. Mikheyev; Colin J. Bates; Valery M. Molochnikov (Russia): On some characteristics of wall turbulence in separated flows IC3—Olivier J. Métais and Martin Salinas-Vasquez (France): Large eddy simulations of the turbulent flow in a rectangular duct with asymmetrical wall-heating IC4—Nikolai V. Nikitin; Sergey I. Chernyshenko (Russia): Regeneration mechanism of the near-wall turbulence IC5—Mosa Chaisi and Derek Stretch; James W. Rottman (South Africa): Turbulence and waves at a gas–liquid interface IC6—K. Johan A. Westin and P. Henrik Alfredsson; Alessandro Talamelli (Sweden): Experimental investigation of a correction procedure for hot-wire x-probes in shear flows ID—Fracture and crack mechanics (jointly with ICF) (Walter J. Drugan, USA, chair) IDl—Viggo Tvergaard (Denmark): Resistance curves for mixed mode interface crack growth between dissimilar elastic–plastic solids ID2—Lew V. Nikitin; Vladimir N. Odintsev (Russia): Tensile fracture of brittle solids under compression ID3—Huajian Gao; Yonggang Huang; Farid F. Abraham (USA): Continuum and atomistic analysis of intersonic crack propagation ID4—Dhirendra V. Kubair and Philippe H. Geubelle (USA): Intersonic crack propagation in homogeneous media under shear-dominated mixed-mode loading

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lvii

ID5—Leon M. Keer and Igor A. Polonsky (USA): Stress analysis of solids with complex crack patterns based on the Mura formula ID6—Jianxiang Wang and Jing Fang; Bhushan L. Karihaloo (China, PRC): Asymptotic bounds on overall moduli of cracked bodies IE—Structural optimization (jointly with ISSMO) (John E. Taylor, USA, chair) IE1—Daniel A. Tortorelli, Nahil A. Sobh, and Robert B. Haber; Robert W. Hyland Jr. (USA): Quench process modeling and optimization IE2—Vassili V. Toropov and Luis F. Alvarez ( U K ) : Selection of structure of response surface approximation models using genetic programming IE3—Satchi Venkataraman and Raphael T. Haftka; John T. Wang and Chauncey K. Wu (USA): Response surfaces for predicting load redistribution in multi-level structural optimizations IE4—Zenon Mróz; Krzysztof Dems (Poland): Damage identification using natural frequency measurement for varying support and mass parameters IE5—Pauli Pedersen (Denmark): Sensitivity of the optimal shapes to changes in material parameters IE6—Uri Kirsch and Panos Y. Papalambros (USA): Accurate displacement derivatives for structural optimization using approximate reanalysis IF—Plates and shells (jointly with IACM) (Christopher R. Calladine, UK, chair) IF1—Arthur W. Leissa (USA): Some observations on plate and shell analysis IF2—Wojciech Pietraszkiewicz (Poland): The nonlinear theory of thin irregular shell structures in terms of rotations IF3—Timothy J. McDevitt; James G. Simmonds (USA): Reduction of the Sanders–Koiter equations for fully anisotropic circular cylindrical shells IF4—J. N. Reddy; C. M. Wang (USA): Relationships between the solutions of the classical and shear deformation plate theories IF5—J. Mark F. G. Holst; Christopher R. Calladine ( U K ) : Surface-inversion phenomena in thin-walled elastic structures IF6—Herbert A. Mang and Christian Schranz (Austria): Ab initio estimates of stability limits of shells by means of arc-length extrapolation IG—Mechanics of thin films and nanostructures (Jürg Dual, Switzerland, chair) IG1—Terry J. Delph; Ralph J. Jaccodine (USA): The mechanics of silicon oxidation IG2—Johannes Vollmann, Dieter Profunser, and Jürg Dual (Switzerland): Femtosecond ultrasonics for the characterization of thin films and microstructures IG3—Zhigang Suo and Min Huang; Qing Ma and Harry Fujimoto (USA): Thin film cracking and ratcheting caused by temperature cycling IG4—Lei Lian and Nancy R. Sottos (USA): Stress effects in ferroelectric thin films IG5—George C. Johnson, Peter T. Jones, Ming Ting Wu, and Zachary Bell (USA): Testing and analysis of thin film strength using micro-electro-mechanical systems 1G6—Kenneth M. Liechti and Kalyan C. Vajapeyahula; Hyacinth Cabibil and John M. White; Robb M. Winter (USA): Interfacial force microscopy studies of polymer / glass interphases

IH—Impact and wave propagation (Werner Goldsmith, USA, chair) IH1—Alexander M. Samsonov, Galina V. Dreiden, Alexey V. Porubov, and Irina V.

Semenova (Russia): Strain solitons in solids—and how to do them IH2—Rodney J. Clifton and Seung-Yong Yang (USA): Computational modeling of stresswave-induced martensitic phase transformations IH3—Ajit Mal; Jungki Lee (USA): Wave propagation in a locally heterogeneous unbounded solid IH4—Federico J. Sabina and Oscar Valdiviezo; Valery M. Levin (Mexico): Thermoelastic waves in randomly particulate composites IH5—Shiming Zhuang and Guruswaminaidu Ravichandran; Dennis E. Grady (USA): Shock wave propagation in layered heterogeneous media IH6—Pablo D. Zavattieri; Horacio D. Espinosa (USA): An investigation of rate dependent decohesion for capturing damage kinetics in brittle materials IK—Flows in thin films (Yves Couder, France, chair)

IK1—Lou Kondic; Javier Diez (USA): Nonlinear dynamics of thin film flows IK2—Jean-Marc Vanden-Broeck ( U K ) : Air blown solitary waves IK3—Christian Ruyer-Quil and Paul Manneville (France): On the introduction of viscous dispersion in the modeling of film flows down inclined planes IK4—Mahesh Tirumkudulu and Andreas Acrivos (USA): Coating flows within a rotating horizontal cylinder: Lubrication analysis and numerical computations

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ICTAM 2000

IK5—Boris Y. Rubinstein, Alexander A. Golovin, and Leonid M. Pismen (Israel): Effect of van der Waals forces on fingering instability of thermally driven contact lines IK6—Takao Yoshinaga and Takeshi Uchiyama (Japan): Nonlinear wave behavior on a liquid

sheet with a temperature difference between both surfaces IL—Multiphase flows (L. Gary Leal, USA, chair) IL1—S. Balachandar and Prosenjit Bagchi (USA): Steady and unsteady forces on a spherical

particle in nonuniform flows IL2—Simon Crispel; Alain Cartellier (France): Hindering effect and velocity variance in a sedimenting suspension IL3—Warren B. Tauber and Grétar Tryggvason (USA): Direct numerical simulations of primary breakup IL4—Alexander Z. Zinchenko and Robert H. Davis (USA): Shear flow of a highly concentrated emulsion of spherical drops by numerical simulation IL5—Alberto Aliseda, Franek Hainaux, and Juan C. Lasheras; Alain Cartellier (USA): Settling velocity and clustering of particles in homogeneous and isotropic turbulence IL6—Shu Takagi, Kazuyasu Sugiyama, and Yoichiro Matsumoto (Japan): New approach to the modeling of sub-grid scale structures of disperse multiphase flows IO—Stability of Structures (Stephen H. Crandall, USA, chair)

IO1—Alexei A. Mailybaev (Russia): Stability domains of non-conservative systems with small parametric excitation IO2—Paolo S. Valvo and Salvatore S. Ligarò (Italy): Stress-concentration in a partly wrinkled elastic membrane IO3—Theodoro Netto; Stelios Kyriakides (Brazil): Dynamic propagation and flip-flopping of buckles in pipelines

IO4—Anthony N. Kounadis and Charis J. Gantes (Greece): Approximate dynamic buckling loads of nonconservative structural systems via energy considerations IO5—Piergiovanni Marzocca and Liviu Librescu; Walter A. Silva (USA): Nonlinear sta-

bility and response of lifting surfaces via Volterra series IO6—Walter V. Wedig (Germany): Strong stability conditions in elastic structures under stochastic parametric loading IR—Willis Symposium (Pierre M. Suquet, France, chair) IR1—Davide Bigoni; Domenico Capuani (Italy): On decay effects in nonlinear elasticity IR2—Sia Nemat-Nasser (USA): Dislocations-based rate- and temperature-dependent deforma-

tion of FCC, BCC, and HCP metals IR3—Xi Chen and John W. Hutchinson (USA): The influence of foreign object damage on fatigue cracking IR4—John W. Morrissey; James R. Rice (USA): Perturbative simulations of crack front

waves IR5—Hansuk Lee and Ares J. Rosakis; L. Ben Freund; Elizabeth Kolawa (USA): Optical

measurements of curvatures of film-substrate systems in the nonlinear deformations range IR6—Jiang Yu Li and Kaushik Bhattacharya (USA): The macroscopic behavior of ferroelectric polycrystals: Variational bounds and estimations IS—Viscoelasticity and creep (John L. Bassani, USA, chair) IS1—Holm Altenbach, Chunxiao Huang, and Konstantin Naumenko (Germany): Numer-

ical creep-damage predictions in thin-walled structures based on standard and refined approaches IS2—Nobutada Ohno, Takaaki Ando, and Shiro Biwa; Takushi Miyake (Japan): Varia-

tional method for fiber stress profile analysis of unidirectional composites with matrix creep IS3—Petros Sofronis; Prasad B. R. Nimmagadda (USA): Stress redistribution in creeping matrix composite materials: Effect of interface slip and diffusion IS4—S. Leigh Phoenix, Chung-Yuen Hui, and David Shia; Leonid Kogan (USA): Time dependent fiber fragmentation in a composite with matrix creep leading to creep-rupture IS5—Ren Wang and Qi-Feng Yu (China, PRC): Life expectation of fiber reinforced composites IV—Vortex dynamics (Dale I. P u l l i n , USA, chair) IV1—Rene Steijl and H. W. M. Hoeijmakers (The Netherlands): Numerical simulation of cooperative instabilities in a counter-rotating vortex pair IV3—Vyacheslav V. Meleshko and Alexandre A. Gourjii; Andreas Doernbrack, Frank Holzaepfel, Thomas Gerz, and Thomas Hofbauer (Ukraine): Interaction of twodimensional trailing vortex pair with a shear layer IV4—Paul Billant; Jean-Marc Chomaz (France): Zigzag instability and the origin of the pancake turbulence

Scientific program

lix

IV5—Christophe Eloy, Patrice Le Gal, and Stéphane Le Dizès (France): Multipolar instability of a vortex in a deformed elastic cylinder IV6—Yasuhide Fukumoto (Japan): Three-dimensional motion of a vortex filament with elliptically deformed core JA—Flow instability and transition (Michio Nishioka, Japan, chair)

JA1—Jonathan J. Healey (UK): Wave envelope steepening effects in the Blasius boundary

layer JA2—Masaharu Matsubara; P. Henrik Alfredsson (Japan): Streaky structure in boundary

layer transition induced by free stream turbulence JA3—Donatella Ponziani, Carlo M. Casciola, and Renzo Piva; Francesco Zirilli (Italy): Formation of structures and regeneration mechanisms in by-pass transitional flows JA4—Maxim V. Ustinov (Russia): Streaks production by free-stream nonuniformity interaction with blunt leading edge JA5—Masahito Asai and Akira Fukuoka; Michio Nishioka (Japan): Experimental investigation of the instability of low-speed streak in a boundary layer JA6—Victor V. Kozlov, Genrih G. Grek, Mikhail M. Katasonov, and Valeriy G. Chernoray (Russia): Study of the development of a lambda-structure and its transformations into a turbulent spot

JB—Convective phenomena (S. Balachandar, USA, chair) JB1—Hao-wen Xi; Xiao-jun Li; James D. Gunton (USA): Phase turbulence in stress-free Rayleigh–Bénard convection JB2—Thierry Passot, Dimitri Laveder, Yannick Ponty, and Pierre-Louis Sulem (France): Scaling of the correlation length in rotating convection JB3—Detlef Lohse; Siegfried Grossmann (The Netherlands): Scaling in thermal convection: A unifying view JB4—Jackson R. Herring and Robert M. Kerr (USA): Prandtl dependence of Nusselt in direct numerical simulation JB5—Richard L. Fernandes and Ronald J. Adrian (USA): Large scale structure of turbulent Rayleigh–Bénard convection: Proper orthogonal decomposition analysis

JB6—Axel Günther and Philipp Rudolf von Rohr (Switzerland):

Visualization of fluid

temperature and velocity in Rayleigh–Bénard convection: Mean and second-order quantities JC—Fluid mechanics of materials processing (Wilhelm Schneider, Austria, chair)

JC1—Kenneth J. Craig, Danie J. de Kock, and Jan A. Snyman (South Africa): Continuous casting optimization using CFD and Dynamic-Q JC2—Valod Noshadi and Wilhelm Schneider (Austria): Internal melt flow in horizontal continuous casting JC3—Yuri M. Gelfgat (Latvia): Influence of combined electromagnetic fields on convective transfer processes in conducting fluids JC4—Christian Resagk, Joerg Grabow, and Ulrich Schellenberger (Germany): Optical characterization of interface waves using particle image velocimetry and laser vibrometer techniques JC5—Jean N. Koster and Hongbin Yin (USA): In-situ solidification and flow visualizations with x-rays

JC6—Taketoshi Hibiya, Takeshi Azami, and Shin Nakamura; Kusuhiro Mukai (Japan): Marangoni flow of molten silicon: Role of oxygen at melt surface JD—Elasticity (Paolo Podio-Guidugli, Italy, chair) JD1—Elena F. Grekova (Russia): On the choice of strain tensors for 3D mechanical polar media, shells, and magnetic materials JD2—Andrew N. Norris (USA): Stress invariance in three-dimensional elasticity JD3—Arthur H. England (UK): Finite elastic deformations of a tyre modelled as an ideal fibre-reinforced shell JD4—Thomas J. Pence and Jose Merodio (USA): Equilibrium shocks in a directionally reinforced nonlinearly elastic material JE—Material instabilities (Ahmed Benallal, France, chair)

JE1—Henryk Petryk (Poland): Uniqueness and stability in materials with multiple mechanisms of inelastic deformation JE2—Henrik M. Jensen (Denmark): Model of delamination buckling instability JE3—Tracy J. Vogler, Sheng-Yuan Hsu, and Stelios Kyriakides (USA): On the initiation

of kink bands in fiber composites JE4—Jean-Claude Grandidier; Sylvain Drapier; Michel Potier-Ferry (France): A structural approach of plastic microbuckling in long fibre composites

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ICTAM 2000

JE5—Karine Siruguet and Jean-Baptiste Leblond (France): Influence of inclusions upon void growth and coalescence in porous ductile solids JE6—Kenneth Runesson and Thomas Svedberg (Sweden): An adaptive finite element algorithm for gradient theory of plasticity __ __

JF—Multibody dynamics (Martin Arnold, Germany, chair) JF1—Lorenzo Trainelli and Marco Borri (Italy): Solution of constrained dynamical systems via an embedded projection method JF2—John M. Hansen (Denmark): Synthesis of kinematically driven systems using dynamics JF3—Jean W. Zu and Friedrich P. Rimrott (Canada): The collinearity principle applied to a gyrosatellite JF4—John J. McPhee, Marcus Scherrer, and Pengfei Shi (Canada): A unified modelling approach for electro-mechanical multibody systems JF5—Sotirios Natsiavas, George Vaklavas, and George Verros (Greece): Dynamics and control of vehicles with dual-rate suspension dampers JG—Computational solid mechanics (jointly with IACM) (Peter Wriggers, Germany, chair)

JG1—Tian Yun Wu and Gui Rong Liu (Singapore): A generalization of the differential quadrature method JG2—Peter Betsch and Paul Steinmann (Germany): Time finite elements for nonlinear elastodynamics JG3—Alain G. Combescure and Anthony Gravouil (France): A time-space multi-scale algorithm for transient structural non linear problems JG4—Pierre Ladevèze, David Dureisseix, and Olivier Loiseau (France): A micro–macro computational strategy including homogenization for analysing heterogeneous structures JG5—Wing Kam Liu and Gregory J. Wagner (USA): Bridging scale hierarchical enrichment

JG6—Hannes Schmidt and Simon D. Guest (UK): Symmetry-adapted equilibrium of repetitive structures JH—Impact and wave propagation (Jan D. Achenbach, USA, chair)

JH1—K. T. Ramesh and Yulong Li; Ernest S. C. Chin (USA): Dynamic response of uniaxial continuous fiber-reinforced metal matrix composites JH2—Werner Goldsmith, Gabrielle Cipparrone, and Benjamin C. Bourne; Dennis L. Orphal (USA): Momentum exchange alteration in projectile / target impact JH3—Tongxi Yu; Jia Ling Yang; Stephen R. Reid (Hong Kong): Energy partitioning between two colliding beams JH4—Igor Ye. Telitchev (Russia): Study of burst conditions of thin-walled pressure vessels under hypervelocity impact JH5—Janusz R. Klepaczko and Ahmed Brara (France): Impact tension of concrete by spalling JH6—Min Zhou (USA): An experimental characterization of the impact failure of mortar JK—Waves (Lev A. Ostrovsky, USA, chair)

JK1—Mei C. Shen and Chia-Chin Chang (USA): Forced nonlinear surface waves in a waterfilled circular basin JK2—Chad M. Topaz and Mary Silber; Anne C. Skeldon (USA): Pattern selection of twofrequency forced Faraday waves JK3—Christian Kharif; Regis Pontier (France): Occurrence of new 3D patterns in water wave fields JK4—Evgeny A. Demekhin and Hsueh-Chia Chang (USA): Wave transitions in a falling film JK5—Gérard M. Iooss (France): Gravity and capillary-gravity travelling waves for superposed fluid layers, one being deep JK6—Triantaphyllos R. Akylas and David C. Calvo (USA): Stability of steep gravity– capillary solitary waves in deep water JL—Drops and bubbles (John R. Blake, UK, chair) JL1—Yulii D. Shikhmurzaev (UK): Coalescence and capillary breakup of liquid volumes JL2—Jerzy Blawzdziewicz, Vittorio Cristini, and Michael Loewenberg (USA): Nearcritical behavior of drops in linear flows JL3—Yuriko Renardy and Jie Li (USA): Breakup of a viscous drop under shear JL4—L. Gary Leal, Jong-Wook Ha, and Derek C. Tretheway (USA): Experimental studies of the deformation and breakup of polymeric drops JL5—David I. Bigio, Richard V. Calabrese, and Charles R. Marks (USA): Drop breakup and deformation in sudden onset strong flows JL6—Craig D. Eastwood and Juan C. Lasheras (USA): The breakup time of a droplet in a fully developed turbulent flow

Scientific program

lxi

JO—Cellular and molecular mechanics (Jüri Engelbrecht, Estonia, chair)

JO1—Sheldon Weinbaum; Jianjun Feng (USA): The mechanics of skiing from red cells to humans JO2—Dennis E. Discher, James Lee, Carol Kwok, and Philippe Carl (USA): From single

molecules to cell membrane mechanics JO3—Michael Elbaum and Hanna Salman (Israel): Dynamics of DNA uptake to the cell nucleus JO4—Christopher R. Calladine (UK): Mechanics of tether formation in liposomes JO5—Anna J. Diaz and Dominique Barthès-Biesel (France): Entrance of a cell into a pore JO6—Alexander A. Spector (USA): Nonlinear electroelastic constitutive relations for the cochlear outer hair cell JR—Willis Symposium (Pedro Ponte Castañeda, USA, chair)

JR1—Norman A. Fleck and Vikram Deshpande (UK): Constitutive models for foams JR2—Renaud Masson; Michel Bornert and André Zaoui; Pierre M. Suquet (France): An affine formulation for the prediction of the overall behavior of nonlinear heterogeneous media JR3—Graeme W. Milton and Sergey K. Serkov (USA): Neutral inclusions and composite assemblages JR4—Walter J. Drugan (USA): A higher-order nonlocal constitutive equation and optimal effective moduli for elastic composites JR5—Pierre M. Suquet, Herve Moulinec, and Jean-Claude Michel (France): A numerical

method for composites with complex micro structure JR6—Nicolas Triantafyllidis (USA): Onset-of-failure surfaces in periodic solids: applications

Theory and

JS—Microgravity mechanics (Vadim I. Polezhaev, Russia, chair) JS1—Constantine M. Megaridis and Kevin Boomsma; Dimos Poulikakos; Vedha Nayagam (USA): Contact angle dynamics of molten solder droplets impacting onto flat

substrates in reduced gravity JS2—Valentin S. Yuferev, Olga N. Budenkova, and Elvira N. Kolesnikova (Russia): Thermal convection under quasi-static component of microgravity field

JS3—Günter Wozniak; R. Balasubramaniam; R. S. Subramanian (Germany): Thermocapillary bubble and drop motion—Experiments versus theory JS4—Alexander L. Yarin; Günter Brenn, Oliver Kastner, and Dirk Rensink; Cameron Tropea (Israel): Evaporation of acoustically levitated droplets of pure liquids and binary liquid mixtures JS5—J. Iwan D. Alexander and Lev A. Slobozhanin (USA): Stability of doubly connected and disconnected free surfaces in a cylinder under microgravity JS6—You-Seop Lee, Hendrik C. Kuhlmann, and Hans J. Rath; Chung-Hwan Chun (Germany): Flow instability in thermocapillary liquid bridges with and without vortex breakdown JV—Topological fluid mechanics (H. Keith Moffatt, UK, chair)

JV1—Pieter G. Bakker and Michael Proot (The Netherlands): Topology of laminar juncture flows JV2—Morten Brøns; Lars K. Voigt and Jens N. Sørensen; Andreas Spohn; Johan Hartnack (Denmark): Two- and three-dimensional streamline topology in the vortex breakdown in a closed cylinder JV3—Konrad Bajer (Poland): Heat flow around a diffusing vortex JV4—Carlo F. Barenghi and David C. Samuels; Renzo L. Ricca (UK): The evolution of vortex knots JV5—Vladimir A. Vladimirov; H. Keith Moffatt; Konstantin I. Iline (Hong Kong): Generalized isovorticity conditions for fluid flows JV6—Robert W. Ghrist; John B. Etnyre (USA): Contact topological techniques for steady

Euler flows KD—Sectional Lecture by Peter A. Monkewitz (Nicholas Rott, USA, chair)

KD1—Peter A. Monkewitz (Switzerland): Flow instabilities and transition in spatially inho-

mogeneous systems KE—Sectional Lecture by Vladimir A. Palmov (Keh-Chih Hwang, China, chair)

KE1—Vladimir A. Palmov (Russia): Stationary waves in elasto–plastic and visco–plastic bodies

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ICTAM 2000

KP—Sectional Lecture by Erwin Stein (Günther R. Kuhn, Germany, chair)

KP1—Erwin Stein (Germany): Error controlled adaptivity for hierarchical models and finiteelement approximations in structural mechanics KQ—Sectional Lecture by Stephen J. Cowley (Roddam Narasimha, India, chair) KQ1—Stephen J. Cowley (UK): Laminar boundary-layer theory: A 20th century paradox? KW—Sectional Lecture by Erik van der Giessen (James R. Rice, USA, chair) KW1—Erik van der Giessen (The Netherlands): Plasticity in the 21st century KX—Sectional Lecture by Charles S. Peskin (Philip J. Holmes, USA, chair) KX1—Charles S. Peskin (USA): Muscle and blood: Fluid dynamics of the heart and its valves LD—Sectional Lecture by Shigeo Kida (Tomomasa Tatsumi, Japan, chair) LD1—Shigeo Kida (Japan): Vortical structure in turbulence LE—Sectional Lecture by R. Narayana lyengar (Jean Salençon, France, chair) LE1—R. Narayana lyengar (India): Probabilistic methods in earthquake engineering LP—Sectional Lecture by Gedeon Dagan (Sol R. Bodner, Israel, chair) LP1—Gedeon Dagan (Israel): Effective, equivalent and apparent properties of heterogeneous media

LQ—Sectional Lecture by Amable Liñán (Moshe Matalon, USA, chair) LQ1—Amable Liñán (Spain): Diffusion controlled combustion LW—Sectional Lecture by Subra Suresh (Johannes Weertman, USA, chair) LW1—Subra Suresh (USA): Nanomechanics and micromechanics of thin films, graded coatings and mechanical/nonmechanical systems LX—Sectional Lecture by Ole Sigmund (Pauli Pedersen, Denmark, chair)

LX1—Ole Sigmund (Denmark): Optimum design of Micro Electro Mechanical Systems (MEMS) MA—Flow instability and transition (Peter A. Monkewitz, Switzerland, chair)

MA1—William D. Thacker; Chester E. Grosch; Thomas B. Gatski (USA): Dynamics of ensemble-averaged linear disturbances in a transitioning boundary-layer MA2—Leonhard Kleiser, Dirk Wilhelm, and Carlos Haertel (Switzerland): A numerical simulation study of the 2D-3D transition in forward-facing step flow MA3—Herbert Steinrück (Austria): Upstream travelling waves in the boundary layer of a horizontal mixed convection flow MA4—Artur Chung-Che Kuo and Heinrich Fiedler (Germany): Weakly nonlinear model for the coherent structures in a curved turbulent mixing layer MA5—Michael G. Olsen; J. Craig Dutton (USA): Stochastic estimation of large structures in incompressible and weakly compressible mixing layers MA6—Jean-Marc Chomaz; Arnaud Couairon; Ivan Delbende (France): Nonlinear global modes in wakes behind bluff bodies MB—Geophysical fluid dynamics (Kolumban Hutter, Germany, chair)

MB1—Andrey N. Salamatin and Dina R. Malikova (Russia): Structural dynamics of ice sheets MB2—Dambaru Raj Baral; Kolumban Hutter (Nepal): Higher order asymptotic theories of large scale motion, temperature and moisture distribution in ice sheets

MB3—Gunter Gerbeth and Frank Stefani; Agris Gailitis, Olgerts Lielausis, and Ernests Platacis (Germany): Magnetic field self-excitation in the Riga dynamo experiment MB4—Sidney Leibovich; Thomas M. Haeusser (USA): Langmuir circulation patterns in a rotating ocean

MB5—Fabrice Veron and W. Kendall Melville (USA): Experiments on the initiation of Langmuir circulations and surface waves MB6—William R. Phillips (USA): Longitudinal vortices in wavy shear flows with a free surface MC—Low-Reynolds-number flow (Antonio Delgado, Germany, chair)

MC1—Jens G. Eggers (Germany): Stability of free surface cusps MC2—Jeremy Teichman, Maria-Isabel Carnasciali, Sahraoui Chaïeb, Gareth H. McKinley, and Lakshminarayanan Mahadevan (USA): Wrinkling of sheared viscous sheets MC3—Taher A. Saif and Chad Sager (USA): Interaction force between two thin solids on the surface of a liquid: Theory and experiment

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1xiii

MC4—Dominique Legendre and Jacques Magnaudet (France): The relation between the

forces acting on a spherical drop and those acting on a solid sphere MC5—Ehud Gavze; Michael Shapiro (Israel): Gravity induced drift of ellipsoidal inertial particles in a simple shear flow MC6—Ludwig C. Nitsche and Uwe Schaflinger; Gunther Machu and Walter Meile (Aus-

tria): Coalescence, torus formation and break-up of settling drops: Experiments and computer simulations MD—Mechanics of foams and cellular materials (Gustavo Gioia, USA, chair) MD1—Vikram Deshpande, Norman A. Fleck, and Michael F. Ashby (UK): Effective prop-

erties of a ”cubic” lattice material MD2—Michael K. Neilsen and William M. Scherzinger; Wei-Yang Lu; Andrew M. Kraynik (USA): Mechanics of aluminum honeycomb

MD3—Patrick R. Onck (The Netherlands): Modeling of notch effects in metallic foams MD4—Thomas Daxner, Franz G. Rammerstorfer, and Helmut J. Boehm (Austria): Adap-

tation of the density distribution in weight-efficient metal foam structures MD5—Mark W. Schraad and Francis H. Harlow (USA): A stochastic two-field approach to

modeling the mechanical response of cellular materials ME—Damage and failure of composites (S. Leigh Phoenix, USA, chair) ME1—Alexander E. Bogdanovich (USA): Three-dimensional analysis of inhomogeneous solids

using Bernstein polynomial approximations ME2—Irene J. Beyerlein; S. Leigh Phoenix; Linda S. Schadler (USA): Distributions and size scalings for strength and lifetime of a short-fiber composite

ME3—William A. Curtin (USA): Multiscale models of damage accumulation and failure in fiber-reinforced composites ME4—Zdenek P. Bazant; Drahomir Novak (USA): Energetic probabilistic theory of fracture

scaling of composites ME5—Olivier Allix and Pierre Ladevèze (France): Application of a meso-model of laminates

to low-energy impact ME6—Anthony A. Caiazzo; Francesco Costanzo (USA): Effective constitutive relations for

the structural analysis of components with evolving cracks MF—Structural vibrations (Utz von Wagner, Germany, chair) MF2—James F. Doyle; Elizabeth M. Webster (USA): A dynamic view of static instabilities MF3—Nathan van de Wouw and Dick H. van Campen; Henk Nijmeijer (The Netherlands):

Statistical bilinearization in stochastic nonlinear dynamics MF4—Alexander F. Vakakis; Igor Rozhkov (USA): Energy pumping phenomena in nonlinear

coupled oscillators MF5—Iliya I. Blekhman (Russia): Vibrational mechanics of dry friction and impact systems MF6—Sami F. Masri; Elias B. Kosmatopoulos; Andrew W. Smyth; Anastasios G. Chas-

siakos (USA): Robust neural estimation of internal forces in nonlinear structures under random excitation MG—Plasticity and viscoplasticity (Erik van der Giessen, The Netherlands, chair) MG1—K. Jimmy Hsia; Huajian Gao; Yun-Biao Xin (USA): Effects of dislocation nucleation

on fracture in silicon single crystals MG2—Kazuwo Imai, Satoshi Sugawara, Tsuyoshi Ganbe, and Michio Kurashige; Koji

Sumino (Japan): Dislocation plasticity of semiconductor crystals MG3—Yonggang Huang and Zhenyu Xue; Huajian Gao; William D. Nix (USA): A study

of micro-indentation hardness experiments by strain gradient plasticity MG4—Véronique Favier, Stéphane Berbenni, and Marcel Berveiller; Xavier Lemoine

(France): Coupling between elastic storage and viscoplastic dissipation by a new selfconsistent scheme MG5—Johannes A. W. van Dommelen, W. A. Marcel Brekelmans, and Frank P. T.

Baaijens; David M. Parks and Mary C. Boyce (The Netherlands): A micromechanical

model for the large-strain constitutive behavior of semi-crystalline polymers MG6—Mohammed A. Zikry and W. M. Ashmawi (USA): Microstructurally induced failure evolution and global failure in porous crystalline aggregates MH—Contact and friction problems (jointly with IAVSD) (João A. C. Martins, Portugal, chair) MH1—Joachim Larsson and Bertil Storåkers (Sweden): On inelastic impact and dynamic

hardness MH2—Sinisa Dj. Mesarovic; Norman A. Fleck (USA): Spherical indentation of elastic–plastic

solids

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ICTAM 2000

MH3—Lars-Erik V. Andersson (Sweden): Quasistatic frictional contact problems with finitely many degrees of freedom MH4—Hachmi Ben Dhia and Malek Zarroug; Patrick Massin and Isabelle Vautier (France): Finite elements/points for a hybrid formulation of contact-friction problems in large transformation MH5—Christoph Glocker (Germany): A standard NLCP formulation for rigid body systems with spacial Coulomb friction MH6—James A. Tzitzouris Jr. and Jong-Shi Pang (USA): Dynamic linear complementarity systems MK—Waves (W. Kendall Melville, USA, chair) MK1—Lev A. Ostrovsky (USA): Strongly nonlinear, multisoliton internal waves MK2—Takeshi Kataoka, Michihisa Tsutahara, and Tomonori Akuzawa (Japan): Transverse instability of finite-amplitude internal solitary waves in a weakly stratified fluid MK3—John Grue, Atle Jensen, Per-Olav Rusås, and Johan Kristian Sveen (Norway): On the dynamics of highly nonlinear solitary waves in a stratified fluid MK4—Vladimir E. Zakharov (USA): Weak and solitonic turbulence of gravity waves on the

surface of a finite-depth fluid MK5—Frédéric Dias (France): Wave turbulence in one-dimensional models MK6—Stefan G. Llewellyn Smith; Rupert Ford (USA): Acoustic scattering by vortices in the Born limit ML—Drops and bubbles (Andrea Prosperetti, USA, chair)

ML1—Philip L. Marston, Mark J. Marr-Lyon, and David B. Thiessen (USA): Novel methods for suppressing drop formation from capillary instabilities ML2—Claus-Dieter Ohl; Andrea Prosperetti (The Netherlands): Response of rising bubbles

to a sudden depressurization ML3—Daniel A. Weiss; Alexander L. Yarin; Günter Brenn and Dirk Rensink (Germany): Acoustically levitated drops: Drop oscillation driven by ultrasound modulation ML4—Tatyana P. Lyubimova; Dmitry V. Lyubimov and Sergey V. Shklyaev (Russia): Quasi-equilibrium shape of a drop on oscillating rigid plate ML5—Daniel Attinger, Vincent Butty, Stephan E. Haferl, and Dimos Poulikakos (Switzerland): Numerical modeling of microdroplet impact and solidification with substrate melting ML6—Steve Glod, Dimos Poulikakos, Ziqun Zhao, and George Yadigaroglu (Switzerland): An investigation of microscale explosive vaporization of water on an ultrathin Pt-wire MO—Biological solid mechanics (Lakshminarayanan Mahadevan, USA, chair)

MO1—Jacques M. Huyghe, Menno M. Molenaar, and Marcel W. Wijlaars; Arjan J. H. Frijns (The Netherlands): Mechanics of 3D finite swelling of ionised media MO2—Jüri Engelbrecht and Marko Vendelin; Peter H. M. Bovendeerd and Dick H. van Campen; Theo Arts (Estonia): Contraction of cardiac muscles and ATP consumption MO3—Alexander A. Stein (Russia): Modeling of plant tissue growth on the basis of continuum mechanics MO4—Timothy J. Van Dyke and Anne Hoger (USA): A comparison of two methods for analyzing finite growth of soft biological materials MO5—Alexander Rachev and Miglena Kirilova (Bulgaria): A mathematical model of arterial regeneration over bioresorbable grafts MS—Chaos in fluid and solid mechanics (Bruno Eckhardt, Germany, chair) MS1—Giuseppe Rega; Rocco Alaggio and Francesco Benedettini (Italy): Dimensionality

and reduced-order models for complex dynamics from experimental observations MS2—Frantisek Peterka and Stanislav Cipera; Tadashi Kotera (Czech Republic): Additional impact causes the intermittency chaos of instable subharmonic motions of impact oscillator MS3—Francis C. Moon; Masaharu Kuroda (USA): Complexity measures in large arrays of fluid-elastic oscillators MS4—Timothy J. Burns; Matthew A. Davies (USA): On solid-dynamical chaos in high-speed machining

MS5—Gert van der Heijden ( U K ) : The static deformation of a twisted elastic rod constrained to lie on a cylinder MS6—Philip J. Holmes; John Schmitt (USA): Mechanical models for insect locomotion: Dynamics and stability on the horizontal plane MV—Vortex dynamics (Vyacheslav V. Meleshko, Ukraine, chair)

MV1—Stéphane Le Dizès (France): Two-dimensional breakdown of a vortex subject to a rotating strain f i e l d

Scientific program

lxv

MV2—Henry M. Tufo; Paul F. Fischer (USA): A numerical study of hairpin vortices induced by a hemispherical roughness element MV3—Victor F. Kopiev and Sergey A. Chernyshev (Russia): On possible mechanism of turbulence generation outside the vortex ring core MV4—Helene Politano, Thomas Gomez, and Annick Pouquet; Michele Larcheveque (France): A Lundgren spiral vortex model for compressible turbulence

MV5—Keri A. Aivazis and Dale I. Pullin (USA): Velocity structure functions from the Hill spherical-vortex model for isotropic turbulence MV6—Anthony Leonard (USA): Evolution of localized packets of vorticity and scalar in turbulence NA—Granular flows and complex and smart fluids (Isaac Goldhirsch, Israel, chair) NA01—Frank Spahn and Miodrag Sremcevic (Germany): ”Fingerprints” of the size distribution in planetary rings? NA02—Farzam Zoueshtiagh and Peter J. Thomas (UK): Computational and experimental study of spiral patterns in granular media under a rotating fluid NA03—Robert P. Behringer and Sarath Tennakoon; Guy Metcalfe (USA): The transition to flow in a horizontally shaken system NA04—Milica Medved, Heinrich M. Jaeger, and Sidney R. Nagel (USA): Coupled response modes in horizontally vibrated granular material NA05—Osamu Sano, Akiko Ugawa, and Katsuhiro Suzuki (Japan): Regular and quasicrystal patterns on the vertically vibrated thin granular layer NA06—Clara Salueña and Thorsten Pöschel; Sergei E. Esipov (Germany): From ”solid” to ”fluid”: Time-dependent hydrodynamic analysis of dense granular flows NA07—Tatyana S. Krasnopolskaya; GertJan F. van Heijst and Jan H. Voskamp; Sergei A. Trigger ( U k r a i n e ) : Similarities of patterns in fluid and granular flow inside a horizontally rotating cylinder NA08—Nitin Jain and Julio M. Ottino; Kimberly M. Hill; Alp Akonur and Richard M. Lueptow ( U S A ) : Particle tracking techniques for measuring granular flow NA09—Christine M. Hrenya and Meheboob Alam; Richard Clelland (USA): Observations of inelastic collapse in granular shear flow NA10—Gregory Pacitto, Cyrille Flament, and Jean-Claude Bacri; Mike Widom (France): Rayleigh–Taylor instability at the interface of a ferrofluid NA11—Philippe Corvisier, Cherif Nouar, Rene Devienne, and Michel Lebouché (France): Analysis of the rheological structure evolution for a thixotropic fluid flow in a pipe NA12—Nathanael J. Woo; Eric S. G. Shaqfeh (USA): The rheology of polymer solutions in ultrathin films NA13—Abdulwahab S. Almusallam, Ronald G. Larson, and Michael J. Solomon (USA): Phenomenological model for droplet shape changes and stresses in the flow of immiscible

blends NA14—Stephan E. Haferl, Dimos Poulikakos, and Daniel Attinger; Ziqun Zhao (Switzerland): Fluid dynamics and solidification in the pile-up of picoliter droplets in micromanufacturing NB—Biomechanics (Sheldon Weinbaum, USA, chair) NB01—Jan Rosenzweig and Oliver E. Jensen (UK): Capillary–elastic instabilities and draining flows in buckled lung airways NB02—Hans R. Chaudhry and Bruce G. Bukiet; Arthur B. Ritter (USA): Mathematical analysis of heart reduction surgery NB03—Djenane C. Pamplona; Claudio R. Carvalho (Brazil): Numerical analysis of tissue expansion using finite elements method NB04—Hans I. Weber; Djenane C. Pamplona and Larissa M. Freire (Brazil): A simple experimental technique for the analysis of tissue expansion NB05—Werner Winter (Germany): Trabecular bone behavior and remodeling—based on equations of continuum damage mechanics NB06—Eduard Rohan and Robert Cimrman (Czech Republic): Identification of constitutive parameters for macroscopic modelling of smooth muscle tissues NB07—Jennifer H. Shin and Lakshminarayanan Mahadevan; Paul Matsudaira (USA): Dynamics of an actin spring NB08—Stephen M. Klisch and Anne Hoger (USA): Development of a growth mixture theory for cartilaginous tissues NB09—Henry W. Haslach Jr. (USA): A nonlinear dynamical model for bruit generation by an intracranial saccular aneurysm NB10—Tomasz Lekszycki (Poland): Modeling of bone adaptation process with use of structural optimization methods

NB11—Wendy C. Crone and Kumar Sridharan (USA): Surface modification of NiTi with plasma source ion implantation for biomedical applications

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ICTAM 2000

NB12—Sarah L. Thelen and L. Catherine Brinson (USA): Titanium foam for use in bone implants: Micro structure effects on mechanical properties NB13—Oren Vilnay, Konstantin Yu. Volokh, and Michael Belsky (Israel): Do living cells possess the structure of tensegrity? NC—Transition and turbulence (Gregory p. King, UK, chair) NC01—Luigi de Luca and Giro Caramiello (Italy): On some aspects of transient growth in fluid dynamics stability NC02—Anatoli Tumin and Genxing Han; Israel Wygnanski (Israel): Transition in pipe Poiseuille flow NC04—Alessandro Talamelli and Isabella Gavarini (Italy): Absolute and convective instability of coaxial jets NC05—Jorg Schumacher and Bruno Eckhardt (Germany): Evolution of turbulent spots in a plane shear flow NC06—Osamu Mochizuki, Masaru Kiya, and Hisashi Yamamoto (Japan): Precursor of separation NC07—Valery L. Okulov (Russia): Changes in the helical symmetry of the vorticity field during vortex breakdown into tubes NC08—Katsuhiro Suzuki, Masayuki Hirano, and Osamu Sano (Japan): An experimental study on the reconnection process of vortex tubes NC09—Yasuaki P. Kohama, Yasuhiro Egami, Mitsuyoshi Kawakami, Yoshiyuki Haruta, and Mitsuru Shimagaki (Japan): Transition mechanism, prediction, and control of crossflow field NC10—Yutaka Miyake, Koichi Tsujimoto, and Masaru Nakaji (Japan): Turbulence mixing in a rough-wall channel of two-dimensional square rods NC11—Jie Cui, Virendra C. Patel, and Ching-Long Lin (USA): Large-eddy simulation of

turbulent flow over k-type and d-type rough surfaces NC12—John C. Wells, Yaaufumi Yamamoto, Yuki Yamane, Shinji Egashira, and Hiroji Nakagawa (Japan): Time-resolved particle-image velocimetry in the cross-stream plane of an open channel NC13—Keiji Kishida; Keisuke Araki; Seigo Kishiba; Katsuhiro Suzuki (Japan): Orthonormal divergence-free vector wavelet analysis of nonlinear interactions in turbulence NC14—Honglu Wang, James R. Sonnenmeier, Stephan Gamard, and William K. George (USA): Evaluating DNS simulations of isotropic decaying turbulence using similarity theory ND—Fracture and crack mechanics (jointly with ICF) (Jean-Baptiste Leblond, France, chair)

ND01—Chang-Chun Wu and Zi-Ran Li; Ming Liu (China, PRC): Fracture assessment of piezoelectric material: Bound theorem and numerical implementation ND02—Tov Elperin, Gregory Rudin, and Arkady Kornilov (Israel): Formation of surface microcrack in non-metallic materials using laser thermal shock ND03—Giorgio Novati, Roberta Springhetti, and Antonio Cazzani; Attilio Frangi (Italy): 3D fracture analysis by the symmetric Galerkin boundary element method ND04—Martin L. Dunn and Paul E. W. Labossiere (USA): Fracture initiation at threedimensional bimaterial interface corners ND05—Benjamin A. Gailly; Horacio D. Espinosa (USA): A new model of ceramic based on crack propagation and motion of fragments ND06—Maria Kashtalyan and Costas Soutis ( U K ) : Multi-ply matrix cracking in multidirectional fibre-reinforced composite laminates ND07—Su Hao and Dong Qian; Patrick Klein (USA): A multi-scale damage model ND08—Jürgen Molter, Theo Fett, and Dietrich Munz; Dietrich Munz (Germany): Mixed mode fracture of interface cracks determined by a new fracture test and finite element method ND09—Michael A. Sutton, Fashang Ma, and Xiaomin Deng (USA): Mixed mode I/II cracktip fields ND10—Jerzy T. Pindera; Friedrich W. Hecker; Yuanpeng Zhang (Canada): Advanced experimental mechanics in three-dimensional stress determination ND11—Leon L. Mishnaevsky Jr. and Siegfried Schmauder; Nils Lippmann (Germany): Simulation of fracture in real structures of tool steels as a basis for the steel design NE—Contact and friction problems (jointly with IAVSD) (Anders I. Klarbring, Sweden, chair) NE01—Feodor M. Borodich (UK): Contact problems for fractal punches NE02—Ekaterina E. Pavlovskaya (Russia): Dynamic contact between rough bodies NE04—John M. Golden; G. A. C. Graham (Ireland): The problem of a viscoelastic cylinder rolling on a rigid half-space NE05—Paul M. Rightley, Robert A. Pelak, and James E. Hammerberg (USA): Interfacial dynamics of metal-on-metal sliding at high speeds and loads NE06—Kilian Funk, Andreas Stiegelmeyr, and Friedrich Pfeiffer (Germany): A compact time stepping algorithm for planar friction problems

Scientific program

lxvii

Some of the posters presented during the seminar presentation sessions on

29 August and 31 August 2000. NE07—Nikolay A. Lavrov (Russia): Dynamic interaction between elastic bodies having a slender area of contact NE08—Haralambos G. Georgiadis; Louis M. Brock; George Lykotrafitis (Greece): Thermoelastodynamic Green’s functions for problems of rapid sliding contact NE09—Faruk Yigit; Louis G. Hector Jr. (Saudi Arabia): A thermomechanical model of pure metal solidification on a deformable mold with sinusoidal surface NE10—Przemyslaw Zagrodzki (USA): Transient solutions of a thermoelastic contact problem

with fractionally excited instability NE11—Dong Qian, Shaofan Li, and Jian Cao (USA): 3D simulation of manufacturing process

by a meshfree contact algorithm NE12—Paolo Decuzzi; Giuseppe Monno (USA): The effect of disc thickness ratio on hot spotting in multiple-disc clutches and brakes NE13—Stephane Pagano, Karim Ach, Mikael Barboteu, and Pierre Alart (France): Modelling of multi-jointed structures NE14—Sirpa S. Launis and Erno K. Keskinen; Michel Cotsaftis (Finland): Mechanics of wood motion in groundwood manufacturing system NE15—Michelle Schatzman; Laetitia Paoli (France): Numerics for vibroimpact in many

degrees of freedom NF—Damage and failure of composites (Dusan P. Krajcinovic, USA, chair)

NF01—Qing-Jie Zhang, Peng-Cheng Zhai, Li-Sheng Liu, and Run-Zhang Yuan; ShinIchi Moriya and Masayuki Niino (China, PRC): Mechanics of functionally graded materials: Recent and prospective development NF02—Hodrigue Desmorat (France): Quasi-unilateral conditions in anisotropic elasticity NF03—James J. Mason and Carlos Rubio-Gonzalez (USA): Closed form solutions for dynamic stress intensity factors in composites NF06—Paolo M. Mariano, Giuliano Augusti, and Furio L. Stazi (Italy): Finite elements for a multifield model of microcracked bodies NF07—Chi L. Chow and Xianjie Yang (USA): An anisotropic damage model for forming limit diagram under nonproportional loading NF08—Isaac M. Daniel, Emmanuel E. Gdoutos, and Kuang-An Wang (USA): Failure modes in composite sandwich beams under three-point bending NF09—Anthony M. Waas and JungHyun Ahn (USA): Damage initiation and failure of notched composite laminates

lxviii

ICTAM 2000

NF10—George Z. Voyiadjis; Babur Deliktas (USA): Bridging length scales from meso to macro through gradient damage NF11—Yufu Liu, Yoshihisa Tanaka, and Chitoshi Masuda (Japan): Experimental and numerical study of transverse cracking in cross-woven fiber-reinforced composites NF12—Thomas A. Godfrey; John N. Rossettos (USA): The onset of tear propagation at large slits in biaxially stressed woven fabrics NF13—Federico M. Sciammarella and Cesar A. Sciammarella; Bartolomeo Trentadue (USA): Effect of interfaces in the mechanical properties of particle reinforced rubber composites NF14—Tungyang Chen (Taiwan): Inclusions with variable interfacial parameter and its effect on the effective conductivity tensor NG—Impact and wave propagation (Anders Boström, Sweden, chair) NG01—Avraham Benatar; Alexander L. Yarin and Daniel Rittel (USA): Experimental

measurement of the dynamic moduli of polymers using an instrumented Kolsky bar NG02—Weinong W. Chen and Fangyun Lu; Danny J. Frew (USA): Challenges and novel techniques in dynamic testing of soft materials NG03—Shinichi Maruyama, Toshihiko Sugiura, Akihiro Inoue, and Masatsugu Yoshizawa (Japan): Effects of a flaw in a specimen on ultrasonic detection by an elec-

tromagnetic acoustic transducer NG04—Daniel Gsell and Jürg Dual (Switzerland): Characterization of material properties in thin structures using a finite difference approach NG05—Robert B. Haber and Lin Yin; Amit Acharya (USA): A space–time discontinuous Galerkin method for elastodynamics using implicit–explicit grids NG06—Tobias Leutenegger and Jürg Dual (Switzerland): Numerical modeling of the influence of cracks on wave propagation in cylindrical structures NG07—Miroslav Premrov (Slovenia): An iterative method for solving harmonic elastodynamics of a halfspace NG08—Lev G. Steinberg (USA): Resonance of self-excited defect bar oscillations NG09—Serge N. Gavrilov (Russia): Passage through the critical velocity by a load moving along a string: Nonlinear approach NG10—Leonid I. Manevitch (Russia): Short wavelength asymptotics in nonlinear dynamics NG11—Mihai V. Predoi and Silviu Livescu (Romania): Complex wave numbers for elastic waves at plane interfaces NG12—Bo Yan and Zhanfang Liu; Xiangwei Zhang (China, PRC): Numerical simulation of two types of body waves and Rayleigh wave in fluid-saturated porous media NG13—Joseph A. Turner and Phanidhar Anugonda (USA): Elastic wave propagation and scattering in random, two-phase materials NG14—François Hild; Christophe Denoual (France): A multi-scale model for the dynamic fragmentation of silicon carbide ceramics NH—Smart materials and structures (Yuji Matsuzaki, Japan, chair) NH01—Alief N. Yahya and Wendy C. Crone (USA): Fabrication and characterization of nanostructured NiTi NH02—Chang-Sun Hong, Chi-Young Ryu, and Chon-Gon Kim (Korea): Experiments for smart composite structures using optical fiber sensors NH03—Fumihiro Ashida; Theodore R. Tauchert (Japan): Control of a distribution of transient thermoelastic displacement in a composite circular disk NH04—Reinaldo Rodríguez-Ramos, Raúl Guinovart, Idania Urrutia, and Julián Bravo;

Federico J. Sabina (Cuba): Overall properties in multiphase composite: Applications NH05—Xisheng Cao and Ulrich Gabbert (Germany): Crack analysis of smart composites with piezoceramics NH06—Hiroshi Asanuma, Haruki Kurihara, and Genji Hakoda (Japan): Fabrication of a new actuator material based on continuous SiC fiber reinforced aluminum composite NH07—Tarak Ben Zineb, Shabnam Arbab Chirani, and Etienne Patoor (France): Behavior modeling of piezoelectric ceramics NH08—Xianwei Zeng and R. K. Nimal D. Rajapakse (Canada): Toughening of conducting cracks due to domain switching NH09—Mitsunori Denda; Yook-Kong Yong (USA): Two-dimensional piezoelectric, timeharmonic dynamic BEM for quartz surface wave resonators NH10—Christophe Niclaeys, Tarak Ben Zineb, Shabnam Arbab Chirani, and Etienne

Patoor (France): Interaction between groups of variants in shape memory alloys NH11—Shabnam Arbab Chirani and Etienne Patoor (France): Martensite transformation criterion in shape memory alloys NH12—George J. Weng; Jackie Li (USA): Martensitic transformation of shape-memory alloys

and domain switch of ferroelectric ceramics

Scientific program

lxix

NH13—Hungyu Tsai and Xinjian Fan (USA): Elastic deformations in shape memory alloy fiber reinforced composites NK—Convective phenomena, fluid mechanics of materials processing, and microgravity mechanics (Simon S. Ostrach, USA, chair) NK01—Pierre Carlès (France): The convective stability of supercritical fluids NK02—Sakir Amiroudine; Andrei B. Kogan and Horst Meyer; Bernard Zappoli (Guadeloupe FWI): Convection onset in a near-critical fluid: Comparison between experiment and numerical simulation NK03—Alexander Roxin and Hermann Riecke (USA): Stability of waves coupled to an advected slow field NK05—Alexander A. Nepomnyashchy and Ilya B. Simanovskii; Leonid M. Braverman (Israel): Phenomenon of anticonvection NK06—Naftali A. Tsitverblit (Israel): Bifurcation from infinity in double-component convection driven by different boundary conditions NK07—Cho Lik Chan, Chuan F. Chen, and Wen-Yau Chen (USA): Salt-finger instability generated by lateral heating of a solute gradient NK08—Robert M. Kerr, Eileen M. Saiki, and William G. Large (USA): Numerical simulation of three-dimensional thermohaline fingering NK09—Haruhiko Kohno and Takahiko Tanahashi (Japan): Three-dimensional numerical simulation of unsteady Marangoni flows using GSMAC finite element method NK10—Gennady F. Putin, Igor A. Babushkin, Gennady P. Bogatyrev, and Alexander F. Glukhov; Sergey V. Avdeev, Alexander I. Ivanov, and Marina M. Maksimova (Russia): Measurement of thermal convection and low-frequency microaccelerations aboard orbital station Mir NK11—Haïk Jamgotchian, Nathalie Bergeon, Dominique Benielli, Philippe Voge, and Bernard Billia (France): Transient convective modes during solidification of succinonitrile0.2 wt % acetone alloy NK12—Valeriya A. Brailovskaya and Ludmila V. Feoktistova; Victor V. Zilberberg (Russia): Numerical modelling of nonstationary forced convection in solution at high rate crystal growth NK13—Lev A. Slobozhanin and J. Iwan D. Alexander (USA): The stability margin for zero-gravity liquid bridges NK14—Vadim I. Polezhaev, Oleg A. Bessonov, and Sergei A. Nikitin (Russia): Microgravity mechanics: A bridge between spacecraft’s accelerations and gravity-dependent systems NL—Computational Solid mechanics (jointly with IACM) (Subrata Mukherjee, USA, chair)

NL01—Elio Sacco; Daniela Addessi; Sonia Marfia (Italy): An elasto–plastic nonlocal damage model NL02—Eric Lorentz (France): Constitutive relations with gradient of internal variables NL03—Thomas Warren; Kevin L. Poormon (USA): Penetration of aluminum targets with spherical-nose steel projectiles at oblique angles NL04—Yasser M. Shabana; Naotake Noda (Japan): Thermal stresses in functionally graded materials taking the fabrication process into consideration NL05—Christof Messner and Gerd Reisner; Ewald Werner (Germany): Autocatalysis in the propagation of martensite bands in a shape memory alloy specimen under tension NL07—Magnus Ekh, Kenneth Runesson, and Lennart Mähler (Sweden): Thermoviscoplasticity for porous solids with application to sintering processes NL08—Xiaomin Deng and Chandrakanth Shet (USA): A finite element analysis of residual stresses and strains in orthogonal metal cutting NL09—Sylvie Aubry, Michael Ortiz, and Eduardo A. Repetto (USA): Computation of microstructure

NL10—Laurens P. Evers, W. A. Marcel Brekelmans, and Frank P. T. Baaijens; David M. Parks (The Netherlands): Meso-scale modeling of the elasto-viscoplastic behavior of polycrystal FCC metals NL11—Schalk Kok, Armand J. Beaudoin, and Daniel A. Tortorelli; Paul J. Maudlin (USA): Parameter estimation and texture augmentation of a polycrystal model through identification studies NL12—Claus Oberste-Brandenburg (Germany): Simulation of phase-transformations in steels using tensorial transformation kinetics NL13—Issam Doghri, Svetoslav Nikolov, Serge Munhoven, and Olivier Pierard (Belgium): Micromechanics of the small-strain behavior of polyethylene NM—Functionally graded materials, porous materials, phase transformations, and thin

films (Glaucio H. Paulino, USA, chair) NM01—Joseph Nadeau (USA): Sensitivity of optimal functionally graded systems to variations

in design parameters

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ICTAM 2000

NM02—Marek-Jerzy Pindera; Jacob Aboudi; Steven M. Arnold (USA): Thermal barrier coating stress management using graded bond coat architectures: A critical higher-order theory analysis NM03—Magnus Jinnestrand; Sören Sjöström (Sweden): Stress state in thermal barrier coatings from 3D finite element simulations NM04—Gregory Rudin and Tov Elperin (Israel): Thermal test of functionally graded materials using laser thermal shock method NM05—Ryan S. Elliott, John A. Shaw, and Nicolas Triantafyllidis (USA): Stability of thermally loaded NiTi perfect crystals NM06—Qing-Ping Sun (Hong Kong): On nucleation and propagation of martensite band in superelastic NiTi memory alloys under tension NM07—Christian Hellmich and Herbert A. Mang; Franz-Josef Ulm (Austria): Thermochemomechanics of phase transitions in shotcrete NM08—Toshio Tsuta, Yajun Yin, and Takeshi Iwamoto (Japan): Ductility and micro damage analyses in rigid plastic porous material with mixed hardening property NM09—Karim Beddiar, Yves Berthaud, and André Dupas (France): Thermo-hydroelectromechanics coupling: Application to electro-osmosis in porous media NM10—Robb M. Winter; Jack E. Houston (USA): Nanomechanical properties of polymeric systems measured by interfacial force microscopy NM11—Horacio D. Espinosa and Maximiliano Fischer (USA): Identification of residual stress state in a radio-frequency MEMS device NM12—J. Gregory Swadener; George M. Pharr (USA): Effects of surface stress state on nanoindentation measurements NM13—Sven Strohband and Reinhold H. Dauskardt (USA): Interface debonding under mixed mode loading in the presence of bridging tractions NM14—Takuya Uehara and Tatsuo Inoue (Japan): Molecular dynamics simulations of nucleation and crystallization processes in multi-layer thin films NN—Vehicle dynamics and control of Structures (Dick H. van Campen, The Netherlands,

chair) NN01—Gregory M. Hulbert and Fan-Chung Tseng (USA): A manifold orthogonal projection method for constrained dynamical systems NN02—Tony Postiau and Jean-Didier Legat; Paul Fisette (Belgium): Generation and parallelization of multibody system equations of motion by a symbolic approach NN03—Vaasilios Valtetsiotis; Robin S. Sharp (Germany): Car driving as an optimal linear preview control problem

NN05—All Haj-Fraj and Friedrich Pfeiffer (Germany): Optimization of gear shift operations in automatic transmissions NN06—Alex F. A. Serrarens, Roell M. van Druten, and Bas G. Vroemen (The Netherlands): Mechanical motor assist applied to a vehicle powertrain NN07—Chih-Yu Kuo, Regina A. G. Graf, Ann P. Dowling, and William R. Graham (UK): On the horn effect of a tyre–road interface NN08—Chijoke O. Mgbokwere (USA): Quenching of a ring gear blank NN09—Jan A. Snyman and Willie J. Smit (South Africa): The optimal design of a planar Stewart platform for prescribed machining tasks NN10—Jens Bormann and Heinz Ulbrich (Germany): Active vibration isolation of a hexapod system (Stewart platform) NN11—Anupam S. Ahlawat and Ananth Ramaswamy (India): Design of an optimal hybrid control system for wind excited buildings NN12—Daniil V. Iourtchenko and Mikhail F. Dimentberg; Alexander S. Bratus’ (USA): Optimal control of random vibrations by bounded stiffness variations NN13—Michael J. Winckler and Irena Otasevic (Germany): Fast and reliable computation of sensitivities for valvetrain design using ODEOPT NN14—Katja Dauster and Friedrich Pfeiffer (Germany): Observation of robotic manipulators in automated assembly NN15—Raphael R. Burgmair and Friedrich Pfeiffer (Germany): Modelling and simulation of a rubber seal assembly NS—Viscoelasticity, Creep, and fatigue (Nobutada Ohno, Japan, chair)

NS01—Marina V. Shitikova and Yuriy A. Rossikhin (Russia): A new method for solving dynamic problems of fractional calculus viscoelasticity NS02—Carl R. Schultheisz; Gregory B. McKenna (USA): Measurement and analysis of torque, axial force and volume change in the NIST torsional dilatometer NS03—Harry H. Hilton and Cristina E. Beldica (USA): Piezoelectric control of deformations and failure probabilities in viscoelastic Timoshenko beams

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lxxi

NS04—Ting-Qing Yang, Qunli An, and Wenbo Luo (China, PRC): Craze growth at a crack tip in polymers under creep condition NS05—Koichi Goda (Japan): Semi-discretization analysis for shear lag and creep rupture by the break-influence superposition method NS06—Aleksandra M. Vinogradov, Shane Schumacher, and Iakov M. Klebanov (USA): Creep response of polymers under cyclic loading conditions NS07—Christopher H. M. Jenkins and Zhiyu Liu; Isamu Kitahara and Robb M. Winter; Shane Schumacher and Aleksandra M. Vinogradov (USA): Morphology evolution in polymers subjected to vibrocreep NS08—Pranav Shrotriya and Nancy R. Sottos (USA): Influence of fabric architecture on viscoelastic properties of woven composites NS09—Qi Zhu and Philippe H. Geubelle (USA): Process-induced residual stresses and warpage in polymer composites NS10—Bong-Kyu Kim and Sung-Kie Youn (Korea): A constitutive model of rubber under small vibrations superimposed on large static deformation NS11—David A. Hills; Ludwig Limmer (UK): A theoretical and experimental investigation of fretting fatigue of complete contacts NS12—Richard W. Neu, Dana R. Swalla, and John A. Pape (USA): Plastic deformation in fretting fatigue: Experiments and modeling NS14—Thomas Erber; Sidney A. Guralnick (USA): Novel measurements of damage accumulation and the prediction of fatigue failure

NV—Combustion, compressible flow, and computational fluid dynamics (John Brindley, UK, chair) NV01—Moshe Matalon; Philippe Metzener (USA): Premixed flames in closed cylindrical tubes

NV02—Andy C. Mclntosh and Xinshe Yang; John Brindley (UK): The effect of large step pressure drops on strained premixed flames

NV03—Chun W. Choi and Ishwar K. Puri (USA): Flame stretch effects on partially premixed flames

NV04—Dieter E. Bohn and Joachim Lepers (Germany): Simulation of premixed combustion in highly turbulent flows: Comparison of a joint PDF and an eddy break-up model

NV05—Andrzej P. Szumowski; Gerd E. A. Meier (Poland): Airfoil flow instabilities induced by background flow oscillations NV06—Egon Krause; M. Kharitonov and M. D. Brodetsky (Germany): Supersonic lee-side flow on delta wings NV07—Takashi Adachi and Susumu Kobayashi (Japan): On the transition angle from Mach to regular reflection over concave surface of arbitrary shapes NV08—Piotr P, Doerffer, Jaroslaw Kaczynski, Krystyna Namisnik, and Ryszard Szwaba

(Poland): Flow structure at the interaction of oblique and normal shock waves NV09—Gerd E. A. Meier and Markus Raffel (Germany): Optical flow diagnostics by background oriented schlieren methods NV10—Vit Dolejsi (Czech Republic): Anisotropic mesh adaptation method for viscous compressible flows NV11—Jayandran Palaniappan, Robert B. Haber, and Robert D. Moser; Daniel A. Tortorelli (USA): A space–time discontinuous Galerkin procedure for nonlinear conservation laws NV12—Serge M. Prudhomme (USA): A goal-oriented adaptive strategy to enhance local structures in numerical flows NV13—Noboyuki Satofuka and Mitsuru Ishikura (Japan): Large scale numerical simulation of three-dimensional duct flows using lattice Boltzmann method

OA—Microfluid dynamics (Leen van Wijngaarden, The Netherlands, chair) OAl—John R. Melrose (UK): The transmission of force in flowing colloids OA2—Andreas Acrivos and Marco Marchioro (USA): Shear-induced particle diffusivities from numerical simulations OA3—Jason E. Butler and Eric S. G. Shaqfeh (USA): Dynamic simulations of sedimenting, rigid fibers OB—Biological f l u i d dynamics (Kazuo Tanishita, Japan, chair) OB1 —Hao Liu (Japan): A computational fluid dynamic study of the helical flow in aorta OB2—Olivier Boiron, Valérie Deplano, Boris Wilbois, and Robert Pelissier (France): A numerical and experimental study of physiological flows in a curved tube OB3 —Frans N. van de Vosse, Marco M. Stijnen, and Frank P. T. Baaijens (The Netherlands): Computational and experimental investigation of the flow in the left ventricle

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ICTAM 2000

OC—Turbulent m i x i n g (Anthony Leonard, USA, chair) OC1—Robert E. Breidenthal (USA): Super-exponential Rayleigh–Taylor flow OC2—Yasuhiko Sakai; Shinichi Nakajima; Ikuo Nakamura; Hiroyuki Tsunoda (Japan): The statistical analysis of two-particle structural diffusion in homogeneous isotropic turbulence OC3—William M. Pitts; Cecilia D. Richards; Mark S. Levenson (USA): Interactions between large- and small-scale structures in the scalar mixing field of a turbulent jet OD—Boundary layers (Anatoly I. Ruban, UK, chair) OD1—Yury S. Kachanov (Russia): On a universal nonlinear mechanism of breakdown to turbulence in wall bounded shear flows OD2—Scott E. Hommema and Ronald J. Adrian (USA): Structure of wall-eddies at

106

OD3—Jens M. Österlund and Arne V. Johansson; Hassan M. Nagib and Michael H. Hites (Sweden): Spectral characteristics of the overlap region in turbulent boundary layers

OE—Elasticity (Thomas J. Pence, USA, chair) OE1—Alexander N. Guz’ (Ukraine): On foundation of non-destructive method of determination of three-axial stresses in solids OE2—David M. Barnett (USA): One-component surface waves and supersonic non-radiating defects in triclinic linear elastic solids OE3—Ecevit Bilgili, Barry Bernstein, and Hamid Arastoopour (USA): Boundary layer formation in a sheared rubber-like slab subjected to thermal gradients

OF—Damage mechanics (Davide Bigoni, Italy, chair) OF1—Dusan P. Krajcinovic (USA): Statistical damage mechanics OF2—Quanshui Zheng and Danxu Du (China, PRC): An explicit and universal estimate for effective properties of composites and microcracked media OF3—Ron H. J. Peerlings, Marc G. D. Geers, and W. A. Marcel Brekelmans; René de Borst (The Netherlands): A comparative study of nonlocal and gradient-enhanced continuum damage models OG—Mechanics of thin films and nanostructures (Bertil Storåkers, Sweden, chair) OG1—Simon P. A. Gill, Fei Long, and Alan C. F. Cocks (UK): A continuum model for the growth and relaxation of multi-component epitaxial thin film systems OG2—Vijay Shenoy; L. Ben Freund (India): Compositional stability of alloy films OG3—Michael Ortiz, Eduardo A. Repetto, Kaushik Bhattacharya, and Yi-Chung Shu (USA): A continuum model of kinetic roughening and coarsening in thin films OH—Fatigue (Keisuke Tanaka, Japan, chair) OH1—Yukitaka Murakami (Japan): Mechanism of superlong fatigue failure in Nf ³ 107 and disappearance of conventional fatigue limit OH2—Ryuichiro Ebara (Japan): Influence of hydrogen on fatigue strength of structural steels OH3—Kwai S. Chan, Yi-Der Lee, David L. Davidson, and Stephen J. Hudak (USA): Propagation of small cracks under high cycle fretting fatigue

OK—Fluid-Structure interaction (Michael P. Païdoussis, Canada, chair) OK1—Anatol Roshko, Anthony Leonard, and Doug Shiels (USA): Flow induced vibration of a circular cylinder: A unified description OK2—Noel C. Perkins and Wanjun Kim (USA): Coupled slow and fast dynamics of flow excited cables OK3—Morten Huseby and John Grue (Norway): Experiments on higher harmonic wave forces on a vertical cylinder in periodic waves

OL—Drops and bubbles (Carlo Cercignani, Italy, chair) OL1—Grétar Tryggvason and Bernard Bunner (USA): Direct numerical simulations of bubbly flows OL2—Han E. H. Meijer, Frans N. van de Vosse, and Maykel Verschueren (The Netherlands): On the scaling of diffuse interface models for multiphase flows OL3—James P. Ferry; S. Balachandar (USA): A fast Eulerian method for particle-laden flows

OO—Electromagnetic processing of materials (jointly with HYDROMAG) (Peter A. Davidson, UK, chair) OO1—Martin P. Volz and Konstantin Mazuruk (USA): The effect of a rotating magnetic field on flow stability during crystal growth OO2—Hiroyuki Ozoe and Koichi Kakimoto; Masato Akamatsu; Yoo Cheol Won (Japan): Application of various magnetic fields for the melt in a Czochralski crystal growing system

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OO3—Jochen Friedrich and Oliver Gräbner; Bernd Fischer, Daniel Vizman, and Georg

Müller (Germany): Influence of magnetic fields on convection in Czochralski melts: Experimental and numerical results OS—Mechanics of phase transformations (jointly with IACM) (Franz-Josef Ulm, USA, chair) OS1—Franz-Josef Ulm; Luc Dormieux and Eric Lemarchand (USA): A micromechanical approach to the modeling of swelling due to alkali–silica reaction OS2—Gustavo Gioia; Alberto M. Cuitiño ( U S A ) : Two-phase particle rearrangement in cohesive powder densification OS3—Christian Lexcellent and Christophe Bouvet; Sylvain Calloch (France): Experimental investigations under multiaxial loadings on Cu–Al–Be shape memory alloy OW—Turbulence (William K. George, USA, chair) OW1—Tomomasa Tatsumi (Japan): The velocity distributions of homogeneous turbulence under the cross-independence closure hypothesis OW2—Benoit M. Chabaud, Bernard Hebral, Sylvain Pietropinto, and Philippe Roche; Bernard Castaing and Yves Ladam (France): Very high Reynolds and Rayleigh numbers experiments in low temperature gaseous helium OW3—Joseph J. Niemela, Ladislav Skrbek, and Russell J. Donnelly; Katepalli R. Sreenivasan (USA): Cryogenic turbulent convection PA—Microfluid dynamics (Leen van Wijngaarden, The Netherlands, chair) PA1—Todd M. Squires; Eric Dufresne and David G. Grier; Michael P. Brenner (USA): Like charge attraction and hydrodynamic interaction PA2—Mohamed Gad-el-Hak (USA): Flow physics in microdevices PA3—Marion N. Volpert, Carl D. Meinhart, Igor Mezic, and Mohammed Dahleh (USA): An actively controlled micromixer PB—Biological fluid dynamics (Kazuo Tanishita, Japan, chair) PB1—Shigeo Wada, Makoto Kojiya, and Takeshi Karino (Japan): Computational study of LDL transport in stenosed arteries based on a true-to-scale anatomical model PB2—Sylvie Lorthois; Francis Cassot and Jean-Pierre Marc-Vergnes; Pierre-Yves Lagree (France): Maximal wall shear stress in carotid stenoses and functionality of the

circle of Willis PB3—Jennifer S. Stroud and Stanley A. Berger (USA): Numerical simulation of blood flow in the severely stenotic carotid artery bifurcation PC—Turbulent mixing (Anthony Leonard, USA, chair) PCI—Daniel I. Meiron; Paul E. Dimotakis, Ronald Henderson, Branko Kosovic, Dale I. Pullin, and Ravi Samtaney (USA): Direct numerical simulation and subgrid-scale modeling of compressible turbulent mixing PC2—Qing-Zeng Feng (China, PRC): Inertial range scaling of passive scalars mixed by turbulence PC3—Amsini Sadiki (Germany): A new approach to modeling of turbulent mass and heat transport based on extended thermodynamics

PD—Granular flows (Prabhu R. Nott, India, chair) PD1—James T. Jenkins and Birgir Ö. Arnarson (USA): A theoretical analysis of a laboratory debris flow PD2—Joe D. Goddard and Anjani Kumar Didwania (USA): Fluid mechanics of vibrated

granular layers PD3—Michael Shapiro, Victor Royzen, Alexander Goldshtein, and Vladislav Dudko (Israel): Vibrofluidization and mixing in vibrated granular layers: From repacking to doubleimpact regimes

PE—Fracture and crack mechanics (jointly with ICF) (K. Jimmy Hsia, USA, chair) PEl—John R. Willis (UK): Dynamic perturbation of a crack propagating in a viscoelastic medium PE2—Alexander Chudnovsky; Boris Nuller; Michael Ryvkin (USA): Problem of crack interaction with bimaterial interface revisited PE3—Alexander B. Movchan ( U K ) : Asymptotic models of multi-structures with cracks PF—Material instabilities (Davide Bigoni, Italy, chair) PF1 —Kristina Nilsson (Sweden): Effects of inertia on dynamic neck formation during highrate tension PF2—Niels Sorensen (Sweden): Unstable neck formation in impact problems PF3—John L. Bassani (USA): Strain localization in ductile crystals

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ICTAM 2000

PG—Plasticity and viscoplasticity (Alan C. F. Cocks, UK, chair) PGl—João A. C. Martins and Fernando M. F. Simoes (Portugal): Flutter instabilities in non-associative elastic–plastic solids PG2—Nicolae D. Cristescu and Oana Cazacu (USA): Compaction and flow of granular materials on chutes PG3—Ann Bettina Richelsen; Erik van der Giessen (Denmark): Size effects of plasticity in thin sheets

PH—Fatigue (Keisuke Tanaka, Japan, chair) PH1—Reinhard Pippan; Franz O. Riemelmoser (Austria): The effect of discrete nature of plasticity on the fatigue crack propagation behavior PH2—Toshihiko Hoshide (Japan): Statistical aspect on simulated failure life of notched components in multiaxial fatigue PH3—Thomas N. Farris and Ganapathy Harish (USA): Coupled thermoelastic evaluation of fretting stresses: Application to life prediction

PK—Fluid–Structure interaction (Michael P. Païdoussis, Canada, chair) PK1—Sergey V. Sorokin (Russia): Nonlinear vibrations of elastic thin-walled structures in heavy fluid loading conditions PK2—Anthony D. Lucey, Gerard J. Cafolla, and Peter W. Carpenter ( U K ) : Evolution of flexible-wall disturbances in a boundary-layer flow PK3—Marco Amabili; Francesco Pellicano; Michael P. Païdoussis (Italy): Nonlinear dynamics and stability of circular cylindrical shells with flow P L — M u l t i p h a s e flows (Carlo Cercignani, Italy, chair) PL1—Dmitri O. Pushkin and Hassan Aref (USA): Self-similarity theory of stationary coagulation PL2—Ehud Yariv; Itzchak Frankel (Israel): Kinetic theory for bubbly flows PL3—Élisabeth Guazzelli, Paul Duru, and Maxime Nicolas; E. John Hinch (France): Constitutive laws in liquid fluidized beds

PO—Electromagnetic processing of materials (jointly with HYDROMAG) (Peter A. Davidson, UK, chair) POl—Toshio Tagawa; René J. Moreau and Guillaume Authié (Japan): Buoyant flow in long vertical enclosures under the presence of a horizontal magnetic field PO2—Valdis Bojarevics and Koulis Pericleous; Janis Freibergs (UK): Nonlinear wave models for aluminum reduction cells PO3—Ulrich Burr; Ulrich Müller (Switzerland): Rayleigh–Bénard convection under the influence of magnetic fields PS—Mechanics of porous materials (Wolfgang Ehlers, Germany, chair) PS1—Anjani Kumar Didwania; Reint de Boer (USA): Capillarity in porous solids—a continuum thermo-mechanical approach PS2—Martti J. Mikkola and Juha Hartikainen (Finland): Finite element solution of soil freezing problem by using bubble functions PS3—Pietro Cornetti, Alberto Carpinteri, and Bernardino M. Chiaia (Italy): Scaling laws in the mechanics of porous and damaged materials

PW—Flow instability and transition (William K. George, USA, chair) PW1—Stephane Leblanc (France): Destabilization of a vortex by acoustic waves PW2—Gilles Bouchet and Eric Climent; Agnès Maurel (France): An experimental investigation of free surface instabilities induced by an upward impinging jet PW3—Franck Plouraboué; E. John Hinch (France): Kelvin–Helmholtz instability in a HeleShaw cell QA—Turbulence (Hassan M. Nagib, USA, chair) QA1—Thomas S. Lundgren (USA): The pdf of the velocity difference between two points in homogeneous isotropic turbulent flow QA2—Susumu Goto and Shigeo Kida (Japan): A turbulence closure theory based on sparseness of nonlinear coupling QA3—Javier Jiménez (Spain): Imperfect multiplicative cascades in turbulence QB—Biological fluid dynamics (John O. Kessler, USA, chair) QB1—Matthias Heil and Joseph P. White (UK): Time-dependent non-axisymmetric instabilities of liquid-lined elastic tubes QB2—Sarah L. Waters; Caterina Guiot ( U K ) : Hæmodynamic aspects of the umbilical circulation

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QB3—Poul S. Larsen; Hans Ulrik Riisgård (Denmark): Chimney spacing in encrusting bryozoan colonies

QC—Electromagnetic processing of materials (jointly with HYDROMAG) (J. Iwan D.

Alexander, USA, chair) QC1—Nagy El-Kaddah; Thinium Natarajan; Ashish Patel (USA): Theoretical study of the magnetohydrodynamic flow around a sphere in crossed electric and magnetic fields QC2—Behrouz Abedian and Livia M. Racz; Roberts W. Hyers; Gerardo Trapaga (USA): Transition to turbulence in an electromagnetically levitated droplet QC3—Ramesh K. Agarwal (USA): A novel algorithm for computing combined free- and forced

convection magnetohydrodynamic flows QD—Fracture and crack mechanics (jointly with ICF) (John R. Willis, UK, chair) QDl—Leonid I. Slepyan (Israel): Distinctive solutions for fracture and phase transition in lattices QD2—Thomas H. Siegmund (USA): Local versus global failure: Mechanical integrity of thin walled structures QD3—John E. Dolbow; Nicolas Moes (USA): Modeling crack growth and fractional contact

with the extended finite element method QE—Functionally graded materials (Marek-Jerzy Pindera, USA, chair) QE1—Cornelius O. Horgan; Alice M. Chan (USA): Boundary-value problems for functionally graded linearly elastic materials QE2—Glaucio H. Paulino and Zhi-He Jin (USA): Viscoelasticity theory of functionally graded materials QE3—Fazil Erdogan, Serkan Dag, and M. A. Guler (USA): Contact and crack problems in functionally graded materials QF—Control of structures (Ali H. Nayfeh, USA, chair) QF1—Hans Irschik and Uwe Pichler (Austria): Dynamic deformation control of structures

by shaped piezoelectric actuators QF2—Martin Ruskowski and Karl Popp; Hans-Kurt Tönshoff and Richard Kaak (Germany): Modal control of a magnetically guided machine axis QF3 —Jon J. Thomsen; Dmitri M. Tcherniak (Denmark): Is Chelomei’s pendulum a beam? QG—Continuum mechanics (Donald E. Carlson, USA, chair) QG1—Francis J. Rooney, Stephen E. Bechtel; Qi Wang; M. Gregory Forest (USA): Models for thermal expansion: Ill-posedness of constrained theories and constitutive limits QG2—Gérard A. Maugin (France): Universality of the thermomechanics of forces driving singular sets in continuum mechanics QG3—James Casey (USA): On internally constrained thermoplastic continua QH—Smart materials and structures (Richard D. James, USA, chair) QH1—Yuji Matsuzaki (Japan): Thermomechanical studies of shape memory alloys based on interaction energy function QH2—Mark A. ladicola and John A. Shaw (USA): Parametric study of localized thermo-

mechanical behavior in shape memory alloy wire QH3—L. Catherine Brinson; Michele Brocca and Zdenek P. Bazant (USA): Three dimensional constitutive model for shape memory alloys based on microplane model QK—Drops and bubbles (Lev Shemer, Israel, chair) QK1 —Antoine W. G. de Vries and A. Biesheuvel; Leen van Wijngaarden (The Netherlands): Relation between path and wake of free rising and bouncing bubbles QK2—Charles D. Eggleton; Tse Min Tsai; Kathleen J. Stebe (USA): Tip streaming from a drop in the presence of surfactants QK3—Olga M. Lavrenteva, Alexander M. Leshansky, and Avinoam Nir (Israel): Effects of unsteady convective transport on the thermocapillary interaction of drops and bubbles

QL—Computational solid mechanics (jointly with IACM) (Robert B. Haber, USA, chair) QLl—Roman Lackner, Jürgen Macht, Christian Hellmich, and Herbert A. Mang (Austria): Adaptive analysis of shotcrete shells QL2 —Subrata Mukherjee and Mandar K. Chati; Glaucio H. Paulino (USA): Adaptive meshless boundary node methods for three-dimensional problems QL3—Ted Belytschko, Jingxiao Xu, Hao Chen, and Jack Chessa (USA): Meshfree methods and level sets

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ICTAM 2000

QO—Fluid–Structure interaction (John Grue, Norway, chair) QO1—Dean T. Mook and Ali H. Nayfeh; Sergio Preidikman; Benjamin D. Hall (USA): Nonlinear modeling of unsteady aeroelastic behavior QO2—Earl H. Dowell and Deman Tang (USA): Effects of steady angle of attack on nonlinear flutter of a delta wing plate model QO3—Yosuke Watanabe and Nobumasa Sugimoto (Japan): Nonlinear flexural waves on an elastic beam traveling along its axis in an air-filled tube QR—Compressible flow (Jean-Paul Bonnet, France, chair) QR1—Andrew J. Szeri and Hao Lin (USA): Shock formation in the presence of entropy gradients QR2—Francesco Grasso and Sergio Pirozzoli (Italy): Interactions of shock waves with compressible vortex pairs

QR3—Jeff Eldredge, Tim Colonius, and Anthony Leonard (USA): Particle methods for compressible flows QS—Solid mechanics in manufacturing (Tatsuo Inoue, Japan, chair) QS1—Vinay S. Rao; Krishna Garikipati (USA): Mathematical modeling of silicon oxidation QS2—Wojciech K. Nowacki and Nguyen Huu Viem (Poland): Strain localization in dynamic simple shear at high strain rates QS3—Louis G. Hector Jr.; John A. Howarth; Woo-Seung Kim (USA): Mold microgeometry effect on contact pressure evolution in pure metal solidification QV—Complex and smart fluids (Roger I. Tanner, Australia, chair) QV1—Boris Khusid; Andreas Acrivos and Anne D. Dussaud (USA): Particle separation in a flowing suspension subject to high-gradient electric fields QV2—Stephen K. Wilson, Andrew B. Ross, and Brian R. Duffy (UK): Thin-film flow of a viscoplastic material round a large horizontal cylinder QV3—Patrick Bontoux and Isabelle Raspo; Sakir Amiroudine; Bernard Zappoli (France): Rayleigh-Benard instability in a near critical fluid RA—Flow control (Hassan M. Nagib, USA, chair) RAl—Dale I. Pullin (USA): Large-eddy simulation of passive-scalar mixing by the stretchedvortex subgrid stress model RA2—Sergei V. Manuilovich (Russia): Semi-active control of laminar–turbulent transition in three-dimensional boundary layer RA3—John C. Lin and Chung-Sheng Yao; Steven M. Klausmeyer (USA): Micro-vortex generators for flow separation control and high-lift performance enhancement RB—Biological fluid dynamics (John O. Kessler, USA, chair) RB1—Takeshi Sugimoto (Japan): Aeroelastic reason for asymmetry in primary feathers of flying birds RB2—Michel Versluis, Anna von der Heydt, and Detlef Lohse; Barbara Schmitz (The Netherlands): On the sound of snapping shrimp: The collapse of a cavitation bubble RB3—Junji Seki (Japan): Pulse wave propagation in microvessels studied by a dual laserDoppler anemometer microscope RC—Fluid mechanics of materials processing (J. Iwan D. Alexander, USA, chair) RC1—Tomasz A. Kowalewski and Andrzej Cybulski; Janusz Szmyd and Marek Jaszczur (Poland): Experimental and numerical investigations of buoyancy driven instability in a vertical cylinder RC2—Shoichiro Nakamura, Robert S. Brodkey, and Yang Zhao (USA): Investigation of mixing processes by numerical and experimental approaches

RC3—Markus Scholle, Nuri Aksel, and Andreas Wierschem (Germany): Gravity driven film flows in the presence of side walls, edges and wavy bottom profiles RD—Fracture and crack mechanics (jointly with ICF) (John R. Willis, UK, chair) RD2—Roberta Massabò; Brian N. Cox (Italy): Mixed mode delamination and large scale bridging RD3—Erin Iesulauro, Anthony R. Ingraffea, Paul A. Wawrzynek, and Sanjay R. Arwade (USA): Simulation of grain boundary decohesion and crack propagation in aluminum microstructure models RE—Functionally graded materials (Marek-Jerzy Pindera, USA, chair) RE1—Hareesh V. Tippur and Carl E. Rousseau (USA): Evaluation of dynamic crack-tip f i e l d s and fracture parameters in functionally graded materials

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RE2—Rowland M. Cannon, Terrence L. Becker, and Robert O. Ritchie (USA): Statistical fracture modeling of functionally graded materials RE3—Antonios E. Giannakopoulos and Subra Suresh (USA): Contact-damage resistant graded surfaces

RF—Vehicle systems dynamics (jointly with IAVSD) (Ali H. Nayfeh, USA, chair) RFl—Daniele Casanova and Robin S. Sharp; Pat Symonds ( U K ) : On minimum time opti-

mization of Formula One cars: The influence of vehicle mass RF2—Philipp Heinzl, Manfred Plöchl, and Peter Lugner (Austria): Different actuation strategies for a yaw moment control of a passenger car RF3—Axel Fritz (Germany): Lateral and longitudinal control of a vehicle convoy

RG—Continuum mechanics (Donald E. Carlson, USA, chair) RG1—Marcelo Epstein (Canada): Material geometry of functionally graded bodies RG2—Paul Steinmann, Dirk Ackermann, and Franz-Josef Barth (Germany): Theory and computation of material forces in geometrically nonlinear fracture mechanics RG3—Samuel Forest (France): Strain localization phenomena as elementary deformation mechanism of different materials classes

RH—Smart materials and structures (Richard D. James, USA, chair) RH1—S. Mark Spearing and Catherine L. Sanders; Michael C. Shaw (USA): The use of piezo-mechanical loading to investigate the thermomechanical fatigue of solder joints RH2—Eric N. Brown and Nancy R. Sottos; Scott R. White and Philippe H. Geubelle; Jeffrey S. Moore (USA): Self-heating composites using embedded microspheres: Fracture toughness of epoxy matrix RH3—Sathya V. Hanagud, Steven Griffin, and Maxime Bayon de Noyer (USA): Active structural control for smart violins and guitars RK—Waves (Lev Shemer, Israel, chair) RK1—Chiang C. Mei and Zhenhua Huang (USA): Wind, waves and currents in shallow seas RK2—W. Kendall Melville, Fabrice Veron, and Christopher White (USA): Coherent structures and turbulence under breaking waves RK3—Alexander B. Ezersky and Vladislav V. Papko; Innocent Mutabazi (Russia): Laboratory modeling of impurity transport in a bottom boundary layer of surface waves

RL—Computational solid mechanics (jointly with IACM) (Robert B. Haber, USA, chair) RL1—David Littlefield, J. Tinsley Oden, and Serge M. Prudhomme (USA): A posteriori error estimation for highly nonlinear impact problems RL2—Tina Liebe, Andreas Menzel, and Paul Steinmann (Germany): Computational aspects of higher gradient softening materials RL3—Natarajan Sukumar and Brian Moran (USA): Natural neighbor Galerkin methods

RO—Fluid–Structure interaction (John Grue, Norway, chair) RO1—W. Geraint Price and Jing Tang Xing (UK): A mixed finite element–finite difference method for nonlinear fluid–solid interaction dynamics RO2—Allan Larsen (Denmark): Experimental and numerical study of fluid–structure interaction of an H-shaped cross section RO3—Jørgen J. Jensen, Preben T. Pedersen, and Peter F. Hansen (Denmark): Hydroelastically induced ship hull vibrations

RR—Compressible flow (Jean-Paul Bonnet, France, chair) RR1—Victor V. Golub, Tatiana V. Bazhenova, and Tatiana A. Bormotova (Russia): Post diffracted shock waves 3-D flow RR2—Max Elena and Joel Deleuze (France): Some turbulence characteristics in a shock wave-boundary layer interaction RR3—Bruno Auvity, Mark B. Huntley, and Alexander J. Smits; Michael R. Etz (USA): Boundary layer control at Mach 8 using helium injection

RS—Solid mechanics in manufacturing (Tatsuo Inoue, Japan, chair) RS1—Friedrich Pfeiffer and Günther Prokop (Germany): Optimization of assembly processes with manipulators RS2—Witold Gutkowski; Krzysztof Dems (Poland): Manufacturing tolerances and multiple loading conditions in structural configuration optimization RS3—Claus Ropers and Eckart Doege (Germany): Finite element simulation of sheet metal forming processes considering the elastic nature of the tools

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RV—Low-Reynolds-number flow (Roger I. Tanner, Australia, chair) RV1—Dudley A. Saville and John R. Glynn Jr. (USA): Electrohydrodynamics in non-

homogeneous fluids—studies in a Hele-Shaw cell RV2—Antoine Sellier (France): Electrophoresis of solid particles in an unbounded electrolyte SA—Flow instability and transition (Robert D. Moser, USA, chair) SA1—Yves Couder; Eric Lajeunesse (France): Tip-splitting instability of Saffman–Taylor fingers SA2—Igor N. Veretennikov; Alexandra E. Indeikina and Hsueh-Chia Chang (USA): Air entrainment at low viscosities SA3—Itai Cohen and Sidney R. Nagel (USA): Snakes eating elephants: The selective withdrawal problem and its uses in coating technology

SB—Biological solid mechanics (Hassan Aref, USA, chair) SB1—Jeffrey E. Bischoff, Ellen M. Arruda, and Karl Grosh (USA): An anisotropic constitutive law for human skin SB2—Albrecht Eiber and Hans-Georg Freitag (Germany): Simulation models of the human middle ear SB3—Lakshminarayanan Mahadevan (USA): Limbless undulatory locomotion on land

SC—Geophysical fluid dynamics (William R. Phillips, USA, chair) SC1—Scott Wunsch (USA): Density layer formation in stably stratified turbulence SC2—James W. Rottman; Derek Stretch; Charles W. Van Atta (USA): Using rapid distortion theory to interpret experiments on stably stratified turbulent shear flows SD—Continuum mechanics (James Casey, USA, chair) SD1—Giancarlo Losi (Italy): A discrete constitutive model for nonlinearly viscoelastic solids with structurally dependent viscosity SD2—Michel E. Jabbour; Kaushik Bhattacharya (USA): A continuum model of growth of thin solid films by chemical vapor deposition SE—Damage mechanics (Jean Lemaitre, France, chair) SE1 — Bhushan L. Karihaloo; Jianxiang Wang (UK): Material instability in the tensile response of short-fibre-reinforced quasi-brittle composites SE2—Milena Vujosevic; Dusan P. Krajcinovic (USA): Cooperative effects in brittle deformation of disordered materials SE3—Raimondo Luciano; John R. Willis (Italy): Non-local effective relations for composites subject to random body forces SF—Structural vibrations (Iliya I. Blekhman, Russia, chair) SF1—Utz von Wagner; Jörg Wauer ( G e r m a n y ) : On the optimization of the vibration in dental tools by compensation masses SF2—Haiyan Hu and Dongping Jin (China, PRC): Dynamics of a traveling cable subject to transverse fluid excitation SF3—Toshihiko Sugiura, Tsuyoshi Tanaka, Takuo Kugiya, and Masatsugu Yoshizawa (Japan): Nonlinear interaction between translation and rotation of a magnet above a highsuperconductor

SG—Mechanics of thin films and nanostructures (Scott R. White, USA, chair) SGl—Hiroshi Kitagawa, Tomotsugu Simokawa, and Akihiro Nakatani (Japan): Deformation mechanism of nano-crystalline material under uniaxial tension/compression SG2—Harley T. Johnson; L. Ben Freund (USA): The influence of strain on confinement in semiconductor quantum dots SG3— Moises E. Smart and Alberto M. Cuitiño; Gustavo Gioia; Antonio DeSimone; Michael Ortiz (USA): The energetics of folding in anisotropically compressed thin-film diaphragms SH—Smart materials and structures (Wendy C. Crone, USA, chair) SH1—Hisaaki Tobushi, Kazuyuki Takata, and Kayo Okumura; Shunichi Hayashi (Japan): Thermomechanical properties of polyurethane-shape memory polymer SH2—Masataka Tokuda, Koichi Sugino, and Tadashi Inaba (Japan): New training of shape memory alloy for smart function by multi-axial loading S H3—Wei Yang and Fei Fang (China, PRC): Domain switching versus fracture of ferroelectrics controlled by poling

Scientific program

1xxix

SK—Chaos in fluid and Solid mechanics (Francis C. Moon, USA, chair) SK1—Jonathan Kobine ( U K ) : Nonlinear sloshing modes in shallow fluid layers with horizontal time-periodic forcing SK2—Ken Kiyono, Tomoo Katsuyama, Takuya Masunaga, and Nobuko Fuchikami (Japan): The dripping faucet as a chaotic dynamical system SK3—Mark A. Stremler; Hassan Aref (USA): Quasi-periodic vortex arrays

SL—Plates and Shells (jointly with IACM) (Herbert A. Mang, Austria, chair) SL1—Walter K. Vonach and Franz G. Rammerstorfer ( A u s t r i a ) : A general solution to the local instability problem of sandwich plates SL2—Khaled W. Shahwan; Anthony M. Waas (USA): Delamination buckling and growth incorporating unilateral contact SL3—Wolfgang A. Wall, Michael Gee, and Ekkehard Ramm; Manfred Bischoff (Germany): Tuning of a 3D shell model in nonlinear statics and dynamics SO—Plasticity and viscoplasticity (Petros Sofronis, USA, chair) SO1—Alan C. F. Cocks; Frederick A. Leckie ( U K ) : The thermal loading of ceramic matrix composites

SO2—Piotr Perzyna; Wojciech Dornowski (Poland): Localized fracture phenomena in thermo-viscoplastic flow processes under cyclic dynamic loadings SO3—Igor K. Senchenkov and Yaroslav A. Zhuk (Ukraine): Thermomechanical coupling effects in the elastic-viscoplastic bodies with internal defect

SR—Acoustics (Ann P. Dowling, UK, chair) SR1—Paul W. Hammerton (UK): Propagation of sonic booms through a stratified atmosphere SR2—Nobumasa Sugimoto (Japan): Mass, momentum and energy transfer by the propagation of the acoustic solitary wave SS—Solid mechanics in manufacturing (Friedrich Pfeiffer, Germany, chair) SS1—Tatsuo Inoue and Hiroyuki Inoue; Fumiaki Ikuta and Takashi Horino (Japan): Simulation of dual frequency induction hardening process of a gear wheel SS2—Alan T. Zehnder and Yogesh Potdar; Xiaomin Deng ( U S A ) : Measurement and sim-

ulation of temperature and strain fields in orthogonal metal cutting SS3—Spandan Maiti and Philippe H. Geubelle ( U S A ) : Simulation of damage mechanisms in high-speed grinding of ceramics

SV—Flows in thin films (Jean-Marc Vanden-Broeck, UK, chair) SV1—Hsueh-Chia Chang, Pavlo V. Takhistov, and Alexandra E. Indeikina (USA): Electrokinetic displacement of air bubbles in microchannels SV2—Peter Vorobieff and Marc S. Ingber; Robert E. Ecke (USA): Two-dimensional cylinder wakes: An experimental and numerical study SV3—Igor L. Kliakhandler (USA): Inverse cascade in film flows

TA—Flow instability and transition (Robert D. Moser, USA, chair) TA1—Jitesh S. Gajjar and Andrew Gibson; Pradeep K. Sen ( U K ) : Instability of compliant pipe flows TA2—James P. Denier; Andrew P. Bassom (Australia): Nonlinear short waves in buoyant boundary layers TB—Multibody dynamics (Hassan Aref, USA, chair) TB1—Andrés Kecskemethy, Christian Lange, and Gerald Grabner (Austria): Robospine:

Simulating cervical vertebrae motion using elementary contact pairs TB2—Harry J. Dankowicz; Jesper Adolfsson, Arne Nordmark, and Petri Piiroinen (USA): Stability analysis of passive, bipedal gait in a three-dimensional environment TB3—Jean H. Heegaard and Matt L. Kaplan (USA): Fast optimal control algorithm for multibody models of the human musculoskeletal system TC—Rock mechanics and geomechanics (loannis Vardoulakis, Greece, chair) TC1—Bernhard A. Schrefler and Lorenzo Sanavia; Paul Steinmann ( I t a l y ) : Geometrical

and material nonlinear analysis of partially saturated porous media TC2—Wolfgang Ehlers (Germany): Dilatant and compressive shear bands in saturated geomaterials TC3—Corey S. O'Hern and Andrea J. Liu; Stephen A. Langer; Sidney R. Nagel (USA): Comparing force distributions and chains in glasses and granular materials

TD—Lighthill Memorial Session (Stanley A. Berger, USA, chair) TD1—John E. Ffowcs Williams ( U K ) : Active flow-avoidance

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ICTAM 2000

TD2—I. David Abrahams; Andrew N. Norris (UK): On the existence of edgewaves on fluid loaded elastic plates TD3—Vladimir N. Shtern and Fazle Hussain (USA): Buoyancy as a driving mechanism for

bipolar jets TE—Damage mechanics (Jean Lemaitre, France, chair) TE1—Beate Lauterbach and Dietmar Gross; Michal Basista (Germany): Influence of microcracks on brittle solids' response under compression TE2—Nicola Alberti; Fabrizio Micari (Italy): Application of damage mechanics approach to predict material formability in deep drawing TE3—Kayleen L. E. Helms and David H. Alien (USA): A model for predicting grain boundary crack growth in viscoplastic polycrystalline materials

TF—Control of Structures (Iliya I. Blekhman, Russia, chair) TF1—Ali H. Nayfeh, Ziyad N. Masoud, and Dean T. Mook (USA): Control of cargo pendulation for ship-mounted cranes TF2—Bala Balachandran and Ying Yong Li (USA): Control of nonlinear crane-load oscillations TG1—Yakov Benveniste; Tungyang Chen (Israel): Torsion of composite bars with imperfect interfaces and neutral inhomogeneities in torsion problems

TG—Damage and failure of composites (Scott R. White, USA, chair) TG2—Fabrizio Greco and Domenico Bruno (Italy): Delamination failure of layered composite plates TG3—Reaz A. Chaudhuri (USA): Delocalization/percolation of inelastic kinkband instability in a thick laminated cylindrical shell

TH—Smart materials and structures (Wendy C. Crone, USA, chair) THl—Prashant K. Purohit and Kaushik Bhattacharya (USA): Dynamics of strings made of phase transforming material TH2—Chih-Kung Lee and Yu-Hsiang Hsu (Taiwan): Theory and experiment of autonomous

phase-gain piezoelectric optimal sensing device TH3—Andrew W. Smyth; Sami F. Masri (USA): Representation and transmission of stochas-

tic loads for nonlinear systems response and control TL—Experimental methods in solid mechanics (James W. Phillips, USA, chair) TLl—Martin Haueis, Rudolf Buser, and Jürg Dual; Claudio Cavalloni and Marco Gnielka (Switzerland): Single crystalline microresonator for force sensing with on-chip vibration excitation and detection

TL2—Markus M. Conrad and Mahir B. Sayir (Switzerland): Detection of flaws in metalceramic compound plates with flexural waves and holography TL3—Arun Shukla and Vikas Srivastava (USA): Strength of adhesive bonded lap joints under high loading rates TV—Flow in porous media (Jean-Marc Vanden-Broeck, UK, chair) TV1—Anthony J. C. Ladd and Rolf Verberg (USA): A lattice-Boltzmann model with sub-grid

scale boundary conditions TV2—Robert A. Beddini and Corwyn E. Low (USA): Hydrodynamic stability of flow in homogeneous porous media

UJ—Closing Lecture by H. Keith Moffatt (Grigory Isaakovich Barenblatt, Russia/USA, chair) UJ1—H. Keith Moffatt (UK): Local and global perspectives in fluid dynamics

NEW PERSPECTIVES ON CRACK AND FAULT DYNAMICS James R. Rice Division of Engineering and Applied Sciences and Department of Earth and Planetary Sciences Harvard University, Cambridge, Mass., USA [email protected] Abstract

1.

Recent observations on the dynamics of crack and fault rupture are described, together with related theory and simulations in the framework of continuum elastodynamics. Topics include configurational instabilities of tensile crack fronts (crack front waves, disordering, sidebranching), the connection between frictional slip laws and modes of rupture propagation in earth faulting, especially conditions for formation of self-healing slip pulses, and the rich faulting and cracking phenomena that result along dissimilar material interfaces due to coupling between slippage and normal stress alteration.

INTRODUCTION

The dynamics of cracking and faulting has seen much recent progress, with implications for structural mechanics, materials physics, tribology, and seismology. In this brief review, the following topics will be discussed: Crack front waves Limiting rupture speeds Mode of rupture on faults: enlarging shear crack or self-healing slip pulse? Interfacial fracture dynamics, slip, and opening To put things in context, we will be using linear isotropic elasticity everywhere except along slip or crack zones. The governing equations of that theory are the equations of motion where or u or a combination are given on the boundary, together with stress-displacement gradient relations 1 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 1–24. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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leading to the Navier equations for the displacement field:

Those equations admit a family of waves. There are primary (or dilational) and secondary (or shear) body waves, with respective speeds and typically with and also Rayleigh surface waves with typical speed As we review recent progress, we shall learn of a newly discovered type of wave that lives along a moving crack front, and also see the significance of a little known generalization of Rayleigh waves for the dynamics of dissimilar materials. Much of the numerical simulation that is reviewed here has been done by a spectral elastodynamic method [l]–[3], which is of limited general utility but is very well tailored to produce good numerical solutions, without grid dispersion, for crack or fault problems that can be con-

sidered as events at the border between two homogeneous elastic half spaces. The border at y = 0, which is the fault or crack plane, undergoes displacement discontinuities

The spectral method treats these problems by writing as a large but finite Fourier sum of terms in form the rupture domain is replicated periodically in x and z. The elastodynamic equations are solved exactly in each half space, for terms of that same space dependence, so that in the end the traction stress components are given as a corresponding Fourier sum of terms in the form Each can then be determined in terms of the in an expression involving a convolution over prior That, effectively, lets us relate the history of the at fast-Fourier-transform sample points along the interface, to the history of the and to any given loading stresses, in such a way that the elastodynamic equations are satisfied in the two half spaces. The system is then closed, so that a definite solution can be computed, by specifying another relation between the and the that is a constitutive relation for crack opening [2, 3] (cohesive model) or frictional slip [1]. Different forms will be noted as we go along.

2.

CRACK FRONT WAVES

Figure 1 shows a tensile crack growing in a 3D solid, along the plane y = 0. Earlier work [4, 5] addressed a simplified version of this problem for a model elastic theory with a single displacement component u,

New perspectives on crack and fault dynamics

3

Figure 1 Crack propagating along a plane in an unbounded elastic solid. Inset shows cohesive stress versus crack opening for non-singular crack model.

satisfying a scalar wave equation. It showed that there was a long lived response to a local perturbation of the crack front, e.g. by the crack passing through a region where the fracture energy was slightly different than elsewhere, although in that scalar model the crack ultimately recovered a perfectly straight front. That led to great interest in solving such problems in the context of actual elasticity. Willis and Movchan [6] produced the corresponding small perturbation solution, although that was difficult to interpret and the fuller implications of their solution were revealed later [7], confirming what had been suggested from spectral numerical simulations [8, 9]: for crack growth in a perfectly elastic solid with a constant fracture energy, perturbation of the crack front leads to a wave that propagates laterally, without attenuation or dispersion, along the moving crack front. The wave is called a crack front wave. Two different fracture formulations have been used for these investigations. The first is a singular crack model, in which one sets on the mathematical cut y = 0, which is the crack surface. That leads to a well known singular field of structure

where r, are polar coordinates at the crack tip and the are universal functions for a given cracking mode. The strength has been normalized in terms of G, which is the energy release rate (energy flow to crack tip singularity, per unit of new crack area), expressed by

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Here, W is the strain energy density, coordinate points in the direction of crack growth, is a circuit that loops around the crack tip at the place of interest, and s is arc length along whose outer normal is n. Owing to the great advances on unsteady crack motion in 2D elasticity by Kostrov [10] and Eshelby [11] for mode III (anti-plane shear), Freund [12] for mode I (tensile cracking), and Fossum and Freund [13] for mode II (in-plane shear), we know that G = g(v(t)) Here g(v) is a universal function of crack speed v for each mode, normalized to and diminishing to at a limiting speed (at least for fully subsonic fracturing), which is for modes I and II, and for mode III. The variable is a complicated and generally untractable functional of the prior history of crack growth and of external loading, but is independent of the instantaneous crack speed v(t). Because of that structure for G, it is possible for cracks to instantaneously change v if the requisite energy which must be absorbed for fracture, changes discontinuously along the fracture path. Also, for a solid loaded by a remotely applied stress, it will generally be the case that increases as the crack lengthens, and increases quadratically with the intensity of the applied stress. Thus, if is bounded, then in a sufficiently large body g(v) will be driven towards 0, which means that v will accelerate towards For the crack growing on a plane in 3D, as in Fig. 1, it has so far been possible to solve for the elastic field [6] only for a crack whose front position x = a(z,t) is linearly perturbed from that is, from a straight front moving at uniform speed An alternative fracture formulation, often more congenial to numerical calculation, explicitly accounts for a gradual decohesion, imposing a weakening relation between stress and displacement-discontinuity as a boundary condition on the potential crack plane. (See the inset diagram of Fig. 1.) Thus, the singularity at the crack tip is smeared into a displacement weakening zone. Its width R scales [14] as, roughly, with being the maximum cohesive strength and the displacement at which cohesion is lost, and with f(v) being universal for a given mode, and with f(0) = 1 and The latter limit poses a challenge for numerical simulation of fracture at speeds very close to When all length scales in problem (crack length, distance of wave travel, etc.), predictions of the displacement-weakening model agree with those of the singular crack model, with identified (Fig. 1) as the area under the cohesive relation [14]. The spectral methodology was used with the cohesive model [8, 9] to study what happens when a crack front, moving as a straight line across y = 0 with uniform rupture speed suddenly encounters a localized

New perspectives on crack and fault dynamics

5

“asperity” region for which is modestly higher than its uniform value, say, prevailing elsewhere. The resulting perturbation in the crack front propagation velocity, v(z, t) is shown in Fig. 2. There, is the periodic replication distance in the z direction, as required within the spectral method. The asperity has diameter and has

Figure 2 Numerical simulation results for perturbation of rupture propagation velocity v(z,t) when the crack front, moving at uniform speed encounters a small region of altered fracture energy. Adapted from Ref. [9].

both and were increased by 5% within it. The motivation to examine the response to such a small localized perturbation arose from understanding of the scalar model [5], for which the form of response to such isolated excitation was critical to understanding the response to (small) random excitation. Quite remarkably, when those calculations were done (Fig. 2) for the mode I crack in true elastodynamics, the perturbation of the crack front seemed to propagate as a persistent wave, for as long as it was feasible to do the numerical calculation. This is the crack front wave. Returning to the singular crack model, the existence of the wave was proven, as discussed, by application [7] of the solution [6] for arbitrary, but sufficiently small (linearized) perturbation Writing the critical fracture energy as

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and understanding that its perturbation at the crack front is within a strictly linearized formulation, one must have an expression of the form

where

The function has a simple zero [7, 15] at a certain real value of That proves the existence of the wave, whose speed in the direction parallel to the moving crack front is Figure 3 shows the resulting wave speed measured relative to a place on the fracture surface from which the wave was launched, or

Figure 3 Crack front wave speed, as a function of unperturbed crack speed different values of Possion ratio From Ref. [15].

for

through which it passed; is the speed parallel to the front. The speed is very near to and consistent with the simulation result in Fig. 2. By analyzing the small perturbation problem for the case

New perspectives on crack and fault dynamics

in which

7

it was shown [7] that the wave attenuates if

The existence of the crack front wave leads to rapid disordering in crack propagation through a small random fluctuation about mean of fracture energy on the plane y = 0. Suppose that the variation is statistically stationary and isotropic, with correlation function

and that a sample of that variation is first encountered by the crack at t = 0, with the crack front being straight and moving at unperturbed velocity for t < 0. Within the linearized perturbation formulation, remains the mean velocity after perturbation begins. We consider the random variables A(z, t) and the slope S(z, t) = of the crack front. Then, using the method of Ref. [5] for the scalar case, applied in way outlined in Ref. [9], one derives (for length scales in the correlation function) that

and that

where is a certain function. Thus the variance in crack location grows in direct proportion to distance of propagation into the nonuniform region, whereas the two-point correlation between the positional fluctuations will vanish when has become large enough that there is zero correlation at such distances. This conveys a picture of rapid disordering of the crack front. That is also expressed by the associated result

Note that such second order statistic will not exist if the correlation function has a non-zero slope at separation. Samples of perturbed crack growth histories, within the linearized ideally elastic framework show that strong perturbations cluster in space and time along wave fronts [15]. Many features remain to be understood. These include saturation due to nonlinearities (one must, of course, have damping due to and to material viscoelasticity, and, most especially, to interactions with non-planarity of growth.

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LIMITING RUPTURE SPEED

The theoretical prediction, according to the singular crack model, for ruptures along a plane has already been mentioned. Under typical remote loading conditions we expect that for modes I and II, and that for mode III, as limiting rupture speeds. However, to summarize observations [16]–[18] for tensile (mode I) cracks, from laboratory tests: The crack speed in brittle amorphous solids (glass, PMMA) has an upper limit of order 0.50–0.60 (i.e. 0.55–0.65 The fracture surface is mirror-smooth only for v < 0.3–0.4 The crack surface roughens severely, and becomes severely oscillatory, at higher (average) speeds. The crack forks at the highest speeds.

There are exceptions for which (or a large fraction of brittle highly anisotropic single crystals (W, mica); and incompletely sintered solids [19]. Up to very recently, the only observations for shear ruptures (modes II and III) have come from inversions of seismograms for slip histories on faults: The range is a commonly inferred range of rupture speeds, although it is not well constrained.

Rarely, bursts of intersonic rupture with inferred [20, 21].

have been

In the case of tensile cracks in brittle materials, interferometric studies [16] of propagating fractures in PMMA sheets showed that the propagation process became quite intermittent as the limit speed was approached. That was evidenced by strong stress waves being radiated from the tip in distinctly separated pulses. (It is actually the mean value of , averaged over a few such pulses, and not necessarily the local itself, which should be thought to have a well defined limit, much less than Later studies [17] clearly tied the strongly intermittent , and intensity of fluctuation of about the mean, to a side branching process. In that, small fractures seem to have emerged out of the walls of the main fracture near to its tip, such that the main crack and one or a pair of such side-branching cracks coexisted for a time, although the branches were ultimately outrun by the main fracture, at least at low mean propagation speeds. It may instead [22] be that the side branched cracks began their life as part, not well aligned with the main crack plane, of a damage cluster of microcracks developing ahead of the main crack tip

New perspectives on crack and fault dynamics

9

(Fig. 4(c)), and grew into the main crack walls. In any event, the density of cracks left as side-branching damage features increases substantially

Figure 4 (a) Yoffe consideration of hoop stress near tip. (b) Eshelby question of when can one fracture, by slowing down, provide enough energy to feed two. (c) Possibility that profuse microcrack nucleation, with some clusters linking with the main fracture, may be a more appropriate mechanism than the side-branching from the tip in (b).

with increase of the loading that drives the fracture. That process also results in the well-known increased roughening of the fracture surface with increasing crack speed. Ultimately, the response of the fracture to further increase in loading is no longer to increase the average , which is the result expected from the theoretical analysis of a crack moving on a plane, but rather to absorb more energy by increasing the density of cracks in the damage cluster formed at the tip of the macroscopic fracture. Thus the energy adsorption rises steeply with crack speed [17]. The deviation of the fracture process from a plane is tied yet more definitively to the intermittency and low limiting crack speeds by studies [19] of weakly sintered plates of PMMA. The resistance to fracturing along the joint was much smaller than for the adjoining material, and that led to smooth propagation of the fracture (no evidence of jaggedness of interferograms by stress wave emission), with reaching 0.92 The attempts to explain low limiting speeds and fracture surface roughening begin with the first paper, by Yoffe [23], in which the elastodynamic equations were solved for a moving crack. That was done for a crack that grew at one end and healed at the other, so that the field depended on x — t and y only (2D case), but sufficed to reveal the structure of the near tip singular field, including the dependence

of the functions above. Yoffe found that (Fig. 4(a)), at fixed small r within the singularity-dominated zone, reached a maximum with

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respect to for when > 0.65 That gives a plausible explanation of limiting speeds and macroscopic branching, but does not explain the side-branching and roughening that sets in at much smaller speeds. Eshelby [24] posed the question of what is the at which a fracture must be moving so that, by slowing down, it can provide enough energy to drive two fractures (Fig. 4(b)). The extreme is to slow down to That question still has no precise answer when the angle (Fig. 4(b)) is non-zero, but Eshelby answered it for mode III—Freund [12] had not yet solved the mode I case—in the limit which reduces to sudden change of speed of a fracture moving along the plane. The condition is g( ) < 1/2 (assuming no dependence of which gave > 0.60 for mode III. Freund addressed the same issue based on his mode I solution [12], again in the limit that, with recalculation [25], leads to > By working out the static version of the problem, which has at least approximate relevance since is near 0, and using approximate considerations of wave travel times to limit the interaction of one branch with the other, Adda-Bedia and Sharon [25] optimize with respect to to estimate that the lowest u for branching is Material response with will reduce that threshold but, at present, it seems unlikely that the inferred onset of roughening at speeds in the range 0.3–0.4 can be explained by a branching instability at a crack tip in a solid that is modeled, otherwise, as a linear elastic continuum. Nevertheless, it is interesting that some discrete numerical models of cracking do give branching at roughly realistic speeds. A transition to zig-zag growth was shown to set in around 0.33 in molecular dynamics simulations [26]. It is not yet clear to what extent the at its onset may depend on details of the force law. Cohesive finite elements [27] or cohesive element interfaces [28] that fully model the separation process allow, within constraints of the mesh, for self-chosen fracture paths, and have shown off-plane fracturing. Xu and Needleman [28] nicely reproduce the observations that when the crack is confined to a weak plane, it accelerates towards whereas for the uniform material, side branches form at 0.45 or at lower speed [29] if statistical variation in cohesive properties is introduced. Recent work [30] suggests that such propensity for low-speed side branching may not be universal for all cohesive finite element models. In particular, models that involve linear behavior until a strength threshold is reached [31], after which there is displacementweakening (much like for the inset in Fig. 1) seem less prone to development of side-branching. The procedure of Ref. [28] allows substantial nonlinear deformation at the element boundaries before achieving their peak cohesive strength. Thus local nonlinear features of the pre-peak

New perspectives on crack and fault dynamics

11

deformation response may be critical to understand the onset of roughening [32]. In contrast to tensile cracks, earthquake ruptures (large scale shear cracks) do seem to approach much more closely to and, as mentioned, bursts of intersonic rupture have been inferred [20, 21]. In fact, theoretical prediction [33, 34] of intersonic rupture from rupture simulations with slip-weakening models (mode II or III versions of the tensile

displacement-weakening model discussed above) preceded the observations. A concentration of stress in a shear wave peak develops ahead of a mode II shear rupture. In a model with a finite strength to get slip started, that allows for the possibility that slip-weakening will initiate ahead of the main crack tip. As part of the linking up with the main slip-weakening zone, that allows a fracture to emerge at a high speed in the range of (which is the unique intersonic speed at which a mode II rupture can propagate in the singular crack model [18]). Such had never been seen in the laboratory until Rosakis et al. [35] demonstrated that for weakly sintered Homalite-100 plates, impact loaded in shear, a mode II fracture propagated intersonically with a speed that fluctuated to high values, near to but approached at greater propagation distances a speed near The importance of the weak channel for the rupture is that typical attempts to form a mode II (or III) rupture

in the laboratory, except under quite high pressure [36], lead to mode I cracking from the rupture tip.

4.

MODE OF RUPTURE ON FAULTS: ENLARGING SHEAR CRACK OR SELF-HEALING SLIP PULSE WHEN THERE IS VELOCITY-WEAKENING FRICTION?

In an influential paper, Heaton [37] argued that the mode of rupture in large earthquakes, as inferred from seismic inversion studies for well recorded events, was such that the duration of slip at a point on the fault was much shorter than the overall duration of the rupture. He argued that a point on the fault begins to slip as the rupture front arrives but that this slipping phase is of short duration and that the fault “heals” (by which is meant that it stops slipping) after a time that is much less than the overall duration of the rupture. That is called the “self-healing” rupture mode. That is to be contrasted with what may be called the “crack-like” mode. For it, a point on the fault plane is assumed to slip for a significant fraction of the overall rupture duration, i.e. beginning when the rupture

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front arrives and continuing until waves generated in the arrest of the front, at a barrier of some sort, carry back signals to stop slipping. The crack mode has been widely observed in numerical simulations of spontaneous rupture. These include cases of prescribed uniform strength drop on the fracture surface in singular crack models [38]–[41], nonsingular slip-weakening models [33, 42, 43, 44], and simple, if grid-size dependent, models with a critical stress failure condition at the rupture front [45]–[47]. Heaton’s paper thus went against a widely accepted view of how rupture occurs. It launched an active body of theoretical research, to understand what does indeed determine the mode of rupture, and what could lead to the self-healing mode, which seems to be supported by observations. There are now a few candidates [48], one to be discussed in this section and one in the next. Here we focus on recent theoretical understanding of a possibility already suggested by Heaton, namely that strong velocity-weakening of friction could allow self-healing. In fact, simulations of rupture with strong velocity weakening sometimes showed crack-like rupture and sometimes self-healing [1, 49, 50], and it has been an important goal to understand what controls. One requirement is that the friction strength increase with time on slipped portions of the fault that are momentarily in stationary contact [1]. Another is that the overall driving stress be low [48] in a way that we analyze here. Velocity-weakening friction on faults is interpreted in the rate and state framework, which includes laboratory-based state evolution features. Those also regularize ill-posed or paradoxical features of models of sliding between two identical elastic solids [51]. Thus, as in Fig. 5, we shall think of the heavy solid line as giving the basic response between shear stress and slip rate V; here we regard normal compressive stress as constant. A full constitutive description, which must be used in numerical simulations, involves strength expressed as where the state variable is —e.g. representing lifetime of current population of asperity contacts, or of current gouge packing— and where (note path of instantaneous change, i.e. change at constant in Fig. 5), and The state variable follows an evolution law, e.g. of form although other forms are sometimes used instead [1, 48]. The characteristic slip distance L for state evolution is typically found to be in the 1–50 range. The evolution law has the property that (= L/V for the law above) in sustained slip at fixed V, so that where and, for the cases of interest here, That is a somewhat complicated formulation, and the state evolution is over in micron range changes of slip, which would be very small

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Figure 5 Velocity weakening friction. Interpreted in the rate and state framework, including laboratory-based state evolution features that also regularize ill-posed or paradoxical features in models of sliding between two identical elastic solids. A stress level is also shown, below which crack-like ruptures become impossible.

for applications to tectonic faulting. So the question arises, why use rate- and state-dependent friction instead of a pure rate-dependent friction law, say, The answer is that in addition to attaining

consistency with laboratory evidence and with the microphysical understanding of friction, we eliminate the following [51]: Ill-posedness of the pure rate-dependent formulation when An perturbation of a steady sliding state, with V = then elicits response > 0; see the discussion of a similar issue in the next section. Supersonic propagation of rupture fronts [52] when < An perturbation elicits response V(x,t) —

where r >

for mode II and r >

for mode III

slip. (While supersonic propagation of rupture fronts seems to be

precluded in the rate and state formulation, phase velocities at sufficiently low within the range for which there is unstable response to perturbations, do become supersonic [53]). These shortcomings of the pure rate-dependent model are not widely known, probably because friction studies directed to machine technology

have often focused on sliding rigid blocks rather than deformable elastic continua. The rate and state formulation provides a regularization. Now consider a fault surface, which we treat as the boundary y = 0 between two identical half spaces (Fig. 6). An initial shear stress

on

a constant level too small to cause failure, acts everywhere (the xz plane) except in small nucleation region which will

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Figure 6 Identical elastic half-spaces meeting on a fault plane y = 0; for discussion of crack-like versus self-healing rupture mode.

be overstressed to start the rupture. An important stress level is given by as marked in Fig. 5. It is understood as the highest value that could have if we were to require for all Suppose that As will be seen, that effectively precludes the possibility that rupture could occur on in the form of an indefinitely expanding shear crack [48]. Note that

Important along the way is an elastodynamic conservation theorem [48]

which holds throughout the rupture. This can be derived from relations involving spatial Fourier transforms of elastodynamic fields [2] in the zero wavelength limit. As an aside, it provides an interpretation for the seismic moment release rate,

which has apparently not appeared before in seismology. Let us now assume that with the loading rupture has been locally nucleated and grows on in the form of an indefinitely expanding shear crack. We shall try to develop a contradiction. The integrand everywhere along the rupturing surface except for and for small regions at the rupture front affected by the rate/state regularization, is equal to

where the inequality follows from the above consequence of Thus, letting = region of lying outside the

New perspectives on crack and fault dynamics

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rupture at t, and noting that V = 0 there, we must then have

That seems to be a contradiction: We expect cracks to increase the net force carried outside themselves, or at least to not decrease it. (However,

such non-decrease of force is a proven result, thus far, only for 2D antiplane shear crack solutions [48].) The result does nevertheless suggest that the crack-like rupture mode should not occur if That is consistent with a range of calculations [48, 54] for different stress levels

and forms of slip pulses—when

Crack-like ruptures are not found—only self-healing In the loading regime one

may focus on a dimensionless measure T of the strength of the velocity weakening at a representative slip rate [48]. When T is very small, the rupture surface could reasonably be expected to respond somewhat like

that for a model with constant stress drop, which is in a crack-like mode; that is what is found in simulations.

5.

MATERIAL PROPERTY DISSIMILARITY ACROSS A FAULT PLANE Here we consider two dissimilar solids in sliding frictional contact of

mode II type (Fig. 7). We consider first the simplest case of a constant friction coefficient f, independent of slip rate V or its history. Thus

Figure 7 Frictional sliding of dissimilar elastic solids on one another.

whenever V(x,t) > 0 and normal stress the shear stress The general effect that can drive highly unstable response in this case is that spatially inhomogeneous slip induces a local change in

and such reduction of

allows easier slip.

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Interesting results have been found in the following two types of analyses: Dynamic stability of steady sliding [55]–[59] on the surface between dissimilar elastically deformable materials, loaded with Slip rupture dynamics on faults between dissimilar materials that are loaded below threshold [60]–[65], For the first type of problem, we assume that the two solids are sliding at a uniform rate and are loaded by remote stresses that just meet the sliding conditions, and that the stress state is uniform within each solid. We then consider the response of this system to perturbation, such that the friction law remains valid everywhere along the interface and there is no opening. Solutions to the Navier equations are then sought in the form

For real k, one finds that p is of the form p = – ick, where a and c are real. Most importantly, over a wide range of parameters [56] (values of f, ratios of elastic constants, and densities), it turns out that a > 0 for at least one such solution. That remarkable fact not only implies that the dissimilar material system is modally unstable, but also implies ill-posedness of response to general perturbation. That is, the response to a perturbation that has Fourier strength in that unstable mode is, formally,

which diverges when t > 0 for generic if a > 0. Response in a given mode does not diverge in finite time, but the solution ultimately loses validity because either V < 0 or is predicted. The following has been shown [59] for dissimilar materials: Response is unstable (a > 0) for all f > 0 when a generalized Rayleigh (GR) wave1 exists, and for f > > 0 when a GR wave does not exist. For sufficiently large f, two unstable solutions (i.e. both with a > 0) may exist, with different growth rates a and propagation 1

A GR wave corresponds to a motion with free slip, = 0, but no opening gap at the interface. It exists when there is a real solution c (the wave speed to

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speeds c of different magnitudes and signs (directions), and neutrally stable solutions (a = 0) with supersonic propagation speeds may exist. The second type of problem envisions that the two solids in contact (Fig. 7) are loaded below the friction threshold, and that an event is somehow nucleated (e.g. by local reduction of ) and the focus is on how it propagates into previously locked locations along the interface. Weertman [60] suggested that steady rupture propagation, in a form of what is now called a self-healing pulse, might exist in that circumstance, and such has been confirmed [61]. That solution exists only when the GR wave exists. Rupture propagates at (its possible significance for rupture propagation had been noted earlier [67]), in a unique direction along the interface, which is that of slip in the slower material—i.e. in the +x direction in Fig. 7, for slip as shown. The solution has the remarkable feature that remains unaltered from and that slip occurs because reduces in the slipping region so as to just meet . The speed V is constant within the pulse, and proportional to . Its stability and possibility of emergence from initial conditions remain unclear. Transient dynamic analysis of locally nucleated rupture has also been addressed, first in finite difference simulations [62]–[64]. They did indeed find rupture in the form of a self-healing pulse but results indicated problems of convergence with grid refinement, or with time of rupture propagation, which is now understood to be related to the ill-posedness mentioned above. That ill-posedness was confirmed [65] by using the spectral numerical formulation, as generalized to bimaterials [3]. For a fixed replication distance along strike, the numerical results became progressively more jagged with increase from 256 to 2048 terms in the underlying Fourier series representation of in a way that was consistent with theoretical understanding of the instability. Some possible regularizations of the problem have been discussed [58], e.g. basing the friction law on the average of over a finite patch size around the position of interest. Another approach [59, 65] was based

with k = 1,2 to denote the different materials. GR waves were discovered independently by several investigators [66]–[68]. They satisfy and

and exist only for modestly different materials, typically for

< 1.30–

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on results of oblique shock wave experiments [69, 70]. In those, a shock reflected from the back surface of the target specimen causes an abrupt decrease of normal stress on the sliding interface. As shown schematically in Fig. 8, there is then only a gradual evolution of shear strength (over a few micrometers of slip, or few tenths of microseconds time)

Figure 8 Schematic of graduate evolution of shear strength (over a few micrometers of slip, or few tenths of microseconds time) in oblique shock wave experiments of Refs. [69, 70]. A shock reflected from the back surface of the target specimen causes an abrupt decrease of normal stress.

towards the new level consistent with the altered normal stress. A simple representation of this result is to replace with [59]

assuming where L = const > 0. This regularizes the first group of problems above, i.e. perturbation of steady frictional sliding, as follows [59] (now a and c in p = – ick do depend on k, at least when L/V is of order 1 or larger): When a GR wave exists, the unstable response approaches neutral stability, as (i.e. as wavelength In practical terms, that means stability at length scales

When a GR wave does not exist, there is a critical wavenumber such that response is stable (a < 0) when (or when where scales in proportion to Nothing in those results would change if V/L was replaced by where with at least one > 0, or indeed by any positive factor. The same regularization was then used [65] to address the second class of problems above, systems that are frictionally locked with but for which rupture is nucleated somewhere by local unclamping. With

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the regularization, the problem becomes well posed, in that convergent numerical solutions occur upon refinement of the Fourier basis set. Further, the rupture is in the form of a self-healing pulse propagating at a speed close to In cases for which does not exist, the propagation speed seems to be close to These are all problems of slip rupture along bimaterial interfaces, but there are also challenging problems, with remarkable features seen experimentally and rationalized theoretically, involving combined tensile fracture and slip rupture along bimaterial interfaces. For example, for polymer–metal systems (PMMA–steel [71] and Homalite 100– aluminum [72], respectively), impact loading created fractures that began their life as shear slippage over a millimeter scale at the rupture front, and subsequently opened. Their propagation speed was faster than for the polymer, leading to shock wave structures in high speed photographs of a photoelastic fringe pattern [72]. The essential features of those complex near tip slip and opening features, and the propagation speed, were rationalized in spectral numerical calculations [3], based on a non-singular numerical model with combined displacement weakening in tension and shear.

6.

CONCLUSION

It is hoped this set of short examples conveys some idea of the new discoveries and scientific excitement in understanding the dynamics of rupture. We have seen the discovery of previously unsuspected waves along crack fronts, of reasons why a previously inferred speed limit for fracturing (the Rayleigh speed) is sometimes too high and sometimes too low, of unsuspectedly ill-posed problems in frictional slip dynamics, and of rupture modes, such as the self-healing mode, that were largely unsuspected a relatively short time ago. These are a small sample from a large and vigorously growing body of science.

Acknowledgment The studies discussed were supported by the Office of Naval Research, the U.S. Geological Survey, and NSF through the Southern California Earthquake Center.

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References [1] Perrin, G., J. R. Rice, and G. Zheng. 1995. Self-healing slip pulse on a frictional surface. Journal of the Mechanics and Physics of Solids 43, 1461–1495. [2] Geubelle, P., and J. R. Rice. 1995. A spectral method for 3D elastodynamic fracture problems. Journal of the Mechanics and Physics of Solids 43, 1791– 1824. [3] Breitenfeld, M. S., and P. H. Geubelle. 1998. Numerical analysis of dynamic debonding under 2D in-plane and 3D loading. International Journal of Fracture

93, 13–38. [4] Rice, J. R., Y. Ben-Zion, and K. S. Kim. 1994. Three-dimensional perturbation solution for a dynamic planar crack moving unsteadily in a model elastic solid. Journal of the Mechanics and Physics of Solids 42, 813–843. [5] Perrin, G., and J. R. Rice. 1994. Disordering of a dynamic planar crack front in a model elastic medium of randomly variable toughness. Journal of the Mechanics and Physics of Solids 42, 1047–1064. [6] Willis, J. R., and A. B. Movchan. 1995. Dynamic weight functions for a moving crack—I. Mode I loading. Journal of the Mechanics and Physics of Solids 43, 319–341. [7] Ramanathan, S., and D. Fisher. 1997. Dynamics and instabilities of planar tensile cracks in heterogeneous media. Physical Review Letters 79, 877–880.

[8] Morrissey, J. W., and J. R. Rice. 1996. 3D elastodynamics of cracking through heterogeneous solids: Crack front waves and growth of fluctuations (abstract). EOS Transactions of the American Geophysical Union 77, F485. [9] Morrissey, J. W., and J. R. Rice. 1998. Crack front waves. Journal of the Mechanics and Physics of Solids 46, 467–487.

[10] Kostrov, B. V. 1966. Unsteady propagation of longitudinal shear cracks. Journal of Applied Mathematics and Mechanics 30, 1241–1248 (in Russian). [11] Eshelby, J. D. 1969. The elastic field of a crack extending non-uniformly under general anti-plane loading. Journal of the Mechanics and Physics of Solids 17, 177–199. [12] Freund, L. B. 1972. Crack propagation in an elastic solid subject to general

loading—I, Constant rate of extension, II, Non-uniform rate of extension. Journal of the Mechanics and Physics of Solids 20, 129–152. [13] Fossum, A. F., and L. B. Freund. 1975. Non-uniformly moving shear crack model of a shallow focus earthquake mechanism. Journal of Geophysical Research 80, 3343–3347. [14] Rice, J. R. 1980. The mechanics of earthquake rupture. In Physics of the Earth’s Interior, Proceedings of the International School of Physics ‘Enrico Fermi’ (A. M. Dziewonski and E. Boschi, eds.). Italian Physical Society and North-Holland,

555–649. [15] Morrissey, J. W., and J. R. Rice. 2000. Perturbative simulations of crack front wave. Journal of the Mechanics and Physics of Solids 48, 1229–1251. [16] Ravi-Chandar, K., and W. G. Knauss. 1984. An experimental investigation into dynamic fracture—I, Crack initiation and crack arrest, II, Microstructural aspects, III, Steady state crack propagation and crack branching, IV, On the interaction of stress waves with propagating cracks. International Journal of Fracture 25, 247–262; 26, 65–80, 141–154, 189–200.

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[17] Sharon, E., S. P. Gross, and J. Fineberg. 1995. Local crack branching as a mechanism for instability in dynamic fracture. Physical Review Letters 74, 5096–

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[24] Eshelby, J. D. 1970. Energy relations and the energy-momentum tensor in con-

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[25] Adda-Bedia, M., and E. Sharon. 2000. Private communication. [26] Abraham, F. F., D. Brodbeck, R. A. Rafey, and W. E. Rudge. 1994. Instability dynamics of fracture: A computer simulation investigation. Physical Review Letters 73, 272–275. [27] Johnson, E. 1992. Process region changes for rapidly propagating cracks. International Journal of Fracture 55, 47–63. [28] Xu, X.-P., and A. Needleman. 1994. Numerical simulations of fast crack growth in brittle solids. Journal of the Mechanics and Physics of Solids 42, 1397–1434. [29] Xu, X.-P., A. Needleman, and F. F. Abraham. 1997. Effect of inhomogeneities on dynamic crack growth in an elastic solid. Modeling and Simulation in Materials Science and Engineering 5, 489–516.

[30] Falk, M. L., A. Needleman, and J. R. Rice. 2001. A critical evaluation of dynamic

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fracture simulations using cohesive surfaces. Submitted to 5th European Mechanics of Materials Conference (Delft, 5–9 March 2001). Camacho, G. T., and M. Ortiz. 1996. Computational modelling of impact damage in brittle materials. International Journal of Solids and Structures 33, 2899– 2938. Gao, H. 1996. A theory of local limiting speed in dynamic fracture. Journal of the Mechanics and Physics of Solids 44, 1453–1474. Andrews, D. J. 1976. Rupture velocity of plane strain shear cracks. Journal of Geophysical Research 81, 5679–5687. Burridge, R., G. Conn, and L. B. Freund. 1979. The stability of a rapid mode II shear crack with finite cohesive traction. Journal of Geophysical Research 84, 2210–2222. Rosakis, A. J., O. Samudrala, and D. Coker. 1999. Cracks faster than the shear wave speed. Science 284, 1337–1340.

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[36] Broberg, K. B. 1987. On crack paths. Engineering Fracture Mechanics 28, 663– 679. [37] Heaton, T. H. 1990. Evidence for and implications of self-healing pulses of slip in earthquake rupture. Physics of the Earth and Planetary Interiors 64, 1–20.

[38] Kostrov, B. V. 1964. Self-similar problems of propagation of shear cracks. Journal of Applied Mathematics and Mechanics 28, 1077–1087 (in Russian). [39] Madariaga, R. 1976. Dynamics of an expanding circular fault. Bulletin of the Seismological Society of America 66, 639–666. [40] Freund, L. B. 1979. The mechanics of dynamic shear crack propagation. Journal of Geophysical Research 84, 2199–2209. [41] Day, S. M. 1982. Three-dimensional simulation of spontaneous rupture: The effect of nonuniform prestres. Bulletin of the Seismological Society of America 72, 1889–1902. [42] Ida, Y. 1972. Cohesive force across the tip of a longitudinal-shear crack and Griffith’s specific surface energy. Journal of Geophysical Research 77, 3796–3805. [43] Andrews, D. J. 1985. Dynamic plane-strain shear rupture with a slip-weakening friction law calculated by a boundary integral method. Bulletin of the Seismological Society of America 75, 1–21. [44] Harris, R., and S. M. Day. 1993. Dynamics of fault interactions: Parallel strike-

slip faults. Journal of Geophysical Research 98, 4461–4472. [45] Das, S., and K. Aki. 1977. A numerical study of two-dimensional spontaneous rupture propagation. Geophysical Journal of the Royal Astronomical Society 50, 643–668. [46] Das, S. 1980. A numerical method for determination of source time functions for general three-dimensional rupture propagation. Geophysical Journal of the Royal Astronomical Society 62, 591–604. [47] Das, S. 1985. Application of dynamic shear crack models to the study of the earthquake faulting process. International Journal of Fracture 27, 263–276. [48] Zheng, G., and J. R. Rice. 1998. Conditions under which velocity-weakening

friction allows a self-healing versus cracklike mode of rupture. Bulletin of the Seismological Society of America 88, 1466–1483. [49] Cochard, A. and R. Madariaga. 1996. Complexity of seismicity due to highly rate dependent friction. Journal of Geophysical Research 101, 25321–25336. [50] Beeler, N. M., and T. E. Tullis. 1996. Self-healing slip pulse in dynamic rupture models due to velocity-dependent strength. Bulletin of the Seismological Society of America 86, 1130–1148. [51] Rice, J. R., N. Lapusta, and K. Ranjith. 2001. Rate and state dependent friction and the stability of sliding between elastically deformable solids. Submitted to Journal of the Mechanics and Physics of Solids. [52] Weertman, J. 1969. Dislocation motion on an interface with friction that is dependent on sliding velocity. Journal of Geophysical Research 74, 6617–6622.

[53] Lapusta, N., J. R. Rice, and R. Madariaga. 2000. Research in progress. [54] Nielsen, S. B., and J. M. Carlson. 2000. Rupture pulse characterization: Selfhealing, self-similar, expanding solutions in a continuum model of fault dynamics. Bulletin of the Seismological Society of America, in press. [55] Renardy, M. 1992. Ill-posedness at the boundary for elastic solids sliding under Coulomb friction. Journal of Elasticity 27, 281–287. [56] Adams, G. G. 1995. Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction. Journal of Applied Mechanics 62, 867–872.

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[57] Martins, J. A. C., J. Guimarães, and L. O. Faria. 1995. Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions. Journal of Vibration and Acoustics 117, 445–451. [58] Simões, F. M. F., and J. A. C. Martins. 1998. Instability and ill-posedness in some friction problems. International Journal of Engineering Science 36, 1265–1293.

[59] Ranjith, K., and J. R. Rice. 2001. Slip dynamics at an interface between dissimilar materials. Journal of the Mechanics and Physics of Solids 49, 341–361. [60] Weertman, J. 1980. Unstable slippage across a fault that separates elastic media of different elastic constants. Journal of Geophysical Research 85, 1455–1461. [61] Adams, G. G. 1998. Steady sliding of two elastic half-spaces with friction reduction due to interface stick-slip. Journal of Applied Mechanics 65, 470–475. [62] Andrews, D. J., and Y. Ben-Zion. Wrinkle-like slip pulse on a fault between different materials. Journal of Geophysical Research 102, 553–571. [63] Ben-Zion, Y., and D. J. Andrews. 1998. Properties and implications of dynamic rupture along a material interface. Bulletin of the Seismological Society of America, 88, 1085–1094. [64] Harris, R., and S. M. Day. 1997. Effects of a low velocity zone on a dynamic rupture. Bulletin of the Seismological Society of America 87, 1267–1280. [65] Cochard, A., and J. R. Rice. 2000. Fault rupture between dissimilar materials: Ill-posedness, regularization and slip-pulse response. Journal of Geophysical Research 105, 25891–25907. [66] Weertman, J. 1963. Dislocations moving uniformly on the interface between isotropic media of different elastic properties. Journal of the Mechanics and

Physics of Solids 11, 197–204. [67] Gol’dshtein, R. V. 1967. On surface waves in joined elastic media and their relation to crack propagation along the junction. Prikladnaya Matematika i Mekhanika 31(3), 468–475 (English translation, Journal of Applied Mathematics and Mechanics 31, 496–502). [68] Achenbach, J. D., and H. I. Epstein. 1967. Dynamic interaction of a layer and a half-space. Journal of Engineering Mechanics 5, 27–42.

[69] Prakash, V., and R. J. Clifton. 1992. Pressure–shear plate impact measurement of dynamic friction for high speed machining applications. Proceedings of VII International Congress on Experimental Mechanics. Bethel, Conn.: Society for Experimental Mechanics, 556–564. [70] Prakash, V. 1998. Frictional response of sliding interfaces subjected to time varying normal pressures. Journal of Tribology 120, 97–102. [71] Lambros, J., and A. J. Rosakis. 1995. Development of a dynamic decohesion criterion for subsonic fracture of the interface between two dissimilar materials.

Proceedings of the Royal Society of London A 451, 711–736. [72] Singh, R. P., J. Lambros, A. Shukla, and A. J. Rosakis. 1997. Investigation of the mechanics of intersonic crack propagation along a bimaterial interface using coherent gradient sensing and photoelasticity. Proceedings of the Royal Society

of London A 453, 2649–2667.

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Dr. Richard M. Christensen (left), one of the ICTAM 2000 Introductory Lecturers, jokes with Prof. L. Ben Freund, Treasurer of IUTAM, during a poster session. ICTAM 2000 featured 19 Introductory Lectures in 6 Mini-Symposia, and 18 Sectional Lectures.

A SURVEY OF AND EVALUATION METHODOLOGY FOR FIBER COMPOSITE MATERIAL FAILURE THEORIES Richard M. Christensen Department of Aeronautics and Astronautics Stanford University, Stanford, Calif., USA [email protected] Abstract

1.

The long-standing problem of characterizing failure for fiber composite materials will be reviewed. Emphasis will be given to the lamina level involving nominally aligned fibers in a matrix phase. However, some consideration will also be given to laminate failure using the lamina form as the basic building block along with the concept of progressive damage. The many different lamina-level theories will be surveyed along with the commitment necessary to produce critical experimental data. Four particular theories will be reviewed and compared in some detail, these being the Tsai–Wu, Hashin, Puck, and Christensen forms. These four theories are reasonably representative of the great variety of different forms with widely different physical effects that can be encountered; also, for comparison, the rudimentary forms of maximum normal stress and maximum normal strain criteria will be given. The controversial problem of how many different individual modes of failure are necessary to describe general failure will receive attention. A specific and detailed methodology for evaluation of all the various theories will be formulated.

INTRODUCTION

Theories of failure for anisotropic materials have been propounded for at least the past forty years. The advent of high strength, highly anisotropic fiber composite materials has accelerated the activity and accentuated the importance of the search. The lack of agreement on a single, best theory has not been for lack of activity. If one includes all forms of theoretical failure characterization, there are probably well over one hundred different theoretical forms, sometimes applicable over widely different conditions. To put some scope and limits on the present considerations, only reasonably comprehensive theories (not individual 25 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 25–40. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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mechanism theories) will be considered, meaning theories applicable to fully three-dimensional states of stress and strain. That the consideration and evaluation of such theories of behavior is a difficult proposition should be self evident. Let it just be said that even in the case of isotropic materials, the theoretical characterization of failure is not a settled issue. The corresponding problem for highly anisotropic materials could reasonably be expected to be much more difficult than that for isotropy. A sampling of some fully three-dimensional theories of failure is given in Table 1. The sources for these theories are given in the list of references. By no means are the theories limited to the case of just 10 adjustable parameters. There are theories with 15 or more parameters. This situation immediately raises the issue of the degree of practicality for theories with large numbers of parameters. It would appear that somewhere in the range of 7–9 parameters is pushing the upper limit. There is a corresponding problem at the other end of the scale. What is the fewest number of parameters that can capture the immensely complicated interactive physical effects that occur at the threshold of failure? Additionally, in developing theories of failure, one must first make the decision of the basis type, namely stress-based or strain-based. By far the more common are the stress-based forms, and they will receive primary emphasis here. Most of the consideration here will be given to the lamina-level form of aligned fiber composites. After examining this level at some length, consideration will be given to the laminate form, which is used in most applications. Nevertheless the emphasis here is upon the lamina level, since it is the basic building block for most composites applications. At the lamina level, the fiber composite material will be idealized as being

Fiber composite materials failure theories

27

transversely isotropic, and all theories to be considered will be of that type. Since these are macroscopic theories of failure, the micro-scale failure mechanisms could be due to a wide array of processes related to fiber degradation, matrix degradation, interface failure and all manner of interactive complications. When matrix-controlled modes of failure are designated, these will be implicitly taken to include interface-failureprecipitated events. It is important to acknowledge the existence of an ongoing fiber composite failure theory evaluation program. This commendable effort, organized by Hinton and Soden (1998) and Soden, Hinton and Kaddour (1998) was initiated some years ago and is nearing completion. It considers many more theories than just the sampling shown here in Table 1, and some theories included here are not part of their evaluation. The present evaluation examination is not coordinated with the Hinton, Soden, Kaddour study for two reasons. Their evaluation interests and procedures are entirely of a 2-D, plane-stress nature, whereas the present interests are entirely of a three-dimensional nature. Secondly, their aim is to evaluate 2-D lamina-level theories primarily from laminate-level behavior. Theirs is a rather complex undertaking because of the lamina-to-lamina interaction that occurs in a laminate. Even though the ultimate objective must be to predict laminate-level behav-

ior, it is here felt that the most firm ground for evaluating lamina-level theories is from procedures and results obtained at the same level, that of the lamina. Four of the theories of failure behavior shown in Table 1 are selected here for examination. These are the Tsai–Wu, Hashin, Puck, and Christensen theories, all expressed in terms of stresses. These four theories are quite representative of the great variety of different forms with widely different physical effects that can be encountered. Also, for comparison, the rudimentary forms of normal-stress and normal-strain criteria will be considered. All four of the fully 3-D theories are at the quadratic level of representation. After outlining these theories and showing some aspects of the varied behavior, we will formulate a specific evaluation methodology. The experimental commitment needed to effect the evaluation will receive consideration. Finally, some aspects of laminate-level behavior will be considered.

2.

SPECIFIC THEORIES OF FAILURE

The four theories of failure to be considered here are those of Tsai–Wu (1971), Hashin (1980), Puck (Puck and Schürmann (1998), Kopp and Michaeli (1999)) and Christensen (1997, 1998). Table 2 shows some char-

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acteristics of these four theories. All the theories except the Tsai–Wu form are decomposed into matrix-controlled modes of failure or fibercontrolled modes of failure. The Tsai–Wu form is said to be fully interactive, with all effects folded into one overall mode of failure. In the present context the number of modes of failure is defined as the number of distinct branches that are in evidence in the failure envelope in stress space. These branches may or may not intersect, but if they do, the slopes are discontinuous. Prom Table 1 it is seen that the number of modes of failure range from one to a number equal to the number of parameters in the theory. This variation perhaps as much as anything illustrates the great diversity that is encountered when comparing these theories. An even more graphic form of the differences between the theories is as shown in Fig. 1, which are examples of the failure envelopes in a subspace of stress. The matrix-controlled modes of failure are schematically shown in space, where axis 1 is always in the fiber direction. The forms for the various theories are representations taken from related references. They are not specifically calibrated to identical properties, but rather serve only to illustrate the great variety of different behaviors that are possible. A similar comparison between fiber-controlled modes of failure would likely be even more divergent; however such forms are not readily available in the literature. The four individual theories of failure will now be displayed for comparison purposes. For the most part, the terminology followed will be that of the authors.

2.1.

Tsai–Wu criterion

The Tsai–Wu (1971) theory is often called the tensor polynomial theory since that is exactly what it is. The stress invariants for transversely isotropic symmetry are used in a polynomial expansion up to terms of

Fiber composite materials failure theories

29

Figure 1 Biaxial stress failure examples.

second degree. The following form is then the failure criterion:

where and are the seven parameters that are to be evaluated from data. Five of the parameters are evaluated directly from the relations

where and and and are the uniaxial tensile and compressive failure stress magnitudes in the axial and transverse directions, respectively, and is the axial shear failure stress. The remaining two parameters, and must be evaluated from more complicated stress condition testing, or they must be estimated; the latter procedure is normally followed (Hahn and Kallas 1992). This Tsai–Wu theory is by far the simplest of the four theories; yet it contains all the same threedimensional terms. To the extent allowed by the form of the invariants,

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it has all stress components interacting with each other through (1) and (2).

2.2. Hashin criteria The Hashin (1980) criteria begin with the second-degree polynomial expansion in the invariants. The failure modes are then decomposed into matrix-controlled and fiber-controlled forms, depending upon which stress components act upon the failure planes, these planes being taken parallel and perpendicular to the fiber direction, respectively. Also, interaction parameter in (1) is taken to vanish. Next each mode is further decomposed into tensile-controlled and compressive-controlled forms, introducing 4 additional parameters. Finally, 4 separate assumptions or conditions are imposed, bringing the total parameter count to 6.

Thus, the final forms are composed of 4 modes of failure along with 6 parameters, as follows: Tensile matrix mode,

Compressive matrix mode,

Tensile fiber mode,

Compressive fiber mode,

The 6 parameters have been evaluated from the fiber failure normal and shear stresses in the axial direction,

and the matrix-controlled normal and shear stresses in the transverse direction,

Fiber composite materials failure theories

31

As seen in (3) and (4), the tensile- and compressive-type matrix modes of failure are differentiated by the sign of the transverse direction mean normal stress.

2.3.

Puck criteria

The Puck criteria have evolved over a period of many years. The works by Puck, his students, and colleagues are here represented by Puck and Schürmann (1998) and Kopp and Michaeli (1999). Whereas the Hashin criterion was somewhat motivated by the Coulomb–Mohr approach for isotropic materials, the Puck procedure goes further along this direction and follows the Coulomb–Mohr procedure quite strictly, at least insofar as matrix-controlled failure modes are concerned. The Puck criteria are the most complicated forms considered here, ultimately resulting in numerical procedures. First consider the matrix-controlled modes of failure involving failure planes parallel to the fiber direction, with the corresponding normal and shear stresses upon the failure plane. A failure criterion as shown in Fig. 2 is taken in terms of the axial shear stress, the transverse shear stress, and the normal stress due to the transverse normal stresses. This

Figure 2 Puck theory failure surface.

procedure introduces 7 parameters. Next, all failure plane orientations are scanned to find the failure plane orientation with the worst combination of normal and shear stresses that produces failure, when compared with the failure criterion of Fig. 2. The end result of this numerical scanning program is the generation and display of failure surfaces in stress space. Seven different modes of failure are identified by this process.

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With regard to the fiber-controlled modes of failure, these are written in plane stress form with fiber-direction strain as

for tension, and a similar expression for compression. The two parameters are the strains to failure, and Symbols and are the fiber-phase Poisson’s ratio and Young’s modulus, respectively. The “magnification” factors are said to be known for different fiber types.

2.4.

Christensen criteria

The procedure followed by Christensen (1997, 1998) starts with starts with the general 7-parameter polynomial expansion of second degree. Micromechanics is used to decompose the polynomial form into 2 separate criteria—matrix- and fiber-controlled. Then independence of both these criteria from failure under hydrostatic pressure is imposed. This independence reduces the parameter count from 7 to 5. The matrix-controlled criterion is given by

where

with and being the transverse normal stress failure levels and the axial shear stress failure level. Condition applies here. The fiber-controlled criterion is given by

where

and where are the fiber-direction failure stresses. This fibercontrolled criterion decomposes into two separate branches, as seen in particular applications.

Fiber composite materials failure theories

3.

33

EVALUATION METHODOLOGY

Probably the most contentious argument that ensues when discussing composite material failure theories is how many different modes of failure should be expected. The sampling of different theories shown in Tables 1 and 2 ranges from 1 mode of failure through 9 modes of failure. This discussion or argument probably does not have a simple, easily established answer. Arguing the merits or demerits of various theories on a conceptual basis seems particularly open-ended and nonproductive. The approach to be sought here avoids this argument by seeking a critical comparison of the theories with experimental data. The question then becomes one of what type of experimental data could be used for such an important purpose. The types of experiment to be used to evaluate the theories is intimately connected with the character of the experiments used to evaluate the determining parameters in each and every theory. An informal survey of experimentalists has indicated a consensus that the maximum number of independent strength-property experiments that is practical under most circumstances is about five. Furthermore, the most practical explicit set of experiments is that used to determine the set of failure properties given by

which are the fiber-direction tensile and compressive strengths, the transverse tensile and compressive strengths, and the axial shear strength, respectively. These are the one-dimensional tests that embody current, standard practice. The method for evaluating the various theories is that they should be evaluated in the immediate, near region of the five data points used to calibrate the theories. If a given theory cannot give a successful prediction of behavior in the neighborhoods of the data points used for the calibration of the theory, then that theory is unlikely to be generally successful in the stress-state regions far removed from the data calibration points. In selecting a criterion to be used for the evaluation, there could be several options, but one particularly attractive one, and the one followed here, is that of the effect of superimposed hydrostatic pressure. For example, if the test used to calibrate a given theory were to give a value of the transverse tensile strength as then the evaluating experiment would involve a second testing procedure to determine by how much the transverse tensile strength is changed by the presence of a superimposed pressure. And, the superimposed pressure should be small in magnitude so that one is probing the near neighborhood of

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the data point. Testing under superimposed pressure is a well developed and well validated technique that has been extensively used with isotropic materials. The technique is particularly appropriate for composite materials, since the common characteristics of and are direct manifestations of the effect of and importance of the mean-normal-stress effect. The pressure-effect forms corresponding to the properties (10) that are to be determined experimentally are

where p is the superimposed pressure. After the experimental database for these is established, any theory calibrated by the properties data (10) would then be used to predict the values for (11). Examples of this procedure will be given shortly. It may be noted that the procedure just stated corresponds to examining the first two terms of a Taylor series. The theory calibration data (10) correspond to the first term; the derivatives in (11)—both theory and measured—correspond to the second term. It should be recognized that this proposed approach would not constitute a comprehensive evaluation, but rather would be the first step in such a direction. An overall evaluation would likely be a graduated process. For example, the possible coupling of the fiber-controlled modes of failure with the axial shear stress is of potential significance and possible controversy. Nevertheless, the present approach could be an important first step through which many or most theories could be eliminated from consideration. It also should be noted that any theory with any number of parameters could be evaluated by this method. If the number of parameters were greater than 5, the extra parameters would need to be evaluated by auxiliary information beyond that of (10), as has always been done in such cases. It is helpful to change the notation slightly before illustrating the method with examples. We write the total stress tensor as

where p is the applied pressure and is the difference between the total stress and the pressure-induced stress. With (12) the derivatives are related by

Fiber composite materials failure theories

35

Rather than using the total stress derivatives in (11) for the evaluation, we use the derivatives of the stress difference given in (13). Let

The advantage of the notation in (14) is that a value of is indicative of a Mises-like behavior involving independence of mean normal

stress. Three examples will now be given, the first two being quite simple, but still relevant. Two of the oldest failure criteria are those imposed upon normal stress and normal strain. We take these criteria for the fiber direction stress as

The derivatives in (14) are easily shown to be, from (15),

These results do not even differentiate between different composite materials with different properties. Using the data from Parry and Wronski (1982, 1985) for graphite–epoxy composites, one finds to

– 1.1 and

to 0.42. Based upon these rather old data sets,

even the sign of (16) is incorrect, while the scale of (17) appears to be too large. Now consider normal strain in the fiber direction as the failure criterion:

It can be shown that for (18) the derivatives (14) are given by

where is the axial Poisson’s ratio. Compared with the data of Parry and Wronski mentioned above, relation (19) appears to be of the incorrect sign. These two examples show that this evaluation method is highly discriminating. But before any conclusions can be drawn, it would be necessary to have modern, highly reliable data. Now a more comprehensive example will be given to show that the procedure is well posed and entirely practical. Using the failure criteria of Christensen, (8) and (9),

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one can determine the derivatives in (14) in a straightforward manner, giving

Typical data for graphite–epoxy composites are given by

Using data set (22) in the derivatives (21) gives

It is the specific results such as in (23) that are to be compared with modern experimental measurements, permitting the formal evaluation. These examples illustrate the well grounded and basic nature of this program for composite failure-theory evaluation. If an advocate of a particular failure theory should allege that it is too difficult to evaluate the derivatives in (14), then there could be cause to doubt the utility and practicality of the theory under question. The vital need for well characterized failure data will be considered in the last section. These four theories plus the simple normal stress and strain criteria are employed here as samples, albeit significant ones. Any theory could be subjected to this evaluation methodology. It is possible to build up a picture of overall behavior through the combination of individual modes of failure, although these four theories do not illustrate that possibility. Such a procedure has been recommended by Hart-Smith (1993), and any overall theory of behavior so formed could also be evaluated by this method.

Fiber composite materials failure theories

4.

37

LAMINATES Treating the failure of laminates is inherently much more complicated

than treating the failure of a lamina. Nevertheless, there is a widely used approach for the case of laminate failure. This is usually called progressive damage, and it is a natural extension of lamina-level failure theory. The simplest form is that of first-ply failure, but that approach is too conservative for most purposes. The decomposition of moduli-type and failure-type properties into matrix-controlled and fiber-controlled

modes is the motivation for progressive damage. With strain compatibility in adjacent lamina within a laminate, the damage spreads from lamina to lamina. Matrix failure occurs at much lower strains than fibercontrolled effects, so the matrix-controlled degradation usually starts the process. As loading continues, the properties must be degraded selectively or collectively. Then, when fiber-controlled failures occur over a

sufficiently large region, structural failure follows. One could use fracture mechanics to motivate and describe a critical scale of local failure beyond which uncontrolled fracture follows. This properties-degradation process works fairly well; it is semi-empirical, but well established. The transverse cracking at the lamina level can lead to delamination between lamina. Although appealing in its step-by-step approach, the method can become quite complicated, with many pitfalls and traps. The status of the progressive-damage methodology is probably best described by Rohrauer (1999) in a Ph.D. thesis of unusual scope and gravity. He

states: “The quest to ascertain what is happening inside a failed lamina and how it affects the continued existence or catastrophic destruction of

the whole laminate is a continuing one. Few subjects are as confusing and complex. The search for answers has lead to voluminous publications; few if any definitive methods useful to the designer exist as of yet.”

Certainly significant steps have been taken to define the lamina-level damage problem in precise terms that admit extension and generalization. These works include the essentially lamina-level matrix-cracking problem by Hashin (1996) and Nairn and Shu (1994). An alternative to the lamina-to-laminate failure sequencing through progressive damage can be seen. This alternative would be to treat

failure entirely at the laminate level and thereby obtain a complete, self-contained treatment. This approach apparently has not been taken in the past. It has more-or-less been taken as obvious truth that the laminate level is too difficult to approach directly, and that it must be approached incrementally through progressive damage. That view notwithstanding, the full problem should be approached full on; with

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the proper metrics and invariants, there is a reasonable case to be made for optimism.

5.

CONCLUSION

The overall conclusion is that it is possible and practical to compare and evaluate fiber composite material failure theories at the lamina level. This approach would avoid the complications of evaluating lamina-level theories from behavior at the laminate level, which inter-

mixes all the assumptions and idealizations of the lamina-level theory with all the assumptions and idealizations of any particular progressive damage scenario. In reality, two separate and distinct evaluations should be conducted—that for the lamina-level failure theory, and that for the

progressive damage treatment of laminates. The present work has been focused upon the lamina-level evaluation. The specific recommendations are as follows: (i) One should collect the various failure theories that are to be

considered—four of them have been discussed here—and reduce them to the form that predicts the quantitative effect of a superimposed pressure upon the five basic strength characteristics for the aligned fiber lamina form.

(ii) To calibrate and normalize the theories, one should experimentally determine the five basic strength properties for the fiber composite systems of interest, repeat the experimental determination of the five strength properties under the effect of a superimposed pressure, and combine results from (i) and (ii) to evaluate the laminalevel theories. It is of great importance that the experimental investigation receive the highest possible priority. The detailed

evaluation will be no better than the quality and reliability of the data. An enormous effort has been expended on the development of the many different theoretical forms over many years. A commensurate effort should be dedicated to the experimental investigation in order to complete a meaningful evaluation process.

(iii) As a completely separate matter, the progressive damage methodology should be more highly developed; then the various proposed steps and procedures should be evaluated. (iv) As another completely separate but related matter, failure characterization should be formulated directly at the laminate level, and ultimately compared with projections and predictions obtained from lamina-level theory combined with progressive damage.

Fiber composite materials failure theories

39

Acknowledgments The author is appreciative of many helpful discussions with Dr. S. J. DeTeresa of LLNL. The author is also appreciative of support from the Office of Naval Research, Dr. Y. D. S. Rajapakse, contract monitor. This work was performed under the auspices of the U.S. Department of Energy, University of California Lawrence Livermore National Laboratory, under contract W-7405-Eng-48.

References Boehler, J. P., and M. Delafin. 1979. Failure criteria for unidirectional fiber-reinforced composites under confining pressure. Proceedings of the Euromech Colloquium 115. The Hague: Martinus Nijhoff, 449–470. Christensen, R. M. 1988. Tensor transformations and failure criteria for the analysis of fiber composite materials. Journal of Composite Materials 22, 874–897. Christensen, R. M. 1997. Stress based yield/failure criteria for fiber composites. Inter-

national Journal of Solids and Structures 34, 529–543. Christensen, R. M. 1998. The numbers of elastic properties and failure parameters for fiber composites. Journal of Engineering Materials and Technology 120, 110–113. Cuntze, R. G. 1999. Progressive failure of 3d-stressed laminates: Multiple nonlinearity treated by the failure mode concept. Duracosys Conference Manuscript (from the author). Feng, W. W. 1991. A failure criterion for composite materials. Journal of Composite Materials 25, 88–100. Gosse, J. 1999. Boeing Aerospace. Private communication.

Hahn, H. T., and M. N. Kallas. 1992. Failure criteria for thick composites. Ballistic Research Laboratory Report CR-691, Aberdeen Proving Ground, Md. Hart-Smith, L. J. 1993. Should fibrous composites failure modes be interacted or superimposed? Composites 24, 53–55. Hashin, Z. 1980. Failure criteria for undirectional fiber composites. Journal of Applied Mechanics 31, 223–232. Hashin, Z. 1996. Finite thermoelastic fracture criterion with application to laminate cracking analysis. Journal of the Mechanics and Physics of Solids 44, 1129–1145. Hinton, M. J., and P. D. Soden. 1998. Predicting failure in composite laminates: The background to the exercise. Composites Science and Technology 58, 1001–1010. Kopp, J., and W. Michaeli. 1999. The new action plane related strength criterion in comparison with common strength criteria. Proceedings of the International Conference on Composite Materials 12, Paris. Nairn, J. A., and S. Shu. 1994. Matrix microcracking. In Damage Mechanics of Composites (R. Talreja, ed.). New York: Elsevier Science, 187–243. Parry, T. V., and A. S. Wronski. 1982. Kinking and compressive failure in uniaxially aligned carbon fiber composite tested under superimposed hydrostatic pressure. Journal of Materials Science 17, 893–900. Parry, T. V., and A. S. Wronski. 1985. The effect of hydrostatic pressure on the tensile properties of pultruded CFRP. Journal of Materials Science 20, 2141–2147. Puck, A., and H. Schürmann. 1998. Failure analysis of FRP laminates by means of physically based phenomenological models. Composites Science and Technology

58, 1045–1067. Rohrauer, G. 1999. Ultra-high pressure composite vessels with efficient stress distributions. Ph.D. thesis, Concordia University, Montreal, Canada.

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Soden, P. D., M. J. Hinton, and A. S. Kaddour. 1998. A comparison of the predictive capabilities of current failure theories for composite laminates. Composites Science and Technology 58, 1225–1254. Tsai, S. W., and E. M. Wu. 1971. A general theory of strength for anisotropic materials. Journal of Composite Materials 5, 58–80.

DAMAGE MECHANICS OF METAL-MATRIX COMPOSITES: VARIOUS LEVELS OF APPROACH Jean-Louis Chaboche Département de Mécanique du Solide et de l’Endommanagement (DMSE) Office National d’Etudes et de Researches Aérospatiales (ONERA) Châtillon Cedex, France and Université de Technologie de Troyes [email protected]

Abstract

1.

Constitutive and damage mechanics of composite materials is discussed, with a special attention to unidirectional metal-matrix composites. Macroscopic thermo–elasto–visco–plastic and damage models based on micromechanical approaches are presented that offer various modeling capabilities between a two-phase and a multi-subvolume model. A micromechanics analysis of transverse creep, using periodic homogenization and finite elements demonstrates the role of the fiber–matrix interphase in the creep resistance of the composite. Such a numerical approach can be used to deliver directly the constitutive response at the component level by a multiscale structural analysis using the FE2 imbricated finite-element method.

INTRODUCTION

Composite materials and structures differ from metallic ones in the following aspects: their low weight, and the corresponding design improvement perspective offered by their high specific mechanical properties; their versatility—in many applications, the possibility to optimize the material (volume fraction, sizes, positions, orientations of reinforcements) during the same design stage as the structural component itself; their ability to support such inserted systems as optical fibers, piezo-electric sensors, and actuators, leading to many future “smart structure” capabilities, for instance in active or passive in situ health monitoring systems; 41 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 41–56. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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Figure 1

Schematic of the various scales in composite materials and structures.

their cost, which is often a serious obstacle to their extensive use, as well as the relatively unstable material sources and material manufacturers; their multiscale structure, showing many different successive levels, as illustrated schematically in Fig. 1 for structural composites used in aeronautical applications—this is one of the difficulties in the mechanical analysis associated with the use of composites; metals also have several levels of microstructures, but at lower scales, and with a much more random organization; their damaging processes, during manufacturing and during operation: in metals initiation of any microcrack of some size could have a significant impact on the remaining lifetime, leading often to an accelerated process under load; contrarily, in composite structures the microcracks can appear very early, leading to a certain deterioration of mechanical properties, but without really affecting the component lifetime—damage in that case is one of the mechanisms by which the heterogeneous material accommodates the applied strain.

In this paper the two last aspects mentioned above will be illustrated and discussed in terms of modeling methodologies, as applied to unidirectional metal-matrix composites (MMCs). The specific structural

Damage mechanics of metal-matrix composites

43

applications underlying the developed methods are the bling components (bladed rings) that should replace in some aerojet engines the classical technologies for compressor and turbine discs. The material in the corresponding experimental studies is a SiC/Ti MMC, but many aspects of the methods are still applicable to other classes of metallic composites (at least long fibers). In 2 we present a micromechanics-based set of constitutive and damage equations, exploiting Dvorak’s transformation field analysis (TFA) method (Dvorak 1992, Dvorak and Benveniste 1992), which offers several variants for the damage-growth modeling—in the matrix, in the fiber, and at the fiber–matrix interface. The models are set in such a way that they can be considered and exploited as macroscopic constitutive laws. Section 3 summarizes a specific study of creep behavior of the composite (essentially transverse creep), in which is shown the duality between the matrix elasto–visco–plasticity and the interfacial damage, in order to explain experimental results of creep tests realized under vacuum. Other aspects are treated elsewhere, for instance the component inelastic and damage analysis, based on a true “multiscale analysis”, the FE2 method, using imbricated finite-element models in order to replace the macrolevel constitutive equations directly (Feyel 1999).

2.

DEVELOPMENT OF A SET OF MICROMECHANICS-BASED CONSTITUTIVE EQUATIONS

For the inelastic and damage modeling of composite materials, within a micromechanics based approach, we have various possibilities:

the tools using averaged quantities in each constituent, similar to those employed for random microstructures, with Eshelby-based operators: in this context localization rules are often based on a secant or tangent writing of the local constitutive equations; the more-or-less analytical models based on a mixture of tangent and initial elastic operators, such as the ones developed by Voyiadjis and co-workers (Voyiadjis and Kattan 1993) or by Baptiste (Guo et al. 1997);

the class of model developed by Aboudi and co-workers (Aboudi 1985, Pindera and Aboudi 1988), which defines a geometrically more-or-less sophisticated unit cell, with local fields approximated through a weak compatibility of constituent elasticity equations; the transformation field analysis (TFA) developed by Dvorak (Dvorak 1992, Dvorak and Benveniste 1992), which introduces a

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systematic way of defining concentration and influence tensors, but writing a purely elastic rule, treating plastic strain and thermal expansion as eigenstrains of the same nature.

We decide to concentrate on the last type of approach, because it continues to treat elasto–plasticity (or elasto–visco–plasticity) of constituents through the correct strain partitioning. Moreover, within a single and consistent methodology, it offers various refinement possibilities, from the two-subvolume model (two-phase model) to the multisubvolume model, which eventually degenerates to the finite-element periodic unit cell.

2.1.

Thermo–elasto–visco–plastic model

The model is built up by combining two kinds of equation. The first is the constitutive equation of each constituent:

in which s denotes the subvolume numbering (there are possibly several subvolumes for each phase), Ls is the initial elastic stiffness operator, and the functional will not be defined more precisely—all kinds of local constitutive equations can be used, with internal state variables symbolized here as “ISV”. The thermal expansion can be isotropic or not. The second kind of equation is the localization rule that relates overall to local quantities, taking into account field interactions with eigenstrains. Using TFA, we have for the local strain in the subvolume s:

where is the average strain (in which cr denotes the volume fraction of subvolume r), As and Dsr being respectively the elastic strain concentration tensor and the influence tensor (between subvolume r and subvolume s). Averaging in (1) leads to the overall elastic constitutive equation

where tration tensor by L = =

with B the stress concenand L the overall elastic stiffness defined The reciprocal formulation can also be defined

Damage mechanics of metal-matrix composites

Figure 2

45

Correction with the asymptotic tangent stiffness for various macroscopic

strain rates (volume fraction 35%).

consistently for the stress localization (not needed here). In the case of a two-phase composite system, with one subvolume each, the operators A s , Bs can be obtained from the Eshelby tensor, either through the Mori–Tanaka method (Mori and Tanaka 1973) or through the selfconsistent method. The influence tensors are also easily determined for the two-subvolume model, with exact closed-form relations (Dvorak 1992). When using a more sophisticated model, with several subvolumes per phase, the initial elastic concentration tensors A s , B s and the influence tensors Dsr are determined once, by a prior set of numerical analyses (using for instance periodic homogenization and finite-element method, as indicated in Dvorak et al. 1994 or Pottier 1998). Considering the overly stiff results generally obtained for the twosubvolume model (see Suquet 1997, Zaoui and Masson 1998), we have proposed an asymptotic correction method (Pottier 1998, Chaboche et al. 2000) that modifies the localization rule into

where the 4th-rank tensors Kr are determined by identification of the rate form of (5) with its “tangent” format in which the tangent concentration tensor is determined from the knowledge of the asymptotic tangent stiffness of each subvolume, directly related with

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the linear kinematic hardening model used in the matrix. More details are given in (Chaboche et al. 2000). The improvement of using this correction factor instead of the original TFA is illustrated in Fig. 2 for the transverse traction at three overall strain rates (the matrix is visco–plastic) by comparison with a periodic unit cell finite-element calculation similar to the ones that will be reported in 3. Other comparisons are reported elsewhere (Chaboche et al. 2000).

2.2.

Various damage modeling approaches

In terms of delivering a micromechanics-based overall constitutive equation of composite materials, three approaches could be used, depending on the kind of deterioration that develops:

If microcracks are large enough compared with the heterogeneous microstructure (fiber size) we could proceed in two steps: define first an effective homogeneous behavior of the undamaged composite, then input microcracks in it and deliver the overall behavior of the damaged material via a second micro-to-macro analysis. This is the method implicitly used in laminate composites when introducing transverse cracking as a meso-damage in the ply volume elements (Allix et al. 1990). If microcracks are small enough, in the matrix or in the fiber, compared with the fiber size, we can use a continuum damage mechanics (CDM) methodology in each of these phases, then first modify the initial local constitutive equations through a micromechanicsbased effective stress concept, and second, make the micro–macro averaging of the composite with its damaged constituents. This is for example the method used by Voyiadjis (Voyiadjis and Kattan 1993) for modeling damage in MMCs.

If microcracks and fiber are of the same size, especially when the crack develops at the fiber–matrix interface, none of the above approaches is acceptable (i.e. there is no scale separability). One simplification consists in approximating the presence of an interface crack by a transverse stiffness reduction in the fiber (Guo et al. 1997). It is also the solution adopted in the direct simplified model developed below. A better approximation is sought by generalizing the TFA method, with a large number of subvolumes, including interphase subvolumes. We present below the corresponding model, called the generalized eigenstrain model.

Damage mechanics of metal-matrix composites

47

Figure 3 Direct simplified model and finite-element computation for transverse tensile loading–unloading: case of an elastic matrix.

Direct simplified model (DSM). We consider only two phases— the matrix and the fiber—and assume a CDM approach to introduce a damage ds in each phase (either a scalar or a tensorial variable). Now the constitutive equation is modified as

The visco–plastic strain rate depends on damage, as well as the elastic stiffness . Figure 4 explains the principle of the direct simplified model. Knowing damage ds after integration (i), we define (e) the effective damaged elastic stiffness (not expressed here). Using closedform Eshelby solutions and the Mori–Tanaka scheme, we can define the effective strain concentration tensor . For a two-phase system, we know explicitly the corresponding from and (Dvorak 1992). The localization rule (l), Eqn. (3) (with and instead of and D sr ), delivers from the controlled overall strain E. Then, we obtain successively by the elastic constitutive equation (ce) and by averaging (h). In this model, the localization tensors have to be defined at each time step of a given loading. It is the reason that the model cannot be generalized to a refined discretization (more than two subvolumes). Figure 3 gives an example of comparison between this direct simplified model and a complete analysis of a unit cell by finite-element and periodic homogenization. This is a theoretical example of transverse tension–compression of a unidirectional elastic metal-matrix composite. The matrix is elastic, the fiber is elastic damageable, and damage deactivation is taken

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Figure 4. Flowchart of the two damage models.

into account (closure of microcracks under compression). The comparison appears as excellent in that case, as well as in the case where the visco–plasticity of the matrix is taken into account.

Generalized eigenstrain model (GEM). This model is developed in order to refine the discretization. Subvolumes are used in order to refine the discretization, to describe the interphase behavior (with damage) and the stress redistributions in the matrix. The model uses only the initial (undamaged) concentration and influence tensors of TFA, obtained initially by a number of elastic finite-element calculations. Its key specificity is to transform the elastic changes due to damage into an additional eigenstrain. We replace (6) by

where the damage strain

is given as

Damage mechanics of metal-matrix composites

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Figure 4 indicates the corresponding flowchart of the model. It uses the

following localization equation in place of (5):

We remark that the approach is also able to take into account the variations of the elastic stiffness with temperature (without modifying A s ,

D sr ). Its flexibility and its comparison with experiments appear to be very satisfactory. When applied in the particular case of two subvolumes, the GEM and DSM models deliver exactly the same stress–strain response.

3.

3.1.

MICROMECHANICAL NUMERICAL MODELING OF TRANSVERSE CREEP BEHAVIOR Experimental study

Silicon carbide–titanium MMCs will be used in the future bling technology as the reinforcement to sustain the high hoop stresses generated by centrifugal effects. The essential characteristics (tensile strength, fatigue, creep) have been studied mainly in the axial direction of uni-

directional composites (Ohno et al. 1994). However, transverse creep is a condition that may be of some significance, associated with secondary centrifugal radial stresses generated by external parts and the blades. An experimental study has been performed at ONERA, in order to characterize the creep resistance of the composite under transverse loads (Carrère et al. 2000). The material is a titanium (Ti6242) matrix reinforced with long unidirectional SiC (SCS6) fibers with a volume fraction of 35%. The specimens used for the experiments are taken from an 8-ply plate manufactured by Snecma, using the fiber-foil technique. The behavior of the titanium matrix is elasto–visco–plastic, the fibers are supposed to remain elastic, and the protecting coat around the fibers (to prevent a reaction between the titanium and the SiC) is a brittle zone where the decohesion and damage in the transverse direction of the composite will take place. Creep tests were performed at 500°C, under vacuum to avoid environmental problems, for several loads (Carrère et al. 2000). Results obtained are shown in Fig. 5 for the secondary creep rate, and can be summarized as follows:

Below a critical stress, estimated here around 225 MPa, the creep rate increases gradually and the lifetime is long—there is no rupture even for long creep time (> 1500 h).

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Figure 5 Measured secondary creep rates of SCS6/Ti6242 at 500°C.

For applied stresses over 225 MPa, the creep rate increases drastically, leading to short failure times.

3.2.

Micromechanical analysis

A micromechanical analysis based on the periodic homogenization assumption and the finite-element method is performed in order to explain the experimentally observed behavior. The regular position of the fibers

Figure 6 Composite microstructure, definition of a unit cell, and axial visco–plastic strain field after 1000 hours creep under 270 MPa.

inside the matrix, as shown in Fig. 6, allows us to select a unit cell that defines the representative volume element of the composite.

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The constitutive behaviors considered for each constituent, fiber, matrix and interphase, are as follows: The fiber is elastic and transversely isotropic. Its characteristics are given by literature data and independent measurements.

The matrix is elasto–visco–plastic, obeying a classical isotropic cyclic constitutive equation that incorporates two back stresses, one of which obeys the nonlinear kinematic hardening rule, the second being linear. This constitutive equation has been determined previously from tension–compression tests performed on pure Ti6242 over the whole temperature domain (20–870°C) (Baroumes and Vincon 1995). The effects of viscosity observed at high temperature are described with a power-law equation determined from relaxation (Malon 2000) and creep tests (Carrère et al. 2000). The carbon interphase is treated using interface elements that obey Tvergaard’s progressive debonding model (Tvergaard 1990). In fact we use a modified model (Chaboche et al. 1997), which introduces friction effects during the debonding phase, in order to eliminate the complete shear unloading when debonding is complete. The model combines mode I and mode II decohesion, with a unique damage parameter which is the maximum value of the quadratic norm of relative mode I and mode II openings (opening displacement divided by its value at complete debonding). It uses 4 parameters (for each mode a maximum stress and a fracture energy), in addition to the Coulomb friction parameter . The mode II and friction parameters are determined by push-out tests (Guichet 1998) and the complete numerical analysis of this test. The mode I is adjusted from the transverse tensile test. The corresponding simulations allow us to describe fairly well all the important features: onset of decohesion before plasticity takes place, progressive evolution of the unloading stiffness, then appearance of plasticity and irreversible strains (with hysteretic effects under cyclic loads), interfacial closure effects when reverse loadings are performed, etc.

The numerical simulation performed on the unit cell presented in Fig. 6(b) follows as closely as possible the experimental procedure. The main steps of the calculation are summarized as follows: (i) The first step consists in determining the residual stresses induced by the manufacturing process, and applying the load/temperature history undergone by the composite (referred to as “manufacturing” in the following curves).

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(ii) The creep load is imposed, giving the response of the primary state (referred to as “loading”), (iii) Finally, the creep test itself is simulated (secondary stage). The third stage, directly linked to the final failure of the specimen, is not modeled because, in this analysis, the crack initiation and propagation in the matrix are not taken into account. Figure 7 presents simulated results for two loadings: the first one, 200 MPa, below the critical stress, and the second one, 270 MPa, above this stress. The figure shows the evolution of the damage parameter around the interface between the fiber and the matrix, at different out-

Figure 7 Local interfacial damage evolution for creep under 200 and 270 MPa.

put times corresponding to the steps described before. It can be seen that the manufacturing process does not induce significant damage. Once the load is applied, damage starts and increases progressively. The comparison between the damage evolution induced by the two loads shows that for low levels, damage increases slowly (and sometimes saturates) during the creep without breaking the interface, though for higher loads the interface begins to break and this process continues until the complete interface debonds. Additional results are discussed in (Kruch et al. 2000). Therefore, the micromechanical analysis confirms that the matrix controls the global deformation of the composite, but the load carried by the matrix is controlled by the strength of the interface. For high stress levels (near the critical stress), the load is carried totally by the matrix (the interface being completely broken), leading to the failure of the composite when a crack initiates and propagates in the matrix. For lower stress levels the interfacial damage is not very important and the load is carried partly by the fibers and the matrix, leading to long lifetimes. The calculated secondary creep rates underestimate the experimental ones by a signigicant factor. There are two main explanations: (1) the

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interface debonding model was determined at room temperature and applied without change for high temperatures (taking into account only the different thermally induced residual stresses), and (2) the titanium matrix was identified from tests on a standard monolithic material whose microstructure could be different from the one of the foil material used

to manufacture the composite. The anisotropic texture of the matrix is right now analyzed at ONERA using the electron back-scattering diffraction (EBSD) technique. Numerical computations are under way, taking into account the measured textures in the matrix visco–plastic behavior, by using a polycrystalline aggregate model (Pilvin and Cailletaud 1990).

4.

CONCLUSIONS

The considered damage modeling approaches have shown the following capabilities for MMCs or similar composite situations: Micromechanics-based constitutive equations using the TFA approach are able to describe both thermo–elasto–visco–plasticity of the matrix and the damage evolution in each constituent. Two damage approaches—the direct simplified model and the generalized eigenstrain model—with different capabilities in terms of

degrees of freedom, are shown to be consistent with periodic homogenization results. Their application in true components as overall constitutive and damage equations is underway.

The finite-element micromechanics analysis of transverse creep loading conditions is able to predict quite well the experimental results. It assumes periodicity and uses a thermo–elasto– visco–plastic constitutive equation for the matrix and an interface debonding model for the fiber–matrix interphase (determined independently by push-out and transverse tensile tests). Improvements in the quantitative prediction of the secondary creep rate is expected by taking into account the true matrix visco– plastic behavior, including texture effects. Such an analysis is underway, using a polycrystalline aggregate model. A higher-level multiscale analysis of components (not presented) has also been developed (Feyel 1999). The method uses imbricated finite elements in order to replace the overall constitutive equations in a finite-element structural analysis. This method has the objective to deliver all the stress–strain redistributions at the microscale, especially in regions where the two scales do interact, very often the most critical regions of the component.

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References Aboudi, J. 1985. Inelastic behaviour of metal-matrix composites at elevated temperature. International Journal of Plasticity 1(4), 359–372. Allix, O., P. Ladevèze, E. L. Dantec, and E. Vittecoq. 1990. Damage Mechanics for composite laminates under complex loading. In Yielding, Damage and Failure

of Anisotropic Solids (J. Boehler, ed.). London: EGF5, Mechanical Engineering Publications, 551–569. Baroumes, L., and I. Vincon. 1995. Identification du comportement de l’alliage Ti6242. Technical Report contrat Snecma/LMT 762 593F, LMT-Cachan. Carrère, N., S. Kruch, R. Valle, and A. Vassel. 2000. Transverse creep behaviour of a unidirectional SCS-6/Ti-6242 composite. In Composites from Fundamentals to Exploitation, Proceedings of the 9th European Conference on Composite Materials. Brighton. Chaboche, J. L., R. Girard, and A. Schaff. 1997. Numerical analysis of composite systems by using interphase/interface models. Computational Mechanics 20, 3–11. Chaboche, J. L., S. Kruch, J. F. Maire, and T. Pottier. 2000. Towards a micromechanics based inelastic and damage modeling of composites. International Journal of Plasticity, to appear.

Dvorak, G. 1992. Transformation field analysis of inelastic composite materials. Proceedings of the Royal Society of London A 437, 311–327. Dvorak, G., Y. Bahei-El-Din, and A. Wafa. 1994. Implementation of the transformation field analysis for inelastic composite materials. Computational Mechanics 14, 201–228. Dvorak, G., and Y. Benveniste. 1992. On transformation strains and uniform fields in multiphase elastic media. Proceedings of the Royal Society of London A 437,

291–310. Feyel, F. 1999. Multiscale FE2 elastoviscoplastic analysis of composite structures. Computational Materials Science 16, 344–354. Guichet, B. 1998. Identification de la loi de comportement interfaciale d’un composite SiC/Ti. Doctorat d’Université, Ecole Centrale de Lyon.

Guo, G., J. Fitoussi, D. Baptiste, N. Sicot, and C. Wolff. 1997. Modelling of damage behaviour of a short-fiber reinforced composite structure by the finite element

analysis using a micro–macro law. International Journal of Damage Mechanics 6(3), 278–316. Kruch, S., J. L. Chaboche, and N. Carrère. 2000. Micromechanics based creep damage analysis of unidirectional metal matrix composites. In IUTAM Symposium on Creep in Structures, Nagoya, Japan.

Malon, S. 2000. Caractérisation des mécanismes d’endommagement dans les composites à matrice métallique de type SiC/Ti. Doctorat d’Université, ENS Cachan. Mori, T., and K. Tanaka. 1973. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica et Materialia 21, 597–629. Ohno, N., K. Toyoda, N. Okamoto, T. Miyake, and S. Nishide. 1994. Creep behavior of a unidirectional SCS-6/Ti-15-3 metal matrix composite at 450°C. Journal of Engineering Materials Technology 116, 208–214. Pilvin, P., and G. Cailletaud. 1990. Intergranular and transgranular hardening in

viscoplasticity. In IUTAM Symposium on Creep in Structures (M. Zyczkowski, ed.) (Cracow, Poland). Berlin: Springer-Verlag, 171–178. Pindera, M. J., and J. Aboudi. 1988. Micromechanical analysis of yielding of metal matrix composites. International Journal of Plasticity 4(3), 195–214.

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Pettier, T. 1998. Modélisation multiéchelle du comportement et de l’endommagement

de composites à matrice métallique. Doctorat d’Université, Ecole Nationale des Ponts et Chaussées. Suquet, P. 1997. Effective properties of nonlinear composites. In Continuum Micromechanics (P. Suquet, ed.) (Volume 377 of CISM Lecture Notes). New York: Springer-Verlag, 197–264. Tvergaard, V. 1990. Effect of fibre debonding in a whisker-reinforced metal. Materials Science and Engineering A 125, 203–213. Voyiadjis, G. Z., and P. Kattan. 1993. Damage of fiber reinforced composite materials with micromechanical characterization. International Journal of Solids and Structures 30(20), 2757–2778. Zaoui, A., and R. Masson. 1998. Modelling stress-dependent transformation strains of heterogeneous materials. In IUTAM Symposium on Transformation Problems in Composite and Active Materials (Cairo, Egypt). Dordrecht: Kluwer Academic Publications, 3–15.

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University of Illinois at Urbana-Champaign Department of Theoretical and Applied Mechanics administrative secretary Carol J. Porter (left) assembles submitted papers with assistant Lori C. Melchi and Prof. James W. Phillips for distribution to the various reviewing bodies and the International Papers Committee. Nearly 2000 papers were submitted to ICTAM 2000.

METALLIC FOAMS: STRUCTURE, PROPERTIES, AND APPLICATIONS Lorna J. Gibson Department of Materials Science and Engineering Massachusetts Institute of Technology, Cambridge, Mass., USA [email protected] Abstract

1.

Recently, a number of novel processes have been developed for making metallic foams. Their combination of properties make them attractive in a variety of engineering applications. Their low weight and ability to be formed into complex shapes make them attractive for use in structural sandwich panels. Their capacity to undergo large deformations at almost constant load can be exploited in energy-absorption devices. And the high thermal conductivity combined with the connected porosity in open-cell metallic foams makes them attractive for heat sinks. Here, we summarize the current understanding of the uniaxial behavior of metallic foams and relate their properties to micromechanical models.1

INTRODUCTION

Metallic foams can be made by a number of novel processes. Although aluminum foams are the most common, nickel, copper, zinc and steel foams are also available. Micrographs of four closed-cell and one opencell aluminum foams are shown in Fig. 1. Metallic foams have a combination of properties that make them attractive in a number of engineering applications. In structural sandwich panels, they offer lower weight than conventional honeycomb or stringer-stiffened designs for some, although not all, loading configurations. Recently developed processing techniques allow the manufacture of panels of complex shape with integrally bonded faces at relatively low cost. The capacity of metal foams to undergo large strains (up to about 60–70%) at almost constant stress allows significant energy absorption without generating damaging peak

1 This article first appeared in a more extended form [45], and appears by permission of Annual Review of Material Science.

57 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 57–74. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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stresses, making them ideal for energy-absorption devices. Open-cell metallic foams make excellent heat dissipation devices due to their high thermal conductivity, their high internal surface area, and the connectivity of the voids, which allows a cooling gas to pass through. This article reviews the uniaxial behavior of metallic foams. The results of micromechanical models for the moduli and strength of openand closed-cell foams are presented and compared with data for metallic foams. The effect of imperfections on the uniaxial properties is discussed. A description of further aspects of the mechanical behavior of metallic foams, such as size effects, yield criterion, creep, and fatigue, is available in the book Metal Foams: A Design Guide [1].

2.

MICROSTRUCTURE

Micrographs of a several aluminum foams are shown in Fig. 1. The microstructure is made up of an interconnected network of struts or

Figure 1 Micrographs of metallic foams, reproduced from [2] (with permission from Elsevier Science).

plates, which form the edges and faces of polyhedral cells. The single most important microstructural feature affecting the mechanical properties of foams is the volume fraction of solid, or relative density, the ratio

Metallic foams: Structure and properties

59

of the density of the foam to that of the solid, Metallic foams typically have relative densities in the range of 0.03 to 0.3. Foams are described as open- or closed-celled. Those shown in Fig. l(a)–(d) are all closed-cell, with faces separating the voids of each cell; that shown in Fig. l(e) is open-celled. Micromechanical models for open- and closedcelled foams are described in the next section. The cell size of most metallic foams is in the range of 2–10 mm [2, 3]. Although the mechanical properties of foams are sensitive to the ratio of the cell wall thickness to length, most do not depend on the absolute cell size. The cell shape of metallic foams ranges from equiaxed to ellipsoidal, with the ratios of major to minor axis lengths up to 1.5 [3]. The elongation of the cells corresponds to the rise of gas in the liquid (or mushy) state during processing. Foams are typically stiffer and stronger when loaded parallel to the major principal direction. Finally, the cell wall has an influence on the mechanical behavior of foams. The cell walls of some metallic foams are curved, possibly as a result of partial cell collapse during solidification [2, 3, 4] (Fig. 2). Cell wall curvature reduces the mechanical properties below the values that would be expected if the walls were

Figure 2 Micrograph showing curvature of cell walls in a closed-cell aluminum foam (Alporas), reproduced from [3] (with permission from Elsevier Science).

2

Throughout this paper, a superscipt asterisk refers to a property of the foam, while a

subscript s refers to a property of the solid from which the foam is made.

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planar. The composition of the cell wall also affects the properties of the foam. The additives and foaming agents used to process metallic foams often result in unconventional alloys. For instance, in one process the use of calcium as a thickening agent and titanium hydride as a blowing agent introduces Al–Ca–Ti precipitates into the microstructure [3, 4]. In this case, the yield strength of the cell wall material is measured directly using indentation techniques. Measured values for microstructural features of the aluminum foams shown in Fig. 1 are summarized in Table 1.

3.

MICROMECHANICAL MODELS FOR FOAMS

A schematic stress–strain curve for a foam in uniaxial compression is shown in Fig. 3. Initial linear elasticity is followed by a roughly constant stress plateau, which continues up to large strains beyond which the stress increases sharply. Previous studies of polymer foams have found that the linear elastic response is related to cell edge bending in opencell foams and to the edge bending and face stretching in closed-cell foams [5]–[12]. As the stress increases, the cells begin to collapse at a roughly constant load by elastic buckling, yielding or fracture, depending on the nature of the cell wall material [5, 6, 9, 10, 13, 14, 15, 16, 17, 18, 19]. Once all of the cells have collapsed, further deformation presses opposing cell walls against each other, increasing the stress sharply: this final regime is referred to as densification. In this section we review models for the mechanical behavior of ideal open- and closed-cell foams. The complex cell geometry of foams is difficult to model exactly. Several approaches are described in the literature. The simplest technique is to use dimensional arguments to model

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Figure 3 A schematic compressive stress–strain curve for a foam, showing the linear elastic, stress-plateau, and densification regimes.

the mechanisms of deformation and failure [5, 6]. The results give the dependence of the properties on relative density and on cell wall properties but not on cell geometry; the constants relating to cell geometry are found by fitting the model equations to experimental data. A second method is to analyze a repeating unit cell, such as a tetrakaidecahedron, using structural mechanics or finite elements [7, 9, 11, 12, 18, 20]. This method gives the dependence of the properties on the geometry of the unit cell; this can be used as an estimate of the geometrical constants of the foam microstructure. A third approach is to model a spatially periodic arrangement of several random Voronoi cells using finite-element analysis [8]. The advantage of this approach is that it gives the best representation of the cell geometry of foams, although it is computationally intensive. Here we review the use of dimensional arguments to show the way in which the properties depend on the relative density of the foam and the solid cell wall properties. We refer to the results of unit-cell and finite-element models to estimate the magnitude of the constants relating to the cell geometry. In the following section we describe the deformation and failure of metallic foams, modify the models to account for imperfections in their cell structure, and compare the results with data for metallic foams. In open-cell foams, the cell edges initially deform by bending. The Young’s modulus can be calculated from a dimensional analysis of the edge bending deflection. The relative modulus (that of the foam, divided by that of the solid, is proportional to the square of the relative density,

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where is a constant related to the cell geometry. Analysis of a tetrakaidecahedral unit cell with the cell cross section defined by Plateau borders finds = 0.98 [12, 20]. This value is confirmed by data for a wide variety of open-cell polymer foams, which give In shear, the cell edges also respond by bending, so that

with The Poisson’s ratio is simply the ratio of two strains and is independent of relative density; measured values suggest that for many foams it is about 1/3.

Closed-cell foams are more complicated. When the foam is loaded, there is stretching of the planar cell faces in addition to bending of the cell edges, adding a linear density dependence to Eqn. (1):

Finite-element analysis of a closed cell tetrakaidecahedral unit cell and of random, Voronoi closed cells suggests that For low-density foams, the second, face stretching, term dominates so that

The plateau stress for foams made from an elastic–plastic material is reached when the cells begin to collapse plastically. For open-cell foams, cell collapse corresponds to the formation of a mechanism of plastic hinges in the bent cell edges. Dimensional arguments give the plastic collapse stress relative to the yield strength of the solid cell edge material as

where the constant is related to the cell geometry. Data for a wide range of foams suggests that

Closed-cell foams have an additional contribution to their strength: yielding of the stretched cell faces. The stress required for this is linearly proportional to the relative density of the foam, so that

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Finite-element analysis of a tetrakaidecahedral unit cell with flat faces suggests that [21]

For low relative densities, the second term, corresponding to axial yielding of the faces, dominates, so that

The Deshpande and Fleck yield surface suggests that the shear strength is 0.69 times the uniaxial strength [22]:

For open-cell foams, using Eqn. (5), we find

and, for closed-cell foams, using Eqn. (7), we find

In the next section we describe the deformation and failure of metallic foams, modify the models to account for imperfections in their cell structure, and compare the results with data for metallic foams.

4.

DEFORMATION AND FAILURE IN METALLIC FOAMS

Compressive stress–strain curves for several aluminum foams are shown in Fig. 4(a). The shape of the curves is similar to that of the schematic of Fig. 3. The initial responses of an open-cell Duocel foam and a closed-cell Alporas foam are shown in more detail in Fig. 4(b). The Duocel foam is linear elastic: the slopes of the loading and unloading curves are identical. Although the Alporas foam exhibits an initial linear regime at strains of up to about 0.01, the slope of the loading curve is much lower than that of the unloading curve, indicating that there is some plastic deformation even at low strains; similar behavior is observed in the other closed-cell foams of Fig. 4(a). Surface strain

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(a) Behavior up to densification, from [2]

(b) Curves for open-cell Durocel and closed-cell Alporas, showing the slopes of the loading and unloading curves, from [24] Figure 4 Compressive stress–strain curves (reproduced with permission from Elsevier Science).

mapping reveals that the Duocel material deforms relatively homogeneously until the plateau stress is approached, while the Alporas material deforms heterogeneously, yielding locally by deformation banding, at stresses of about half the plateau stress (Fig. 5) [23]. The Alporas material strain hardens up to the peak stress and then softens. Tensile stress–strain curves for three aluminum foams are shown in Fig. 6. As in compression, the loading and unloading moduli of the Duocel ERG foam are the same, while the loading modulus of the Alporas foam is lower than the unloading modulus. The lower loading modulus in the Alporas foam again reflects plastic deformation even within the initial linear region. As the foams yield they strain harden and frac-

Metallic foams: Structure and properties

Figure 5

65

Stress–strain curves and surface strain maps of the incremental principal

strains: (a) Duocel and (b) Alporas. (Reproduced from [23], with permission from Elsevier Science.)

ture at strains of less than 1.5%. The Alcan foam is particularly brittle, fracturing at a strain of about 0.25%. The tensile strength is defined as the maximum stress reached before fracture. A shear stress–strain curve for Alporas aluminum foam is shown in Fig. 7. The shear strength is defined as the peak stress. Data for the unloading Young’s modulus from a number of studies are shown in Fig. 8, along with lines representing Eqns. (1) and (3) for open- and closed-cell foams, respectively [2]–[4], [23]–[29]. The modulus and density data have been normalized by the values for solid aluminum = 70 GPa and The data for the open-cell ERG Duocel foam falls close to the line representing the model for open-cell foams (Eqn. (1)), as expected. Nearly all the data for the closed-cell

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Figure 6 Tensile stress–strain curves for aluminum foams. (Reproduced from [2], with permission from Elsevier Science.)

Figure 7 Shear stress–strain curve for Alporas aluminum foam. (Reproduced from [24], with permission from Elsevier Science.)

foams with relative densities less than 0.3 lie below the line representing the model for closed-cell foams (Eqn. (3)). Most are closer to the line representing the open-cell model (Eqn. (1)), suggesting that bending contibutes significantly to the cell wall deformation. Data for the shear moduli are limited: the only available study is that of Dubbelday [30], who found moduli slightly lower than expected from Eqn. (2), perhaps due to anisotropy in the specimens. Data for the compressive and tensile strengths of aluminum foams are plotted in Figs. 9 and 10, along with lines representing Eqns. (5) and (7) for open- and closed-cell foams, respectively [2]–[4], [23]–[25], [27]–[29], [31]–[37]. The strengths have been normalized by the value for the solid

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Figure 8 Relative Young’s modulus plotted against relative density for aluminum

foams. Beals and Thompson [25] data for loading modulus, Dubbelday [26] and Weber [29] data for vibration test, all other data for unloading modulus. (Reproduced from [45], with permission from Annual Review of Materials Science.)

alloys (Table 2) [3, 4, 38]. Again, the data for the open-cell ERG Duocel foam lie close to the line representing the model for open-cell foams (Eqn. (5)), as expected, while the data for the closed-cell foams lie well below the line representing the model for closed-cell foams (Eqn. (7)). Most are closer to Eqn. (5), suggesting that plastic bending contributes significantly to the failure. The compressive and tensile strengths are, in general, similar in magnitude. The shear strengths of aluminum foams are plotted in Fig. 11, along with lines representing Eqns. (10) and (11) for open- and closed-cell foams, respectively [24, 37]. The shear strength of the closed-cell foams, like the uniaxial strength, lies close to the line representing the open-cell model, suggesting that plastic bending is again significant. Bending deformations can arise in closed-cell metallic foams in several ways. Curvature in the cell faces induces bending from both axial and shear forces. Finite-element analysis of closed-cell foams with curved cell walls suggests that, for the typical measured cell wall curvatures, the modulus is 60% and the strength is 30% of those for planar faces,

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Figure 9 Relative compressive strength plotted against relative density for aluminum foams. Compressive strength is defined in slightly different ways in different stud-

ies; generally it is either the plateau stress or the compressive stress at 20% strain. (Reproduced from [45], with permission from Annual Review of Materials Science.)

accounting for most of the discrepancy between Eqns. (3) and (7) and the data [2, 39, 40]. Irregularities in the cell shape can also induce bending. For instance, observations of the deformations within a closed-cell aluminum foam using computed tomography indicate that the presence of a highly elliptical cell with T-shaped cell wall intersections can induce bending in the neighboring cell walls, initiating the collapse of a deformation band [23]. Fractured cell walls can produce bending [41]. Local variations in relative density within a single specimen also reduce the properties. For instance, Beals and Thompson [25] report variations in relative density through the thickness of their specimens from 0.033 to 0.14—the highest values, measured at the bottom of the specimens, result from drainage during processing.

5.

CONCLUSIONS

Metallic foams have a combination of properties that make them attractive for lightweight structural sandwich panels, energy-absorption devices, and heat-dissipation devices. The recent development of a vari-

Metallic foams: Structure and properties

Figure 10

69

Relative tensile strength plotted against relative density for aluminum

foams. (Reproduced from [45], with permission from Annual Review of Materials Science.)

ety of processes for making metallic foams with improved properties at lower cost has led to increased interest in their use in engineering components. A number of studies have led to an improved understanding of the mechanisms of deformation and failure in metallic foams. Their response to uniaxial, multiaxial, creep, and fatigue loading has recently been characterized for the first time. This improved understanding of the mechanical response of metallic foams, combined with structural optimization design strategies, provides a foundation for their use in engineering applications.

Acknowledgments The author would like to acknowledge the financial support of the ARPA MURI Ultralight Metal Structures Contract Number N00014-96-1-1028. Discussions with a number of collaborators on that project have been insightful. I would particularly like to thank Profs. M. F. Ashby and N. A. Fleck of the Cambridge University Engineering Department, Prof. J. W. Hutchinson of the Division of Engineering and Applied Sciences, Harvard University, Prof. A. G. Evans of the Materials Research Institute, Princeton University, Prof. R. E. Miller of the Department of Mechanical Engineering, University of Saskatchewan, Dr. P. Onck of Micromechanics of Materi-

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als, Delft University of Technology, Dr. E. Andrews, Dr. L. Crews, Dr. A. E. Simone, Mr. G. Gioux, Mr. T. M. McCormack, and Mr. W. Sanders of the Massachusetts

Institute of Technology.

References [1] Ashby, M. F., A. G. Evans, N. A. Fleck, L. J. Gibson, J. W. Hutchinson, and H. N. G. Wadley (eds.). 2000. Metal Foams: A Design Guide. London: Butterworth-Heinemann. [2] Andrews, E. W., W. Sanders, and L. J. Gibson. 1999. Compressive and tensile behaviour of aluminum foams. Materials Science and Engineering A 270(2), 113–124. [3] Simone, A. E., and Gibson, L. J. 1998. Aluminum foams produced by liquid-state processes. Ada Materialia 46(9), 3109–3123. [4] Sugimura, Y., J. Meyer, M. Y. He, H. Bart-Smith, J. Grenestedt, and A. G. Evans. 1997. On the mechanical performance of closed cell Al alloy foams. Acta Materialia 45(12), 5245–5259. [5] Gibson, L. J., and M. F. Ashby. 1982. On the mechanical performance of closed cell Al alloy foams. Proceedings of the Royal Society A 382, 43–59. [6] Gibson, L. J., and M. F. Ashby. 1997. Cellular Solids: Structure and Properties, 2nd ed. Cambridge: Cambridge University Press.

[7] Ko, W. L. 1965. Deformation of foamed elastomers. Journal of Cellular Plastics 1, 45–50.

Metallic foams: Structure and properties

Figure 11

71

Relative shear strength plotted against relative density for aluminum

foams. (Reproduced from [45], with permission from Annual Review of Materials Science.)

[8] Kraynik, A. M., M. K. Neilsen, D. A. Reinelt, and W. E. Warren. 1999. Foam micromechanics: Structure and rheology of foams, emulsions and cellular solids. In Foams and Emulsions (J. F. Sadoc and N. Rivier, eds.) Dordrecht: Kluwer Academic Publishers, 259–286. [9] Menges, G., and F. Knipschild. 1975. Estimation of mechanical properties for rigid polyurethane foams. Polymer Engineering and Science 15(8), 623–627. [10] 10. Patel, M. R., and I. Finnie. 1970. Structural features and mechanical properties of rigid cellular plastics. Journal of Materials 5(4), 909–932. [11] Warren, W. E., and A. M. Kraynik. 1988. Linear elastic properties of open-cell foams. Journal of Applied Mechanics 55(2), 341–346. [12] Zhu, H. X., J. F. Knott, and N. J. Mills. 1997. Analysis of the elastic properties of open-cell foams with tetrakaidecahedral cells. Journal of the Mechanics and Physics of Solids 45(3), 319–343.

[13] Barma, P., M. B. Rhodes, and R. Salover. 1978. Mechanical properties of particulate-filled polyurethane foams. Journal of Applied Physics 49(10), 4985– 4991. [14] Chan, R., and M. Nakamura. 1969. Mechanical properties of plastic foams. Journal of Cellular Plastics 5(2), 112–118. [15] Christensen, R. M. 1986. Mechanics of low density materials. Journal of the Mechanics and Physics of Solids 34(6), 563–578.

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[16] Gent, A. N., and A. G. Thomas. 1959. Deformation of foamed elastic materials. Journal of Applied Polymer Science 1(1), 107–113. [17] Matonis, V. A. 1964. Elastic behavior of low density rigid foams in structural applications. Society of Plastic Engineers Journal 20(9), 1024–1030. [18] Mills, N. J., and H. X. Zhu. 1999. High strain compression of closed-cell polymer foams. Journal of the Mechanics and Physics of Solids 47(3), 669–695. [19] Zhu, H. X., N. J. Mills, and J. F. Knott. 1997. Analysis of the high strain compression of open-cell foams. Journal of the Mechanics and Physics of Solids 45(11–12), 1875–1904. [20] Warren, W. E., and A. M. Kraynik. 1997. Linear elastic behavior of a low density Kelvin foam with open cells. Journal of Applied Mechanics 64(4), 787–794. [21] Simone, A. E., and L. J. Gibson. 1998. Effects of solid distribution on the stiffness and strength of metallic foams. Acta Materialia 46(6), 2139–2150. [22] Deshpande, V. S., and N. A. Fleck. 2000. Isotropic constitutive models for metallic foams. Journal of the Mechanics and Physics of Solids 48(6), 1253–1283. [23] Bart-Smith, H., A.-F. Bastawros, D. R. Mumm, A. G. Evans, D. J. Sypeck, and H. N. G.Wadley. 1998. Compressive deformation and yielding mechanisms in cellular Al alloys determined using X-ray tomography and surface strain mapping. Acta Materialia 46(10), 3583–3592. [24] Andrews, E. W., G. Gioux, P. Onck, and L. J. Gibson. 2000. Size effects in ductile cellular solids—Part II: Experimental results. International Journal of Mechanical Sciences 43, 701–713. [25] Beals, J. T., and M. S. Thompson. 1997. Density gradient effects on aluminum foam compression behaviour. Journal of Materials Science 32(13), 3595–3600. [26] Dubbelday, P. S. 1992. Poisson’s ratio of foamed aluminum determined by laser Doppler vibrometry. Journal of the Acoustical Society of America 91(3), 1737– 1744. [27] Gradinger, R., F. Simancik, and H. P. Degischer. 1997. Determination of mechanical properties of foamed metals. Proceedings of the International Conference on Welding Technology, Materials and Materials Testing, Fracture Mechanics and Quality Management 2. Vienna University of Technology: Chytra Druck and Verlag GmbH, 701–722. [28] McCullough, K. Y. G., N. A. Fleck, and M. F. Ashby. 1999. Toughness of aluminum alloy foams. Acta Materialia 47(8), 2323–2330. [29] Weber, M., J. Baumeister, J. Banhart, and H.-D. Kunze. 1994. Selected mechanical and physical properties of metal foams. 1994 Powder Metallurgy World Congress—Vol. 1, Editions de Physique, 585–588. [30] Dubbelday, P. S., and K. M. Rittenmyer. 1985. Shear modulus determination of foamed aluminium and elastomers. Proceedings of the 1985 IEEE Ultrasonics Symposium 2, 1052–1055. [31] Banhart, J., J. Baumeister, and M. Weber. 1995. Powder metallurgical technology for the production of metallic foams. Euro Powder Metallurgy ’95 (Light Alloys), 201–208. [32] Banhart, J., and J. Baumeister. 1998. Deformation characteristics of metal foams.

Journal of Materials Science 33(6), 1431–1440. [33] Gioux, G., T. M. McCormack, and L. J. Gibson. 2000. Failure of aluminum foams under multiaxial loads. International Journal of Mechanical Sciences 46(6), 1097–1117.

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[34] Prakash, O., H. Sang, and J. D. Embury. 1995. Structure and properties of Al–

SiC foam Materials Science and Engineering A 199(2), 195–203. [35] Thornton, P. H., and C. L. Magee. 1975. Deformation characteristics of zinc foam. Metallurgical Transactions A 64(9), 1801–1807.

[36] Triantafillou, T. C., J. Zhang, T. L. Shercliff, L. J. Gibson, and M. F. Ashby. 1989. Failure surfaces for cellular materials under multiaxial loads. II. Comparison of models with experiment. International Journal of Mechanical Sciences 31(9), 665–678. [37] von Hagen, H., and W. Bleck. 1998. Compressive, tensile, and shear testing of melt-foamed aluminium. Proceedings of the Materials Research Society (MRS) Symposium 521. Warrendale, Penn.: MRS, 59–64. [38] Davis, J. R. 1993. Properties of cast aluminum alloys. ASM Specialty Handbook: Aluminum and Aluminum Alloys. Materials Park, Ohio: American Society for Metals. [39] Grenestedt, J. L. 1998. Influence of wavy imperfections in cell walls on elastic stiffness of cellular solids. Journal of the Mechanics and Physics of Solids 46(1),

29–50. [40] Simone, A. E., and L. J. Gibson. 1998. Effects of solid distribution on the stiffness and strength of metallic foams. Acta Materialia 46(6), 2139–2150. [41] Chen, C., T. J. Lu, and N. A. Fleck. 1999. Effect of imperfections on the yielding of two-dimensional foams. Journal of the Mechanics and Physics of Solids 47(11),

2235–2272. [42] McCullough, K. Y. G., N. A. Fleck, and M. F. Ashby. 2000. The stress-life fatigue behaviour of aluminium alloy foams. Fatigue and Fracture of Engineering

Materials and Structures 23(3), 199–208. [43] Banhart, J., J. Baumeister, and M. Weber. 1996. Damping properties of aluminium foams. Materials Science and Engineering A 205(1–2), 221–228. [44] Banhart, J., and J. Baumeister. 1998. Production methods for metallic foams. Proceedings of the Materials Research Society (MRS) Symposium 521. Warrendale, Penn.: MRS, 121–132. [45] Gibson, L. J. 2000. Mechanic behavior of metallic foams. Annual Review of Materials Science 30, 191–227.

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To select papers for ICTAM 2000, the International Papers Committee (IPC) met for 5 days in March 2000 on the campus of the University of Illinois at UrbanaChampaign. Gathering for lunch in this photo are (l–r) ICTAM 2000 SecretaryGeneral James W. Phillips; IUTAM Congress Committee secretary Niels Olhoff (Denmark); IPC members Sol R. Bodner (Israel), Alan Needleman (USA), Dick H. van Campen (The Netherlands), Timothy J. Pedley (UK), and L. Gary Leal (USA); and ICTAM 2000 President Hassan Aref.

LINEAR VISCOELASTICITY OF CONCENTRATED EMULSIONS Martin Nemer, Jerzy Blawzdziewicz, and Michael Loewenberg Department of Chemical Engineering, Yale University, New Haven, Conn., USA [email protected]

Abstract

1.

The linear viscoelastic response of an ordered dense emulsion is explored by numerical simulation. At concentrations below maximum packing, the stress relaxation is dominated by a single time scale associated with lubrication, which diverges at maximum packing. For concentrations above maximum packing, the stress relaxation is dominated by fast time scales of the order of the drop relaxation time. A slow time scale appears but does not dominate.

INTRODUCTION

The rheology of emulsions and foams is important for a wide range of applications, including polymer processing, personal care products, food processing, textile finishing, and oil recovery. Emulsions have non-Newtonian rheology [l]–[4], including shear thinning and nonzero normal stress differences, and may exhibit a nonzero yield stress. In small-amplitude oscillatory flows, emulsions exhibit a linear viscoelastic response [5]. Experiments and simulations on the low-frequency elastic behavior have focused on the static elastic modulus because it is easier to define and measure than the yield stress [6]. The static properties of ordered concentrated emulsions and foams have been investigated theoretically [7]–[9]. In three dimensions the equilibrium behavior of small ordered systems has been studied numerically using minimal-energy simulations [10]. Large systems have been explored in two dimensions using minimal-energy simulations [11], and in three dimensions using simplified effective interparticle potentials to model bubble-bubble interactions [12]–[14]. The dynamical properties of foams and compressed emulsions are less understood. Analytical results for the dynamics of thin inextensible films [15] have been incorporated in theoretical models [16] and simulations [13] of the dynamics of two dimensional systems. In three dimen75 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 75–84. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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sions, the effective-interparticle-potential model has been supplemented by simple bubble–bubble frictional dissipation [5, 17]. However, these models cannot describe the effect of detailed drop-scale dynamics on the evolution of the system. In general, the development of theories for the rheology of foams and emulsions will require a detailed understanding of the relevant drop-level dynamics. The focus of this paper is on phenomena that occur on the drop relaxation time scale In this paper we explore the linear viscoelastic response of concentrated ordered two-dimensional emulsions and foams in the frequency regime where the dynamics of drop shape relaxation is important. The response of the system was explored using numerical simulations that incorporate detailed drop-level hydrodynamics.

2.

PROBLEM FORMULATION We consider the linear viscoelastic behavior of a two-dimensional

hexagonal array of drops in which both phases are incompressible Newtonian fluids governed by the Stokes equations. Uniform interfacial tension is assumed. At high volume fractions, the drops are in a state of compression. In response, the drop forms flat films and Plateau borders of curvature higher than that of the undeformed drop. In this work we assume that the Plateau borders contain all the continuous phase liquid, and the thin films have zero thickness; thus the hydrodynamics of the thin films simplifies. Details of the boundary conditions at the contact point are discussed in Appendix A. The magnitude of drop deformation in a flow with shear rate scales with the capillary number Ca = where a is the (volume equivalent) drop radius, and is the continuous phase viscosity. The dynamics of the system also depends on the dispersed-to-continuous phase viscosity ratio drop volume fraction and the external forcing frequency Herein, dimensionless variables are used, where length is scaled with a, time is scaled with drop relaxation time and stress is scaled with This work focuses on linear viscoelastic behavior; thus We consider a hexagonal lattice in which the linear relation between stress and strain is isotropic [18]. Accordingly our system is characterized by a single frequency-dependent linear viscoelastic modulus

Linear viscoelasticity of concentrated emulsions

The drop contribution to the stress, the stress relaxation function

77

can be expressed in terms of

where is the dimensionless shear rate and is the dimensionless effective viscosity of the system in the high-frequency limit. For volume fractions below the critical value (i.e. maximum packing of undeformed drops), the stress relaxation function decays to zero as At the limiting behavior is For a two-dimensional hexagonal lattice, the static elastic modulus has been evaluated [8]. In our simulations, the stress relaxation function is obtained by analyzing the response of the system to a step strain. Accordingly the evolution equation is linearized around the equilibrium state

where is a shape perturbation. In our simulations, the linearized velocity operator A is obtained by solving the Stokes equations using the boundary-integral method [19]. By projecting the resulting normal modes,

onto a step strain perturbation,

the corresponding contribution to the shear stress can be calculated:

The relation between coefficients and is where is the contribution to the stress from the eigenmode The step strain perturbation is affine with the strain only for for determining the step strain perturbation requires a calculation of the hydrodynamic field due to the viscosity contrast, in the absence of surface tension. The results presented in the next section have been obtained for

3.

RESULTS

Selected results for the viscoelastic behavior of a monodisperse hexagonal system of drops are presented in Figs. 1–4. Figure l(a) illustrates

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Figure 1 Stress relaxation function nondimensionalized by mensionalized by drop relaxation time: (a) for method described in

(solid line); (b) for

versus time nondiobtained from the

obtained from direct step

strain simulations—zero-thickness films (solid line), films with finite average thickness (dashed line), static elastic modulus (dotted line).

Figure 2 Relaxation spectra as amplitudes

normalized by

by drop relaxation time, for

versus time scaled

Volume fractions

1.05, 1.03 were obtained from direct step strain simulations.

the evolution of the stress relaxation function volume fraction below critical, where

for dispersed-phase

for a hexagonal array of cylinders. The results indicate that the stress relaxation function decays to zero at long times. The initial decay occurs

on the capillary relaxation time scale however, evolution on a much slower time scale is evident at long times. The relaxation times and amplitudes corresponding to the eigenmode decomposition (5) are plotted in Figs. 2(a,b) for two subcritical values of According to our

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79

Figure 3 Selected modes for a subcritical and a supercritical volume fraction. (a) Equilibrium drop configuration at (b), (c) normal displacements versus angle defined in (a) (solid lines) for modes with (b) and (c) (d) Equilibrium drop configuration at (e) normal and (f) tangential displacements for mode with Labels a–c in figures (d)–(f) denote contact points. Dashed line in (b) represents analytical result for

results, a small number of modes is sufficient to represent accurately. The dominant mode corresponds to the largest relaxation time associated with the lubrication resistance between closely spaced drops. This slowest time scale diverges as

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Figure 4 Average time scale

ET AL .

given by Eqn. (7) versus

(solid line); aver-

age relaxation time from Eqn. (8) (dotted line). Error bars represent the difference between results obtained from zero-thickness film simulations and finite-thickness film simulations. For results were obtained from simulations with

finite-thickness films.

Figure l(b) shows qualitatively different behavior in the stress relaxation function for At long times, decays to the static elastic modulus In contrast to subcritical volume fractions, the stress relaxation function is dominated by decay on the capillary relaxation time scale a long time scale is evident but with small amplitude. The relaxation times and amplitudes are plotted in Figs. 2(c,d) for two supercritical values of The stress relaxation function obtained from zero-thickness film simulations was compared with the corresponding result for a system with slowly draining films of finite thickness. For average film thickness the results differ by about 1%, as shown in Fig. l(b). Drop-shape perturbations associated with the two slowest modes for a subcritical value of are shown in Fig. 3(b,c). The slowest mode (solid line plotted in Fig. 3(b)) has the largest amplitude according to Fig. 2(b), and results from the lubrication resistance in the near-contact regions between drops. The dashed line shows the limit as

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81

In this limit, drop shape is associated with point forces acting in the lubrication regions of vanishing extent; between the lubrication regions the drop shape is in quasi-static equilibrium due to the separation of the lubrication time scale and the drop relaxation time where is the minimum gap between drops. As the volume fraction decreases, the dominant mode tends continuously to A mode with the relaxation time is plotted in Fig. 3(c). In this case, the shape of the drop interface in the Plateau border outside the lubrication regions is significantly perturbed from equilibrium. Oscillations are observed in the near-contact region that corresponds to matching of the time scales for the relaxation inside and outside the lubrication regions. For the dominant mode (cf. Fig. 2(c)) is shown in Figs. 3(e,f). The central points of the thin films do not move by symmetry, and the films are only weakly deformed in this region. Oscillations observed near contact point a have diminishing amplitude and wavelength. The result shown in Fig. 3(e) indicates that the film a–a increases length, and films b–b and c–c decrease length. These changes in film lengths occur without significant lubrication resistance; thus the mode evolves on drop-relaxation time Figure 4 displays the average relaxation time

where is the zero-frequency viscosity, and is the drop contribution to the stress immediately after a step strain; for the initial stress The results indicate that for the average relaxation time diverges as This behavior is associated with the lubrication singularity of the zero-frequency viscosity at For Eqns. (7) and (B.8) indicate that

In Fig. 4 this result is represented by the dotted line. The results in Fig. 4 suggest that for does not diverge. A more detailed analysis of this phenomenon is underway.

4.

CONCLUSIONS

In conclusion, we have performed boundary-integral simulations of the linear viscoelastic response of monodisperse ordered two-dimensional emulsions. Our results indicate that for volume fractions below close

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ET AL .

packing of undeformed drops, the evolution is dominated by a single slow mode associated with lubrication resistance. For volume fractions above close packing, the system is dominated by evolution on the drop relaxation time, because thin films between drops can stretch without significant lubrication resistance. There also exists a slow mode that is probably associated with squeezing in the lubrication region where drop interfaces meet, but this mode contributes only weakly to the stress. As the current work progresses, the effects of system parameters, including volume fraction, viscosity ratio, and contact angle, will be explored. The effects of an insoluble surfactant layer will be also be studied.

Appendix A. Zero-thickness films In our model the contact point (where three films meet) is convected by the interfacial velocity. A tangential force balance is maintained at the contact point at all times:

we also assume a continuity relation

for the normal traction jumps Here is the cosine of the contact angle, is the interface curvature, is the unit tangent vector to interface i, and is the unit normal vector. The thin film (subscript 0) has two interfaces and is assumed to have negligible thickness compared with the drop size. Subscripts 1 and 2 denote the interfaces of the Plateau border that connect to the thin film.

Appendix B. Low-frequency viscosity for near-critical volume fractions Here we consider the stress in an ordered emulsion subjected to a linear flow in the limit of low frequency and from below. Under these conditions, the drop contribution to the stress is dominated by the lubrication forces at the near-contact regions centered at positions

where is the relative velocity between drops normal to the interface at and R is the pairwise lubrication resistance between undeformed drops. Thus,

Linear viscoelasticity of concentrated emulsions

where

83

is the drop volume,

E is the dimensionless rate-of-strain tensor, are the lattice vectors. Combining (B.2) and (B.3) gives

is the strain rate, and

where

and m is the dimensionality of the system. For two-dimensional clean drops in near-contact motion with moderate viscosity ratios we find

where is the minimum distance between undeformed drops. For a two-dimensional hexagonal lattice,

Thus, a single transport coefficient

characterizes the rheology, where critical volume fraction.

is the distance from the

References [1] Princen, H. M. 1985. Rheology of foams and highly concentrated emulsions:

II—Experimental study of the yield stress and wall effects for concentrated oilin-water emulsions. Journal of Colloid and Interface Science 105, 150–171.

[2] Princen, H. M., and A. D. Kiss. 1986. Rheology of foams and highly concentrated emulsions: III—Static shear modulus. Journal of Colloid and Interface Science 112, 427–437. [3] Princen, H. M., and A. D. Kiss. 1989. Rheology of foams and highly concentrated emulsions: IV—An experimental study of the shear viscosity and yield stress of concentrated emulsions. Journal of Colloid and Interface Science 128, 176–189.

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ET AL .

[4] Mason, T. G., J. Bibette, and D. A. Weitz. 1996. Yielding and flow of monodisperse emulsions. Journal of Colloid and Interface Science 179, 439–448. [5] Mason, T. G., M. D. Lacasse, G. S. Grest, D. Levine, J. Bibette, and D. A. Weitz. 1997. Osmotic pressure and viscoelastic shear moduli of concentrated emulsions. Physical Review E 56, 3150–3166. [6] Tewari, S., D. Schiemann, D. J. Durian, C. M. Knobler, S. A. Langer, and A. J. Liu. 1999. Statistics of shear-induced rearrangements in a two-dimensional model foam. Physical Review E 60, 4385–4396. [7] Princen, H. M. 1979. Highly concentrated emulsions. Journal of Colloid and Interface Science bf 71, 55–66. [8] Princen, H. M. 1983. Rheology of foams and highly concentrated emulsions. Journal of Colloid and Interface Science 91, 160–175.

[9] Buzza, D. M. A., and M. E. Cates. 1994. Uniaxial elastic modulus of concentrated emulsions. Langmuir 10, 4503–4508. [10] Reinelt, D. A., and A. M. Kraynik. 1996. Simple shearing flow of a dry Kelvin soap foam. Journal of Fluid Mechanics 311, 327–342. [11] Weaire, D., T. L. Fu, and Kermode. 1986. On the shear elastic constant of a

two-dimensional froth. Philosophical Magazine B 54, L39–L43. [12] Lacasse, M. D., G. S. Grest, D. Levine, T. G. Mason, and D. A. Weitz. 1996. Model for the elasticity of compressed emulsions. Physical Review Letters 76,

3448–3451. [13] Okuzono, T., and K. Kawasaki. 1993. Rheology of random foams. Journal of Rheology 37, 571–586. [14] Durian, D. J. 1995. Foam mechanics at the bubble scale. Physical Review Letters

75, 4780–4783. [15] Mysels, K. J., K. Shinoda, and S. Frankel. 1959. Soap Films: A Study of Their Thinning and a Bibliography. Tarrytown, N.Y.: Pergamon Press. [16] Reinelt, D. A., and A. M. Kraynik. 1990. On the shearing flow of foams and concentrated emulsions. Journal of Fluid Mechanics 215, 1235–1253. [17] Durian, D. J. 1997. Bubble-scale model of foam mechanics: Melting, nonlinear behavior, and avalanches. Physical Review E 55, 1739–1751. [18] Frisch, U., B. Hasslacher, and Y. Pomeau. 1986. Lattice-gas automata for the Navier–Stokes equation. Physical Review Letters 56, 1505–1508. [19] Pozrikidis, C. 1992. Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge: Cambridge University Press.

ALUMINUM: APPROACHING THE NEW MILLENNIUM Peter A. Davidson Department of Engineering, University of Cambridge, Cambridge, United Kingdom [email protected]

Abstract

1.

The amount of energy currently used to reduce alumina to aluminum in electrolysis cells is staggering, around 1011 kW·h per annum. Yet much of this energy (almost one half) is lost in the form of I2 R heating of the highly resistive electrolyte. Strenuous efforts have been made to minimize these losses by reducing the volume of electrolyte in the cells. However, the aluminum industry has come up against a fundamental problem: when the depth of the electrolyte is reduced below a critical threshold (around 4–5 cm), the liquids in the cell start to ‘slosh around’ in an uncontrolled fashion. This is an instability, fueled by the intense currents that pass through the liquids. At present, cells operate just above the critical electrolyte depth, but if this depth were reduced from, say, 4.5 cm to 4.0 cm, then the annual savings would exceed £100 million. After a number of false starts, we now have a clear understanding of the physical mechanisms that underpin the instability, and it turns out that these are remarkably simple.

INTRODUCTION

Today, virtually all aluminum is produced by reducing alumina in electrolysis cells. There are two remarkable features of these cells: first, the process has remained virtually unchanged for over a century, and second, the amount of energy consumed by these cells is staggering. Aluminum was first isolated in the 1820s by Oersted and Wöhler. They used chemical, rather than electrical, means. By 1827, Wöhler was producing small quantities of aluminum powder by displacing the metal from its chloride using potassium. Later, in the 1850s, potassium was replaced by sodium, which was cheaper, and aluminum fluoride was substituted for the more volatile chloride. However, those chemical processes were all swept aside by the revolution in electrical technology initiated by Faraday’s discoveries. The electrolytic route was first proposed by Robert Bunsen (he of the ‘burner’ fame) in 1854, but it was not until 1886 that a continuous commercial process was developed. It 85 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 85–98. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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was a young college student from Ohio, Charles Martin Hall, and the Frenchman Paul Héroult, who made the breakthrough. Working independently, they realized that molten cryolite, a mineral composed of fluorine, sodium, and aluminum, readily dissolves alumina and that a current passed through the solution will decompose the alumina, leaving the cryolite almost unchanged. Full commercial production began on

Figure 1

A reduction cell at the turn of the century.

Thanksgiving Day in 1888 in Pittsburgh in a company founded by Hall. An example of an early Hall-Héroult reduction cell is shown in Fig. 1. Remarkably, over a century later, the process is virtually unchanged (Fig. 2).

Figure 2 A modern reduction cell.

Aluminum: Approaching the new millennium

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Modern reduction cells consume prodigious amounts of energy. Worldwide, around 18 million tonnes of aluminum are produced annually, and this requires ~ 230 × 109 kW·h per annum (about 2% of all generated electricity). At a notional cost of 3 cents per kW·h, this represents an annual bill of around $10 billion (see, for example, et al. 1999). Yet much of this energy, around 40%, is wasted in the form of I2 R heating of the electrolyte! Needless to say, strenuous efforts have been made to reduce these losses, largely centered around reducing the depth of the electrolyte. (Note the difference in electrolyte depth between Figs. 1 and 2.) However, the aluminum industry is faced with a fundamental problem. When the depth of electrolyte is reduced below some critical threshold, the reduction cell becomes unstable. This instability, which manifests itself as a sloshing of the liquids within the cell, has been the subject of intensive research for over two decades. After several false starts, we have finally reached the point where we understand the cause of the instability, not just in some abstract mathematical sense but in a simple, intuitive, way. It is now clear how to design new cells that are inherently stable, and this is likely to be the next major change in the industry. The potential benefits of such a change would be a rise in productivity (of around 20%) plus a reduction in energy cost from, say, 14 kW·h/kg to around 11 kW·h/kg. At a notional cost of 3 cents per kW·h, this translates to worldwide savings of $1 billion per annum. Given that energy consumption accounts for between 20 and 30% of the total production cost of aluminum, there is scope for substantial savings.

2.

THE STABILITY OF MODERN REDUCTION CELLS

Consider the cell shown in Fig. 2. A large vertical current, perhaps 300 kA, flows downwards from the carbon anode blocks, passing first through the electrolyte (where it reduces the alumina) and then through a liquid aluminum pool before finally being collected at the carbon cathode at the base of the cell. The liquid layers are broad but shallow, perhaps 4m by 10m in plan, yet only a few centimeters deep. (The electrolyte layer might be 4–5 cm thick and the aluminum pool perhaps 20–30 cm deep.) There are two quite distinct efficiencies used to characterize the per-

formance of a cell: the efficiency of electrolysis (usually called the current efficiency) and the energy efficiency. The first is a measure of how many electrons do what they are meant to do, i.e. reduce alumina molecules. This efficiency is very high, typically around 90–95%. The energy effi-

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ciency, on the other hand, which measures the difference between useful and wasted power, is extremely low. Although the aluminum is an excellent conductor, and the carbon a moderate one, the electrolyte (cryolite) is a very poor conductor. Typically, the cell voltage is around 4.5 V, of which only 1.6V is needed for the electrolysis and a staggering 2V is lost in the electrolyte. Thus, around 40% of the input power is lost to Ohmic heating of the thin (4.5 cm) layer of cryolite. There is considerable incentive, therefore, to lower the resistance of the electrolyte layer by reducing its thickness. In fact, worldwide, each millimeter of electrolyte costs around $60 million per year in lost profits! There is a fundamental problem, however. It turns out that unwanted disturbances are readily triggered at the electrolyte–aluminum interface (Fig. 3). In effect, these are interfacial gravity waves, modified by the

Figure 3

The idealized cell geometry.

intense magnetic fields that pervade the cell. When a certain stability threshold is exceeded, these waves can grow, absorbing energy from the ambient electric and magnetic fields. The wave period is measured in minutes, and its growth rate in hours; yet once such a wave takes hold it can disrupt the electrolysis to such an extent that the cell must be withdrawn from operation. These waves represent the primary impediment to increasing both productivity and energy efficiency. In particular, the cell current I must be maintained below a certain critical value in order to ensure stability. Similarly, there is a critical value of the cryolite depth h, below which instability sets in and waves can grow. Most cells operate just on the verge of instability, at a current just below and at an electrolyte depth marginally above To some extent, the mechanism of the instability is clear. Let J0 be the nominal (unperturbed) current density in the cell and j be any

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additional current density arising from movement within the cell. Now consider a tilting of the interface as shown in Fig. 4. Excess current

Figure 4

Long-wavelength disturbances in a cell.

is drawn into the aluminum at points where the thickness of the highly resistive cryolite is reduced, and less current is drawn at points where the electrolyte depth is increased. Since the carbon cathode is much more resistive than the aluminum, these perturbations in vertical current feed into the aluminum pool but do not penetrate the cathode block. The perturbation in current is as shown schematically in Fig. 4. On the left, where excess current is drawn, j points downward. On the right, where there is a deficit in current, the perturbation is negative and so j points upward. Since the perturbed current j must form a closed circuit, and since the fluctuations do not penetrate into the cathode block, the perturbations of vertical current in the electrolyte must short through the aluminum as shown. Now the waves commonly seen in cells have wavelengths comparable with the horizontal cell dimensions. (Higher frequency components are destroyed by friction.) Thus the typical wavelength is much greater than h. There are several consequences of the long-wavelength limit. First, it may be shown that, to leading order in the small quantity the perturbed current density in the cryolite is purely vertical, while that in the aluminum, is horizontal (see Davidson and Lindsay 1998). More importantly, it takes only a minute tilting of the interface to cause a large rush of horizontal current in the aluminum. For example, suppose

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we have a tilt of only 1°, which corresponds to a maximum change in h of around 2% (1 mm). This will induce a 2% change in local resistance (at the crest and valley of the wave) and hence in fluctuation in cryolite current of, say, ~ 2% × 300 kA = 6kA. However, from geometrical considerations, we can show that this leads to a horizontal current of 100 kA! It is this surprisingly large horizontal current that lies at the root of the instability. This perturbation in current density is important because it causes a change in the Lorentz force, F = J × B, which acts on the fluids. That is, as J goes from to + j, and B from to + b, F changes from to F = + × b + j × b. (Here is the ambient magnetic field in the absence of a wave.) In the long-wavelength limit, the dominant contribution to =F– can be shown to be where is the vertical component of the ambient magnetic field (see Davidson and Lindsay 1998). In short, the dominant contribution to the perturbation in force is the interaction of horizontal current in the aluminum with the ambient, vertical magnetic field. This results in a force that acts only on the aluminum and that is directed normal to the plane of Fig. 4. The key question, of course, is whether or not this change in Lorentz force will tend to amplify the initial motion, driving an instability. At first sight it might seem that the answer to this must be ‘no’. Why should there be any interaction between a motion in one plane and a force acting in the perpendicular direction? However, suppose we arrange to have two waves, both of which have the same frequency, and which slosh back and forth in mutually perpendicular planes, say one in the xz plane and the other in the yz plane. (The coordinate system is shown in Fig. 3.) Each wave then produces a force that acts on its companion wave. If we, or rather nature, now choose the relative timing of these waves appropriately, then we have the possibility that the force produced by one wave is in phase with the other wave (and hence does work on that wave) and vice versa. In short, we end up with a kind of resonance. It is this resonance that causes cells to become unstable. There have been many detailed mathematical models of cell stability. These try, in various ways, to capture the instability mechanism described above and to make predictions of and It has to be said, however, that most of these models are rather idealized, simplifying the cell geometry to the point where the quantitative predictions are not very accurate. However, from the perspective of an engineer, this does not matter. The important point is that we now understand the basic mechanisms that cause the waves to amplify, and so we are finally in a position to redesign these cells and engineer the problem away.

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Perhaps the instability mechanisms are best understood by using a simple mechanical analogue. This is due to Davidson and Lindsay (1998) and relies on the fact that, in the long-wavelength limit, the motion in the aluminum is almost purely horizontal (Fig. 4).

3.

A SIMPLE MECHANICAL ANALOGUE FOR THE INSTABILITY

Suppose we replace the liquid aluminum by a thin, rigid, aluminum plate attached to the center of the anode by a light rigid strut. The strut is pivoted at its top and so the plate is free to swing as a compound pendulum about two horizontal axes x and y (see Fig. 5). Let the plate

Figure 5 The compound pendulum shown above contains all the essential physics of the reduction cell instability.

have thickness H, lateral dimensions and density The gap h between the plate and the anode is filled with an electrolyte of negligible inertia and poor electrical conductivity. A uniform current density passes vertically downward into the plate and is tapped off at the center of the plate. Finally, suppose that there is an externally imposed vertical magnetic field Evidently, we have replaced one mechanical system (the cell), which has an infinite number of degrees of freedom, with another that has only two degrees of freedom. However, electrically the two geometries are alike. Moreover, the nature of the motion in the two cases is not dissimilar. In both systems we have movement of the aluminum associated with tilting of the electrolyte–aluminum interface. In a cell this takes the form of a ‘sloshing’ back and forth of the aluminum as the interface tilts first one way and then the other (Fig. 4). Let and be the angles of rotation of the plate measured about the x and y axes. Then it is not difficult to show that, as a result of

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these rotations, the perturbed current in the aluminum plate is

The perturbation in Lorentz force j × can be calculated from these expressions, and the equations of motion for the compound pendulum then follow. Note that a tilting of the plate about the y axis induces both motion and a current flow in the x direction. This then interacts with to create a y-directed force acting on the plate. Thus, movement in one direction, say x, tends to induce a force that drives motion in the perpendicular direction (in this case, y). It turns out that the equations of motion for the pendulum are

where and are the conventional gravitational frequencies of the pendulum. Note the cross-coupling of and If we look for solutions of the form ~ exp(iwt) we find constant amplitude oscillations for small values of and exponentially growing sinusoids (unstable solutions) for large The transition occurs at

Figure 6 Variations of

with

for the compound pendulum.

Figure 6 shows the movement of in the complex plane as is increased. The two natural frequencies and move along the real axis until they meet. At this point, they move off into the complex plane and an instability develops. The important points to note are:

The tendency for instability depends only on the magnitude of and on the natural gravitational frequencies and

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To minimize the danger of an instability it is necessary to keep low and the gravitational frequencies well apart. The closer the natural frequencies are, the lower the threshold value of at which an instability appears. (In the absence of friction, circular and square plates are unstable for vanishingly small values of The system is unstable whenever

When is large the unstable normal mode corresponds to a rotating, tilted plate (see Davidson and Lindsay 1998). Very similar behavior is seen in reduction cells. In particular, the sensitivity of reduction cells to the destabilizing influence of depends on the initial separation of the gravitational frequencies. The closer the gravitational frequencies the lower the stability threshold. Moreover, unstable waves frequently correspond to a rotating, tilted interface (Sele 1977). The physical origin of the instability of the pendulum (and of a cell) is now clear. Tilting the plate in one direction, say produces a horizontal flow of current in the aluminum that interacts with to produce a horizontal force which is perpendicular to the movement of the plate and in phase with This tilting also produces a horizontal velocity, which is out of phase with the force and mutually perpendicular to it. Two such tilting motions in perpendicular directions can reinforce each other, with the force from one doing work on the motion of the other. This is the instability mechanism of the pendulum and essentially the same thing happens in a reduction cell.

4.

CAN WE MODEL THE INSTABILITY?

The instability mechanism in a reduction cell is essentially the same as that of the plate. The main differences are that there is friction in a cell, which tends to stabilize the waves, and that there are many different wave patterns (sloshing modes), each of which may have a different natural frequency. There are many mathematical models of cell stability around. The primary aim of these models is to identify all the possible sloshing modes, and to determine which pair are the most dangerous in the sense that they may interact and go unstable in the way that the pendulum does. Perhaps the best known of these models are due to Sneyd and Wang (1994), Bojarevics and Romerio (1994),

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Urata (1985), Sele (1977), and Davidson and Lindsay (1998). Given the complexity of the dynamics involved, it is remarkable that one can produce a mathematical model at all, and so these models represent a significant achievement. However, they all make a variety of simplifying assumptions, particularly about the cell geometry. For example, the channels at the edges of the cell are usually ignored, and this can influence the accuracy of the predicted gravitational frequencies. Worse still, the background steady-state motion in the fluids is often overlooked in the stability analysis. As a consequence, there is considerable doubt as to the accuracy of the model predictions for particular cells. Nevertheless, the broad trend tends to be captured by these analyses, and perhaps we should be grateful for that. In general, these models tend to end up with a stability criterion of the form (see Davidson and Lindsay 1998)

where is the nominal current density in the cell, some typical measure of the ambient, vertical magnetic field, is the density difference between the two fields, h and H are depths of the two fluid layers, and are the horizontal dimensions of the cell, and F(~, ~)

is some function. The left-hand side is the critical value of at which the cell goes unstable, normalized by the gravitational force The right-hand side includes the cell aspect ratio, because the natural frequencies of the sloshing modes depend critically on the ratio For a given it can be seen that large values of and are destabilizing, as is a small value of h. This is why a high-amperage cell with a thin cryolite layer is prone to instability. The field arises from the bus-bars, which carry current to and from the cell. (The current within the fluids is largely vertical, and so produces only a horizontal magnetic field.) Thus a well designed cell has a bus-bar arrangement that minimizes Another traditional approach to cell design is to choose such as to make F, and hence as large as possible. This is achieved by keeping the natural frequencies of the sloshing modes as far apart as possible through an appropriate choice of aspect ratio. As suggested by (5), this will produce a high critical value of

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A TYPICAL STABILITY MODEL

We shall briefly summarize the stability model of Davidson and Lindsay (1998). We start with conventional shallow-water theory. It is not difficult to show that, to second order in kH, k being wavenumber, the pressure in each layer is hydrostatic. As a consequence, we may apply the conventional shallow-water equation to each layer in turn. This is a two-dimensional equation for the horizontal motion:

Here is the aluminum depth, is the interfacial pressure, and is the horizontal body force in each layer. We now linearize our equation of motion about a base state of zero background motion. Taking we obtain

Although is a two-dimensional velocity field, vertical movement of the interface means that the two-dimensional divergences of and are both non-zero. In fact, it is readily confirmed that

Next, we replace and by the volume fluxes and Also, we may take (to leading order in kH). This is valid because, as we have seen, the current perturbation in the cryolite is an order of magnitude smaller than that in the aluminum. The governing equations become

We now perform a Helmholtz decomposition on q: q= appropriate decomposition is

An

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Evidently, is zero in the electrolyte, while is the same in both layers: We now rewrite (7) and (8) in terms of and eliminate by adding the equations, and use (10) to express in terms of The resulting equation of motion is

where The subscript P on the bracket implies that we take only the irrotational component of the corresponding term. Note that, when the Lorentz force is zero, we recover the conventional equation for interfacial waves in the shallow-water limit. The wavespeed for such waves is We now evaluate and hence using the longwavelength approximation. In the cryolite we have, to leading order in kH, This current passes into the aluminum, and so the boundary conditions on are

It is readily confirmed that the conditions of zero divergence and zero curl, as well as the boundary conditions given above, are satisfied by

Here is the horizontal component of the current density in the aluminum, which satisfies

Comparing equation (14) with (10) we find

This is the key relationship that allows us to express the Lorentz force in terms of the fluid motion and therefore deserves some special attention. The physical basis for (15) is contained in Fig. 4. When the interface tilts, there is a horizontal flow of current from the high to the low side of the interface. Simultaneously, there is a horizontal rush of the aluminum in the opposite direction. It is this coupling that lies at the heart of the instability, and which is expressed by (15).

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We now note that the leading term in the Lorentz force arises from the background component of Substituting for in (12) and introducing

we find, after a little algebra,

This equation is readily solved in any shape of domain and it predicts the unstable growth of waves whenever exceeds a critical value, which depends on the geometry. For rectangular domains the end result is a neutral stability curve that leads to an equation of the form of (6).

6.

CONCLUSIONS

It is over a century since Hall first started aluminum production in Pittsburgh. Reduction cells have been carefully optimized over the years, but basically they are much the same as those envisaged by Hall. If aluminum is to compete for new markets, then its cost of production must fall. Given that energy consumption constitutes a large fraction of this cost, and that 40% of this energy is wasted in I2R heating of the electrolyte, it makes sense to focus our efforts on eliminating this inefficiency. However, if the cryolite losses are to be reduced, then we must somehow neutralize the cell instability. After a number of false starts we now finally understand the mechanisms that underlie the instability and so engineering solutions cannot be far off.

References Bojarevics, V., and M. V. Romerio. 1994. Long wave instability of liquid metal– electrolyte interface in reduction cells. European Journal of Mechanics B 13, 33– 56. Davidson, P. A., W. R. Graham, and H. L. O’Brien. 1999. Control of instabilities in aluminium reduction cells. Light Metals 1999, 327–331.

Davidson, P. A., and R. I. Lindsay. 1998. Stability of interfacial waves in aluminium reduction cells. Journal of Fluid Mechanics 362, 273–295. Moreau, R. J., and D. Ziegler. 1986. Stability of aluminium reduction cells: a new approach. Light Metals 1986, 359–364. Øye, H. A., N. Mason, R. D. Peterson, E. L. Rooy, F. J. Stevens McFadden, R. D. Zabreznik, F. S. Williams, and R. B. Wagstaff. 1999. A history of aluminium production. Journal of Metals 51(2), 29–41. Sele, T. 1977. Instabilities of the metal surface in electrolyte alumina reduction cells. Metallurgy Transactions B 8, 613–618. Sneyd, A. D., and A. Wang. 1994. Instabilities of the metal surface in electrolyte alumina reduction cells. Journal of Fluid Mechanics 263, 343–359.

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Urata, N. 1985. Magnetics and metal pad instability. Light Metals 1985, 581–589.

ELECTROMAGNETIC PHENOMENA IN STEEL CONTINUOUS CASTING Takehiko Toh and Eiichi Takeuchi Technical Development Bureau, Nippon Steel Corporation, Chiba, Japan [email protected] Abstract

1.

The application of magnetohydrodynamics (MHD) in the continuous casting process of steel has now advanced to electromagnetic stirring in the mold and the control of molten steel flow by an in-mold directcurrent (DC) magnetic field brake. These applied MHD technologies are designed to improve further the continuous casting (CC) process capability. They improve the surface quality of cast steel by homogenizing the meniscus temperature, stabilizing the initial solidification, and cleaning the surface layer. They also improve the internal quality of steel slab by preventing the inclusions from penetrating deep into the strand pool and promoting the flotation of argon bubbles. This paper describes the characteristics of the phenomena appearing in these technologies on the basis of MHD. Additionally, a description is given of a technique for controlling the initial solidification of steel in the mold by imposing an alternating-current (AC) magnetic field, which enables the production of a surface-defect-free slab at a higher casting speed.

INTRODUCTION

In the past decade, the continuous casting (CC) process has progressed markedly, has solved many technological problems, and has spread widely as a result. In the meantime, attempts have been made to develop novel casting processes for further innovation in the coming century. Among them, the application of magnetohydrodynamics (MHD) has aroused a great deal of interest as a powerful elementary technology to develop casting processes with higher productivity and better cast steel quality. Nippon Steel started the application of MHD with the development of a strand pool electromagnetic stirrer in the mid-1960s for the purpose of improving the internal quality of the cast steel. The equipment, which has the merit of raising the equiaxed zone ratio, was installed on various continuous casters to prevent center segregation, porosity, and ridging in the late 1970s. The 1980s saw the development of in-mold 99 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 99–112. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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electromagnetic stirring for continuously casting pseudo-rimmed steel. The in-mold electromagnetic stirring technique has proved effective in suppressing the formation of CO bubbles as originally planned, as well as in preventing longitudinal surface cracks and reducing subsurface inclusions. It is now extensively utilized to improve the surface quality of cast steel. In the late 1980s, in-mold electromagnetic braking drew attention as a useful technique for high-speed casting. Nippon Steel also worked on a new type of electromagnetic braking technique that uniformly applies a direct-current (DC) magnetic field across the width of the strand being cast. This resulted not only in preventing the penetration of inclusions deep into the pool, but also in discovering a new function of preventing the mixing in the strand pool. The technique of controlling the initial solidification of continuously cast steel by utilizing electromagnetic pressure has been a subject of ongoing research and development in recent years. In this paper, the details of each application, including the important MHD phenomena, are introduced and are followed by the description of principal results in each plant application.

2.

NUMERICAL SIMULATION METHOD FOR THE INVESTIGATION AND CONTROL OF MHD APPLICATION

The analysis of electromagnetic fields is performed by using simplified scalar or vector potential methods for simple cases, such as round billet CC, or the finite-element method (FEM) for more complicated cases, such as in-mold stirring of a slab caster to obtain the distributions of magnetic field, induced electric current field, and Lorentz force as their product. The basic equations are Maxwell's equations, Ohm’s law, and the Biot–Savart law, which are used as the support equations in the simple treatment of the electromagnetic field. In the case of the electromagnetic brake, a potential equation is used. For the axisymmetric case, a vector potential method, which leads to an integral equation of induced current, is used. Once the distribution of induced electric current is found, the magnetic field and the Lorentz force distribution can be calculated. For the numerical simulation of electromagnetically driven flow and the temperature field, the Navier–Stokes equations, along with the mass conservation and energy conservation equations, are solved by using a finite-difference method, where the Lorentz force is introduced as an external force and Joule heating as source term in the energy conservation equation.

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For the trajectory calculation of particles, which is necessary for the prediction of inclusion behavior in molten steel, a Lagrangian method is used.

3.

IN-MOLD ELECTROMAGNETIC STIRRING BY TRAVELING MAGNETIC FIELD

In-mold electromagnetic stirring is schematically illustrated in Fig. 1. A linear motor is installed in the upper part of the water box in the

Figure 1

Schematic illustration of an in-mold electromagnetic stirrer.

wide face of the mold. The linear motor horizontally stirs the molten steel near the meniscus by generating a traveling magnetic field that covers the full width of the mold [1]. Several new techniques are incorporated to enhance the electromagnetic stirring efficiency. For example, using stainless steel-clad copper plates or light-gage copper plates of low electrical conductivity, and adopting a 10 Hz or lower frequency traveling magnetic field, minimize the attenuation of magnetic flux density. Low-melting-point alloy experimentation and numerical analysis showed that the distance between the linear motor and the meniscus has a great effect on the meniscus velocity and flow pattern.

3.1.

Effect of in-mold electromagnetic stirring on slab quality

In-mold electromagnetic stirring was first introduced to suppress the formation of CO blowholes during casting of pseudo-rimmed steel by

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reducing the concentration of solute elements at the solidification front by the flow [1]. When in-mold electromagnetic stirring was applied to the continuous casting of aluminum-killed steel, it sharply reduced the amount of aluminum oxides within 10mm of the slab surface [2]. As shown in Fig. 2, by applying electromagnetic stirring, the number of inclusions is reduced drastically, even in the unsteady state of the cast

Figure 2 Reduction in subsurface inclusions by in-mold electromagnetic stirring.

(bottom and inter-charge). A longitudinal crack on the slab surface is reduced when the in-mold electromagnetic stirrer causes the molten steel to flow at the front of the solidifying shell in the initial stage of solidification during the continuous casting of medium-carbon steel [2]. The reduction of longitudinal surface cracks is attributable to the washing action of the electromagnetically stirred flow of molten steel. The washing action should level the shell thickness through the promotion of temperature uniformity in the mold pool and reduce the interdendritic segregation.

3.2.

Numerical simulation of in-mold molten steel flow controlled by traveling magnetic field

A magnetic field traveling through a molten steel pool induces an electric current in the molten steel. The interaction of the traveling magnetic field and the induced current produces a Lorentz force, which in turn drives the molten steel. The induced current path is greatly affected by

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the surrounding electric boundary conditions, and the electromagnetic force does not always have a component in the mold width direction alone. The discharge stream from the submerged entry nozzle causes the flow of molten steel in the mold. This flow interferes with the flow driven by the linear motor. As numerical analysis techniques, the FEM and the finite-difference method (FDM) with large-eddy simulation (LES) were used for analyzing the magnetic field and the flow field, respectively. The mold width and thickness were 1,830 and 280 mm, respectively, and the casting speed was 1.3m/min. Figure 3 shows the flow velocity distributions in the mold meniscus position on the transverse section and in the solidifying

Figure 3 Effect of in-mold electromagnetic stirring on the molten steel flow in the mold pool.

shell front position on the longitudinal section without and with in-mold electromagnetic stirring, respectively, when the nozzle port angle is 35°. Without in-mold electromagnetic stirring, the dominant flow is not seen at the solidification front. When electromagnetic stirring is conducted, the molten steel is accelerated to form a relatively uniform rotating flow in the vicinity of the meniscus. The temperature field calculated by the simplified treatment of boundary condition as a Dirichlet condition shows that the stirring increases the uniformity of the temperature field. With regard to the inclusion-removal effect, there are many factors affecting the inclusion behavior, such as flotation, Saffman’s effect, the Magnus effect, and interfacial tension. However, from an order estimation, it can be shown that the flow field itself and Saffman’s effect

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are dominant in the boundary-layer region except within a few dozen micrometers from the solidifying shell. Here, only Saffman’s effect is discussed by using boundary-layer analysis and an estimation of the force acting on a particle. The boundary layer is analyzed by using LES and by introducing fine meshes near the wall. Figure 4 shows the result of numerical calculation. It is noted that with EMS, which generates a flow of about 0.5 m/s inside the mold, the boundary layer becomes thin-

Figure 4

Velocity distribution near the boundary with electromagnetic stirring.

ner and then the Saffman’s lift force becomes higher in the EMS-driven flow than in the one generated by the bulk flow itself.

4.

CONTROL OF MOLTEN STEEL FLOW IN A MOLD BY A LEVEL DC MAGNETIC FIELD

The in-mold electromagnetic braking technique started as a localized MHD application [3] to brake directly the molten steel stream discharged from the submerged entry nozzle. At that time, however, the technique had several problems concerning the stability of the braking effect and the resultant metallurgical benefits. A new technique of stably controlling the molten steel flow in the mold was developed by applying a level DC magnetic field across the mold width; a new function—that of suppressing mixing—was then discovered, in addition to the conventional functions ensuring the slab quality [4, 5, 6, 7].

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Effect of level magnetic field brake on slab quality

Reduction in the number of inclusions. When molten steel was continuously cast on a 1-ton test machine, a level magnetic field (LMF) brake was operated below the mold of the caster, and the metallurgical benefits of this molten steel flow control were investigated. To show the effect of LMF on the penetration depth of injected flow from the submerged entry nozzle, a sulfur tracer wire was fed to the meniscus in the mold. The nozzle discharge stream melted the wire, and the tracer was carried deep into the pool by the downward flow. When a magnetic flux density of 0.55 T was applied, the penetration depth of the tracer was approximately halved, compared with the case in which no electromagnetic braking was applied. This result is similar to the effect of electromagnetic braking in preventing the introduction of inclusions and bubbles into the pool [8]. Minimizing the transition region at ladle change. To study the control of the transition region during a ladle change by imposing the level magnetic field, a medium-carbon steel was cast successively after a low-carbon steel. When the magnetic flux density was 0T, the mixing of the melt proceeded in the transition region mainly by the nozzle discharge stream. As a result, the solute concentration is widely distributed in the cast slab, as shown in Fig. 5. When a 0.55 T magnetic field was applied at the bottom of the mold, the upper part of the pool

Figure 5 Minimizing transition region at ladle change.

was separated by the LMF from the lower pool, thereby suppressing the mixing and minimizing the transition region, as shown in Fig. 5.

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Numerical simulation of in-mold molten steel flow control by level DC magnetic field

When an external DC magnetic field is applied to a moving, electrically conductive fluid, an electric current is induced in the fluid. The current flowing through the fluid establishes a circuit satisfying the law of conservation of electric current, according to the electric boundary condition, and the velocity distribution and magnetic field at right angles induce electromagnetic braking forces on the fluid. The current path formed in this way contains components in the direction opposite to the braking direction. The actual electromagnetic braking phenomenon

is not simple. Hence, substantial understanding of the electromagnetic braking mechanism is indispensable for controlling the flow of molten steel in complex systems, such as the mold of a continuous caster. Concerning the electromagnetic field, a simplified potential equation

is introduced. The electric potential density distribution was obtained and was used to calculate the Lorentz force. For the electrical boundary condition, the solidifying shell drawn at the casting speed was taken into

account. As inlet boundary conditions, a fully developed turbulent flow was given in the upper part of the nozzle tube, and a free jet was formed at the nozzle port. The effect of the free surface was not taken into account.

Figure 6 show the calculated velocity distribution without and with a level magnetic field. Without the magnetic field, the nozzle discharge

Figure 6 Change in velocity distribution in CC mold with LMF.

stream impinges on a narrow face of the pool, and penetrates deep into the pool. On the other hand, with a level DC magnetic field, the nozzle

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stream crossing the DC field is decelerated by an electromagnetic braking force, in which the downward stream at the narrow face is retarded. Figure 7 shows the induced current distributions both in the molten

steel and in the solidifying shell. These three-dimensional current dis-

Figure 7 Electric current distribution in the cross-section of the strand pool with solidifying shell at the level of LMF.

tributions are generated not only by the magnetic and flow fields but also by the complicated electric boundary conditions. Figure 8 shows an example of the calculated results, which demonstrate the suppression

Figure 8 change.

Change of solute distribution at the half width section, 120s after the ladle

of mixing by LMF in the sequential cast of different steel grades. The trajectory calculation for 100 diameter inclusions (Fig. 9) shows the prevention of inclusion movement deep into the strand by the flow. The important factor for setting the amplitude of magnetic field is what kind

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Figure 9 Change in trajectory of inclusions of 100

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without and with LMF.

of defect is to be controlled. For the control of smaller size inclusions, since flotation cannot be expected in plug flow, a choice of proper magnetic field is necessary. In order to control the inclusions and argon bubble behavior by LMF, it is important to know not only the particle behaviors themselves but also the typical phenomena caused by the MHD. The numerical and experimental results showed that the reverse flow is formed around the jet from the submerged entry nozzle. The behavior of argon bubbles is largely affected by this flow.

5.

CONTROL OF INITIAL SOLIDIFICATION BY AN AC MAGNETIC FIELD

The surface of a continuous cast slab is characterized by oscillation marks. The oscillation marks are considered to result from the reciprocal motion of the oscillating mold and the solidifying shell being withdrawn through the layer of mold flux that flows as lubricant between the mold wall and the solidifying shell. These marks are a troublesome obstacle to improving the surface quality of continuously cast steel. In their basic research on the behavior of molten metal in an alternating current (AC) magnetic field, the authors clarified that the free-surface profile of the liquid metal and the static pressure near the meniscus can be controlled by electromagnetic pressure [9]. This phenomenon can be utilized to control the initial solidification of steel being continuously cast. In other words, an electromagnetic force is used to control the meniscus shape and the mold flux channel shape, to relax the dynamic pressure produced in

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the channel by the mold oscillation, and to stabilize the mold lubrication, thereby improving the steel surface quality.

5.1.

Results of plant test

The phenomenon near the initial solidification part and controlling effect of an AC electromagnetic field is illustrated schematically in Fig. 10. A solenoid coil is installed in the mold water-cooling box [10].

Figure 10 Schematic illustration of apparatus for controlling initial solidification of molten steel by electromagnetic force.

Figure 11 shows the relation between imposed magnetic flux density and surface roughness of the cast in the case of electromagnetic casting of stainless steel round billet. The axial component of magnetic flux density at the level of the coil top and on the axis is chosen as the representative value. The bloom surface became smooth and the segregation of nickel, which is typically observed at the bottom of the oscillation mark of the cast, disappeared when the magnetic flux density became more than 0.1 T.

5.2.

Mechanism of initial solidification control

The meniscus shape is calculated from the balance between the electromagnetic force acting on the molten steel pool and the ferrostatic pressure of the molten steel. The velocity distribution of the molten steel in the pool is also obtained. Then, the temperature distribution in the pool is determined by coupled steady-state analysis of heat transfer and

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Figure 11 the cast.

BP2—INTRODUCTORY LECTURE: TAKEHIKO TOH

AND

EIICHI TAKEUCHI

Relation between imposed magnetic flux density and surface roughness of

fluid flow by the finite-difference method (Fig. 12). By analyzing the flux

flow in the gap between the solidifying shell and mold wall through two-

Figure 12 Distribution of flow velocity and temperature (a) without and (b) with electromagnetic pressure.

dimensional viscous flow field calculations, the dynamic pressure of the

flux channel is evaluated [11]. The static pressure brought about by the electromagnetic pressure generated in the molten steel pool in the mold

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induces the convex shape of meniscus, and the static pressure acting on the solidifying shell is diminished. The amount of heat generated in the molten steel by electromagnetic induction is almost negligible compared with the amount of heat extracted. When compared with the case in which no electromagnetic force is applied, the flux channel thickness is increased, and the dynamic pressure acting on the top of the solidifying shell is decreased; the decrease in the depth of oscillation marks formed

on the bloom surface can be explained in this way. The expansion of the channel thickness also decreases the withdrawal resistance in the mold.

6.

CONCLUSIONS

Examples of the application of MHD in the continuous casting process—the electromagnetic stirring of the strand pool with a traveling magnetic field, the control of molten steel flow by an in-mold DC magnetic field brake, and the initial solidification control by an electromagnetic pinching force—have been introduced, along with results of plant tests. For the interpretation and optimization of these applied MHD processes, numerical simulations have played an important role since the phenomena, which are important to the total control of the

system, are both complex and remarkable. With the aid of numerical simulations, these applied MHD techniques have been designed to improve further the continuous casting process capability. They improve the surface quality of cast steel by homogenizing the meniscus temperature, stabilizing initial solidification, and cleaning the surface layer. They also improve the internal quality of the cast steel by preventing inclusions from penetrating deep into the pool and by promoting the flotation of argon bubbles. The application of a pinching force to control initial solidification will lead to the simultaneous achievement of high productivity and high surface quality in continuous casting of steel.

References [1] Takeuchi, E., H. Fujii, T. Ohhashi, H. Tanno, S. Takao, K. Furugaki, and O. Kitamura. 1983. Continuous casting of pseudo-rimmed steel with in-mold electromagnetic stirring. Tetsu-to-Hagane 69(14), 1615–1622.

[2] Pukuda, J., Y. Ohtani, A. Kiyose, T. Kawase, and K. Tsutsumi. 1998. Improvement of slab quality with in-mold electromagnetic stirrer. Proceedings of the 3rd European Conference on Continuous Casting, 437–445. [3] Nagai, J., K. Suzuki, S. Kojima, and S. Kollberg. 1984. Steel flow control in a high-speed continuous slab caster using an electromagnetic brake. Iron and Steel Engineer 61(5), 41–47.

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[4] Takeuchi, E., H. Harada, H. Tanaka, and H. Kajioka. 1992. Suppression of mixing in the pool of continuous casting strand by DC magnetic field. Magnetohydrodynamics in Process Metallurgy, Minerals, Metals and Materials Society

(TMS), 261–266. [5] Zeze, M., H. Harada, E. Takeuchi, and T. Ishii. 1993. Application of a DC magnetic field for the control of flow in the continuous casting strand, Iron and Steelmaker 20(11), 53–57. [6] Harada, H., E. Takeuchi, M. Zeze, N. Ishii, A. Uehara, and T. Okazaki. 1996.

Control of the molten steel flow in CC with a level magnetic field. Current Advances of Materials and Processes, The Iron and Steel Institute of Japan 9, 205. [7] Ishii, N., N. Konno, T. Okazaki, A. Uehara, E. Takeuchi, H. Harada, T. Kikuchi, and K. Watanabe. 1996. The electromagnetic technique with level DC magnetic field—1, Current Advances of Materials and Processes, The Iron and Steel Institute of Japan 9, 206. [8] Yoneyama, Y., E. Takeuchi, K. Matsuzawa, I. Sawada, Y. Hattori, and Y. Kishida. 1989. Study on the electromagnetic brake of molten steel flow. Seit-

etsu Kenkyu, No. 335, 26–34. [9] Miyoshino, I., E. Takeuchi, T. Saeki, H. Kajioka, H. Yano, and J. Sakane. 1989.

Influence of electromagnetic pressure on the early solidification in a continuous casting mold. The Iron and Steel Institute of Japan International 29(12), 1040– 1047. [10] Toh, T., E. Takeuchi, M. Hojo, H. Kawai, and S. Matsumura. 1997. Electromagnetic control of initial solidification in continuous casting of steel by low frequency alternating magnetic field. The Iron and Steel Institute of Japan International 37 (11), 1112–1119. [11] Takeuchi, E., and J. K. Brimacombe. 1984. The formation of oscillation marks in the continuous casting of steel slabs. Metallurgical Transactions B 15(3), 493–509.

RAIL VEHICLE DYNAMICS FOR THE 21ST CENTURY Ronald J. Anderson Department of Mechanical Engineering, Queen’s University, Kingston, Ontario, Canada [email protected]

John A. Elkins RVD Consulting, Inc., Pueblo, Colorado, USA [email protected]

Barrie V. Brickle Transportation Technology Center, Inc., Pueblo, Colorado, USA [email protected] Abstract

1.

Although railway vehicles were first used in the 18th century, it was only in the 20th century that engineers began to understand their dynamics and were able to write down, and solve, their equations of motion. The single most fundamental property of a railway vehicle is the use of a steel wheel running on a steel rail with all the traction, braking and guidance forces being transmitted through a small contact area between the wheel and the rail. While the inherent guidance provided by a conical wheelset running on rails was known even to George Stephenson in 1821, who described its kinematic oscillation, it is only in the last 40 years that an adequate theory has been available for vehicle suspension designers to use. It is well known that a mechanical system that is subject to non-conservative forces may become dynamically unstable under certain conditions. In a railway vehicle the non-conservative forces arise at the contact point between the wheel and rail due to creepage, and the sustained lateral and yawing oscillation that results in some vehicles is known as hunting. This paper traces the developments in railway dynamics up to the end of the 20th century and discusses the latest developments, with predictions as to what the future might hold.

INTRODUCTION

It is usually the case that predictions made as centuries end are incorrect and not of much use. While this paper has a title that indicates that it addresses the topic of rail vehicle dynamics for a century that is just beginning, it is in fact more of a historical document. The authors feel 113

H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 113–126. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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that this paper provides an opportunity to look at where the field has been in the last 100 years, point out some of the major results during the 20th century, and then provide only a very brief glimpse into the near future. Having taken this approach to writing the paper, it was also felt that, for the readers’ benefit, a significant body of written material on the topic of rail vehicle dynamics should be listed in this paper even if the individual papers there are not all referenced. The result is that a relatively extensive chronological bibliography has been included subject to the constraints on the length of this paper. It is impossible to list all the papers on the subject of rail vehicle dynamics, and the bibliogra-

phy contains only those papers that are important and easily accessible through most technical library systems. The authors are keenly aware of the wealth of literature that is missing due to restricting the list to the English language. With our apologies to those readers from non-English speaking countries, we present the bibliography as a list of core material from which researchers can build a more comprehensive library in all languages by following the leads provided there.

2.

THE LEGACY OF THE CONED WHEELSET

At the beginning of the 20th century, vehicles supported by steel wheels running on steel rails had already been a well-accepted part of the transportation infrastructure worldwide for many years. Rail vehicles had developed rapidly since 1825 when George Stephenson (1781–1848) persuaded the directors of the first public railway between Stockton and Darlington in Northern England to abandon their plans for horse traction and use instead a steam locomotive.

Rigid coned wheelsets were integral to the design of steam trains, as they had been to horse-drawn rail vehicles. This was because of their natural self-centering tendency on tangent track and their ability to negotiate curves by moving outward on the track, thereby developing a differential rolling radius between inner and outer wheels. Even with the low speeds of trains during the early 19th century, Stephenson observed and described a “kinematic oscillation” in 1821. Much later this would be recognized as a “wheelset hunting motion”, which, due to the nonconservative forces acting at the wheel–rail interface, tends to become unstable with increasing speed and increasing conicity. The story of rail vehicle dynamics is a story of two contradictory design goals. Rail vehicles should:

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be designed to travel comfortably at high speeds on straight or “tangent” track, and

be designed to traverse curved track without excessive noise or wear, which arise from misalignment of the wheelsets with the track. The first of these goals requires low conicity to ensure stability. The second requires high conicity to promote large rolling radius differences between the inner and the outer wheel on a curve. As it turns out, the 19th century practice of using coned wheels on rigid wheelsets has carried on with minor modifications throughout the 20th century and the conicity of the wheels has a profound effect on both the stability of running on tangent track and the ability to negotiate curves freely. In most conventional railway vehicles, pairs of wheels are mounted rigidly on a common axle, providing a structure known as a “wheelset”. The wheelset evolved by trial and error during the early 19th century and has changed very little to the present day. It consists of two wheels having a coned or profiled tread with a flange on the inside. The tread cone angle is about 2° while the flange cone angle is about 70°. The wheelset runs on rails that are usually canted inwards at 1 in 20. Figure 1

shows sample wheel and rail profiles in contact.

Figure 1

Wheel and rail profiles.

It is now known that the frequency of Stephenson’s “kinematic oscillation” is proportional to the forward speed of the wheelset and that a free single wheelset is unstable, as shown in Fig. 2. The single wheelset can be stabilized if it is attached elastically to a hypothetical infinite mass that moves along the tangent track at a forward speed V. This is shown schematically in Fig. 3. The longitudinal and lateral primary suspension stiffnesses and respectively, are design variables that can be chosen to achieve some desired level of stability as the wheelset moves along the track. Realistically, the infinite mass concept cannot be achieved and rail–vehicle design tended toward a situation in which a relatively long carbody was supported by single

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Figure 2 The instability of a single wheelset.

Figure 3

A wheelset elastically suspended from an infinite mass.

wheelsets at each end. The carbody and its load approximated the infinite mass, especially for fully loaded cars, but could not totally remove the tendency of the wheelsets to “hunt” at high speed regardless of the suspension stiffnesses employed. In fact, if the stiffnesses were made very large, the entire structure including the two wheelsets and the carbody could still oscillate in an unstable mode, which came to be called “body hunting”. The design trade-off between stability and curving became evident with these early two-axle vehicle designs. Simple geometry (Fig. 4) shows that the carbody, on a curve, must be a chord of that curve since the leading and trailing wheelsets are constrained to remain essentially centered on the track. If the primary suspension is relatively stiff and retains the angular positions of the wheelsets relative to the car-

body, then there must be a misalignment of the wheelsets to the track. This angular misalignment a is called the “angle of attack” and from it results most of the wheel–rail wear and noise generated as a rail vehicle traverses a curve.

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The angle of attack developed by a long carbody.

There are two possible remedies to the curving problem. One is to use softer primary suspensions so that the wheelsets can exercise their natural tendency to align themselves with the rails on the curve. Alternatively, the carbody can be reduced in length so that the angle of attack is reduced through geometry. The first of these possibilities reduces the restraint against instability imposed by the stiff suspension envisioned for the “infinite mass” case (Fig. 3) and can therefore not be exploited to the extent desired without making the vehicle have less than adequate stability. The second option, reducing the carbody length, would seem to imply a reduction in the payload capability of the vehicle. Exploration of this option led to the introduction of small two-axle structures that support the ends of the carbody and are connected to it by another level of

suspension (the “secondary suspension”). These are termed “trucks” or “bogies” and a sample is shown schematically in Fig. 5.

Figure 5 A “truck” or “bogie”.

The bogies are short in comparison with the carbody and are able to swivel relatively freely under it so that they can be better aligned with the track. Geometrically they remain similar to the two-axle carbody

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shown in Fig. 4 and a stiff primary suspension holds the wheelsets parallel to each other, thereby producing angles of attack. Nevertheless, the angles of attack for a carbody fitted with bogies are substantially smaller than those of the same carbody fitted with two wheelsets. Rail vehicles fitted with bogies are found to experience two distinct types of wheelset hunting motion. Figure 6 shows the two modes—one with the wheelsets oscillating out-of-phase with each other (a “bending”

Figure 6 Bending and shear modes.

mode) and the other with the wheelsets oscillating in phase with each other (a “shear” or “lozenging” mode). The displacement in Fig. 6 represents the interaxle shear, which develops as the wheelsets oscillate in phase with each other. In the 1970s it became clear to designers that the bending mode would promote good curving whereas the shear mode would not. Further, it was postulated that vehicles could be designed for the same level of stability with more than one combination of primary stiffnesses. The goal, then, was to design a stable vehicle that was soft in bending and stiff in shear so that it would simultaneously be stable and perform well in curves. It is relatively easy to show that a conventional bogie (Fig. 5) has an effective bending stiffness that can be written as

and an effective shear stiffness

given by

Substituting Eqn. (1) into Eqn. (2) and simplifying yields the expression for as a function of and

Similarly, from Eqn. (1), an expression for be developed:

as a function of

can

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Equations (3) and (4) now provide the ability to specify the actual primary stiffnesses, and in order to achieve the desired bending and shear stiffnesses, and The goal was to choose a combination of and that used high shear stiffness to achieve stability and low bending stiffness to promote good curving. In fact, the designers found that they were not totally free to choose the design parameters because—see the denominator of Eqn. (3)—values of less than required that the stiffness be less than zero, and this was physically impossible. It became common to produce contour plots showing lines of constant critical speed on axes of and Also shown on these plots was the so-called “limit line” above which one could not go without producing a suspension component with a negative stiffness. The design area above the limit line was attractive to many people and the conventional bogie began to change when it was realized that direct interaxle connections could be used to add shear stiffness without affecting the bending stiffness. This led to a generation of bogies variously called “cross-braced” (to represent the direct interaxle connections), “radial” (to indicate that the bending stiffness is small and the wheelsets are expected to align themselves radially on curves to minimize the angles of attack), or “self-steering” (to indicate that the wheelsets are not restrained from allowing their conicity to steer them to a radial position on the curve). These designs remain with us today. Forced-steering bogies were another development that attempted to overcome the trade-off between curving and stability. This is a concept in which a kinematic linkage between the wheelset, the bogie, and the carbody is used to steer the wheelsets in response to the swivel angle of the bogie. The swivel angle is directly related to the curve radius, the length of the carbody, and the length of the bogie. The required steer angle is related to the length of the bogie and the radius of the curve. Using the known lengths, it is a relatively simple matter to derive a relationship between swivel angle and wheelset steer angle that can be realized with a simple linkage. The advantage of such a system is that the linkage, which acts as the longitudinal primary suspension, can be made very rigid to promote stability on tangent track while still guiding the axles to radial positions on the curves. Two terms have been coined for these designs—“forced-steering” for bogies that retain their original primary suspensions and require that the linkage overcome the suspension forces to steer, and “guided-steering” for designs in which the primary suspension has been removed and the linkage acts in its place

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while guiding the wheelsets to radial positions on curves without any resistance.

3.

DEVELOPMENTS IN THE 20TH CENTURY

The development of the ability to write and solve the equations of motion for complex rail vehicles hinges on a few critical developments in

the 20th century. All the authors in the bibliography and many who are not there have made contributions to the field, but some contributions were pivotal to success. These are described in this section.

3.1.

The wheel–rail interface

It was Carter [1, 2, 3, 4]1 who first brought to the field an understanding of the relationship between the creepages and forces acting in the contact patch between the wheel and the rail. Creepage is a function of

the motion of the wheelset, and Carter’s development of a mathematical expression relating the forces acting on the wheelsets to their motion was a fundamental first step in the process of being able to write the

equations of motion for the complete rail vehicle. The development of refined rolling contact theories continued throughout the century with particular achievements being made by Vermeulen and Johnson [16] and, especially, by Kalker [22, 24, 30, 40, 42, 52, 59].

Kalker’s work, while mathematically complex, has produced highly accurate and practical models of wheel–rail forces, which are used almost exclusively in modern simulations of rail vehicle dynamics. Kalker’s models range from linear to fully nonlinear.

3.2.

Linear dynamic models

Wickens [17, 19, 28, 33, 36, 37, 39, 73] is credited with first applying a linear form of the rolling contact theory to a vehicle model. His 1965– 66 paper entitled “The dynamics of railway vehicles on straight track:

Fundamental considerations of lateral stability” [19] is a classic, which provided the impetus for many other authors to develop models of rail vehicle dynamics from which much knowledge was gained. From these linear models came the understanding of vehicle stability and critical speed that is widely in use even today. The introduction of digital computers provided an opportunity for researchers to develop

1 References are arranged by date of publication. Not all entries in the References are cited in the text.

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computer codes capable of calculating the eigenvalues and eigenvectors for rail vehicles and, from them, interpreting the frequencies, damping ratios, and shapes of the dynamic modes of the vehicle. This provided very useful design information that had not previously been available.

3.3.

Multibody dynamics

Once the procedure for including the wheel–rail forces in the equations of motion had been developed and computers were available to solve large numbers of simultaneous equations, the most difficult part of creating dynamic models of new vehicles became the process of deriving the multi-degree-of-freedom equations manually. In many cases, analysts resorted to using only single-bogie or half-body models and accepted the inaccuracy in exchange for gains in efficiency. This all changed with the introduction of multibody dynamics to various fields, including rail vehicle dynamics, in the 1980s and 1990s (see Schiehlen [57]). Analysts were able to exploit these techniques to develop very complex vehicle models, which included nonlinear wheel–rail forces, suspension elements, and geometry. The equations of motion for these complex models can be solved only through numerical integration, and few multibody software packages retain the ability to linearize the equations and produce the eigensolutions that were so useful to designers in the past.

3.4.

Nonlinear dynamic models

Clearly the equations of motion of a rail vehicle are nonlinear and therefore belong to the class of nonlinear dynamical equations that may exhibit chaotic motion. The last two decades of the 20th century saw much effort directed toward these problems in general, with only a few researchers looking at chaotic motions of rail vehicles (see True and Kaas-Petersen [58] and True [65]) but obtaining interesting results. In particular, it has been shown that the critical speed established by linear theory is suspect. More developments in this area are expected.

4.

THE 21ST CENTURY

The ability to analyze and predict the motions of rail vehicles has seen enormous change in the last century. The field developed from one in which vehicle designers had, without a scientific basis, some understanding of the dynamic behavior of the vehicles to one in which we now have the knowledge and tools to make highly accurate predictions. The result of this knowledge base has been the development over time of rail vehicles that are more comfortable, faster, and more efficient. Although the

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basic idea of a vehicle that runs on conical wheelsets rolling on steel rails has not changed, the ability to design suspensions has moved rail vehicle technology ahead in step with all the other technological innovations of the 20th century. Rail vehicle dynamics has now reached the point where a number of general computer programs, such as NUCARS and A’GEM, exist, which include all the latest developments described above and can simulate the dynamic behavior of any type of railroad vehicle, or vehicles, negotiating curves, transitions, or tangent track. These programs have been validated against the results from many tests and benchmarks. As long as an accurate set of vehicle characteristics are input into the programs, they can provide a precise simulation of the dynamic response of the vehicle. The problem of ensuring that accurate vehicle characteristics are used in the computer programs has become a priority for rail vehicle dynamicists. The authors have used a number of different techniques for measuring these characteristics, such as vehicle resonance tests, stiffness tests, and impulse “hammer” tests. It is not sufficient to accept manufacturers’ specification data for components, as these are often found to be different from the actual component characteristics as fitted on the vehicle. As we move to the 21st century, the “mechatronic train” is approaching quickly. Mechatronics exploits a synergy between mechanics and electronics to design, construct, and maintain improved products and processes. The application to rail vehicles is through the use of sensors, actuators, and communication to tune the suspension parameters to achieve the highest possible levels of comfort and safety. It is probable that the operating speeds of even the highest speed trains of today will be significantly increased through this marriage of mechanics and electronics. The development and refinement of mechatronic trains will likely consume the first two decades of the 21st century. Beyond that the authors deem it unwise to predict except to say that some form of the conical wheelset will still be evident, perhaps even in the 22nd century.

References [1] Carter, F. W. 1916. The electric locomotive. Minutes of Proceedings of the Institution of Civil Engineers 201, 221–300. [2] Carter, F. W. 1926. On the action of a locomotive driving wheel. Proceedings of the Royal Society of London A 112, 151–157.

[3] Carter, F. W. 1928. On the stability of running locomotives. Proceedings of the Royal Society of London A 121, 585–611.

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[4] Carter, F. W. 1930. The running of locomotives with reference to their tendency to derail. The Institution of Civil Engineers Selected Engineering Papers 81, 1– 25. [5] Cain, B. S. 1935. Safe operation of high-speed locomotives. Transactions of the American Society of Mechanical Engineers 57, 471–479. [6] Langer, B. F., and J. P. Shamberger. 1935. Lateral oscillations of rail vehicles. Transactions of the American Society of Mechanical Engineers 57, 481–493. [7] Giesl-Gieslingen, A. 1936. Riding qualities of passenger cars. Railway Age 101, 214–216. [8] Davies, R. D. 1938–39. Some experiments on the lateral oscillation of railway vehicles. Journal of the Institution of Civil Engineers 11, 224–288.

[9] Cain, B. S. 1940. Vibration of Rail and Road Vehicles. Toronto: Pitman Publishing Company. [10] Cain, B. S. 1941. Notes on the dynamics of electric locomotives. Transactions of the American Society of Mechanical Engineers 63, A-30–A-36. [11] Newberry, C. W. 1945. A study of the riding and wearing qualities of railway carriage tyres having various profiles. Proceedings of the Institution of Mechanical Engineers 153, 25–40. [12] Davies, R. D., and A. F. Cook. 1948. The motion of a railway axle. Proceedings of the Institution of Mechanical Engineers 158, 426–436.

[13] Johnson, K. L. 1958. The effect of spin upon the rolling motion of an elastic sphere on a plane. Transactions of the American Society of Mechanical Engineers 80, 332–338. [14] Johnson, K. L. 1958. The effect of tangential contact force upon the rolling motion of an elastic sphere on a plane. Transactions of the American Society of Mechanical Engineers 80, 339–346. [15] Katz, E., and A. D. DePater. 1958. Stability of lateral oscillations of a railway vehicle. Applied Scientific Research A 7, 393–407. [16] Vermeulen, P. J., and K. L. Johnson. 1964. Contact of nonspherical elastic bodies transmitting tangential forces. Journal of Applied Mechanics E 31(2), 338–340. [17] Wickens, A. H. 1965. The dynamic stability of railway vehicle wheelsets and bogies having profiled wheels. International Journal of Solids and Structures 1, 319–341. [18] Bishop, R. E. D. Some observations on the linear theory of railway vehicle instability. 1965–66. Interaction Between Vehicle and Track, Proceedings of the Institution of Mechanical Engineers 180(3F), 15–28. [19] Wickens, A. H. 1965–66. The dynamics of railway vehicles on straight track: Fundamental considerations of lateral stability. Interaction Between Vehicle and Track, Proceedings of the Institution of Mechanical Engineers 180(3F), 29–44. [20] Matsudaira, T. 1965–66. Hunting problem of high-speed railway vehicles with

special reference to bogie design for the new Tokaido line. Interaction Between Vehicle and Track, Proceedings of the Institution of Mechanical Engineers 180(3F), 58–72. [21] Gilchrist, A. O., A. E. W. Hobbs, B. L. King, and V. Washby. 1965–66. The

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Between Vehicle and Track, Proceedings of the Institution of Mechanical Engineers 180(3F), 99–119. [22] Kalker, J. J. 1966. Rolling with slip and spin in the presence of dry friction. Wear 9, 20–38. [23] Hobbs, A. E. W. 1967. A survey of creep. Dynamics Section, A.P.T. Group, Research and Development Division, Railway Technical Centre, Derby.

[24] Kalker, J. J. 1968. The tangential force transmitted by two elastic bodies rolling over each other with pure creepage. Wear 11, 421–430. [25] Newland, D. E. October 1968. Steering characteristics of bogies. The Railway Gazette 124(19), 745–750. [26] Matsudaira, T., N. Matsui, S. Arai, and K. Yokose. 1969. Problems on hunting of railway vehicle on test stand. Journal of Engineering for Industry B 91(3), 879–890. [27] Newland, D. E. 1969. Steering a flexible railway truck on curved track. Journal of Engineering for Industry B 91(3), 908–923.

[28] Wickens, A. H. 1969. General aspects of the lateral dynamics of railway vehicles. Journal of Engineering for Industry B 91(3), 869–878.

[29] Boocock, D. 1969. Steady-state motion of railway vehicles on curved track. The Journal of Mechanical Engineering Science 11(6), 556–566. [30] Kalker, J. J. 1970. Transient phenomena in two elastic cylinders rolling over each other with dry friction. Journal of Applied Mechanics E 37(3), 677–688. [31] Law, E. H., and N. K. Cooperrider. 1974. A survey of railway vehicle dynamics research. Journal of Dynamic Systems, Measurement and Control G 96(2), 132– 146. [32] Gilchrist, A. O., and B. V. Brickie. 1976. A re-examination of the proneness to derailment of a railway wheel-set. Journal of Mechanical Engineering Science 18(3), 131–141. [33] Wickens, A. H. 1976. Steering dynamic stability of railway vehicles. Vehicle System Dynamics 5, 15–46. [34] Sheffel, H. January 1976. Self-steering wheelsets will reduce wear and permit higher speeds. Railway Gazette International 132(1), 453–456. [35] Pollard, M. C. 1977. Studies of the dynamics of vehicles with cross-braced bogies. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 5th VSD– 2nd IUTAM Symposium, 518–526. [36] Wickens, A. H. 1977. Static and dynamic stability of a class of three-axle railway vehicles possessing perfect steering. Vehicle System Dynamics 6, 1–19.

[37] Wickens, A. H., and A. O. Gilchrist. July–August 1977. Vehicle dynamics—A practical theory. Railway Engineer 2(4), 26–34. [38] Elkins, J. A., and R. J. Gostling. 1978. A general quasi-static curving theory for railway vehicles. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 5th VSD–2nd IUTAM Symposium, 388–406. [39] Wickens, A. H. 1978. Stability criteria for articulated railway vehicles possessing perfect steering. Vehicle System Dynamics 7, 165–182. [40] Kalker, J. J. 1979. The computation of three-dimensional rolling contact with dry friction. International Journal for Numerical Methods in Engineering 14(7), 1293–1307.

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[41] Pollard, M. G. September 1979. The development of cross-braced freight bogies. Rail International (9), 736–758.

[42] Kalker, J. J. 1979. Survey of wheel-rail rolling contact theory. Vehicle System Dynamics 8, 317–358. [43] Sweet, L. M., J. A. Sivak, and W. F. Putman. 1979. Nonlinear wheelset forces in flange contact. Journal of Dynamic Systems, Measurement and Control 101(3), 247–255. [44] Scheffel, H. August 1979. The influence of the suspension on the hunting stability

of railway vehicles. Rail International (8), 662–696. [45] Horak, D., C. E. Bell, and J. K. Hedrick. 1981. A comparison of the stability and curving performance of radial and conventional rail vehicle trucks. Journal of Dynamic Systems, Measurement and Control 103(3), 191–200. [46] Bell, C. E., and J. K. Hedrick. 1981. Forced steering of rail vehicles: Stability and curving mechanics. Vehicle System Dynamics 10, 357–386.

[47] Newland, D. E. 1982. On the time-dependent spin creep of a railway wheel. Journal of Mechanical Engineering Science 24(2), 55–64. [48] Scheffel, H. 1982. The dynamic stability of two railway wheelsets coupled to each other in the lateral plane by elastic and viscous constraints. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 7th IAVSD Symposium, 385–400. [49] Cooperrider, N. K., and E. H. Law. 1982. The nonlinear dynamics of rail vehicles in curve entry and negotiation. The Dynamics of Vehicles on Roads and on

Tracks, Proceedings of the 7th IAVSD Symposium. 357–370.

[50] Horak, D., and D. N. Wormley. 1982. Nonlinear stability and tracking of rail passenger trucks. Journal of Dynamic Systems, Measurement and Control 104(3), 256–263. [51] Elkins, J. A., and B. M. Eickhoff. 1982. Advances in nonlinear wheel/rail force prediction methods and their validation. Journal of Dynamic Systems, Measurement and Control 104(2), 133–142.

[52] Kalker, J. J. 1982. A fast algorithm for the simplified theory of rolling contact. Vehicle System Dynamics 11, 1–13.

[53] Elkins, J. A., and R. A. Allen. 1982. Verification of a transit vehicle’s curving behavior and projected wheel/rail wear performance. Journal of Dynamic Systems Measurement and Control 104(3), 247–255.

[54] Johnson, K. L. 1982. One hundred years of Hertz contact. Proceedings of the Institution of Mechanical Engineers 196, 363–378. [55] Clark, R. A., B. M. Eickhoff, and G. A. Hunt. 1982. Prediction of the dynamic response of vehicles to lateral track irregularities. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 7th IAVSD Symposium, 535–548. [56] Goodall, R. M., and W. Kortüm.

1983.

Active controls in ground

transportation—A review of the state-of-the-art and future potential. Vehicle System Dynamics 12, 225–257. [57] Schiehlen, W. O. 1984. Modelling of complex vehicle systems. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 8th IAVSD Symposium, 548–563.

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[58] True, H., and C. Kaas-Petersen. 1984. A bifurcation analysis of nonlinear oscillations in railway vehicles. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 8th IAVSD Symposium, 655–665. [59] Kalker, J. J. 1984. A simplified theory for non-Hertzian contact. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 8th IAVSD Symposium, 295–302. [60] Johnson, K. L. 1987. Contact Mechanics. Cambridge: Cambridge University Press. [61] Evans, J. R. 1987. An investigation into the interaction between vehicle ride and tilt systems. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 10th IAVSD Symposium, 101–112. [62] Anderson, R. J., and C. Fortin. 1987. Low conicity instabilities in forced-steering railway vehicles. The Dynamics of Vehicles on Roads and on Railway Tracks, Proceedings of the 10th IAVSD Symposium, 17–28. [63] Smith, R. E., and R. J. Anderson. 1988. Characteristics of guided-steering railway trucks. Vehicle System Dynamics 17, 1–36.

[64] DePater, A. D. 1988. The geometrical contact between track and wheelset. Vehicle System Dynamics 17, 127-140. [65] True, H. 1989. Chaotic motion of railway vehicles. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 11th IAVSD Symposium, 578–587. [66] Prederich, F. 1989. Dynamics of a bogie with independent wheels. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 11th IAVSD Symposium, 217–232. [67] Elkins, J. A. 1989. The performance of three-piece trucks equipped with independently rotating wheels. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 11th IAVSD Symposium, 203–216. [68] Elkins, J. A. 1991. Prediction of wheel/rail interaction: The state-of-the-art. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 12th IAVSD Symposium, 1–27. [69] Sheffel, H., R. D. Fröhling, and P. S. Heyns. 1993. Curving and stability analysis of self-steering bogies having a variable yaw constraint. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 13th IAVSD Symposium, 425–436. [70] Scheffel, H. 1995. Unconventional bogie designs—their practical basis and historical background. Vehicle System Dynamics 24, 497–524. [71] Harris, N. R., F. Schmid, and R. A. Smith. 1998. Introduction: Theory of tilting train behaviour. Journal of Rail and Rapid Transit 212(F1), 1–6. [72] Gilchrist, A. O. 1998. The long road to the solution of the railway hunting and curving problems. Proceedings of the Institution of Mechanical Engineers F 212, 219–226. [73] Wickens, A. H. 1998. The dynamics of railway vehicles—from Stephenson to Carter. Proceedings of the Institution of Mechanical Engineers F 212, 209–217.

FUNDAMENTALS OF THE LATERAL DYNAMICS OF ROAD VEHICLES Robin S. Sharp School of Mechanical Engineering, Cranfield University, United Kingdom [email protected] Abstract

1.

Road vehicle steering control is discussed as a man–machine problem. A conceptual framework for thinking about vehicle handling qualities issues is put forward. Fixed control and free control ideas are introduced. Rolling contact between tire and ground is central to understanding of the engineering of the vehicle and a section is devoted to an explanation of it. The steady turning behavior of road vehicles is extremely significant. The interactions of suspensions and chassis have a strong influence on the system qualities and these are highlighted. The full dynamics are then described by reference firstly to the well-known yaw–sideslip linear model and then further by generalization of the results to include nonlinear behavior of the simple car and of more complex, higher-order systems representing vehicles with much design detail. Linearization about a trim equilibrium cornering condition, and the intimate relationship between steady turning behavior and static stability properties, are important topics. Vehicle design alternatives are exposed and the use of the tire longitudinal force capability, via electronic chassis controls, to avoid compromise are mentioned. The paper concludes with a brief description of contemporary vehicle modeling and simulation issues.

INTRODUCTION

The paper is a tutorial review of some of the fundamental issues in the steering behavior of road vehicles. Since the controller in the road vehicle system is a human driver or rider, the problem involves both man and machine and interactions between them. Some discussion of driving is included, leading to ideas on the behavioral qualities necessary in a vehicle. Important aspects of the vehicle engineering are then considered, mainly from the viewpoint of two-track vehicles. These include the external forces acting through the tire–road friction interfaces. Also included 127 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 127–146. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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are the suspension kinematic and compliant chain comprising the linkages between tires and main structure. The suspension and steering links interact with the tire forces by influencing the presentation of the tires to the road surface and the distribution of the total weight of the vehicle between the wheels. The interactions are demonstrable at the first level by considering in detail the steady-state turning behavior. Dynamic properties are revealed via solutions of low-order models of simplified vehicles. Stability in particular is of interest, since the main vehicle classes have characteristic modes of instability that must be understood and guarded against in design. The extension of simple models to more fully represent real vehicles is considered. The main influences come from the additional freedoms afforded by the suspension system and steering system details. Computer-aided model building becomes almost essential for the construction of detailed models. Numerical and symbolic/numerical schemes are mentioned.

2.

MAN, MACHINE, AND INTERACTIONS BETWEEN THEM

The lateral behavior of road vehicles involves strong interactions between man and machine. Contexts vary widely over vehicle type and driver skill level. The human-factors part of the problem is much more complex than the machine part and, correspondingly, it is much less understood. However, it is possible to put the vehicle engineering into a particular context by considering first some human factors issues. Driving skill is acquired by learning, involving repeated small adjustments to mental processes, based on trial and observation. Learned mental processes can be stored and recalled when needed. The capacity to learn is a universal human property. Precisely what is learned is personal and not completely definable. It depends on the machine properties. The main lateral control mechanism on a road vehicle is the steering system. The steering control input can be thought of as a torque, involving so-called free control, or a rotational displacement, involving fixed control. Elements of both free and fixed control are present in any real steering task. The person is free to choose that combination which suits him/her best for the conditions. If the operating mode selected is principally fixed control, the person will provide a local closed-loop steering displacement controller, such that whatever torques are needed to create

the desired steering wheel angle will be provided. Throttle and brake controls will often have some modest influence on lateral behavior. Such influences imply the need for coordination of lateral and longitudinal

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controls, as opposed to completely decoupled actions. In any case, the driver is automatically exposed to both the control device displacement and to its feel. The throttle control is normally arranged to move significantly but not visibly, with spring loading to the closed condition; the brake control moves through a non-executive distance against a spring restraint and then becomes primarily a force control device, moving little once the brakes are properly applied; while the steering wheel moves substantially, is easily observable in doing so, and requires significant driver effort in order to make it move. Clearly, there are man–machine interaction consequences from these different operating modes. Typically, a driver can operate as a feedback controller, subject to the following restrictions. Firstly, he/she must be able to observe any departure from a desired condition. Secondly, there must be an effective control action that will correct the observed error. Thirdly, there must be enough time available for the control to be effective. The comparative slowness of human beings in observing, processing, decision making, and actuation makes the last quite a severe restriction. It implies that rapidly divergent systems and oscillatorily unstable systems with vibration frequencies greater than about 3 Hz will not be controllable by human subjects by active intervention.

2.1.

Steering control

Steering control involves preview of the road ahead. Without sight of the road ahead, safe travel at any reasonable speed is not feasible. Linear optimal preview control theory can be used to find how the steering wheel angle can be derived from road preview information to enable precise path following (Valtetsiotis 1999). The structure of the timeinvariant optimal path following control is shown in Fig. 1. The linear theory can be extended into a nonlinear, force-saturation framework to deal effectively even with limit conditions (Sharp et al. 2000). In providing the necessary steering control, the driver is required to produce appropriate torques and naturally has knowledge of those torques. The steering torque is needed to overcome frictional and hysteretic system losses, to accelerate the massive parts and, in particular, to work against the self-steering effects possessed by a normal system, a large contribution to which comes from the tire–ground interaction forces at the front road wheels. Such torques are modified by torqueassistance systems (power steering), which need to be designed not to spoil the driver’s access to knowledge of the tire–road interface conditions. In practice, these are bound to change substantially from time

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Figure 1 Time invariant optimal steering control structure for previewed path tracking (Sharp et al. 2000).

to time and it is imperative for at least the more adventurous driver to be well informed of them, so that the forward plan can be made with proper recognition of the circumstances.

2.2.

Requirements of the vehicle

The vehicle needs to be steerable and preferably it will be stable. If it is not itself stable, it will rely on the driver to stabilize the system, which may or may not be possible, as discussed above. Ideally, the driver’s observations of the path ahead will be from a stable platform to minimize the complexity of the view, implying the need for minimum pitch response to road roughness input. Minimum response to aerodynamic disturbance is also an ideal, since the driver will need to regulate against the effects of such response (Macadam et al. 1990), adding to the workload. Underdamped oscillatory modes will, in general, be more demanding in terms of driver control, than well damped ones. Decoupled control–response relationships are simpler than cross-coupled ones, as above. Time lags in the controlled system, between control input and significant response, should be very short compared with the corresponding human lags (the latter of the order of 0.3 s) and there should not be several conflicting responses all requiring to be controlled via a single control input.

Learning is most effective when systems behave most consistently. Road vehicle operating conditions are bound to vary according to speed, operating circumstances (loading or wear, for example), and weather. Small changes in conditions should not make large changes to behavior.

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Changes in the weather especially can make radical differences to the contact between tire and ground, dramatically altering the main external force production system. Complete consistency of vehicle behavior is not possible, without designing for worst case conditions, which is not reasonable. The best that can be done is that changes in behavior

are progressive and slow and that adequate warning signals are available to the driver that changes are occurring. In this connection, the steering system is vital. With a good system, it is possible for the continuously observed relationship between steering torque and steering angle, together with potentially confirmatory information in the vehicle responses, to keep the driver more or less fully informed of prevailing tire–road friction levels, even though only mild maneuvering is being done (Sharp 2000).

3.

TIRE SHEAR FORCE GENERATION

In the contact between a pneumatic tire and the road, the road can be considered rigid, with all necessary deformations taking place in the tire. As the tire rolls along, the rubber in the finite contact length continually leaves the contact patch at the back and is replaced by “new” rubber at the front. The simplest view of the shear force generation process comes from the isotropic brush model (Pacejka and Sharp 1991). In this model, the longitudinal and lateral compliances in the tire’s structure are confined to the individually acting tread rubber blocks. The carcass deforms radially to allow conformity of the blocks in the contact region with the flat road surface, and the blocks (or bristles) may also deform laterally and/or longitudinally, with isotropic stiffness, depending on the running conditions. The simplest running condition is free rolling. In this case, each rubber block in turn comes into contact with the ground, adheres to the ground while it is in the contact region, and leaves again. In steady free rolling, the travel direction is along the intersection of the wheel plane and the ground plane and the spin velocity is equal to the forward speed divided by the effective rolling radius (Fig. 2). For a greater spin velocity, the tread bristles need to bend forwards after entering the contact region in order to grip the ground, where they first touched down. The tread base moves backwards relative to the ground. With infinite friction available, increasing forward bending of the bristles would occur through the contact length to the rearmost point. The distribution of the normal loading through the contact length is usually considered to be independent of the running conditions and is parabolic in shape. This implies that the normal pressure between bristle and ground falls continuously to zero at the rear of the contact

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Figure 2 Representation of rolling contact with continuous renewal of contact elements.

patch. With only finite friction available, there must come a point where the elastic forces on the bristle tip due to its bending distortion are too strong for the shear force by friction at the ground contact. Then the bristle tip will slide across the road to relieve the bending strain and the shear force. The greater is the spin velocity, the earlier in the contact region will a bristle start to slide across the road. The front part of the region is characterized by adhesion, while the rear involves sliding (Fig. 3).

Figure 3 A brush model tire in straight running, spinning faster than for free rolling. Only infinite friction will prevent sliding at the rear of the contact region. Then the shear force intensity will be proportional to the distance from the front edge of contact.

The distortion of the bristles depends on the ratio of the tread base velocity, to the modulus of the wheel center velocity, For this special quantity, the term “slip” is reserved. At a certain value of the

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slip, the sliding region will extend over the whole of the contact—the bristle tips will start to slide across the road as soon as they enter the contact region—and the tire then generates the maximum shear force of which it is capable, for the load at which it is operating. This is on the basis of a finite friction coefficient independent of contact pressure and sliding speed, both of which are simplifications of reality. An important consequence of this mechanism is that the gradation from free rolling to full sliding is continuous. The shear force is a continuous function of the slip, being proportional to it for small slips. In the brush model, no inertial effects are considered and the wheel velocity does not matter in an absolute sense. The force–slip relationship is depicted, for various contact lengths and loads, in Fig. 4.

Figure 4 Isotropic brush model driving force as a function of slip for different contact lengths.

The isotropy of the brush model implies similar shear forces when the spin velocity is the free rolling velocity (no longitudinal slip) and the slip is purely lateral slip. In this event, the bristle distortion measure is the (tangent of the) angle between the wheel-center velocity vector and the intersection line common to the wheel plane and the ground plane (Fig. 5). For pure lateral slip with moderate slip angles, Fig. 5 shows that the center of action of the side force developed is behind the geometric center of the contact length. The tire generates an aligning moment. For small slip angles, the ratio of the aligning moment to the side force, called the pneumatic trail, is 1/6 of the contact length. As the lateral slip increases and sliding extends further forward in the contact region,

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Figure 5 Bristle distortions for pure lateral slip with no sliding. Successive positions of the tread base structure at times zero, and are shown in plan view. Bristle distortion is proportional to distance back from the first point of contact; tan(slip) =

the pneumatic trail decreases, reaching zero when sliding extends over the whole contact patch and the tire is generating its peak side force (Fig. 6).

Figure 6 Side force and aligning moment for isotropic brush model tire in pure sideslip for a particular load.

The compliance of real tires, which has been represented as belonging entirely to the bristles of the brush model, resides substantially in the tire carcass. The structure is very different longitudinally from laterally, so that the real tire is non-isotropic. The pure slip properties are qualitatively similar but quantitatively different, and the rather simple combined slip properties of the isotropic brush are substantially more complex for real tires. The simplest way of thinking about the carcass flexibilities is to imagine that the circumferential ring of the tire is rigid, taking the place of the undeformable base structure of the brush, and that it is connected to the wheel hub by an elastic restraint system allowing it all six degrees of freedom, relative to the hub. Such a tire has the same pure longitudinal properties in steady state as the corresponding brush. When it is generating longitudinal force, the carcass is simply wound up such that the carcass torsion balances the moment of the shear

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force about the wheel center. However, when sideslipping, the carcass will twist in yaw, by virtue of the aligning moment developed, and the slip angle will be shared between the carcass and the bristles. In the brush tire, of course, the bristles see all the sideslip, not just a part of it, so the carcass twist compliance acts to reduce the side force for a given slip angle roughly in proportion to the aligning moment generated. The carcass deflections also cause the contact friction forces to be displaced relative to the wheel center, so that an overturning moment is developed and a longitudinal force may contribute, via a lateral offset, to the aligning moment. The geometry of the rolling and slipping tire in contact with even flat terrain becomes quite complex if wheel camber is included, so that fundamental tire models become very complex before they are accurate. Rubber-to-road friction is variable also, according to contact pressure and sliding speed, with rules that are difficult to know with any precision. Empirical representation of laboratory measured steady-state force and moment data is the norm in simulation (Pacejka and Besselink 1996). The flexible carcass of the tire causes the force and moment system to lag the motions giving rise to them, under transient conditions. For example, if the straight running, free rolling tire is given a step steer input, the carcass/bristle structure is twisted initially in yaw and no

sideforce will come from that. On rolling forwards though, the bristle tip-ground forces will alter the distortion of the structure until it gets into its steady sideslipping state. The tire carcass relaxes into this state depending on the distance traveled. The relevant tire parameter is called the relaxation length. Again, for reasonable speeds, tire mass effects are not important and the transient response of the tire approximates to that of a first-order lag. The time constant of the lag is the relaxation length divided by the forward speed.

4.

AERODYNAMIC FORCES

Under typical road vehicle conditions, there is little flow dependency on Reynolds number, and compressibility effects are negligible. Forces and moments are roughly proportional to relative velocity squared, so that they become significant to the vehicle dynamics only at higher speeds of travel. For a vehicle running straight in still air, symmetry indicates no side force, rolling moment, or yawing moment. Lift may be a problem, especially at the rear (see below) and downforce is deliberately generated with inverted wings and vehicle-hull–ground interactions in many classes of competition car. The effects are to alter the tire loadings and thereby to modify the shear forces developed by the tires. In

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crosswinds, side forces and rolling and yawing moments will, in general, occur. They will influence the directional motion of the vehicle and oblige the driver to regulate against the disturbance. The typical problem is that the lateral center of pressure is well forwards in the vehicle and the yawing moment causes the vehicle to veer to leeward. The influence is particularly strong with rear-heavy cars. This is a smallperturbation problem and the tire forces are simple. The location of the center of action of the tire forces, the so-called neutral steer point, influences the steady-state yaw rate response to a crosswind excitation, which is a first-order measure of the quality of this aspect of the vehicle (MacAdam et al. 1990).

5.

SUSPENSIONS AND STEADY TURNING VEHICLE ATTITUDES Lateral suspension geometry can be thought of on any one of about

four different levels. At the lowest level, the track change property for

the static, symmetric car is captured in a parameter called the rollcenter height. At the second level, the wheel lateral, camber, and steer motions can be considered geared to the bump motion by constant gear ratios or suspension derivatives (Hales 1964). At the third level, the displacements can be considered analytic functions of bump displacement, determined by some off-line geometric analysis (Mousseau et al. 1992), while at the fourth level, the mechanical details can be included in a multibody analysis, with mechanism and vehicle-dynamics problems being solved concurrently (Orlandea et al. 1977). When a car maneuvers, its attitude is influenced by its suspension geometry, while its geometry is in turn influenced by the attitude. Understanding this interaction problem is not easy and, to simplify the mechanics, it is common to treat the car-cornering attitude problem with a “level one” suspension description. For a single-end car, the simplified problem is illustrated in Fig. 7. The car body is treated as transmitting a torque from one end to the other, with each end then considered to operate independently. By taking moments for the body about the roll center and then for the whole system about a ground point, the roll angle and the load transfer can be found and it can be established how these things depend on the suspension geometry. If the unsprung mass is neglected in comparison with the sprung mass, if the front and rear ends of the vehicle are joined together by a torsionally rigid chassis, and if small

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Figure 7 Equilibrium of single end, simplified car in steady turning.

angles are assumed, it can be shown that

where The impression is gained from these results that raising the roll center is an effective way of restricting the body roll angle and consequently the road wheel camber angles. It may not be appreciated that the system is constrained to disallow vertical movement of the roll center and it fails to represent in any way the geometric changes that are consequent on the body attitude changing. A fuller treatment will reveal the dangers in believing too literally in the results of a simplified analysis. In fact, high roll centers encourage jacking in cornering and the resulting geometry changes may have very strong consequences (Sharp and Segel 1983). Hales (1964) showed how to analyze cornering equilibrium accurately by virtual work, but this work has not been fully developed and the method has been surpassed by the advent of multibody mechanics computer packages (Schiehlen 1990). Notwithstanding that the problem can be solved straightforwardly with modern numerical methods, to the

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author’s knowledge there is still no definitive material in the literature on the suspension–attitude interaction problem. An illustration of how the suspension geometry and the vehicle body attitude interact in cornering is given in Fig. 8. The figure was generated via a multibody model and a simple animation of the behavior in cornering with slowly increasing

Figure 8 Illustration of suspension geometry and vehicle body attitude interactions in cornering at increasing lateral accelerations.

lateral acceleration. The lateral tire forces are shared between the inside and outside tires in proportion to the loads carried by them.

6.

LATERAL DYNAMICS

The simplest useful model of the lateral dynamics of a car is a yaw– sideslip model (Whitcomb and Milliken 1957, Wong 1993, Dixon 1996). The car is represented as a single rigid body moving on a flat plane and having a rigid steering linkage. The input is a steer angle, implying fixed control. Inertial effects from steering the front wheels are ignored but the tire forces due to steering are described as linearly dependent on the lateral slip. Usually, the braking and drive systems distribute torques equally to left and right sides, and the steering and turning geometries are such that there are hardly any differences between right and left sides except for the wheel loads. If the influences of load transfer on the tire forces of a pair of wheels belonging to one axle are ignored, which is approximately the case for mild maneuvering, it is as if the two wheels of an axle were joined together at the vehicle center plane. Capitalizing on the absence of car-position-based forces in the problem, one can describe the motions conventionally by longitudinal, lateral, and yaw velocities, as seen from the car (Fig. 9). These are sufficient to describe tire slip angles and car accelerations and to allow the derivation of the equations of motion. In the absence of the kind of differential longitudinal tire forces used in traction control and active stability control systems (Abe 1998), the equation for the longitudinal translation

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Figure 9 Plan view of longitudinal, lateral, yaw car model.

is only weakly coupled to the lateral and yaw equations, and the latter two describe the handling problem at this level. In linear form, they are (Dixon 1996)

Solving the equations of motion for steady turning reveals that the curvature response to steer angle ratio is given by

the character of which depends on whether K is positive or negative (Fig. 10). The sign of K depends on the relationship between and representing the respective moments of the axle forces about the car mass center for unit slip angle. The “understeering” car, with K positive, requires more steer angle for a given turn as the speed increases. In the “oversteering” case—when the front axle is “too strong” for the rear axle —the curvature response goes to infinity at a certain speed, called the critical speed (Fig. 10). The understeer level, for K > 0, can be characterized by the characteristic speed which is that speed for which the curvature response is half its low-speed limit value. This value is 1/l, being determined by simple geometry. Above the critical speed for the oversteering car, the curvature response is negative, corresponding to the “opposite lock” case most commonly

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Figure 10 Curvature responses of understeering, neutralsteering and oversteering cars, from linear theory, in steady state.

observed in competition motoring on low-friction surfaces. It should be clear, however, that too close a parallel between real behavior involving close approach to friction limits and the results of the linear theory should not be drawn. The characteristic equation of the yaw–sideslip car is of the second order form and the analogy with a mass–spring–damper system indicates the basic car to have a normalized directional stiffness

which is always positive for the understeering car but which becomes zero for the oversteering car at the critical speed. Zero directional stiffness corresponds to neutral stability. Above the critical speed, the oversteering car is divergently unstable. The infinite steady-state response at the critical speed and the neutral stability at the same condition are intimately related. They both depend on the directional stiffness becoming zero. The stiffness S appears in the denominator of the steady-state response relations and is also the zero-order term in the characteristic equation, determining the static stability of the system (Sharp 2000). The damping factor of the simple car is always positive. At low speeds it is near unity and it remains close to unity for neutral steering for all speeds. The understeerer, for high speeds, has a reduced damping fac-

tor and its yaw response may show a significant resonance at the yaw– sideslip natural frequency. The bandwidth of the car is typically less

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than that of the driver (3 Hz say). It is extended by high axle-cornering stiffnesses, small mass, and small yaw inertia. To get a proper understanding of the basic fixed control car, it is

necessary to extend the ideas from the linear yaw–sideslip model first into the nonlinear, tire force saturating, region. It is fundamental that the tire forces will saturate and that the process towards saturation is progressive. Correspondingly, the car behavior changes progressively. Altering only the tire force description from linear to nonlinear in the simple model, one can linearize for small perturbations from any steadyturn equilibrium (trim) condition. The small perturbation equations are the same as the original linear ones, except that the tire cornering stiffnesses are replaced by local stiffnesses, representing the tangents to the

tire side-force–slip-angle curves at the trim state operating point (Hales

Figure 11 Normalized axle force to sideslip relationships, illustrating a nonlinear vehicle understeering or oversteering depending on the trim, from which small perturbations are considered. At a particular trim, the relative slopes determine the understeer.

1970, Sharp 1973). The trim states that the car adopts as the lateral acceleration changes from zero to the limit possible can be represented as in Fig. 11, showing each normalized axle side force as a function of the axle sideslip angle. This is effectively the cornering compliance concept

of Bundorf and Leffert (1976). If the rear axle limits, in the sense of Fig. 11, at lower slip than the front, the steady-turn performance limit of the car is set by the rear

axle. The front axle will have spare force capability at the limit, when the car will be statically unstable. Conversely, if the front axle limits first, the limit behavior is stable but the car is not steerable. To use the

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full force capability in a trim state, the car needs to be neutral at the limit. Figure 11 need not be interpreted in the context of a yaw-sideslip car with nonlinear tire forces. Rather it can be thought of in the more general context of a real vehicle with many extra degrees of freedom and design details. A connection can be made with Fig. 8. As the trim changes, the body attitude alters, the suspension geometry changes, compliant bushes deflect, etc. Even in this quite general context, it remains the case that the steady-turn behavior is related to the stability as before, both being dependent on the static stability coefficient in the characteristic equation—the normalized directional stiffness S (Sharp 2000). Whereas the simple model contains few contributions to S, a complex model will have many, from suspension and steering elastokinematics, body attitude changes, tire properties, load transfers, etc. The relationship between lateral acceleration and steering wheel angle for a car at a fixed speed can be illustrated in a handling diagram (Fig. 12). Again, it is the incremental changes in steer input and lateral

Figure 12 Steady state handling diagram for a particular speed, showing the lateral acceleration as a function of steer angle for three types of car. One is always neutralsteering, while the others change with lateral acceleration level.

acceleration from one trim to a neighboring one that relate to understeering, oversteering, and static stability via a local linearization of the cornering problem. In terms of the vehicle chassis design, static stability can be guaranteed by ensuring that the front axle will limit first in any steady turn. Since the operating conditions vary widely, over vehicle loading, road friction level, tire type and condition, etc., this design strategy may be very conservative and will imply under-utilization of the rear tire forces

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and a relatively narrow performance envelope for the car. On the other hand, designing for a more neutral limit behavior will introduce the risk of the vehicle becoming statically unstable at its limits and relying on the driver to maintain stability of the man–machine system. The major limitations of drivers in this context are twofold. Firstly, they are slow in sensing, processing, decision making, and actuation, with reaction times to expected events of about 0.3–0.5 s. Secondly, they need to learn what to do for advantage by repeated trials and observations. This is the province of electronically controlled vehicle dynamics intervention systems. Sensors can be used to detect events and conditions and their outputs can be used to actuate engine, transmission, and braking controls, such that the above designer’s dilemma can be avoided. Differential longitudinal tire forces, as between left and right sides, can be employed to introduce yawing moments designed to prevent the car from spinning (Abe 1998). In the design and realization of such intervention systems, continuity is an important property, in view of the man–machine nature of the total system. In principle, accurate detailed predictions of the motions of a car in response to a prescribed set of control inputs can be made. Handbuilt models have largely been superseded by computer-built models (Sharp 1994, 1998), some of which are available freely or commercially

(http://www.trucksim.com . ). The difficult areas in relation to complex vehicle modeling are parametric data and interpretation of results in terms of quality. Tire and aerodynamic force systems can be effectively measured only with special facilities (http: //www.MTS. com) .. Special installations are also advantageous for the determination of suspension and steering elasto-kinematics(http://www.abd.uk.com and http: //www. MIRA. co. uk) and inertial properties (Heydinger et al. 1995, Sharp 1997). Some complex and elaborate studies have simple objectives. Understanding the circumstances under which rollover or spinning occurs, establishing the “maximum challenge” set of control inputs (Ma and Peng 1998), or solving a minimum-time maneuver problem (Casanova et al. 2000a, 2000b) are not so difficult in terms of interpretation. There are many other circumstances, however, when interpretation is both difficult and crucial. In the author’s view, the more complex quality judgments are best attempted through reference to the basis set out in 2.

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CONCLUSION

In the above discussion, a perspective on contemporary road vehicle handling dynamics has been offered. Space restrictions have mitigated against a more complete coverage.

References Abe, M. 1998. Vehicle dynamics and control for improving handling and active safety: from 4WS to DYC. Multibody Dynamics: New Techniques and Applications (Institution of Mechanical Engineers, 1998–13). London: Professional Engineering Publishers, 229–248.

Bundorf, R. T., and R. L. Leffert. 1976. The cornering compliance concept for the description of vehicle directional control properties. Society of Automotive Engineers 760713. Casanova, D., R. S. Sharp, and P. Symonds. 2000a. Minimum time manoeuvring: The significance of yaw inertia. Vehicle System Dynamics 34(2), 77–115.

Casanova, D., R. S. Sharp, and P. Symonds. 2000b. On minimum time optimisation of formula one cars: The influence of vehicle mass. Proceedings of Symposium on

Advanced Vehicle Control (AVEC’2000) (Ann Arbor, Mich.), 585–592. Dixon, John C. 1996. Tires, Suspension and Handling, 2nd ed. Warrendale, Penn.: Society of Automotive Engineers.

Hales, F. D. 1964–65. A theoretical analysis of the lateral properties of suspension systems. Proceedings of the Institution of Mechanical Engineers 2A 2 and 3, 73–91.

Hales, F. D. 1969–70. Vehicle handling qualities. Proceedings of the Institution of Mechanical Engineers 2A 12, 233–244. Heydinger, G. J., N. J. Durisek, D. A. Coovert, D. A. Guenther, and S. J. Novak. 1995. The design of a vehicle inertia measurement facility. Society of Automotive Engineers 950309. Ma, W., and H. Peng. 1998. Worst case evaluation methodology—Examples on truck rollover/jackknifing and active yaw control systems. Proceedings of Symposium on Advanced Vehicle Control (AVEC’98), Society of Automotive Engineers of Japan, 299–305. MacAdam, C. C., M. W. Sayers, J. D. Pointer, and M. Gleason. 1990. Crosswind sensitivity of passenger cars and the influence of chassis and aerodynamic properties on driver preferences. Vehicle System Dynamics 19(4), 201–236. Mousseau, C. W., M. W. Sayers, and D. J. Fagan. 1992. Symbolic quasi-static and dynamic analyses of automobile models. In The Dynamics of Vehicles on Roads and on Tracks (G. Sauvage, ed.). Lisse: Swets and Zeitlinger, 446–459. Orlandea, N., M. A. Chace, and D. A. Calahan. 1977. A sparsity oriented approach to the dynamic analysis and design of mechanical systems—Parts I and II. Journal of Engineering for Industry 99, 773–784.

Pacejka, H. B., and R. S. Sharp. 1991. Shear force development by pneumatic tyres in steady state conditions: A review of modelling aspects Vehicle System Dynamics 20(3–4), 121–176. Pacejka, H. B., and I. J. M. Besselink. 1996. Magic Formula tyre model with transient properties In Tyre Models for Vehicle Dynamic Analysis (F. Boehm and H.-P.

Willumeit, eds.) (Supplement to Vehicle System Dynamics 27). Lisse: Swets and Zeitlinger, 234–249.

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Schiehlen, W. (ed.). 1990. Multibody Systems Handbook. Berlin: Springer-Verlag.

Sharp, R. S. 1973. Relationship between the steady-handling characteristics of automobiles and their stability. Journal of Mechanical Engineering Science 15(5), 326– 328. Sharp, R. S., and L. Segel. 1983. Digital simulation of rollover of a military vehicle. Proceedings of the Institution of Mechanical Engineers/Motor Industry Research

Association Conference on Road Vehicle Handling. London: Mechanical Engineering Publications, 23–30. Sharp, R. S. 1994. The application of multibody computer codes to road vehicle dynamics modelling problems. Proceedings of the Institution of Mechanical Engineers D1 208, Journal of Automobile Engineering, 55–61. Sharp, R. S. 1997. The measurement of mass and inertial properties of vehicles and components. In Automotive Vehicle Technologies 1997–7. London: Mechanical Engineering Publications, 209–217.

Sharp, R. S. 1998. Multibody dynamics applications in vehicle engineering. In Multibody Dynamics: New Techniques and Applications, Institution of Mechanical Engineers 1998–13. London: Professional Engineering Publishers, 215–228. Sharp, R. S., D. Casanova, and P. Symonds, 2000. A mathematical model for car steering, with design, tuning and performance results. Vehicle System Dynamics

33(5), 289–326. Sharp, R. S. 2000. Some contemporary problems in road vehicle dynamics. Proceedings of the Institution of Mechanical Engineering C1 214, Journal of Mechanical Engineering Science, 137–148. Valtetsiotis, V. 1999. A discrete time optimal preview control driver model. M. Sc.

thesis, Cranfield University School of Mechanical Engineering. Whitcomb, D. W., and W. F. Milliken. 1957. Design implications of a general theory

of automobile stability and control. Proceedings of the Institution of Mechanical Engineers (a.d.), 367–391. Wong, J. Y. 1993. Theory of Ground Vehicles, 2nd ed. New York: John Wiley and

Sons.

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More than 1400 scientists, engineers, and students from 51 countries participated in ICTAM 2000.

The UIUC Office of Continuing Education, directed by Dr.

James C. Onderdonk, coordinated meeting registration and tours, and assisted with hotel arrangements.

A MECHATRONICS APPROACH TO ADVANCED VEHICLE CONTROL DESIGN Masato Abe Department of System Design Engineering Kanagawa Institute of Technology, Kanagawa, Japan

J. Karl Hedrick Department of Mechanical Engineering University of California, Berkeley, Calif., USA Abstract

1.

This paper presents a “mechatronic” approach to design vehicle chassis controls as well as active suspensions using preview information. The term mechatronics is used to denote a combined mechanical and electronic design to achieve improved vehicle safety and performance. In the chassis control area, two alternative approaches are analyzed to provide compensation for vehicle lateral instability due to nonlinear tire characteristics. First, a direct yaw moment control (DYC) approach that is based upon sensing side-slip and yaw rate and directly controlling the longitudinal tire forces is analyzed. Next, a four-wheel steering (4WS) approach to control the tire lateral forces is analyzed and compared with the DYC method. Simulations are provided to support the conclusion that the DYC approach provides superior performance. In the second half of the paper an active suspension system is analyzed that has access to road preview information. A controller design method called model predictive control (MPC) is developed to provide a real-time constrained optimization procedure, which optimizes the vertical ride quality subject to inequality constraints on the suspension deflection.

INTRODUCTION

The general purpose of chassis control is to control a vehicle’s lateral, vertical, and longitudinal motions in order to improve its handling performance, ride comfort, and traction/braking performance. The first part of this paper focuses on vehicle chassis control, primarily to improve vehicle handling performance and safety. One way to improve handling performance is through chassis control using four-wheel steering (4WS). Four-wheel steering control depends upon the tire lateral forces, which are proportional to the steer angle 147 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 147–164. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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in a range where the lateral acceleration is small. In this range, the control law can rather easily be introduced by using a two-degree-offreedom linear vehicle model. However, at higher lateral acceleration, the lateral force is not necessarily proportional to the steer angle due to its saturation as a function of side-slip angle. The lateral force also becomes strongly dependent upon the tire vertical load and longitudinal force as well, so that the control law becomes rather sensitive to the vehicle motion and its environmental conditions. A direct yaw moment control (DYC), which controls vehicle motion by a yaw moment that is actively generated by the intentional distribution of the tire longitudinal forces, is becoming one of the most promising means of chassis control. One of the major advantages of this control method is that the tire longitudinal force has no feedback from the vehicle lateral motion as long as it is within the limit of the tire capacity due to the vertical load. Therefore, a precise yaw moment (required to control the vehicle lateral motion) can be developed, and the vehicle motion with the control becomes more robust to fluctuations in the vehicle running condition and its environment. It is obvious that any chassis control aimed at improving vehicle handling performance must rely on the tire lateral force, and that the tire longitudinal force can also be used for the chassis control. These two forces each rely upon the vertical load and are interdependent. This relationship makes the vehicle dynamic characteristics very complex, and a comprehensive study is required to develop more effective chassis control strategies. The second part of this paper discusses the use of preview information to improve the ride quality of off-road vehicles. It has long been known that an active suspension system that had access to upcoming road disturbances can significantly improve the vehicle’s ride quality. This paper presents an interesting control methodology called model predictive control (MPC), which uses real-time, on-line constrained optimization to optimize ride quality subject to such constraints as available suspension clearances.

A mechatronic approach to advanced vehicle control design Table 1

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Nomenclature

INDIVIDUAL VEHICLE CHASSIS CONTROL Chassis control to compensate for loss of stability

The basic nature of the lateral dynamics is described by the following equations of motion for a two-degree-of-freedom vehicle plane model:

where m is the vehicle mass, etc., as noted in Table 1. Equation (1) can be rewritten for small perturbation around an equilibrium point as follows:

According to the stability theorem of nonlinear systems, the roots of the characteristic equation with respect to the singular point determine stability. The characteristic equation of the system described by Eqn. (2) is with

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where represent the slopes of the tire lateral forces versus side-slip angle at the equilibrium point considered. These coefficients represent the equivalent cornering power at the front and rear, respectively. For the motion to be stable, A and B must be positive. For A < 0, the instability condition

arises. It is interesting to find from this equation that the nonlinear vehicle motion tends to unstable motion predominantly by a decrease of the equivalent cornering power of the rear tire with an increase of the side-slip angle—the nonlinear tire characteristics. The equation of motion (2) yields the following description for the ratio of steady-state side-slip gain to yaw rate gain:

This result shows that if the equivalent cornering stiffness of the rear tire is reduced due to low friction coefficient between tire and road surface, large slip angle, load transfer from rear to front during braking, etc., then the negative value of the side-slip angle to generate a definite yaw rate increases. This observation suggests that adopting a control to achieve a definite yaw rate yields a larger side-slip response when the vehicle is subjected to the deterioration of rear tire characteristics. On the other hand, if the side-slip motion is controlled, the yaw rate response gain is automatically restricted with the deterioration of rear tire characteristics. This is a reason why, from the stability point of view, it is better to adopt the side-slip control for the chassis control rather than to adopt the yaw rate control directly. Equation (6) also shows that the side-slip angle needed to maintain a definite yaw rate is determined by the rear tire and not by the front tire. Therefore, in order to control the vehicle side-slip angle by active wheel steering, a rear wheel steer has to be used, as in 4WS. Thus, deterioration of control ability due to the load transfer during braking is inevitable, and the vehicle motion is likely to become unstable even though it is due to the side-slip control, not the yaw rate control. The above discussion may lead to the view that side-slip control by DYC is more effective than other controls in preventing the vehicle from falling into unstable motion due to nonlinear tire characteristics. This

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view will be corroborated through experimental tests at the proving ground, as shown below.

3.

CONTROL STRATEGY TO STABILIZE

VEHICLE MOTION In order to compensate for loss of stability due to nonlinear tire characteristics, we propose a model-following control to follow the response of a linear two-degree-of-freedom vehicle plane plant model with constant speed. Then there will be no control requirement so long as the vehicle motion remains within a linear characteristic region. The model responses of side-slip angle and yaw rate are described as follows:

where P, Q, are the response parameters of the side-slip angle determined by the vehicle parameters as follows:

The response error of the vehicle from the model response caused primarily by the tire nonlinear characteristics and acceleration or braking would require the control to compensate the error and stabilize the vehicle motion.

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Side-slip control by DYC

A sliding control is adopted for the model-following control. The model response of side-slip angle, Eqn. (7), is rewritten in the form of a sliding surface for the sliding control as follows:

The sliding condition is and equations for the vehicle motion are

From these equations, the control yaw moment for the DYC is derived as:

where (i = 1,..., 7) are expressed in terms of P, Q, and k. In order to calculate the lateral forces, their slopes, and the equivalent

cornering stiffness in the above equation, we use the tire model described by Eqn. (20) below.

3.2.

Yaw rate control by DYC

The model response of yaw rate, Eqn. (8), is rewritten in the form of a sliding surface for the sliding control as follows:

Using Eqn. (13) instead of Eqn. (9), we can derive the control yaw moment for the yaw rate model-following control in the same manner as the control yaw moment for the side-slip control:

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Side-slip control by 4WS

Putting equal to zero in Eqn. (11), one can solve for Eqns. (9), (10), and (11):

by using

where (i = 1,..., 6) are expressed in terms of P, Q, and k. This is the required lateral force of the rear axles to be produced by rear wheel steer in order to follow the model response of the side-slip angle. Once the lateral force is given, the rear tire side-slip angle can be obtained by inverse use of the tire model described by Eqn. (3) and then the rear wheel steer angle is found to be

If the required lateral force is greater than the saturated value in the tire model, then the rear wheel steer angle is adjusted so that the tire side-slip angle remains at the saturated point by satisfying the equation

3.4.

Estimation of side-slip angle

It is essential to know the side-slip angle to be used in the control. The authors have already proposed an estimate of the vehicle side-slip angle by a model observer for use in the control. The side-slip motion is described by the equation

and side-slip angle is defined as

These two equations are valid no matter how the vehicle motion changes with acceleration or braking. In Eqn. (18), and are given by the following simple tire model:

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depending on whether or respectively. In Eqn. (20), the subscript T corresponds to f and r, representing front and rear, respectively; and the cornering stiffness is regarded as a function of the vertical load By on-line real-time integration of Eqn. (18), the side-slip angle defined by Eqn. (19) can be obtained.

3.5.

Experimental validation

Validation of side-slip angle estimation. Before studying the effects of the control, the estimated side-slip angle was compared with the real side-slip angle measured by an optical side-slip sensor. The configuration for the side-slip angle estimation is shown in Fig. 1 and the results are shown in Fig. 2. It is found that the side-slip angle can be estimated precisely even under such severe running conditions as evasive lane change maneuvering with quick and large steering angle,

turning with near-limit lateral acceleration with large step steer input, and brake-in-a-turn with steering correction. The experimental results of the side-slip estimation on a low-friction road prove that the estimation method introduced is effective during running on a low-friction road as well. Some estimation errors are recognized during maneuvering with near-limit lateral acceleration when a high friction coefficient is used in the on-board-tire model; nevertheless, no catastrophic failure can be seen in the estimation.

Vehicle and control configurations. In order to compare the effects of DYC and 4WS, as well as to compare the effects of the sideslip control and the yaw rate control, a small-size passenger car on the market, equipped with the automatic torque transfer system and the rear wheel steering system for DYC and 4WS, respectively, was used as the experimental test vehicle. Figure 3 shows the vehicle configuration with side-slip and yaw rate controls by DYC. The vehicle configuration with side-slip control by 4WS is shown in Fig. 4. Effects of the controls. The experimental results are shown in Fig. 5, in which the effects of the controls during a double lane change are

A mechatronic approach to advanced vehicle control design

Figure 1

155

Estimation of side-slip angle.

compared. The vehicle response without control seems almost unstable. On the other hand, the side-slip controls, as well as the yaw rate control, stabilize the vehicle motion and it is found that the effect of the sideslip control by DYC is almost the same as that of the control by 4WS.

However, the vehicle response with the yaw rate control shows larger side-slip angle to follow the reference yaw rate. The experimental results of the single-lane-change test with braking are shown in Fig. 6. The vehicle motion without control seems com-

pletely unstable with large side-slip angle. It is found that the side-slip control by DYC can successfully stabilize the driver–vehicle system; however, the vehicle motions even with the side-slip control by 4WS or the yaw rate control by DYC are almost unstable, showing large side-slip angle. For the side-slip control by 4WS, the instability is likely due to the excessive load transfer from rear to front during braking, which dete-

riorates the rear tire characteristics; the control cannot keep the vehicle stable. In order to follow the model yaw rate strictly, the vehicle with the yaw rate control shows an excessively large side-slip angle, which causes the unstable motion.

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Figure 2 Experimental results of side-slip angle estimation.

4.

ACTIVE SUSPENSIONS WITH PREVIEW INFORMATION

The benefits of active suspensions have been known for several decades. An active suspension has the ability to add power as well as to dissipate power (as is done by passive suspensions). Active suspensions have not achieved commercial success due to the fact that the cost/benefit ratio is not yet low enough to warrant their wide-scale introduction into commercial passenger vehicles. The use of preview information with active suspensions has also been studied and its benefits are clearly understood. Due to the natural frequencies associated with most ground vehicles, it can be shown that road preview information less than or equal to 0.3 seconds provides significant ride quality improvement over fully active suspensions with no preview information. A recent study at UC–Berkeley on the use of active suspensions combined with a preview sensor for application on an off-road military vehicle is described in this paper. Figure 7 shows a schematic of an HMMVW

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Figure 3 Vehicle side-slip and yaw rate controls by DYC.

with active suspensions added at each wheel. Drivers of off-road vehicles

generally feel capable of increasing their speed until all the suspension clearances are used; once metal-to-metal contact has been reached due to the combination of road roughness and vehicle speed, the driver needs to reduce speed in order to maintain control of the vehicle. In this paper a control methodology called model predictive control (MPC) is applied to design a broad-bandwidth active suspension system that has access to upcoming road disturbances from a sensor located at the front of the vehicle. Model predictive control solves a constrained optimization problem in “real time” that optimizes a quadratic performance index subject to constraints on the suspension deflections at the four corners of the vehicle.

4.1.

Vehicle model

Figure 8 shows a quarter car model with an active force element in parallel with the passive suspension. Figure 9 shows a full car model

with four active elements at each wheel. If one considers the force of each

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Figure 4 Vehicle with side-slip control by 4WS.

actuator to be the controlled variable, the linear equations of motion for the quarter car model can be expressed as

If we let

then the discretized versions of Eqns. (21) become

where u(k) represents the control force at time k and v(k) represents the road profile disturbance at time k. The MPC method seeks to minimize

A mechatronic approach to advanced vehicle control design

Figure 5 Vehicle responses during double lane change.

a quadratic performance index

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Figure 6 Vehicle responses during single lane change with braking.

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Figure 7 HMMVW–Active suspension schematic.

Figure 8 Quarter car model.

where the index P represents the total preview horizon and N represents the number of time steps in the future at which we will choose the control

variables u(k). The suspension travel constraints can be expressed as

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Figure 9 Full car model.

The constraint over a time horizon

can express as

The minimization problem can be expressed in vector form as

with constraints where

The MPC method uses a predictor to forecast the output over a receding time horizon. It determines the control variables u(k) by minimizing a

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quadratic cost function of the future output and control inputs subject to constraints on the suspension deflection magnitudes. The algorithm implements only the first control variable calculated and then starts the optimization over at each step. The output predictor equation is given

by The predictor for the constrained suspension travel is given by

where

are the future inputs over the constrained horizon:

Using the output predictor equation, one can express the cost function

as

Using the output predictor equation, one can express the suspension constraints as

Thus the final quadratic programming problem can be expressed as

subject to

The MPC algorithm described above was implemented at the Berkeley Active Suspension laboratory. It was determined from the experimental results that the hydraulic actuator dynamics needed to be considered to achieve accurate force tracking results. A simple nonlinear dynamic model was used to design an inner loop. The following equation for the hydraulic actuator was used:

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where u is the spool valve displacement and is the by-pass spool valve displacement, which was kept fixed. A simple first-order lag model was used to model the dynamics between the spool valve displacement and the input voltage v: The sliding control method was used to design a fast inner loop to track the computed MPC desired force command. In the experimental work the MPC force command was computed every 10ms and the inner force tracking loop was updated every 1 ms. The experimental laboratory results show that significant acceleration reductions can be achieved by active suspensions using road profile preview information. Experiments have been conducted in both a laboratory situation as well as on an HMMWV as shown in Fig. 7. The HMMWV field tests are still ongoing but preliminary results are consistent with the laboratory results.

5.

CONCLUSION

It is conclusively found that, compared with the other controls, the side-slip control by DYC has a prominent effect on compensation of vehicle lateral dynamics during severe maneuvering for a loss of stability due to nonlinear tire characteristics. Active suspensions can be designed to provide significant ride quality improvements in a variety of road situations by the use of preview information and such advanced control methodologies as MPC, which can take into consideration such hard operating constraints as available suspension clearances.

References Abe, M., Y. Kano, Y. Shibahata, and Y. Furukawa. 1999. Improvement of vehicle handling safety with vehicle side-slip control by direct yaw moment. Proceedings

of 16th IAVSD Symposium, 665–679. Abe, M. 1999. Vehicle dynamics and control for improving handling and active safety: from four-wheel steering to direct yaw moment control. Proceedings of Institution of Mechanical Engineers K 213, 87–101. Abe, M., Y. Kano, K. Suzuki, Y. Shibahata, and Y. Furukawa. 2000. An experimental validation of side-slip control to compensate vehicle lateral dynamics for loss of stability due to nonlinear tire characteristics. Proceedings of 5th International Symposium for Advanced Vehicle Control, 179–186. Sharp, R. S., and S. A. Hassan. 1986. The relative performance capabilities of passive, active, and semi-active car suspension systems. Proceedings of the Institution of Mechanical Engineers D 203(3), 219–228. Alleyne, A., and J. K. Hedrick. 1995. Nonlinear adaptive control of active suspensions. IEEE Transactions on Control Systems Technology 3(1), 94–102.

MIXING: KINETICS AND GEOMETRY Emmanuel Villermaux Institut de Recherche sur les Phénomènes Hors Équilibre Université de Provence, Aix–Marseille 1, France [email protected]

Abstract

1.

We review the different facets of the phenomenon of mixing, including its geometrical, temporal, and structural aspects. Then we suggest that a complex mixture can be viewed as the superposition of independent sources. Kinetics and geometry are shown to be closely linked to each other when following the transient mixing of an isolated scalar source in a turbulent flow. The composition law between multiple interacting sources is established experimentally, therefore allowing one to reconstruct any scalar field from well defined elementary contributions.

WHY MIXING?

Mixing1 is a subject that suffers from the Bourgeois Gentilhomme complex. Like Monsieur Jourdain in Molière’s play (1670), scientists, engineers, and indeed all of us often “do mixing without even knowing it” (just think about yourself trying to prepare mayonnaise … ) . As an operation, it consists simply in putting together two or more initially segregated constituents and stirring, in order to attain uniformity, or a new product, or the complete disappearance of one of the constituents, etc.; mixing thus is indeed at the crossroads of many different areas of science. One often needs to mix elements in order to make a new product, a homogeneous blend, or to make possible a chemical reaction or an efficient combustion. One also needs to understand how nature mixes or has mixed to gain information on, e.g., the size of a pollutant spot in a valley, the rate of destruction of ozone in the atmosphere, or the dynamics of the earth’s mantle. More than a fascinating subject in itself, mixing is thus ubiquitous and a key process in many complex man-made or natural operations. 1 The text in this section is inspired from the foreword of Mixing: Chaos and Turbulence, Chaté, H., E. Villermaux, and J. M. Chomaz (eds.), Kluwer Academic/Plenum Publishers, New York, 1999.

165 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 165–180. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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It is no doubt because of its universality that mixing, as noted with regret by J. Ottino (1989), “… does not enjoy the reputation of being a very scientific subject …” as is frequently the fate of interdisciplinary topics. The dendritic nature of the subject matter is revealed by the different angles from which the various scientific communities attack the problem, according to their needs. These approaches can be roughly grouped into three naturally overlapping categories, each of them of interest to different schools of thought: Geometry Kinetics Structures Beyond the tools and attitudes developed by different scientific groups regarding the problem of mixing, one might appreciate that mixing, as suggested by common sense, is the operation by which a system evolves from one state of simplicity (the initial segregation) to another state of simplicity (the complete uniformity). Between these two extremes, complex patterns emerge and die. Questions then naturally arise: how can the geometry of complex patterns be characterized; what is the clock, the time-scale of the process; and what are the structures involved in the flow?

1.1.

Geometry

Very early on, emphasis was placed on the geometry of the mixing zone in the combustion context. Indeed the involved, multiscale geometry of the interface that separates two streams being mixed is not only a spectacular facet of the process, but also sometimes at the core of the physical problem. Exothermic reactions in gases, or reactions with a fast kinetics in liquids confine the reaction zone to a thin region of space and the flame appears as a sharp interface for most scales in the flow. The total extent of the flame area dictates the propagation speed of the front in premixed reactants, as noted by Damköhler (1940), and the net combustion rate in diffusion flames (see the celebrated experiments of Hawthorne et al. (1949)). Knowing how and why the front is distorted by the underlying motions is thus crucial for predicting the flame extent. The multiscale structure of the contour of a scalar blob immersed in a disordered flow was recognized by Welander (1955), who suggested how a connection could be made between the internal structure of the underlying motions and the complexity of the blob shape. Welander even

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made reference to the Koch curve, long before fractals were popularized in this and other contexts by Mandelbrot (1975). The program, still very active today, for investigating the geometry of the scalar support, or the iso-concentration surfaces in flows was certainly already contained in Reynolds (1894). Reynolds was probably even more ambitious, suggesting that watching the dynamics of “coloured bands” in a flow was a route towards understanding the driving motions.

1.2.

Kinetics

Mixing is, in the strict sense, a transient process from initial segregation to ultimate homogeneity. This calls for the understanding of the kinetics and timescales of the process. Studies on dispersion address the problem of the growth rate of the radius of a tracer blob in a prescribed displacement field. In addition to pure molecular diffusion in the medium at rest, fluid motion usually enhances dispersion (Taylor 1921, 1953) and alters not only the diffusion law, by a renormalization of the diffusion coefficient, but also the structure of the law itself. Due to persistent ballistic movements and ever larger jumps in turbulent flows, dispersion laws exhibit, in the absence of traps or slow recirculating motions, a faster-than-linear growth of the mean-squared radius of the blob (Prandtl 1925, Richardson 1926). The presence of bypasses or dead-ends in complex geometries alter, in continuous flow systems (a river, a valley through which wind blows, an open chemical reactor), the residence time distribution of a tracer introduced at the inlet of the system. The novel features of the distribution are spikes at short times if a short circuit is present, and/or long tails caused by traps and slow motions in confined cavities. This was first described by Danckwerts (1953). The kinetics of mixing does not only refer to dispersion. Dispersion may result solely from a spatial reorganization of the quantity to be mixed, with no interpenetration with the substrate at the molecular level. Mixing, as opposed to stirring, actually means homogenization at the smallest, i.e. diffusive, scales; and the appropriate quantity to define and measure the mixing time, namely the “intensity of segregation,” was introduced by Danckwerts (1952). The mixing time (for instance in a tank stirred with an impeller, familiar in the chemical industry) is found to be solely determined by the large-scale features of the flow (integral scale, root-mean-square velocity; see, e.g., Nagata (1975)), regardless of the intimate structure of turbulence, provided the Reynolds number is

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large enough (typically larger than in a wide variety of flows. This absence of a role for turbulence and its structure in the characteristic time of the inhomogeneities decay is an important point, which has been somewhat overlooked. The intimate structure of the local rearrangements in the flow does, however, play a role when chemical reactions with a nonlinear kinetics, or consecutive–concurrent chemical reactions occur within the mixture (Danckwerts 1953, Epstein 1990).

1.3.

Structures

Disordered flows develop a broad hierarchy of scales of motion that convect and distort the scalar field. Notwithstanding the fact that mixing is, like the decay of the energy-containing eddies, a transient phenomenon, the problem of the interaction of the scalar field with the underlying turbulent flow has focused on hypothetical “stationary conditions.” This approach parallels the Kolmogorov quasi-equilibrium picture of turbulence (Corrsin 1951, Oboukhov 1949, Batchelor 1959, Batchelor et al. 1959) and it is in this assumed limit that the timescales of the stirring motions that distort the scalar field are all shorter than the global mixing time (i.e. the variance of the scalar fluctuations is stationary as seen on the timescale of these motions), allowing the possibility of resorting to cascade arguments, and spectral analysis. It is customary, in turbulence, to oppose the statistical approaches, the cascade representation being one of those, to the approaches focusing on “structures” more or less coherent or permanent in the flow. This opposition is somewhat artificial. Modern statistical models of turbulence rely in fact on a description of the flow as a hierarchical structure in the real space, possibly fractal, the structure of the hierarchy inducing the statistics. The cascade picture, notably, is thus in a sense a “structural approach.” In this cascade picture, the mixing time is the time needed for the scalar to travel, by a process of successive reductions of scale imposed by the hierarchy of pre-existing scales of motion (eddies) in the flow, from its initial size to the dissipation scale, prescribed by the Reynolds and Schmidt numbers (Corrsin 1964). This description presents many shortcomings, the first one being its inconsistency with the observed fact that the mixing time depends linearly on the initial size of the blob to be mixed, instead of on its size raised to the power 2/3. If it is clear that mixing macroscopic objects implies a reduction of their transverse size and a multiplication of ever smaller scales for diffusion to act efficiently, there is no direct proof that a scalar blob should follow the

“Kolmogorov cascade” step by step to be ultimately erased by molecular

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Figure 1 Start-up vortex. A fast chemical reaction between the ring and the ambient fluid traces the spiraling interface between the two media.

diffusion. Here is a sign that, in turbulence, even many basic properties escape our precise understanding; the future should therefore be rich in forthcoming works (see the program reviewed by Shraiman and Siggia 2000). The will to reduce a complicated problem such as turbulence to a set of elementary objects containing, presumably, all of the desired information, is in keeping with a long tradition in physics and fluid mechanics. Coherent structures, filaments, worms, and sheets are frequentlyinvoked paradigms. In the context of mixing, the discovery of largescale vortical structures sustained by shear flows has prompted a surge of activity (Brown and Roshko 1974). Originating essentially from a Kelvin–Helmhotz type of instability, these intense, long-lived structures are serious candidates to compete with the cascade representation since they directly couple injection scales with dissipation scales, eliminating intermediary steps. Their existence alone is not, however, sufficient to ensure a rapid mixing. When formed by a given velocity difference across a shear layer, their size (equivalent to the Reynolds number) has to grow beyond a critical value to allow the onset of the fine-scales activity, thereby hastening the uniformization within the layer. This is the so-called “mixing transition” (Breidenthal 1981). Although its ubiquitous character is striking and goes far beyond the context of mixing layers, as recently reviewed by Dimotakis (2000), little is known in detail about what occurs during and after this transition. It is, nevertheless, a known fact that the deformation tensor in turbulent flows bears, in the mean, two directions of stretching and one direction of compression as first analyzed by Betchov (1956). This property

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Figure 2

Cuts in the longitudinal and transverse directions of the flow of a coaxial

jet showing the rolling-up structures resulting from the shear instability between the streams (from Villermaux and Rehab 2000).

leads to an increase in the length and area of material lines and surfaces (Batchelor and Townsend 1956). The stretched sheet is thus likely to be the elementary brick of mixing, whether turbulent or not (Ranz 1979,

Ottino 1989).

1.4.

Towards a new paradigm

Restricting our field of sight to the particular limit of mixing in homo-

geneous fluids, and keeping the Pandora’s box of mixing in complex, non-Newtonian fluids, multiphase flows, granular flows, etc., carefully sealed, we may wonder what the minimal set of ingredients should be to reach a satisfactory description of the above three facets of the mixing phenomenon. Let us examine mixing patterns obtained in different flow conditions, going from laminar (Fig. 1) to strongly turbulent (Fig. 3), via transitional shear instabilities (Fig. 2).

Figure 1 shows a start-up vortex (a kind of a smoke ring), several turns after its formation. The fluid constitutive of the ring develops a chemical reaction with the ambient medium in which it is rolled up. This is a fast chemical reaction and the reaction diffusion zone, very thin in

this case, traces the spiraling interface between the two media. The ring is a non-turbulent object. It directly couples its own injection scale with the diffusive scale via the spiraling motion and the accretion of sheets of ambient fluid at each turn. However, the ring presents many scales, even a continuous spectrum of scales, due to the continuous increase of

the striation thickness from the core of the spiral to its edge. This is a fractal whose covering dimension is close to 3/2. This is a multiscale, dissipative object; but where is the cascade?

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Figure 3 Blob of dye deposed in a large scale sustained turbulent flow converted into disjointed sheets which dilute in the surrounding medium (from Villermaux and Innocenti 1999).

Figure 2 represents cuts in the longitudinal and transverse directions of the flow of a coaxial jet. The inner jet is slower than the outer annular jet, and the pictures show the structures that result from the instability of the shear between the streams. One recognizes in the longitudinal structures formed from the shear instability the rolling-up vortices similar to those in Fig. 1. But the coherence of these structures gets lost and the resulting mixture finally looks like Fig. 3. Figure 3 illustrates what happened to a small compact blob of scalar deposed in a large-scale sustained turbulent flow. The blob has been progressively converted into disjointed sheets that dilute in the surrounding turbulent medium. Except for loose, diffuse sheets of dye, no obvious structures can be distinguished. The striking observation is nevertheless the broad fluctuations in the transverse size of the sheets, and in the concentration level they carry. Although captured at the same instant of time in the snapshot of Fig. 3, those sheets have not all experienced the same history. Some are still thick and dark, while others are already so thin that they almost fade away in the diluting medium. What is the paradigm, the elementary structure of mixing in disordered flows? The coherent rolling-up vortex, the cascade, the stretched sheet? It seems that none of these caricatures is fully satisfactory and could give way to a new paradigm that would account for the parallel, distributed histories of cumulated stretchings experienced by the material elements in the course of time. This new paradigm could reconcile the three facets of mixing by unifying geometrical, temporal, and structural aspects of the process: its intrinsic transient nature, its evolutive geometry as a set of particular structures, namely sheets with distributed histories.

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Figure 4 A snapshot of the scalar distribution in the region 8 < x/d < 12 downstream of the injection point, with d = 1 cm and Re = = 6000.

2. 2.1.

FROM ELEMENTARY SOURCES TO COMPLEX MIXTURES The single source

Experiments (Villermaux et al. 1998, Villermaux and Innocenti 1999) have been conducted aiming at following an initially smooth and compact blob of dye released in a turbulent flow along its transient evolution, from the initial segregation, towards uniformity. The scalar to be mixed is injected continuously in the far field and on the axis of a turbulent jet via a small tube whose diameter d is smaller than the local integral scale L (typically The injection point behaves neither as a source nor as a sink of momentum, in the mean, and the properties of the flow (stirring scale L, r.m.s. velocity are constant during the uniformization period of the scalar. These experiments involve three types of scalars: temperature in air temperature in water (Sc = 7), and the concentration of disodium fluorescein in water (Sc = 2000), thus allowing an investigation of the quantitative role of the intrinsic diffusive molecular properties of the scalar being mixed on the process.

Mixing time. The blob is progressively converted into a set of stretched sheets, possibly coalescing as they fade away (Fig. 4), and the scalar concentration fluctuations PDF, where denotes the injection concentration, exhibits rapidly a self-preserving shape, whose

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Figure 5 Fluctuations PDFs, normalized by the initial concentration (temperature), recorded for three Schmidt numbers 20 diameters d downstream from the injection point. The PDFs exhibit an exponential decay of the form Insert: The argument of the exponentials at different times ut/d and three Schmidt numbers. Sc = 2000, Re = 6000 and 12000, d/L = 0.05, 0.1, 0.6. Re

= 6000, d/L = 0.05, 0.1, 0.16. 0.7, d/L = 0.08, Re = 45000.

Sc = 0.7, d/L = 0.08, Re = 23000;

tail is an exponential with an argument increasing linearly in time t as

with the mixing time

being found to be

as shown in Fig. 5. The factor f(Sc) is a slowly increasing function of the Schmidt number. A fit consistent with the data is f(Sc) ln(Sc), although a weak power-law dependence of the form is not inconsistent as well. The use of three different injection diameters indicates that the mixing time of a blob of size d scales like ln(5Sc), as opposed to according to the vision proposed by Corrsin (1964) and Oboukhov (1949). The spectrum of the mixture is also found to decay like in the inertial range of scales, and for scales smaller than the injection scale d, as opposed to Moreover, the mixing time

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Figure 6 Left: Contours of the level sets of the scalar field of Fig. 4, for four different concentration thresholds levels. From left to right and top to bottom, 0.3, 0.4 and 0.5. Right: Corresponding box-counting relationships. Continuous lines fit by Eqn. (5).

has been found to be smaller than the cascade time We have suggested (Villermaux et al. 2000) that the process of mixing does not follow the sequential route expected from cascade arguments, but is on the contrary “bypassed” by a strong and constant stretching rate acting at the injection scale. Transient geometry. The changes in the blob morphology in the course of its dilution reflect closely the uniformization process itself. This is best illustrated by following the transient shape of a particular isoconcentration contour Figure 6 shows four contours corresponding to four different concentration levels Consider a blob of dye initially smooth and segregated. The number of segments, or square boxes of linear size r needed to cover its contour at a threshold level is on a two-dimensional cut such as those of Figs. 3 or 4. If the blob is immersed in a prescribed displacement field whose stationary velocity increments give rise to a scale-dependent stretching rate then the number of segments constitutive of the contour at time t is

where the amplification factor expresses the net increase in the contour corrugations length, all the more rapid that is intense, and where the exponential factor reflects the disappearance of the scalar by molecular diffusion according to the global temporal evolution of the concentration histogram (Eqn. (1)).

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The maximal rate of shear, giving the separation rate of two material

points initially close to each other, is of the order of as suggested by the scale dependence of the mixing time in Eqn. (2). Considering two material points belonging, for instance, to the same isoconcentration contour, initially separated by a distance r not necessarily infinitely small, the rate at which their separation distance increases is smaller than Velocity gradients in the flow tend to vanish, in the mean, for separation distances r larger than the scale of the mean gradient support, namely the injection size d in the present case. A possible model for the elongation rate for any scale r is

The number of boxes are represented, for the corresponding threshold levels by (see Fig. 6)

according to (3) and (4). Accounting for the initial smoothness of the contour, namely the covering relationship (5) is a combination of the trivial 1 / r factor, times a corrective factor, increasing in magnitude with time, and whose weight depends on scale: it is, at a given instant of time, a decreasing function of scale, expressing the

fact that small scales have, in proportion, more contributed to the corrugation of the contour than larger scales, precisely because shearing motions are less efficient at large scales than at smaller ones (Eqn. (4)). The covering relationship (5) thus exhibits a curvature, whose direct

consequence is the scale dependence of the fractal dimension of the contour This fact has been recognized in a number of related instances (see Villermaux and Innocenti 1999 and references therein) where the local (in scale) fractal dimension is found to increase from 1 at small scale, to larger values, possibly reaching 2, characteristic of space-filling objects in two dimensions. The above scenario provides a mechanism for this continuous transition, its origin lying in the close interplay between kinetics and geometry.

2.2.

Multiple sources

A complex mixture in the real world is obviously the result of the

superposition of many different sources. If we think of a hot jet discharging in a quiescent cold environment, the temperature fluctuations on its centerline at a given distance from the nozzle are certainly likely

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VILLERMAUX

Figure 7 Two nearby sources discharging in a turbulent medium in which they mix.

to be the result of cumulated contributions from different parcels of cold fluid entrained at the boundary of the jet in the course of its development. The “signature” of one scalar source presented in the previous section is likely to be altered if another source is present in its vicinity (as shown in Fig. 7), and the interaction between those two has to be considered. If and are the bare PDFs of each of the two sources disposed in the vicinity of each other, the compound PDF is shown in Fig. 8. As long as the plumes emanating from each of the sources do not interfere, the sources develop in an anticorrelated manner: the concentration measurement point is either in one plume, or in the other (see also Warhaft 1984). Then, as soon as the plumes merge, it is observed experimentally that is very close to the convolution of and The concentration C in the resulting mixture is the sum of the concentration from source 1 and of concentration from source 2, and being chosen independently in each of the original distributions provided that

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Figure 8 Composition of sources of heat in water (Sc = 7). Solid line: experiment with two sources. Dashed line: reconstruction of the scalar field from and under the anticorrelated rule. Circles: reconstruction of the scalar field by convolution.

This result is extended to the turbulent jet problem (Duplat and Villermaux 2000), for which it is shown that the scalar fluctuation PDF on the centerline is the result of well defined elementary sources whose size is given by the local width of the jet. These findings, bridging the kinetics and statistical aspects of the phenomenon of mixing, also suggest that a complex mixture can be understood as the superposition of well defined elementary contributions, whose spatial distribution reflects the entrainment properties of the flow.

References Batchelor, G. K. 1959. Small-scale variation of convected quantities like temperature in a turbulent fluid—Part 1. General discussion and the case of small conductivity. Journal of Fluid Mechanics 5, 113–133. Batchelor, G. K., I. D. Howells, and A. A. Townsend. 1959. Small-scale variation of convected quantities like temperature in a turbulent fluid—Part 2. The case of large conductivity. Journal of Fluid Mechanics 5, 134–139.

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Batchelor, G. K., and A. A. Townsend. 1956. Turbulent diffusion. In Surveys in Mechanics, Batchelor, G. K., and R. M. Davis (eds.). Cambridge: Cambridge University Press, 352–399. Betchov, R. 1956. An inequality concerning the production of vorticity in isotropic turbulence. Journal of Fluid Mechanics 1, 497–504. Breidenthal, R. 1981. Structure in turbulent mixing layers and wakes using a chemical reaction. Journal of Fluid Mechanics 109, 1–24. Brown, G. L., and A. Roshko. 1974. On density effects and large scale structures in turbulent mixing layers. Journal of Fluid Mechanics 64(4), 775–815. Corrsin, S. 1951. On the spectrum of isotropic temperature fluctuations in an isotropic turbulence. Journal of Applied Physics 22, 469–473. Corrsin, S. 1964. The isotropic turbulent mixer: Part II. Arbitrary Schmidt number. American Institute of Chemical Engineers Journal 10(6), 870–877. Damkohler, D. 1940. Der Einfluss der Turbulenz auf die Flammen-geschwindigkeit in Gasgemischen. Zeitschrift fur Elektrochemie 46(11), 601–652. Danckwerts, P. V. 1952. The definition and measurement of some characteristic mixtures. Applied Scientific Research A 3, 279–296. Danckwerts, P. V. 1953. Continuous flow systems. Distribution of residence times. Chemical Engineering Science 2, 1–13. Dimotakis, P. E. 2000. The mixing transition in turbulent flows. Journal of Fluid Mechanics 409, 69–98. Duplat, J., and E. Villermaux. 2000. In preparation. Epstein, I. R. 1990. Shaken, stirred—but not mixed. Nature 346, 16–17. Hawthorne, W. R., D. S. Wendell, and H. C. Hottel. 1949. Mixing and combustion in turbulent gas jets. In Third Symposium on Combustion and Flame and Explosion Phenomena. Baltimore: Williams and Wilkins, 266–288. Mandelbrot, B. B. 1975. On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars. Journal of Fluid Mechanics 72(2), 401–416. Nagata, S. 1975. Mixing, Principles and Applications. New York: John Wiley & Sons. Oboukhov, A. M. 1949. Structure of the temperature field in a turbulent flow. Izvestiia Academii Nauk USSR, Seriia Geograficheskaia i Geofizicheskaia 13, 58–69.

Ottino, J. M. 1989. The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge: Cambridge University Press. Prandtl, L. 1925. Uber die ausgebildete Turbulenz. Zeitschrift für angewandte Mathematik und Mechanik 5, 136–139. Ranz, W. E. 1979. Application of a stretch model to mixing, diffusion and reaction in laminar and turbulent flows. American Institute of Chemical Engineers Journal 25(1), 41–47. Reynolds, O. 1894. Study of fluid motion by means of coloured bands. Nature 50, 161–164. Richardson, L. F. 1926. Atmospheric diffusion shown on a distance-neighbour graph. Proceedings of the Royal Society of London A 110, 709–737.

Shraiman, B. I., and E. D. Siggia. 2000. Scalar turbulence. Nature 405, 639–646. Taylor, G. I. 1921. Diffusion by continuous movements. Proceedings of the London Mathematics Society 20, 196–212. Taylor, G. I. 1953. Dispersion of a soluble matter in a solvent flowing slowly through a tube. Proceedings of the Royal Society of London A 218, 44–59.

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Villermaux, E., C. Innocenti, and J. Duplat. 1998. Histogramme des fluctuations scalaire dans le mélange turbulent transitoire. Comptes Rendus à l'Académic des Sciences, Paris 326, Série IIb, 21–26.

Villermaux, E. and C. Innocenti. 1999. On the geometry of turbulent mixing. Journal of Fluid Mechanics 393, 123–145. Villermaux, E., H. Chaté, and J. M. Chomaz. 1999. Why mixing? In Mixing: Chaos and Turbulence, Chaté, H., E. Villermaux, and J. M. Chomaz (eds.). New York: Kluwer Academic/Plenum Publishers, 1–8. Villermaux, E., C. Innocenti, and J. Duplat. 2001. Short circuits in the CorrsinOboukhov cascade. Physics of Fluids 13(1), 284–289. Villermaux, E., and H. Rehab. 2000. Mixing in coaxial jets. Journal of Fluid Mechanics 425, 161–185. Warhaft, Z. 1984. The interference of thermal fields from line sources in grid turbu-

lence. Journal of Fluid Mechanics 144, 363–387. Welander, P. 1955. Studies on the general development of motion in a two-dimensional, ideal fluid. Tellus 7(2), 141–156.

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ICTAM 2000 early registrants enjoy conversation and a spectacular view of Chicago's skyline from the 9th floor of the Chicago Marriott Downtown hotel during the Get-Together Party on Sunday, 27 August 2000. In the background is Chicago's landmark Hancock Building.

STATISTICAL APPROACH TO THE MECHANICAL BEHAVIOR OF GRANULAR MEDIA Stéphane Roux Unité Mixte de Recherche CNRS/Saint-Gobain, “Surface du Verre et Interfaces” Aubervilliers Cedex, France [email protected]

Farhang Radjaï Laboratoire de Mécanique et de Génie Civil, Université de Montpellier II Montpellier Cedex, France [email protected] Abstract

1.

We discuss the quasistatic rheology of ideal granular media consisting of rigid discs interacting via the Coulomb law of friction and perfectly inelastic collisions. The macroscopic description of the rheology of quasistatic deformation of such media is rigid–plastic with hardening laws parameterized with internal variables that have to characterize the geometry of the assembly. A phenomenological approach is proposed along these lines. An outline of a microscopic–macroscopic derivation of the required characteristics is presented. Finally, we list some possible effects of fluctuations that may limit the precise quantitative success of this approach.

INTRODUCTION

The quasi-static behavior of granular materials is already a mature field in which a number of elasto–plastic models reproduce very accurately the available experimental tests. They allow us to design civil engineering structures with confidence. However, this description is essentially based on extensions of the elasto–plasticity of other materials rather than a microscopic modeling of a granular assembly (Mróz 1998). On the other hand, by focusing on the details of particle interactions, considerable progress has been made in recent years through various algorithms that allow us to describe different regimes of granular flows. (See e.g. Bardet 1998, Kishino 1999 for recent reviews.) The discrete 181 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 181–196. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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modeling of granular media allows nowadays very accurate simulation of very stiff particles up to particles for a significant cumulative strain. In cases where the accurate modeling of the contact and of the particle properties is less stringent, more than one million grains can be taken into account. The interest of this approach is that it allows us also to follow the response of a granular assembly subjected to uniform stress or strain. In this context, the use of bi- or tri-periodic boundary conditions (in two or three dimensions, respectively) is an important achievement allowing us to reduce significantly the role of walls. We are thus now in a suitable position to answer questions pertaining to the characterization of the geometry of the packing (and hence eventually to answer such unsolved issues as the orientation of localization bands), to address the role of fluctuations, and to measure the effective stress carried by a specific granulometric class in a packing (relevant for fragmentation)— issues that are clearly out of reach within today’s continuum modeling. With this motivation in mind, one is naturally invited to have a fresh look at the continuum modeling and to progress in the direction of introducing geometric information in the macroscopic modeling, even if such an approach will inevitably lead first to a deterioration in the accuracy of the macroscopic modeling. The hope is that, after some time and effort, one may achieve a more satisfactory description in terms of connections to microscopic reality and still with a fair account for experimental tests.

2.

A MODEL SYSTEM

In the following, we will focus on the simple model of a granular assembly of rigid discs in two dimensions. These particles interact only via a hard-core potential (without adhesion) and the Coulomb friction law with a single coefficient of friction. When numerical simulations are used, a slight polydispersity is introduced in order to avoid the crystallization of the system that is a specific feature of two-dimensional systems. However, in our theoretical description we ignore the polydispersity and the particles will be characterized by a single average radius R.

2.1.

General remarks concerning the macroscopic modeling

Since the particles are considered as rigid and with no adhesion, the macroscopic behavior has to be rigid–plastic. Therefore, the domain of admissible stresses where the medium is rigid, with its boundary where plastic flow may take place, has to be specified. The absence of a stress scale imposes moreover that is a cone in stress space.

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The Coulomb friction law itself can be seen as a rigid–plastic behavior. Within this framework, the Coulomb friction does not obey the normality rule, and hence macroscopic modeling has to be a non-associated plastic behavior. Therefore, the direction of the plastic strain rate has to be specified independently from The plastic flow rule, as well as the yield stress, has to be defined as a function of the internal variables that characterize the state of the medium. When considered at the microscopic scale, it is obvious that this state is defined only through its geometry, i.e. the spatial organization of particles with respect to each other, and hence it is natural to require that internal variables have a simple geometrical interpretation. Although this statement seems quite obvious, it readily disqualifies a number of macroscopic descriptions used at present. Stated simply, the fundamental difference between a microscopic modeling and a macroscopic one lies in the number of internal variables. Going from the micro- to the macro-scale, we wish to preserve only a few of them, enough to characterize precisely the geometrical state of the assembly and not too many, so as to be able to have an efficient formulation and a limited number of parameters to adjust through fitting rheological responses or microscopic considerations. As usual, the choice of pertinent state variables is the crucial issue, which often results from a compromise between accuracy and simplicity. In order to highlight this compromise, we proceed in three steps. First, we apply the above stated constraints to the formulation of a (trivial) rigid–plastic description with no internal variable. Then, we go one step further and incorporate a single scalar internal variable. Finally, we add the fabric to the description. We will see that the incorporation of more and more variables leads to a progressively richer description. We will sketch the outline of a systematic procedure. Coming back to the macroscopic description, in order to arrive at a complete formulation we have to specify three points: (1) a yield stress parameterized by the internal variables, (2) a plastic flow rule parameterized by the internal variables, and (3) a “hardening” law that describes the incremental evolution of internal variables with plastic strain.

3.

NO INTERNAL VARIABLE

The most elementary choice is to assume that no internal variable is necessary. In this case, there is obviously no need for a “hardening law”. The medium has to be assumed isotropic—otherwise the anisotropy parameters would constitute internal variables. In two dimensions, the requirement of objectivity implies that the domain is a function of

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stress invariants. The easiest case refers to the principal stresses, i.e. a two-dimensional description of the stress space. Then, is a cone whose axis has to lie along the isotropic pressure direction. Thus, a single parameter is required, which quantifies the relative magnitude of the deviatoric stress with respect to the average stress (trace). This single parameter can be rephrased in terms of the classical friction angle in the Mohr–Coulomb framework. For similar reasons, the plastic flow rule consists of defining the relative amount of dilation/contraction with respect to the deviatoric part of the plastic strain rate. Since the macroscopic description should hold for an arbitrarily large cumulative strain, the only physically admissible choice is to require that the plastic strain be isochoric, or This corresponds to the fact that the internal state of the granular medium is assumed to be unique, and thus it has a well-defined packing fraction that is independent of its prior deformation history. Moreover, along the plane where the Mohr–Coulomb criterion is reached, the axial strain has to be zero. This defines uniquely the relative orientations of the principal axes of the strain rate and stress. The above considerations are sufficient to determine fully the macroscopic bahavior of the medium in two dimensions. In particular, this shows that a single scalar parameter is needed—the Mohr–Coulomb friction angle. It is obvious that the resulting description is very crude. Nevertheless, in cases where the cumulative shear strain is large (e.g. surface flow in avalanches), this level of description may prove quite sufficient. The unique state of the medium corresponds to the “critical state” of soil mechanics reached after a sufficiently large deformation (Schofield and Wroth 1968).

4.

ONE INTERNAL VARIABLE

Let us now try to add some information about the geometrical state of the medium. The first and most obvious property pertaining to the geometry of a granular packing is the packing fraction c, defined in two dimensions as the fraction of surface covered by the particles. A number of equivalent variants, such as void ratio, density of particle centers, global density, and porosity may be used instead. We can proceed as in the previous case but using c as an internal variable for the yield surface and for the plastic flow direction. The yield surface is still parameterized by a friction angle, but now the latter is a continuous (increasing) function Concerning the direction of the incremental plastic strain, dilation and contraction are now admissible.

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The ratio of the spherical part to the deviatoric part of the plastic strainrate tensor is denned to be the dilation angle satisfying

where are the two eigenvalues of In the absence of internal variables we argued that = 0. Following the same argument, one infers that there should exist a specific packing fraction such that

This packing fraction characterizes then the critical state. Now a hardening law has to be specified. It should describe the way c evolves under an increase of plastic strain. In the present case, this evolution law is dictated by the very definition of our internal variable, namely

This description provides already a more detailed description of the transient stages leading to the critical state. One also recovers the previous model if only large strains are considered. The price to pay for this more accurate description is that we have to specify two functions and There we have different routes at our disposal: either we consider a purely phenomenological approach and thus we try to identify these functions from simple tests (Schofield and Wroth 1968), or we may try to relate these functions to a microscopic description of the packing. Eventually, we could also combine both approaches by using part of the information from the micro-scale and identify only a few parameters from experiments. Along the latter direction, one could for instance use Taylor’s hypothesis (1948) in order to relate the two functions and together, and measure then only one of them experimentally. We will come back to this issue after the following section devoted to a richer description of the geometrical state of the medium. In order to simplify the analysis and the number of parameters to be introduced, we may focus on the neighborhood of the critical state and expand the functions of interest in Taylor series. This leads to a simple three-parameter description, based on and (Roux and Radjaï 1999). An interesting point to make is that this approach naturally leads to the occurrence of localization in a dense granular medium. We will however not enter this issue, which would require lengthy developments.

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FABRIC AS STATE VARIABLES

The previous modeling predicts that once the system has reached its critical state, it remains in this state for all directions of shearing. This is not what is observed experimentally. In particular, if the shear is simply reversed, one typically observes a long transient deformation where the system evolves towards a new critical state. This discrepancy clearly indicates that a scalar internal variable such as c (which is unable to encode a specific anisotropy of the medium) is insufficient for the description of the state of the medium. From symmetry arguments, the most elementary object that can characterize such information is a second-order tensor. The latter has to be built from statistical information describing pairs of particles. Moreover, assuming that the required information is local, one should focus on particles in contact. These remarks point to the internal variables pertaining to the distribution of contact normals. Let us consider the probability distribution of contact normals . This is the probability that a given particle has a contact along the direction parameterized by the polar angle of the contact normal n. The function is -periodic and it can be Fourier expanded as (Rothenburg and Bathurst 1989)

Truncation of the Fourier expansion after the second term provides the most salient features of the texture of the medium. In a totally equivalent fashion, one could simply construct the classical fabric tensor where the brackets denote averaging over all particles in a representative element of volume, and is the dyadic (tensor) product (Satake 1982). While the fabric tensor is often normalized with respect to the total number of contacts, here we choose to normalize it by the number of particles. In other words, in our case A = where z is the coordination number. Assuming that the latter is simply related to the packing fraction, we see that the three descriptions considered above can be seen as retaining more and more terms (0, 1, or 2) in the Fourier expansion of p. Let us now characterize all the required information to obtain a complete mechanical description of the behavior using A, B, and as internal variables:

Yield stress: Since the medium has some anisotropy, a single friction angle is no longer appropriate. However, in order to keep close to the concepts used within the previous cases, we consider all potential slip planes characterized by a polar angle of the

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normal to the slip plane. The largest ratio of the tangent to the normal stress that can be supported by this plane naturally defines a friction angle that is a function of and the above three internal variables. Galilean invariance implies that only the difference is meaningful. We thus have to specify the function

Plastic strain rate: Similarly, along each potential slip plane we can characterize the orientation of the relative velocity of two points aligned perpendicular to the slip plane. This defines a dilation angle which as before depends on and the internal variables, i.e. Hardening rule: A hardening rule here means that and are functions of their current values and the value of the incremental plastic strain as well as the rotation i.e. the antisymmetric part of velocity gradient, because of the induced anisotropic texture of the medium. This hardening law should account for the advection of contacts in the plastic flow, as well as the creation and opening of contacts. A last constraint to be taken into account comes from the quasistatic nature of the loading we are interested in. This implies that the rheology should be rate independent, i.e. time as such should not play a role in these equations. Thus (as well as all other such rates) should depend on through a positively homogeneous function of degree 1. As in the previous case, there are now different routes to follow according to different strategies. Again, one possible route is a purely phenomenological approach with the perspective of identifying all the required information from tests. Some basic principles should still be used to constrain the hardening rule. However, due to the crowding of internal variables, this task is quite challenging. A second route is to circumvent first the form of the possible dependencies by means of expansions around special values of the internal variables, such as the isotropic case B = 0 or, alternatively, around values that would correspond to the “critical states”. We use the latter in plural form because the critical state may well depend, for instance, on the relative rotation rate with respect to the deviatoric strain rate, i.e. the critical state under pure shear may differ from the critical state under simple shear. To be more precise, we may assume that, if the deviatoric part of the fabric tensor is reasonably small, may be expanded as

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so that the determination of the full function of three parameters is reduced to four scalar parameters for i = 0 , . . . , 3. Another possible strategy is to resort to the microscopic world in order to identify the above-listed unknown functions. We now discuss some elements along this direction.

6.

MICRO–MACRO TRANSITION

At the macroscopic level, we deal with stress and strain as continuous fields. However, at the particle level, we have a discrete set of forces transmitted by the interparticle contacts and velocities of particles. The connection between these two levels of description has been the subject of intensive work. A clear review of different proposed relations has been presented by Bardet (1998). Let us first recall very briefly a few essential results.

6.1.

From discrete to continuous

The first key point is to define the average stress over any domain, which encompasses an arbitrary number of particles. This can be done consistently down to the scale of one single particle. Let us label each contact around one particle by an index i, and denote the unit normal vector by and the force transmitted at the contact by The stress that characterizes the particle is (Cundall and Strack 1979)

where S is an area associated with the particle from a Voronoï construction from particle centers. This expression exploits the fact that the particles are at rest and no torque is transmitted at the contacts. We have assumed that the particles are circular with a radius R. We see that the transition from forces to stress requires the definition of a representative environment of a particle where each neighbor is specified,

i.e. a “node” of the contact network. The analogous treatment of the displacement field is somewhat more subtle. The main point is that the object in which we are interested at the macroscopic scale is the displacement field particle center. Whether a particle rotates or not will affect the velocity of a material element in the grain, but not the overall displacement. Thus, the strategy usually adopted in this respect is to construct a continuous field from an interpolation of the velocity field that matches exactly the particle velocity at the particle center.

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The standard technique is again to resort to a tesselation of space with polygons whose vortices are the particle centers, and whose edges connect the contacting particles. Once the problem is reduced to the estimation of the average strain inside an elementary polygon whose boundary velocity is prescribed, the solution is standard (Kruyt 1996). As in the case of the force–stress relation, here we need a representative elementary structure—a polygon formed by contiguous particles—which will be referred to as a “cell” in the sequel.

6.2.

Environments: Nodes and cells

A major difficulty is that, as discussed above, the macroscopic description of the geometrical state of a granular medium is based on the fabric represented by (including a more or less severe truncation). However, in order to construct the two elementary tensors, we have to deal with nodes or cells and this requires richer information. For instance, a node is characterized by its coordination number z and the orientations of its contact normals. Thus, we need the probability distributions The corresponding issue is trivially written in similar terms for the cells. In order to construct these representative environments, we have to propose an “educated guess” for these multicontact distributions. An easy solution is to assume the most “disordered” situation, i.e.

In our case, this solution is obviously wrong since a contact in the direction with a given particle impedes other contacts to be established in a direction such that hence

Here, the strategy that we propose is still to resort to a similar “most disordered” situation, provided these steric constraints are taken into account. Operationally, this translates into the maximization of an entropy functional under constraints. The entropy S is classically defined as

and S is maximized over the set of functions that (1) fulfill the steric hindrance conditions, Eqn. (11), and (2) whose partial summation over all but one angle gives back the known The second constraint

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is imposed through Lagrange multipliers. Let us introduce a periodic function such that = 0 for and H = 1 otherwise. It can be shown that takes the form

The unknown function g is then determined from an implicit equation resulting from the resummation condition (2), and whose solution can be obtained from a simple iterative scheme. Note that without steric hindrance, the simple solution is recovered. We note that not all values of A and B in the truncated form of p admit a solution. The condition is that

which simply states that no more than one contact can be found in any sector of opening angle Using the simple truncated form Eqn. (5), we obtain

which together with the condition (no negative probability) gives the domain of physically accessible values of the fabric.

7.

YIELD STRESS AND PLASTIC STRAIN DIRECTIONS

The previous section proposed an operational way of generating representative nodes (and using a similar construction, cells) with acceptable statistics, i.e. consistent with the known information concerning fabric. Thus, we have now a key that will allow us to compute the yield stress. From the previous discussion, we have to compute the maximum allowed deviatoric stress for all orientations of the principal axes of stress with respect to that of the fabric. We propose a Monte-Carlo procedure, which consists in generating a large collection of node configurations with the computed statistics. Then, for fixed orientation and trace of the stress tensor we compute the maximum value of its deviatoric component that can be obtained from the contact forces with the following constraints: (1) the forces are balanced, (2) they yield the imposed stress when using Eq. (9), and (3) each force fulfills Signorini (no traction) and Coulomb conditions. A simple average over the configurations gives an estimate of the sought yield surface.

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A similar procedure can be designed for the strain. Cells are generated, and for each configuration the minimal dilation angle for a principal strain orientation is computed. This is done by computing an admissible set of particle center velocities that is consistent with the average strain, and with the steric exclusion conditions. An average over configurations gives the sought dilation angle.

8.

HARDENING RULE

We still have to address the final point, which is the hardening rule, i.e. the evolution of the fabric, in terms of or fabric tensors as a function of strain. The two effects to take into account are (1) the advection of contacts by the plastic flow and (2) the induction (gain or loss) of contacts (Roux and Radjaï 1999). We can write the time evolution of as a balance equation:

where the divergence operator is simply J is the contact “current”, and I the induction term. The mean velocity field with respect to a particle center in polar coordinates is written For contacting particles, where r = 2R, the mean velocities are written The expression of the contact current is thus simply written as

The Signorini condition requires that Galilean invariance dictates that the effect of a rotation rate should come into play only in the tangential component, as and We note that current can also be split into two contributions, one part from the rotation, and one from the pure deformation term. Extracting the rotation term from the current, we can rewrite the left-hand side of the balance equation as

where the total time derivative includes the rotation term. With this in mind, we can now safely ignore the rotation effect. The velocity field for non-contacting particles can be deduced simply from a mean-field assumption since no direct steric constraints are at play. Thus u and are directly related to The induction term I consists of a creation of contacts, and an opening term with The creation of contacts involves

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the non-contacting normal velocity when the latter is negative, and the probability that a given particle lies in the region sufficiently close to reach the reference particle. This involves the packing fraction, through the areal density of centers The creation term is written

where [ … ] _ denotes the negative part of the velocity. The contact

opening term is Up to now, we have specified all terms except the contacting velocities They are obviously related to through a geometric function that is positively homogeneous of degree 1, because of the rate independence. This could be computed together with the direction of the plastic strain rate, because at this stage we generate a representative set of local configurations where we have access to the relative particle velocities. Alternatively, we may consider two arbitrarily remote particles along

a given direction, which can be reached through a path of contacting particles. When the particles are sufficiently far from each other, their relative velocity is equal to the macroscopically determined one. However, it is also equal to the sum of the relative contacting particles along the path. This constrains to be equal to when averaged over an interval of angles that allows for the existence of a path. This

allows the deficit in negative U values to be compensated by a larger

value of V at a different angle. The difference between and i.e. the deviation from the mean field, may be interpreted as some kind of diffusive contribution to the current due to the steric hindrance transmitted through particles outside our “shell” description. This section has been written directly in terms of the entire distribution and not its truncated Fourier expansion. In order to express the hardening equations in terms of A, B, and it suffices to consider weak formulations of the latter by multiplying Eqn. (16) by 1,

and

and by integrating over all angles

Then, a term-by-term

identification gives a direct transcription corresponding to the reduced description of the fabric.

Let us finally mention an additional important point. We pointed

out that the fabric is restricted by geometrical constraints (see e.g. Eqn. (15)) at the particle level. However, these constraints do not appear in the present hardening rules, so they might be violated if one proceeds along the proposed route. A consistent way to circumvent this difficulty is to require that the fabric always lies in the admissible domain. This

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will in turn generate an admissible set of plastic strain increments. The latter may thus be used to define the plastic flow rule. This would constitute an alternative way to end up with a consistent rigid–plastic formulation, and it would bypass the stage that consists in generating representative “cells”. In other words, it would exploit different ways of relating the relative velocities of contacting particles to the mean strain field, through paths rather than elementary cells.

9.

EFFECTS OF FLUCTUATIONS

In this section, we briefly discuss potential limitations to the scheme proposed above, as inferred from numerical observations. Most of these difficulties result from fluctuations either in time or in space that have received in the past a much more limited attention than average behaviors. Indeed, the above scheme is based at each time step on rest configurations. However, when running numerical simulations we observe that from time to time the system reaches unstable points where a dynamical rearrangement of particles has to take place. In this dynamical phase, inelastic collisions dissipate the potential energy drop (i.e. the work of the loading forces), due to the reorganization of the assembly. The cumulative effect of these restructuring events may have paradoxical effects. In particular they can produce an effective mean dissipation that is “Coulomb”-like even for ideal frictionless particles. Indeed, we can imagine that averaging the instantaneous dilation rate during a long steady shear strain (once the steady state has been reached) over all rest configurations, we may obtain a positive value that is exactly counterbalanced by the compression taking place during these unstable events. The dynamics of restructuring being much faster than any external loading (quasistatic condition), the (time averaged) dissipation appears to be rate independent. Moreover, the dissipation at each event is equal to the external work accumulated prior to the event. Therefore, the dissipation is simply proportional to the external loading. Both these features characterize a solid “Coulomb” friction. It is amusing to note that a similar explanation based on elastic asperity interactions is at the heart of microscopic physical modeling of friction (Caroli 1996, Tanguy 1998). Trying to quantify the above effect in a more realistic case, we quantified the relative part of dissipation due to inelastic collisions, as compared with the total dissipation, i.e. including contact friction (for an interparticle coefficient of friction = 0.5). The

resulting ratio was close to 30% due to restructuring events. This effect being absent from the proposed description, the effective macroscopic friction angle is at best off by this amount if a global dissipation balance

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is performed. In contrast, an analysis of experimental results based on instantaneous values of the stress may be in much closer agreement. Fluctuations may be important for basically two reasons. The first one is their amplitude relative to the mean, and the second is the existence of long-range correlations, which may be present either in time or space and which may affect very significantly the validity of the proposed approach. In order to investigate the latter, we studied the trajectory of particles in a simple shear test. The analysis of the fluctuating part of the displacement and of the distance between two neighboring particles reveals the existence of long-range temporal fluctuations that give rise to an anomalous diffusion regime. Similarly, the instantaneous strain field also displays large scale inhomogeneity, which was unexpected. These observations, which are not taken into account in the approach we presented, may potentially invalidate a quantitative agreement between theory and numerical or experimental tests.

10.

CONCLUSIONS

We presented a systematic approach to progressively enrich a description of the mechanical behavior of a granular medium from both macroscopic and microscopic (particle level) points of view. This constitutes a template, and a number of these elementary steps have still to be validated in particular from detailed numerical simulations. We proposed an original scheme based on entropy maximization that allows us to generate representative environments around particles—an essential step for determination of the yield stress or the plastic strain rate direction. We also proposed a simple “shell model” description of the velocity field around a particle, which allows us to obtain an evolution equation for the fabric. Finally, we pointed out some potential difficulties of this kind of approach, where salient aspects of fluctuations, not included in the theoretical approach, may affect the accuracy of the description. Nevertheless, the global framework, i.e. the form of the equations, should be unaffected. A macroscopic description of these fluctuations might thus appear a necessary step for a complete description, and that is, in our view, a major challenge for the future.

Acknowledgments The authors wish to thank R. Behringer, J. C. Charmet, M. Jean, J. J. Moreau, and H. Troadec for valuable discussions on the subject covered in this chapter.

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References Bardet, J. P. 1998. Introduction to computational granular mechanics. In Behaviour of Granular Media (B. Cambou, ed.). Vienna: Springer, 99–169. Caroli, C., and P. Nozieres. 1996. Dry friction as a hysteretic elastic response. In Physics of Sliding Friction (B. N. J. Persson and E. Tosatti, eds.). Dordrecht: Kluwer Academic Publishers, 27–49. Cundall, P. A., and O. D. L. Strack. 1979. The distinct element method as a tool for research in granular media. National Science Foundation Report 76-20711, University of Minnesota, Minneapolis, Minn. Kishino Y., and C. Thornton. 1999. Discrete element approaches. In Mechanics of granular materials (M. Oda and K. Iwashita, eds.). Rotterdam: Balkema, 147–223. Kruyt, N. P., and L. Rothenburg. 1996. Micromechanical definition of the strain tensor for granular materials. Journal of Applied Mechanics 118, 706–711. Mróz, Z. 1998. Elastoplastic and viscoplastic constitutive models for granular materials. In Behaviour of Granular Media (B. Cambou, ed.). Vienna: Springer, 269–337. Rothenburg, L., and R. J. Bathurst. 1989. Analytical study of induced anisotropy in idealized granular materials. Geotechnique 39, 601–614. Roux, S., and F. Radjaï. 1999. Texture-dependent rigid-plastic behaviour. In Physics of Dry Granular Media (H. J. Herrmann, J. P. Hovi, and S. Luding, eds.). Dordrecht: Kluwer Academic Publishers, 229–235. Satake, M. 1982. Fabric tensor in granular materials. Proceedings of the IUTAM Symposium on Deformation and Failure of Granular Materials (P. A. Vermeer and H. J. Luger, eds.). Balkema: Delft, 63–68. Schofield, A. N., and C. P. Wroth. 1968. Critical State Soil Mechanics. New York: McGraw-Hill.

Tanguy, A. 1998. Un modèle de frottement solide sec par multistabilité de milieux elastiques. Ph.D. dissertation, University of Paris VII, France. Taylor, D. W. 1948. Fundamental of Soil Mechanics. New York: John Wiley and Sons.

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Ukranian participants Tatyana S. Krasnopolskaya and her husband, Vyacheslav V. Meleshko, receive ICTAM 2000 “stress relief” apples from session aide Gareth Block as they enter the Opening Ceremony. The instructions on the apple read: “Archimedes, float! Galileo, drop! Newton, throw! all others, squeeze!”

DAMAGE ANALYSIS AND PREVENTION IN COMPOSITE MATERIALS George J. Dvorak Rensselaer Polytechnic Institute, Troy, New York, USA [email protected] Abstract

1.

A new model for incremental analysis of distributed damage evolution in heterogeneous solids is developed with the Transformation Field Analysis. Stress changes caused by local debonding under increasing overall loads are described by a selected model of imperfectly bonded inhomogeneity and represented by equivalent eigenstrains that act together with the applied loads and prescribed local transformation strains on an undamaged aggregate. Interaction between the still bonded and partially debonded phases at any damage state is described by transformation influence functions. Damage rates are derived from the local fields, in terms of a prescribed probability distribution of interface strength and local energy released by debonding. An incremental procedure is outlined that predicts the extent of damage and its effect on local and overall response under variable loads. Also, potential applications of fiber prestress in damage control and prevention in laminated structures are illustrated by two examples. One shows how prestress release can expand the tensile damage-free region of symmetric laminates. Another derives the fiber prestress magnitudes needed to eliminate free edge stress concentrations caused in any laminated plate by cooling from the processing temperature.

INTRODUCTION

A significant effort has been devoted in recent years to modeling of interface decohesion in heterogeneous solids, which is responsible for most of the distributed damage preceding failure of composite materials and polycrystals. In most models, a single cylindrical or spherical inhomogeneity is bonded to the surrounding matrix by an imperfect interface or interphase. Elastic or inelastic interlayers or spring-type interfaces, and also sliding and frictional interfaces in contact, have been used in analytical studies of the stress and deformation fields in partially debonded inhomogeneities. Moreover, both numerical unit cell models and analytical tools have been developed that incorporate separation laws for imperfect interfaces consistent with fracture mechanics 197

H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 197–210. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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concepts. While such models are essential building blocks of micromechanical damage evolution models, their utilization has been rather limited. Available studies tend to predict only uniform or periodic damage distribution in the representative volume, which do not correspond to observations. The next chapter of this paper briefly outlines a new analytical framework for modeling of distributed damage evolution that can incorporate different local debonding models, together with extensive information on local material properties. The damage process is assumed to start in a few weakly bonded interfaces at a certain initial level of applied load, and proceed under increasing load or time by affecting ever larger internal surface areas and volume fractions. Interaction between both the still bonded and partially or completely debonded inhomogeneities at each loading step is described using certain transformation influence functions. To this end, the local average stress changes in the debonded phase are simulated by equivalent transformation strains applied to an otherwise undamaged aggregate. Local thermal and variable transformation strains are also included in the applied load set. Local stresses and strains, overall deformations and compliance changes are found in terms of applied loads and damage density, and then updated under increasing load according to a prescribed probability distribution of local debonds, and an energy criterion for debonding of each individual volume. The third chapter describes some recent results on using selected magnitudes of fiber prestress, applied prior to and released after matrix consolidation, to create residual stress fields that help prevent damage initiation in laminates. In particular, fiber prestress is shown to change the position of the initial damage envelopes in the overall loading space, allowing for higher tensile stresses to be applied in a damage-free region. Another example illustrates how prestress release can be used to cancel free edge stresses caused in laminates by cooling from processing temperature.

2.

2.1.

DAMAGE EVOLUTION IN A TWO-PHASE COMPOSITE SYSTEM Local stresses in current damage state

A representative volume V of a composite aggregate contains a statistically homogeneous distribution of dispersed particles or fibers of stiffness initially bonded to a matrix of stiffness Incremental loading is applied by a superposition of surface tractions on 5 of V, derived from a uniform overall stress with thermal and other transformation strains (r = 1, 2) in the phases. Under increasing loads,

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starting at some of the interfaces between the matrix and dispersed reinforcements may undergo partial or complete debonding. At any stage k of the damage process, the volume V contains a constant matrix volume while the total volume of the reinforcements is divided into fully bonded (r = b) and debonded (r = d) parts; the respective volume fractions are related by After partial debonding, certain interface tractions are transmitted by friction or by interlocking asperities; hence an average stress is supported by each debonded particle or fiber. If the overall loading path is reversed, parts of the debonded interfaces may reconnect under local compression. Therefore, in general, some or all components depend on the loading set except under complete debonding, when In what follows, debonding-induced changes in local stresses, local and overall strains are simulated by introducing equivalent local eigenstrains in the r = d subvolumes of the undamaged composite system. Since the eigenstrain currently present in the bonded reinforcements must be present in the entire volume the total eigenstrain in is and will be decomposed again in below. At any stage of the process, the components of the current load set are regarded as independent loads acting on a fully bonded aggregate. All local eigenstrains are assumed to be uniform, or to represent certain averages of the actual local fields. Also, all reinforcement particles or fibers are assumed to have the same shape and orientation, but each can have a different “diameter” or size. Local stress averages caused in a heterogeneous medium by a uniform overall stress and a piecewise uniform distribution of local transformation strains are written in the form (Dvorak 1990)

where are Hill’s (1965) mechanical stress concentration factors of the undamaged system, which depend only on the local and overall stiffnesses. The local stresses satisfy the requirement hence and Since there is and The are transformation stress concentration factor tensors that evaluate the residual stresses contributed by the equivalent eigenstress in the transformed and debonded phase volumes to stresses in all phases r. Phase definition is expanded here to mean homogeneous subvolumes of the same

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stiffness in a fixed overall space, subjected to the same average eigenstrain. Thus the debonded and still bonded reinforcements are regarded as different phases of the same stiffness loaded by different eigenstrains and Evaluation of the must take into consideration all three phases in (2); this is reflected by adopting the two equivalent forms valid for multiphase systems (Dvorak and Benveniste 1992)

where r, s = 1, b, d, the local and overall compliances and also and but

M =

Both and are evaluated in a perfectly bonded system, where the overall compliance M has been estimated by a selected averaging method. The stress concentration factor where is the compliance of the cavity containing in a large volume of a certain homogeneous comparison medium S is the Eshelby tensor of a transformed homogeneous inclusion in According to the above assumption, all reinforcements have the same shape and orientation; hence only a single S or are needed. The choice of defines the averaging method used in evaluation of M, and Preferred here is the choice which implies the Mori–Tanaka method. The S and are then evaluated in the matrix material and M can be found using the expression (M + = Other choices of are possible, providing that the estimate of M remains within the Hashin–Shtrikman bounds. In any case, selecting as a certain function of only is required to render the stress field independent of in the limiting case of porous medium, when and Since the choice of affects only the evaluation of S and , any such choice can be easily implemented in what follows. The local stress averages can now be found as functions the current total load set and the debonded phase volume fraction Substituting from (4) into an expanded form of (2) and using the notation (Hill 1965)

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201

provides after some

Since one can readily verify the requirement As expected, the stresses depend only on the differences between the local eigenstrains in the reinforcement and matrix; application of a uniform eigenstrain in the entire volume of any solid has no effect on its stress state. Also, it turns out that

At the current state of debonding the is an unknown quantity that needs to be found from additional information. This is usually provided by a local decohesion model that relates the local stress field in the partially debonded inhomogeneity to that in the surrounding matrix. A suitable general form connects the averages of the two fields

by means of a partial stress concentration factor It should provide an interface for implementation of many different models that simulate interface decohesion modes. For example, those employing an elastic interphase (Christensen and Lo 1979, Chen et al. 1990) or a spring-type interface (Hashin 1991a,b) provide fairly simple forms of that may change with values of the interface parameters, but do not depend on the debond angle or area. A more realistic class of models that evaluate the extent of debonding either by numerical or analytical methods (Achenbach and Zhu 1989, Hutchinson and Jensen 1990, Karihaloo and Vishvanathan 1985, Gao 1995), and rely on specific models of interface decohesion (Needleman 1987, Tvergaard 1990) would yield dependent also on as would models that incorporate interface sliding and friction (Steif and Dollar 1988, Furuhashi, et al. 1992). Implementation of the latter class may rely on a separate numerical routine for evaluation and updating of Using (8) in yields the equivalent eigenstrain that, when applied within together with on S of V, and within and respectively, creates the local stresses (6) and (8), where

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Substituting this into (6) readily yields the local stress in terms of and eliminating this with (7) and (8), in favor of the prescribed quantities

and current value of

Again,

finally gives

As one would expect, in contrast to (6), the

introduction of in (8) has rendered the fields dependent on the damage parameters and even under purely mechanical loading. The former property is retained in the case of complete debonding, where and

2.2.

Local and overall strains in current damage state

The relevant expressions evaluating the local and overall strain averages are

where, according to the Levin formula and (5), the overall eigenstrain is

Noting that

it follows from (9)

that the overall strain (12) can then be reduced to the form where the damage-induced addition to the overall elastic compliance of the damaged aggregate is found as

and the overall eigenstrain induced by the applied local eigenstrains

becomes

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Since both Q and are diagonally symmetric, is also diagonally symmetric in the case of complete debonding, where and when The latter symmetry has been established for the Hashin (1991) spring layer model of an imperfect interface, applied to an isotropic inclusion or an isotropic and transversely isotropic fiber, each embedded in an isotropic matrix. Other debonding models can be used, providing that they yield the said diagonal symmetry of Note that for the overall strain (14) and eigenstrain (16) both revert to their respective values in the undamaged state, where and Finally, we recall that where the is the actual equivalent eigenstrain representing the effect of damage by local debonding on the average local stresses and strains and on the overall strain in the composite aggregate. A similar decomposition of small strains can be introduced in the matrix, where the total eigenstrain may consist of thermal and inelastic components. Since the local stresses (10) and strains (12) depend only on the total transformation strains in the respective volumes, such decompositions do not alter the above results. Of course, a somewhat different form of the governing equations would result if inelastic strains or other local eigenstrains were denned as functions of the local stress history (Chaboche et al. 2000).

2.3.

Energy changes

Progressive debonding of the reinforcements from the matrix under either changing or constant overall stress and local eigenstrains applied to the representative volume involves release of potential energy. For the damage process to proceed, the amount of energy released by partial or complete debonding of each particle or fiber must be equal or exceed the work required for interfacial separation. In the context of the loading and damage process described in the change in total potential energy or Gibbs free energy of the representative volume V is caused by changes in both the total load set and local debonded volume during transition from the current state k to the next state k + 1. This can be recorded as

with the understanding that is applied in the current debonded volume and in and that and are applied in and respectively. However, the energy change that can be attributed to debonding alone depends only

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on changes in the damage-related variables under a constant total load set, selected as This can be written in the form

where the

in the second term is now applied only in

Specific evaluation utilizes the expression

where

is the local elastic strain field and the surface integral is

The local stress and strain components in the first integral (19) are taken from (10) and (11), and the overall strain in the second integral from (14), with values of the respective parameters indicated in (18). The actual volume of the reinforcing particles or fibers that experience debonding in the transition from current state k to the next state k + 1, is reflected in the evaluation of the energy change by substituting in the fields (10), (11) that enter the integrals (18)–(20).

2.4.

Damage evolution

Two distinct conditions need to be satisfied for interfacial decohesion to proceed. Local stress at the interface of a bonded fiber or particle has to reach a certain minimum level needed to nucleate an interfacial crack. For this crack to propagate, the rate of energy release must exceed the interfacial toughness. Both complete and partial debonding with asperity contacts require potential energy release equal or greater than the total surface energy dissipated by interface decohesion. Specific forms of the two criteria depend on the material system considered and on the debonding model employed in evaluation of (8). For example, a stress-based debonding criterion that estimates the volume fraction of debonded reinforcements as a function of the local stresses can be selected as a Weibull distribution

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where, depending on the available resolution of the local fields, can be a function of the tension or shear stress maxima or invariants,

such as the isotropic tension component. The

is the value of

at

initial debonding, which may be detected experimentally. The p and q

are scale and shape parameters; no damage occurs but it becomes widespread

= 0) for

when the second term in (21)

tends to zero. In principle, the in (21) can be found as a function of using (13) and (14), from unloading compliances measured in an experiment under incrementally increasing overall stress. In the decohesion models that estimate the extent of local debonding

as a function of applied stress, the energy criterion is typically satisfied during the entire separation process. In the imperfect interface models that estimate only the average stress in the debonding inhomogeneity, a simple criterion can be written as

where

taken from (18)–(20) is an estimate of the total energy

released by partial or complete decohesion of the surface area one or more particles or fibers in the same loading step, and

of is the

interface toughness. Under constant overall stress and typical reinforcement volume, the energy released by debonding of each particle or fiber increases as the

growing number of debonds reduces overall stiffness. Also, the average stress in the stiffer and still bonded reinforcements increases with increasing porosity of the matrix. Therefore, in agreement with observations, the present damage process model suggests acceleration of the

debond rate with increasing volume fraction of debonded reinforcements. An iterative solution procedure under overall stress control starts with (i) selecting trial values of the next load increments and (ii) These are added to the current values at step k, and the updated totals substituted into (10), which yield estimates of the

updated local stress averages, (iii) A new value of is then found from (21) and the increment found by subtracting The is rounded to correspond to debonding an integer number of reinforcement volumes, (iv) The energy criterion (22) is examined for compliance,

and the increments (i) are adjusted as needed for debonding of a preselected number of reinforcements, (v) The debonding model may yield an updated matrix for the new value of in (8). (vi) Outcomes of steps (iii) to (v), substituted into (10), yield updated local stress aver-

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ages, (vii) Steps (iii) to (vi) are repeated until selected error criteria are satisfied by the updated local stress and magnitudes. Under overall strain control, one selects the increments of and and finds from (14)–(16). The updated local stress and Cd values are then found by retracing (ii)–(v). Next, the strains (13)–(16) are updated and adjusted as required by the selected Again, these iterations continue until selected error criteria are satisfied, as in (vii).

3.

DAMAGE CONTROL BY FIBER PRESTRESS

Damage in fiber composites can be regarded, in part, as a constrained fracture or failure process in the matrix, impeded by undamaged fibers that bridge the prospective cracks and thus limit their extension. Another part of the process involves tensile or compressive fiber failure that can be contained to some extent by the surrounding matrix or plies. In both cases, damage can be prevented or retarded by redistribution of internal stresses induced by fiber prestress, applied prior to and released after matrix consolidation. Prestressing of structural components for improvement of internal stress distributions and overall deflections is widely used in concrete structures, and to a lesser degree in steel structures. However, prestressing of fibers in composite plies and laminates, which offers a similar method for stress redistribution, has been apparently used only in reduction of fiber waviness for enhancement of ply compressive strength. Since the forces needed to apply significant stresses to fiber tows are relatively small, within the capacity of conventional equipment used in filament winding or fiber placement, fiber prestress could be used in many composite structures for damage retardation or control. Examples of several such applications that have been recently identified are briefly described below. In all cases studied, this method of damage control by fiber prestress relies on creating fixed residual stress states that keep local ply stresses within allowable ranges in composite structures exposed to prescribed thermomechanical loading programs. One application of fiber prestress in damage control has been developed for laminated plates, where release of prestress after matrix consolidation is equivalent to application of fixed compressive stresses in the fiber direction of each ply (Dvorak and Suvorov 2000). Analysis of these problems involves application of available laminated plate theories, with the released prestress acting as fixed load, in superposition with applied thermomechanical loads. An example of the prestress effect on estimated onset of damage is shown in Fig. 1. A 9-layer

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S-glass/epoxy laminate is loaded by biaxial normal stresses and with the 0° fibers aligned in the x direction. Initial damage envelopes are constructed from interior branches that reflect ply strength magnitudes estimated by the critical stress criterion on the basis of experimentally measured longitudinal and transverse tensile or compressive and shear strengths of the individual plies. Application of different fiber prestress magnitudes to all plies causes mostly rigid body translation of the envelope in the direction of the prestress force resultant, although different ply strength branches may assume the interior position at different levels of prestress. The outcome is reminiscent of the shakedown effect on positions of loading surfaces of laminates with

kinematically hardening matrices. In a more recent study, we have developed a procedure for evaluation of fiber prestress distributions in individual plies that reduce to prescribed strength ranges the free edge stress concentrations caused in homogenized plies by thermal changes and variable mechanical loads. Also, the procedure accommodates the average stresses in the laminate interior within translated initial damage envelopes. The useful fiber prestress magnitudes are typically limited by the requirement that ply com-

Figure 1 Effect of fiber prestress applied in all plies, and cooling by and position of initial damage envelopes for the 9-ply laminate.

–150°C, on shape S-glass/epoxy

pressive strengths are not exceeded after mechanical unloading, which

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implies that the damage envelopes must contain the origin of overall stress coordinates (Fig. 1). These values are much lower than typical fiber strengths, hence the effect of prestress on fiber stresses is relatively small, especially in systems with compliant matrices. As an example, consider a laminated plate of any layup that is subjected to a uniform thermal change applied during cooling from the fabrication temperature. In-plane moduli and CTEs of all plies need to be identical, and as is the case in most composites, the expansion coefficient of the matrix is assumed to be larger than that of the fiber; hence during cooling a free ply contracts more in the transverse direction. This allows application of fiber prestress that cancels the thermal stress concentrations at the free edges of the laminate, and thus creates a uniform stress and strain field in each ply. To evaluate the required prestress magnitude, the plies are separated, and each ply is subjected to the uniform, stress-free thermal strain. In general, these strains are different in the plane of each ply, so the laminate cannot be reassembled; however, the plies remain plane and parallel to each other. Next, suppose that a certain fiber prestress has been released; this is a product of the fiber volume fraction and tensile stress applied to

the fibers in the ith ply prior to matrix consolidation; the local coordinate system in each homogenized ply is selected such that the fibers are aligned with the direction in the plane of the ply. The prestress magnitude is selected such that after its release, and in superposition with the thermal strain, the elastic strains create an isotropic in-plane strain

where the are ply compliances and the linear coefficients of thermal expansion. The required prestress value in each ply is thus found as

and the uniform strain in all plies after cooling and prestress release as

Note that

the assumption is that > 0 and < 0. Since all plies are deformed by the same isotropic in-plane strain, and are free of tractions, the laminate can be reassembled. The strain (25) is preserved through laminate thickness. Note that neither this strain nor the prestress (24) depends on ply orientation or laminate layup. Of course, local concentrations at the free fiber ends are not eliminated by the prestress.

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CONCLUSIONS

Space limitations prevent description of specific applications of the damage evolution model, of its extension to inelastic composites, and of implications for stability of the damage process. These and other subjects will be described elsewhere. Many other fiber prestress applications in preventing initiation of both distributed damage and free edge debonding also await exposure in a different setting. However, the examples shown should provide some insight into the two research trends that open the way both to a better understanding of distributed damage evolution in heterogeneous systems, and to active control and prevention of damage in laminated composite structures.

Acknowledgments This work supported by a grant from the Army Research Office; Dr. Mohammed Zikry served as program monitor. The author appreciates contributions by Mr.

Alexander P. Suvorov and Mr. Jian Zhang, who also kindly produced the manuscript file.

References Achenbach, J. D., and H. Zhu. 1989. Effect of interfacial zone on mechanical behavior and failure of fiber-reinforced composites. Journal of the Mechanics and Physics of Solids 37, 381–393. Chaboche, J. L., S. Kruch, J. F. Maire, and T. Pottier. 2000. Towards a micromechanics based inelastic and damage modeling of composites. International Journal of Plasticity, to appear. Chen, T., G. J. Dvorak, and Y. Benveniste. 1990. Stress fields in composites reinforced by coated cylindrically orthotropic fibers. Mechanics of Materials 9, 17–32. Christensen, R. N., and K. H. Lo. 1979. Solutions for effective properties of composite materials. Journal of the Mechanics and Physics of Solids 27, 315–330. Dvorak, G. J. 1990. On uniform fields in heterogeneous media. Proceedings of the Royal Society of London A 431, 89–110. Dvorak, G. J., and Y. Benveniste. 1992. On transformation strains and uniform fields

in multiphase elastic media. Proceedings of the Royal Society of London A 437, 291–310. Dvorak, G. J., and A. P. Suvorov. 2000. Effect of fiber prestress on residual stresses and onset of damage in symmetric laminates. Composite Science and Technology 60, 1129–1139. Furuhashi, R., J. H. Huang, and T. Mura. 1992. Sliding inclusions and inhomogeneities with frictional interfaces. Journal of Applied Mechanics 59, 783–788.

Gao, Z. 1995. A circular inclusion with imperfect interface: Eshelby’s tensor and related problems. Journal of Applied Mechanics 62, 860–866. Hashin, Z. 1991a. The spherical inclusion with imperfect interface. Journal of Applied Mechanics 58, 444–449. Hashin, Z. 1991b. Thermoelastic properties of particulate composites with imperfect interface. Journal of the Mechanics and Physics of Solids 39, 745–762.

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Hill, R. 1965. Continuum micromechanics of elastoplastic polycrystals. Journal of the Mechanics and Physics of Solids 13, 89–101. Hutchinson, J. W., and H. M. Jensen. 1990. Models of fiber debonding and pullout in brittle composites with friction. Mechanics of Materials 9, 139–163. Karihaloo, B. L., and K. Vishvanathan. 1985. Elastic field of a partially debonded elliptical inhomogeneity in an elastic matrix. Journal of Applied Mechanics 52, 835–840. Needleman, A. 1987. A continuum model for void nucleation by inclusion debonding. Journal of Applied Mechanics 54, 525–531. Steif, P. S., and A. Dollar. 1988. Longitudinal shearing of a weakly bonded fiber composite. Journal of Applied Mechanics 55, 618–623. Tvergaard, V. 1990. Micromechanical modeling of fiber debonding in a metal reinforced by short fibers. In Inelastic Deformation of Composite Materials, G. J. Dvorak (ed.). Berlin: Springer, 99–111.

ELECTROMAGNETIC PHENOMENA IN CRYSTAL GROWTH John S. Walker Department of Mechanical and Industrial Engineering University of Illinois at Urbana-Champaign, USA [email protected]

Abstract

1.

Integrated circuits and optical devices are produced on wafers sliced from single crystals that are grown from a body of liquid semiconductor or melt. The crystal must have few defects, such as dislocations, and must have uniform distributions of dopants, which are added to the melt to give the crystal the desired electrical or optical properties. Since molten semiconductors have large electrical conductivities, magnetic fields can be used to eliminate hydrodynamic instabilities in the melt and to tailor the convective dopant transport. For silicon, controlling the convective transport of oxygen is important, and a particular nonuniform axisymmetric magnetic field is optimal for this purpose. For compound semiconductors, such as gallium-arsenide, eliminating instabilities in the buoyant convection is necessary for the growth of large crystals with few defects.

INTRODUCTION

Computer processors and memory chips, high-speed optical-fiber communications, lasers and light-emitting diodes, wireless communications, electronic controls for automotive systems, and many other new technologies begin with wafers or substrates sliced from single crystals of semiconductors. These crystals are grown from a body of liquid or melt, and the crystal properties needed for each application depend on the melt motion during crystal growth. Dramatic advances in these new technologies will require radical improvements in the quality of semiconductor crystals through melt-motion control. Most molten semiconductors behave as liquid metals: their electrical and thermal conductivities are large and their viscosities are small. With the large electrical conductivity, (1) a steady magnetic field can be used to eliminate deleterious hydrodynamic instabilities and to control the steady laminar melt motion, or (2) a periodic magnetic field can be used to drive a beneficial melt motion.

211 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 211–224. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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While some effects of the melt motion are common to all semiconductors, other effects are important only for certain materials. There are three types of semiconductor crystals: single-material, compound and solid-solution. A single-material semiconductor crystal consists of an element with four valence electrons, such as silicon or germanium. Silicon is currently the basis of computer processors and memory chips. Single crystals with diameters of 30 cm and lengths of 1 m are currently produced commercially by the Czochralski process. Each crystal yields hundreds of substrates, and many integrated circuits are produced on each substrate. Dopants with three valence electrons, such as boron, may be added to the melt in order to create a missing electron or hole at each location of a dopant atom in the crystal. An electron can jump to this hole, so these dopants are called acceptors and produce a p-type semiconductor. Alternately dopants with five valence electrons, such as phosphorus, may be added to the melt to create an extra electron at the location of each dopant atom in the crystal. Such dopants are called donors and produce an n-type semiconductor. With adjacent pieces of n-type and p-type, a small voltage can cause the donor electrons to jump to the holes or back. If the substrate is n-type, sections of the surface are converted to p-type by various processes to create transistors. Other sections of the surface are coated with metal to carry electrical signals between the transistors or are converted to SiO2, which serves as an electrical insulator. Currently a silicon-based integrated circuit contains about 8 million transistors, while the next generation will contain about 240 million on the same area. With smaller transistors, the time for signals to travel between transistors is reduced, making the computer faster. Optical applications, such as optical-fiber data transmission, optical data storage and retrival, other applications of lasers, light-emitting diodes and photodetectors, are based on compound or solid-solution semiconductor crystals because silicon lacks the optoelectronic properties required for these applications. Compound semiconductors involve equal amounts of elements with three and five valence electrons (IIIV’s), such as gallium-arsenide (GaAs) and indium-phosphide (InP), or of elements with two and six valence electrons (II-VI’s), such as mercurytelluride (HgTe) or cadmium-telluride (CdTe). The elements in compound semiconductors maintain their molecular bond in the liquid state as long as the temperature is below the dissociation temperature, which is a different function of pressure for each material combination. Therefore compound semiconductors grow like single-material semiconductors. On the other hand, solid-solution crystals are grown from liquid mixtures of unassociated elements or molecules. Examples include

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95% silicon and 5% germanium (SiGe), 95% germanium and 5% silicon (GeSi), and 80% mercury-telluride and 20% cadmium-telluride (HgCdTe or Hg0.8Cd0.2Te). Since the elements are not bound in the melt, the melt motion must be controlled so that the convective mass transport during crystal growth produces an axially and radially uniform distribution of both species in the crystal. Compound and solid-solution crystals also provide the substrates for the integrated circuits in cell phones and other wireless communication devices, and for high-temperature applications. For example, on automotive engines, silicon-based sensors and control circuits require expensive cooling, while GaAs-based devices operate well at high temperature and require no cooling. It is much more difficult to produce compound and solid-solution crystals with uniform dopant and species distributions, and with very few dislocations or other crystal defects than it is for silicon crystals. The poorer quality of these crystals places severe limits on many important technologies. Ten years ago, GaAs was expected to replace silicon in the fastest supercomputers because an integrated circuit on a GaAs substrate would perform at roughly ten times the speed of the same circuit on a silicon substrate. However, the same integrated circuit cannot be produced on both materials because of nonuniformities and defects in current GaAs crystals. At about the time silicon-based integrated circuits reach 240 million transistors, GaAs-based integrated circuits may reach 1.5 million transistors. With the shorter distance between transistors, silicon-based computer chips remain faster than GaAs-based ones. However, silicon cannot be used for the technologies which are currently revolutionizing communications, so that there is a great need to improve the quality of compound and solid-solution crystals. Without flow control, hydrodynamic instabilities lead to unsteady melt motions in virtually all crystal-growth processes. In the Czochralski growth of 30 cm diameter silicon crystals, the initial melt volume is very large and the temperature difference in the melt is 50–100 K, so that the uncontrolled buoyant convection is turbulent. For the smaller melt volumes involved in the growth of compound and solid-solution crystals, the uncontrolled melt motion is generally not turbulent, but it is cer-

tainly unsteady and involves enough different frequencies that the effects of melt-motion oscillations appear to be rather chaotic. The convective heat transfer associated with unsteady melt motions produces fluctuations in the heat flux from the melt to the crystal-growth interface, and these heat-flux fluctuations produce temporal oscillations in the solidification rate. Oscillations in the solidification rate: (1) are a major cause of dislocations in the crystal (Kuroda et al. 1984) and (2) produce spatial oscillations of the dopant or species concentration in the crystal

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(Garandet 1993, Wilson 1980). Most dopants are rejected into the melt during solidification, and the amount of excess dopant in the melt adjacent to the growth interface depends on the balance between the rate of rejection and the rate of diffusion through the melt. The part of the crystal grown during an increase in the heat flux from the melt has a lower dopant concentration because the rejection rate decreases with the solidification rate, so that diffusion gains in the balance and carries more rejected dopant away from the interface. The part of the crystal grown during a decrease in the heat flux from the melt has a higher dopant concentration because the rejection rate increases with the solidification rate, so that rejection gains in the balance and less rejected dopant can escape by diffusion before being solidified. When a section of the crystal is etched, the spatial oscillations in dopant or species concentration appear as lighter and darker bands, so they are called striations. Both the wavelength and amplitude in these oscillations often appear to be rather chaotic, due to the superposition of several frequencies in the periodic laminar melt motion. Without melt-motion control, a local maximum dopant concentration may be twice the local minimum, while an average distance between adjacent maxima on an etched section is generally a few micrometers. When there were only a few thousand transistors in an integrated circuit, each transistor spanned many striations, so that its function was determined by a local average and striations were not a major problem. Now each transistor is so small that it may

fall on a maximum or minimum in the local striation, and this can drastically alter its performance, which depends on the density of donors or acceptors. When a steady magnetic field is applied during crystal growth, any motion of the electrically conducting melt across the magnetic field produces an electric current, which interacts with the magnetic field to produce an electromagnetic (EM) body force opposing the melt motion. A sufficiently strong steady magnetic field can eliminate hydrodynamic instabilities and can thus eliminate the striations produced by unsteady melt motions. In addition a properly tailored steady magnetic field may control the convective mass transport in the melt in order to produce axially and radially uniform dopant and species distributions in the crystal. While a steady magnetic field damps the melt motion, a periodic magnetic field may produce an EM body force that drives a melt motion. With a cylindrical melt, a rotating magnetic field (RMF) produces an azimuthal EM body force, while a travelling magnetic field (TMF) produces a meridional EM body force with radial and axial components. In the next two sections we discuss some applications of steady and periodic magnetic fields.

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STEADY MAGNETIC FIELDS

When the Sony Corporation obtained a patent on the Czochralski growth of silicon with a steady magnetic field in 1980, there was a general hope that any strong steady magnetic field would produce wonderful crystals with no striations and very low dislocation densities. Experiments revealed that the magnetic damping that eliminates the hydrodynamic instabilities may also produce undesirable effects in the crystal. There are two types of deleterious effects of a steady magnetic field. First a magnetic field may produce a deviation from axisymmetry in the melt motion, temperature, and dopant distribution, or it may amplify the effects of an intrinsic deviation from axisymmetry. Since the crystal is rotated about its vertical centerline, a steady nonaxisymmetric temperature distribution leads to temporal oscillations in the solidification rate at a point on the rotating crystal-growth interface, and thus leads to striations for a steady melt motion. Since one or two striations are produced with each revolution of the crystal, the spacing between light and dark bands in an etched section of the crystal is very uniform, and these striations are called rotational striations. The second type of deleterious effect from a steady magnetic field is that it may damp a beneficial part of the melt motion more that it damps a deleterious part, and thus leads to radial or axial variations in the dopant or species concentration in the crystals, called radial or axial macrosegregation, respectively. For examples of both types of deleterious effects and the tailoring of the steady magnetic field to achieve the beneficial elimination of hydrodynamic instabilities without any negative effects, we focus on the Czochralski (CZ) growth of silicon crystals. In the CZ process, the melt is contained in an open cylindrical crucible. Crystal growth begins when a single-crystal seed touches the center of the melt’s free surface. The diameter is increased from the seed diameter to the desired crystal diameter, and then the crystal is moved vertically upward so that the diameter remains constant and so that the growth-interface remains close to the horizontal plane of the free surface. The unique aspect of the CZ growth of silicon crystals is oxygen contamination because the melt at more than 1700 K dissolves the quartz (SiO2) crucible. To achieve acceptably low oxygen concentrations in the crystal, 95–98% of the oxygen from the crucible must evaporate as SiO from the free surface. The buoyant convection involves upward flow near the vertical crucible wall, radially inward flow beneath the free surface, downward flow beneath the growth interface, and radially outward flow near the crucible bottom. The melt accumulates oxygen as it travels outward along the crucible bottom and upward along the vertical cru-

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cible wall. Since the diffusion coefficient for oxygen in molten silicon is very small, namely 10–8 m2/s, the oxygen-rich melt must flow very close

to the free surface in order to lose enough oxygen to keep the crystal concentration low. The thermocapillary convection due to the increase in the surface tension of the free surface from a minimum at the hot vertical crucible wall to a maximum at the colder periphery of the crystal-growth

interface is very important because it pulls the radially inward streamlines very close to the free surface. If the crystal is not rotated about its vertical centerline, there is much more oxygen at the center of the crys-

tal than at its periphery where oxygen concentrations are low because of evaporation from the free surface. Therefore the crystal is rotated at 5–25 rpm so that the axial variation of the azimuthal velocity overwhelms the radially inward buoyant convection and produces a radially outward melt motion over the entire crystal-growth interface. The associated convective mass transport leads to radially uniform oxygen and dopant distributions in the crystal. The first experiments and models for the CZ growth of silicon with a steady magnetic field focused on uniform fields that were either vertical (axial) or horizontal (transverse). Since the EM body force opposes the velocity components perpendicular to the magnetic field but not the

component parallel to the field, a uniform axial magnetic field provides more damping of radial velocities than of axial ones. Since the beneficial thermocapillary convection and radially outward flow due to crystal

rotation are primarily perpendicular to an axial magnetic field, they are strongly damped, while the upward buoyant convection near the vertical

crucible wall is not damped as much and continues to convect oxygen from the crucible to the melt and crystal.

Therefore silicon crystals

grown by the CZ process with an uniform axial magnetic field have no striations, but they have average oxygen concentrations that are several times the maximum acceptable level and they have extreme radial macrosegregation with much higher oxygen and dopant concentrations near the center than at the periphery (Ravishankar et al. 1990). Hurle and Series (1994) estimated that a crystal rotation rate of 700 rpm would be required to produce a radially uniform oxygen distribution in a silicon crystal grown in an axial magnetic field with a magnetic flux density B = 0.2 T, while 30 rpm is roughly the maximum practical rotation rate.

A uniform transverse magnetic field produces less damping of the ther-

mocapillary convection and of the radially outward flow due to crystal rotation and more damping of the upward buoyant convection near the vertical crucible wall. Hence oxygen levels in the crystal are very low and can be adjusted by varying the crucible rotation rate. Unfortunately

any steady uniform transverse magnetic field that is strong enough to

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eliminate hydrodynamic instabilities also produces large deviations from axisymmetry in the melt motion and hence in the temperature, dopant, and oxygen through nonaxisymmetric heat and mass transport. Thus crystals grown with uniform transverse magnetic fields have severe rotational striations (Ravishankar et al. 1990). A certain nonuniform axisymmetric magnetic field called a cusped field was first proposed independently by Series (1989) and by Hirata and Hoshikawa (1989). This field is produced by two identical solenoids placed symmetrically above and below the horizontal plane of the free surface and crystal-growth interface. The two solenoids carry electrical current in opposite azimuthal directions so that their axial magnetic fields cancel at the plane of the free surface and growth interface. Since the local magnetic field is radial, it provides only weak damping of the beneficial thermocapillary convection and radially outward flow due to crystal rotation. The fringing magnetic field lines in the melt involve significant normal components at both the bottom and vertical wall of the crucible, so that the local buoyant convection is strongly damped, which reduces the transport of oxygen from the crucible to the melt and crystal. A cusped field leads to low oxygen concentrations in the crystal, radially uniform oxygen and dopant distributions, and no striations. As the diameter of a commercial silicon crystal increased from 8 cm in 1985 to the current 30 cm, steady magnetic fields became progressively more important for the elimination of turbulence and all other unsteadiness in the melt motion during the CZ growth of silicon. Compound III-V semiconductor crystals are also grown with the CZ process. Two differences from silicon are that: (1) the sizes of the crystal and crucible are much smaller since the diameter and length of a large InP crystal are both 10 cm, and (2) there is a layer of liquid boron oxide in the annular region between the periphery of the crystal and the vertical crucible wall above the melt. This liquid encapsulant prevents the evaporation of the volatile component, i.e., arsenic in GaAs and phosphorus in InP. The melt does not dissolve the quartz crucible so that there is no contamination. The primary concerns are (1) dislocations and other crystal defects and (2) the uniformity of the dopant distribution in the crystal. All of the dopant in the melt before the start of crystal growth ends up in the crystal or in the small amount of melt left in the crucible at the end of the process. Compared with silicon, compound semiconductor crystals are much more prone to the development of dislocations and other defects, such as twinning. Twinning occurs when a part of the crystal begins to grow with a different crystallographic orientation, i.e., one crystal twins into two crystals. For GaAs and InP and other III-V crystals grown by the liquid encapsulated Czochralski

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(LEC) process, most experiments with magnetic fields have focused on the dramatic reduction in dislocation densities, the eliminating of twinning, and the elimination of unsteady melt motions with a uniform axial magnetic field with B = 0.2–0.5 T. Bliss et al. (1993) have developed a process to grow large InP crystals by a slight modification of the LEC process with a uniform axial magnetic field. The crystals have very low and radially uniform dislocation densities and no twinning. Unfortunately many GaAs and InP crystals grown by the LEC process with an axial magnetic field have rotational striations. While a uniform axial magnetic field does not produce any deviations from axisymmetry, it can amplify the deviations from axisymmetry in the melt motion and dopant distribution due to deviations from axisymmetry in the heat flux from the external heater. Many heaters have a gap at one azimuthal position where the leads are connected and there is less heat flux at this location. With induction heating, a slight eccentricity between the induction coil and the graphite susceptor holding the crucible can lead to a large deviation from axisymmetry in the heat flux into the melt. Without a magnetic field, crucible rotation at a few rpm is considered sufficient to eliminate the adverse effects of nonaxisymmetric heaters. For the purposes of explanation we split the three-dimensional buoyant convection in the melt into an axisymmetric convection, i.e., the azimuthal average, and a nonaxisymmetric convection, i.e., the threedimensional convection minus its azimuthal average. A uniform axial magnetic field produces strong damping of the axisymmetric convection because the radial components of this motion produce azimuthal electric currents, which in turn produce large EM body forces opposing the radial motion. The nonaxisymmetric convection can involve upward and downward flows near the hotter and colder azimuthal locations, respectively, and large azimuthal velocities near the vertical crucible wall from the hot to the cold azimuthal positions in the upper part of the crucible and from cold to hot in the lower part. There is very little EM damping of this nonaxisymmetric convection because the upward and downward flows are parallel to the magnetic field, and the electrically insulating vertical crucible wall blocks the radial electric currents that would produce the EM body force opposing the large azimuthal velocities near this wall. Therefore, with a uniform axial magnetic field, a small deviation from axisymmetry in the heat flux to the melt can lead to a large deviation from axisymmetry in the magnetically damped melt motion, leading in turn to rotational striations through nonaxisymmetric dopant transport. It appears that these rotational striations can be eliminated by insuring that the heat input is axisymmetric.

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Most solid-solution crystals are grown by the vertical Bridgman process, which uses a three-zone furnace with a vertical centerline. The top hot zone has a temperature above the solidification temperature, and the bottom cold zone has a temperature well below the solidification temperature. The middle zone is often thermally insulated so that it carries an axial heat flux from the hot zone to the cold zone. The two species and any dopant are contained in a sealed cylindrical ampoule which is initially moved along the furnace centerline into the hot zone until all the semiconductor material is melted except the single-crystal seed at the bottom of the ampoule. Then the ampoule is moved slowly downward through the middle adiabatic zone and into the cold zone, so that the crystal grows upward through the ampoule from the seed at the bottom. A conical insert guides the increase in diameter from that of the seed to that of the ampoule. During solidification, the heavier of the two species is generally rejected into the melt, i.e., Ge in GeSi or SiGe and HgTe in HgCdTe. Since the difference between the thermal conductivities of the melt and crystal leads to a crystal-growth interface that is concave into the crystal, the heavier rejected species sinks to the lower center of the growth interface. Since the solidification temperature decreases as the concentration of the heavier species increases, the growth interface becomes more concave into the crystal with a higher solidification temperature at the periphery than at the center. The increase in the slope of the growth interface accelerates the sinking of the rejected heavier species. As a result, almost all solid-solution crystals grown without a magnetic field have severe radial macrosegregation with much more of the heavier species at the center and much more of the lighter species at the periphery. Watring and Lehoczky (1996) and Becla et al. (1992) grew solid-solution crystals by the vertical Bridgman process with very strong axial magnetic fields with B = 4–5 T. Such a strong field provides enough EM damping of the radially inward melt motion with the higher concentration of the heavier species that the rejected species is solidified before there is significant radial migration. There was far less radial macrosegregation in these crystals than in crystals grown by the

same process without a magnetic field. While good solid-solution crystals are harder to produce than good compound semiconductor crystals, the potential advantages of solid-solution crystals are great. Each compound semiconductor has a limit on how much its properties can be changed by varying dopant concentrations, while the properties of solidsolution crystals can be varied much more by changing the percentages of the two species. The application of strong steady magnetic fields during the vertical Bridgman growth of solid-solution crystals appears to be the key to achieving both radially and axially uniform compositions and

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to reducing the number of dislocations, but far more research is needed to realize the potential benefits of magnetic damping for solid-solution crystals.

3.

PERIODIC MAGNETIC FIELDS

A rotating magnetic field (RMF) is produced by connecting the successive phases of a multiphase AC power source to a number of inductors at equally spaced azimuthal positions around a crystal-growth process. An RMF is a transverse magnetic field with an essentially fixed spatial pattern that rotates about the centerline of the process with an angular velocity which depends on the frequency of the AC power source and the ratio of the number of inductors to the number of phases in the power source. Davidson and colleagues (1987, 1992, 1995, 1999) have defined the fundamental phenomena in the melt motions driven by an RMF. The characteristic ratio of the induced magnetic field produced by the electric currents in the melt to the applied magnetic field produced by the external inductors is the shielding parameter where and are the magnetic permeability and electrical conductivity of the melt, while R is the melt radius, i.e., the inside radius of the crucible or ampoule. For a 50 or 60 Hz power source, the radius of a vertical Bridgman ampoule is small enough that is small and the induced magnetic field can be neglected. On the other hand, for the CZ growth of 30 cm diameter silicon crystals, the crucible radius is large, so that is not small, and the induced magnetic field is important. Two other important parameters are the magnetic Taylor number, and the interaction parameter, where and are the density and dynamic viscosity of the melt, while B is the characteristic magnetic flux density of the RMF. For all current applications of RMFs in crystal-growth processes, the magnetic fields are very weak, B = 1–10 mT, the values of N are extremely small, and the values of are large. An important combination of these two parameters is which is small for all current crystal-growth applications. The first consequence of the assumption that 1 is that the induced electric field due to the melt motion across the instantaneous magnetic field is negligible in Ohm’s law. As a result, the electric current, electric field, and magnetic field are independent of the melt motion, and their solutions determine the EM body force on the melt. Therefore the melt motion is not a magnetohydrodynamic flow with an intrinsic coupling between the electric current and the velocity. Instead it is an ordinary hydrodynamic flow that is driven in part by a known body force. The EM body force consists of a steady axisymmetric

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body force, plus a force that varies as where t is time and is the azimuthal coordinate. The second consequence of the assumption that is that the temporally and azimuthally periodic part of the EM body force can be neglected, so that the only important part of the EM body force is steady and axisymmetric. For the only non-zero component of this force is in the azimuthal direction. Since the magnetic flux density in an RMF is very small, the RMF has no direct effect on hydrodynamic instabilities. Since an RMF creates an azimuthal flow in the melt, there is a stabilizing effect through the axial variations of the centripetal acceleration created by the convection of angular momentum due to the radially inward and outward flows in the buoyant or thermocapillary convection. Experiments have indicated that this may increase the frequency of the periodic melt motions, so that the distance between striations and the amplitude of the dopant concentration variations in the striations both become smaller, which is certainly beneficial (Dold and Benz 1999). If an RMF is applied and the crystal is not rotated, the axial variation of the centripetal acceleration drives a radially inward melt motion near the crystal-growth interface, which is generally undesirable. This is easily corrected by rotating the crystal in one azimuthal direction and applying the RMF in the opposite azimuthal direction. For the vertical Bridgman process, this involves rotating the ampoule about its vertical centerline. An RMF for drives an azimuthal melt motion, and the axial variations of the centripetal acceleration drive the radial and axial velocities. For which is almost always true for crystal-growth applications, the azimuthal velocity is nearly independent of the axial coordinate except inside Ekman or Bödewadt layers, so that meridional motion due to the RMF becomes relatively weak. A traveling magnetic field (TMF) is also a periodic magnetic field, but it is created by connecting the successive phases of the multiphase AC power source to inductors at successively higher or lower axial locations outside the ampoule or crucible. The assumptions for a TMF are very similar to those for an RMF, but now the steady axisymmetric part of the EM body force is concentrated near the periphery of the melt and is axially upward or downward. Therefore a TMF drives a meridional motion directly and does not create any azimuthal melt motion. The use of TMFs in crystalgrowth processes has only recently been proposed.

4.

CONCLUSIONS

Since molten semiconductors are good electrical conductors, a magnetic field can be used to improve the quality of semiconductor crys-

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tals through electromagnetic control of the melt motion during crystal growth or by driving a beneficial melt motion. Different uses involve

strengths from 0.001 T to 5 T, steady and periodic fields, and uniform and nonuniform fields. The spatial variation of the strength and direction of a magnetic field can be tailored in many different ways through the placement of coils, ferromagnetic materials, and shields. More experimental and modeling studies are needed in order to further optimize the benefits of magnetic fields in various crystal-growth processes.

Acknowledgment The author’s research is supported by the National Aeronautics and Space Administration under Grant NAG8-1453.

References Becla, P., J. C. Han, and S. Motakef. 1992. Application of strong vertical magnetic fields to growth of II-VI pseudo-binary alloys: HgMnTe. Journal of Crystal Growth

121, 394–398. Bliss, D. F . , R. M. Hilton, and J. A. Adamski. 1993. MLEK crystal growth of large diameter (100) indium phosphide. Journal of Crystal Growth 128, 451-445. Davidson, P. A., and J. C. R. Hunt. 1987. Swirling recirculating flow in a liquid-metal column generated by a rotating magnetic field. Journal of Fluid Mechanics 185,

67–106. Davidson, P. A. 1992. Swirling flow in an axisymmetric cavity of arbitrary profile,

driven by a rotating magnetic field. Journal of Fluid Mechanics 245, 669–699. Davidson, P. A., D. J. Short, and D. Kinnear. 1995. The role of Ekman pumping in confined electromagnetically driven flows. European Journal of Mechanics

B/Fluids 14, 795–821. Davidson, P. A., D. Kinnear, R. J. Lingwood, D. J. Short, and X. He. 1999. The role of Ekman pumping and the dominance of swirl in confined flows driven by Lorentz forces. European Journal of Mechanics B/Fluids 18, 693–711. Dold, P., and K. W. Benz. 1999. Rotating magnetic fields: Fluid flow and crystal growth applications. Progress in Crystal Growth and Characterization of Materials 38, 7–38. Garandet, J. P. 1993. Microsegregation in crystal growth from a melt: An analytical approach. Journal of Crystal Growth 131, 431–438. Hirata, H., and K. Hoshikawa. 1989. Silicon crystal growth in a cusp magnetic field. Journal of Crystal Growth 96, 747–755. Hurle, D. T. J., and R. W. Series. 1994. Use of a magnetic field in melt growth. In Handbook of Crystal Growth 2, Bulk Crystal Growth. Amsterdam: North-Holland, 259–285. Kuroda, E., H. Kozuka, and Y. Takano. 1984. The effect of temperature oscillations at the growth interface on crystal perfection. Journal of Crystal Growth 68, 613–623. Ravishankar, P. S., T. T. Braggins, and R. N. Thomas. 1990. Impurities in commercialscale magnetic Czochralski silicon: Axial versus transverse magnetic fields. Journal of Crystal Growth 104, 617–628.

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Series, R. W. 1989. Effect of a shaped magnetic field on Czochralski silicon growth. Journal of Crystal Growth 97, 92–98. Watring, D. A., and S. L. Lehoczky. 1996. Magnetohydrodynamic damping of convection during vertical Bridgman–Stockbarger growth of HgCdTe. Journal of Crystal

Growth 167, 478–487. Wilson, L. O. 1980. The effect of fluctuating growth rates on segregation in crystals grown from the melt, I. No backmelting, II. Backmelting. Journal of Crystal Growth 48, 435–458.

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Between sessions, ICTAM 2000 participants mingle while enjoying coffee and pastries served by Chicago Marriott Downtown wait staff. The Congress filled three

floors of the conference hotel.

SOFTWARE TOOLS: FROM MULTIBODY SYSTEM ANALYSIS TO VEHICLE SYSTEM DYNAMICS Willi Kortüm DLR German Aerospace Center, Institute of Aeroelasticity, Wessling, Germany [email protected]

Werner O. Schiehlen Institute B of Mechanics, University of Stuttgart, Stuttgart, Germany [email protected]

Martin Arnold DLR German Aerospace Center, Institute of Aeroelasticity, Wessling, Germany [email protected] Abstract

1.

After successful application to spacecraft and appropriate theoretical examinations, multibody system (MBS) approaches, their formalisms, and software became of interest to the vehicle system dynamicists both for rail and road vehicles. This introductory paper sketches a few important milestones in the MBS general development related to vehicle system dynamics. While at the beginning the absence of system-specific force laws was the major stumbling block, later on the numerical methods and the efficiency of the formalisms became of prime focus. More recently, the transition from system analysis to system design and optimization, as well as the integration into multidisciplinary computer aided engineering (CAE) of vehicle systems, was and still is a challenge.

MULTIBODY DYNAMICS—THE BEGINNING

The origin of modern multibody dynamics is closely related to the beginning of the space age in the ’50s and ’60s of the 20th century. The development of such space vehicles as spacecraft and orbiting satellites was essentially based on a computer-aided dynamic analysis, in particular for the rotational behavior, see e.g. [1]. To this end multibody systems were reconsidered from a more algorithmic point of view that took into account the increasing efficiency of computers. 225 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 225–238. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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Since then, the term multibody system (MBS) has been used more precisely for a system of a finite number of rigid or flexible bodies and their interconnections. It is supposed that the mass of the system is concentrated in the bodies that are connected by such idealized massless elements as joints, springs, dampers, and actuators [2]. The state of an MBS is described by a finite-dimensional vector q(t) of position coordinates and the corresponding velocities v(t). Based on principles of classical mechanics, such as d’Alembert’s principle, the equations of motion are derived in a systematic way [3]. For a single body these equations may be written down straightforwardly, but for complex MBS the derivation becomes technically very complicated and is most conveniently performed by computers using

multibody formalisms. These formalisms are algorithms to generate the equations of motion for a user-specified MBS in a form that allows an efficient numerical solution. The multibody system approach, multibody formalisms, and the derived software packages became a powerful basis for the kinematic and dynamic analysis of mechanisms. Shortly after the first applications in space technology and beyond its prime academic appeal, multibody dynamics was found to be useful also in vehicle system dynamics both for rail (including magnetically levitated) and road vehicles. This introductory paper sketches a few important milestones in the development of multibody dynamics in general, as well as their particular evolution in the vehicle system dynamics area. In the final part of the paper some new challenges are summarized, which result from the concurrent engineering approach to vehicle system dynamics software.

2.

MULTIBODY SYSTEMS AND VEHICLE SYSTEM DYNAMICS

Essential contributions to multibody dynamics and to the efficient numerical solution of the equations of motion were made during and after the late 1980s—see 3. Nevertheless, as described by two of the authors in a first survey paper in 1985 [4], the MBS approach had already been used successfully in vehicle system dynamics much earlier. All these early approaches were adapted to the restricted computing power in the early 1980s; they were not designed to handle fully nonlinear spatial MBS models with loop-closing constraints that represent today’s stateof-the-art. A typical representative of these early attempts is the software package MEDYNA developed at DLR [5]. It is based on kinematically linearized equations of motion, a restriction that was motivated by the applica-

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tions to rail-guided vehicle system dynamics that includes both rail and magnetically levitated vehicles. Ten years later this type of software reached a deadlock because of the inherently linearized kinematics, but in the beginning this restriction was very helpful since the implementation was substantially simpler compared with a fully nonlinear formalism and furthermore the handling of closed loops in the MBS was strongly simplified. Despite its shortcomings, MEDYNA met a number of nontrivial needs that are typical of applications in vehicle system dynamics: flexible bodies were considered on the basis of a modal approach; the package offered reasonable system specific library elements for force laws and joints (wheel–rail interaction, magnetic levitation force models, tire models); and there was a basic interface to other simulation packages, such as ACSL. Furthermore, MEDYNA offered from the early beginning not only solvers for (nonlinear) time integration but also linear system analysis methods in the time and frequency domain that are especially important for applications in vehicle system dynamics. MEDYNA and other early commercial MBS packages are the starting point of an ongoing development that made MBS software one of the basic analysis tools of modern vehicle system dynamics. This application

to real-life vehicle system dynamics is a complex issue that has been addressed in a number of surveys and reviews, which were necessarily supported by extensive benchmarking activities—see e.g. [6, 7]. There are basic requirements on general-purpose MBS software, such as MBS modelling elements, typical modes of analysis, robust and efficient solvers, software aspects, and user-friendly interfaces. Nevertheless, the application to industrial vehicle dynamic simulation definitely requires a number of additional qualities and functionalities, such as:

a) Additional modeling requirements as, for instance, the consideration of elastic deformations in ride/comfort analysis, tire and wheel–rail contact models, and such other important nonlinear effects as friction, nonlinear and discontinuous force laws, b) Powerful solvers to deal with the high complexity of the models

characterized by many degrees of freedom, highly nonlinear phenomena, stiff equations with or without discontinuities, and control systems that may include digitally sampled data, c) Complex geometries of track and rail profiles (including switches), as well as deterministic, stochastic, and measured irregularities,

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d) The reliability of simulation results that is determined by the status of verification and validation of the models used and of the MBS software itself.

New challenges resulted from the introduction of electronics and control in vehicle systems. Control circuits had to be included into the simulation software, and engineers had to be enabled to work in an MBS simulation environment as well as in a control system analysis and design environment using computer-aided control system design

(CACSD) [8, 9, 10].

3.

ADVANCES IN MULTIBODY SYSTEM THEORY

In parallel with the progress in vehicle system dynamics and its associated simulation software, the basic problems of multibody systems as a topic in applied and computational mechanics attracted and motivated quite a number of research-oriented mechanical engineers as well as applied mathematicians. Also specific problems from such other fields

as robotics, biomechanics, and mechanisms stimulated research groups in MBS. The results of these early efforts and the state of the MBS software were first discussed in a 1977 IUTAM Symposium in Munich [11] and then during the 1985 lUTAM/IFToMM Symposium in Udine [12]. Finally, in 1990 the known MBS software packages of the 1980s were put together in the Multibody System Handbook [13] with contributions from 17 research groups. The topics discussed within the MBS community ranged from the choice of coordinates (absolute, relative, mixed) and the basic mechanical principles in the derivation of the equations of motion (Newton– Euler, d’Alembert, Jourdain), to the most efficient numerical solution strategies (O(N) formalisms, explicit vs. residual formalisms), to software aspects (analytical vs. numerical representation of the equations of motion, user interfaces, standardization of input data). Many important issues in these areas were examined in the research program “Dynamics of Multibody Systems” of the German Research Council (DFG) [14]. A key problem of multibody dynamics is the handling of joints that connect bodies in the MBS and restrict their relative motion. These joints define the topology of the MBS that forms the backbone of all multibody formalisms. The topology is represented by a graph with N knots representing the N bodies of the MBS. The branches of the graph represent the joints in the MBS: two knots are connected by a branch if and only if the corresponding bodies are connected by a joint.

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If this graph has tree structure (tree-structured MBS), then the translational and rotational degrees of freedom of all joints define a natural set of generalized coordinates q that allows the analytical elimination of all constraints such that the equations of motion take the general form of ordinary differential equations (ODEs)

with the symmetric positive-definite inertia matrix M and the vector f of generalized applied and gyroscopic forces [3]. If Cartesian coordinates q are used instead, then the equations result in differential–algebraic equations (DAEs)

The

constraints generate the constraint forces with (t, q) and Lagrangian multipliers At first sight (2) has a higher dimension and a more complicated structure than (1) but the topology of the MBS implies a characteristic sparsity structure in (2) that may be exploited in the numerical solution

[15]. If the graph of the MBS contains loops (MBS with closed loops), then the generalized coordinates have to be evaluated using complex kinematics while the Cartesian coordinates are subject only to additional constraints—see e.g. [16, 17]. By exploiting the MBS topology, the right-hand sides of (1) and (2) may be evaluated with O(N) arithmetic operations (O(N) formalisms) in contrast to a complexity of O for a standard matrix inversion, which is a characteristic of early multibody formalisms. The numerical solution of DAEs (2) was an unsolved problem in the early days of multibody dynamics, so the explicit or implicit transformation to ODEs according to (1) found much interest in the MBS community. Nowadays the time integration methods for DAEs (2) are as highly developed as the ones for ODEs (1) [18, 19, 20]. Note that these robust DAE integrators require also the evaluation of hidden constraints, such as

so the multibody formalisms have to be extended accordingly. Time integration is the most challenging numerical problem in multibody dynamics; but also linearization, linear system analysis in time and frequency domains, as well as kinematic analysis, static analysis, and inverse dynamic analysis, are of importance [19].

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Besides these algorithmic and numerical improvements, multibody dynamics itself extended its classical approach of rigid body systems to flexible MBS with elastic deformation of bodies by using a modal representation [21, 22]. Furthermore the classical concept of equality constraints in (2) was extended to unilateral constraints, which allow a very natural formulation of contact conditions [23].

4.

FROM SYSTEM ANALYSIS TO SYSTEM DESIGN, FROM MECHANICAL MULTIBODY SYSTEMS TO MECHATRONICS

Traditional vehicle dynamics and MBS simulation concentrated on the analysis and the evaluation of given vehicle systems, perhaps including parameter variation and sensitivity studies of purely mechanical systems. More recently, much work has been done to overcome these limitations, grasping the full system with its multidomain (multiphysical) properties and supporting the optimization of system parameters with respect to performance criteria [24] (simulation-based design or design by simulation). Optimization of MBS models has been studied for more than a decade [25], but even today it is not yet available with all standard industrial MBS packages. A fully automated numerical multi-objective parameter optimization of industrial MBS models is provided by SIMPACK, the general purpose MBS software being developed at DLR [26]. With the advent of mechatronics thinking in vehicle systems—see [27] for a survey of current trends and future possibilities—the request for a higher integration of mechanical and control/electronics software packages became more demanding [28]. A key feature of modern MBS software is therefore an interface to the leading CACSD packages MATLAB/Simulink and Matrixx/System Build. Figure 1 illustrates the advanced interface SIMAT [29] between SIMPACK and MATLAB/Simulink, that provides a simple export of system matrices A, B, C, D, a bidirectional function-call interface, code export and code import facilities, and an advanced co-simulation interface including an option for parallel computing using inter-process communication. Originally, these interfaces were developed for and tailored to the coupling of MBS and CACSD software, but they have been extended to the more general integration of mechanical system properties and other physical domains. Simulator coupling [30] is the method of choice for simulating the interaction of mechanical and hydraulic or pneumatic sys-

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Figure 1 Interface between SIMPACK (MBS) and MATLAB/Simulink (CACSD).

tem components. Typical examples are the interfaces of SIMPACK to AMEsim [31] and DSHplus [32]. Furthermore, DLR is currently developing fluid–structure–MBS interfaces to connect aeroelastic and MBS

models [33]. Last but not least, the importance of computer-aided design (CAD) for vehicle system design in general, and in relation to MBS software in particular, should be mentioned. Computer-aided design not only provides high quality graphical output in 3D, but also—even more important—comprises all geometrical data and such other data as material properties, masses, and inertias. Therefore CAD may be considered as the parent database for such other computations as CFD, FE, and MBS analysis. As a consequence, connections between CAD and MBS have been developed [34]. Industrial MBS packages provide advanced interfaces that consider both views: the MBS view to derive MBS data from the CAD source as well as the CAD engineer’s view, which allows MBS functions to be called from the CAD user interface [35].

5.

CURRENT AND CONCURRENT PROBLEM AREAS—A CRITICAL REVIEW AND OPEN PROBLEMS

Industrial MBS software, such as ADAMS, SIMPACK, and DADS, provides powerful tools for vehicle system dynamics. Nevertheless, from time to time, they leave the frustrated user unsatisfied, both in ordinary

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and high-end applications. Therefore, a critical assessment of current problems may be helpful. Fundamental problems result from the fact that by 1985 the basic conceptions were already established for all the MBS packages that are today extensively used in industry. Later developments in multibody dynamics and MBS numerics had to be added afterwards, and these additions proved to be extremely difficult and tedious to manage if their development had not been foreseen in the conception phase. Other problems result from the rapidly increasing demands and increasing complexity of real-life applications in vehicle system dynamics. MBS formalisms and time integration methods. Today it is well known that a numerically stable and robust time integration of DAE (2) requires the evaluation of hidden constraints, such as (3) [18]. In the early days of multibody dynamics [15], nobody knew about that and the multibody formalisms were not prepared to evaluate (3) during time integration. A later redesign to include the necessary information proved to be nontrivial because of the large set of already existing library elements and user routines. For similar reasons, modern non-stiff time integration methods for (2) are not yet available in industrial MBS packages since they require the separate evaluation of M, f and [36] but the formalisms evaluate M, f and simultaneously. Efficiency. There are mainly two sources for efficiency and low simulation times in MBS software. In the widely used MBS packages

the time integration methods have been adapted to the special structure of (1) and (2) and the equations of motion are evaluated efficiently by O(N) formalisms. For MBS with small displacements, further improvements may be achieved by linearizing the kinematics. For nonlinear problems the O(N) formalism may be extended by a pre-processor that eliminates for a fixed MBS topology all arithmetic operations with vanishing terms. See [37] for a comprehensive overview of new and adapted algorithms in SIMPACK, which is today known among the commercial packages for its high efficiency. Further improvements are necessary to handle MBS models with a

few hundred degrees of freedom, which are more and more often used in industry. Flexible MBS.

Elastic bodies in MBS models may cause major difficulties or even wrong simulation results with MBS packages that

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were originally designed for rigid bodies. Only a few industrial packages have taken into account the elastic deformation from the outset. Typical examples are DADS and SIMPACK, which use a modal approach with eigenmodes as well as static modes. Second-order terms have to be included because of geometrical stiffening effects [21, 22, 37]. Eigenmodes and static modes are obtained in a separate pre-processing step using industrial FE packages and interfaces between FEM and MBS. More recent developments are dealing with the inverse connection from MBS to FE analysis, called LOADS in SIMPACK. The results of MBS simulation (states, forces) are transferred to the FE packages to perform a load analysis including such post-processing evaluations as life prediction [38]. Semi-symbolic equations of motion may be also derived using NEWEUL and FE codes [39].

Open software structures. Generally speaking, earlier software packages were designed as closed systems that compute for given input data the simulation results, without any additional output. By using numerical MBS formalisms to evaluate the equations of motion, there was no need to generate these equations explicitly in source code form. Later it was recognized that open software structures, including code export, facilitate interfaces to other CAE packages and real-time applications. Symbolic MBS formalisms have been developed to generate

FORTRAN or C statements representing the equations of motion. Code export for MBS models has the inherent problem that the MBS topology has to be exploited for an efficient numerical solution. This may be done by using such symbolic formalisms as NEWEUL [40, 41] or the Symbolic Code Module of SIMPACK [37], which eliminate all constraints resulting from tree-spanning joints during the generation of symbolic code. Other packages supplement their exported symbolic code by special sparse-matrix solvers, but this approach is less straightforward and is nontrivial to handle. With the increasing importance of hardware-in-the-loop simulations, there is an increasing interest in efficient symbolic code running in real time in a digital signal processor environment [42]. The robust numerical solution of DAEs (2) in real time is currently a field of active research, since standard DAE solvers use iterative methods to satisfy the constraints 0 = (t, q) but real-time simulation has to be based on noniterative algorithms. Underestimating the mechanical problems. Visualization and graphical user interfaces are definitely important features, since they support users in their daily work and allow a presentation of simula-

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tion results in a form that is also easily understandable for non-experts. But some packages overemphasize the “show” to impress potential customers and seem to neglect the difficulties in the modeling and solution of complex problems. A typical example is the wheel–rail problem with its intrinsic and complex contact geometry and creep-force laws—and the resulting numerical problems. The difficulties caused by the strongly coupled problems of modeling and numerical solution are discussed in [43], the background for the Wheel-Rail Module of SIMPACK, including multiple contact problems and switching maneuvers of railroads. Automotive. The automotive industry is one of the main customers of MBS software. To create accurate and reliable vehicle models quickly and easily, the packages offer special parameterized substructures for such model components as wheel suspensions, tires, and engines. Furthermore the pre- and post-processing is supported by specialized model setup and evaluation tools that allow easy access to standardized driving maneuvers and other typical analysis methods [44]. Without such extensions, an MBS package is of very restricted use for engineers in the automotive industry, and further developments are necessary to meet the demands of these users even better. Open problems and future developments. The solution of the above-mentioned problems, such as robustness, efficiency, and handling of MBS packages, is of prime importance for industrial vehicle system dynamics. But there are also a number of more basic ongoing developments that will definitely have a strong impact on these analysis tools. Multibody system software has been used in industry for more than 15 years. There is a huge knowledge of how to set up MBS models so that they may be analyzed efficiently, how to handle error messages and problems, and how to use these tools most efficiently. This expert knowledge has to be built in. Solver-independent pre- and post-processors, such as MotionView for applications in the automotive industry, make the user independent of the selected MBS package. To support this strategy, standards have to be defined and implemented for all basic MBS modeling elements and data structures. More and more often, engineers use pre-defined substructures to set up complex MBS models. Further improvements in numerical solvers are necessary to guarantee nevertheless acceptable simulation times. The analysis of nonlinear elastic structures with large rigid-body motion is a topic of very active academic research that may result in a much

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closer connection between structural dynamics and multibody dynamics in the near future—see e.g. [45, 46, 47]. Today it is an open question if there will be software that provides the full functionality of FEM as well as MBS software in one and the same industrial package. Simulator coupling is a promising approach for incorporating MBS simulations in a general CAE environment. Standard coupling strategies may, however, reduce the efficiency of time-integration methods; they may even become unstable. Improved numerical coupling strategies and extended co-simulation interfaces are a topic of actual research—see

e.g. [30, 48]. A natural extension of coupled software tools in simulation is a coupling to optimize a complex technical system with respect to some performance criteria, i.e. multidisciplinary design optimization [49, 50].

6.

SUMMARY

The developments in multibody dynamics and vehicle system dynamics have been intertwined for more than 20 years. Today, industrial MBS packages are powerful and important analysis and design tools in vehicle dynamics that are well prepared for the demands of concurrent engineering processes in industry. Further improvement of the software tools requires fundamental research, which is provided by the multibody dynamics community.

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satellite attitude control. Proceedings of the 3rd IFAC Congress, London. [2] Roberson, R. E., and R. Schwertassek. 1988. Dynamics of Multibody Systems. Berlin: Springer–Verlag.

[3] Schiehlen, W. O. 1985. Technische Dynamik. Stuttgart: Teubner Studienbücher.

[4] Kortüm, W., and W. Schiehlen. 1985. General purpose vehicle system dynamics software based on multibody formalisms. Vehicle System Dynamics 14, 229–263.

[5] Duffek, W., C. Führer, W. Schwartz, and O. Wallrapp. 1986. Analysis and simulation of rail and road vehicles with the program MEDYNA. Proceedings of the 9th IAVSD Symposium (O. Nordström, ed.). Amsterdam: Swets and Zeitlinger, 71–85. [6] Kortüm, W., and R. S. Sharp, eds. 1993. Multibody Computer Codes in Vehicle System Dynamics (supplement to Vehicle System Dynamics 22), Amsterdam:

Swets and Zeitlinger. [7] Iwnicki, S., ed. 1999. The Manchester Benchmarks for Rail Vehicle Simulation (supplement to Vehicle System Dynamics 31). Lisse: Swets and Zeitlinger.

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[8] Schiehlen, W. O. 1992. Multibody dynamics software for controlled vehicle simulation. Proceedings of the International Symposium on Advanced Vehicle Control, AVEC’92, 37–42. [9] Grüibel, G., and W. Kortüm. 1992. Towards a coherent technology for computational vehicle system dynamics. Proceedings of the International Symposium on Advanced Vehicle Control, AVEC’92, 48–55. [10] Kortüm, W., W. Schiehlen, and M. Hoffmann. 1994. Progress in integrated system analysis and design software for controlled vehicles. In The Dynamics of Vehicles on Roads and Tracks, Proceedings of the 13th IAVSD Symposium (Z. Shen, ed.) (supplement to Vehicle System Dynamics 22). Amsterdam: Swets and Zeitlinger, 274–296.

[11] Magnus, K., ed. 1978. Dynamics of Multibody Systems, Proceedings of the IUTAM Symposium, Munich, Aug. 29–Sept. 3, 1977. Berlin: Springer–Verlag. [12] Bianchi, G., and W. Schiehlen, eds. 1986. Dynamics of Multibody Systems, Proceedings of the lUTAM/IFToMM Symposium, Udine, Sept. 16–20, 1985. Berlin: Springer–Verlag. [13] Schiehlen, W. O., ed. 1990. Multibody Systems Handbook. Berlin: Springer– Verlag. [14] Schiehlen, W. 1992. Prospects of the German multibody system research project on vehicle dynamics simulation. In The Dynamics of Vehicles, Proceedings of the 12th IAVSD Symposium (G. Sauvage, ed.) (supplement to Vehicle System Dynamics 20). Amsterdam: Swets and Zeitlinger, 537–550. [15] Orlandea, N. 1973. Development and application of node-analogous sparsityoriented methods for simulation of mechanical dynamic systems. PhD thesis, University of Michigan.

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geschlossenen kinematischen Schleifen. Fortschritt-Berichte VDI Reihe 11, Nr. 167. Düsseldorf: VDI–Verlag. [17] Blajer, W., W. Schiehlen, and W. Schirm. 1994. A projective criterion to the coordinate partitioning method for multibody dynamics. Archive of Applied Mechanics 64, 86–98.

[18] Hairer, E., and G. Wanner. 1996. Solving Ordinary Differential Equations—II. Stiff and Differential–Algebraic Problems, 2nd ed. Berlin: Springer–Verlag.

[19] Eich-Soellner, E., and C. Führer. 1998. Numerical Methods in Multibody Dynamics. Stuttgart: Teubner–Verlag.

[20] von Schwerin, R. 1999. MultiBody System SIMulation—Numerical Methods, Algorithms, and Software. Berlin: Springer. [21] Wallrapp, O. 1991. Linearized flexible multibody dynamics including geometric stiffening effects. Mechanics of Structures and Machines 19, 385–409.

[22] Schwertassek, R., and O. Wallrapp. 1999. Dynamik flexibler Mehrkörpersysteme. Braunschweig: Vieweg.

[23] Pfeiffer, F., and Ch. Glocker. 1996. Multibody Dynamics with Unilateral Contacts. New York: John Wiley and Sons.

[24] Wentscher, H., and W. Kortüm. 1996. Multibody model-based multiobjective parameter optimization of aircraft landing gears. Proceedings of the

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GRANULAR SEGREGATION IN COLLISIONAL SHEARING FLOWS Michel Y. Louge, James T. Jenkins, Haitao Xu, and Birgir Ö . Arnarson Cornell University, Ithaca, New York, USA [email protected] Abstract

1.

This paper considers segregation in collisional granular shearing flows from the experimental, computational, and theoretical standpoints. We focus on a phenomenon of segregation where the separation of grains by size or mass is driven by spatial gradients in the fluctuation energy of the grains. We report experiments carried out in microgravity with a shear cell shaped as a race track and containing a mixture of two types of spherical grains. In those experiments, a gradient of fluctuation energy was produced between an inner moving boundary driving collisions among the grains and an outer, more dissipative boundary at the periphery of the cell. The grain segregation and the velocity statistics were captured by a rapid video camera and analyzed using computer-vision software. We briefly outline a kinetic theory and simulations for these flows. We compare the corresponding profiles of granular concentration, mean velocity, and fluctuation energy with the experimental results.

INTRODUCTION

The size segregation of flowing or shaken grains is a commonly observed phenomenon in industrial processes and in nature. In a gravitational field, flows are generally dense and dominated by enduring contacts; their segregation mechanisms are diverse and complex (Savage and Lun 1988, Haff and Werner 1986, Rosato et al. 1986, 1987). In reduced gravity, collisional flows become possible and the segregation is driven mainly by spatial gradients in the energy of granular velocity fluctuations. In steady, fully-developed flows, the balance of momentum exchanged in collisional interaction among different species of grains requires that gradients of particle fluctuation energy be balanced by concentration gradients, thus giving rise to segregation (Jenkins and Mancini 1989). 239 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 239–252. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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To observe collisional segregation driven by a fluctuation energy gradient in the direction perpendicular to the flow, we constructed a shear cell in the form of a race track in which the segregation of a sheared binary mixture of two different types of sphere is maintained by the motion of the inner boundary. We conducted experiments on NASA’s KC-135 microgravity aircraft. To analyze the resulting images, we developed appropriate vision software. We then compared the profiles of volume fraction and mean and fluctuation velocities with computer simulations of the apparatus. The link between the computer simulations and the physical experiments was achieved by measuring the parameters that characterize individual impacts (Foerster et al. 1994). To interpret the results of the simulations and experiments, we solved equations governing the spatial variation of the mixture fluctuation energy and velocity. For simplicity, we employed the approximate treatment for the mixture segregation and the transport coefficients of Arnarson and Jenkins (2000), and carried out the averaging of Jenkins and Arnarson (2000) to capture the effects of side walls. We begin with a brief outline of the theory. Then, we summarize the principle of the simulations, describe the experiments, compare the results, and discuss their significance.

2.

SKETCH OF THE THEORY

Because physical experiments inevitably involve boundaries, their interpretation must be conducted with a theory involving these. We briefly outline such a theory for a steady, fully-developed, rectilinear, unidirectional flow in a rectangular cross section bounded by two bumpy boundaries and two flat, frictional walls. In the absence of an interstitial gas, the steady, fully-developed momentum balance in the flow direction is

where is the granular stress tensor. The directions x, y, and z are shown in Fig. 1. The origin is located midway between the flat side walls at an ordinate above the centers of the stationary boundary bumps, where is the sum of the two species radii and d is the bump diameter. Because the kinetic theory follows the center of interior spheres, its domain occupies a narrower range than the physical experiment, namely and With bumps of uniform size, and

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Figure 1 Sketch of the bottom half of the cell. Its straight section (I) has length L, width Y between centers of cylindrical boundary bumps and depth Z between the two flat side walls. Bumps are affixed to both boundaries, including the curved regions (II) and (IV). The upper straight section (III) and the other halves of regions (II) and (IV) are omitted for clarity. Fully developed flows are simulated using a periodic boundary condition in the observation region with to scale.

Dimensions are not

The equation governing spatial variations of the mixture fluctuation energy is

where is the volumetric rate of collisional dissipation and energy flux. The mixture velocity u along the flow is given by

where and velocity of species is given by

is the

are the material density, volume fraction, and mean respectively. Similarly, the mixture temperature T

where is (2/3) the average fluctuation kinetic energy of species (Jenkins and Mancini 1989), is its number density, is its mass, and The shear stress is related to the velocity gradient through the viscosity In fully-developed flow,

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Similarly, the flux is given by

where is the coefficient of thermal conduction. In this work, we adopt the simplified forms of the transport coefficients calculated by Arnarson and Jenkins (2000) for slight differences and in the radii and masses of the two species,

and

where n is the mixture number density, is the sum of masses, and G, J, and M are functions of the mixture volume fraction

In Eqns. (7) and (8), is the number fraction of species A and the overbar denotes its average value in the cross section. For nearly elastic spheres, the rate of dissipation per unit volume is

where, following Zhang (1993), we incorporate the dissipation due to a modest friction in collision between grains using an effective restitution coefficient on the order of

Finally, the mixture pressure is

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Our experience with the simulations is that the flat side walls play a relatively minor role in this flow. Thus, rather than carrying out a numerical integration of Eqns. (1) and (2), we follow Jenkins and Arnarson (2000) by first integrating these along z. For simplicity, we assume that the mixture kinetic energy and species volume fractions are uniform in that direction, that the average along z of the square of the shear stress is equal to the square of its average, and that the shear stress is proportional to z and vanishes at the centerline by symmetry. In that case, Eqns. (1) and (2) reduce to

and

where In these expressions, brackets denote the averaging along z; they are omitted for those variables which, from our assumptions, are constant along that direction. The superscripts + and – represent quantities at the flat side walls at and respectively. Jenkins (1992) and Jenkins and Louge (1997) derived expressions for, respectively, the granular stress and the granular flux of fluctuation kinetic energy. If all spheres are sliding at the side wall,

and

where is the Coulomb friction coefficient of spheres impacting the walls and is a known function of volume fraction and wall impact parameters. When the spheres roll rather than slide, the stresses and fluxes are functions of the mean relative velocity of the contact point of the grains with the wall, which we evaluate in terms of the mean center of mass velocity assuming that the granular spin equals half the granular vorticity. At the bumpy boundaries, we employ the conditions derived by Jenkins, Myagchilov, and Xu (2000), who refined the conditions calculated by Richman and Chou (1988) for smooth bumps by adding nonlinear corrections in the slip velocity that remain accurate when the latter becomes large. We incorporate frictional interactions on the bumps in the simple way proposed by Jenkins and Arnarson (2000). Then, the

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boundary conditions for the temperature are found from a balance of fluctuation energy at the stationary wall,

and at the moving wall,

where and are complicated functions of the parameters of impact with the bumps, the stress ratio and the wall bumpiness

which is an average measure of the penetration of a sphere of either species in bumps of spacing s between adjacent cylinder edges. Similarly, the velocity boundary conditions are derived from a momentum balance near the stationary boundary,

and the boundary moving with speed U,

where and are complicated functions of bump impact parameters, wall bumpiness and relative velocity. To capture the segregation, we adopt the one-dimensional form of the simplified transport equation for proposed by Arnarson and Jenkins (2000). In the absence of gravity,

where

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and

Here, we determine lateral profiles of T, and by solving Eqns. (15), (16), and (24) numerically subject to conditions (19), (20), (22), and (23), and for given values of the mean volume fraction and mean number fraction

3.

SIMULATIONS

The simulations follow the dynamics of an ensemble of two species of spheres interacting with the boundaries and among themselves through individual impacts. They are based on the algorithm described by Hopkins and Louge (1991). In that algorithm, collisions occur when a sphere overlaps slightly with another sphere or with the wall. The algorithm adjusts its time step periodically to ensure that the mean overlap is kept below a negligible tolerance. In addition, a search grid is superimposed on the flow domain to permit fast identification of near neighbors. Because this method makes it superfluous to maintain a list of future impacts, its computing time is merely proportional to the number of spheres and, consequently, it can simulate the entire shear cell on a relatively small workstation. Profiles of solid volume fraction, velocity, and temperature for the two grain species are measured by dividing the flow domain into a number of averaging slices (Fig. 1). The average value of an intrinsic grain property in a slice is calculated by considering a number of instantaneous realizations of the flow, by summing all contributions from spheres passing through the slice over all realizations, and by dividing the result by and This center-averaging is consistent with the theory outlined earlier. In the absence of gravity, all grain velocities scale with the velocity U of the boundary. In addition, we make the fluctuation energy dimensionless with and Dimensionless quantities are denoted by a dagger (†). Because any rigid boundary tends to order spheres in its neighborhood, the transverse profiles of solid volume fraction exhibit spatial oscillations near the bumpy inner and outer walls with wavelength on the order of a sphere diameter. Because the kinetic theory ignores these fluctuations, their presence can hinder comparisons of the measured segregation with the corresponding theoretical predictions. In grain mixtures, we alleviate this difficulty by focusing on the relative number fraction rather than the volume fractions of each individual species. For example,

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for species A,

Values of above unity indicate a local surplus of species A spheres. As long as the radii of the two species are not greatly different, the relative number fractions incorporate ratios of volume fractions that nearly oscillate in phase and, consequently, their own spatial fluctuations are considerably reduced. In the simulations, impacts are characterized in terms of three coefficients (Walton 1988). The first is the coefficient of normal restitution e. The second is the coefficient of friction for sliding impacts. The last is the coefficient of tangential restitution for impacts that do not involve sliding. For collisions between two free spheres or between a sphere and one of the fiat side walls, we measured these parameters with the facility described by Foerster et al. (1994); for collisions with a cylindrical boundary bump, we adopted the method of Lorenz et al. (1997). Table 1 summarizes the impact parameters.

4.

EXPERIMENTS

The apparatus is shaped as a race track. Its straight sections have dimensions L = 419 mm, Y = 29 mm and Z = 40 mm; they permit granular segregation in a rectilinear shearing flow without significant body forces. The grains are recycled through circular regions of a similar cross section and an inner radius of 62 mm. The moving boundary consists of a chain to which closely spaced hemi-cylindrical stainless steel rods with d = 3.2mm are affixed. The chain is driven by a direct-current motor connected to one of the sprockets through a timing belt.

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The surface of the outer stationary boundary is made bumpy with similar hemi-cylinders. In the curved regions, these cylinders are closely

spaced (s = 0). In the upstream half of the straight regions, the separation gradually increases to s = 1.6mm, and it remains at this value in the downstream half where observations were carried out. The texture of the stationary boundary is meant to maintain the grains as agitated as possible in the curved sections, while providing a smooth transition to a steep temperature profile in the neighborhood of the observation window.

Flat side walls provide lateral containment and allow the flow to be viewed through glass windows. A digital Kodak EktaPro R0 camera images a 25 mm wide region of the cell with abscissa from the upstream sprocket axis in the range where = 292 mm and

326 mm. The spheres exhibit excellent sphericity and narrow size distribution. Their finish allows the vision software to track their movement

with accuracy. Louge et al. (2000) describe the computer image analysis. A typical image is shown in Fig. 2. As long as the displacement between two consecutive frames remains less than a sphere radius, this method

Figure 2 A typical image for the conditions of Test I. Circles and lines are superimposed to indicate the location and trajectory of detected spheres. The moving boundary can be discerned at the bottom of the picture.

permits unambiguous tracking of the moving spheres. Adequate lighting is an important condition for success of the computer vision. The field of view is illuminated with two optical-fiber light guides. The shutter speed of the camera, the opening of the lens, and the illumination level

are tuned by trial and error until grains are reliably detected.

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The vision algorithm typically detects grains located within a sphere diameter from the window. The two species are then distinguished according to their distinct gray scale or size. Each frame is subdivided into ten horizontal strips of constant width spanning the entire length of the image. By incrementing the observed sphere cross-sectional area intersecting each strip, the vision algorithm calculates the fraction of the strip surface occupied by each species within the field of view and

estimates the corresponding relative number fraction by substituting this fraction for in Eqn. (28). The simulations showed that transverse profiles of produced by this method closely represent the state of segregation in the interior. Sphere velocities are then calculated from the positions of each sphere center in two consecutive images and the corresponding center-averaging statistics are incremented as in the simulations. They are made dimensionless with the chain speed directly measured from the image sequence. Table 2 summarizes experimental conditions for the flight tests in the KC-135 microgravity aircraft. A typical KC-135 trajectory included a gravitational pull of order followed by a parabolic flight yielding 20 s of reduced gravity. We began each test by starting the motor at the onset of the parabola. Peak-to-peak gravity fluctuations remained typically below The simulations helped us prescribe mixtures for which these fluctuations had minimal effects on the flow, and provided estimates of the dimensionless time required for each experiment to re-establish steady segregation after canceling that level of residual acceleration. Based on those estimates, we waited a time after the onset of reduced gravity to acquire images with the camera. Electrostatic charging of the dielectric flow spheres was mitigated by maintaining high relative humidity in the apparatus. With this precaution, no evidence of such charging could be observed.

5.

RESULTS AND DISCUSSION

The opacity of the grain assembly and the flow development in the straight section make it challenging to compare experiments and theory

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directly. While the latter predicts average quantities in the interior of a fully-developed flow, the former observe these quantities through side walls in a developing flow. Although flow variables vary little in the z direction, thus making lateral observations representative of the state in the interior, the flow development remains a more serious impediment to direct comparisons between experiments and theory (Fig. 3).

Figure 3 Development of the cross-sectional averaged solid volume fraction along the cell for the conditions of Test I.

In contrast, the simulations can be matched separately with experiments or theory. For comparisons with experiments, we simulate the flow in the entire cell while reproducing the method of imaging, as follows. Because the vision software detects spheres within a depth of focus

on the order of

the simulations calculate observable quantities only

from spheres centered within that distance from the window. To evaluate the corresponding statistics, the simulations ignore their knowledge of sphere velocities. Instead, they collect the locations (x, y) of visible

sphere centers from virtual images generated at the frame rate of the camera. From these, they infer two components of the center velocity

and proceed to calculate strip statistics in the same manner as in the experiments. As Figs. 4 and 5 illustrate with Test II, the simulations reproduce well the fluctuation velocities and segregation observed in the experiments.

Louge et al. (2000) reported similar results with Test I. This agreement demonstrates the utility of the simulations as an equal partner to theory and physical experiment. In this event, fewer physical experiments need be done, and the simulations can be used to evaluate the accuracy of

the theoretical modeling.

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Transverse profiles of the dimensionless fluctuation velocities for the con-

ditions of Test II. The symbols and lines are experimental measurements and predictions of the simulations, respectively. The open symbols and thin lines refer to the x direction, while the closed symbols and thick lines refer to the y direction. The squares and circles represent small and large acrylic spheres, respectively.

Figure 5 Transverse profiles of the relative number fraction for the conditions of Test II. For symbols and lines, see Fig. 4.

Such an evaluation is shown in Fig. 6. Here, after simulating the entire cell, we extracted an observation region in the range to which we imposed the periodic boundary condition sketched in Fig. 1. The spheres contained in this region thus experienced a fully developed flow, which could be matched with results of the theory. Note that, because the periodic simulations were carried out with an actual sample of the developing flow, their species volume fractions were not necessarily equal to those in the entire cell. As this figure illustrates, the theory captures segregation better at low solid volume fraction. Our conjecture is that higher particle number densities introduce geometric obstacles that the continuum kinetic theory ignores.

Acknowledgments This research was sponsored by NASA’s Microgravity Science and Applications

Division under contracts NCC3-468 and NAG3-2112.

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Figure 6 Transverse profiles of dimensionless mixture mean velocity, mixture fluctuation velocity, and relative number fraction for the conditions of Test I (left) and II (right). Triangles and lines represent simulations and theoretical predictions, respectively. In the bottom graphs, circles, diamonds, and squares are simulations results for large acrylic, ceramic, and small acrylic spheres, respectively. In these fullydeveloped flows, Test I has volume fractions = 12.0% and = 27.5%, and Test II has = 25.0% and = 4.3%.

The authors are indebted to Anthony Reeves for designing the computer vision algorithm and for producing the corresponding velocity statistics; to Gregory Aloe,

Roshanak Hakimzadeh, Jeffrey Larko, and Elaina McCartney for participating in the KC-135 flights; and to the engineering team at the NASA–Glenn Research Center: Joe Balombin, Christopher Gallo, Frank Gati, Roshanak Hakimzadeh, Jeffrey Larko, Pamela Mellor, Emily Nelson, Enrique Ramé, Leon Rasberry, John Yaniec and Gary Wroten; and to the KC-135 flight crews and ground support team.

References Arnarson, B.Ö., and J. T. Jenkins. 2000. Binary mixtures of inelastic spheres: simplified constitutive theory. In preparation.

Foerster, S., M. Y. Louge, H. Chang, and K. Allia. 1994. Measurements of the collision properties of small spheres. Physics of Fluids 6, 1108–1115.

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Haff, P. K., and B. T. Werner. 1986. Computer simulation of the mechanical sorting of grains. Powder Technology 48, 239–245. Hopkins, M. A., and M. Y. Louge. 1991. Inelastic microstructure in rapid granular flows of smooth disks. Physics of Fluids A 3, 47–57. Jenkins, J. T. 1992. Boundary conditions for rapid granular flows: flat, frictional walls. Journal of Applied Mechanics 59, 120–127. Jenkins, J. T., and B. Ö. Arnarson. 2000. The influence of side walls on a collisional shearing flow. In preparation.

Jenkins, J. T., and M. Louge. 1997. On the flux of fluctuation energy in a collisional grain flow at a flat, frictional wall. Physics of Fluids 9, 2835–2840. Jenkins, J. T., and F. Manciní. 1989. Kinetic theory for binary mixtures of smooth, nearly elastic spheres. Physics of Fluids A 1, 2050–2057. Jenkins, J. T., S. V. Myagchilov, and H. Xu. 2000. Nonlinear boundary conditions for collisional grain flows. In preparation.

Lorenz, A., C. Tuozzolo, and M. Y. Louge. 1997. Measurements of impact properties of small, nearly spherical particles. Experimental Mechanics 37, 292–298. Louge M. Y., J. T. Jenkins, A. Reeves, and S. Keast. 2000. Microgravity segregation in collisional granular shearing flows. In Segregation in Granular Flows (A. D. Rosato, ed.). Dortrecht: Kluver Academic Publishers, in press. Richman, M. W., and C. S. Chou. 1988. Boundary effects on granular shear flows of smooth disks. Zeitschrift für angewandte Mechanik und Physik 39, 885–901. Rosato, A., K. J. Strandburg, F. Prinz, and R. H. Swendsen. 1986. Monte Carlo simulation of particulate matter segregation. Powder Technology 49, 59–69. Rosato, A., K. J. Strandburg, F. Prinz, and R. H. Swendsen. 1987. Why the Brazil nuts are on top: Size segregation of particulate matter by shaking. Physical Review Letters 58, 1038–1040.

Savage, S. B., and C. K. K. Lun. 1988. Particle size segregation in inclined chute flow of dry cohesionless granular solids. Journal of Fluid Mechanics 189, 311–335. Walton, O. R. 1988. Granular solids flow project. Quarterly report, January–March 1988, UCID-20297-88-1, Lawrence Livermore National Laboratory. Zhang, C. 1993. Kinetic theory for rapid granular flows. Ph.D. thesis, Cornell University.

NONLINEAR COMPOSITES AND MICROSTRUCTURE EVOLUTION Pedro Ponte Castañeda Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania, Philadelphia, Penn., USA [email protected]

Pierre Suquet Laboratoire de Mécanique et d’Acoustique, CNRS, Marseille, France [email protected]

Abstract

1.

Recently developed methods for estimating the effective behavior of nonlinear composites are reviewed. The methods follow from variational principles expressing the effective behavior of the given nonlinear composites in terms of the behavior of suitably chosen “linear comparison” composites. These methods allow the use of classical bounds and estimates (e.g. Hashin–Shtrikman, effective medium approximations) for linear materials to generate corresponding information for nonlinear ones. Comparisons are made with numerical simulations for metalmatrix composites, showing that the new methods are significantly more accurate than earlier ones, especially at high nonlinearity and heterogeneity contrast. The methods can be extended to incorporate evolution of the microstructure and its influence on the effective response under finite-strain conditions. An application to a forming process involving a porous metal is considered for illustrative purposes.

INTRODUCTION

In the context of linear elasticity, rigorous and reliable methods have been available for quite some time to estimate the effective or overall behavior of heterogeneous materials. These so-called homogenization methods include the variational methods of Hashin and Shtrikman (1962) and Beran (1965), both of which are particularly well suited to composites with particulate random microstructures. There is also the self-consistent approximation, which is known to be fairly accurate for polycrystals and other materials with granular microstructures. For a review, see e.g. Willis (1981). 253 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 253–274. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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For heterogeneous materials with nonlinear (e.g. plastic, viscoplastic) properties, rigorous methods have not been available until fairly recently, even though efforts along these lines have been going on for a long time, particularly in the context of ductile polycrystals. Following an extension of the Hashin–Shtrikman (HS) variational principles by Willis (1983), the first bounds of the HS type for nonlinear composites were derived by Talbot and Willis (1985). A more general approach consisting in the use of an optimally chosen “linear comparison composite” was proposed by Ponte Castañeda (1991), and, independently for the special case of power-law materials, by Suquet (1993). This approach not only is capable of delivering bounds of the HS type for nonlinear composites, but also can be used to generate bounds and estimates of other types, by making use of the corresponding bounds and estimates for the linear comparison composite. More recently, Ponte Castañeda (1996a) proposed an alternative approach making use of a more sophisticated linear comparison composite, which while not yielding bounds, appears to give more accurate results. In particular, this method gives the only general homogenization estimates to date capable of reproducing exactly to second order in the contrast the asymptotic expansions of Suquet and Ponte Castañeda (1993). While the idea of using linear composites to estimate the effective behavior of nonlinear ones is quite old, the key feature in these novel linear comparison methods is the use of rigorous variational principles to determine the best possible choice of the linear comparison composite of a given type. Within the context of the “secant” approximation, first used by Chu and Hashin (1971), it follows from the work of Suquet (1995) (and, independently, Hu (1996)) that the optimal choice is that made in the variational method of Ponte Castañeda (1991), i.e. the secant moduli of the phases evaluated at the second moments of the relevant fields in the phases, and not at the phase averages, or first moments, as had been done previously in the classical secant approaches. Alternatively, within the context of “tangent”-type approximations, first used by Hill (1965) in his popular incremental method, the work of Ponte Castañeda and Willis (1999) shows that the optimal choice is, in some restricted sense, that made in the second-order procedure of Ponte Castañeda (1996a), i.e. the tangent moduli of the phases evaluated at the phase averages of the relevant fields in a more general thermoelastic comparison composite. In this short review paper, we attempt to summarize these recent developments, focusing on applications and on comparisons with recent numerical simulations (Moulinec and Suquet (1998), Michel et al. (1999), Michel et al. (2000)). For more detailed reviews, concentrating on the

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more theoretical aspects, the reader is referred to Ponte Castañeda and Suquet (1998) and Willis (2000).

2.

EFFECTIVE BEHAVIOR

The assumption is made that the material is composed of N different phases, which are distributed randomly in a specimen occupying a volume at a length scale that is much smaller than the size of the specimen and the scale of variation of the loading conditions. The constitutive behavior of the nonlinear phases will be characterized by convex energy functions (r = 1 , . . . , N), such that the local stress–strain relation (Fig. l(a)) is defined by

where the function is equal to 1 if the position vector x is inside phase (i.e. and zero otherwise. The relations (1) can be used to describe several constitutive models, including deformation theory of plasticity, in which case and

Figure 1 Constitutive relation.

are identified with the infinitesimal strain and stress, respectively. The relation applies equally well to viscoplastic materials, in which case the associated deformations are finite and and are associated with the Eulerian strain rate and Cauchy stress, respectively. A commonly used

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form for

is the isotropic, incompressible power-law form

where m is the rate-sensitivity parameter, such that is the flow stress, is a reference strain rate, and is the equivalent von Mises strain or strain rate. Making use of the symbols and to denote volume averages over the composite and over phase respectively, we determine the effective behavior of the composite by the effective energy function

where the scalars denote the volume fractions of the given phases and denotes the set of kinematically admissible strains such that there is with in Thus, physically corresponds to the energy stored in the composite when subjected to an affine displacement on the boundary with prescribed average strain It can be shown (Hill (1963)) that the average stress is then related to the average strain via

3.

HOMOGENIZATION VIA LINEAR COMPARISON COMPOSITES

In this section, a brief introduction is given to the “linear comparison composite” methods. A linear composite is introduced with potential

where the are symmetric, positive definite, constant tensors. Following Ponte Castañeda (1996b), we make use of the basic result that

where the assumption has been made that the functions (or functionals) f and are bounded below within the set A.

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Noting that under the hypotheses associated with expression (2) the function is bounded below, and applying the result (6) to the function we find that

and therefore that

where is the effective potential associated with the linear elastic comparison composite with local potential given by (5):

Here is the effective modulus of the linear comparison composite. Next, relaxing the constraint in the second term on the right of expression (8), we arrive at

Observing that this result must hold for any choice of the tensors and bringing the infimum over inside of the averages, we find that

It is noted that if the phases of the nonlinear composite are isotropic, the optimal choice of the comparison moduli is also isotropic with bulk and shear moduli and respectively. Expression (11) is the bound first proposed by Ponte Castañeda (1991) in the context of nonlinear composites with general isotropic phases. A generalization for polycrystals was given by deBotton and Ponte

Castañeda (1995). When it is used in conjunction with the upper bound of Hashin and Shtrikman (1963) for the bounds of Talbot and Willis (1985) are recovered. However, there are pathological cases (not including the standard models of plasticity) for which the bounds generated by the Talbot–Willis procedure are superior (see Willis (1992); Talbot and Willis (1992)). On the other hand, the result (11) can be used together with any other bound (e.g. three-point bounds) or estimate (e.g. the

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self-consistent estimates) for linear composites to generate corresponding estimates for nonlinear composites. For example, Ponte Castañeda (1992) used this procedure to compute three-point bounds. In addition, for the special case of power-law composites, an alternative, but equivalent, form has been given by Suquet (1993) using Hölder-type inequalities. The result is given by

The effective constitutive relation for the nonlinear composite is obtained by making use of expression (11) for in expression (4) and enforcing the optimality condition in the variables to obtain

This means that the effective constitutive relation for the nonlinear composite is precisely the same as that of the linear comparison composite, where the local properties of the linear comparison composite are determined as the solution of the procedure (11) for the optimized variables Clearly, since these variables depend on the applied strain the above relation is nonlinear in this variable, as expected. It is emphasized, however, that the microstructure of the linear comparison composite is identical to that of the nonlinear composite. Finally, it is noted that the optimal choice of the variables can be given an interpretation in terms of the second moment of the strain field in the linear comparison composite. Full details about this derivation can be found in Suquet (1995), Suquet (1997), Ponte Castañeda and Suquet (1998); therefore, we limit ourselves here to simple heuristic arguments when the phases are isotropic. The condition for the optimal choice of the variables in the inner infimum problem in (11) is given by

which physically means that should be chosen to be the secant modulus tensor of nonlinear phase (Fig. l(b)). Assuming stationarity

with respect to we generate the condition for the optimal choice of these variables by substituting the expression (9) for in (11) to obtain

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In other words, the modulus of phase r in the linear comparison composite is the nonlinear secant modulus evaluated at some “effective” strain defined through relation (15), where the right-hand side is the second-order moment of the strain field over phase r of the linear comparison composite. It is emphasized that the expression (15) may not be satisfied in general. However, Suquet (1995) has remarked that, for the case of isotropic phases, this tensorial relation is not needed and that the optimal linear comparison composite can be defined using the second moment of the von Mises strain The more general case was considered in Ponte Castañeda and Suquet (1998) by making use of a suitable extension of the above ideas. The important point to retain here is the following. As mentioned earlier, many attempts have been made to estimate the effective behavior of nonlinear composites in terms of the effective behavior of linear comparison composites—in particular, making use of secant-type approximations. However, because the strain field in the composite is highly nonuniform, it is not obvious what strain to use in the evaluation of the secant moduli for the phases of the linear comparison composite. In the past, ad hoc prescriptions have been tried, mostly making use of the average, or first moment, of the strain field in the phases of the composite. Expression (11) shows that the best choice—within the context of a rigorous variational principle—is the second moment of the strain field over the phases. As will be seen in the next section, this approach gives much better results than the classical schemes. It should be noted that Buryachenko and Lipanov (1989) were apparently to be the first to use second-moment type quantities in the context of their “multi-particle effective field scheme.” However, as already noted in the Introduction, the estimates (11) are not able to recover the perturbation estimates of Suquet and Ponte Castañeda (1993) for weakly inhomogeneous nonlinear materials, which are exact to second order in the heterogeneity contrast. This is in contrast with the Hashin–Shtrikman and self-consistent estimates for linear composites, which are known to be exact to second-order in the contrast. A homogenization method that has the capability to recover small-contrast results exactly to second order in the contrast was introduced recently by Ponte Castañeda (1996a). A full derivation of this result cannot be given here, but the result can be stated succinctly as

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where is the average of the strain field over phase r in a linear thermoelastic composite with phase r defined by the constitutive relation

where the modulus tensor

is defined by

It was shown by Ponte Castañeda and Willis (1999) that this “secondorder” procedure follows from suitable approximations in the context of a rigorous variational principle. However, the principle is only stationary and therefore leads only to stationary estimates and not to bounds. It is further noted that this procedure leads naturally to the choice of the tangent modulus tensors (Fig. 1(b)) for the linear comparison composite. But these should be evaluated at the phase averages of the strain—not at the second moments. As will be seen later, although the second-order estimates are not bounds, they are complementary to the bounds discussed earlier and appear to be more accurate. Other recent developments in the field of nonlinear composites, which will not be reviewed here for lack of space, include the computation of the “hard” bounds by Talbot and Willis (1997), who made use of suitably chosen nonlinear comparison composites; bounds on the strain fields (as opposed to the energies) by Milton and Serkov (2000); an affine procedure due to Masson et al. (2000) that is closely related to the second-order method, but that also works for elastic–viscoplastic composites; new self-consistent estimates for polycrystals (Bornert and Ponte Castañeda (1998), Nebozhyn et al. (1999) and Gilormini et al. (2001)) showing dramatic improvements over the classical estimates; and applications of the second-order method in finite elasticity (Ponte Castañeda and Tiberio (2000)), where the relevant potentials are nonconvex, leading to the failure of other methods.

4.

SAMPLE RESULTS

Sample results are presented here for various special cases, including particle-reinforced composites and porous metals. For simplicity, all constituents are assumed to be governed by relation (2) with the same exponent m, but with different flow stresses The microstructures are assumed to be statistically isotropic, so that the effective potentials of the nonlinear composite are isotropic functions of the (traceless)

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strain thus depending only on its second and third invariants. They can therefore be written in the form

where the effective flow stress

is seen to be a function of the plastic

phase angle which in turn is related to the two invariants of through the relation cos = 4 det Note that the extreme values of the variable correspond to uniaxial tension and simple shear Some of the results presented here are for two-dimensional composites with transverse isotropy subjected to plane-strain conditions, in which case a result analogous to (19) is generated, but where the equivalent strain has been suitably redefined. The corresponding effective flow stresses are constant (independent of in this case. In the figures and discussion below, the following abbreviations will be used for simplicity: Hashin–Shtrikman (HS), self-consistent (SC), upper bound (UB), and lower bound (LB). Similarly, the following terminology will be used: “Variational” refers to the relation (11) of Ponte Castañeda (1991), or, equivalently, for power-law materials, relation (12) of Suquet (1993). “Second-order” refers to relation (16) of Ponte Castañeda (1996a). “Incremental” refers to the procedure first proposed by Hill (1965), in the form developed by Hutchinson (1976) for power-law viscous materials. “Secant” refers to the classical secant procedure, used by various authors, starting with Chu and Hashin (1971). “Tangent” refers to the procedure first proposed by Molinari, Canova, and Ahzi (1987) in its full anisotropic form (no further approximations). “Voigt” and “Reuss” will be used to denote the classical microstructureindependent upper (uniform strain) and lower (uniform stress) bounds. Finally, FEM and FFT will be used to denote the the results of the finiteelement method and fast-Fourier-transform simulations of Moulinec and Suquet (1998), Michel et al. (1999), and Michel et al. (2000).

4.1.

Particle-reinforced composites

In Fig. 2, the variational HS UB and the second-order HS estimate are compared against the classical Voigt UB and Reuss LB, the incremental, secant, and tangent procedures, and the FEM simulations of Michel et al. (1999) for two-phase, isotropic, rigid ideally plastic composites with 15% concentration of the harder phase, subjected to uniaxial tension. The effective flow stress is thus plotted as a function of the flow stress ratio of the two phases. It is emphasized that the FEM simulations are for composites with periodic microstructures and cylindrical unit cells, while all the HS estimates correspond to random microstructures with

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Figure 2 Rigidly reinforced, ideally plastic composites with statistically isotropic microstructures: Influence of the heterogeneity contrast.

overall isotropy. Therefore, the HS and FEM estimates are not expected to be in good agreement for general values of the volume fraction of the reinforcing phase. (In particular, the FEM predictions will not lead to exactly isotropic behavior.) However, for small enough volume fractions, the HS estimates would be expected to be in good agreement with the FEM simulations and hence the comparisons shown the figure at 15% concentration are considered to be meaningful. The main observation in this figure is that the second-order HS esti-

mate gives the best overall agreement with the FEM simulations, even at large contrast (within 1%). The variational HS result, which is an upper bound for all other HS estimates, is satisfied by the second-order and tangent estimates, but not by the secant and incremental estimates. However, the tangent prediction in this limiting case agrees exactly with the Reuss lower bound, which is believed to be too soft. In conclusion, for this extreme nonlinearity, essentially all the classical schemes break down and we are left with the variational bound and second-order estimates, which are complementary to each other and can be used to estimate the effective behavior of the composite fairly accurately.

4.2.

Fiber-reinforced composites

Figure 3 (a) depicts a comparison between yield surfaces computed with the variational procedure using the HS lower bounds to estimate

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Figure 3 Effective flow surfaces and deformation maps for fiber-reinforced composites.

the properties of the linear comparison composite (continuous lines) and FFT simulations (various symbols) for generalized plane strain loading of a fiber-reinforced, ideally plastic composite. The fibers, which are aligned in the out-of-plane direction, are taken to be 2, 5, and 10 times stronger than the matrix. The horizontal axis corresponds to inplane shear loading of the form and the vertical axis to out-of-plane axial loading. It can be seen that the agreement between the analytical predictions and the FFT simulations is quite good, at least in qualitative terms. The variational procedure is able to capture the “bimodal” character of the yield surfaces, which has also been observed experimentally (Dvorak and Bahei-El-Din (1987)). Figure 3(b) presents strain intensity maps for transverse shear loading of the plastic composite. Note that the microstructure in the

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FFT simulations is taken to be periodic, with 64 fibers thrown at random in the unit cell. The shear bands (white lines) that develop through matrix channels between the hard fibers (black circles) are the source of the relatively “weak” behavior in this mode of deformation. For comparison purposes, results are also given in Figs. 3(c) and (d), for the same mode of deformation in linear elastic composites with the same microstructure and shear moduli ratios of 6 and 100, respectively. It is interesting to note that while strain intensification is also observed in the regions of close proximity between the fibers, shear bands going from one end of the specimen to the other end are not observed. It may then be surprising that the variational procedure, which uses the solution of a linear elastic problem to estimate the effective behavior of the nonlinear problem, does so well, in particular, capturing the strongly nonlinear “flat” sector on the yield surface. However, the important thing to remember here is that the variational principle is designed to select the optimal properties of the linear comparison composite to model as accurately as possible the effective behavior of the nonlinear composite, which is characterized in this case by the effective yield surfaces.

4.3.

Effect of the third invariant: Particle-reinforced composites

In this section, the effect of the third invariant of the loading, as measured by the plastic angle on the effective flow stress of two-phase, ideally plastic composites with overall isotropy is considered. Thus, in Fig. 4(a), a comparison is given between the variational and second-order procedures, used in conjunction with the HS lower bounds for the relevant linear comparison problem, and the FFT simulations carried out by Moulinec and Suquet (unpublished) for “quasi-random” microstructures, as defined by the unit cell with 15% reinforcement shown in Figure 4(b). The FFT simulations show dependence on as expected from general considerations. This dependence is also captured by the second-order procedure quite well near the axisymmetric end and less well near the pure shear limit where it tends to the flow stress of the matrix. The variational HS estimate is independent of but on the other hand provides an upper bound for the effective flow stress. Figures 4(c) and (d) provide deformation maps in the FFT simulations for the two extreme cases, axisymmetric tension and pure shear, respectively. As can be seen in the transverse diagonal planes highlighted in

the figures, the deformation is distributed more uniformly in the matrix for axisymmetric tension, while it tends to become localized on shear bands passing through the matrix for pure shear, thus elucidating the

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Figure 4 Rigidly reinforced, ideally plastic composites with statistically isotropic microstructures: Influence of the third invariant of the applied stress.

physical source of the dependence on the third invariant for strongly nonlinear composites.

4.4.

Cellular microstructures

In this section, two-phase composites with transversely isotropic cellular microstructures, subjected to plane strain loading, are considered. Comparisons are shown in Figs. 5(a) and (b) between the predictions

of the second-order procedure, using the SC estimates for the linear

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comparison composite, and FFT simulations for composites with ran-

Figure 5 Two-dimensional composites with cellular microstructures: Effect of volume fraction and nonlinearity.

dom distributions of hexagons defined by unit cells of the type shown in Fig. 5(c) (25 configurations were used; the circles denote the aver-

age value, and the error bars the maximum and minimum values). It is known from earlier work in the context of linear elastic composites that these microstructures are well approximated by the self-consistent model, at least at finite heterogeneity contrast. It is thus seen in Fig. 5(a)

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that the second-order SC estimates predict with fair accuracy the dependence on volume fraction at this moderate level of nonlinearity. (The results are not as good as the nonlinearity increases for intermediate values of the volume fraction.) Similarly, Fig. 5(b) shows relatively good agreement even at infinite contrast and high values of the nonlinearity (m tending to zero), for a volume fraction of 25% of the rigid phase. (Also shown in this figure are the corresponding second-order

HS estimates, which are compared with periodic FFT results for a perfect lattice consisting of a hexagonal distribution of rigid hexagons in the matrix material.) Finally, Fig. 5(d) shows the deformation maps for the configuration defined by Fig. 5(c) with an ideally plastic matrix

and reinforcement phase 5 times as strong. (The soft phase is shown in black in Fig. 5(c).) Note, once again, that shear bands develop passing through channels of the weaker phase.

4.5.

Microstructure evolution in porous materials

In forming processes involving porous metals, for example, the strains are large enough to cause the microstructure to evolve as a function of

the deformation. An isotropic porous plasticity model that has been shown to work extremely well under nearly hydrostatic loading conditions was proposed by Gurson (1977). However, for processes involving lower triaxiality conditions, such as rolling and extrusion, the material

is expected to develop anisotropy, even if its initial state is isotropic. An anisotropic model, which is based on the use of the HS variational estimates of Ponte Castañeda (1991) and Michel and Suquet (1992) for porous media, was proposed by Ponte Castañeda and Zaidman (1994) and Kailasam et al. (2000) to account for the evolution of pore shape and orientation. In Fig. 6, a comparison is made between the predictions of the aniso-

tropic model of Ponte Castañeda and Zaidman (1994) and finite-strain FEM simulations for a periodic composite with axisymmetric unit cell, as depicted in the figure. In addition, the predictions of an extension of the isotropic Gurson model, due to et al. (2000), for a powerlaw viscoplastic matrix material are also shown. The initial porosity is the strain-rate sensitivity is m = 0.2, and uniaxial tension is applied with triaxiality X = 1/3. It can be seen that the predictions of the anisotropic model—in contrast with those of the isotropic model—

are at least in qualitative agreement with the FEM simulations at this level of triaxiality. In particular, the anisotropic model captures not only the evolution of the average shape of the voids but, in addition,

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Figure 6 Porous materials subjected to uniaxial tension.

also predicts the gentler increases in the porosity f (which is found to saturate at large enough strain ), as well as the overall hardening of the porous material (positive slope in the uniaxial stress–strain relation, as opposed to the negative slope predicted by the isotropic model). Analogous observations have been made for other low-triaxiality situations, including uniaxial compression, where the anisotropic model is found to predict full densification at around 70% strain, which is in much better agreement with numerical and experimental evidence than the value predicted by the Gurson-type models (around 250% strain). On the other hand, it should be emphasized that the Gurson-type models give the most accurate predictions for high triaxialities. Finally, in Fig. 7, contour plots are given (Kailasam et al. (2000)) for the distribution of the porosity predicted by the anisotropic and Gurson models, as well as for the pore aspect ratios predicted by the anisotropic model, in a disk compaction experiment (Parteder et al.

Nonlinear composites and microstructure evolution

Figure 7 Compaction of a tapered disk with a height reduction of 37.5%.

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(1999)). The initial porosity of the tapered axisymmetric disk (only one quarter of the cross-section of which is shown) is = 15%, and the macroscopic height reduction is 37.5%. The main observation in the context of this figure is that the porosity reduction predicted by the anisotropic model (Fig. 7(a)) is considerably larger than that predicted by the Gurson model (Fig. 7(b)). More specifically, the size of the region where the porosity is less than 1% (inner dark region) is much larger for the anisotropic model than for the Gurson model. Also, in this region, the anisotropic model predicts that the pores have become nearly flat (i.e. they have become cracks) and are aligned with the horizontal direction (Fig. 7(c) and (d)). For more details on these simulations, the reader is referred to Kailasam et al. (2000).

5.

CONCLUDING REMARKS In this review, two nonlinear homogenization methods have been

briefly described, the main ingredient in both being the use of linear com-

parison composites in the context of suitably designed variational principles. Their predictions for various types of composite material have been compared with the corresponding predictions of the classical schemes, as well as with numerical simulations. It is found that the “second-order” method, which makes use of the tangent moduli evaluated at the average strain in the phases, usually leads to the most accurate estimates, while the less accurate “variational” method, which makes use of secant moduli evaluated at the second moments of the strain, gives rigorous bounds. Thus, in some sense, they provide complementary information. The main advantage of these methods is their relative simplicity and computational efficiency. Explicit (or nearly so) expressions are available for most of the cases considered here, including rigidly reinforced composites and porous materials—see Ponte Castañeda and Suquet (1998) for the most comprehensive set of results. These results compare favorably with FEM and FFT simulations, which, while very accurate for the specific configurations chosen, are much more computationally intensive. It is important to emphasize that, by their very nature, these homogenization methods cannot be more accurate than the estimates that are used as input in the computation of the effective behavior of the relevant linear comparison composites. Thus far, reasonable accuracy has been achieved for composites that are modeled well by the Hashin–Shtrikman and self-consistent approximations. More accurate estimates, incorporating higher-order statistics, are likely to be needed for other types of microstructure and for improved performance. Concerning the nonlinear schemes themselves, improvements are still needed to be able to handle

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the most extreme circumstances, usually involving very highly nonlinear behavior.

Acknowledgments PPC acknowledges the support of the NSF through grants DMS-99-71958 and CMS-99-72234. We are grateful to J. C. Michel and H. Moulinec for the FFT results shown in Figs. 3 to 5, which are taken from the joint works of PS with them. The financial support of NSF (grant INT-97-26521) and CNRS through the joint project “Micromechanics of nonlinear composites and polycrystals” is also acknowledged.

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Bornert, M., and P. Ponte Castañeda. 1998. Second-order estimates of the selfconsistent type for viscoplastic polycrystals. Proceedings of the Royal Society of London A 356, 3035–3045. Buryachenko, V., and A. M. Lipanov. 1989. Prediction of nonlinear flow parameters for multicomponent mixtures. Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki 4, 63–68. Chu, T., and Hashin, Z. 1971. Plastic behavior of composites and porous media under isotropic stress. International Journal of Engineering Science 9, 971–994. deBotton, G., and P. Ponte Castañeda. 1995. Variational estimates for the creep behavior of polycrystals. Proceedings of the Royal Society of London A 448, 421– 442. Dvorak, G., and Y. Bahei-El-Din. 1987. A bimodal plasticity theory of fibrous com-

posite materials. Acta Mechanica 69, 219–241. M., J. C. Michel, and P. Suquet. 2000. A micromechanical approach of damage in viscoplastic materials by evolution in size, shape and distribution of voids. Computer Methods in Applied Mechanics and Engineering 183, 223–246. Gilormini, P., M. Nebozhyn, and P. Ponte Castañeda. 2000. Accurate estimates for the creep behavior of hexagonal polycrystals. Acta Materialia 49, 329–337. Gurson, A. 1977. Continuum theory of ductile rupture by void nucleation and growth: Part I—Yield criteria and flow rules. Journal of Engineering Materials and Technology 99, 1–15. Hashin, Z., and S. Shtrikman. 1962. On some variational principles in anisotropic and nonhomogeneous elasticity. Journal of the Mechanics and Physics of Solids 10, 335–342. Hashin, Z., and S. Shtrikman. 1963. A variational approach to the theory of the elastic behavior of multiphase materials. Journal of the Mechanics and Physics of Solids 11, 127–140. Hill, R. 1963. Elastic properties of reinforced solids: Some theoretical principles. Journal of the Mechanics and Physics of Solids 11, 357–372. Hill, R. 1965. Continuum micro-mechanics of elastoplastic polycrystals. Journal of the Mechanics and Physics of Solids 13, 89–101. Hu, G. 1996. A method of plasticity for general aligned spheroidal voids or fiberreinforced composites. International Journal of Plasticity 12, 439–449.

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Hutchinson, J. W. 1976. Bounds and self-consistent estimates for creep of polycrystalline materials. Proceedings of the Royal Society of London A 348, 101–127.

Kailasam, M., N. Aravas, and P. Ponte Castañeda. 2000. Constitutive models for porous metals with developing anisotropy and applications to deformation processing. Computer Modeling in Engineering and Sciences 1, 105–118. Masson, R., M. Bornert, P. Suquet, and A. Zaoui. 2000. An affine formulation for nonlinear composites and polycrystals. Journal of the Mechanics and Physics of

Solids 48, 1203–1227. Michel, J. C., and P. Suquet. 1992. The constitutive law of nonlinear viscous and porous materials. Journal of the Mechanics and Physics of Solids 40, 783–812.

Michel, J. C., H. Moulinec, and P. Suquet 1999. Effective properties of composite materials with periodic microstructure: A computational approach. Computer Methods in Applied Mechanics and Engineering 172, 109–143. Michel, J. C., H. Moulinec, and P. Suquet. 2000. A computational method based on augmented Lagrangians and fast Fourier transforms for composites with high contrast. Computer Modeling in Engineering and Sciences 1, 79–88. Milton, G., and S. K. Serkov. 2000. Bounding the current in nonlinear conducting composites. Journal of the Mechanics and Physics of Solids 48, 1295–1324. Molinari, A., G. R. Canova, and S. Ahzi. 1987. A self-consistent approach of the large deformation polycrystal viscoplasticity. Acta Metallurgica Materialia 35, 2983– 2994. Moulinec, H., and P. Suquet. 1998. A numerical method for computing the overall response of nonlinear composites with complex microstructure. Computer Methods in Applied Mechanics and Engineering 157, 69–94. Nebozhyn, M. V., P. Gilormini, and P. Ponte Castañeda. 1999. Variational selfconsistent estimates for viscoplastic polycrystals with highly anisotropic grains. Comptes Rendus de l’Académie des Sciences, Paris IIB 328, 11–17. Parteder, E., H. Riedel, and R. Kopp. 1999. Densification of sintered molybdenum during hot upsetting experiments and modeling. Materials Science and Engineering A 264, 17–25. Ponte Castañeda, P. 1991. The effective mechanical properties of nonlinear isotropic composites. Journal of the Mechanics and Physics of Solids 39, 45–71. Ponte Castañeda, P. 1992. New variational principles in plasticity and their application to composite materials. Journal of the Mechanics and Physics of Solids 40, 1757–1788. Ponte Castañeda, P. 1996a. Exact second-order estimates for the effective mechanical properties of nonlinear composites. Journal of the Mechanics and Physics of Solids 44, 827–862. Ponte Castañeda, P. 1996b. Variational methods for estimating the effective behavior of nonlinear composite materials. In Continuum Models and Discrete Systems (CMDS 8) (K. Z. Markov, ed.). Singapore: World Scientific, 268–279. Ponte Castañeda, P., and P. Suquet. 1998. Nonlinear composites. Advances in Applied Mechanics 34, 171–302. Ponte Castañeda, P., and E. Tiberio. 2000. A second-order homogenization method in finite elasticity and applications to black-filled elastomers. Journal of the Mechanics and Physics of Solids 48, 1389–1411. Ponte Castañeda, P., and J. R. Willis. 1999. Variational second-order estimates for nonlinear composites. Proceedings of the Royal Society of London A 455, 1799–

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Ponte Castañeda, P., and M. Zaidman. 1994. Constitutive models for porous materials with evolving microstructure. Journal of the Mechanics and Physics of Solids 42, 1459–1497. Suquet, P. 1993. Overall potentials and extremal surfaces of power-law or ideally plastic materials. Journal of the Mechanics and Physics of Solids 41, 981–1002.

Suquet, P. 1995. Overall properties of nonlinear composites: A modified secant moduli theory and its link with Ponte Castañeda’s nonlinear variational procedure. Comptes Rendus de l’Académie des Sciences, Paris IIB 320, 563–571. Suquet, P. 1997. Effective properties of nonlinear composites. In Continuum Micromechanics (P. Suquet, ed.), CISM Lecture Notes 377. New York: Springer, 197–264. Suquet, P., and P. Ponte Castañeda. 1993. Small-contrast perturbation expansions for the effective properties of nonlinear composites. Comptes Rendus de l’Académie des Sciences, Paris II 317, 1515–1522. Talbot, D. R. S., and J. R. Willis. 1985. Variational principles for inhomogeneous nonlinear media. IMA Journal of Applied Mathematics 35, 39–54.

Talbot, D. R. S., and J. R. Willis. 1992. Some simple explicit bounds for the overall behavior of nonlinear composites. International Journal of Solids and Structures 29, 1981–1987. Talbot, D. R. S., and J. R. Willis, 1997. Bounds of third order for the overall response of nonlinear composites. Journal of the Mechanics and Physics of Solids 45, 87– 111. Willis, J. R. 1981. Variational and related methods for the overall properties of composites. Advances in Applied Mechanics 21, 1–78. Willis, J. R. 1983. The overall response of composite materials. Journal of Applied Mechanics 50, 1202–1209. Willis, J. R. 1992. On methods for bounding the overall properties of nonlinear composites: correction and addition. Journal of the Mechanics and Physics of Solids 40, 441–445. Willis, J. R. 2000. The overall response of nonlinear composite media. European Journal of Mechanics A/Solids 19, S165–S184.

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The technical sessions were kept running smoothly by these dedicated session aides, most of whom were graduate students from Midwestern schools—front row (l–r): Zhiqun Deng (UIUC*), Gareth Block (UIUC), Carmen Hernandez (Northwestern), Zhengyu Zhang (UIUC), Susan Olson (Notre Dame), Almila Guvenc (UIC†); back row (l–r): Mihajlo Golubovic (UIC), Michael J. Rodgers (Northwestern), Russell Todres (UIUC), Michael R. Kessler (UIUC), John Aldrin (Northwestern), Igor Kouznetsov (UIUC), Amine Benzerga (Brown), Prof.

Hassan Aref, and Ilker S. Bayer (UIC). Not pictured is Brett E. Soltz (UIUC). *University of Illinois at Urbana-Champaign

†University of Illinois at Chicago

HARD PROBLEMS WITH SOFT MATERIALS: THE MECHANICS OF FOAMS Denis L. Weaire and Stefan Hutzler Physics Department, Trinity College, Dublin, Ireland [email protected], [email protected] Abstract

1.

The static properties of foams with low liquid fraction are now well understood. We review current topics of foam research, which include the search for appropriate boundary conditions for drainage models, and dynamic effects, such as convective bubble motion. The formation of metal foams poses interesting problems with regard to the solidification of draining metal during the cooling process.

NEW FRONTIERS IN FOAM RESEARCH

Many aspects of the behavior of foams (both liquid and solid) have come to be well understood in the last twenty years [1]. The lowestorder description of their properties is virtually complete. Nevertheless the remaining terra incognita is large, varied, and essential to many practical applications [2]. In the case of liquid foam (Fig. 1), a new frontier is approached as the foam becomes very wet, or is subjected to rapid deformation. The problems presented to theory and experiment are not merely those of mapping out quantitative corrections to established models, but include new phenomena, such as convection and avalanches. Our intention here is not to dwell too long on what is by now accepted as common knowledge. Rather we shall concentrate on sketching the boundaries of the unknown. Of the four key aspects of the physics of liquid foams, indicated in Fig. 2, our focus will be concentrated on drainage and rheology. From the present perspective these are closely interlinked fields of inquiry.

2.

STATIC EQUILIBRIUM

The local laws of equilibrium of a foam together with their consequences have been well understood (with some exceptions) for a long time. A recent flurry of interest in problems that relate to static equi275 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 275–288. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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Figure 1

Foam as seen through the camera lens of art photographer Michael Boran.

Figure 2

The physics of liquid foams includes at least four key phenomena, which

may occur in combination. The physicist’s approach is to separate these for study in isolation.

librium has been provoked not so much by fresh insights as by fresh computing power, and the software to exploit it. In three dimensions, Ken Brakke’s Surface Evolver [3] gives an accurate representation of static structures, within the usual idealized model. Figure 3 shows such a computed element of a foam structure. It may also be useful in the more demanding context of dynamic effects, at least at the outset.

3.

FOAM DRAINAGE

Whenever a foam is freshly created, perhaps by shaking up a liquid, and is then left to come into equilibrium under gravity, much of the liquid

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Figure 3

277

Junction of four Plateau borders.

will be seen to drain away. It does so primarily through the network of Plateau borders, which are the liquid-filled interstices at the edges of the polyhedral bubbles (Fig. 3). Provided that the foam is stable, this is almost all that happens over the first few minutes. The “wet” foam

Figure 4

Sketch of the equilibrium density profile of a liquid foam.

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is said to become “dry”, but not uniformly so. There is an equilibrium profile of density or liquid fraction, such that capillarity balances gravity at all points. The sketch of Fig. 4 makes use of static two-dimensional computer simulations, for purposes of illustration. The equilibrium density profile was described by Henry Princen [4, 5], one of the modern pioneers of the subject. Another was Robert Lemlich [6, 7, 8], who understood the essentials of the drainage process, but failed to arrive at a palatable version of the theory, or to stumble on the “solitary wave” phenomenon described below. More recently our own group, together with Guy Verbist, has discovered, rediscovered, and rendered more coherent various components of the study of drainage [9, 10]. The resulting theory neatly describes drainage under various circumstances, including the usual case of “free drainage” and that in which liquid is continually added, which we call “forced drainage”. When drainage is initiated, the downward progress of the wet region takes the form of a solitary wave. This means that the partial differential equation (the drainage equation) for the vertical profile of the liquid fraction has a solution in which the profile is of constant shape, moving steadily downward. There are many attractive features of this experiment and theory, not least of which is the existence of an analytic solution for the solitary wave. Once established, forced drainage creates a column of essentially uniform liquid fraction The corresponding volume flow rate of liquid varies as

for steady flow. This might be regarded as the primary law of foam drainage within such a theory. It is to be contrasted with electrical conduction, for which current is proportional to for a given applied voltage. Equation (1) rests on very strong assumptions (particularly that of Poiseuille flow) and approximations that might seem to limit its validity to relatively dry foams. Nevertheless, it has been repeatedly found to agree with data from detergent foams, over quite a large range, as have the more exotic solutions of the drainage equation. Recently a body of fresh experimental results was presented by the group of Howard Stone, which deviated significantly from Eqn. (1) [11]. The new data were more consistent with

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which can be justified by the assumption that flow is not of Poiseuille character in the Plateau borders, but closer to plug flow, so that most of the dissipation takes place in the hitherto neglected junctions. In due course the apparent discrepancy has been resolved in an amusing manner: the two groups used leading brands of dishwasher detergent (Fairy Liquid and Dawn) from their respective sides of the Atlantic, and it would seem that their surface rheology is different. This comes down to the magnitude of the surface viscosity, which must be sufficiently high to establish Poiseuille flow.

4.

A COMPREHENSIVE THEORY?

In order to put such conclusions on a firmer basis, we need to capture many effects in a more comprehensive theory of drainage. It is a formidable task, even if one is prepared to adopt a semi-empirical approach. We require a description of the bulk flow through the borders and the junctions, coupling this with the surface flow. Various additional complications may also be necessary, including coupled motion of the adjoining films, the distortion by fluid pressure of the shape, and even the arrangement of junctions and borders. All this is called for without invoking any more exotic effects from physical chemistry (Marangoni effect, etc.) [12], and at every multidisciplinary conference we are told by the physical chemists that these are important. All this might seem too daunting, were it not for the challenge of new qualitative effects that are observed at high drainage rates (and liquid fractions), such as the convective instability described in the following section. These are not just numerical corrections to the basic theory. We have taken only tentative first steps towards reliable calculations in an augmented theory. Specifically, we have calculated the correction due to the inclusion of junctions that maintain the shape dictated by static equilibrium, while keeping Poiseuille flow. The challenges involved in further progress are taken up again in

5.

CONVECTIVE INSTABILITY

Convective instability at high rates of forced drainage, as sketched in Fig. 5, was first reported by our group [13], and confirmed by Douglas Durian and co-workers [14], with some additional features. It is found that, whenever forced drainage is impressed on a vertical column, uniform steady drainage becomes unstable when the liquid fraction exceeds about one half its maximum possible value for a foam (about 0.35, at which point the bubbles come apart). The instability

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Figure 5

Forced drainage at large flow rates leads to the establishment of a convective

bubble motion.

takes the form of a slow convective circulation of bubbles, the speed of which increases as the flow rate is further increased. That this instability awaits a satisfactory explanation is a good indication of the limited state of our success so far—it does not extend to combining drainage and rheology (or bubble motion), as seems required. Cilliers et al. have made a start on this in the context of foam flotation [15].

6.

FOAM RHEOLOGY

How does a foam respond to stress? Attempts to answer this have had to cope with awkward experiments and difficult theory, and there remains a good deal of uncertainty about anything beyond the most basic aspects of foam rheology [16]. Let us begin with those basic properties. A foam remains solid under low stress (Fig. 6), so it has bulk and shear elastic moduli. The bulk modulus is rarely significant, and is almost entirely attributable to the constituent gas. While surface tension or energy makes only a tiny contribution to the bulk modulus, it is entirely responsible for the shear rigidity, so that the shear modulus G is proportional to the surface tension On dimensional grounds we

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Figure 6

281

Sketch of the stress–strain relation for a liquid foam, subjected to increasing

strain. Shear modulus and yield stress depend strongly on the liquid fraction of the foam.

might expect

where c is a numerical constant (depending on and d is the mean diameter of bubbles. Such an equation, with the constant c being largely independent of the details of structure, is well supported by experiment and simulations. Indeed there is an old argument due to Stamenovic, which provides an excellent estimate for c for the dry foam, by considering a single junction and its adjoining Plateau borders, in the spirit of a mean field theory. There is a strong dependence of the shear modulus on the liquid fraction. For the usual case of a disordered foam, the shear modulus goes continuously to zero as the limit of stability is reached. Exactly how it does so is a tricky problem, not yet quite resolved. Of course, the elastic regime is quite limited: plasticity sets in at higher stresses and beyond a certain yield stress, the foam yields continuously, conforming to its designation as a “soft solid” (Fig. 6). At the heart of this plasticity are topological rearrangements, in which the local configuration of Plateau borders or bubbles is changed. At the dry limit, Plateau’s rules of equilibrium tell us to expect these sudden changes, whenever more than four Plateau borders come together in a single junction. Extensive simulations in both 2D and 3D have given us a good feeling for the role of these topological changes. Two kinds of simulation should

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be distinguished. In the first of these, the surface area of a large sample is minimized at all times, so that the effect of stress is simulated quasistatically [17, 18]. Such accurate simulations are very demanding in computer resources, at least in the case of 3D [19]. Accordingly they are usefully complemented by less accurate models, in which the contact of bubbles is roughly represented by simple forces. Using these, much larger samples can be used, and viscous effects at high strain rates can be included, at least in a heuristic style [20]. Again, the justification for the pursuit of such a task should be seen to lie in the promise of new phenomena. Indeed it became evident at an early stage of the simulations, that the conventional picture of the yielding process in terms of individual uncorrelated topological events was wrong, at least for wet foams. Instead they suffer large avalanches of these events [21]. However, no coherent physical picture of these avalanches has yet emerged (and not even a coherent definition), as the scale of the rearrangements seems to be strongly dependent on the simulation method [20, 22]. Experiments by Abd el Kader and Earnshaw [23] on two-dimensional bubble rafts under strain showed that bubble rearrangements may indeed occur in the form of correlated clusters. Whether the clusters can be system-wide if the foam is wet enough is still unanswered.

7.

FOAM DYNAMICS AT THE CELL LEVEL

In both drainage and rheology we are eventually drawn to detailed consideration of foam dynamics at the cell level. While this is part of the frontier of today’s research, it should not be thought that it has not been addressed before. In the case of the drainage process, Leonard and Lemlich [6] already recognized the role of surface viscosity and the need for a model that described the coupling of bulk and surface motions. Indeed they framed just such a model, and arrived at a criterion for the neglect of surface motion, which has been the prevailing assumption since that time. The criterion is

Here is the bulk viscosity, is the surface viscosity, and is the radius of curvature of a Plateau border. Note that where d is the average bubble diameter and is the liquid fraction. Since this criterion has been cited without any experimental verification (as far as we are aware) it is worth examining its basis.

The model used by Leonard and Lemlich [6] for this purpose uses only a single Plateau border. Surface flow velocity is set equal to zero where

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the adjoining films are attached. This condition entails surface shear if the surface is mobile. This shear is resisted by surface viscosity and driven by the forces associated with bulk shear. From today’s perspective, following the line of Koehler et al. [11], this physical model seems questionable. One would rather expect the junctions to play an important role, and the films to be compliant, that is, exerting no significant longitudinal force on the borders and moving freely with them. Such a revised model will lead to a different criterion for the neglect of surface viscosity, and experimental tests need to be made, by analyzing drainage experiments for various bubble sizes and liquid fraction. In the case of rheology, the history of ideas on local dynamics was reviewed by Kraynik in 1988 [25], and it would hardly be unfair to say that little has changed, on the fundamental level. A more comprehensive review has recently been prepared by H. M. Princen [26]. Again, we are confronted by reasonable but largely unconfirmed models, and again must hope for more systematic experiments. The models address the following question. What additional forces, beyond the quasi-static description, are induced by the local distortion of foam structure? Khan and Armstrong [27] made an estimate of the viscous resistance to the stretching of films, with no direct reference to

Plateau borders. However, in 1987 Schwartz and Princen introduced a different model, in which the pulling of films out of Plateau borders is assumed to be the dominant effect [28]. This model has since been used extensively in simulations by Kraynik and collaborators [19].

8.

SOLID FOAM FABRICATION

Many solid foams result simply from the freezing of a liquid foam, so that they have the same structure. As stiff, insulating, structural, energy-absorbing or cushioning materials, they have many familiar uses. While industrial processes are highly sophisticated, the underlying science is not yet well developed. Among those who have pioneered the science are Gibson and Ashby [24] and Kraynik [25]. In the formation of a solid foam, the processes shown in Fig. 7 are interlinked. It is generally expanded by blowing agents and solidified (by freezing or chemical reaction) before drainage can create inhomogeneity and collapse. Drainage is of particular concern for metallic foams, a relatively new class of materials presently under active assessment. In Fig. 8 some results of a simulation of metallic foam formation by our own group are presented [29].

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Figure 7 In one process for formation of a metal foam the metal is expanded by a blowing agent at its melting point and must be solidified before drainage leads to collapse.

Figure 8 Drainage occurs during the solidification process of a metal foam, resulting in a peculiar density profile [29].

Again, further progress may depend on better models of local dynamics.

9.

SOLID FOAM MECHANICS

A classic book on solid foams is Cellular Solids—Structure and Properties, now in its second edition [24]. A distinction must be made between closed-cell solid foams, where the cell faces of the parent liquid foam have been left intact, and open-cell foams, where they have been removed by some mechanism. For example, closed-cell foams find applications in heat insulation. The qualitative behavior of solid foams under an applied load is strongly dependent on the type of solid material. If it is elastomeric,

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the deformation is entirely elastic, that is, the foam recovers its initial shape once the stress is removed. A ductile material will result in some unrecoverable deformation, while foams made of a brittle material will simply fracture at some point of deformation. Analytic calculations performed on simple space-filling structures have now given way to finite-element computations of Surface Evolver generated foam structures, for both open- and closed-cell foams [19].

10.

CONCLUSIONS

Let us delineate the challenges that arise out of the work described in the previous sections (Fig. 9). (a) Present theories using approximations appropriate to dry foams need to be extended to wet foams (Fig. 2).

(b) Dynamic effects need to be added to static and quasi-static models. (c) We do not yet have realistic formulations that simulate simultaneous drainage and flow, wherever they are strongly coupled.

(d) It is likely that (b) and (c) require extensive studies of local dynamics, in which the roles of Plateau borders, films, and junctions are

all recognized.

Figure 9 State of the art of foam physics, frontiers (a) and (b). While static effects in dry foams are well understood, dynamic effects in wet foams evade an understanding. Question marks hint at areas that have in parts been tackled successfully but are still full of unresolved issues.

All these theoretical objectives will be greatly furthered by more and better experiments. In the face of continuing uncertainty about much of the underlying physics and chemistry, theory may diverge from reality, without systematic comparisons with data. Some readers may recognize within the stated desiderata certain problems already addressed in the literature—indeed we have cited references

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from the mid-1980s, to quite advanced ideas. But in reviewing such contributions, one is constantly reminded of the story of the blind men and the elephant .... Beyond the most elementary level, a comprehensive picture is needed, rather than isolated contributions. It may require another decade to accomplish this. In the meantime, foam scientists can enjoy the satisfaction of certain agreed basic principles, a luxury not to be found in most other branches of the study of soft matter. Our colleagues who devote their time to granular matter cannot even agree on the fundamental equations of their subject!

Acknowledgments This research was supported by Enterprise Ireland, Shell Amsterdam, and the European Space Agency, through a Topical Team and a contract under the Prodex program. Thanks are due to S. J. Cox for providing various diagrams.

References [1] Weaire, D., and S. Hutzler. 1999. The Physics of Foams. Oxford: Oxford University Press. [2] Zitha, P. L. J. (ed.). 2000. Proceedings of EuroFoam 2000, the EuroConference on Foams, Emulsions and Applications. Delft: Kluwer. [3] Brakke, K. 1992. The Surface Evolver. Experimental Mathematics 1, 141–165. [4] Princen, H. M. 1986. Osmotic pressure of foams and highly concentrated emulsions—I. Theoretical considerations. Langmuir 2, 519–524. [5] Princen, H. M., and A. D. Kiss. 1987. Osmotic pressure of foams and highly concentrated emulsions—II. Determination from the variation in volume fraction with height in an equilibrated column. Langmuir 3, 36–41. [6] Leonard, R. A., and R. Lemlich. 1965. A study of interstitial liquid flow in foam—Part I. Theoretical model and application to foam fractionation. American Institute of Chemical Engineers Journal 11, 18–25. [7] Leonard, R. A., and R. Lemlich. 1965. A study of interstitial liquid flow in foam—Part II. Experimental verification and observations. American Institute of Chemical Engineers Journal 11, 25–29. [8] Shih, F. S., and R. Lemlich. 1967. A study of interstitial liquid flow in foam— Part III. Test of theory. American Institute of Chemical Engineers Journal 13, 751–754. [9] Weaire, D., S. Hutzler, G. Verbist, and E. A. J. F. Peters. 1997. A review of foam

drainage. Advances in Chemical Physics 102, 315–374. [10] Cox, S. J., D. Weaire, S. Hutzler, J. Murphy, R. Phelan, and G. Verbist. 2000. Applications and generalizations of the foam drainage equation. Proceedings of the Royal Society of London A 456, 2441–2464. [11] Koehler, S. A., S. Hilgenfeldt, and H. A. Stone. 1999. Liquid flow through aqueous foams: The node-dominated drainage equation. Physical Review Letters 82, 4232–4235. [12] Exerowa, D., and P. M. Kruglyakov. 1998. Foam and Foam Films. Amsterdam: Elsevier.

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[13] Hutzler, S., D. Weaire, and R. Crawford. 1998. Convective instability in foam drainage. Europhysics Letters 41, 461–465. [14] Vera, M. U., A, Saint-Jalmes, and D. J. Durian. 2000. Instabilities in a liquid–

fluidized bed of gas bubbles. Physical Review Letters 84, 3001–3004. [15] Neethling, S., and J. J. Cilliers. 1999. Visualization and drainage of coalescing,

flowing foams. In Foams and Films (D. Weaire and J. Banhart, eds.). Bremen: MIT–Verlag. [16] Weaire, D., and M. A. Fortes. 1994. Stress and strain in liquid foams. Advances in Physics 43, 685–738. [17] Bolton, F., and D. Weaire. 1991. The effects of Plateau borders in the twodimensional soap froth—I. Decoration lemma and diffusion theorems. Philosophical Magazine B 63, 795–809. [18] Bolton, F., and D. Weaire. 1992. The effects of Plateau borders in the twodimensional soap froth—II. General simulation and analysis of rigidity loss transition. Philosophical Magazine B 65, 473–487. [19] Kraynik, A. M., M. K. Neilsen, D. A. Reinelt, and W. E. Warren. 1999. Foam micromechanics. In Foams and Emulsions (J. F. Sadoc and N. Rivier, eds.). Dordrecht: Kluwer Academic Publishers. [20] Durian, D. J. 1997. Bubble-scale model of foam mechanics: Melting, nonlinear behaviour, and avalanches. Physical Review E 55, 1739–1751. [21] Hutzler, S., D. Weaire, and F. Bolton. 1995. The effects of Plateau borders in the two-dimensional soap froth—III. Further results. Philosophical Magazine B 71, 277–289. [22] Jiang, Y., P. J. Swart, A. Saxena, M. Asipauskas, and J. A. Glazier. 1999. Hysteresis and avalanches in two-dimensional foam rheology simulations. Physical Review E 59 5819–5832. [23] Abd el Kader, A., and J. C. Earnshaw. 1999. Shear-induced changes in twodimensional foam. Physical Review Letters 82, 2610–2613. [24] Gibson, L. J., and M. F. Ashby. 1997. Cellular Solids—Structure and Properties, 2nd ed. Cambridge: Cambridge University Press. [25] Kraynik, A. M. 1988. Foam flows. Annual Review of Fluid Mechanics 20, 325– 357. [26] Princen, H. M. 2000. To be published.

[27] Khan, S. A., and R. C. Armstrong. 1986. Rheology of foams: I. Theory for dry foams. Journal of Non-Newtonian Fluid Mechanics 20, 1–22. [28] Schwartz, L. W., and H. M. Princen. 1987. A theory of extensional viscosity for flowing foams and concentrated emulsions. Journal of Colloid Interface Science 118, 201–211. [29] Cox, S. J., G. Bradley, and D. Weaire. 2000. Modelling metallic foam formation: The competition between heat transfer and drainage. Submitted.

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Tony Carwardine, Group Marketing Manager for John Wiley & Sons, displays his company’s materials while ICTAM 2000 Sectional Lecturer Hamda BenHadid examines a book. In all, eight exhibitors hosted booths, and others provided promotional materials for participants’ registration packets. (See pp. 561–562.)

MAGNETOHYDRODYNAMIC DAMPED BUOYANCY-DRIVEN CONVECTION Hamda BenHadid Laboratoire de Mécanique des Fluides et d’Acoustique, UMR-CNRS 5509 Université Lyon1/Ecole Centrale de Lyon, France [email protected]

Abstract

1.

We are particularly interested in the flows occurring in metallic liquids confined in cylindrical cavities and subjected to a temperature gradient and a constant or rotating magnetic field. We review previous experimental, numerical, and theoretical works about this topic and summarize the basic phenomena and the scaling laws for such characteristic quantities as velocities and boundary layers. The main features of the action of a constant magnetic field are analyzed, considering the effects of its orientation, intensity, and spatial uniformity on the nature and structure of convective flows. For the combined buoyancy and a rotating magnetic field-driven flow, the stability diagram for the onset of the oscillatory flow is given for a moderate aspect ratio Az = 4, and a representative value of the Prandtl number Pr = 0.026.

INTRODUCTION

In materials processing facilities, constant or rotating magnetic fields are used for a contactless control of fluid flow and thus mass transport during the processing. Both static and rotating magnetic fields can lead to a suppression of the temperature fluctuations. However, the two fields act on the fluid in different ways: for a static field the strength of the Lorentz forces (body forces) depends on the vigor of the convective flow and diminishes as the intensity of the convective flow decreases, whereas the rotating magnetic field can induce a flow even in the melt initially at rest. In the last decades the use of magnetic fields in the crystal-growth process from the melt was limited mainly to stationary magnetic fields, owing to their damping effects on the melt motion and thus the reduction of temperature fluctuations. The strength of the applied stationary field is generally in the range of several hundred millitesla (Utech and Fleming 1966, Series and Hurle 1991, Hurle 1993), but rotating fields of 289 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 289–306. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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a few millitesla are sufficient to affect the melt motion considerably and eliminate time-dependent flows (Dold and Benz 1995, 1997). Experimental observations of Dold and Benz (1995) indicate that forced convection in circular cylindrical cavities generated by a rotating magnetic field produces very complicated flows. They also show that above a critical imposed magnetic field or rotation speed, the flow fields exhibit several complicated oscillatory modes. This timedependent oscillatory convection is not only of inherent phenomenological significance but also of prime practical interest to materials processing. For example, in crystal growth, time-dependent oscillatory convection implies unavoidable crystal–liquid interface oscillations that influence solute segregation in the crystal. Thus, understanding the nature and origin of the oscillatory forced convection becomes necessary for applying an efficiently rotating magnetic field to control the fluid motion. In this paper, the influence of static or rotating magnetic fields on buoyancy-driven convection is investigated. We present results for two different geometries and several physical situations. At first we consider the influence of an externally imposed, static magnetic field on natural convection and investigate the flow in electrically insulating parallelepipedic or cylindrical cavities that are heated and cooled at the opposite side walls. Then we investigate the influence of a rotating magnetic field on the flow in circular cylindrical cavities. Natural convection is driven by a horizontal temperature gradient while forced convection is driven by spatially uniform, transverse, rotating magnetic field. The dynamical behavior of the forced convection is analyzed and its influence on the regime and structure of the convective natural flow is presented.

2.

MATHEMATICAL MODEL

Consider a container of length L and height H (or diameter D for a circular cylinder), containing an electrically conducting metallic liquid. The temperature of the right and left side walls are uniformly maintained at and respectively. The container lateral walls are thermally insulating. The material properties of the fluid—kinematic viscosity thermal diffusivity density thermal expansion coefficient magnetic permeability and electrical conductivity —are assumed to be constant. The container is subjected to an external magnetic field acting on the melt. Neglecting dissipation and Joule heating in the energy equation and using the Boussinesq approximation, we obtain

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the following system of nonlinear partial differential equations:

where t is the time, v = (u,v,w) is the velocity, p is the pressure, T is the temperature, and B is the magnetic induction. Since the magnetic Reynolds number (where V is the characteristic velocity) is very small with regard to unity, the unsteady induced field b is negligible (see e.g. Moreau 1990). In typical melt zones of semiconductor crystal growth, does not exceed Thus we can assume that the magnetic induction B is that of the applied field The Lorentz force is given by where the current density J is determined by Ohm’s law:

Conservation of electric current density gives

These equations are subject to no-slip conditions at all the walls. Because of the electrically insulating walls, the normal component of the electric current vanishes at the wall. Thus the boundary condition for the electric currents is where n denotes the normal to the boundary. For an externally imposed, stationary magnetic field, the induced electric field can be expressed as the gradient of an electric potential and thus, applying Ohm’s law to the conservation of the electric current density, one obtains

with a boundary condition With such a condition, the value of the potential is not unique and needs thus to be fixed at one point of the cavity. Note that the dimensionless form of the system (1)– (6) contains three dimensionless quantities: the Prandtl number Pr = which represents a physical property of the melt; the Hartmann number Ha = which represents the square root of the ratio of electrodynamic to viscous forces; and the Grashof number Gr = which represents the ratio of buoyant energy release to viscous energy dissipation.

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For an externally applied rotating magnetic field, the magnetic field may be described by

where is the magnetic field amplitude, denotes the frequency of the applied current used to generate the magnetic field, and t is the time. We assume that the well-known skin effect is neglected and thus the skin depth We also assume that the frequency is much larger than any typical velocity in the melt; e.g. The electric and magnetic fields are coupled through the Maxwell equations

Note that for a rotating magnetic field the specific governing parameters are the rotating Reynolds number and the magnetic Taylor number

3. 3.1.

DISCUSSION Stationary magnetic field

Theoretical approaches. The control of fluid flows with the help of external magnetic fields seems to be promising. Therefore, current research has intensified in the field of magnetohydrodynamic (MHD) convection, and several new results have been obtained in the last decade. In this context we refer to papers of Garandet et al. (1992) and Alboussière et al. (1993, 1996) for a theoretical approach. These authors proposed an analytical model based on the assumption of a parallel core flow, derived expressions for both the velocity and the induced current density in the presence of a horizontal transverse magnetic field, and predicted that for a rectangular cavity, these should vary as and respectively. They also showed that the analytical model is in good agreement with the numerical results over a limited parameter range, provided that the cavity has an aspect ratio larger than 3. The influence of the cross-section shape on the magnetic damping in the case of long horizontal cavities has been studied analytically by Alboussière et al. (1996). In the case of electrically insulating boundaries, the nature of the symmetry is found to govern the magnitude and structure of the damped velocity. In fact, with electrically insulated walls, the magnetically damped convection velocity varies as when the cross section has a horizontal plane of symmetry, and for nonsymmetrical shapes. When the walls are perfectly conducting, the damped veloc-

ity always varies as

The stability of the parallel core flow has

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been investigated by Bojarevics (1995), who assumed the existence of a two-dimensional solution in the rectangular cross section. Numerical simulations. Ozoe and Okada (1989) reported the results of a three-dimensional numerical investigation of the directional effect of a steady magnetic field on side wall convection in a cube of molten silicon. They found that the longitudinal orientation of the magnetic field is most effective for suppressing convection, whereas the transverse direction is least effective. The results of an experimental study of the flow of molten gallium in a cube are reported in Ozoe and Okada (1992), where they confirmed qualitatively their earlier numerical finding. BenHadid and Henry (1996, 1997) carried out numerical calculations of three-dimensional flows of mercury (Pr = 0.026) in both cylindrical and rectangular cavities of aspect ratio 4, with different orientations of the magnetic field. They found good agreement with the analytical estimates of Garandet et al. (1992) and Alboussière et al. (1993, 1996) for the damping of the velocity field. For the rectangular cavity, the investigation included free-surface effects. Interesting changes in the flow structure are reported and these appear to be closely linked to the distribution of the induced electric currents and their interaction with the applied magnetic field. Three-dimensional numerical simulations of melt convection carried out by Baumgartl and Müller (1992) in a cylindrical geometry subjected to a constant magnetic field show that only the models that include the electric potential equation agree well with the experimental results. The numerical results of Baumgartl et al. (1993) show that for flow fluctuations with typical amplitudes of about 4% of the mean values are suppressed, and the resulting steady flow has essentially the same features as the unsteady one. Möbner and Müller (1999) investigated three-dimensional natural convection driven by a horizontal or vertical temperature gradient under a stationary magnetic field.

Experimental approches. The pioneering experimental work on this problem was carried out by Utech and Fleming (1966) and Hurle, Jakeman, and Johnson (1974). They observed thermal oscillations in the flow and showed that a magnetic field applied orthogonally to the main convective flow can be used to damp the time dependence. Juel et al. (1999) investigated the effect of a horizontal transverse magnetic field on temperature fluctuations in a horizontal Bridgman configuration of aspect ratio four. Their combined experimental and numerical investigations show that the base flow state in a rectangular box filled with liquid gallium is modified into a new two-dimensional configuration. A value

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of Ha = 64 was sufficient to damp the initial temperature fluctuations caused by buoyancy-driven convection. Based on the vertical temperature difference in the center of the cavity, the comparisons between the experimental and numerical results are very good. In agreement with a numerical simulation, the dependence of the vertical temperature difference and similarly the maximum horizontal and vertical velocities, which vary approximately as for Ha > 100, have been experimentally validated. Recent experiments by Davoust et al. (1999) demonstrate the damping of thermogravitational flow of mercury in a horizontal circular cylindrical cavity of aspect ratio 10 by means of a uniform vertical magnetic field. The structure of the steady flow is investigated at large values of Hartmann number up to 235, in the light of current theoretical predictions, and good qualitative agreement is found. A study of the time-dependent convection flow is also performed, which shows in particular that damping is found to occur for small values of the Hartmann number between 1 and 10. Rayleigh–Bénard convection. The three-dimensional numerical simulations performed by Touihri et al. (1999) show that the flow in a vertical circular cylindrical cavity heated from below and subjected to a vertical or horizontal magnetic field keeps the same symmetries as that without a magnetic field. It is also noted that similar convective modes (m = 0, m = 1, and m = 2, where m is the azimuthal wave number) occur and are generally stabilized by the magnetic field. However, they are not equally stabilized. For an aspect ratio Az = 0.5 and for sufficiently large values of the Hartmann number, the mode m = 2 becomes the critical mode in place of m = 0. The horizontal magnetic field breaks some symmetries of the flow. The axisymmetric mode disappears, giving an asymmetric mode m = 02 (which is the combination of the m = 0 and m = 2 modes), whereas the asymmetric modes (m = 1 and m = 2), which were invariant by azimuthal rotation without a magnetic field, now have two possible orientations—either parallel or perpendicular to the applied magnetic field. These five modes are stabilized differently— weakly for those having the axis of the rolls parallel to the direction of the applied magnetic field, and strongly if the axis is perpendicular. The nonlinear evolution of the convection beyond its onset is also modified by the horizontal magnetic field. In fact, the secondary bifurcation, found in the pure thermal case for Az = 0.5, becomes an imperfect bifurcation consisting of two disconnected branches. Thermoelectromagnetic convection. Another potential use of magnetic fields is to create thermoelectromagnetic convection (TEMC).

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This convection is based on the fact that in systems where the solid– melt interface is not isothermal, the temperature gradient produces a Seebeck electromotive force where S is the thermoelectric power of the material. If the two gradients and are not parallel, thermoelectric currents are generated in the fluid. In the presence of an applied magnetic field, these induced electric currents produce a driving thermoelectric body force, which is able to generate flow in the melt. Lehmann et al. (1998a,b) used this possibility to modify the interdendritic convection in directional solidification by adjusting TEMC through the control of the applied uniform magnetic field. Direct numerical simulations of the effect of TEMC on Bridgman-growth semiconductors was carried out by Yesilyurt et al. (1999, 2000). Theoretical studies on the subject were done by Gelfgat (1994). Nonuniform magnetic field. The sensitivity of magnetohydrodynamic convection to a slight nonuniformity of the magnetic field (where b is the magnetic field perturbation) was studied experimentally and analytically by Neubrand (1995). The asymptotic approach confirms the experimental finding in the sense that, for electrically insulating walls, the existence of a slight magnetic field nonuniformity can entail a velocity disturbance of the order

Ha. Thus, it

is clear that for Bridgman-growth systems, there is a more severe relation than expected between efficiency and the magnet size and design. Following the experiments of Neubrand et al. (1995), BenHadid et al. (1996) carried out numerical calculations in a cylindrical cavity subjected to a magnetic field with cusp-shape nonuniformity equal to 10% of the uniform magnetic field. Their results agree rather well with the experimental finding of Neubrand (1995) and show that the MHD flow inside a horizontal cylinder is strongly changed by the nonuniformity in structure as well as in intensity.

Measure of diffusion coefficient.

Based on recent progress in the

knowledge of the damping effect of a uniform constant magnetic field on convective flow, an analytical model for mass transport by molecular diffusion and unsteady solute buoyancy-driven convection under magnetic field was developed by Maclean and Alboussière (2000) in continuation of the previous analytical study of Garandet et al. (1995). Experimental approaches based on the shear-cell technique (see Praizey 1989) were

performed recently by Botton et al. (2000) in order to measure under normal gravity the solute diffusivity in liquid metals and semiconductors for Sn–In (l%at) and Sn–Bi (0.5%at) couples. A vertical magnetic field (0.3 to 0.75 T) was used to damp out the convection that develops within

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a horizontal capillary tube subjected to a longitudinal temperature gradient. Starting from an initial one-dimensional step difference in composition, three phases in the diffusion process were identified and assessed by analytical models. During the first phase, where solutal convection dominates, the apparent diffusion coefficient depends on the duration of the experiment and decreases as In the last phase, thermal convection prevails and the scaling law for the apparent diffusion coefficient is of the form For the intermediate phase, the developed onedimensional semi-integral analytical model provides a good description of the experimental concentration profile evolution. Prom the measure of the apparent diffusion coefficient, the extrapolation to infinite magnetic field value gives the molecular diffusivity value. Note that the use of a magnetic field to measure the diffusion coefficient appears to be a promising method, as it gives results in good agreement with previous microgravity experiments on the same metallic alloys. However, it needs a careful control of the experimental conditions and a strong magnetic field of the order of 2 T or more.

Influence of static magnetic field parameters on the flow.

It

can be generally stated that an increase in the strength of the applied magnetic field leads to several fundamental changes in the properties of thermal convection. The convective circulation progressively loses its intensity, and when Ha reaches a certain critical value, which is found to depend on the direction (longitudinal, vertical, or transverse) of the applied magnetic field, the decrease of the flow intensity takes on an asymptotic form with important changes in the structure of the flow circulation. In order to characterize the effects of a constant magnetic field in three-dimensional parallelepidedic geometry, we plot in Fig. 1 the velocity vectors in the transverse vertical plane at Az/2, and in Fig. 2 the velocity vectors in the longitudinal horizontal plane at H/2. From these figures it appears that the flow structure may be separated into three regions: the core flow; Hartman layers (thickness which develop along walls that are not aligned with the applied magnetic field; and the parallel layers (thickness appearing along walls parallel to the applied magnetic field. Note that viscous effects are confined to these layers, whereas the remainder of the flow, namely the core flow, may be considered as inviscid. When the magnetic field is applied longitudinally, the main effect is observed on the vertical velocity (Fig. 2(a)), which is perpendicular to the direction of the applied field. A strong decrease is obtained in the core region, whereas the largest velocities appear in two (parallel) layers near the lateral walls. The characteristic decrease of the flow

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Figure 1 Horizontal velocity vectors in the transverse vertical (y, z) plane at Az/2

for three orientations of the applied magnetic field. The parameters are Gr = = 0.026, Ha = 100, and Az = 4.

Pr

Figure 2 Vertical velocity vectors in the horizontal (x, z) plane at y = 0.5, for three orientations of the applied magnetic field. The parameters are Gr = Pr = 0.026, Ha = 100, and Az = 4.

with increasing Ha corresponds to an asymptotic variation for the maxima of the absolute values of the longitudinal and vertical velocities. When the magnetic field is applied vertically, the horizontal velocity profile is linear with respect to the vertical coordinate y in the core region. The profile is given by and is connected to the lower and upper rigid boundaries by the classical Hartmann exponential

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profiles. The decrease of the intensity of the flow reveals an asymptotic behavior for large Ha values The velocities vary as and whereas in the parallel layers the velocity follows an law. For the transverse case the flow becomes two-dimensional except for a quick decrease of the velocities near the lateral walls where the noslip condition holds (Figs. l(c) and 2(c)). In this case, as has been shown by Alboussière (1994), the horizontal velocity profile is given by . The decrease of the intensity of the flow with increasing Ha affects more symmetrically the values of and and follows roughly an law for The velocity distributions are easily explained by analyzing the paths of the electric currents and the resulting Lorentz forces. The electric current and the related electric potential fields are presented in Figs. 3 and 4, respectively. As indicated by Eqn. (4), the induced electric cur-

Figure 3 The electric current projection in the transverse vertical (y, z) plane at Az/2, for three orientations of the applied magnetic field. The parameters are Gr = Pr = 0.026, Ha = 100, and Az = 4.

rents have two distinct sources, due to the melt motion through the magnetic field lines, and the electric field part, which appears to allow conservation of electric currents in the melt. The resulting electric currents from E produce two effects: on the one hand they allow the closing of the electric currents along the side walls and on the other hand they compete with the current produced by mainly in the vicinity of the corners. In general, the resulting total electric currents form loops in the sense dictated by the main values of and give rise to Lorentz forces that counteract the buoyancy-induced flow. Therefore, a significant braking effect on the convection is generated in regions where the electric current vectors are perpendicular to the applied magnetic field, whereas a weak braking effect is obtained in regions where the electric current is no longer perpendicular—in par-

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Figure 4 The electric potential field in the transverse vertical (x, z) plane at y = 0.5, for three orientations of the applied magnetic field. The parameters are Gr = Pr = 0.026, Ha = 100, and Az = 4.

ticular near the side walls, where consequently layers of overvelocities develop. It is interesting to note that for the longitudinal case, rather than generating long longitudinal current loops, conservation of the current still occurs mainly in cross-sectional planes with the creation of two superposed counter-rotating loops, with opposite senses of circulation in the right and left parts of the cavity (Fig. 3(c)). Detailed information concerning the scaling laws and the subtle actions of the electric currents in other situations can be found in papers of Alboussière et al. (1993) and BenHadid and Henry (1996, 1997). The damping of the oscillatory flow by a magnetic field is analyzed with respect to the amplitude of the temperature signals that are relevant to crystal growth. This was performed for Gr = 1.5 x Pr = 0.026, and Az = 3 by progressively increasing the Hartmann number up to stabilization. Note that for Ha = 0 the transition from timeindependent to time-dependent flow occurs around under the form of a Hopf bifurcation. It corresponds spatially to a breaking of the longitudinal and center-point symmetries, but keeps the symmetry that concerns the transverse direction. In Fig. 5 we give the time signal for the dimensionless temperature at the middle of the cavity (x = 0, y = 0 and z = Az/2) for the longitudinal direction of the applied magnetic field. From the results, the stabilization of the flow is obtained for and for vertical, transverse, and longitudinal magnetic field orientation, respectively. Thus, the vertical direction is the most efficient orientation for the applied magnetic field to produce a rapid damping on the flow. The least effective one is the longitudinal orientation. In general, as Ha is increased, the time signals become gradually more simple, going through a doubleperiod signal before giving a purely periodic signal that persists with

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Figure 5 Time series of the temperature at (x = 0, y = 0, and z = Az/2) for the longitudinal orientation of the applied magnetic field at Gr = 1.5 × 105 and increasing values of Ha.

a decreasing intensity up to stabilization. We found in addition that the corresponding power spectra show the progressive disappearance of

the low-frequency peaks, whereas the fundamental frequency is maintained up to the stabilization. This finding raises the question of whether the damping of oscillatory flow is due to the modification of the averaged base flow, or to a direct interaction between the magnetic field and the three-dimensional time-dependent flow. However, the original symmetry properties of the steady flow are not restored. Moreover, if Ha is now decreased, a periodic flow is found again as soon as Ha has a value below the critical one. This result does not give any evidence of hysteresis phenomena.

3.2.

Rotating magnetic field (RMF)

Recently, there has been a growing interest in the use of a rotating magnetic field (RMF) to control transport phenomena in the melt and thereby improve the quality of the grown crystals. There have been a number of papers devoted to this subject. Among the earliest significant contributions, Davidson and Hunt (1987) conducted a combined theoretical and experimental study and derived some of the basic scaling

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relationships relating the flow velocity to RMF. More recently, Gelfgat (1994) provided a good review on the possibility of using various types of RMF and combinations of these to control the hydrodynamics and heat/mass transport in the crystal melt. The theoretical works of Moffatt (1965), Richarson (1974), and Witkowski et al. (1998, 1999) provide a framework to study RMF in a liquid-metal column related to growth systems. The solutions obtained in these results established the parametric dependence of the flow field induced by RMF on the controlling nondimensional parameters. Kaiser and Benz (1998) performed a numerical stability analysis for the flow induced by transverse RMF and investigated the effect of the aspect ratio for finite circular cylinders. For the infinite cylinder, their numerical simulation stability results are in good agreement with those of Richardson (1974) and show that instability sets in with Taylor vortex structure. Friedrich et al. (1999) studied by experiments as well as by numerical simulations the interaction of buoyancy and RMF-driven convection in the classical Rayleigh–Bénard configuration. For a cavity of aspect ratio 2, they give

evidence of the existence of thermal waves with wave number k = 2, traveling azimuthally in the same direction as the rotation direction of the RMF. They also conclude that the small-scale temperature oscillations are caused by Taylor instabilities. Barz et al. (1997) in a model cavity with an aspect ratio 2, filled with InGaSn and under the conditions that found experimentally the ratio of the velocity of the secondary flow to that of the primary flow to be 1/5. They also found that, for magnetic Taylor number the boundary-layer regime prevails and the azimuthal velocity follows asymptotically a When buoyancy forces act on the melt, the structure of the flow depends on the relative strength of the forced and natural convection. The azimuthal flow does not influence the heat and mass transport directly; it sustains a secondary recirculating cell, which may interact with the prevailing buoyancy-driven flows or promote mixing of an otherwise quiescent melt. The experiment of Dold and Benz (1995) shows that for a circular cylindrical cavity of aspect ratio 2, filled with liquid gallium and subjected to a vertical temperature gradient, buoyancydriven convection is time-dependent for a Rayleigh number Ra = Magnetic fields in the range of 1 mT and rotation frequency of 50 Hz are sufficient to dominate the buoyancy convection completely and to reduce the amplitude of the buoyancy-caused temperature fluctuations by more than one order of magnitude. At the same time, the fluctuation frequency increases with the field strength.

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Influence of RMF parameters on the flow. In this part we give some numerical results for the stability thresholds of RMF-driven motions of isothermal fluid for a cylindrical geometry of aspect ratio Az = 8. The stability thresholds of buoyancy and RMF-driven motions are also considered. In general for RMF flows, exceeding a critical rotating Reynolds number leads to the generation of Taylor vortices in the middle of the cavity. For fixed Ha, it is interesting to note that the onset of Taylor vortex structure is found to depend essentially on the aspect ratio and on the rotation speed, i.e. From the results, which are in agreement with previous publications, it is clear that, due to the stabilizing influence of the bounding walls, the critical decreases with increasing aspect ratio. In Fig. 6 it is also clear that the number of Taylor vortices increases with the aspect ratio; we observe one pair of

Figure 6 Velocity vectors projected onto the longitudinal horizontal plane for four aspect ratios: Az = 2, 3, 5, and 8. The flow is induced by a rotating magnetic field at Ha = 1.

vortices for Az = 2, whereas eight pairs of vortices exist for Az = 8. At the end regions, due to the existence of the Eckman cell, the last vortices do not develop fully. In addition, we note that the Hartmann number has a significant stabilizing effect and produces a significant increase of the value of the critical magnetic Taylor number for the transition from time-independent to time-dependent flow. Starting from a situation where the buoyancy-driven convection is time-dependent (Gr = 1.5 × 105, Pr = 0.026 and Az = 3), we keep the Hartmann number constant at Ha = 1, and increase stepwise.

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The influence of is investigated from zero to the limiting value corresponding to the transition from time-dependent to time-independent flow. The results show that RMF can be used to damp significantly the temperature fluctuations caused by time-dependent buoyancy-driven convection. The stability diagram in space given in Fig. 7 shows clearly two distinct behaviors in the evolution of the critical Grashof

Figure 7 Stability diagram in the plane for combined buoyancy–rotating magnetic field-driven flow for Az = 3 and Pr = 0.026.

number, When the RMF is considered, increases and reaches a maximum. In this first stage, which illustrates the stabilizing effect of the RMF, the critical Grashof number evolves as Above a certain limit of a stabilizing effect of RMF is no longer valid, and a further increase in the leads to a rapid decrease of the value of which follows an law. Therefore, at moderate Gr numbers, a region of exists where steady state prevails. This region becomes narrower with increasing Grashof number. From the flow patterns given in Fig. 7, we note that in the first stage the flow patterns possess the characteristics of the buoyancy-driven flow, whereas in the second stage significant changes in the flow patterns occur and the flow becomes progressively dominated by RMF convection.

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4.

SUMMARY

We now summarize the basic action of a static or rotating magnetic field on convection. In the case of static magnetic field, for high Ha number the characteristic quantities undergo an asymptotic behavior and significant modifications of the flow structure occur. These modifications are governed mainly by the electric-current circulation. The results show also that a static field can be used to damp significantly the temperature fluctuations caused by time-dependent buoyancy-driven convection. The vertical orientation (parallel to the gravity vector) of the applied magnetic field is the most efficient orientation to damp the flow intensity; the longitudinal orientation is the least efficient. The recent investigations concerning RMF show that RMF also can be used to damp fluctuations induced by buoyancy-driven convection. For Ha = 1, the stabilization of the flow is obtained by RMF when its rotation speed exceeds some threshold value, which is found to depend on the vigor of the buoyancy convection. But this stabilization by RMF of the

initially time-dependent buoyancy-driven melt flow is found to occur in a

narrow range of the values, as destabilization by RMF occurs soon. It can be generally stated that the characteristics of time-dependent melt flows due to RMF at high frequencies and low amplitudes, are less detrimental to real crystal growth processes than the time-dependent flow due to buoyancy convection with typical high-amplitude and low-frequency fluctuations. Thus the interesting features are the relative intensity of the two flow modes—forced or buoyancy-driven flow—in order to know which flow mode dominates and determines the heat and mass transport in the melt.

Acknowledgments The author wishes to thank Prof. R. Moreau from MADYLAM, Dr. D. Henry from LMFA Lyon, and Dr. J. P. Garandet from CEN Grenoble for helpful discussions. This work is supported by the Centre National des Etudes Spatiales. The calculations were carried out on IBM-SP and SGI computers with the support of the CINES.

References Alboussière, T., J. P. Garandet, and R. Moreau. 1993. Buoyancy driven convection with a uniform magnetic field. Part 1. Asymptotic analysis. Journal of Fluid Mechanics 253, 545–563. Alboussière, T., J. P. Garandet, and R. Moreau. 1996. Asymptotic analysis and symmetry in MHD convection. Physics of Fluids 8, 2215–2226. Alboussière, T., A. C. Neubran, J. P. Garandet, and R. Moreau. 1997. Segregation during horizontal Bridgman growth under an axial magnetic field. Journal of Crys-

tal Growth 181(1–2), 133–144.

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Barz, R. U., G. Gerbeth, U. Wunderwald, E. Buhrig, and Y. M. Gelfgat. 1997. Modelling of the isothermal melt flow due to rotating magnetic fields. Journal of Crystal Growth 180, 410–421. Baumgartl, J., and G. Müller. 1992. Calculation of the effects of magnetic field damping on fluid flow. Comparison of magnetohydrodynamic models of different complexity. Proceedings of the 8th European Symposium on Materials and Fluid Sciences in Microgravity (ESA-SP-333), 161–164. BenHadid, H., and D. Henry. 1996. Numerical simulation of convective threedimensional flows in a horizontal cylinder under the action of a constant magnetic field. Journal of Crystal Growth 166, 436–445. BenHadid, H., and D. Henry. 1998. Numerical simulation of convection in the horizontal Bridgman configuration under the action of a constant magnetic field. Part II. Three-dimensional flow. Journal of Fluid Mechanics 333, 57–83. Bojarevics, V. 1994. Buoyancy driven flow and its stability in a horizontal rectangular channel with an arbitrary transversal magnetic field. Proceedings of the Second International Conference on Energy Transfer in Magnetohydrodynamic Flows. Assois, France, 1 (PAMIR). Botton, V., P. Lehmann, R. Moreau, and R. Haettel. 2000. Measurement of solute diffusivities. Part 2. Experimental measurements in a convection-controlled shear cell. Interest of a uniform magnetic field. International Journal of Heat and Mass Transfer, in press. Chandrasekhar, S. 1961. Hydrodynamic and Hydromagnetic Stability. Mineola, N.Y.: Dover Publications. Davidson, P. A., and J. C. R. Hunt. 1987. Swirling recirculating flow in a liquid metal column generated by rotating magnetic field. Journal of Fluid Mechanics 185, 67–106. Davoust, L. 1996. Convection naturelle MHD dans une cavité horizontale élancée. Ph.D. thesis, Institut National Polytechnique de Grenoble, France. Davoust, L., M. D. Cowley, R. Moreau, and R. Bolcato. 1999. Buoyancy-driven con-

vection with a uniform magnetic field. Part2. Experimental investigation. Journal of Fluid Mechanics 400, 59–90. Dold, P., and W. Benz. 1995. Convective temperature fluctuations in liquid gallium in dependence on static and rotating magnetic fields. Crystal Research and Technology 30, 1135–1145. Dold, P., and W. Benz. 1997. Modification of fluid flow and heat transport in vertical Bridgman configurations by rotating magnetic fields. Crystal Research and Technology 32, 51–60.

Friedrich, J., Y. S. Lee, B. Fischer, C. Kupfer, D. Vizman, and G. Müller. 1999. Experimental and numerical study of Rayleigh–Bénard convection affected by a rotating magnetic field. Physics of Fluids 11(4), 853–881. Garandet, J. P., C. Barat, and T. Duffard. 1995. The effect of natural convection in mass transport measurement in dilute liquid alloys. International Journal of Heat

and Mass Transfer 38(12), 2169–2174. Gelfgat, Y. M. 1994. Electromagnetic field application in the process of single crystal growth under microgravity. Proceedings of the 45th Congress of the International Astronautical Federation. Jerusalem, Israel. Oxford: Pergamon. Hurle, D. T. J., E. Jakeman, and C. P. Johnson. 1974. Convective temperature oscillations in molten gallium. Journal of Fluid Mechanics 64, 565–576. Juel, A., T. Mullin, H. BenHadid, and D. Henry. 1998. Magnetohydrodynamic convection in molten gallium. Journal of Fluid Mechanics 378, 97–104.

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Kaiser, Th., and K. W. Benz. 1998. Taylor vortex instabilities induced by a rotating magnetic field: A numerical approach. Physics of Fluids 10, 1104–1110. Lehmann, P., R. Moreau, D. Camel, and R. Bolcato. 1998a. Modification of interdendritic convection in directional solidification by a uniform magnetic field. Acta Materialia 46(11), 4067–4079. Lehmann, P., R. Moreau, D. Camel, and R. Bolcato. 1998b. A simple analysis of the effect of convection on the structure of the mushy zone in the case of horizontal Bridgman solidification—Comparison with experimental results. Journal of Crystal Growth 183(4), 690–704. Maclean, D. J., and T. Alboussière. 2000. Measurement of solute diffusivities. Part 1. Analysis of coupled solute buoyancy-driven convection and mass transport. International Journal of Heat and Mass Transfer 44(9), 1639–1648. Moffatt, H. K. 1965. On fluid flow induced by a rotating magnetic field. Journal of Fluid Mechanics 22, 521–528. Moreau, R. 1990. Magnetohydrodynamics. Dordrecht: Kluwer. Mössner, R., and U. Müller. 1999. A numerical investigation of three-dimensional magnetoconvection in rectangular cavities. International Journal of Heat and Mass Transfer 42, 1111–1121. Neubrand, A. C. 1995. Convection naturelle et ségrégation en solidification Bridgman sous champ magnétique. Ph.D. thesis, Institut National Polytechnique de Grenoble, France. Neubrand, A. C., G. P. Garandet, R. Moraeu, and T. Alboussiére. 1995. Influence of a slight nonuniformity of the magnetic field on MHD convection. Magnetohydrodynamics 31(1–2), 1–17. Ozoe, H., and K. Okada. 1989. The effect of direction of the external magnetic field on the three-dimensional natural convection in a cubical enclosure. International Journal of Heat and Mass Transfer 32, 1939–1954. Ozoe, H., and K. Okada. 1992. Experimental heat transfer rates of natural convection of molten gallium suppressed under an external magnetic field in either the x, y, or z direction. Journal of Heat Transfer 114, 107–114. Praizey, J. P. 1989. Benefits of microgravity for measuring coefficient transport in metallic alloys. International Journal of Heat and Mass Transfer 32, 2395–2401. Richardson, A. T. 1974. On the stability of a magnetically driven rotating fluid flow. Journal of Fluid Mechanics 63, 593–605. Series, R. W., and D. T. J. Hurle. 1991. The use of magnetic fields in semiconductor crystal growth. Journal of Crystal Growth 113, 305–328. Utech, H. P., and M. C. Fleming. 1966. Elimination of solute banding in indium antimonide crystal growth. Journal of Applied Physics 37, 2021–2023. Witkowski, L. M., and P. Marty. 1998. Effect of a rotating field of arbitrary frequency on a liquid metal column. European Journal of Mechanics B/Fluids 17(2), 239– 254. Witkowski, L. M., J. S. Walker, and P. Marty. 1999. Nonaxisymmetric flow in a finitelength cylinder with a rotating magnetic field. Physics of Fluids 11(7), 1821–1826. Yesilyurt, S., L. Vujisic, S. Motakef, F. R. Szofran, and M. P. Volz. 1999. A numerical investigation of the effect of thermoelectromagnetic convection (TEMC) on the Bridgman growth of . Journal of Crystal Growth 207(4), 278–291. Yesilyurt, S., L. Vujisic, S. Motakef, F. R. Szofran, and A. Croll. 2000. The influence of thermoelectromagnetic convection (TEMC) on the Bridgman growth of semiconductors. Journal of Crystal Growth 211(1–4), 360–364.

ADAPTIVE, NONLINEAR, AND LEARNING TECHNIQUES FOR THE CONTROL OF VEHICLE RIDE DYNAMICS Timothy J. Gordon Department of Aeronautical and Automotive Engineering Loughborough University, United Kingdom [email protected]

Abstract

1.

The ride dynamics of road vehicles is concerned with the control of whole-body vibration, to provide comfort and vibration isolation for occupants and transported goods. Ride isolation from road unevenness is conventionally achieved through the pneumatic tire, coupled with a spring and damper in the suspension; however substantial benefits can be derived from active computer control of the suspension system. Active ride control also benefits from on-line adaptation, and similar advantage can be derived via nonlinear feedback control. This paper reviews the fundamental issues and considers the potential for future vehicles, against a background of increasing total system complexity and interaction, as well as the continuing need for robust, safe, and faulttolerant operation. Consideration is also given to the use of “intelligent” control systems that adapt and learn in real-time on the vehicle.

INTRODUCTION

The tire and suspension of a road vehicle provides an interface between the vehicle structure and the road surface, to transmit forces for lateral and longitudinal handling control—braking, acceleration and cornering— and isolate road surface irregularities for vertical ride control. This paper is concerned with the control of ride dynamics, wherein the principal degrees of freedom are body-bounce, pitch, and roll, as well as the relative motion of the wheels to the body. Assuming a rigid vehicle structure, this corresponds to seven principal degrees of freedom for a four-wheeled passenger vehicle. The ride control problem is to minimize accelerations in the three body degrees of freedom, as well as control body attitude (angular deflections) with respect to roll and pitch. This is to be achieved for a large range of road conditions and vehicle speeds, and working within the limited deflections available for both the suspen307 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 307–326. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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sion workspace and the tire structure. A further limitation derives from the safety requirement that sufficient vertical tire loads are maintained, in order to provide in-plane forces for cornering and braking. Active suspension control is achieved through some form of mechanical actuation—typically a servo-hydraulic unit incorporating an electronically controlled spool valve to regulate hydraulic pressure, and hence applied force or torque. The details of such actuators will not be considered here; and while there are significant practical issues relating to actuator technology, reliability and failure effects, cost, and power consumption, it will be sufficient here to regard the suspension actuator as a sub-system that can deliver ‘force on demand’. Also required is a sensor set and a state observer (or equivalent signal processing) to reconstruct real-time dynamic states from measured outputs. Again, we shall assume this has been done, and the details will be skipped (see however [1]). Although this paper has a specific focus on ride control, many of the issues covered apply to a much wider class of dynamic control problems. The general context includes adaptive vs. nonlinear control, effectiveness of feed-forward information, intelligent and learning control, as well as issues of dynamic system integration. In the next section, a review is undertaken of the main issues for automotive ride control. Section 3 considers some aspects of control system adaptation, which are taken further in Section 4 in the form of on-line reinforcement learning. Optimal nonlinear control techniques are then described in Section 5, and the concluding section includes an outline of future research challenges.

2.

OPTIMAL RIDE CONTROL

The quarter-car suspension model, shown in Fig. 1, has been used widely for fundamental investigations into ride control [2, 3, 4]. Though it contains just two of the seven ride degrees of freedom, it represents much of the basic dynamics, and has proved particularly suitable for the evaluation of fundamental control concepts. Vehicle parameters have been based on a medium-sized passenger car, and typical values may be found in the cited references. For the standard passive system, the active force is zero, and there is relatively limited scope for suspension tuning, at least within the context of linear feedback and the quarter-car model. With the introduction of computer controlled actuation, there is considerable scope for delivering additional forces that might improve suspension performance in some way. The most direct way to optimize the ride performance of the suspension is to use linear feedback of suspension states, via such optimal

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Figure 1 Quarter-car suspension model.

design methods as linear quadratic regulator (LQR) and Here we focus on the simpler LQR method, where the time integral or statistical expectation of a quadratic cost function is to be minimized, and it can be rigorously shown that the closed-loop system gives minimum cost response to sudden events and Gaussian white noise inputs. The cost function is commonly taken to be of the form

where T(t) is the dynamic vertical tire load, S(t) is the suspension workspace deflection, and A(t) is the vertical sprung mass (body) acceleration. The positive weighting parameters are adjustable to suit the particular vehicle parameters and specific design requirements. Since it is only the ratio between these parameters that is significant, we may set without any loss of generality. Parameters and may be regarded as Lagrange multipliers, used to impose constraints on the peak or RMS values of T and S, under design conditions. In that case, the LQR optimal controller provides a rigorous minimum RMS value for A(t), or in other words gives a “best comfort” optimal controller. Note that when Gaussian white noise is considered as an input for controller design, it is more for analytical convenience than for real-

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world accuracy. However, the approximation is not unreasonable when the model equations are expressed so that the dynamic input is the vertical velocity v(t) of the road–tire interface. The effectiveness of active control is indicated in Fig. 2, where frequency response gains are shown for both active and passive suspension systems. In each case v(t) is the input, and T(t), S(t), and A(t) are the three outputs. Active 1 has been tuned against the passive sys-

Figure 2 Active and passive frequency response gains.

tem, to give identical peak tire load and suspension deflections under design conditions, here prescribed as unit velocity initial conditions for

the unsprung and sprung masses respectively. For comparison, Active 2 was allowed 25% more tire load variation and suspension deflection under design conditions. From the lower plot, both active systems enjoy substantially improved ride isolation, compared with Passive, with the majority of improvement being shown near the “body bounce” resonance frequency at approximately 1 Hz. Active 2 has the lowest body acceleration gain across the whole frequency range, except for a single frequency, which coincides with the ideal “wheel-hop” resonance frequency for free vibration of the unsprung mass on its tire spring. This is a particular example of an “invariant point”, one of a small number that constrain dynamic

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responses in the quarter-car system; all active and passive suspension system variants have coincident gains at such specific frequencies [6]. The frequency responses in the upper two plots, for the “constraint” variables, show that Active 1 is always at least comparable with Passive, with the exception of increased suspension gain at low frequencies, a feature that is described further below. Active 2 on the other hand allows greatly increased wheel motion at the wheel-hop resonance, giving rise to increased dynamic tire loads and suspension variations at this frequency. However, it is fundamental to what follows to note that this is not always detrimental to system performance; provided the excitation amplitude at these frequencies is sufficiently low, Active 2 can usefully improve ride comfort isolation compared with Passive and Active 1. While the active suspension is an interesting concept for dynamic control, it may not always be considered feasible, due to practical considerations of weight, packaging, cost, and power consumption. In this case, an interesting alternative is the semi-active suspension, which is designed to modulate the dissipation of energy from the suspension, but without any external source of mechanical power [7]. One way this can be achieved is via an electronically controlled spool valve within an otherwise standard hydraulic damper; provided the valve has sufficiently fast transient response, and the range of damping rates is sufficiently large, the semi-active system performance can theoretically approach that of the active suspension. This is demonstrated in Fig. 3, where three road bumps generate significant disturbance for the passive system, while active and semi-active both filter the input approximately equally. The semi-active model is very much idealized (though so is the active system), but the conclusion that a well designed semi-active system can produce significant ride benefits is certainly valid. An interesting problem for active suspensions arises through the use of “skyhook damping”. In the LQR active control, all available suspension states are fed back into the actuator, including the “absolute” vertical body velocity—i.e. the velocity of the sprung mass relative to the original rest frame. In fact, without such skyhook damping, the ride control system reverts to something very similar to a standard passive suspension, and therefore its use is particularly significant. The problem alluded to occurs whenever the active suspension vehicle ascends a road of constant incline; this induces a steady-state velocity that feeds back a non-zero signal to the actuator. This results in a steady-state suspension offset, where the suspension spring force (including any suspension deflection feedback gain in the active control law) cancels the erroneous skyhook damping force. The problem is also apparent from the nonzero suspension gain at 0 Hz shown in Fig. 2. It can be “solved” in a number

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Figure 3 Comparison of passive, active and semi-active systems.

of ad hoc ways, but is fundamentally due to an incomplete description of the road surface geometry in the underlying optimal design. The incompleteness relates to low frequencies generally, not just to steady-state, and a systematic solution to this problem requires a subtle modification to the control optimization [8].

3.

ADAPTATION OF RIDE CONTROL FEEDBACK It is apparent from Fig. 2 that there is a compromise, or trade-off, between tire and suspension deflections on the one hand, and ride comfort on the other. When the road input amplitude is large, there is danger of exceeding available suspension workspace limits, or of losing contact between the tire and road surface; in this case ride comfort must be sacrificed for the suspension, and hence the vehicle, to operate safely. At lower amplitudes, where such limits are not an issue, it is desirable to ‘soften’ the active suspension and improve ride comfort. To achieve this, some form of on-line adaptation is necessary. Conceptually, the simplest approach is “gain scheduling”; a set of LQR gains are designed off-line to accommodate the various expected road inputs types (varying with amplitude and frequency content), and as different conditions

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Figure 4

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Control of vertical tire load for severe events.

are experienced, the vehicle response is used to estimate ‘best-fit’ design conditions, and hence adapt controller gains [9]. Table 1

Test road profile component events (at 15 m/s)

Figures 4 and 5 show suspension responses to an aggressive test road profile that consists of a series of raised sinusoidal bumps and linear inclines—one large one up, followed by four shorter ones down—see Table 1. The road speed is a moderate 15 m/s, but gives rise to severe

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Figure 5 Control of suspension workspace.

vertical inputs, as can be seen from the Passive responses in the following plots. Figure 4 shows vertical tire loads for Passive, Active 1, and Active 2, as well as a further Adaptive system. These tire loads now include the static component, and should therefore remain well above zero for safe handling control. Clearly this is far from the case for Passive; and while the Active systems are greatly improved, the Adaptive system provides a much better and consistent minimum load across the various surface changes. In Fig. 5, unconstrained suspension travel is shown, even though a real vehicle is always subject to workspace limits. For a medium-sized passenger vehicle, workspace of around 0.1 m would be typical, or even

generous, and it is clear that the non-adaptive systems all exceed this limit on occasion. In reality, a suspension will include bump rubbers to reduce the worst effects of metal-on-metal contact, but it is reasonable to expect an active suspension to be designed to control such events as far as possible, and it clear that the adaptive system is superior in this aspect. An important point here is that an adaptive suspension of this type combines the above authority over tire load and workspace, with high levels of ride comfort wherever this is possible. Figure 6 shows the corresponding vertical body accelerations, where Passive gives relatively

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Figure 6 Vertical body accelerations.

poor ride comfort whenever the road generates significant amounts of body bounce—on the initial set of bumps, and on the later ramp events— though between t = 6 s and 15 s, where the input frequencies are higher, the passive ride is comparatively reasonable. Overall, the ride comfort predicted for Active 1 is much improved over Passive; on the other hand, Active 2 is ‘too good to be true’, because the system design has sacrificed suspension and tire load control for the apparent improvement in ride comfort. For example, for 15 s < t < 15.5 s, the ramp induces very little body acceleration, but there is very large suspension compression, as seen in Fig. 5. A more reasonable comparison can be made between Adaptive and Active 1, where it is clear that Adaptive gives equal or lower accelerations, except where the need to meet tire or suspension constraints leads to higher suspension forces, and hence higher body accelerations.

In terms of real-time application, there are a number of remaining issues. The adaptation implemented here is based on a full knowledge of the road profile, where in reality the system must adapt as a function of vehicle response. For example, the sudden events occurring between 19 s and 23 s on the above plots, which cause poor tire load control for the other systems, each require an adaptive ‘switch to hard’ response to occur within around 20 ms. A second issue relates to the integration with

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handling dynamic control, which is important at low lateral accelerations on smooth roads; a very ‘soft’ suspension provides poor control of body motion, and also poor feedback to the driver. Both of these aspects are taken up later.

4.

LEARNING CONTROL

An alternative approach to control system design and adaptation is via reinforcement learning. Such an approach uses a computer system to act on the dynamic controller, and adapt the operation of the controller to reinforce desirable closed-loop performance. As with traditional adap-

Figure 7 System architecture for the CARLA learning process.

tive control, reinforcement learning must operate on a slower timescale than the underlying system dynamics, and normally the timescale is very much slower. The simplest such approach is non-associative reinforcement learning, where the learning system treats the controller and vehicle as a ‘black box’, and applies actions by setting control parameters. Recent work on the application of reinforcement learning to ride control has used stochastic learning automata in both simulation and realtime practical implementation [10, 11, 12]. An effective approach has been to employ a ‘team’ of Continuous Action Reinforcement Learning Automata (CARLA) [13]. Each CARLA defines an action based on probability densities, and in this case each action corresponds to defining a control system parameter, such as a feedback gain (Fig. 7).

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At each iteration n, the ith CARLA has an associated probability density where is the corresponding control parameter with a pre-defined range Using a pseudo-random number generator with uniformly distributed variate, is selected and used to define according to the formula

The full set of control parameters are then implemented, in real time or in simulation, to control the vehicle dynamics for a pre-assigned period of time, and then a cost function, possibly similar to that of Eqn. (1), is used to assess performance cost J(n) and define a reward signal A reward–inaction reinforcement algorithm alters the set of probability density functions whenever ‘improved’ performance is achieved, and no change is made otherwise. The reward and probability update equations are

where and are respectively the median and minimum cost values measured during the previous m samples, is a scale factor, defined to normalize to unit area within and is a symmetric Gaussian neighborhood function, which ‘diffuses’ the reinforcement of probability over neighboring values of the controller gain

The CARLA approach has been applied to a variety of simulation and real-world control problems. In the simulation of a fully active suspension control system, the method reproduces optimal LQR gains with minimal errors. However, being much less limited in its underlying mathematical assumptions, the CARLA approach can successfully optimize controller gains for more general conditions than are possible with LQR—for example, the learning system can directly use real or realistic road surface data in place of Gaussian white noise [12]. Figure 8 shows a typical probability density function, created during real vehicle learning of a semi-active controller; in this case the learning took place using stationary random disturbances from a four-poster road simulator rig. In the figure, the sharp peak created at around = 1000

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Figure 8 Probability density convergence for semi-active ride control.

corresponds to a preferred suspension spring stiffness gain of around 50% of the existing passive value on the vehicle. Thus improved control results from the passive suspension force cancelling 50% of the spring force whenever energy considerations make this possible. Furthermore, this result does not depend on any modeling assumptions, since it was ‘derived’ from the actual vehicle test; the

approach thus provides a valuable diagnostic tool, as well as providing an optimal controller. It is worth noting that learning on real roads is a much more challenging exercise, due to the nonstationary nature of the input, and while

some progress has been made to achieve satisfactory learning and adaptation, further work remains to be done.

5.

NONLINEAR OPTIMAL CONTROL

There are many situations where nonlinear control techniques may be preferred over linear methods, though there are often serious technical difficulties associated with achieving—or even defining—the ‘best’ nonlinear control for any given problem. Particular cases are when the underlying system model is taken to be nonlinear [14, 15] or when the control objectives naturally lead to nonlinearity [16, 17]. For ride control, the former category includes the application of semi-active or other force-limited actuation, while the latter includes the imposition of fixed

limits on suspension workspace and absolute criteria for vertical tire load control. The use of gain scheduling and other forms of adaptive control is essentially a heuristic approach, used in the absence of more powerful nonlinear control design methods.

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Nonlinear optimal control (NOC) is a well-known concept that is under-used, mainly because implementation is often impractical. The results described below are obtained using the famous Pontryagin maximum principle, via numerical techniques outlined in [15, 17]; the approach is computationally intensive and is not suitable for real-time application. And while neural networks offer new opportunities for real-time application of NOC, the underlying ‘curse of dimensionality’ remains—for anything other than low-order systems, large amounts of information must be encoded into the neural network, and excessive training times may result. However, NOC offers interesting opportunities for investigating the theoretical limits of a control system, and hence provides a benchmark for more practical control schemes; here we shall use this approach to assess the active suspension results obtained above. For demonstration, we modify the control objectives, but retain an underlying linear description of the suspension. The dynamic cost for suspension workspace is modified to impose rigid limits at with a = 100 mm in simulation. A term

is added to the cost function (1)—with Active 2 parameters—and clearly

as To improve the tire load control of Active 2, a further term is also added that becomes significant only for large dynamic tire load variations: The resulting suspension and tire components of the cost function (including quadratic terms from Eqn. (1)) are shown in Fig. 9. The resulting dynamic responses on the test road surface are shown in Fig. 10, where the nonlinear results are referred to as NOC, and again compar-

isons are made using Active 1 as a reference. Overall, as expected, the NOC suspension workspace is rigorously maintained within the available limits. There is a single event at t = 15.5 s, at the end of the upward ramp, where NOC tire load is controlled less effectively, but it is easy to understand why. Active 1 exceeds the suspension constraint at this point, in an effort to hold the tire on the road surface; this is simply not possible, and the NOC results reflect this. Elsewhere (e.g. 6 s < t < 10 s) NOC maintains somewhat better tire load control, and where constraints are unimportant (e.g. 10 s < t < 15 s) the body acceleration for NOC is also improved. Thus NOC shares all of the benefits of the previous Adaptive control, but without the associated difficulties of detection and inference.

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Figure 9 Modified cost functions for tire load and suspension workspace.

An interesting result arises from the ability of nonlinear optimal control to exploit a wide range of cost function modifications. The reader may have noticed an anomaly, that the above cost functions—whether quadratic or more general—have been chosen symmetrical with respect to both tire load T and suspension deflection S. While this is certainly sensible for S, it is clear that only reductions in tire load cause concern for handling control, and therefore in place of Eqn. (1) we might substitute the following:

Simulations show that, provided the value of is doubled from the previous value (to reflect the fact that it is used only ‘half of the time’) the dynamic responses are virtually identical to the previous linear system. Far more significant is the effect of using previewed (or ‘look-ahead’) information from the road surface. Although this has already been

included in a very simple manner for the Adaptive system, there are great potential benefits when such information is available as a dynamic input to the controller [4, 15, 18, 19]. Again both linear and nonlinear approaches are available, though linear methods are not valid for

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Comparison of NOC with linear Active 1 responses.

semi-active suspension, or other conditions where the underlying model is nonlinear, or where as above the control objectives force the nonlinearity. Figure 11 compares the previous NOC results with the case where accurate one-second preview is available. Here the improvement in body acceleration is impressive and overwhelming—the active suspension makes effective use of available suspension workspace and tire load variations, without losing control, and to effect excellent ride vibration isolation. Of course there are questions over the degradation of such results due to modeling errors, sensor type, and accuracy, and over signal processing requirements for preview-based control; but the fact remains that ride isolation is very sensitive to the addition of such information, and even imperfect preview is potentially effective. For example, in [18] a linear analysis shows that rear suspension response may be significantly improved by feeding forward the dynamic responses from the front wheels.

6.

CONCLUSIONS

The above work has addressed a number of fundamental issues concerned with the ride dynamics of road vehicles. The literature on the subject is vast, and the references given here provide only a sampled

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Figure 11

Inclusion of preview information for NOC.

introduction. Many of the issues are well known, though the results presented here on nonlinear optimal control are new. In this short paper it is not possible to go very far into the issues that lie beyond the quartercar concept, and indeed much of the literature is also restricted to this simple model. However, in conclusion is seems appropriate to consider current and future research issues involving ride dynamics, and these very definitely go beyond such a limited perspective, and involve interactions with handling, safety, and driver information systems—in other words, are concerned with ‘dynamic systems integration’. In its simplest practical form, systems integration means increasing the use of common hardware, sensors, communication, and data analysis within the vehicle environment, on the basis that duplication is wasteful. A more fundamental aspect of dynamic systems integration is to balance design objectives for various systems, so that interactions and compromises (e.g. between ride and handling) are properly addressed at the design stage. This is essentially the application of multivariable control to a large and complex system—the vehicle—so that multiple objectives can be optimized in a systematic and simultaneous fashion. Unfortunately, a global system approach to vehicle dynamics control is not necessarily practical or even desirable. A basic problem is that vehicle manufacturers and systems suppliers are often not the same com-

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panies, so that complete sharing of design knowledge—especially control algorithms—simply does not occur. Control algorithms are often embedded as ‘black boxes’, and a systems supplier is more likely to ‘tune the system to the vehicle’ rather than share details of embedded algorithms

with the vehicle manufacturer or other systems suppliers. Although this is ‘only’ a commercial problem, it is a very real one, and is associated with a more fundamental problem of vehicle–system combinatorics; as new vehicles and systems are developed, between various different companies, each offering several different options to customers, the number of distinct vehicle variants grows factorially, and it becomes totally

unreasonable to design separate software for every such combination. Also, as vehicles and their control systems become increasingly complex, there is less scope for providing underlying mathematical models for the total system behavior; and where such models do exist, they are

limited by the need for linearity in most existing multivariable control methods. Thus the expected trend for dynamic systems integration on road vehicles is for more intensive real-time computation and on-line optimization, more intelligence and learning within the vehicle systems, more parallelism and modularity in design, and for the integration (including ‘trade-offs’ and compromises) to actually take place dynamically on

the vehicle. Some small progress in this direction has been reported by the author [20, 22] but it is clear that new techniques are required. The work presented here suggests that a minimum requirement for such

dynamic integration is that it should include scope for nonlinear behavior, and make use of adaptation and learning in the real-world operating environment. A further key requirement, that potentially comes ‘for free’, is that of robust and fault-tolerant operation; dynamic integration and optimization in real time can also accommodate dynamic re-optimization once localized failures are detected [20, 21]. Finally there must be suitable interfaces with the driver and passengers—so that individual preferences and driving styles are taken into account in the dynamic integration process. These future challenges are immense, but also immensely exciting.

References [1] Best, M. C. 1995. On the modelling requirements for the practical implementation of advanced vehicle suspension control, Ph.D. Thesis, Loughborough University. [2] Thompson, A. G. 1970. Design of active suspensions. Proceedings of the Insti-

tution of Mechanical Engineers (Part I) 185, 553–563.

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[3] Thompson, A. G. 1976. Active suspensions with optimal linear state feedback. Vehicle System Dynamics 5, 187–203. [4] Hac, A. 1992. Optimal linear preview control of active vehicle suspension. Vehicle System Dynamics 21, 167–195. [5] Yamashita, M., K. Fujimori, C. Uhlik, R. Kawatani, and H. Kimura. 1990. control of an automotive active suspension. Proceedings of the 29th Conference on Decision and Control, 2244–2250. [6] Hedrik, J. K., and T. Butsuen. 1988. Invariant properties of automotive suspensions. Proceedings of the Institution of Mechanical Engineers International Conference on Advanced Suspensions, London. [7] Margolis, D. L. 1982. Semi-active heave and pitch control for ground vehicles. Vehicle System Dynamics 11, 31–42.

[8] Fairgrieve, A., and T. J. Gordon. 2000. On-line estimation of local road gradient

for improved steady-state suspension deflection control. Vehicle System Dynamics 33 (Supplement—Dynamics of Vehicles on Roads and Tracks), 590–603.

[9] ElBeheiry, E. M., and D. C. Karnopp. 1996. Optimal control of vehicle random vibration with constrained suspension deflection. Journal of Sound and Vibration 189, 547–564. [10] Gordon, T. J., C. Marsh, and Q. H. Wu. 1993. Stochastic optimal control of active vehicle suspensions using learning automata. Proceedings of the Institution of Mechanical Engineers—Part I (Journal of Systems and Control Engineering) 207, 143–152.

[11] Marsh, C., T. J. Gordon, and Q. H. Wu. 1995. The application of learning automata to controller design in slow-active automotive suspensions. Vehicle System Dynamics 24, 597–616. [12] Frost, G. P., T. J. Gordon, M. N. Howell, and Q. H. Wu. 1996. Moderated reinforcement learning of active and semi-active vehicle suspension control laws. Proceedings of the Institution of Mechanical Engineers—Part I (Journal of Systems and Control Engineering) 210, 249–257. [13] Howell, M. N., G. P. Frost, T. J. Gordon, and Q. H. Wu. 1997. Continuous action reinforcement learning applied to vehicle suspension control. Mechatronics 7, 263–276. [14] Ono, E., S. Hosoe, H. D. Tuan, and Y. Hayashi. 1996. Nonlinear control of active suspension. Vehicle System Dynamics 25 (Supplement—Dynamics of Vehicles on Roads and Tracks), 489–401. [15] Gordon, T. J., and R. S. Sharp. 1998. On improving the performance of automotive semi-active suspension systems through road preview. Journal of Sound and Vibration 217, 163–182. [16] Gordon, T. J., C. Marsh, and M. G. Milsted. 1991. A comparison of adaptive LQG and nonlinear controllers for vehicle suspension systems. Vehicle System Dynamics 20, 321–340.

[17] Gordon, T. J., and M. C. Best. 1994. Dynamic optimization of nonlinear semiactive suspension controllers. Proceedings of the Institution of Electrical Engineers (IEE) 1994 International Conference “Control 94” (IEE Publication no. 389), Warwick, U.K., 332–337.

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[18] Sharp, R. S., and D. A. Wilson. 1990. On control laws for vehicle suspensions accounting for input correlations. Vehicle System Dynamics 19, 353–363. [19] Louam, N., D. A. Wilson, and R. S. Sharp. Optimization and performance enhancement of active suspensions for automobiles under preview of the road. Vehicle System Dynamics 21, 39–63. [20] Gordon, T. J. 1995. An integrated strategy for the control of complex mechanical systems based on sub-system optimality criteria. Proceedings of the IUTAM Symposium on Optimization of Mechanical Systems (Stuttgart, 1995) (D. Bestle and W. Schiehlen, eds.). Dordrecht: Kluwer, 97–104. [21] Gordon, T. J. 1996. An integrated strategy for the control of a full vehicle active suspension system. Vehicle System Dynamics 25 (Supplement—Dynamics of Vehicles on Roads and Tracks), 229–242. [22] Frost, G. P., M. N. Howell, T. J. Gordon, and Q. H. Wu. 1996. Dynamic vehicle roll control using reinforcement learning. Proceedings of the 1996 United Kingdom Automatic Control Council International Conference on CONTROL 96 (Exeter, U.K.), 1107–1118.

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ICTAM 2000 participants line up at one of several carving tables at the Welcome

Reception on Monday evening, 28 August 2000. University of Illinois president James J. Stukel hosted the event.

RECENT ADVANCES IN TURBULENT MIXING Paul E. Dimotakis Graduate Aeronautical Laboratories California Institute of Technology, Pasadena, Calif., USA [email protected]

Abstract

1.

Experimental, modeling, and direct-numerical-simulation (DNS) studies have advanced our understanding of turbulence and mixing. The mixing transition that occurs across a broad spectrum of flows at outerscale Reynolds numbers in the vicinity of or Taylor Reynolds numbers of will be discussed. Inflow/initial-condition effects in high-Re shear layers will also be discussed. Comparisons between DNS studies of strained diffusion-flame regions and experiments in chemically reacting shear layers provide some new insight in the overall chemical-product formation. DNS studies of the Rayleigh–Taylor instability (RTI) between miscible fluids, with identical boundary conditions and different initial perturbations, reveal an initial-growth regime dominated by diffusion, with a subsequent nonlinear growth that depends on the details of the initial perturbations for as long as the simulations were run. Mixing in RTI flow is found even more sensitively dependent on initial conditions. The discussion concludes with some general comments on high-Re turbulence.

INTRODUCTION

Turbulent mixing dynamics span the full range of flow scales, from the outer spatial scales that define the extent of the turbulent region, or perhaps some internal large scale (such as the mesh in grid turbulence) of the flow, to molecular-diffusion scales where all mixing occurs. Outerscale structure and dynamics dominate the spatio-temporal growth of the turbulent region and the attendant entrainment of fluid, or fluids, that are eventually mixed, as well as the mixed-fluid composition. Intermediate- and small-scale eddies and dynamics are responsible for stirring the entrained fluid(s) and the generation of the high surfaceto-volume ratio of interfacial surface, across which (molecular) mixing occurs. 327 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 327–344. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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The inner (small) scales of the flow are set by viscosity, with the ratio of outer to inner (viscous) scales, scaled by the outer-flow Reynolds number, as proposed by Kolmogorov [1, 2]:

where is the (mixed-fluid) density, the characteristic (outer-scale) velocity that drives the flow, the outer spatial scale, across which is applied, and the (dynamic) viscosity. Diffusion scales can be comparable with, or smaller than, the viscous scales, depending on the fluid Schmidt number Sc. In particular [3],

for gases; for water). For turbulent flows, such as the flow behind a grid, for which an outer-flow Reynolds number is not appropriate, scaling can be referenced to the Taylor Reynolds number

where is the root-mean-square velocity fluctuation and the Taylor microscale [4, 5]. For flows for which both can be defined, one finds empirically that and therefore

Proportionality constants in (1), (2), and (4) are found to be of order unity. Following Kolmogorov (1941) [1], turbulence theories implicitly assume that in high-Re flows, for which inner and outer spatial scales are well separated (1), small-scale dynamics, when scaled by inner-flow parameters, i.e. the mean kinetic-energy dissipation rate and the kinematic viscosity v, may be regarded as universal [4]. While subsequent proposals by Kolmogorov and Oboukhov have refined this assumption by treating the dissipation rate as a stochastic variable [6]–[8], the consequent modifications to spectral measures, for example, are slight and the premise of small-scale universality is held as a good approximation [9]. As (diffusive) mixing is a small-scale phenomenon, small-scale dynamics must be correctly modeled to capture turbulent mixing. Conversely,

experimental mixing measures may be used to interrogate small-scale behavior. Some recent progress in our understanding of turbulence and turbulent mixing will be discussed below.

Recent advances in turbulent mixing

2.

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THE MIXING TRANSITION IN TURBULENT FLOWS

Turbulent flows exhibit a conspicuous transition at outer-scale Reynolds numbers (1) given by

The latter, in terms of a Taylor Reynolds number (3), can be used where an outer-scale Reynolds number is not appropriate. These are necessary but not sufficient criteria for this transition; laminar flows can exist at higher Re yet [10]. This transition was first noted in gas-phase shear layers [11, 12]. For zero-pressure-gradient shear layers, the velocity difference, const and owing to a linearly increasing thickness of the turbulent region, i.e. For this flow, a rather abrupt transition to a more disordered, enhanced-mixing state was noted (Fig. 1), at

Figure 1 Side-view schlieren image of a gas-phase shear layer over a densitymatched mixture of Ar and He). Note abrupt transition in the early part of the flow

([11], Fig. 2f).

Similar behavior was found in liquid-phase shear layers [13], demonstrating that this is not a Schmidt number (2) dependent phenomenon (Fig. 2). At least for high-Sc flow (water), fluid within the shear-layer extent is mostly pure high- and low-speed fluid prior to the mixing transition, with very little mixed. While not as conspicuous for gas-phase shear layers, the mixed-fluid fraction, X (x,t), i.e. the fluid for which for some small increases as for both gas- and liquid-phase shear layers [10]. Moderate-Re shear layers can remain quasi-2-D, forming near-cylindrical interfaces between the entrained fluids (e.g. Fig. 2(a)). Beyond the mixing transition, however, the flow becomes decidedly 3-D. It was unclear at the time whether this was a Re-dependent transition, or a transition in flow dimensionality.

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Laser-induced fluorescence streak images of the scalar field in a liquid-phase

shear layer. Color assignment denotes high-speed-fluid mole fraction ([13], Figs. 7(b) and 9).

Turbulent jets are three dimensional, even at relatively low Reynolds numbers. Experiments in liquid-phase jets also indicate a transition at, This is shown in Fig. 3, which depicts the jet-fluid concentration in the plane of symmetry of liquid-phase turbulent jets [14]. Unmixed reservoir fluid (black) penetrates throughout the turbulent region in the lower-Re jet ((a), This is not the case in the higher-Re case Measurements in chemically reacting, gas-phase jets indicate that, for jets, the transition is completed at i.e. roughly twice as high as for shear layers.1 The length scale for a jet should be the (local) radius and not the diameter—shear is applied across the former—offering a possible explanation for the difference in transition Reynolds numbers between shear layers and jets [10]. That jets also exhibit this transition indicates that this is not a 2-D to 3-D transition. A similar transition occurs in liquid-phase transverse jets in a uniform crossflow (Fig. 4) [17]. Figure 5 depicts the probability density functions (PDFs) of far-field jet-fluid concentration of jets discharging in (a) a quiescent reservoir is pure jet-fluid concentration) [18] and in (b) the far-field of trans1

As opposed to shear layers,

in jets. The whole flow is either below, or above,

the mixing transition. Compare Fig. 1 and Fig. 3; this is a Reynolds number effect.

Recent advances in turbulent mixing

Figure 3

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Jet-fluid concentration in the plane of symmetry of round turbulent jets.

([14], Figs. 5 and 9).

Figure 4

Transverse jets in cross-flow:

Jet-fluid concentration in

the plane of symmetry. Intensity compensated for downstream concentration decay

verse jets [17]; Reynolds numbers are based on jet-nozzle parameters in both cases. Both flows are liquid phase with all PDFs

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Figure 5 Jet-fluid concentration PDFs: (a) Jets in a quiescent reservoir with lines of increasing solidity denoting increasing and ([18], Fig. 8); (b) Transverse jets with increasing PDF at associated with increasing and

normalized to differential area fraction, i.e. [19]

where A(c) is the area portion where the concentration is less than PDF normalization is a subtle issue, with differences between PDFs normalized in time, i.e. relative frequency, vs. space, i.e. line/area/volume fraction. (Space limitations do not permit its discussion here.) The differences are interesting. Jets in a quiescent reservoir exhibit a preferred composition at low Re, whereas transverse jets in a cross flow develop a preferred concentration that develops as Re increases. This can probably be attributed to the difference in entrainment vs. stirring/mixing rates between the two flows and indicates that the shape and Re-dependence of scalar PDFs may not be treated as possessing universal behavior. In addition to flows discussed above, transitions in the qualitative behavior of turbulence and the consequent mixing have been documented in a variety of other flows, spanning both low- and high-Sc fluids, freeshear and wall-bounded flows, uniform-density and stratified flows, as well as incompressible and (weakly) compressible flows, all occurring at values of Re, or as indicated in (5) [10]. The wide variety of flows that exhibit this transition argues for reasons not associated with the large-scale structure, which varies considerably across the spectrum of flows. Its apparent universality suggests a quasiuniversal cause. It was proposed that the reason lies in the requirement for an inviscid range of eddies in the flow. This translates into a separation between the Liepmann–Taylor scale, and the (inner) viscous

Recent advances in turbulent mixing

scale,

333

(1), estimated as

where is the Kolmogorov scale. The former represents the thickness of an internal layer formed as the result of a plunging motion across The factor of 5 is for a Blasius boundary layer but is also appropriate for free shear layers at near-unity velocity ratios. The latter represents the inner scale where viscous effects begin to act, as diagnosed by where power-law spectra deviate for a near-5/3 exponent—this occurs at As the scale range between and can be spanned by viscous action and is the inner viscous scale, inertial size eddies require, at a minimum,

in accord with observation [10].

3.

HIGH-Re SHEAR LAYERS: EFFECTS OF INITIAL/INFLOW CONDITIONS

It is a tenet of turbulence theory that, for high-enough Reynolds numbers, sufficiently far from inflow boundaries, and for times long after the initial conditions are established, the flow, at least in a statistical sense, is characterized by local flow parameters [4]. In free-shear layers and some other flows, however, inflow, or “initial” conditions in a convective frame, are known to exert a significant influence on such far-field properties as growth rate, even at high Re [20]. The question is whether such an influence can be felt at the inner scales of the flow, where mixing occurs, far enough downstream and at high enough Re. Experiments in gas-phase shear layers employed undisturbed or tripped initial boundary layers. The Reynolds numbers were high, with measurements at The Reynolds number for the high-speed boundary layer at the splitter plate trailing edge, for which tripping transitions a laminar boundary layer to turbulence. The experiments employed low-heat-release chemical reactions between hydrogen and nitric oxide, in the high-speed stream, and fluorine in the low-speed stream. They were in the fast-kinetic (high Damköhler number) regime, where chemical-product formation is limited by the mixing rate. Schlieren visualization covered the whole flow, with local measurements of temperature rise (product formation) at in particular, further than has been stated as required for self-similar behavior [21]–[24] by some margin [20].

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Tripping changed large-scale attributes, increasing the visual growth rate, and decreasing spanwise coherence throughout. Figure 6 summarizes the mixing results and plots the temperature rise profile, normalized by the adiabatic temperature rise, i.e.

where is the freestream temperature and is the adiabatic flame temperature, for each reactant blend. The kinetic rate for the slowest reactant blend at a stoichiometric mixture ratio in the untripped-flow case (a) was reduced further by lowering the reactant

Figure 6 Normalized temperature-rise data ([20], Figs. 3 and 6). Diamonds:

Triangles:

Asterisks:

reduced chemical-kinetic rate.

concentrations. The resulting normalized temperature rise (asterisks) is within experimental reproducibility of the kinetically faster, but otherwise identical run (diamonds), confirming that all runs in this series were in the fast-kinetic (diffusion-/mixing-limited) regime [20]. Tripping had a major effect on mixing and mixed-fluid composition, as evidenced in the data for the “flip” experiments in the limit of high and low stoichiometric mixture ratios, for the two freestreams. The width of the mixed-fluid region decreased with tripping, as opposed to the visual thickness, which increased. More importantly, the location of peak chemical production does not shift, for the untripped initial boundary layers, as the stoichiometry changes from lean to rich, indicating a “non-marching” PDF, i.e. one characterized by a distribution of mixed-

fluid compositions that is nearly the same across the shear layer (e.g. [25], Fig. 11, and [13], Figs. 10 and 20). Tripping increases and shifts peak production towards the lean freestream, as changes, as expected for a “marching PDF”, i.e. one whose peak composition changes across

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the shear layer. Tripping and the resulting changes in shear-layer structure permeate to the smallest scales of the flow, altering the mixing environment [20]. In this flow, the Reynolds number and kinetic-energy dissipation, did not change with tripping. While it may not be conceptually difficult to attribute the effects of tripping to entrainment/stirring/mixing environment changes imposed by changes in largescale structure and geometry, this is not the classical view. The documented changes in mixing behavior, even at high Reynolds numbers, indicate that small-scale behavior may not be regarded as universal, in the conventional sense.

4.

HIGH-Re SHEAR LAYERS: MIXING

Entrained fluid mixes, residing over a spectrum of scales, and is subjected to a broad range of strain-rates. In the model proposed by Broadwell and Breidenthal [26], further refined by Broadwell and Mungal [27], mixed fluid is treated as residing in either a homogeneously mixed state, at a composition set by entrainment, or in strained diffusion layers, subject to a strain rate associated with the Taylor microscale, e.g. as modeled by a Kolmogorov cascade. In an alternative model by Dimotakis [28], mixed fluid is treated as residing over the spectrum of outer-to-inner scales and subject to strain rates associated with each scale size, again as modeled by a Kolmogorov cascade. In modeling nonpremixed, chemically reacting flows, it is of some interest to associate the amount of mixed fluid that contributes to chemical-product formation, as a function of the strain rate to which it is subjected. Chemical production in thin, strained, diffusion zones, can be described in terms of “flamelet” models [29], if the amount in each strain-rate state is known. The highest strain rates in turbulent free-shear flows are associated with the smallest (viscous) scales, i.e. (assuming Kolmogorov scaling),

The prefactor is for shear layers at such as the ones described in for which We then have for the expected strain-rate values in such high-Re shear layers,

The higher values are the more relevant; a surface-to-volume weighting of strained diffusion interfaces will favor smaller scales [28, 30]. Non-premixed, opposed-jet diffusion flames were simulated numerically (by DNS) with the same reactant/diluent blends used in shearlayer experiments. The simulations were for a jet of diluent,

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against a jet of diluent, for both rich and lean reactants, for blends. Chemical production for the blends simulated is not limited by chemical kinetics, at least to an experimentally discernible extent (see discussion above, Fig. 6(a)). The peak normalized adiabatic temperature rise in the chemical-reaction zone (9), i.e. was calculated as a function of the imposed strain rate, For the reactants investigated, which are both hypergolic (require no ignition source) and kinetically fast, the results indicate that, for strain rates above or so (Fig. 7), the expected contribution from internal strained diffusion layers is negligible [31]. Considering the

Figure 7 Calculated peak scaled temperature rise in non-premixed strained flame region for

and

(diluted) reactants (Ref. [31], Fig. 1).

strain-rate range for such flows (11), we conclude that, at least for such flows, for which experimentally the contribution from internal strained diffusion layers is negligible. Substantial reactant leakage must occur in internal strained layers, setting an environment for chemical production best described as occurring in small-scale, (partially) premixed regions. Considering the Lagrangian time for the mixing/reaction sequence, the time delay for this leakage to be completed, is relatively small and overall chemical production is negligibly affected by it.

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5.

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RAYLEIGH–TAYLOR FLOW: EFFECTS

OF INITIAL CONDITIONS Rayleigh–Taylor instability (RTI) occurs whenever fluids of different density are subjected to acceleration in a direction opposite that of the density gradient [32]–[35]. These flows eventually become turbulent, as secondary, Kelvin–Helmholtz instabilities (KHI) develop and grow. If the fluids are miscible, species diffusion and mixing, which reduce density differences and hence forcing, can play a dynamic role.

Models of the Rayleigh–Taylor instability mixing zone have it growing quadratically in time, following an initial, linear-stability evolution [36]– [38]. For constant acceleration, g, the vertical penetration, of the light fluid into the heavy fluid (dubbed “bubbles”), and of the advancing heavy into light fluid (“spikes”), are modeled as

where and s denote “bubbles” and “spikes”, with and model constants, and the Atwood number, with and the densities of the (pure) heavy and light fluids, respectively. The extent h(t) of the RTI mixing zone is then given by

where

Empirical estimates are in the range Such growth models implicitly assume that the

density difference across the advancing fronts remains constant and that the flow structure remains self-similar [40]. Direct numerical simulation of the Navier–Stokes equations, aug-

mented by a species-conservation equation (binary, Fickian diffusion) was performed for this flow, on a grid with matched fluid viscosities. Initial error-function density (mole-fraction) profiles were perturbed by displacing the intermediate isosurface, nominally

at by an amount in units of the initial profile scaling length [40]. Three simulations, Cases A, B, and C, were performed, each with a different initial perturbation spectrum, but otherwise identical in every other respect. The initial

spectra are plotted

in Fig. 8, along with the linear-stability exponential-growth coefficient, (k) [34]. High-k perturbations are initially damped by viscosity and diffusion. The perturbation energy is the same for all cases:

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Spectra of initial interfacial perturbations (14) for Cases A, B, and C.

The exponential growth coefficient from viscous and diffusive linear stability theory (LST) [34] is also plotted. Left axis: Perturbation spectra scale. Right axis: Lineargrowth rate coefficient, with units of inverse time [40].

The evolution for Case C is depicted in Fig. 9. Pure heavy fluid is red. pure light fluid is blue, and the intermediate

Figure 9 Time evolution of intermediate heavy fluid is red, pure light fluid

isosurface (green) for Case C. Pure is blue [40].

isosurface (1:1 molar ratio) is green. The initial

isotropic

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perturbations are barely discernible on the interface. By mushroom-like structures have formed on the upper side of the isosurface is the transverse extent of the RTI cell). By mushrooms have merged into larger structures and a long, thin protuberance (“spike”) can be seen penetrating deep into the light fluid [40]. Figure 10 depicts the extent, h(t), of the RTI mixing zone (13), along with indicatrices for and growth. An initial, diffusion-dominated

Figure 10 Mixing-zone extent,

vs.

for the three cases [40].

regime is the same for all cases, i.e. independent of the initial perturbations, and in good agreement with analytical predictions for purely diffusive growth. A faster-growth stage occurs at a different time for each of the three cases, reflecting differences in initial modal seeding. Case C growth is well represented as quadratic. Growth for Cases A and B is different. Even greater differences between the three cases are found in mixing, as measured by the amount of product that would be formed from a stoichiometric chemical reaction between the two fluids being mixed. These differences persist as far as the simulations could be run, to [40]. While final Reynolds numbers attained are below the anticipated mixing transition, the flow evolved over a significant multiple of the initial transverse extent, which is greater than that for the high-Re shear layers, i.e. described above.

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SOME COMMENTS ON HIGH-Re TURBULENCE The equations for the turbulent flow, at least for incompressible flow are the (scaled) Navier–Stokes equations,

which depend on one parameter, the Reynolds number, Re. Of course, solutions also depend on initial and boundary conditions; we cannot

envisage The Book of Turbulence, simply indexed by Re. If initial conditions were eventually “forgotten”, one could conceive of a volume for turbulent shear layers—one for boundary layers, one for pipes, etc.—a finite project, at least as regards the classical set of flows. The potential geometric complexity of boundary conditions, however, as encountered in internal flows in many engineering devices, for example, quickly elevates this to an unmanageable project. To make matters worse, experimental and computational evidence indicates that the influence of initial conditions can persist, not only as discussed above for shear layers and Rayleigh–Taylor flows, but also in other flows, including grid turbulence [42], which has been treated as the universal flow par excellence. Once the flow has been launched into a particular dynamic state, or strange attractor, there is no “entropic spring” that will return it to some universal, dynamic-equilibrium state; the prerequisites that render this the expected behavior in statistical mechanics and thermodynamics, for example, are not at work here. Conversely, the evidence indicates that Re effects on the dynamics and mixing become substantially weaker as Re increases and, in particular, for i.e. beyond the mixing transition. This is illustrated in the image sequence in Fig. 11, which includes (a) a buoyant flame at the pair of jet images (b, c) at and 104, respectively, and (d) a rocket exhaust at an estimated The first three differ considerably, even though separated from each other by a factor of, roughly, four in Re. In contrast, the difference between the last two is not great, even though their Reynolds numbers are separated by four orders of magnitude. Consequently, experimental/DNS results at should be viewed with care, with extrapolation to the higher Reynolds numbers typically of interest of dubious validity. Figure 11 illustrates the caveat.

7.

CONCLUSIONS

Laboratory and DNS experiments indicate that turbulence can transition to a fully developed state, characterized by enhanced mixing, at

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Figure. 11

341

Turbulent-jet flows at increasing Re. [*Photo (d) from Los Angeles (Menlo

Park, Calif.: Sunset Lane, 1968), 246–247.]

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or At least for flows away from solid boundaries, the Re-dependence of the dynamics for is found to be weak. This suggests that DNS at just about within reach today for many flows, may serve as a surrogate for simulations of higher-Re flows, at least for free-shear flows. There can be no question that, if not DNS, then large-eddy simulation (LES) and/or sub-grid scale (SGS) simulations will serve as the solution to the “turbulence problem” in the long run. At this time, however, LES/SGS approaches to mixing and chemically reacting flows have had only limited success. To my knowledge, only one recent SGS scalar-mixing model appears to have captured elements of the mixing transition [43], indicating decreasing scalar variance, until and essentially constant beyond that. The effects of variable density and compressibility, Schmidt number, and differential or multicomponent diffusion are not in the offing in SGS mixing models at this time, however. In closing, and considering that this is part of the ICTAM 2000 Proceedings, it is worth noting that of all the classical physics problems bequeathed as such by the 19th century to the 20th, turbulence is the only one we are handing the baton to the 21st. I would like to hope that some lecturer at the transition of the next century will not be making a similar observation.

Acknowledgments I would like to acknowledge the extensive and expert support and contributions by D. Lang in the Caltech experiments reviewed, in all matters regarding digital imaging, electronic, computing, and networking, as well as the JPL imaging team, led by M. Wadsworth. Discussions and contributions by the many collaborators cited, as well as R. D. Henderson, S. V. Lombeyda, D. I. Meiron, and J. M. Patton of Caltech, and P. L. Miller and T. A. Peyser of the Lawrence Livermore National Laboratory (LLNL) are also gratefully acknowledged. The recent Caltech work on mixing reviewed was sponsored by the Air Force Office of Scientific Research Grants F49620–94–1–0353 and F49620–98–1–0052, and the DOE/Caltech ASCI/ASAP subcontract B341492. The collaborative work with LLNL was performed under U.S. Department of Energy contract W–7405–Eng–48.

References [1] Kolmogorov, A. N. 1941. Local structure of turbulence in an incompressible viscous fluid at very high Reynolds numbers. Akademiia Nauk SSSR Doklady 30, 301–305. [2] Kolmogorov, A. N. 1941. Dissipation of energy in locally isotropic turbulence. Akademiia Nauk SSSR Doklady 32, 19–21. [3] Batchelor, G. K. 1959. Small-scale variation of convected quantities like temperature in turbulent fluid. Part I—General discussion and the case of small conductivity. Journal of Fluid Mechanics 5, 113–133.

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[4] Batchelor, G. K. 1953. The Theory of Homogeneous Turbulence. Cambridge, U.K.: Cambridge University Press. [5] Tennekes, H., and J. L. Lumley. 1972. A First Course in Turbulence. Cambridge, Mass.: MIT Press. [6] Kolmogorov, A. N. 1962. A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. Journal of Fluid Mechanics 13, 82–85. [7] Oboukhov, A. M. 1962. Some specific features of atmospheric turbulence. Journal of Fluid Mechanics 13, 77–81. [8] Monin, A. S., A. M. Yaglom, and J. Lumley (eds.). 1975. Statistical Fluid Mechanics, Vol. 2: Mechanics of Turbulence. Cambridge, Mass.: MIT Press. [9] Hinze, J. O. 1975. Turbulence, 2nd ed. New York: McGraw-Hill. [10] Dimotakis, P. E. 2000. The mixing transition in turbulence. Journal of Fluid Mechanics 409, 69–97. [11] Konrad, J. H. 1976. An experimental investigation of mixing in two-dimensional turbulent shear flows with applications to diffusion-limited chemical reactions. Ph.D. thesis, California Institute of Technology. [12] Bernal, L. P., R. E. Breidenthal, G. L. Brown, J. H. Konrad, and A. Roshko. 1979. On the development of three-dimensional small scales in turbulent mixing layers. Proceedings of the 2nd International Symposium on Turbulent Shear Flows. New York: Springer-Verlag, 305–313. [13] Koochesfahani, M. M., and P. E. Dimotakis. 1986. Mixing and chemical reactions in a turbulent liquid mixing layer. Journal of Fluid Mechanics 170, 83–112. [14] Dimotakis, P. E., R. C. Miake-Lye, and D. A. Papantoniou. 1983. Structure and dynamics of round turbulent jets. Physics of Fluids 26, 3185–3192. [15] Gilbrech, R. J. 1991. An experimental investigation of chemically-reacting, gasphase turbulent jets. Ph.D. thesis, California Institute of Technology. [16] Gilbrech, R. J., and P. E. Dimotakis. 1992. Product formation in chemicallyreacting turbulent jets. AIAA 30th Aerospace Sciences Meeting, Paper 92–0581. [17] Shan, J. W. 2001. Mixing and isosurface geometry in turbulent transverse jets. Ph.D. thesis, California Institute of Technology.

[18] Catrakis, H. J., and P. E. Dimotakis. 1996. Mixing in turbulent jets: Scalar measures and isosurface geometry. Journal of Fluid Mechanics 317, 369–406. [19] Dimotakis, P. E., and H. J. Catrakis. 1996. Turbulence, fractals, and mixing. NATO Advanced Studies Institute series on Mixing: Chaos and Turbulence. Report FM97–1, Graduate Aeronautical Laboratory, California Institute of Technology. [20] Slessor, M. D., C. L. Bond, and P. E. Dimotakis. 1998. Turbulent shear-layer mixing at high Reynolds numbers: Effects of inflow conditions. Journal of Fluid Mechanics 376, 115–138. [21] Bradshaw, P. 1966. The effect of initial conditions on the development of a free shear layer. Journal of Fluid Mechanics 26(2), 225–236.

[22] Dimotakis, P. E., and G. L. Brown. 1976. The mixing layer at high Reynolds number: Large-structure dynamics and entrainment. Journal of Fluid Mechanics 78, 535–560 + 2 plates.

[23] Ho, C.-M., and P. Huerre. 1984. Perturbed free shear layers. Annual Review of Fluid Mechanics 16, 365–424. [24] Karasso, P. S., and M. G. Mungal. 1996. Scalar mixing and reaction in plane liquid shear layers. Journal of Fluid Mechanics 323, 23–63.

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[25] Mungal, M. G., and P. E. Dimotakis. 1984. Mixing and combustion with low heat release in a turbulent mixing layer. Journal of Fluid Mechanics 148, 349–382. [26] Broadwell, J. E., and R. E. Breidenthal. 1982. A simple model of mixing and chemical reaction in a turbulent shear layer. Journal of Fluid Mechanics 125, 397–410. [27] Broadwell, J. E., and M. G. Mungal. 1991. Large-scale structures and molecular mixing. Physics of Fluids A 3(5), Part 2, 1193–1206. [28] Dimotakis, P. E. 1987. Turbulent shear layer mixing with fast chemical reactions. In Turbulent Reactive Flows (R. Borghi and S. N. B. Murthy, eds.), Lecture notes in Engineering 40. New York: Springer-Verlag, 417–485. [29] Peters, N. 1984. Laminar diffusion flamelet models in non-premixed turbulent combustion. Progress in Energy and Combustion Science 10, 319–339. [30] Dimotakis, P. E. 1991. Turbulent free shear layer mixing and combustion. Chapter 5 in High Speed Flight Propulsion Systems. Progress in Astronautics and Aeronautics 137, 265–340. [31] Egolfopoulos, F. N., P. E. Dimotakis, and C. L. Bond. 1996. On strained flames with hypergolic reactants: The H2/NO/F2 system in high-speed, supersonic and subsonic mixing-layer combustion. Proceedings of the 26th International Combustion Symposium, 2885–2893. [32] Rayleigh, Lord. 1883. Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proceedings of the London Mathematical Society 14, 170–177. [33] Taylor, G. I. 1950. The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proceedings of the Royal Society of London A 201, 192–196. [34] Duff, R. E., F. H. Harlow, and C. W. Hirt. 1962. Effects of diffusion on interface instability between gases. Physics of Fluids 5, 417–425. [35] Sharp, D. H. 1984. An overview of Rayleigh–Taylor instability. Physica D 12, 3–18. [36] Annuchina, N. N., Yu. A. Kucherenko, V. E. Neuvazhaev, V. N. Ogibina, L. I. Shibarshov, and V. G. Yakovlev. 1978. Turbulent mixing at an accelerating interface between liquids of different densities. Izvestiia Akademiia Nauk SSSR, Mekhanika Zhidkosti i Gaza 6, 157–160. [37] Read, K. I. 1984. Experimental investigation of turbulent mixing by Rayleigh– Taylor instability. Physica D 12, 45–58. [38] Youngs, D. L. 1984. Numerical simulation of turbulent mixing by Rayleigh– Taylor instability. Physica D 12, 32–44. [39] Dimonte, G., and M. Schneider. 2000. Density ratio dependence of Rayleigh– Taylor mixing for sustained and impulsive acceleration histories. Physics of Fluids 12, 304–321. [40] Cook, A. W., and P. E. Dimotakis. 2000. Transition stages of Rayleigh–Taylor instability between miscible fluids. Report No. UCRL–JC–139044, Lawrence Livermore National Laboratory. Journal of Fluid Mechanics (submitted). [41] Dimotakis, P. E. 1997. Non-premixed hydrocarbon flame. Nonlinearity 7, 1–2. [42] George, W. 2000. Decay of isotropic turbulence. Physics of Hydrodynamic

Turbulence (31 Jan–30 June, 2000, University of California, Santa Barbara), http://online.itp.ucsb.edu/online/ hydrot00/george/. [43] Pullin, D. I. 2000. A vortex-based model for the subgrid flux of a passive scalar. Physics of Fluids 12, 2311–2319.

KINETIC AND CONTINUUM DESCRIPTIONS OF GRANULAR FLOWS Isaac Goldhirsch Department of Fluid Mechanics and Heat Transfer, Tel-Aviv University, Tel-Aviv, Israel [email protected] Abstract

1.

Unlike the basic units of molecular systems, the elementary constituents of granular matter (the grains) experience dissipative interactions. This fact is the root cause of many of the difficulties encountered in the study of granular materials and among its major consequences are the existence of unique states and instabilities (e.g. collapse and clustering) as well as the multistable/metastable nature of most granular states. Another central consequence is the inherent lack of scale separation. The latter is responsible e.g. for the prominent normal stress differences (anisotropic pressures), long-range correlations, scale-dependent stress (and other) fields and, in general, the rheological nature of these materials. Constitutive relations and boundary conditions for dilute and near-elastic rapid granular flows have been derived from the pertinent Boltzmann equation with extensions to moderate densities obtained by employing the Enskog–Boltzmann equation or (systematically, via) response theory. Unlike the rapid flows, the dense, static and quasistatic regimes have not been treated in a systematic fashion heretofore. Preliminary results on elasticity in the static regime are presented.

INTRODUCTION

There are at least three major reasons to study granular materials: they are abundant and of great importance on Earth (sand, gravel, soil) and in space (Saturn’s and other planets’ rings, interstellar dust), they are of central importance in a large number of industries (chemical, construction, food) and in the environment (pollutants, snow avalanches, mud and rock slides), and they exhibit numerous unusual and unexpected phenomena whose understanding poses exciting challenges [1]. The storage, transport, and, in general, handling of granular matter encounters serious difficulties [2], which range from undesired clogging, through unwanted mixing or segregation, to the structural collapse of silos. Granular matter is often described as ‘unpredictable’, ‘irreproducible’ or ‘erratic’. These and other adjectives used to characterize 345 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 345–358.

© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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granular materials are a clear sign that much is still lacking in our understanding of these materials. A variety of approaches have been proposed and applied to the description of the statics and dynamics of granular matter, ranging from tables (e.g. Jenike tables [3]) for engineering purposes, through phenomenological/empirical constitutive relations [4], to equations of motion that are based on fundamental considerations and systematic derivations. The latter are usually limited to the fully fluidized regime of granular matter, otherwise known as “rapid granular flow” [5], and they are restricted to the near-elastic regime as explained below. This paper has two major goals. The first is to identify some of the root causes for the difficulties encountered in the study of granular materials. The second is to report some recent progress in the theoretical description of these materials.

2.

SOME CONSEQUENCES OF DISSIPATIVE INTERACTIONS

When granular materials are strongly forced (e.g. by vibration or shear) they can become fully fluidized, the grains interacting by (inelastic) collisions. This ‘rapid granular flow’ is strongly reminiscent of the classical model of a gas. The analogy led to the definition of granular temperature [6] as a measure of the velocity fluctuations in ‘granular gases’, and later to the development of kinetic, Boltzmann-based theories [7]–[14] for these ‘gases’. The other extreme, i.e. the static state of granular media, is clearly reminiscent of the solid state of matter; as typical granular solids are not in lattice configurations, the molecular analogue would be amorphous or disordered solids. These observations suggest that methods borrowed from statistical mechanics (such as the Boltzmann equation mentioned above) should be useful for the description of granular systems. However, in importing these methods to the study of granular materials one should exercise extreme care as the dissipative nature of grain interactions (inelastic collisions, friction) is the source of significant differences between ‘molecular’ and ‘granular’ materials. Practically all states of granular matter are metastable. For instance, the ground state of a sand pile is one in which all grains reside on the ground; common piles owe their very existence to the frictional forces among the grains. Notice also that static granular piles cannot spontaneously rearrange themselves; thus their configurations are history dependent. Granular gases are different from classical gases not only in the typical dimensions of the constituents. For instance, a granular gas possesses

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no equilibrium state at finite temperature, as any initial state must lose energy to inelasticity (i.e. to internal degrees of freedom of the grains); hence its granular temperature must decay to zero (the only possible equilibrium state). The only way one can prevent this energy loss is by external replenishment, but then the gas is in a non-equilibrium state. The absence of equilibrium is the source of some difficulties in the development of a perturbative (Chapman–Enskog [15]) approach to the kinetics of granular gases, as this state is usually the zeroth order in the Chapman–Enskog expansion. Typical grains are (by definition) macroscopic. Hence grain/container size ratios are always far larger than the corresponding ratio in molecular systems. This ‘practical’ aspect of the lack of scale separation in granular systems is not a fundamental property of these systems. It turns out that granular systems lack scale separation in a fundamental way, i.e. even in the ideal ‘thermodynamic’ limit. This fact has far reaching consequences [16].

2.1.

Lack of scale separation

Granular systems possess correlations whose typical scales are comparable with the scale of the system itself; as such they may be coined mesoscopic [16]. For instance, static granular systems may possess arches that span the entire size of the system (or a significant part thereof). Recent experiments [17] revealed the existence of ‘stress chains’ or ‘force chains’ in static and quasistatic granular systems, which are fractallike strings of particle contacts that span distances comparable with the size of the system and along which the ‘large’ forces (beyond a cutoff) ‘propagate’. Interestingly, rapid granular flows lack scale separation as well [16]. Consider the case of a linearly sheared monodisperse granular system in which the particle interactions are collisions characterized by a fixed coefficient of normal restitution e. Let the macroscopic velocity field V be given by (a well known solution of the pertinent equations of motion), where is the shear rate, y is the spanwise coordinate and is a unit vector in the streamwise direction. Both phenomenological [18, 19] (even dimensional) and kinetic theoretical calculations [10]–[12] show that the granular temperature, T (defined as the mean square fluctuating velocity), in this system approaches a steady state value given by

where C is a volume fraction dependent prefactor, whose value at low volume fractions can be shown to equal approximately 0.6 in two dimen-

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sions [10, 11] and 3 in three dimensions [12], is the ‘degree of inelasticity’ defined by and is the mean free path of the particles. The change of the macroscopic velocity over a distance of a mean free path, in the y direction, is given by A shear rate can be considered small if is small with respect to the thermal speed Here, following i.e. the shear rate is not small unless the system is nearly elastic. (Notice that for , for example, This result alone implies that the Chapman–Enskog expansion of the constitutive relations (which is truly a gradient expansion, formally an expansion in the Knudsen number K; in the discussed case, must be carried out beyond the Navier–Stokes order (linear in K), the lowest next orders being the Burnett [10, 11, 12, 20] and super-Burnett orders. This is why the normal stress differences, a Burnett order effect, are prominent in granular gases. As the Burnett (and super-Burnett) equations are ill-posed [21, 22], resummation techniques [22, 23] may have to be used to tame the ill-posedness. Another problem associated with the gradient expansion is that higher orders in the gradients may be non-analytic [24], indicating non-locality. (A second argument for non-locality is presented below.) Still another consequence of the above findings is that granular flows are typically supersonic [16]. This can be understood on the basis of the following argument: consider two particles moving (approximately) in the same direction; as they collide their relative velocity decreases but their total momentum remains unchanged, which amounts to a decrease in the fluctuating component of their velocities, i.e. the granular temperature. Moreover, the same mechanism is responsible for the emergence of correlated strings of particles—an effect which is also known [25] in molecular systems at high shear rates; here it is generic—hence to a violation of the hypothesis of molecular chaos [26]. The mean free time equals the ratio of the mean free path and the thermal speed: Clearly, is the microscopic time scale characterizing the system at hand and is the macroscopic time scale characterizing it; hence the ratio is a measure of the degree of temporal scale separation. Employing (1), one obtains an O(1) quantity except in the near-elastic case. Thus typically the considered system lacks temporal scale separation irrespective of its size or the size of the grains. Consequently, one cannot a priori employ the assumption of “fast local equilibration” and/or use local equilibrium as a zeroth order distribution function unless the system is nearly elastic. The latter condition severely limits the applicability of the hydrodynamic description. For instance, consider the application of these equations to a stability study. As expected (and is well known [19, 27]) some of the eigenvalues of the granular stability problem (includ-

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ing those corresponding to instabilities) must be of the order of the only “input” inverse time scale (in the absence or irrelevance of gravity), i.e. Since, as explained above, one obtains instabilities whose characteristic times are comparable with the mean free time. Thus one encounters a paradoxical situation in which the time constants corresponding to the development of instabilities (the inverse values of the corresponding growth rates) predicted on the basis of the hydrodynamic equations may be shorter than the time scales that the hydrodynamic

equations themselves resolve. Another conclusion from this observation is that one cannot distinguish between microscopic and macroscopic fluctuations; for instance, pressure fluctuations which are merely results of single particles impacting on a wall are measurable on macroscopic time scales and appear as ‘intermittent events’ [28]. A similar phenomenon has been reported in the dense case [29]. The free path of a particle, i.e. the distance it moves in between consecutive collisions, depends on its absolute velocity; hence it is not Galilean invariant. It follows that the mean free path is not a Galilean invariant entity either (one can define a Galilean invariant mean free path in a co-moving, Lagrangian frame; this entity is denoted above by Denote the instantaneous velocity of a particle by v. Let where V denotes the macroscopic velocity at the particle’s position and is the fluctuating part of its velocity. It follows (assuming statistical independence of and V) that the average (denoted by an overbar) of is hence the typical velocity u of a particle is or in the specific case considered, Multiplying u by the mean free time (as mentioned, a Galilean invariant entity), one obtains the mean free path At values of y at which the speed is subsonic (following the above considerations this happens when one can neglect with respect to T, in which case However, when , in particular when the thermal speed is far smaller than the average speed (i.e. the flow is supersonic) and in this

case, i.e. the ‘true’ mean free path is (much) larger than the equilibrium mean free path. Moreover it is of macroscopic dimensions, being an O(1) quantity times This implies that the considered system has long-range correlations, unless is small enough for a given system size, a fact that may invalidate the hydrodynamic equations, unless additional fields are used [16].

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Scale dependence of stress

In the realm of molecular fluids (when they are not under very strong thermal or velocity gradients) there is a range, or plateau, of scales that are larger than the mean free path and far smaller than the scales characterizing macroscopic gradients, and that can be used to define “scale independent” densities (e.g. mass density, momentum density, energy density or temperature) and fluxes (e.g. stresses, heat fluxes). Such

plateaus are virtually non-existent in systems in which scale separation is weak, and therefore these entities are scale dependent. By way of example, the “eddy viscosity” in turbulent flows is a scale-dependent (or resolution-dependent) quantity, since in the inertial range of turbulence there is no scale separation. There is a plenitude of “rheological materials” in which the lack of scale separation is associated with scale dependence of stresses and other fields. The scale-dependent entity discussed below [28] is the stress tensor. For simplicity we shall mostly discuss the kinetic part of the stress tensor,

which dominates at low volume fractions. The kinetic theoretical expression for this tensor is where is the ensemble average of A, v' is the fluctuating part of the velocity, and is the mass density. It can be shown [28, 30, 31] that the stress field can be defined for single realizations (hence one does not need to invoke the notion of an ensemble; the latter may not be known) in such a way that the standard continuum equation of motion, holds. The general theory of [28] allows for a variety of coarse graining functions (or averaging weights). Here we choose to study the simplest version, which also corresponds to standard practice in computer simulations. Let the macroscopic velocity field V(r, t) be defined as the center of mass velocity, at time t, in (say) a cube of side w of which r is the geometrical center. The fluctuating velocity of a particle i residing in this cube at time t is defined as The stress tensor at point r at time t is defined as where denotes the average over all particles in the cube. Define next to be the fluctuation of the velocity of a particle with respect to the average velocity at its instantaneous position and let be the difference between the average velocity at and the coarse graining center r. Clearly The above decomposition yields two contributions to the kinetic stress tensor: the first is a ‘standard’ kinetic contribution, and the second is a contribution which is proportional to the square of the velocity gradient tensor and which stems from the fact that the variation of the macroscopic velocity in the coarse graining cell

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is a contribution to the velocity fluctuations. The latter contribution is negligibly small in molecular systems as changes of velocity over typical coarse graining scales (of the order of the mean free path) are small. In contrast, the second contribution can be dominant in the realm of granular flows. In the case of a linear shear flow, the contribution of the second term to the normal stress is proportional to whereas the kinetic contribution (in the dilute case) is The ratio of these contributions, is typically a very small quantity in molecular gases; in contrast, in granular gases this ratio is proportional (following (1)) to a large quantity for The above (and other, related) results have been corroborated by simulations [28].

2.3.

Collapse and clustering

Clustering [19, 32] and collapse [33] are effects that are specific to granular systems; they are results of the inelastic nature of these systems and have no equivalent in atomic gases.

Collapse. The essence of the collapse phenomenon [33] can perhaps be explained as follows. Consider a hard sphere bouncing off a floor. Assume that when the sphere hits the floor with velocity v, it bounces off with velocity ev, where is the coefficient of restitution. An elementary exercise shows that if the sphere is dropped from rest at an initial height it reaches, upon rebouncing n times, a maximal height of Let denote the time that elapses between consecutive maximal height positions of the particle. It is easy to show that Since the sum of is finite it follows that an infinite number of collisions occurs in a finite time. A similar effect occurs in many-body granular systems where the “floor” is replaced by a collection of particles and where the relative velocities of the colliding particles become very small, leading (via a theoretically infinite number of collisions) to the emergence of strings of particles whose relative velocities vanish [33]. Clustering. Granular materials have a tendency to form dense clusters even when the ‘initial condition’ is homogeneous. Unlike the collapse phenomenon, clustering is a hydrodynamic effect (but it may precede collapse [9]). An intuitive explanation [19] of the clustering phenomenon proceeds as follows. Consider a region in a granular gas in which the density is increased due to a fluctuation, without a change in the granular temperature T. In this region the frequency of collisions is larger than in neighboring, less dense regions. Since the collisions are inelastic, the granular temperature in the dense region decreases faster than in the neighboring

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regions, causing the pressure in this region to be lower than that in the neighboring regions. The resulting pressure gradient leads to a migration of particles into the dense region, thus further increasing its density and decreasing its pressure. Thus, once commenced, a density fluctuation leads to the formation of a dense cluster, provided the (hydrodynamic) mechanisms that may disperse the agglomeration of particles are slower than the clustering process. A more quantitative analysis of the clustering phenomenon requires a detailed study of the hydrodynamic equations for granular gases. Here, a simplified derivation is presented. Consider a monodisperse collection of spheres of radius a whose collisions are characterized by a fixed coefficient of normal restitution e. The pressure in this dilute granular gas is given by where is the (mass) density. Let m be the mass of a particle and n be the number density. Recall the standard kinetic (or mean free path theory) formula for the mean free path, where is the total collision cross section of two particles. It follows that i.e. the pres-

sure is inversely proportional to the number density n, in accord with the above physical considerations. Numerical simulations of unforced as well

as of sheared granular gases [19, 27] corroborate this result. An analysis [19] of the equation for the momentum density, reveals that the clustering time is shorter the shorter the scale of the fluctuation, an intuitively plausible result (it takes less time for mass to move a shorter distance). On the other hand, an inspection of the equation of motion of the temperature field reveals that the above neglect of the diffusive terms in the hydrodynamic equations is justified only when Hence the fastest, and consequently dominant, cluster formation process occurs at i.e. the typical cluster separation distance is When is larger than the system size, one does not expect clustering to occur. Instead, one expects a shear mode and an unstable density fluctuation mode of the smallest wavenumber allowed

by the system’s geometry [19]. Consider a system under time-dependent shear [27]. As energy is pumped into this system at the boundaries, it heats up near the boundaries, increasing the pressure there; the result is the creation of dense clusters (or plugs) ‘far’ from the boundaries. Other states of sheared flows are possible as well. In general, the clustering mechanism is responsible for many of the hysteretic/multistable properties of rapid granular flows.

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KINETIC THEORY OF GRANULAR GASES

It is easy to modify the Boltzmann equation to account for inelastic collisions [8, 10, 11, 12]. Pioneering studies [7] of granular kinetics employed the Enskog equations of motion for the moments of the velocity and obtained closures by assuming specific forms of the distribution function. Recent progress includes the development of a systematic approach to the problem, which is based on the Chapman–Enskog expansion [10, 11, 12, 14]. Further developments include the use of response theory [34] to extend the validity of the Boltzmann constitutive relations to moderately dense granular gases. The rational approaches to the granular kinetic problem have led to the discovery of novel terms in the constitutive relations; there are also some quantitative differences between the phenomenological results and those obtained systematically. Another discovery is that the normal stress differences are truly Burnett effects, which are prominent in the granular case because of the lack of scale separation. As mentioned above, these recent studies have revealed limitations of the constitutive relations, in particular the fact that they are restricted to the near-elastic case.

3.1.

Developing a Chapman—Enskog expansion

Following Eqn. (1) it is convenient to consider the mathematical limit in which both the shear rate (more generally, the Knudsen number) and the degree of inelasticity tend to zero in such a way that the temperature is kept fixed (i.e. In this (elastic, unforced) limit the system evolves toward a state of equilibrium. It can be shown that this limit is not singular. Consequently, one can scale the shear rate by define an expansion (of the distribution function) in powers of and employ this expansion to solve the pertinent Boltzmann equation in powers of this small parameter. This expansion is limited to steady states alone [10, 11]. In order to obtain constitutive relations that are not restricted to steady state situations, one can define a double expansion in powers of the Knudsen number K and the degree of inelasticity [12]. As expected, this expansion reduces to that described above in the steady case. The leading order (elastic) viscous contribution to the stress tensor is O ( K ) (the Navier–Stokes order) and the leading inelastic correction is Since in the steady state, this correction is also hence one needs to calculate the super-Burnett, i.e. for consistency. Some of the conclusions stemming from this finding have been stated above. Explicit constitutive relations can be found in [12] for three dimensions and in [10, 11] for two. It is appropriate to mention the

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existence of a competing method [14] for deriving constitutive relations for rapid granular flows; the latter is based on an expansion around the ‘decaying’ unforced, homogenous granular system.

3.2.

Boundary conditions

Except for the study of (idealized) infinite systems, the constitutive relations are useless without corresponding boundary conditions [35]. Even in the realm of molecular systems, there have been no systematic derivations of boundary conditions at gas–solid interfaces heretofore. The basic problem encountered in attempts to derive such boundary conditions is the lack of a small parameter. However, it turns out that one can derive an expansion of the boundary conditions in which the zeroth order corresponds to the Maxwell condition (in which it is assumed that the distribution function of the particles colliding with the boundary is Maxwellian) and the nth order corresponds to a particle that has collided n times (in between collisions with the boundary); hence the incoming distribution function is affected by the presence of the boundary [36]. While this expansion does not have a formal small parameter, it does converge as (due to the spectrum of the linearized Boltzmann operator) it takes only a few collisions per particle to achieve local equilibration. The latter assumption restricts the theory to near-elastic systems, much like the restriction on the Boltzmann-based constitutive relations. One of the results obtained using this formulation is that for a boundary orthogonal to the z direction that is characterized by a degree of inelasticity the boundary condition for the z component of the velocity in the presence of a thermal gradient in the z direction is to lowest order in the collisions. This boundary condition is quite surprising as it implies a violation of mass conservation in steady states when The resolution of this “paradox” is found by noting that in a steady sheared state, the order (which is the order of the above expression for and the order are the same; hence a super-Burnett correction should be added to the above expression for to obtain the vanishing of at the boundary in a steady state.

3.3.

Moderately dense gases

The Enskog–Boltzmann equation, in which a phenomenological, density-dependent collision kernel represents the effect of finite density, has been useful for deriving constitutive relations for moderately dense molecular and granular gases alike. This equation does not contain information on dynamic correlations (in particular repeated collisions, also

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known as ring events). A general approach to this problem is offered by the (nonlinear) response theory, originally developed for molecular systems. A direct generalization of this theory to granular systems is inapplicable as the theory requires energy conservation to hold. However, a variant of the response method, proposed by Ronis [37], can be modified for the derivation of constitutive relations for granular gases [34]. For sake of this derivation it is useful to measure time in units of the number of collisions accumulated (on average) by a particle and consider the (unstable) homogeneous cooling state as the zeroth order in a gradient expansion. (In this case, unlike that in which the Boltzmann equation is employed, there is no a priori restriction on the density.) Among the results obtained by employing this formulation are Green–Kubo formulae for the dissipative coefficients, which, similarly to the molecular case, assume the form of time integrals of correlations of dissipative fluxes. A contribution to the energy sink term (due to inelasticity) in the equation of motion for the granular temperature, which is proportional to the divergence of the velocity field, is one of the new finite-density terms obtained by employing the response formulation.

4.

ELASTICITY OF GRANULAR SYSTEMS

Dense granular systems in three dimensions and polydisperse granular systems in two and three dimensions are disordered. The derivation of constitutive relations for these systems requires the identification of relevant characterizations of their configurations. A sufficient characterization is not known at present [1]. Surprisingly, even the elastic state of amorphous solids or granular systems is not well characterized or understood. (The only elastic theories truly derived from microscopic descriptions pertain to lattice configurations or very specific non-lattice ones.) As a matter of fact, even the definition of such a basic entity as strain is under debate in the literature [38]. It is the goal of this short section to shed some light on the latter problem [39]. A common method for determining strain in a granular (or random) system is to fit the displacements of the particles to a linear or piecewise linear function of their coordinates, the ‘prefactor’ being identified as the strain. The mean field methods that are defined by this procedure (e.g. [38]) are based on the (implicit or explicit) assumption that ‘to lowest order’ a particle’s displacement is determined by the local strain field at its position. This assumption obviously breaks down when strain fluctuations are significant or when (another implicit assumption) the number density of the particles is significantly non-constant. It is claimed here that continuum mechanics actually teaches us how to

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resolve this problem in a manner that is consistent with general principles [39]. Indeed, the displacement field is defined as the Lagrangian integral of the (macroscopic) velocity field. The latter can be expressed as where n is the number density (assuming here a monodisperse system) and is a coarse graining (averaging) function. This definition is compatible with the continuum mechanics equations of motion. Upon integrating this expression from time 0 to time t and assuming, for sake of simplicity, that the system is unstrained at one obtains, following a straightforward integration by parts, that the displacement field u(r, t) is given, up to a quadratic error in the strain, by where are the displacements of the particles. This formula has been corroborated by extensive numerical simulations. One of the findings of the simulations

(and the theory) is that the more random the system is the smaller the range of strain values for which linear elasticity holds [39].

5.

CONCLUSION

Granular gases, as well solids, are generically mesoscopic and require rheological descriptions except in the case of near-elastic granular gases. The lack of scale separation, which follows from the dissipative nature of the grain interactions seems to be one of the most important factors underlying the difficulties encountered in developing theories of granular matter. The lack of scale separation also offers certain advantages, e.g. one is in a unique situation to ‘see’ the interior of a shock wave, as the latter may have a macroscopic width. Granular materials possess states and instabilities specific to them; in addition they amplify properties of molecular systems (such as normal stress differences) from nearly negligible values to O(1), thus serving (among other things) as a macroscopic laboratory for the study of molecular systems (provided the differences between the two classes of systems are understood). Dense granular matter bears some similarities to amorphous solids. The properties of the latter are not well understood and thus the study of dense granular materials forces one to consider fundamental questions that are not limited to this field alone.

Acknowledgments This work has been partially supported by the National Science Foundation, the Department of Energy, the United States–Israel Binational Science Foundation (BSF) and the Israel Science Foundation (ISF).

References [1] —. 1999. The entire issue of Chaos 9(3) is devoted to granular matter.

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[2] Knowlton, T. M., J. W. Carson, G. E. Klitzing, and W.-C. Yang. April 1994. Particle technology: The importance of storage, transfer and collection. Chemical Engineering Progress 44, 44–54. [3] Jenike, A. W. 1964. Storage and flow of solids. Bulletin of the University of Utah Engineering Experiment Station 123. [4] Johnson, P. C., P. Nott, and R. Jackson. 1990. Frictional–collisional equations of motion of particulate flows and their application to chutes. Journal of Fluid Mechanics 210, 501–535. [5] Campbell, C. S. 1990. Rapid granular flows. Annual Reviews of Fluid Mechanics 22, 57–92, and references therein. [6] Ogawa, S., A. Unemura, and N. Oshima. 1980. On the equations of motion of fully fluidized granular materials. Zeitschrift für Mechanik und Physik 31, 482– 493, and references therein. [7] For pioneering works in this field, see [5] and references therein, and Lun, C. K. K. 1991. Kinetic theory for granular flow of dense, slightly inelastic, slightly rough spheres. Journal of Fluid Mechanics 223, 539–559, and references therein. [8] Goldshtein, A., and M. Shapiro. 1995. Mechanics of collisional motion of granular materials, Part I: General hydrodynamic equations. Journal of Fluid Mechanics 282, 75–114. [9] Sela, N., and I. Goldhirsch. 1995. Hydrodynamics of a one-dimensional granular medium. Physics of Fluids 7(3), 507–525. [10] Goldhirsch, I., N. Sela, and S. H. Noskowicz. 1996. Kinetic theoretical study of a simply sheared granular gas—to Burnett order. Physics of Fluids 8(9), 2337– 2353.

[11] Goldhirsch, I., and N. Sela. 1996. Origin of normal stress differences in rapid granular flows. Physical Review E 54(4), 4458–4461 (1996). [12] Sela, N., and I. Goldhirsch. 1998. Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. Journal of Fluid Mechanics 361, 41–74, and references therein. [13] Brey, J. J., F. Moreno, and J. W. Dufty. 1996. Model kinetic equations for low density granular flow. Physical Review E 54(1), 445–456. [14] Brey, J. J., J. W. Dufty, C. S. Kim, and A. Santos. 1998. Hydrodynamics for granular flow at low density. Physical Review E 58, 4638–4653.

[15] Chapman, S., and T. G. Cowling. 1970. The Mathematical Theory of Nonuniform Gases. Cambridge: Cambridge University Press. [16] Tan, M.-L., and I. Goldhirsch. 1998. Rapid granular flows as mesoscopic systems. Physical Review Letters 81(14), 3022–3025. [17] Jaeger, H. M., S. R. Nagel, and R. P. Behringer. 1996. Granular solids, liquids and gases. Reviews of Modern Physics 68(4), 1259–1273, and references therein. [18] Haff, P. K. 1983. Grain flow as a fluid mechanical phenomenon. Journal of Fluid Mechanics 134, 401–430. [19] Goldhirsch, I., and G. Zanetti. 1993. Clustering instability in dissipative gases. Physical Review Letters 70, 1619–1622. See also Goldhirsch, I., M.-L. Tan, and G. Zanetti. 1993. A molecular dynamical study of granular fluids I: The unforced granular gas in two dimensions. Journal of Scientific Computation 8(1), 1–40. [20] Burnett, D. 1935. The distribution of molecular velocities and the mean motion in a nonuniform gas. Proceedings of the London Mathematical Society 40, 382– 435.

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[21] See, for example, Bobylev, A. V. 1984. Exact solutions of the nonlinear Boltzmann equation and the theory of Maxwell gas relaxation. Theoretical and Mathematical Physics 60(2) 280–310. See also Gorban, A. N., and I. V. Karlin. 1992. Structure and approximations of the Chapman–Enskog expansion for linearized Grad equations. Transport Theory and Statistical Physics 21, 101–117. [22] Rosenau, P. 1989. Extending hydrodynamics via the regularization of the Chapman–Enskog expansion. Physical Review A 40, 7193–7196. [23] Slemrod, M. 1999. Constitutive relations for monoatomic gases based on a generalized rational approximation to the sum of the Chapman–Enskog expansions. Archive for Rational Mechanics and Analysis 146, 73–90. See also Jin, S., and M. Slemrod. Regularization of the Burnett equations for fast granular flows via relaxation. 2000. Preprint. [24] Ernst, M. H., and J. R. Dorfman. 1976. Nonanalytic dispersion relations for classical fluids II: The general fluid. Journal of Statistical Physics 12(4), 311– 357. [25] Lutsko, J. F. 1996. Molecular chaos, pair correlations and shear induced ordering

of hard spheres. Physical Review Letters 77, 2225–2228. [26] Luding, S., M. Miiller, and S. McNamara. 1998. The validity of molecular chaos in granular flows. World Congress on Particle Technology, Brighton. See also

Soto, R., unpublished. [27] Tan, M.-L., and I. Goldhirsch. 1997. Intercluster interactions in rapid granular

shear flows. Physics of Fluids 9(4), 856–869, and references therein. [28] Glasser, B. J., and I. Goldhirsch. 1999. Scale dependence, correlations and fluctuations of stresses in rapid granular flows. Preprint. [29] Miller, B., C. O’Hern, and R. P. Behringer. 1996. Stress fluctuations for continuously sheared granular materials. Physical Review Letters 77, 3110–3113. [30] Babic, M. 1997. Average balance equations for granular materials. International Journal of Engineering Science 35, 523–548. [31] Murdoch, A. I. 1998. On effecting averages and changes of scale via weighting functions. Archives of Mechanics 50, 531–539, and references therein. [32] Hopkins, M. A., and M. Y. Louge. 1991. Inelastic microstructure in rapid granular flows of smooth disks. Physics of Fluids A 3(1), 47–57.

[33] McNamara, S., and W. R. Young. 1992. Inelastic collapse and clumping in a one-dimensional granular medium. 1992. Physics of Fluids A 4, 496–504. Recent work is cited in Kadanoff, L. P. 1999. Built upon sand: Theoretical ideas inspired by granular flows. Reviews of Modern Physics 71(1), 435–444. [34] Goldhirsch, I., and T. P. C. van Noije. 2000. Green–Kubo relations for granular fluids. Physical Review E 61(3), 3241–3244. [35] For previous work on boundary conditions for granular gases, see Jenkins, J. T., and E. Askari. 1991. Boundary conditions for rapid granular flows. Journal of Fluid Mechanics 223, 497–508, and references therein. [36] Goldhirsch, I. 1999. Scales and kinetics of granular flows. Chaos 9(3), 659–672. [37] Ronis, D. 1979. Statistical mechanics of systems nonlinearly displaced from equilibrium I. Physica 99A, 403–434. [38] See, for example, Liao, C.-L., T.-P. Chang, D.-H. Young, and C. S. Chang. 1997. Stress–strain relationship for granular materials based on the hypothesis of best fit. International Journal of Solids and Structures 34, 4087–4100. [39] Goldenberg, C., and I. Goldhirsch. 2000. Elasticity of microscopically inhomogeneous systems. Preprint.

STATIONARY WAVES IN ELASTO-PLASTIC AND VISCO-ELASTO-PLASTIC BODIES Vladimir A. Palmov St. Petersburg State Technical University, St. Petersburg, Russia [email protected]

Abstract

1.

Attention is paid to the analysis of longitudinal vibration of a semiinfinite rod loaded by a harmonic force at one end. A rod with different constitutive equations is considered, and vibration decay with distance from the loaded end is sought. In order to overcome problems caused by nonlinearity of the constitutive equations, the method of harmonic linearization is used. As a result simple analytical solutions are obtained and analyzed. If the material of a rod is rigid-plastic and has linear kinematical hardening, the vibration occupies only finite domain near the loaded end. The same is true for an elasto-plastic material with a bilinear diagram, while elastic waves are traveling in the remaining part of the rod. In the case of so-called amplitude-dependent internal friction, vibration occupies the entire rod and possesses a saturation property. This property implies that there exists a limit of attainable level of vibration, which depends on the material properties, the distance from the loaded end, and frequency. One special sort of continuous media with complex structure is formulated in a latter section. It is shown that dynamical structure, rather than input power and constitutive equations of components, influences the wave propagation.

FORMULATION OF THE PROBLEM

We consider the semi-infinite rod longitudinal vibration is given by

The governing equation for

where Q denotes a tensile axial force in the cross section with coordinate x, u is axial displacement, and m is the mass per unit length. The dot and prime indicate differentiation with respect to time t and spatial coordinate x, respectively. The axial strain of the rod is

359 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 359–372. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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The constitutive equations for the rod material is assumed to be given by the equation

where is the visco-elasto-plastic functional. We formulate boundary conditions at the cross section can take one of the forms

They

or

At the Sommerfeld condition should be satisfied: it is possible to have waves traveling only in the x direction and not coming back. We intend to consider different constitutive equations (3) and to analyze the peculiarity of traveling and standing waves in a semi-infinite rod.

2.

WAVES IN VISCO-PLASTIC MATERIAL

For determinacy we use the generalized Maxwell rheological model. Acting force Q and strain are connected by the integral functional

where

is the so-called relaxation function. A steady-state vibration of the rod is sought in the form

where Re denotes real part of the complex function and const. Substituting of Eqn. (8) in Eqn. (6) and then in Eqn. (1) leads to the equation for the determination of the unknown quantity

Only that root of this equation having the form

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should be used; only in this case will we have a wave traveling in the x direction:

For pure elastic material of the rod we should put for all relaxation times T except infinite ones. In this case the right hand side of Eqn. (9) becomes real-valued. So, and Eqn. (11) gives a wave that travels in the x-direction without any spatial decay. The simplest analysis of this section is well known. But we pay attention to this analysis for the convenience of comparison with future results.

3.

WAVES IN RIGID-PLASTIC MATERIAL WITH LINEAR HARDENING Let the constitutive equation of the rod material have the form

where c and H are constants. One can easily recognize the deformation law of a rigid-plastic material with linear hardening and the Bauschinger effect being taken into account. Equations (1), (2) and (12) exhibit a nonlinear problem. The method of harmonic linearization (see Ziegler 1995, Palmov 1998) will be applied to solve this problem. This method is one of the simplest and most effective methods of nonlinear vibration theory and is actively used in the theory of automatic control, theory of vibration protection, etc. An essential feature of the method of harmonic linearization is its simplicity and clearness of physical results obtained. In this paper this method is applied to the analysis of wave propagation in a rod whose material is governed by a nonlinear equation. Using the method of harmonic linearization, the strain ε is assumed to be a nearly harmonic process in each cross-section:

where a(x) is the amplitude and is the phase. Now we replace the nonlinear function (12) by the linearized version

Substituting Eqn. (14) in equations (2) and (1), and equating sine and cosine coefficients, we get the system of equations

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Although we have not succeeded in constructing a general solution with four integration constants, we have obtained a solution that has two integration constants and satisfies the Sommerfeld conditions. This solution has the form

Direct substitution into (15) and (16) shows that the constants have the values

and

The constants and are still arbitrary and must be determined from the boundary condition at Solution (17) is valid only for those values of x for which i.e.

For

it should be taken that

A direct substitution of the latter solution into the system of equations (15) and (16) shows that this system is satisfied. Besides, the solutions (17) and (16) meet. Inserting the obtained expressions for amplitude and phase into (17), we get the following expression for the deformation:

Let Eqn. (4) be the boundary condition at Satisfying it by means of a linearized expression for force Q, we obtain the following values for the integration constants:

It follows from this equation that the constructed solution makes sense only if the amplitude of the force in the cross section at is larger then From a physical perspective, this means that force must exceed the yield stress H in law (12). If the force is smaller than then no motion in the rod occurs.

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Displacement in this problem is obtained by integration of the equation for deformation (21), to give

It is seen that in the region near the driven end the displacement is a superposition of running and standing waves. For there is no wave motion. The region of wave motion is obtained by substitution of Eqs. (22) and (18) into (19) and is given by

The lower the material hardening and the higher the frequency of vibration, the smaller is the zone of motion. Further, decreasing the yield stress H or increasing the amplitude enlarges the region of motion. In addition, as Hence, in a rod made of a rigid-plastic material without hardening, the vibration localizes in the cross section where the source of vibration is located.

4.

WAVES IN ELASTO-PLASTIC MATERIAL WITH LINEAR HARDENING

Let the rod material be an elasto-plastic material with a linear hardening. For developed plastic deformation the rod behavior is governed by the following system of equations (see Palmov 1998):

Here denotes the plastic strain. The method of harmonic linearization can be applied directly to Eqn. (25). Following this method, we put

where b and are the amplitude and phase of the plastic strain, respectively. Next, the nonlinear function in (25) is approximated by a linear one

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Inserting this approximation into (25) we find the following expressions for and Q in terms of

Differentiating the equation of dynamics (1) with respect to x and using eqs. (2) and (28), we obtain an equation for the plastic strain:

Substituting the expression for due to (26) into (25) and combining coefficients in sine and cosine, we arrive at following system of two differential equations for the amplitude and phase of plastic strain:

This system of equation has a solution where and are integration constants, and and are easily obtained by substitution of (32) into the system of equations (30) and (31). These are given by

Solution (32) makes sense only in that part of the rod where the amplitude b is positive, i.e. only for x satisfying the inequality

For

one must take

Direct substitution of solution (35) into the system of equations (30) and (31) shows that these equations are satisfied. In addition, the solutions

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(32) for and (35) for meet. Indeed, at the amplitude and phase of the plastic strain are continuous. By virtue of (28) the total strain and force in the rod are continuous as well. Therefore, the plastic strain is equal to zero for The solution constructed satisfies the Sommerfeld radiation condition at infinity, as as Let us determine the integration constants and find the main variables of the problem, namely the total strain and the displacement u. A convenient way to do this is to transform to a complex exponential form in Eqn. (26): which is allowed since Eqs. (28) are linear in It goes without saying that only the real parts of (36) and others have a physical meaning. Inserting (36) into (28) yields the equations

Using the first equation in (37) and the boundary condition in (4) at we easily obtain the amplitude and phase of the plastic strain in the plastic zone:

The solution obtained is seen to be meaningful only if the amplitude of the external force satisfies the inequality

This inequality is a condition of initiation of plastic deformations in the rod. This condition is an approximate one and differs from the exact condition only in an inessential factor in As seen from the second Eqn. (37) the strain amplitude is

or, accounting for the exact expression for b,

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It follows from these equations that the strain amplitude decreases from its maximum value at to the value and then remains constant and is equal to maximal elastic strain. Let us determine the displacement u. To this end, we insert expressions for b and from (32) and (41) into (37), and integrate over x taking into account the continuity of u at and the radiation condition as The result is

The parameters that appear in this equation have been determined above. As seen from (42) a wave of constant amplitude is outgoing at infinity for For the rod vibration is a superposition of propagating and standing waves, where amplitude of the running wave decreases with increasing distance from the source of vibration. Let us draw some conclusions. The cross-section is a border between a part of the rod with plastic deformation and part with pure elastic deformation. An expression for is obtained by substitution of the expressions for (38) and and (33) into (34):

Thus, the higher the material hardening is and the lower the mass per unit length and vibration frequency are, the larger the zone of plastic strain is.

5.

WAVES IN ELASTO-PLASTIC MATERIAL WITH POWER-LAW HARDENING

In this section we make use of the generalized Prandtl model. In Russian literature this model is called sometimes as Ishlinsky model (Palmov 1998). The constitutive equations are as follows:

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where C is longitudinal rigidity, is the plastic strain for a Prandtl element with dimensionless plastic limit h, and p(h) is the probability distribution of plastic limits h. For small h we assume

In this case we have a simple expression for the loading function:

It is widely used in the theory of internal and structural friction. Considering the behavior of material under harmonic deformation in time,

we again exploit harmonic linearization:

Substituting this approximate representation in Eqn. (44) we get

Since Eqns. (44) are linear with respect to and we can use the general complex exponential representations for the variables

where A, a, and are amplitudes, and and are phases. After long-term calculations we can get the simple formula

where

In what follows, is referred to as the complex rigidity. As follows from (52), the complex rigidity does not depend on the deformation frequency for any density p(h). It is easy to see that depends on the amplitude of deformation a. These properties of the model correspond to the laws of internal friction observed by most metals and structural materials. In this sense, the model considered here successfully describes internal friction under intensive stresses.

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Differentiating of Eqns. (1) with respect to x and using Eqns. (2) and (51) renders the equation

Using this equation and first Eqn. (50), we get equations for amplitude B and phase of traction force:

where For example, for the power function p(h) given by (45), we have

Equations (54) and (55) are identical with the equations for dynamics of a point mass in a plane non-central field. The established analogy is useful from two points of view. Firstly, it gives an idea of the character of the difficulties one faces when integrating the system (54), (55). Secondly, it allows the exact and approximate methods developed in the dynamics of a point mass—in particular, in the dynamics of satellites and spacecraft—to be applied. We apply one of these methods: we neglect radial acceleration. In our problem it means that we should neglect the second derivative in Eqn. (54). This gives

Substituting the expression of complex rigidity given by (52) into Eqn. (58) and retaining only terms of the first order in effects of plastic deformation yields

For the power function (45) this equation takes the form

Integrating this equation we obtain

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It is seen that vibration occupies the whole rod, decaying from the value at to zero. The following peculiarity of the vibration field is of interest. For large values of x we can neglect the value of unity in the parentheses of (61) to obtain

The strain amplitude at the loaded end is absent in this equation. Hence it indicates an upper limit of vibration at any point. This limit does not depend on the power of the vibration source, but depends on the properties of material, distance x, and frequency. In particular, it means that amplitude of the excitation cannot be reliably determined by measurements from the far field of elasto-plastic vibrations.

6.

WAVES IN VISCO-ELASTO-PLASTIC MATERIAL

Let us consider now a more complicated rheological model. The model consists of the generalized Prandtl model and generalized Maxwell model, connected in parallel. Repeating the derivation of the previous section we find that the strain

amplitude a(x) decreases and is governed by the following ordinary differential equation: where

Here describes the energy dissipation due to viscosity in the generalized Maxwell part of the model. The integral of Eqn. (63) is given by

As seen from Eqn. (65), vibration occupies the whole rod, decaying from the value at to zero. The vibration distribution has the same qualitative feature as in the previous example. Indeed, an increase of the strain amplitude leads to an increase in the amplitude a at any x. However, this increase at a given x has a limit, which is formally obtained from Eqn. (65) as

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Hence the following inequality holds:

which is similar to inequality (62). Again this indicates the existence of an upper limit of vibration that does not depend on the power of the source of vibration, but is a function of frequency, the rod parameters, and the distance from the source. The effect of vibration saturation observed in the two last examples may be explained as follows. As the power of the source of vibration increases, the density of vibrational energy increases in the cross section near the loaded end. The energy dissipation in this zone increases even more intensively. This results in an absorption of vibration in a region adjoining the loaded end, while the vibrational energy in remote cross sections increases negligibly.

7.

WAVES IN MEDIUM WITH COMPLEX STRUCTURE

We postulate a carrier medium and assume that it is governed by the equation of an elastic rod. A set consisting of an infinite number of noninteracting oscillators with continuously distributed eigenfrequencies is attached to each cross section of the carrier rod (Palmov 1998). The equations of dynamics of this rod with complex structure are as follows:

where u is the displacement of a rod cross section, is the displacement of the oscillator mass with eigenfrequency k, E is Young’s modulus of the carrier rod, and is the density of the carrier rod. The quality of is equal to mass of the oscillators having eigenfrequencies in the interval per unit volume. Thus the total mass density of the oscillators attached to the carrier structure is given by

The rigidity of an oscillator’s suspension is denoted in Eqn. (69) by C(k), which is given by

The term is introduced in Eqn. (69) in order to take into account the energy dissipation in the oscillator suspension. Account of

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damping will be achieved firstly by a visco-elastic model, and then will be generalized by an elasto-plastic or visco-elasto-plastic one. The term with is assumed to be small for any oscillator motion in both cases. Lastly we note that K, Q, exhibit external forces applied to the carrier rod and to the attached oscillators. We formulate the boundary condition at the cross section in the form For we formulated the Sommerfeld condition. If K, Q, equal to zero, we are seeking a wave solution in the form

where

are

is governed by the equation

in which We should select only the root of this equation that has the form

Substitution of this representation in Eqn. (73) renders the simple formula for the displacement of carrier rod

Spatial decay of this wave is determined by the factor We intend to prove that the vibration decay does not depend primarily on the oscillator damping and that it remains finite even for vanishingly small damping. To this end, it is sufficient to show that the expression for remains complex when Using the Sokhotsky–Plemelj formula we obtain the following limit in Eqn. (74) as

As the imaginary part in (78) does not vanish, the spatial decay factor has a finite value indeed, and is determined by the dependence of the oscillator masses m(k) on their eigenfrequency k! This effect is typical only for a model that accounts for the complex structure of the medium, i.e. existence of the suspended oscillators. From the physical point of

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view the effect can be explained by the fact that suspended oscillators act as dynamic absorbers. At first glance, this conclusion seems to be paradoxical. However, a careful analysis shows that the energy dissipation has a finite value even for small damping of the oscillators, which is explained by large amplitudes of vibration for the resonating oscillators. The theory presented here may be used for the analysis of vibration in such complex dynamical systems as vessels, spacecraft, and submarines.

8.

SUMMARY

The laws of vibration decay with distance from the source of vibration were investigated in this paper. The character of the decay turned out to be essentially dependent on the rheological properties of rod material. For a purely elastic rod, spatial decay is absent. The more difference of material law from pure elastic, the more spatial decay there is. There exists, however, another important factor that influences the decay character, namely the dynamical structure of the medium. We considered one of the simplest media of this sort. It was an elastic carrier rod with suspended oscillators. The medium under consideration

possesses the following interesting peculiarity: Though the medium is “built” from high-quality elements, such as an ideally elastic carrier rod and slightly damped oscillators, the coefficient of spatial decay has a finite value. The latter model can be successfully used for the description and analysis of vibration of complex mechanical systems.

Acknowledgments This work made possible in part by a grant from the Russian Foundation for Basic Research (N 98-01-01073).

References Ziegler, F. 1995. Mechanics of Solids and Fluids. New York, Vienna: Springer-Verlag. Palmov, V. 1998. Vibrations of Elasto-Plastic Bodies. Berlin, Heidelberg, New York: Springer-Verlag.

HIERARCHICAL MODEL- AND DISCRETIZATION-ERROR ESTIMATION OF ELASTO–PLASTIC STRUCTURES Erwin Stein, Stephan Ohnimus, and Marcus Rüter Institute of Structural and Computational Mechanics (IBNM) University of Hannover, Hannover, Germany [email protected], [email protected], [email protected]

Abstract

1.

A posteriori error estimators of the discretization and model errors are presented for finite-element solutions of small-strain elasto–plasticity and large-strain elasticity problems. The described posterior equilibrium method (PEM) also yields anisotropic error indicators that are more efficient for thin-walled structures than isotropic ones. The model error in a sequence of hierarchical mathematical models and related dimensions can be as important as the discretization error. Therefore, integrated error-controlled adaptivity of finite-element approximations and of models in relevant subdomains, such as thickness jumps, is a challenging task for complex engineering structures. The posterior equilibrium method yields an effective estimation of discretization errors and model errors in general. Based on this approach, an upper bound of the error in a local quantity of interest can be obtained by solving an auxiliary local Neumann problem for the error of the dual problem. Some illustrative examples show the numerical and physical features of the methodologies.

INTRODUCTION

An important task for engineers is the reliable prediction of local and global mechanical behavior of structures under various loading conditions, including damage and failure. This prediction needs adequately simplified physical structures and systems and related mathematical models, based on experimental evidence. Engineering models result from physical idealizations and related mathematical homogenizations on different scales of consideration (from nano- to macro-scales), following the rule “as simple as possible and as complicated as necessary”. For a certain class of problems we define a mathematical master model, e.g. 3D elasto–plasticity, from which a sequence of more simplified or reduced models can be introduced stepwise in a hierarchical or parallel way, determined by further kinematical and/or physical hypotheses (e.g. 373 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 373–388. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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the 2D elastic Reissner–Mindlin plate theory). The analytical solutions of such a nested sequence of models should provide consistent transitions by asymptotic analysis, e.g. concerning the types of operators and singularities depending on constitutive equations, topology, geometry and loading. The global discretization error of the approximated solution of a specific model is estimated by locally computed error estimators of local Neumann type. This estimator was introduced by Bank and Weiser [1], proven by Ainsworth and Oden [2], and extended by Stein and Ohnimus [3, 4]. Since we do want convergence not only with respect to a specific simplified model but for the hierarchically expanded master model, model adaptivity has to be applied in certain subdomains integrated with solution adaptivity for each related model. The goal of hierarchical model- and solution-error analysis therefore is the automatic detection of these subdomains [3, 4]. However, these estimators are based on a global energy norm or norm control of the error. To estimate the error in a local quantity of interest, local error measures based on duality techniques (see Johnson [5] and Becker and Rannacher [6]) were systematically developed in recent years for a wide range of mostly linear problems, e.g. in linear elasticity by Rannacher and Suttmeier [7], Cirak and Ramm [8] and Ohnimus et al. [9], here also with local Neumann problems. In this paper we present model-error estimators as well as implicit a posteriori error estimators of local Neumann type for both the initialboundary-value problem of elasto–plasticity and the boundary-value problem of finite elasticity.

2.

2.1.

VARIATIONAL FORMULATION AND A POSTERIORI ERROR ANALYSIS IN LINEAR ELASTICITY

The weak form and its discretization

Let us begin with a brief outline of the linear elasticity problem and its discretization with the finite-element method. The elastic body occupies a bounded open set with piecewise smooth and Lipschitz continuous boundary such that and where and are the Dirichlet boundary and the Neumann boundary, respectively. If the elastic body is in equilibrium, the displacement field fulfills the equilibrium condition in and the boundary conditions u = 0 on and on where are prescribed body forces, are prescribed surface tractions, is the unit outward normal, is the symmetric

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stress tensor, and is the constitutive equation. Here, C is the symmetric and positive definite elasticity tensor, and is the strain tensor. The variational problem now reads: find a solution such that

where

denotes the Sobolev space and and are bilinear and linear forms, respectively,

given by

In the finite-element method, the Galerkin method for finite subdomains, we subdivide the domain into elements, i.e. with boundary and introduce finite dimensional subspaces Hence, we search for such that which means that we restrict the variational calculus in its action. Replacing by in (1) and subtracting (3) from (1) result in the Galerkin orthogonality for all where we introduced the solution error From (3) it follows with Céa’s lemma that the Galerkin method is optimal in the energy norm Furthermore, subtracting the Galerkin orthogonality condition from yields the main equation of error analysis Note that only by introducing the further test function it is possible to get the upper bound estimate for the interpolation constant of the important Babuška–Miller upper and lower estimate—see Babuška and Miller [10].

2.2.

Equivalent representations of residual and equilibrated error estimators

Upon introducing the local bilinear and linear forms and respectively, on the local space the restrictions of a and l to an element such that

by

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we may present three equivalent forms for the estimated error for all and as follows:

where the residuals

and

are given by

and in respectively, and equilibrated tractions fulfilling Cauchy’s lemma, i.e. and the Neumann boundary condition Further requirements are discussed in 3.1.

3.

are on on

DIRECTIONAL ERROR ESTIMATORS WITH EQUILIBRATED BOUNDARY TRACTIONS, CALLED POSTERIOR EQUILIBRIUM METHOD (PEM)

The implicit “posterior equilibrium method” (PEM) for quantitative local computations of directional (anisotropic) global or local as well as model-error estimates is based on an equivalent problem for direct or explicit residual-error estimation and yields sharper upper bounds and thus much better effectivity indices than the direct “residual error estimation method” (REM), especially for locking problems [3].

3.1.

Introduction of residual error estimators with equilibrated tractions

Let us now consider the following local Neumann problem derived from (6c): find the approximated error such that

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where is an enriched finite-dimensional test space by at least two further hierarchical test modes. For the solvability of (8) we have to ensure that the equilibration condition

is fulfilled, where is the space of rigid-body motions. Hence, a simple averaging ansatz of the form

is possible with linear polynomials

This results in

We remark that the are not really consistent to the original local finiteelement problem with the property if Hence, anisotropic error estimation might not be reliable. In order to get around this problem, we introduce an extended space for the equilibration instead of This leads to a local Neumann problem if Integration by parts yields the necessary identity (see also (7))

The consistency condition

is fulfilled, such that the global equivalence of the weak form

holds. A condition for the equivalence with the Babuška–Miller residual error estimator is that with C > 0 holds. Thus, we choose g such that which is the same regularization condition as

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The determination of is clearly not unique. Hence, different approaches exist. The basic idea of improving the stresses by a post process goes back to Bufler and Stein [11] and Stein and Ahmad [12]. Ladevèze and Leguillon [13] were the first who used this concept for a posteriori error estimation. Different strategies are discussed e.g. in Ainsworth and Oden [14]. Here, we follow Stein and Ohnimus [3], who use the energy-equivalent nodal forces to determine the equilibrated tractions by least-squares regularizations on patches instead of averaging the neighboring tractions. The main part of the error analysis is performed only once on unit elements and then transformed energy equivalently onto real elements. Additional elementwise variational problems are solved for each element in order to get improved displacements from the new tractions using an enriched space The improved solution follows from

but only if is reduced by the rigid-body motions. The locally computed equilibrium discretization error follows from the energy norm of the difference

3.2.

Model adaptivity based on equilibrated boundary tractions (PEM)

The concept of model adaptivity is similar to the one of solution adaptivity, but uses an additional hierarchically expanded model 2, e.g. the expansion from a 2½ D-plate to a 3D-continuum theory. Our goal is to get the global error between the considered model 1 and the expanded model 2 with solutions and respectively. Obviously, the model error should not be larger than the solution error. In conjunction with the model expansion we also need dimensional adaptivity of the ansatz functions, especially in the thickness direction (denoted by such that we realize an integrated solution–dimension–model adaptivity, necessary in boundary layers, thickness jumps, and concentrated loads. We want to compute the dimensionally expanded solution of the new model 2 with the enhanced bilinear form for all

where denotes the dimensionally expanded test space. We remark that the dimensions of and the solution of the reduced model 1, i.e. have to be equal as well as the dimensions of the related strains.

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The equilibrium model error estimator in the energy norm follows from the difference of the improved stresses (from PEM) obtained by

the hierarchically expanded model, by the prolongation from

and the transformed stresses to obtained by the

investigated model as follows:

Note that the using of instead of is mathematically inconsistent and would generally result in an overestimation of the error due to deformation constraints, i.e. locking effects. In order

to suppress influences on the model error caused by large discretization errors, the model is adapted only if the global model error is larger than the global discretization error. Furthermore, the model error is activated only in certain subdomains after reaching a prescribed tolerance of the

discretization (solution) error. The following model-error estimators are possible: The estimator of the model displacement error, i.e. which is not reasonable due to the discussed thicknesslocking effects. The error estimator using stresses of the current and the expanded model, i.e. Here, the locking phenomena do not appear. This can be shown by a mixed variational formulation. A relative-error estimator, taking into account the neglected deformation modes of the reduced model, i.e. This estimator is useful for the expansion of 2D-plate and shell models. For a pure displacement method, an energy-oriented estimator

can be used, too. But in most investigated cases, this indicator tends to zero.

3.3.

Error estimators for Prandtl–Reuss elasto–plasticity by PEM

We show next how to derive a posteriori error estimators for perfect elasto–plasticity (Prandtl–Reuss elasto–plasticity) based on PEM. Therefore, let us first briefly outline the basic theory of elasto–plasticity. The von Mises yield condition at the process time is given by with yield stress The associated

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flow rule is given by with the plastic multiplier and the flow direction The Kuhn–Tucker conditions are and To satisfy these conditions, we introduce the additive split of the strain rates into elastic and plastic parts by with

Differentiation of the specific strain energy with respect to the elastic strains results in the stresses Further, we assume that the time integration is sufficiently exact (quasi exact). Hence, we get with the analytical Prandtl–Reuss tangent n(t). The elasto–plastic problem is now solved approximately with the finite-element method and yields the approximations and In order to estimate the error in the equilibrium and the plastic strains, we follow a two-step strategy. First, we assume that the flow direction and the plastic strain field are exact. Thus, we have only to control the error of the equilibrium for all Hence, we get the residual formulated in rate equations for the error

with the test space on and the definition of the solution space with the important property For the semi-discrete approximation method, we introduce the spatial finite solution and test subspaces and respectively. Thus, we have The residual rates have to be equilibrated in the same way as we have seen above, i.e.

with the equilibrium condition for all In order to determine the equilibrated tractions, one has to solve for each element separately the following problem for all

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since the equilibrated tractions have to represent the correct tractions that solve the local discrete variational problem. Furthermore, the consistency condition has to be satisfied for the rates of the tractions for all For this elliptic problem, we can prove that we get an upper bound error estimator for the current tangential space as follows:

where we defined the total energy norm by

For further details we refer to Stein and Ohnimus [4]. Note that this estimator holds also for the pure (linear) elastic case. We further remark that we estimate the error at time T. If we want to get an error estimator in time that also includes the accumulation error, we have to use a different norm, namely

An upper bound for the error measured in this norm is carried out in the same way as the upper bound (22), and is explained in [4]. From the asymptotical relation, our estimator will lead to an upper error estimator since we describe the plastic domain more and more accurately, such that the error estimator is asymptotically exact. In the second step we consider an error indicator for the plastic strains. Thus we test the local solution e* of the error with respect to the current elasto–plastic tangent. The motivation is that we have to control whether the stress rates and the strain rates are orthogonal to the flow direction Therefore, we introduce the plastic strain error indicator in the energy related norm as

The exact solution of the local variational problem

yields an upper error bound. Since an exact solution is not possible for realistic engineering problems, we solve the local problem in an expanded

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approximation space with additional hierarchically expanded polynomial test and solution shape functions. The polynomial order of the current test space is p, and the local extension is k, such that we get the local polynomial order p + k with at least k = 2. (We tested until k = 5.)

We then define the isotropic error estimator

by

The

corresponding effectivity index is

3.4.

An error estimator in finite elasticity

In what follows, we present an error estimator for the geometrically

as well as materially nonlinear problem of finite elasticity. Therefore, we first sketch below the weak form of the boundary-value problem of finite elasticity. Material points in the reference configuration are mapped to material points in the current configuration by the deformation mapping Let with denoting the deformation gradient; then the mapping is orientation-

preserving if the Jacobian Since it proves convenient to formulate the problem in the current configuration (i.e. an Eulerian formulation), we introduce the left Cauchy–Green tensor as a strain measure. From the second law of thermodynamics we thus obtain the Kirchhoff stresses by where is the specific strain-energy function of an isotropic homogeneous body. For the sake of simplicity, we make the choice of a neo-Hooke material given by with positive Lamé constants and This leads to the Kirchhoff stresses where 1 denotes the second-order unity tensor. Let us now consider the total potential energy of the elastic body

with

Clearly, the minimizer

fulfills the variational formulation

where the semi-linear form which is linear only with respect to its second argument, and the linear form as an element of the dual space are now given by

with

Since this is a nonlinear problem, we apply Newton’s

method and solve within a recursive procedure the linearized (around

Hierarchical model- and discretization-error estimators

variational problem of finding increments

where the functional weak form of the residual, and

given by

383

such that

is the is a symmetric bilinear

form given by the second variation of the energy functional (27), that is,

with the spatial elastic moduli

Here, I

denotes the fourth-order unity tensor.

Let us now assume that we iterated a finite-element solution of the corresponding discrete analogue to (30) with sufficient accuracy. A linearization of the variational problem (28) around the finite-element solution thus leads to the error representation formula

where we assumed that the error is small. The weak residual R takes now the form for all Replacing by

leads to the nonlinear formulation of the Galerkin orthogonality R(v) = for all Furthermore, the tangent form aT induces a norm, the tangential-energy norm which is the appropriate norm to measure the error e. Analogously to the local forms (5), we define the restriction of to an element given by

Recalling (32), we may now formulate the corresponding local Neumann

problem as follows: find a solution

where the weak residual

such that

contains now the nonlinear finite-element solution—cf. (8). We can now show that the solutions of the local Neumann problems (34) yield an upper bound of the error given by

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However, the local Neumann problems (34) may become negative definite long before a global stability point is reached. Possibilities to get around this problem are discussed in Rüter et al. [15]. Brink and Stein derive in [16] a similar estimator for the stress error. Nonlinear anisotropic problems are discussed in Rüter and Stein [17].

3.5.

A local a posteriori error estimator in finite elasticity

Following the pioneering works of Johnson [5], Eriksson et al. [18], and Becker and Rannacher [6], we show next how to estimate the error in a local quantity of interest, such as local displacements or stresses. Therefore, let us first introduce the dual problem: find a solution such that

where is an arbitrary bounded linear functional that defines the local quantity of interest within an adequate local support. Note that, since the associated linear operator to the bilinear form is self-adjoint, only the right-hand side of the primal problem (30) changes. Substituting in (36), we obtain (Betti’s theorem)

Next, we subtract the Galerkin orthogonality from (37) and apply the Cauchy–Schwarz inequality. We thus arrive at the estimator

The norms can now be estimated by the solutions of the corresponding local Neumann problems, using improved boundary tractions for both the primal and the dual problem. Note that this nonlinear error estimator also includes the linear elasticity problem as analyzed by Ohnimus et al. [9]. Another approach of local error estimators yielding upper and lower error bounds was suggested by Prudhomme and Oden [19].

4.

NUMERICAL EXAMPLES

In the first example (Fig. 1), the automatic error-controlled expansion from a 2D clamped square plate in bending to a 3D-continuum theory is shown as well as the dimensional and model adaptivity. The second example treats the elastic–perfectly-plastic tension of a plate with a hole (only a quarter is shown) in the plane-strain state. The system data and results are given in Fig. 2.

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Figure 1 Refined meshes and isoplots of and for coupled model and solution adaptivity of an elastic square plate (length l = 10 m, thickness t = 0.01 m) with a central hole (radius R = 0.01 m) in bending and constant transverse loading. System data: E = 206900 MPa, and anisotropic polynomial order results for mesh 10.

The last example, depicted in Fig. 3, shows the results of a finite elastic problem, here a plate clamped on the left and subjected to a tension force on the opposite side.

References [1] Bank, R. E., and A. Weiser. 1985. Some a posteriori error estimators for elliptic partial differential equations. Mathematics of Computation 44, 283–301.

[2] Ainsworth, M., and J. T. Oden. 1993. A unified approach to a posteriori error estimation based on element residual methods. Numerische Mathematik 65, 23– 50.

[3] Stein, E., and S. Ohnimus. 1997. Coupled model- and solution-adaptivity in the finite-element method. Computer Methods in Applied Mechanics and Engineering 150, 327–350. [4] Stein, E., and S. Ohnimus. 1999. Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems. Computer Methods in Applied Mechanics and Engineering 176, 363–385.

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Figure 2 Elasto–plasticity: Square plate (l = 200 mm, t = 1 mm) with a hole (R = 10mm) in the plane-strain state. System with E = 206900 MPa, 95% of critical load, elements. [5] Johnson, C. 1993. A new paradigm for adaptive finite element methods. Proceedings of The Mathematics of Finite Elements and Applications (MAFELAP) 93. New York: Wiley. [6] Becker, R., and R. Rannacher. 1996. A feed-back approach to error control in finite element methods: Basic analysis and examples. East-West Journal of Numerical Mathematics 4, 237–264.

[7] Rannacher, R., and F.-T. Suttmeier. 1997. A feed-back approach to error control in finite element methods: Application to linear elasticity. Computational Mechanics 19, 434–446. [8] Cirak, F., and E. Ramm. 1998. A posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem. Computer Methods in Applied Mechanics and Engineering 156, 351–362.

[9] Ohnimus, S., E. Stein, and E. Walhorn. 2000. Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems. International Journal of Numerical Methods in Engineering, submitted. [10] Babuška, I., and A. Miller. 1987. A feedback finite element method with a posteriori error estimation: Part I. The finite element method and some basic proper-

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Figure 3 Finite elasticity: Tension test, neo-Hooke material, plane strain, elements; upper half of the specimen is discretized due to symmetry conditions.

ties of the a posteriori error estimator. Computer Methods in Applied Mechanics and Engineering 61, 1–40. [11] Bufler, H., and E. Stein. 1970. Zur Plattenberechnung mittels Finiter Elemente.

Ingenieur-Archiv 39, 248–260. [12] Stein, E., and R. Ahmad. 1977. An equilibrium method for stress calculation using finite element methods in solid- and structural-mechanics. Computer Methods in Applied Mechanics and Engineering 10, 175–198. [13] Ladevèze, P., and D. Leguillon. 1983. Error estimate procedure in the finite

element method and applications. SIAM Journal of Numerical Analysis 20, 485–509. [14] Ainsworth, M., and J. T. Oden. 1997. A posteriori error estimation in finite-

element analysis. Computer Methods in Applied Mechanics and Engineering 142, 1–88. [15] Rüter, M., S. Ohnimus, and E. Stein. 2001. On global and local a posteriori error estimation in finite elasticity. In preparation. [16] Brink, U., and E. Stein. 1998. A posteriori error estimation in large-strain elasticity using equilibrated local Neumann problems. Computer Methods in Applied Mechanics and Engineering 161, 77–101.

[17] Rüter, M., and E. Stein. 2000. Analysis, finite element computation and error estimation in transversely isotropic nearly incompressible finite elasticity. Computer Methods in Applied Mechanics and Engineering 190, 519–541. [18] Eriksson, K., D. Estep, P. Hansbo, and C. Johnson. 1995. Introduction to adaptive methods for differential equations. Acta Numerica, 106–158. [19] Prudhomme, S., and J. T. Oden. 1999. On goal-oriented error estimation for local elliptic problems: Application to the control of pointwise errors. Computer Methods in Applied Mechanics and Engineering 176, 313–331.

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At the Welcome Reception, the Grady Johnson Orchestra provided Congress participants with a sample of the jazz and blues for which Chicago is famous.

LAMINAR BOUNDARY-LAYER THEORY: A 20TH CENTURY PARADOX? Stephen J. Cowley Department of Applied Mathematics and Theoretical Physics University of Cambridge, Cambridge, United Kingdom [email protected]

Abstract

1.

Boundary-layer theory is crucial in understanding why certain phenomena occur. We start by reviewing steady and unsteady separation from the viewpoint of classical non-interactive boundary-layer theory. Next, interactive boundary-layer theory is introduced in the context of unsteady separation. This discussion leads onto a consideration of large-Reynolds-number asymptotic instability theory. We emphasize that a key aspect of boundary-layer theory is the development of singularities in solutions of the governing equations. This feature, when combined with the pervasiveness of instabilities, often forces smaller and smaller scales to be considered. Such a cascade of scales can limit the quantitative usefulness of solutions. We also note that classical boundary-layer theory may not always be the large-Reynolds-number limit of the Navier–Stokes equations, because of the possible amplification of short-scale modes, which are initially exponentially small, by a Rayleigh instability mechanism.

INTRODUCTION

Sectional lecturers were invited ‘to weave in a bit more retrospective and/or prospective material [than normal] given the particular [millennium] year of the Congress’. This invitation is reflected in the current, possibly idiosyncratic, article. For alternative viewpoints the reader is referred to Stewartson (1981), Smith (1982), Cowley and Wu (1994), Goldstein (1995), and Sychev et al. (1998). We begin with a deconstruction of the components of the title. Boundary-layer theory. Prandtl (1904) proposed that viscous effects would be confined to thin shear layers adjacent to boundaries in the case of the ‘motion of fluids with very little viscosity’, i.e. in the case of flows for which the characteristic Reynolds number, Re, is large. In a more general sense we will use ‘boundary-layer theory’ (BLT) to refer to any 389 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 389–412.

© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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large-Reynolds-number, asymptotic theory in which there are thin shear layers (whether or not there are boundaries). 20th century. Prandtl (1904) published his seminal paper on the foundations of boundary-layer theory at the start of the 20th century, while ICTAM 2000 was held at the end of the same century. Laminar. Like Prandtl (1904) we will be concerned with laminar, rather than turbulent, flows. Flows that are in the process of laminarturbulent transition will be viewed as unstable laminar flows. A paradox. Experimental flows at large Reynolds numbers are turbulent; yet useful comparisons with laminar-flow experiments at moderately large Reynolds numbers can sometimes be made with largeReynolds-number asymptotic theories. We view as a paradox this seemingly contradictory result, i.e. that useful comparisons with laminar flow can be made with expansions made about Reynolds numbers when flows are almost invariably turbulent. ?. The question this paper will discuss is whether the final ‘?’ is needed in the title. A subjective conclusion is given at the end.

2. 2.1.

CLASSICAL BOUNDARY LAYERS Formulation

We consider incompressible flow of a fluid with constant density and dynamic viscosity past a body with typical length We assume that a typical velocity scale is and that the Reynolds number is given by

For simplicity we will, for the most part, consider two-dimensional incompressible flows, although many of our statements can be generalized to three-dimensional flows and/or compressible flows.

The key approximations. Boundary-layer theory applies to flows where there are extensive inviscid regions separated by thin shear layers, say, of typical width For one such shear layer take local dimensional Cartesian coordinates and along and across the shear layer respectively. Denote the corresponding velocity components by and respectively, pressure by and time by On the basis of scaling arguments (e.g. Rosenhead 1936) it then follows that

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391

Further, it can also be deduced that the key approximations in classical BLT are that the pressure is constant across the shear layer, i.e.

and that streamwise diffusion is negligible, i.e. if (·) represents any variable, then The former approximation is more significant dynamically.

The governing equations.

and taking the limit the Navier–Stokes equations:

Using the transformations

we can deduce the BLT equations from

For flow past a rigid body the appropriate boundary conditions are

where U(x,t) is the inviscid slip velocity past the body. Further, from (6) evaluated at the edge of the boundary layer,

We define the ‘viscous blowing’ velocity out of the boundary layer to be

The velocity indicates the strength of blowing, or suction, at the edge of the boundary layer induced by viscous effects. It is a good diagnostic for dynamically significant effects within the boundary layer—much better than, say, the wall shear which can remain regular while becomes unbounded.

2.2.

Steady flows

Steady flow past an aligned flat plate: A success. Probably the most famous solution to (6) and (7) is that of Blasius (1908) for flow past an aligned flat plate. A comparison between this similarity solution and Wortmann’s visualization of that flow (Van Dyke 1982) demonstrates that BLT seems to work in this case—at least for where

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Steady flow past a circular cylinder: A failure. On the assumption that far from the boundary the velocity field is irrotational and inviscid to leading order, the inviscid slip velocity for steady flow past a circular cylinder is given by U = 2 sin x. Terrill (1960) showed numerically that the solution to (6)–(8) with this slip velocity terminates in a Goldstein (1948) singularity at with

for some constant k. This singularity occurs at the point where reversed flow is about to develop, i.e.

A serious problem. The occurrence of a singularity often indicates that there is a significant development in the flow physics, e.g. the formation of small-scale structure. In such circumstances new physics can usually be included in the model by introducing an asymptotic scaling

close to

thus enabling a solution to be found for However, Stewartson (1970) showed that, in general, there is no inner rescaling that ‘smoothes out’ the Goldstein singularity. (Smith and Daniels (1981) discuss an exception.) As a result, no BLT solution exists for Moreover this result implies that both the BLT solution for and the inviscid solution far from the cylinder, are incorrect. Boundary-layer theory does not always work.

What has gone wrong? The short answer is that the assumption that the flow far from the wall is irrotational is incorrect. Experimentally it is observed that, other than at very small Reynolds numbers,

there is a rotational eddy behind the cylinder that is at least as large as the cylinder; moreover the flow is steady and symmetric only for (For larger Reynolds numbers the flow is unsteady and asymmetric.) Further, is not an asymptotically large Reynolds number! Steady symmetric solutions for obtained numerically, by specifically excluding the possibility of unsteadiness and asymmetry, suggest that the asymptotic regime for steady symmetric solutions is reached only for (Fornberg 1985), i.e. at Reynolds numbers far larger than those at which the steady flow is stable. One way forward is to study unsteady flows on the basis that these are what are observed experimentally—see Another is to seek the asymptotic form of the steady symmetric solution (even if it is experimentally unobservable) in the hope that the solution will shed light on the failure of classical BLT (although the true justification for studying

Laminar boundary-layer theory: A paradox?

393

the problem may be closer to that of mountaineers for climbing Everest, i.e. it’s hard and it's there).

Steady symmetric flow past a circular cylinder: An answer. An asymptotic solution for steady symmetric separated flow past a bluff body at large Reynolds number must contain at least two important ingredients. First there must be a local solution at the point of separation of the boundary layer from the body surface. Second, an asymptotic model of the global wake is needed. Local separation is described by Sychev’s (1972) ‘triple-deck’ analysis (see also Smith 1977). This analysis is based on the premise that, at the point of separation on a smooth surface, the pressure gradient is In the context of Kirchhoff free-streamline theory this means that the appropriate free-streamline solution satisfies the Brillouin–Villat condition to leading order (see e.g. Sychev et al. (1998) for a discussion). There have been a number of attempts to fit the above local description of separation into a consistent large-Reynolds-number asymptotic global solution for the flow past a circular cylinder. Building on the work of others, Chernyshenko (1988) proposed an asymptotic structure based on the special Sadovskii (1971) vortex where there is no velocity

jump at the edge of the vortex. While this structure, in which both the length and the width of the wake are O(Re) in magnitude, may not be unique, it overcomes the technical shortcomings, especially as regards wake reattachment, of other proposals (see e.g. Chernyshenko 1998).

2.3.

Unsteady separation

While the large-Reynolds-number asymptotic solution for steady, symmetric laminar flow past a bluff body is not experimentally realizable,

this is not the case for fast starting flow past a smooth bluff body. In particular, when the unsteady term in (6) is much larger than the nonlinear term, and is balanced by the diffusive term (so that the boundary layers are very thin with Hence when each point of the boundary layer looks locally like Rayleigh’s solution for starting motion over a flat plate, and hence separation does not take place at sufficiently early times (see e.g. Goldstein and Rosenhead 1936). Impulsively started flow past a circular cylinder. There have been a number of visualizations of this flow, e.g. Prandtl (1932) and Coutanceau and Bouard (1977). Prandtl’s (1932) film is particularly instructive as regards where separation of the boundary layer starts.

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For many years research into unsteady separation focused on the rear stagnation point (see e.g. Robins and Howarth 1972), since it is there that reversed flow first sets in. However, reversed flow is not the same as separation/breakaway of the boundary layer from the body surface— there is plenty of reversed flow in Stokes’s solution for flow over an oscillating plate, and yet this unsteady boundary layer remains attached to the plate for all Reynolds numbers. In contrast, Prandtl (1932) focused attention on a region approximately from the front stagnation point, because it is approximately there that the boundary layer clearly first separates from the body surface. It is arguable that conventional wisdom, i.e. that the rear stagnation point was the place to look, delayed an understanding of unsteady separation by fifty years. Unsteady separation: Physics. On the rear-side of an impulsively moved cylinder, fluid particles are decelerating. These particles will tend to be squashed in the streamwise direction, with a compensating expansion in the direction normal to the boundary. In Navier–Stokes (NS) or Euler flows it is not possible to squash a particle to zero thickness in one direction and an infinite length in another because the rapid stretching of the fluid particles leads to the generation of a pressure gradient that inhibits the stretching. However, in classical BLT, (see (7)), and hence no pressure gradient can be induced in the direction normal to the wall. Unsteady separation occurs when a fluid particle is squashed to zero thickness in the direction parallel to the wall, thus ejecting the fluid above it out of the boundary layer (van Dommelen 1981).

Unsteady separation: Mathematics. Since unsteady separation is connected with the deformation of a particle, it is natural to seek a mathematical description in terms of Lagrangian coordinates, say (Shen 1978, van Dommelen and Shen 1980). Then with and the momentum equation (6) becomes

while kinematics and mass conservation yield

Van Dommelen and Shen (1980) made the key observation that (12) depends only on x and u, and hence that (12) and the first equation in (13) can be solved independently of the equations governing y and v.

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395

The solution for y can then be obtained from the third equation of (13), while the solution for v can be deduced subsequently from the second equation of (13). For a given the Jacobian mass conservation relation in (13) is a hyperbolic equation for

with a unit source term. If at some time, say the solution evolves so that (indicating that a particle has been squashed to zero thickness in the x direction), then ‘shock’ singularities can form in y, and hence v (van Dommelen and Shen 1980).

Impulsively started circular cylinder: Results. Numerical calculations show that a ‘shock’ singularity develops at a time and a position The position where the unsteady singularity forms, is not the position where the Goldstein singularity forms, i.e. As the unsteady singularity results in a rapid thickening of the boundary layer over a streamwise distance where is the center of the singularity structure, while the displacement thickness and blowing velocity vary like and respectively—see Fig. 1 for a schematic.

Figure 1

Schematic of a separating boundary layer

Three-dimensional unsteady separation. The above analysis can be extended to describe three-dimensional separation (e.g. van Dommelen and Cowley 1990). When there are no symmetries, the boundary layer still thickens like and the singularity is quasi-two-dimensional with a slower variation in a direction orthogonal to the more rapid variation. However, if there is a plane or axis of symmetry, then the thickening of the boundary layer is more rapid than

396

2.4.

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A singularity . . . surely not?

Conventional wisdom is that finite-time singularities do not spontaneously develop in solutions to the NS equations. (Of course conventional wisdom may prove to be wrong—as illustrated by the aforementioned fifty years of misguided study of unsteady separation.) Assuming that conventional wisdom is correct, the BLT singularity must be an artifact of the BLT approximation. Hence, at times very close to at least one of the terms that are usually asymptotically smaller than those included in the BLT equations, must grow to be so large that it cannot be neglected at leading order; e.g. in order to stop the extension of the squashed particle in the y direction, we might anticipate that the approximation will need to be refined. An upper deck. For it follows from (5) and the above scaling for that the dimensional blowing velocity has magnitude

This blowing velocity causes a perturbation to the inviscid flow in a region just above the boundary layer. This perturbation is both inviscid and irrotational; hence it is governed by Laplace’s equation. The perturbation extends over a region sufficient for a pressure gradient normal to the wall to be felt (and so reduce the normal velocity to zero). Since the Laplacian is a ‘smooth operator’ and the extent of the variation in the direction is the extent of the variation in the direction is also From the continuity equation it follows that the perturbation velocity in the streamwise direction is of the same order of magnitude as the blowing velocity (14). Similarly it follows from the linearized version of the time-dependent Bernoulli equation that the dimensional pressure perturbation is Hence the dimensionless pressure-gradient perturbation has a magnitude

This induced perturbation pressure gradient can have a feedback effect on the boundary-layer flow when it is as large as the acceleration, within the boundary layer. This condition occurs when At such times a new asymptotic problem needs to be formulated involving four distinct asymptotic regions in the y direction (Elliott et al. 1983). For this ‘quadruple-deck’ analysis to be valid, i.e. for the four asymptotic regions to be dist

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Mechanics for a New Millennium Proceedings of the 20th International Congress of Theoretical and Applied Mechanics Chicago, Illinois, USA 27 August – 2 September 2000

Edited by

Hassan Aref and James W. Phillips Department of Theoretical and Applied Mechanics University of Illinois at Urbana-Champaign

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

0-306-46956-1 0-792-37156-9

©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2000 Kluwer Academic / Plenum Publishers New York All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:

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PREFACE The 20th International Congress of Theoretical and Applied Mechanics, ICTAM 2000, took place August 27 – September 2, 2000, in the conference facilities of the Chicago Marriott Downtown on Michigan Avenue in Chicago, Illinois, USA. The Congress was invited by the US National Academy of Sciences on the recommendation of its National Committee on Theoretical and Applied Mechanics. A consortium of university departments and programs acted as the local hosts. The undersigned served, respectively, as President and Secretary-General of the Congress. In this Proceedings volume we have attempted to capture the excitement of that memorable week. We have laid out the book so that it, roughly, tracks the events during the week of the Congress in much the order in which they occurred. We have given addresses and reports in context, although sometimes with embellishments to provide more data than could reasonably be presented orally. We have tried to reproduce essentials of the Opening and Closing ceremonies for the benefit of those who could not attend one or the other. We have tried to assemble an attractive volume that will have lasting value by adding an extensive name index and a briefer keyword index, and by exercising considerable care in the consistency of the layout of the full manuscripts. The international congresses of mechanics are major events of the field—we compare them to the Olympic Games in sports. It is essential that future generations can assess exactly where our field stood in the year 2000. We hope this volume—pre-ordered at the Congress in record numbers—will be found useful for many years to come. Hassan Aref

James W. Phillips

President of ICTAM 2000

Secretary-General of ICTAM 2000

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TABLE OF CONTENTS Committees, Bureau of IUTAM (1996–2000), Congress Committee of IUTAM, Local Organizing Committee, Chairs of mini-symposia and pre-nominated sessions, vii Opening ceremony, xi Congress poster, xxvii Program at a glance, xxxi Scientific program, xxxix Invited papers, 1 JAMES R. RICE: New perspectives on crack and fault dynamics, 1 RICHARD M. CHRISTENSEN: A survey of and evaluation methodology for fiber composite material failure theories , 25 JEAN-LOUIS CHABOCHE: On constitutive and damage modeling in metal matrix composites, 41 LORNA J. GIBSON: Metallic foams: Structure, properties, and applications, 57 MARTIN NEMER, JERZY BLAWZDZIEWICZ, and MICHAEL LOEWENBERG: Linear viscoelasticity of concentrated emulsions, 75 PETER A. DAVIDSON: Aluminum: Approaching the new millennium, 85 TAKEHIKO TOH and EIICHI TAKEUCHI: Electromagnetic phenomena in steel continuous casting, 99 RONALD J. ANDERSON, JOHN A. ELKINS, and BARRIE V. BRICKLE: Rail vehicle dynamics for the 21st century, 113 ROBIN S. SHARP: Fundamentals of the lateral dynamics of road vehicles, 127 MASATO ABE and J. KARL HEDRICK: A mechatronics approach to advanced vehicle control system design, 147 E M M A N U E L V I L L E R M A U X : Mixing: Kinetics and geometry, 165 S TÉPHANE R O U X and F ARHANG R ADJAÏ: Statistical approach to the mechanical behavior of granular media, 181 GEORGE J. DVORAK: Damage analysis and prevention in composite materials, 197 JOHN S. WALKER: Electromagnetic phenomena in crystal growth, 211 WILLI KORTÜM, WERNER O. SCHIEHLEN, and MARTIN ARNOLD: Software tools: From multibody system analysis to vehicle system dynamics, 225

iv MICHEL Y. LOUGE, JAMES T. JENKINS, HAITAO XU, AND BIRGIR Ö. ARNARSON: Granular segregation in collisional shearing flows, 239

PEDRO PONTE CASTAÑEDA and PIERRE SUQUET: Nonlinear composites and microstructure evolution, 253 DENIS L. WEAIRE and STEFAN HUTZLER: Hard problems with soft materials: The mechanics of foams, 275 HAMDA BENHADID: Magnetohydrodynamic damped buoyancy-driven convection, 289

TIMOTHY J. GORDON: Adaptive, nonlinear, and learning techniques for the control of vehicle ride dynamics, 307

P AUL E. D IMOTAKIS : Recent advances in turbulent mixing, 327 ISAAC G OLDHIRSCH : Kinetic and continuum descriptions of granular flows, 345

VLADIMIR A. PALMOV: Stationary waves in elasto–plastic and visco–elasto–plastic bodies, 359

ERWIN STEIN, STEPHAN OHNIMUS, and MARCUS RÜTER: Hierarchical model- and discretization-error estimation of elasto–plastic structures, 373 STEPHEN J. COWLEY: Laminar boundary-layer theory: A 20th century paradox?, 389 ERIK VAN DER GIESSEN: Plasticity in the 21st century, 413

DAVID M. MCQUEEN AND CHARLES S. PESKIN: Heart simulation by an immersed boundary method with formal second-order accuracy and reduced numerical viscosity, 429 S HIGEO K IDA : Vortical structure of turbulence, 445

R. NARAYANA IYENGAR: Probabilistic methods in earthquake engineering, 457 GEDEON DAGAN: Effective, equivalent, and apparent properties of heterogeneous media, 473 A MABLE L IÑÁN : Diffusion-controlled combustion, 487

SUBRA SURESH: Nanomechanics and micromechanics of thin films, graded coatings, and mechanical/nonmechanical systems, 503 OLE SIGMUND: Optimum design of microelectromechanical systems, 505

H. KEITH MOFFATT: Local and global perspectives in fluid dynamics, 521

Closing ceremony, 541 List of attendees, 553 List of exhibitors, 561 Index to authors, co-authors, and session chairs, 563 Subject index, 581

COMMITTEES Bureau of IUTAM Jüri Engelbrecht (Estonia) L. Ben Freund (USA), Treasurer Michael A. Hayes (Ireland), Secretary-General H. Keith Moffatt (UK)

Werner O. Schiehlen (Germany), President Tomomasa Tatsumi (Japan) Ren Wang (China) Leen van Wijngaarden (The Netherlands), Vice-President

Congress Committee of IUTAM Andreas Acrivos* (USA) 2002† Hassan Aref (USA) 2000 Sol R. Bodner* (Israel) 2000 David B. Bogy (USA) 2000 Jüri Engelbrecht (Estonia) 2000 Norman A. Fleck (UK) 2002 L. Ben Freund (USA) 2000 Graham M. L. Gladwell (Canada) 2000 Michael A. Hayes (Ireland) 2002, Rep. of ISIMM Tatsuo Inoue (Japan) 2002, Rep. of ICM Javier Jiménez (Spain) 2002 Alfred Kluwick (Austria) 2002 Anthony N. Kounadis (Greece) 2002 Y. H. Ku (USA) Peter Lugner (Austria) 2000, Rep. of IAVSD Bengt Lundberg (Sweden) 2000 Gerd E. A. Meier (Germany) 2000 H. Keith Moffatt (UK) 2000 René Moreau* (France) 2000

Bahadur C. Nakra (India) 2002 Robert I. Nigmatulin (Russia) 2000 J. Tinsley Oden (USA) 2002, Rep. of IACM Neils Olhoff* (Denmark) 2000, Secretary J. R. Anthony Pearson (UK) 2000, Rep. of ICR Timothy J. Pedley* (UK) 2000 Mahir B. Sayir (Switzerland) 2000 Werner O. Schiehlen* (Germany) 2000, Chairman Bernhard A. Schrefler (Italy) 2002 Kazimierz Sobczyk (Poland) 2002 Pierre Suquet (France) 2000 Roger I. Tanner (Australia) 2002 Tomomasa Tatsumi (Japan) 2000 Eiichi Watanabe (Japan) 2002 Leen van Wijngaarden (The Netherlands) 2000 Fen-Gan Zhuang (China) 2000

*Members of the Executive Committee (1996–2000) †2000, 2002—Year, where stated, indicates end of term (applies to members elected after 1972)

Local Organizing Committee‡ Carol J. Porter, Administrative Secretary Hassan Aref, President of ICTAM 2000 James W. Phillips, Secretary-General of On-site support: Patricia J. Franzen ICTAM 2000 Chicago Marriott Downtown: Willie Clay, Robin Enke James C. Onderdonk, Head, Conferences and Institutes Science press contact: Susan K. Mumm ‡Except as noted, all members of the Local Organizing Committee are affiliated with the University of Illinois at Urbana-Champaign, Urbana, Ill.

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CHAIRS OF MINI-SYMPOSIA AND PRE-NOMINATED SESSIONS Mini-symposium chairs Mini-symposium

Chair

Co-chair

Electromagnetic René Moreau (Prance) processing of materials (jointly with HYDROMAG)

Nagy El-Kaddah (USA)

Granular flows

Stuart B. Savage (Canada)

Robert P. Behringer (USA)

Turbulent mixing

Emil J. Hopfinger (Prance)

Paul E. Dimotakis (USA)

Damage and failure of composites

Zvi Hashin (Israel)

George J. Dvorak (USA)

Mechanics of foams and Andrew M. Kraynik (USA) Norman A. Fleck (UK) cellular materials Vehicle systems Peter Lugner (Austria) J. Karl Hedrick (USA) dynamics (jointly with IAVSD)

Pre-nominated session chairs Topics in fluid mechanics Pre-nominated session

Chair

Co-chair

Biological fluid dynamics Kazuo Tanishita (Japan)

Timothy J. Pedley (UK)

Boundary layers

Hans H. Fernholz (Germany)

Anatoly I. Ruban (UK/Russia)

Combustion and flames

John D. Buckmaster (USA) Moshe Matalon (USA)

Complex and smart fluids Compressible flow

Eric S. G. Shaqfeh (USA)

Roger I. Tanner (Australia)

John-Paul Bonnet (France)

Alexander J. Smits (USA)

Computational fluid John Kim (USA) dynamics (jointly with IACM)

Alien J. Baker (USA)

Convective phenomena

Friedrich H. Busse (Germany)

Nigel O. Weiss (UK)

Drops and bubbles

John R. Blake (UK)

Andrea Prosperetti (USA)

vii

viii

ICTAM 2000

Environmental fluid

Paul F. Linden (USA)

Julian C. R. Hunt (UK)

mechanics Experimental methods in Henrik Alfredsson (Sweden) Helmut Eckelmann fluid mechanics (Germany) Flow control

Mohamed Gad-el-Hak

Roddam Narasimha (India)

(USA) Flow in porous media

George M. Homsy (USA)

Joel Koplik (USA)

Flow instability and

Sidney Leibovich (USA)

Michio Nishioka (Japan)

Yves Couder (France)

Leonard W. Schwartz

transition

Flows in thin films

(USA) Fluid mechanics of materials processing

Wilhelm Schneider (Austria)

J. Iwan D. Alexander

Geophysical fluid

Kolumban Hutter (Germany)

Peter Lynch (Ireland)

Low-Reynolds-number flow

Howard A. Stone (USA)

Antonio Delgado (Germany)

Microfluid dynamics

John F. Brady (USA)

Leen van Wijngaarden (Netherlands)

Multiphase flows

Grétar Tryggvason (USA)

L. Gary Leal (USA)

Topological fluid

Arkady B. Tsinober (Israel) Richard B. Pelz (USA)

mechanics Turbulence

Olivier Métais (France)

dynamics

(USA)

Katepalli R. Sreenivasan

(USA) Vortex dynamics

Dale I. Pullin (USA)

Viatcheslav V. Meleshko (Ukraine)

Waves

Roger H. J. Grimshaw (Australia)

Chiang C. Mei (USA)

Topics in solid mechanics Pre-nominated session

Chair(s)

Co-chair(s)

Biological solid mechanics Computational solid mechanics (jointly with IACM)

Stephen C. Cowin (USA)

Ivars Knets (Latvia)

Pierre Ladevèze (France), Walter Wunderlich (Germany)

Computational strategies J. Tinsley Oden (USA) for multiscale phenomena in mechanics (jointly with IACM)

Erwin Stein (Germany)

Chairs of mini-symposia and pre-nominated sessions

ix

—

Contact and friction problems (jointly with IAVSD)

Anders Klarbring

Control of structures

Dick H. van Campen (Netherlands)

Felix L. Chernousko (Russia)

Damage mechanics

Dusan Krajcinovic (USA)

Jean Lemaitre (France)

Dynamic plasticity of structures

Stephen R. Reid (UK)

Victor P. W. Shim (Singapore)

Elasticity

Raymond W. Ogden (UK)

James K. Knowles (USA)

(Sweden), James R. Barber (USA)

Experimental methods in Isaac M. Daniel (USA) solid mechanics Fatigue Robert O. Ritchie (USA)

Jörg F. Kalthoff (Germany) Keisuke Tanaka (Japan) —

Fracture and crack mechanics (jointly with ICF)

Bhushan L. Karihaloo

Functionally graded materials

Marek-Jerzy Pindera

Impact and wave propagation

Rodney J. Clifton (USA)

Anders Boström (Sweden)

Material instabilities

Viggo Tvergaard

—

(UK), Jean-Baptiste Leblond (France)

Glaucio H. Paulino (USA)

(USA)

(Denmark), Alan

Needleman (USA) Alain Molinari (France)

Franz-Josef Ulm (USA)

Mechanics of porous materials

Alberto Carpinteri (Italy)

Wolfgang Ehlers (Germany)

Mechanics of thin films and nanostructures

L. Ben Freund (USA)

Jürg Dual (Switzerland)

Multibody dynamics

Werner O. Schiehlen (Germany)

John J. McPhee (Canada)

Plasticity and

Zenon Mróz (Poland),

—

Mechanics of phase transformations (jointly with IACM)

viscoplasticity

Bertil Storåkers (Sweden) Herbert A. Mang (Austria)

Arthur W. Leissa (USA)

Rock mechanics and geomechanics

Bernhard A. Schrefler (Italy)

Ioannis Vardoulakis (Greece)

Smart materials and

Richard D. James (USA)

Yuji Matsuzaki (Japan)

Friedrich G. Pfeiffer (Germany)

David B. Bogy (USA), Tatsuo Inoue (Japan)

Plates and shells (jointly

with IACM)

structures

Solid mechanics in manufacturing

x

ICTAM 2000

Stability of structures

Anthony N. Kounadis

George J. Simitses (USA)

(Greece)

Raphael T. Haftka (USA)

Structural optimization (jointly with ISSMO)

Martin P. Bendsøe (Denmark)

Structural vibrations

Peter Hagedorn (Germany), — James F. Doyle (USA)

Viscoelasticity and creep

John L. Bassani (USA)

Nobutada Ohno (Japan)

Topics involving both fluid mechanics and solid mechanics Pre-nominated session

Chair(s)

Co-chair(s)

Acoustics

Ann P. Dowling (UK)

Allan D. Pierce (USA), Leif Bjørnø (Denmark)

Cellular and molecular mechanics

Christopher R. Calladine

Chaos in fluid and solid

Tom Mullin (UK)

Sheldon Weinbaum (USA)

(UK)

Francis C. Moon (USA)

mechanics

Continuum mechanics

Gérard A. Maugiu (France) James Casey (USA)

Fluid–structure interaction

Michael P. Païdoussis (Canada), P. Terndrup

John Grue (Norway)

Pedersen (Denmark) Microgravity mechanics

Simon S. Ostrach (USA)

Hans J. Rath (Germany), Vadim I. Polezhaev

(Russia)

OPENING CEREMONY Monday, August 28, 2000, 10:00 Prof. Hassan Aref, President of ICTAM 2000 “Esteemed colleagues, distinguished guests, ladies and gentlemen, “Welcome to the 20th International Congress of Theoretical and Applied Mechanics, or ICTAM 2000 as we call it. The city of Chicago has an advertising slogan. It says: ‘We are glad you are here!’ I have to concur: We are really, really glad that you are all finally here! Unless you have recently organized a conference of this general magnitude and complexity yourself, you can’t begin to fathom my pleasure and relief that you are really, finally all here. You are registered, we have managed to put all the many papers into a program, your lodging is taken care of, even the audio-visuals seem to be working. This is a truly great moment, believe me!

“To get things off to a proper start, I will first call upon Prof. Werner Schiehlen of the University of Stuttgart, in Germany, the current president of the International Union of Theoretical and Applied Mechanics, to formally open the Congress. Please welcome Professor Schiehlen to the podium.” Prof. Werner O. Schiehlen, President of IUTAM “Dear Mr. President Stukel, dear members of the Host Consortium, dear colleagues from all over the world, ladies and gentlemen, “Mathematics and physics, as well as civil engineering and mechanical engineering, are the roots of mechanics. Theoretical mechanics uses mathematical methods and physical principles, today enriched with computational tools. Applied mechanics deals with constructions and machines subject to new designs and materials. “While theoretical mechanics was established as a branch of science more than 300 years ago by Isaac Newton, applied mechanics dates back to antiquity with famous names like Archimedes. However, the ancient Greeks built not only grandiose temples, mechanically well designed, but they invented the Olympic Games, too. Let us have a short look at Webster’s dictionary. The entry on Olympics reads as follows: ‘An ancient Panhellenic festival held every fourth year and made up of con-

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Prof. Hassan Aref, President of ICTAM 2000, welcomes participants to the Congress

in the Grand Ballroom of the Chicago Marriott Downtown.

tests of sports, music, and literature with the victor’s prize, a crown of wild olive.’ “Thus, the ancient games were not restricted to sports. They involved

literature as well. Literature, in general, comprises belles-lettres and textbooks, which may be also scientific. And there must be a contest. And there must be a prize. All these are essential features of the 20th International Congress of Theoretical and Applied Mechanics, and enable us to call it the Olympics of Mechanics. “But how is the contest performed? What about the prize? Well, it is known in the mechanics community that the International Papers Committee of the mechanics congresses is very selective, and nearly half of the submitted papers are rejected. Nevertheless, the number of submissions is increasing from congress to congress. This means a real contest of scientists. “On the other hand, since 1988 the Bureau of IUTAM has awarded prizes for outstanding presentations by young scientists. In particular, at this millennium congress, the three prizes are especially attractive by the ceramic plaque designed and manufactured by Illinois artist SitiMariah Jackson. The Hellenic prize of a crown of wild olive is finished as an artistic plaque in a unique way to be awarded to young victors of mechanics. “Ladies and gentlemen, it is my great honor to call the 20th International Congress on Theoretical and Applied Mechanics to order and to welcome all of you here in Chicago on behalf of IUTAM. The Marriott on Chicago’s Magnificent Mile is an excellent location for a world congress and we are looking forward to learn about the latest developments in mechanics in a most pleasant environment. After three years of extensive preparation under the guidance of Professor Hassan Aref, your active participation in the Congress will crown it with the success we all expect.

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“I have to admit that the Congress is also the central scientific event of IUTAM, opening the door to the new millennium for mechanics. Despite its long history, mechanics is a very lively subject today. The Mechanics Calender maintained for IUTAM by the University of Illinois at UrbanaChampaign lists for the year 2000 altogether 99 scientific meetings in 24 different countries. “In addition, to look back and ahead IUTAM is publishing a report entitled Mechanics at the Turn of the Century. This report is the result of an initiative of the Bureau of IUTAM to provide some landmarks on the development in mechanics during the 20th century, to report on the 50 years of impulse to mechanics by IUTAM, and to look ahead on a very personal basis to future areas of research, and also to show the broad international involvement of scientists in IUTAM in recent years. “Ladies and gentlemen, enjoy a world congress in a fascinating city on a most challenging subject: Theoretical and Applied Mechanics.”

Professor Aref “Thank you President Schiehlen. ICTAM 2000 came about through a lengthy, competitive proposal process. In fact, last night during the first meeting of the Congress Committee we began the similar process for the next congress in 2004. The formal invitation to host the 20th congress in Chicago was issued by the U.S. National Academy of Sciences on the recommendation of the U.S. National Committee for Theoretical and Applied Mechanics. While the National Academy, then, is the official sponsor, it was a consortium of universities, most of them in the Midwest, that really stepped up to the plate and provided the base of financial backing necessary to embark on this project. The host consortium includes 13 university groups with strong interests in mechanics, including, of course, the two remaining departments in the U.S. that have the name ‘Theoretical and Applied Mechanics’. The university sponsors are listed in the accompanying table. “Several professional societies, all but one represented on the National Committee for Theoretical and Applied Mechanics, also provided support to ICTAM 2000. The financial contributions of the professional societies were used to subsidize our banquet at the Museum of Science and Industry on Thursday so that we could induce as many of you as possible to attend. I am pleased to report that this incentive worked wonderfully and about 1000 delegates and accompanying persons will attend the banquet. I would like to acknowledge them, again in alphabetical order. [See accompanying table.] “Our stalwart supporters within the Federal government provided much appreciated funding for participant support to younger attendees

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and to attendees from countries with weak financial support structures for science. I should like to thank them. [See accompanying table.]

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“As a special feature at this Congress we are conducting a Science Teachers Day on Thursday, August 31. Some 65 teachers from the Northern half of Illinois will come to the Congress and be treated to a special program in the morning before mixing with the rest of us in the afternoon. Illinois artist Billy Morrow Jackson and I will tell you and the teachers a bit about the creation of the Congress poster during lunch on that day in Ballroom I. Many of the teachers will be joining us in the evening for the banquet. This special event was generously underwritten by the Chicago Engineers Foundation of the Union League

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Prof. Werner O. Schiehlen, President of IUTAM (left), and Prof. James J. Stukel, President of the University of Illinois, welcome participants to the Congress.

Club of Chicago. Jerry Moyar, an officer of the Foundation and an alum of my department, was responsible for securing this funding for us. “Finally, I should like to acknowledge a few individuals who contributed personal financial support directly or indirectly to the Congress. They are Professors Andy Acrivos, Stanley Berger, Tom Lundgren, and Franz Ziegler. Thank you all for your generous financial support. “My university, the University of Illinois, is represented by two campuses in the host consortium—my home campus in Urbana and the Chicago campus not too far from here. The president of the University of Illinois system, Professor James J. Stukel, whose faculty background is in mechanical engineering, has taken a personal interest in ICTAM 2000 and has generously underwritten the Welcome Reception that you will enjoy this evening. President Stukel is with us this morning, and he will now welcome you on behalf of the academic institutions hosting ICTAM 2000. Please welcome President James Stukel to the podium.”

Prof. James J. Stukel, President of the University of Illinois “Let me add my welcome to everyone here attending this conference on behalf of the host consortium universities and the academic community. As I looked at this final program, I realized that this really is going to be a very exciting conference. As a mechanical engineer, I fully appreciate the challenges of basic research in the mechanical sciences dealing with elasticity, fracture, plasticity, fatigue, turbulence, and combustion. As president of a large, public research university, I appreciate the contributions of the mechanical sciences to other academic

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disciplines. Examples include research on the mechanics of emulsions, porous materials, bio-materials, thin films, composites, and micro- and nano-technologies. As a citizen of the world, I benefit from the technology transfer from universities, which result in a better understanding of things like oceanic mixing, a better management of the world’s water resources, making available the possibility of space exploration, biotechnology, and recent advances in micro- and nano-technologies. We as a society and as a country would not be experiencing our robust economy were it not for the transfer of technology from the universities to the industrial sector. The same is true around the world. “Another wonderful event that I noticed on the agenda, which I think is really first rate, is bringing world class scientists in contact with high school science teachers. I can’t think of a more inspirational thing for a science teacher than to come in contact with the people who are here today. They need that inspiration in this country, and I suspect it is the case worldwide. So, I would like to thank all the scientists for taking part in this very important program. “I am also very, very proud to welcome you to one of the great cities in the US, in my view one of the great cities of the world. It is a city with superb architecture, wonderful museums, a beautiful arid accessible lake-front, and world class theater and music establishments. I certainly hope you have the chance to take advantage of these wonderful opportunities here in Chicago. And, speaking of art, as we have just indicated, I hope you have enjoyed, as much as I did, the watercolor by Billy Morrow Jackson, entitled ‘Meters of Motion’, depicting the history of your discipline. I thought it was a wonderful piece of art. Don’t you agree? “The Governor has also recognized this wonderful event and how important it is to the State of Illinois. Let me read the Proclamation as it is unveiled by Professor Aref: ‘Whereas the 20th International Congress of Theoretical and Applied Mechanics, ICTAM2000, is being held in Chicago, Illinois, during the week of August 28, 2000, and ‘Whereas, the international congresses of the International Union of Theoretical and Applied Mechanics have been ongoing for more than 75 years, and have visited major cities of the world including several sister cities of Chicago, and ‘Whereas, prior congresses have been held in the United States of America on only two prior occasions in 1938 and in 1968, and ‘Whereas, ICTAM 2000 is invited by the U.S. National Academy of Sciences, the pre-eminent scientific body of this nation, and ‘Whereas, the host university consortium includes several illustrious institutions of higher learning in the State of Illinois, to wit University of Illinois at UrbanaChampaign, University of Illinois at Chicago, University of Chicago, and Illinois Institute of Technology, and

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Left: Proclamation of Theoretical and Applied Mechanics Day in Illinois, 28 August 2001; right: Illinois artist Billy Morrow Jackson (center) with the ICTAM 2000 poster based on his watercolor Meters of Motion, accompanied by his wife, Siti-Mariah Jackson (right), creator of the IUTAM Bureau Prize plaques, and Dr. Gerald J. Moyar, representing the Chicago Engineers Foundation of the Urban League Club of Chicago, sponsors of the ICTAM 2000 Science Teachers Day on

31 August 2000.

‘Whereas, the attendees at ICTAM 2000 represent a gathering of leading researchers and scholars in the mechanical sciences from the international scientific community encompassing more than 50 nations, ‘Therefore, I, George H. Ryan, Governor of the State of Illinois, proclaim August 28, 2000, Theoretical and Applied Mechanics Day in Illinois. In witness whereof, I have hereunto set my hand and caused the Great Seal of the State of Illinois to be affixed. Done at the Capitol, in the city of Springfield, this twenty-first day of August, in the Year of Our Lord two thousand, and of the State of Illinois the one hundred and eighty second. ‘Signed George H. Ryan, Governor.’

“And lastly, I’d like to invite you to the reception this evening to eat on our dollar! So, enjoy it. I know you will have a wonderful conference.” Professor Aref

“Thank you President Stukel, and thanks to Governor Ryan for this nice show of support for our congress and our field. Today, Monday, August 28, 2000, is officially Theoretical and Applied Mechanics Day in the State of Illinois! The declaration is here for your inspection. We will move it to the foyer area following the Opening Ceremony. “From the consortium of universities and from the State of Illinois let me turn next to the city of Chicago. What links are there between our science of mechanics and the city of Chicago? “Well, Chicago has always been a kind of junction within the United States, a hub through which goods passed, by rail and by truck and,

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of course, these days by airplane. Chicago is called the city of the big shoulders, the tool maker, and the nation’s freight handler in the famous 1916 poem by Pulitzer Prize winning poet and biographer Carl Sandburg. This is certainly a most appropriate location for a mini-symposium on vehicle dynamics! “As you approach the city, you immediately see, of course, those huge manifestations of mechanics-in-action called skyscrapers. The skyscrapers in Chicago are among the tallest in the world. The Sears Tower, completed in 1974, with 110 stories, is 1450 ft, or 443 meters, tall to the

so-called structural or architectural top with only the Petronas Towers in Kuala Lumpur, built some 25 years later, being taller. However, the highest occupied floor in the Sears Tower is at 1431 ft, some 200 feet higher than the corresponding point in the Petronas Towers. The top of the roof is also higher, as is the tip of the spiral antenna on the roof. Even taller buildings were at various times planned for Chicago but never built. Between the structural engineers and the architects the rebuilding of Chicago after the Great Fire of 1871, when wooden buildings were banned and land became ever more expensive, produced the first city of skyscrapers. Design pioneers, such as Daniel Burnham and Louis Sullivan, who led what became known worldwide as the Chicago School of Architecture, gave Chicago its unique skyline. As the famous architect Frank Lloyd Wright said, ‘Beautiful buildings are more than scientific. They are true organisms, spiritually conceived; works of art, using the best technology by inspiration rather than the idiosyncrasies of mere taste or any averaging by the committee mind.’ “Since bio-mechanics is prominently featured at ICTAM 2000, I might mention that Chicago’s firm of William Wrigley Jr. is the world’s largest chewing gum manufacturer. “The World’s Columbian Exposition of 1893 in Chicago introduced a couple of mechanical firsts associated with the City. On a small scale the idea of a slide fastener or what we today would call a zipper was exhibited by Whitcomb L. Judson. This invention, later refined by others, was to become a mainstay of clothing for at least the next century. On a large scale, the Ferris Wheel was the engineering highlight of the Exposition and one of the most pervasive, lasting influences of the 1893 fair. The Ferris Wheel was Chicago’s answer to the Eiffel Tower, the landmark of the 1889 Paris exhibition. The wheel was created by bridge builder George W. Ferris. Supported by two 140 foot steel towers, its 45 foot axle was the largest single piece of forged steel at the time in the world. The wheel itself had a diameter of 250 feet and was powered by two 1000 horsepower reversible engines . . .

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Ray Wild as Archimedes (left) and Craig Spidle as Galileo Galilei.

Isaac Newton has entered on the far side of the room and is negotiating a seat with members of the audience. Newton: “Oh! That’s a nice seat! Do you mind if I sit right up there?” Aref: “So, these are some of the historical links between mechanics and Chicago. Now, continuing this grand tradition, we celebrate ICTAM 2000 in the windy city ... Excuse me, sir . . . ” Newton ignores him; continues wandering towards the front of the room. Aref: Newton:

“Sir! Are you registered for the Congress? Sir?” “Who are you? And what, may I inquire, is this gathering?” Galileo has entered from a door on the opposite side of the Ballroom. Galileo: “Isaac! My dearest friend! What are you doing here?” Newton: “Finally a familiar face! Where are we? Who are all these people?” Archimedes enters in a rush, half-running, waving the Congress program. Archimedes: “Eureka! Eureka! We are at an international congress of mechanics!” Galileo: “Slow down my friend!” Newton: “Of what?” Archimedes: “We’re at the 20th International Congress of Theoretical and Applied Mechanics, ICTAM 2000 . . . ”

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Genevieve VenJohnson as ‘Chicago’ (left) and Robert Ayres as Isaac Newton.

Galileo:

“Icky-what?”

Archimedes:

“ . . . a worldwide conference to map out advances in the mechanical sciences for the new millennium. These congresses are held every four years, just like the Olympic

Games.” Galileo: Newton:

“Theoretical and Applied?” “I have always preferred the terms ‘Rational’ and ‘Practical’.”

Archimedes:

“International!”

Newton:

“But what are all these people doing here? In my day it took only a few of us to discover the universal law of gravitation, determine the figure of the Earth and the orbits of the planets, write down my first, second and third laws, of course...”

Galileo:

“Your first law was really mine, Isaac!”

Newton: Galileo:

“Your one and only, I might add!” “Wait a minute, per favore! I started dialogues on two new sciences, and I spent a lot of time paving the way for you, arguing the Earth actually moves...”

Archimedes: “Gentlemen, gentlemen, please! My dear friends, this is all irrelevant. Look at what we have wrought: Hundreds of natural philosophers improving on our discoveries, making new ones, debating all kinds of issues. Look at this massive program of presentations.”

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Newton:

“What? There is a paper here on non-Newtonian fluid mechanics. Preposterous!” Galileo: “I see a paper here that uses Galilean invariance—and one that mentions your principle, Archimedes.” Newton: “Non-Newtonian fluid mechanics! Hypothesis non fingo!” Chicago enters and stands behind the podium in relative darkness. Newton: “Where are we anyway?” Galileo: “Not Roma!” Archimedes: “Certainly not Syracusa. Though, if you gave me a fixed fulcrum and a place to stand I can move you there!” Newton: “Well, it is definitely not grand old London!” Archimedes: “It’s a new world! A city called Chick-a-goo!” Galileo: “Chick-a-goo?” Newton: “Chick-a-goo?” Chicago is now at the podium with a spotlight directed at her. Chicago Chicago! HOG Butcher for the World, Tool Maker, Stacker of Wheat, Player with Railroads and the Nation’s Freight Handler; Stormy, husky, brawling, City of the Big Shoulders:” Archimedes: Chicago: Galileo: Chicago:

Newton: Chicago:

“They say you are wicked!” “And I believe them. For I have seen the painted women under the gas lamps luring the farm boys.” “They tell me you are crooked!” “And my answer is: Yes, for I have seen the gunman kill and go free to kill again.”

“And they tell me you are brutal!” “And my reply is: On the faces of women and children I have seen the marks of wanton hunger. But having answered so I turn once more to those who sneer at this my city, and I give them back their sneer and say to them: Come and show me another city with lifted head singing so proud to be alive and coarse and strong and cunning. Flinging magnetic curses amid the toil of piling job on job, here is a tall bold slugger set vivid against the little soft cities;”

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Newton, Galileo, Archimedes protest. Chicago: “Fierce as a dog, tongue lapping for action, cunning as a savage pitted against the wilderness,” Newton, Galileo, Archimedes (take turns):

“Bareheaded, Shoveling, Wrecking, Planning, Building, breaking, rebuilding,”

Chicago:

“Under the smoke, dust all over his face, laughing with white teeth, Under the terrible burden of destiny laughing as a young man laughs, Laughing even as an ignorant fighter laughs who has never lost a battle, Bragging and laughing that under his wrist is the pulse, and under his ribs the heart of the people,”

Galileo: Archimedes: Newton: Chicago:

“Laughing!” “Laughing the stormy, husky, brawling laughter of Youth...” “ . . . half-naked, sweating,” “Proud to be Hog Butcher, Tool Maker, Stacker of Wheat, Player with Railroads and Freight Handler to the Nation.”

Chicago steps away from the podium; exits at the front of the room. Galileo:

“Eppur si muove! But still it moves! I hope it is not too late to register! I would like to discuss with these people the latest developments in the Galilean invariance.”

Newton: Archimedes: Newton:

“Non-Newtonian fluid mechanics!” “I will give my poster at the session on Thursday.” “I think I shall have to look up the President and address the General Assembly about what needs to happen in natural philosophy. And it will be strictly Newtonian!”

Newton, Galileo and Archimedes are now together at the front of the room. Galileo: “Look at these social events! Have you ever been to a baseball game?” Newton: Archimedes:

“It just doesn’t match the excitement of cricket.” “I think the Art Institute is more my style.”

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Newton:

“It might be interesting to throw an apple from the Sears Tower.”

Galileo:

“Or drop it!”

Archimedes:

“I wonder how they get water to the top floors of these tall buildings.”

Galileo: Newton:

“Is there an open staircase where one can hang a pendulum?” “I am famished. What shall we have for lunch?”

Galileo: Archimedes:

“I hear the Italian food here is excellent.” “The Greek food is very good as well.”

Newton:

“I must immediately find a fruit vendor to secure an

apple.” Archimedes picks up an apple from a member of the audience. Archimedes:

“That’s alright. Wait, wait a minute. Here is an apple! Right here. Thank you!”

Throws it to Newton who catches it. Galileo: “Then it’s off to the tower!” Galileo sets off towards the entry doors at the back of the room. The others follow. Archimedes: “You’re going to drop it from that height?” Newton: “Throw it! Throw it! Dropping is a trivial, special case, hardly warranting the consideration of philosophers.”

Galileo protests; advances towards Newton. Archimedes intervenes.

Galileo:

“How do we get there?”

Archimedes:

“By hydraulic lift, of course! And the view must be magnificent!” “With my telescope you can see forever.”

Galileo: Newton:

“If I have seen farther than others it is from the shoulders of giants!”

Archimedes pats his own shoulder.

Archimedes:

“That is my shoulder.”

Galileo:

“Would it be a good idea for me to stand on your shoulders?” “Not so far as I can see!”

Archimedes:

Galileo: “Okey-dokey.” They have reached the doors and exit the Ballrooms.

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Professor Aref “So this is another, very important aspect of Chicago—culture! Those of you who go to the Art Institute on Wednesday, with Archimedes, will see one important aspect of Chicago’s cultural life. And, surely, the Chicago Symphony and the Lyric Opera of Chicago will be household names to many of you. Of course, when it comes to music, Chicago is particularly well known for blues and jazz. We’ll have a small sampling tonight at the Reception. But there is also a vibrant theater life in the city, practitioners from which you have just met. Please give a warm round of applause to our four actors,

Robert Ayres as Sir Isaac Newton and the director of our group, Craig Spidle as Galileo Galilei, Ray Wild as Archimedes, and Genevieve VenJohnson as ‘Chicago’. “This cultural interlude, as it was called in the program—we had to call it something that didn’t give away what it was—was kindly and generously underwritten by our friends at the Illinois Institute of Technology through the Office of the Chief Scientist of the IIT Research Institute—among friends better known as Hassan Nagib.

I thank the

actors for taking this on and for doing such a great job with it. I suspect the organizers of the next congress in 2004 will be in touch, and you will be off to the bright lights of Manchester, Brussels, Dresden or Warsaw. “Well, following this recital of Carl Sandburg’s poem, albeit in a somewhat unorthodox setting, it is time for a welcome from the City of Chicago. We had originally tried to get Mayor Daley himself, but as you can imagine with his involvement in the presidential race as a major force behind the Democratic ticket, he was not available. In fact, there is so much going on in Chicago that a number of business executives who are enthusiastic about the city volunteer to come to groups like ours and provide a welcome on behalf of the City. With us this morning is Mr. Martin Chasen, President of SMC Photo Promotions, Inc., who will say a few words of welcome on behalf of the City of Chicago.”

Martin Chasen welcomes the attendees on behalf of the City, introduces them to a printed Chicago guide that was made available, and invites them to explore the many attractions of Chicago. “Last, but certainly not least, we would like to extend a welcome to you all from the Chicago Marriott Downtown, one of the great hotel properties in this city, a full service conference hotel that hosts events like ours year in and year out and that will be our main stomping grounds

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Left: Grigory Isaakovich Barenblatt (left), closing lecturer at ICTAM 1992, jokes with H. Keith Moffatt (center), closing lecturer at ICTAM 2000, and Frithiof I. Niordson, IUTAM President 1976–1980, before Niordson introduces the ICTAM 2000 opening

lecturer; right: James R. Rice delivers the ICTAM 2000 opening lecture on “New perspectives in crack and fault dynamics”.

for the next five days. Mr. John Adams, the general manager, was regrettably out of town today, but he has asked Mr. Bob Pierce, director of marketing, to say a few words of welcome on behalf of the Chicago Marriott Downtown. Please welcome to the podium Mr. Bob Pierce.” Bob Pierce welcomes the attendees on behalf of the Chicago Marriott Downtown and promises that the hotel and its staff will do everything

to make the week successful and enjoyable. “Thank you Mr. Pierce. Thank you to all our welcoming speakers, thanks to our actors, and thanks to you for your attention. Have a great time and please let us know if you need assistance with any practical matters. We are here to keep the Congress running smoothly and to assure that you have a good week. This concludes the Opening Ceremony. The Opening Lecture will begin in this room in just a few moments at 11:00 a.m.”

Note: QuickTime™ movies of all parts of the Opening Ceremony are

available on the ICTAM 2000 web site at http://www.tam.uiuc.edu/ICTAM2000/ Program/ Opening-Ceremony/

CONGRESS POSTER Meters of Motion Billy Morrow Jackson, 1998 Meters of Motion, a watercolor by Illinois artist Billy Morrow Jackson, was the basis for the poster announcing the 20th International Congress of Theoretical and Applied Mechanics. Jackson’s image depicts several of the giants of mechanics. In the center we have Isaac Newton (1642–1727) holding the device known as Newton’s cradle, illustrating the laws of impact and momentum, today frequently used as an executive toy because of its pleasing periodic motion. Above Newton the phases of the Moon and the falling apple—one of his key insights was the unification of the orbital motion of the Moon about Earth and the accelerated fall of an apple in Earth’s gravitational field. Newton’s results on his universal law of gravitation were published in 1687 in the Principia, probably the most influential scientific treatise of all time. Newton stands on the word ‘PRINCIPIA’ and his famous ‘second law’, force equals mass times acceleration or appears at top right in the image. To the left of Newton we find his intellectual predecessor Galileo (1564–1642) watching a swinging pendulum, timing its oscillations using his pulse. Galileo made seminal contributions not only to kinematics, dynamics and observational astronomy, but also to the subject of strength of materials. His work on kinematics and strength of materials, entitled Dialogues on Two New Sciences, is represented by the word ‘DIALOGUES’, itself a cantilever beam anchored in a brick wall and carrying a suspended weight. Behind Galileo a ball and feather fall with the same acceleration. Above Galileo, in the uppermost left-hand corner, we see Archimedes (287–212 B.C.), one of the great geniuses of antiquity who gave us the laws of hydrostatics and buoyancy, holding forth the symbol the ratio of the circumference to the diameter of a circle. Archimedes was the first to realize the universality of this ratio and to determine approximations to its numerical value. On the right we find Leonhard Euler (1707– 1783), the great mathematician and mechanician, who cast Newton’s laws in their enduring mathematical form using the calculus, and who

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solved legions of problems in both fluid and solid mechanics. He is shown with the famous elastica curve emanating from his forefinger. The elastica is the curve that gives the shape of a thin, perfectly elastic rod when it buckles under load. This curve also arises in other contexts, for example in the shape of a concentrated vortex, such as the tornado seen in the background between Newton and Euler. Below and between Newton and Euler we see another legend of mechanics and mathematics, Joseph Louis Lagrange (1736–1813), whose famous work Mecanique Analytique, published a century after Newton’s Principia, set out the equations of dynamics in a novel variational formulation that gave a teleological interpretation to all motion in the Universe. Lagrange’s work has served as a paradigm of all physical theories since then, from optics to electromagnetism to modern quantum mechanical field theories. Lagrange’s formulation of dynamics also lends itself to the theory of vibrations. Thus patterns of nodal lines on vibrating plates, so-called Chladni figures, flow like sheets from the manuscript he is holding. The three remaining large figures in the picture are of three of the greats of mechanics in the 20th century, individuals whose initiative

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was important also for the inception of the international congresses of mechanics. In the lower left-hand corner Theodore von Kármán (1881–1963) gazes out at sequences of vortices produced in the wakes of bluff bodies. These organized patterns, today known as Kármán vortex streets, are ubiquitous in the wakes of everything from huge tankers to power lines ‘singing in the wind’. Between Newton and Galileo we see Ludwig Prandtl (1875–1953), von Kármán’s teacher, and a seminal figure in both solid arid fluid mechanics. Prandtl is known for his work on the dynamics of the boundary layer, the thin layer of fluid adjacent to a moving rigid body, be it an airplane wing or a ship hull, a layer of fluid very much responsible for the resistance to motion experienced by the body. Golf balls have dimples precisely so that they can ‘fool’ the boundary layer into offering less resistance to their motion than to that of a smooth sphere. Prandtl is also known for his work on aerodynamics (hence the propeller churning to the right of him), on the strength of materials, and for conducting comprehensive experiments on a variety of mechanical phenomena. In the lower right hand corner is G. I. Taylor (1886–1975), whose many contributions included a theory of dislocations in solids, early studies of turbulence in the atmosphere (hence the weather balloon), a theory of the fireball from a nuclear explosion (hence the series of fireballs depicted—a film record was released of the first nuclear explosion and G. I. Taylor developed a theory of the blast phenomenon that predicted the total, classified yield of the bomb). Taylor was an avid sailor and he was very proud of his invention of a drop anchor (depicted). The mathematician and novelist Sofia Kovalevskaya (1850–1891) discovered a very important solution to the problem of the ‘heavy top’. She made her lasting contributions to mechanics during a time when women were not allowed to attend lecture courses in universities. She is standing between Newton and Galileo, below Prandtl, writing an equation (the famous Korteweg–deVries equation that describes water waves) on a blackboard. Off to the right we have a bridge structure, where the members spell out ‘IUTAM’, the acronym for the International Union of Theoretical and Applied Mechanics, the professional body responsible for the international congresses. On the bridge we find current and past officers of IUTAM: past presidents Sir James Lighthill, Paul Germain, Leen van Wijngaarden and Daniel C. Drucker, and current president Werner Schiehlen. We also see the current and past secretaries of lUTAM’s Congress Committee, Niels Olhoff and H. Keith Moffatt. The president of the 19th Congress in Kyoto, Tomomasa Tatsumi, is

in the middle of this group. Holding up the bridge are the president and

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secretary-general of the 20th Congress, Hassan Aref and James W. Phillips of the University of Illinois at Urbana-Champaign. Billy Morrow Jackson has a website at http://www.soltec.net/ jacksonstudios/.

PROGRAM AT A GLANCE Monday, 28 August 2000 10:00—Opening Ceremony Hassan Aref, President of ICTAM 2000, presiding 11:00—Opening Lecture, plenary session AZ—James R. Rice, Harvard University, USA Frithiof I. Niordson, IUTAM President 1976–80, presiding 12:00—Lunch 14:00—Mini-Symposia introductory lectures, parallel sessions BD—Damage and failure of composites (Christensen, Chaboche) BE—Mechanics of foams and cellular materials (Gibson, Loewenberg) BP—Electromagnetic processing of materials (Davidson, Toh) BQ—Vehicle system dynamics (Anderson, Sharp, Abe/Hedrick) BX—Turbulent mixing (Ottino, Villermaux) BY—Granular flows (Roux, Nagel/Jaeger) 16:00—Break 16:30—Contributed lectures, parallel sessions CA—Large-eddy simulation CB—Biological fluid dynamics—Lighthill Memorial Session CC—Convective phenomena CD—Fracture and crack mechanics (jointly with ICF) CE—Structural optimization (jointly with ISSMO) CF—Multibody dynamics CG—Material instabilities CH—Acoustics CK—Flow control CO—Stability of structures CS—Mechanics of phase transformations (jointly with IACM) CV—Environmental fluid mechanics CW—Elasticity CX—Flow instability and transition

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19:00—Welcome reception Host: James J. Stukel, President, University of Illinois

Tuesday, 29 August 2000 08:00—Mini-Symposia introductory lectures, parallel sessions DD—Damage and failure of composites (Dvorak) DE—Mechanics of foams and cellular materials (Kyriakides) DP—Electromagnetic processing of materials (Walker) DQ—Vehicle system dynamics (Schiehlen) DW—Turbulent mixing (Wunsch) DX—Granular flows (Louge) 09:00—Sectional lectures, parallel sessions ED—Pedro Ponte Castañeda, USA EE—Denis L. Weaire, Ireland EP—Hamda BenHadid, France

EQ—Timothy J. Gordon, UK

EW—Paul E. Dimotakis, USA EX—Isaac Goldhirsch, Israel 10:00—Break

10:30—Contributed lectures, parallel sessions FA—Turbulence FB—Fluid mixing FC—Boundary layers FD—Mechanics of foams and cellular materials FE—Damage and failure of composites FF—Vehicle systems dynamics (jointly with IAVSD) FG—Plasticity and viscoplasticity FH—Impact and wave propagation FK—Electromagnetic processing of materials (jointly with HYDROMAG) FL—Granular flows FO—Computational strategies for multiscale phenomena in mechanics (jointly with IACM) FR—Experimental methods in solid mechanics FS—Mechanics of porous materials FV—Combustion and flames 12:30—Lunch

Program at a glance

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14:00—Seminar presentation session, parallel sessions—three-minute overviews GA—Mechanics of foams and cellular materials

GB—Boundary layers and waves GC—Geomechanics and geophysical fluid dynamics GD—Fracture and crack mechanics (jointly with ICF) GE—Structural optimization (jointly with ISSMO) GF—Structural vibrations GG—Multibody dynamics GH—Computational solid mechanics (jointly with IACM) GK—Mixing and multiphase flow GL—Fluid–structure interaction GM—Plasticity, viscoplasticity, and dynamic plasticity of

structures GN—Continuum mechanics and nonlinear dynamics GR—Elasticity GS—Plates, shells, and stability of structures GV—Low-Reynolds-number flow, microfluid mechanics, and fluid mechanics of materials processing 15:00—Seminar presentation session, plenary session—poster display and discussion 16:00—Break

16:30—Contributed lectures, parallel sessions HA—Convective phenomena HB—Flow control HC—Flow instability and transition HD—Elasticity HE—Structural optimization (jointly with ISSMO) HF—Fatigue HG—Computational solid mechanics (jointly with IACM) HH—Contact and friction problems (jointly with IAVSD) HK—Waves HL—Granular flows HO—Dynamic plasticity of structures HR—Control of structures HS—Rock mechanics and geomechanics HV—Complex and smart fluids 18:30—General Assembly of IUTAM Werner O. Schiehlen, President of IUTAM, presiding

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Wednesday, 30 August 2000 08:00—Contributed lectures, parallel sessions IA—Boundary layers IB—Biological fluid dynamics IC—Turbulence ID—Fracture and crack mechanics (jointly with ICF) IE—Structural optimization (jointly with ISSMO) IF—Plates and shells (jointly with IACM) IG—Mechanics of thin films and nanostructures IH—Impact and wave propagation IK—Flows in thin films IL—Multiphase flows IO—Stability of structures IR—Willis Symposium IS—Viscoelasticity and creep IV—Vortex dynamics 10:00—Break 10:30—Contributed lectures, parallel sessions JA—Flow instability and transition JB—Convective phenomena JC—Fluid mechanics of materials processing JD—Elasticity JE—Material instabilities JF—Multibody dynamics JG—Computational solid mechanics (jointly with IACM) JH—Impact and wave propagation JK—Waves JL—Drops and bubbles JO—Cellular and molecular mechanics JR—Willis Symposium JS—Microgravity mechanics JV—Topological fluid mechanics 12:30—Lunch and Excursions 14:00—General Assembly of IUTAM Werner O. Schiehlen, President of IUTAM, presiding

Program at a glance

Thursday, 31 August 2001 08:00—Sectional lectures, parallel sessions KD—Peter A. Monkewitz, Switzerland KE—Vladimir A. Palmov, Russia KP—Erwin Stein, Germany KQ—Stephen J. Cowley, UK KW—Erik van der Giessen, The Netherlands KX—Charles S. Peskiri, USA 09:00—Sectional lectures, parallel sessions LD—Shigeo Kida, Japan

LE—R. Narayana Iyengar, India LP—Gedeon Dagan, Israel LQ—Amable Liñán, Spain LW—Subra Suresh, USA LX—Ole Sigmund, Denmark 10:00—Break 10:30—Contributed lectures, parallel sessions MA—Flow instability arid transition MB—Geophysical fluid dynamics MC—Low-Reynolds-number flow MD—Mechanics of foams and cellular materials ME—Damage and failure of composites MF—Structural vibrations MG—Plasticity and viscoplasticity MH—Contact and friction problems (jointly with IAVSD)

MK—Waves ML—Drops and bubbles MO—Biological solid mechanics MS—Chaos in fluid and solid mechanics MV—Vortex dynamics

12:30—Lunch 14:00—Seminar presentation session, parallel sessions—three-minute overviews NA—Granular flows and complex and smart fluids NB—Biomechanics NC—Transition and turbulence ND—Fracture and crack mechanics (jointly with ICF) NE—Contact and friction problems (jointly with IAVSD) NF—Damage and failure of composites

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NG—Impact and wave propagation NH—Smart materials and structures NK—Convective phenomena and fluid mechanics of materials processing NL—Computational solid mechanics (jointly with IACM) NM—Functionally graded materials, porous materials, phase transformations, and thin films NN—Vehicle dynamics and control of structures NS—Viscoelasticity, creep, and fatigue NV—Combustion, compressible flow, and computational fluid dynamics 15:00—Seminar presentation session, plenary session—poster display and discussion 16:00—Break

16:30—Contributed lectures, parallel sessions OA—Microfluid dynamics OB—Biological fluid dynamics OC—Turbulent mixing OD—Boundary layers OE—Elasticity OF—Damage mechanics OG—Mechanics of thin films and nanostructures OH—Fatigue

OK—Fluid–structure interaction OL—Drops and bubbles OO—Electromagnetic processing of materials (jointly with HYDROMAG) OS—Mechanics of phase transformations (jointly with IACM) OW—Turbulence 17:30—Contributed lectures, parallel sessions PA—Microfluid dynamics PB—Biological fluid dynamics PC—Turbulent mixing PD—Granular flows PE—Fracture and crack mechanics (jointly with ICF) PF—Material instabilities PG—Plasticity and viscoplasticity PH—Fatigue PK—Fluid–structure interaction PL—Multiphase flows

Program at a glance

PO—Electromagnetic processing of materials (jointly with

HYDROMAG) PS—Mechanics of porous materials PW—Flow instability and transition 19:00—Banquet at the Museum of Science and Industry

Friday, 1 September 2000 08:00—Contributed lectures, parallel sessions QA—Turbulence QB—Biological fluid dynamics QC—Electromagnetic processing of materials (jointly with

HYDROMAG) QD—Fracture and crack mechanics (jointly with ICF) QE—Functionally graded materials QF—Control of structures QG—Continuum mechanics QH—Smart materials and structures QK—Drops and bubbles QL—Computational solid mechanics (jointly with IACM)

QO—Fluid–structure interaction QR—Compressible flow QS—Solid mechanics in manufacturing QV—Complex and smart fluids 09:00—Contributed lectures, parallel sessions RA—Flow control RB—Biological fluid dynamics RC—Fluid mechanics of materials processing RD—Fracture and crack mechanics (jointly with ICF) RE—Functionally graded materials RF—Vehicle systems dynamics (jointly with IAVSD) RG—Continuum mechanics RH—Smart materials and structures RK—Waves RL—Computational solid mechanics (jointly with IACM) RO—Fluid–structure interaction RR—Compressible flow RS—Solid mechanics in manufacturing RV—Low-Reynolds-number flow 10:00—Break

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10:30—Contributed lectures, parallel sessions SA—Flow instability and transition SB—Biological solid mechanics SC—Geophysical fluid dynamics SD—Continuum mechanics

SE—Damage mechanics SF—Structural vibrations SG—Mechanics of thin films and nanostructures SH—Smart materials and structures SK—Chaos in fluid and solid mechanics SL—Plates and shells (jointly with IACM) SO—Plasticity and viscoplasticity SR—Acoustics SS—Solid mechanics in manufacturing SV—Flows in thin films 11:30—Contributed lectures, parallel sessions

TA—Flow instability and transition TB—Multibody dynamics TC—Rock mechanics and geomechanics TD—Lighthill Memorial Session TE—Damage mechanics TF—Control of structures TG—Damage and failure of composites TH—Smart materials and structures TL—Experimental methods in solid mechanics TV—Flow in porous media 12:30—Lunch

14:00—Closing Lecture, plenary session UJ—H. Keith Moffatt, Cambridge University, UK G. I. Barenblatt, Closing Lecturer, ICTAM XVIII, presiding 15:00—Closing Ceremony Werner O. Schiehlen, President of IUTAM, presiding

SCIENTIFIC PROGRAM Listed below are the technical sessions and session chairs, along with the authors and titles of papers presented within each session. For each paper, the country of the first-named author, who was the designated presenter, appears in parentheses following the list of authors. Please refer to the Index to Authors, Coauthors, and Session Chairs at the end of this volume to locate a particular author or chair. AZ—Opening Lecture by James R. Rice (Frithiof I. Niordson, Denmark, chair) AZ1—James R. Rice (USA): New perspectives on crack and fault dynamics BD—Damage and failure of composites—Introductory Lectures (George J. Dvorak, USA, chair) BD1—Richard M. Christensen (USA): A survey of and evaluation methodology for fiber composite material failure theories BD2—Jean-Louis Chaboche (France): On constitutive and damage modeling in metal matrix composites BE—Mechanics of foams and cellular materials—Introductory Lectures (Andrew M. Kraynik, USA, chair) BE1—Lorna J. Gibson (USA): Metallic foams: Structure, properties and applications BE2—Michael Loewenberg (USA): Numerical simulation of dense emulsion flows BP—Electromagnetic processing of materials—Introductory Lectures (René J. Moreau, France, chair) BP1—Peter A. Davidson (UK): Aluminum: Approaching the new millennium BP2—Takehiko Toh and Eiichi Takeuchi (Japan): Electromagnetic phenomena in steel continuous casting BQ—Vehicle system dynamics—Introductory Lectures (Peter Lugner, Austria, chair) BQl—Ronald J. Anderson; John A. Elkins; Barrie V. Brickie (Canada): Rail dynamics for the 21st century BQ2—Robin S. Sharp (UK): Fundamentals of the lateral dynamics of road vehicles BQ3—Masato Abe; J. Karl Hedrick (Japan): A mechatronics approach to advanced vehicle control system design BX—Turbulent mixing—Introductory Lectures (Paul E. Dimotakis, USA, chair) BX1—Julio M. Ottino (USA): The kinematics of mixing flows BX2 —Emmanuel Villermaux (France): Mixing: Kinetics and geometry BY—Granular flows—Introductory Lectures (Stuart B. Savage, Canada, chair) BY1—Stéphane Roux; Farhang Radjai (France): Statistical approach to the mechanical behavior of granular media BY2—Sidney R. Nagel and Heinrich M. Jaeger (USA): Signatures of microstructure in granular shear flows: The use of non-invasive probes CA—Large-eddy simulation (Ronald J. Adrian, USA, chair) CA1—Carlo Cercignani, Antonella Abbà, and Lorenzo Valdettaro (Italy): Analysis of subgrid scale models CA2—Marcel R. Lesieur (France): Large-eddy simulations of turbulence in shear flows CA3—Robert D. Moser, Stefan Volker, Prem Venugopal, and Jacob A. Langford (USA): Optimal models for large eddy simulation of turbulence

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CA4—Daniele Carati; Grégoire S. Winckelmans and Hervé Jeanmart (Belgium): Filtering and discretization in large eddy simulation of turbulent flows CA5—Andrew Pollard and Stuart McIlwain (Canada): Large eddy simulations of a coannular jet with low swirl CA6—James G. Brasseur, Anurag Juneja, and Yong Zhou (USA): Weakest links in practical application of large-eddy simulation: The surface layer

CB—Biological fluid dynamics—Lighthill Memorial Session (Andreas Acrivos, USA, chair) CB1—John R. Blake ( U K ) : Microbiological fluid dynamics: A tribute to Sir James Lighthill CB2—Timothy J. Pedley and Vanesa Magar ( U K ) : Model of nutrient uptake by a spherical squirming micro-organism CB3—Stanley A. Berger and Brian E. Carlson (USA): Flow of sickle cell blood in a microcirculatory bed CB4—Z. Jane Wang (USA): Unsteady aerodynamics of insect flight CB5—John O. Kessler (USA): Turbulence, mixing, and dynamics of bacterial suspensions CC—Convective phenomena (Friedrich H. Busse, Germany, chair) CC1—Alexander V. Getling (Russia): Magnetoconvection: Structuring of the amplified magnetic field and freezing of the flow pattern CC2—Keith Julien; Edgar Knobloch; Steve M. Tobias (USA): Highly supercritical magnetoconvection CC3—Wilhelm Schneider (Austria): Lift force induced by buoyancy at a finite horizontal plate CC4—Sergey A. Suslov; Samuel Paolucci (Australia): Spatially developing instabilities in non-Boussinesq convection flows CC5—Alexander A. Golovin; Alexander A. Nepomnyashchy; Leonid M. Pismen (Israel): Non-potential effects in nonlinear dynamics of Marangoni convection patterns CC6—Valentina M. Shevtsova, Denis E. Melnikov, and Jean-Claude Legros (Belgium): Influence of the temperature dependent viscosity on the onset of instability in liquid bridge

CD—Fracture and crack mechanics (jointly with ICF) (Bhushan L. Karihaloo, UK, chair) GD1—Daniel Rittel (Israel): Transient thermomechamcal effects in dynamic fracture CD2—Yoichi Sumi and Yang Mu (Japan): Thermally induced quasi-static wavy crack propagation in a brittle solid CD3—Asher A. Rubinstein ( U S A ) : Crack path development in materials with nonlinear process zone CD4—Krishnaswamy Ravi-Chandar (USA): Dynamic shear crack propagation

CD5—Giulio Maier, Gabriella Bolzon, and Roberto Fedele (Italy): Parameter identification of cohesive crack models by Kalman filter CD6—Jean-Baptiste Leblond and Joel Frelat (France): Crack kinking from an initially closed crack in the presence of friction

CE—Structural optimization (jointly with ISSMO) (Martin P. Bends0e, Denmark, chair) CE1—Helder C. Rodrigues, Paulo S. Miranda, and José M. Guedes; Christopher Jacobs

(Portugal): Optimization study of the interface stiffness in non-cemented hip prostheses CE2—Andreas Rottler, Hans A. Eschenauer, and Robert Dragos; Guido Bieker and Matthias Flach (Germany): Dynamic behavior and durability of rail vehicle components— modeling, simulation, optimization CE3—Andrzej Osyczka and Krenich Stanislaw (Poland): Structural optimization of robot grippers using genetic algorithms CE4—Niels Olhoff and Jacob P. Foldager; Jorn S. Hansen (Denmark): Optimization of the buckling load for composite shells taking thermal effects into account CE5—Guillaume Renaud and Jorn S. Hansen (Canada): Shape optimization of a composite patch repair CE6—John E. Taylor; Martin P. Bendsøe (USA): A mutual energy formulation for the design of continuum structures for generalized objective CF—Multibody dynamics (John J. McPhee, Canada, chair) CF1—Ahmed A. Shabana (USA): Use of the absolute nodal coordinate formulation in modeling flexible multibody systems

CF2—Wojciech Blajer (Poland): An improved formulation for multibody systems with possible singularities and redundancy

CF3—Peter Eberhard (Germany): Transitions in mixed multibody systems/finite element contact simulation CF4—Jiazhen Hong, Anping Guo, and Yanzhu Liu (China, PRC): A dynamic model with substructures for contact-impact analysis of flexible multibody systems

CF5—Florian Wegmann and Friedrich Pfeiffer (Germany): Couplings with neglectable bending stiffness in multibody systems

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CF6—Olivier A. Bauchau and Yuri G. Nikishkov (USA): An implicit transition matrix approach to stability analysis of flexible multibody systems CG—Material instabilities (Viggo Tvergaard, Denmark, chair) CG1—Ahmed Benallal; Claudia Comi (France): Bifurcation and perturbation of elasticplastic saturated porous media CG2—John W. Rudnicki and Dmitry I. Garagash (USA): Stability of undrained deformation

of fluid-saturated dilating/compacting solids CG3—Matthew R. Kuhn (USA): The response of dense granular materials to large strain gradients CG4—Lambertus J. Sluys, Harm Askes, Alena Pozivilova, and Wei M. Wang (The Nether-

lands): Large strain viscoplasticity for the modelling of shear banding and necking CG5—Yoshihiro Tomita (Japan): Computational prediction of mechanical characteristics of blended polymer under combined loading CG6—Klaus Thermann; Henryk Petryk (Germany): Post-critical plastic deformation pattern in incrementally nonlinear materials CH—Acoustics (Allan D. Pierce, USA, chair) CH1—Fernando Lund (Chile): Interaction of acoustic waves with singularities in solid and fluid mechanics CH2—Hafiz M. Atassi (USA): Propagation of acoustic and vortical disturbances in a duct with swirling mean flow CH3—Mikael A. Langthjem and Niels Olhoff (Denmark): On analysis and minimization of flow noise in a centrifugal pump CH4—William R. Graham (UK): The effect of modal coupling on low frequency vibration and radiation of fluid-loaded plates CH5—Tim Colonius, Clarence W. Rowley, and Richard M. Murray (USA): Simulation, modeling, and control of self-sustained oscillations in the flow past an open cavity CH6—Ted A. Manning and Sanjiva K. Lele (USA): A mechanism for the generation of shock-associated noise

CK—Flow Control (Mohamed Gad-el-Hak, USA, chair) CK1—Peruvemba R. Viswanath and Kinikkara T. Madhavan (India): Separation control by energization of bubble flow: A novel concept CK2—Frank Urzynicok and Hans H. Fernholz (Germany): Flow-induced acoustic resonators for separation control CK3—Peter W. Carpenter and Duncan A. Lockerby; Christopher Davies (UK): Is

Helmholtz resonance important for boundary-layer control by micro-jet actuators? CK4—J. David A. Walker, Chi-Young Kim, and Hediye Atik (USA): Control of boundarylayer separation with localized suction CK5—Sedat Tardu (France): Regularization of the near wall turbulence activity by unsteady local blowing/suction CK6—Junwoo Lim, Sung M. Kang, John Kim, and Jason L. Speyer (USA): Application of a linear controller to turbulent flows CO—Stability of Structures (Anthony N. Kounadis, Greece, chair) CO1—Simon D. Guest and Sergio Pellegrino (UK): Bi-stable shell structures CO2—James G. A. Croll; Seishi Yamada (UK): Numerical validation of analytical lower bounds for buckling of axially loaded cylinders CO3—James H. Starnes; Mark W. Hilburger (USA): Effects of imperfections on the buckling response of compression-loaded composite shells

CO4—Rivka Gilat; Jacob Aboudi (Israel): Parametric stability of nonlinearly elastic composite plates by Lyapunov exponents CO5—Christos C. Chamis (USA): Probabilistic stability of composite stiffened shells CS—Mechanics Of phase transformations (jointly with IACM) (Alain Molinari, France, chair) CS1—Franz D. Fischer; Narendra Simha (Austria): Phase transformations in elastic-plastic solids: Experiments and simulations CS2—Edgar N. Mamiya, Fabio A. Ferreira, and Dianne M. Viana (Brazil): Description of

internal hysteresis loops in pseudoelastic materials CS3—Mohammed Cherkaoui and Marcel Berveiller (France): Nucleation and growth of martensitic domains in TRIP steels CS4—Xiujie Gao and L. Catherine Brinson (USA): Simplified multivariant model and SEM/ EBSD verification of variant formation and switching

CS5—Yves M. Leroy and Joumana Ghoussoub (France): Solid-fluid phase transformation within grain boundaries

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A waiter presents 1988–1992 IUTAM president Paul Germain (center) with an 80th birthday cake during the ICTAM 2000 Welcome Reception at the Chicago Marriott Downtown.

Joining Germain are (clockwise) John R. Willis, René J. Moreau,

Hassau Aref, and Werner O. Schiehlen (back to camera).

Enjoying the Welcome Reception are (from left) Jette Olhoff, 1976–1980 IUTAM president Frithiof I. Niordson, ICTAM 2000 president Hassan Aref, 1992–2000 IUTAM Congress Committee chair Niels Olhoff, and Susanne Aref.

CV—Environmental fluid mechanics (Peter Lynch, Ireland, chair)

CVl—Colm-cille P. Caulfield; Andrew W. Woods; Adrian McBurnie and Jeremy C. Phillips ( U S A ) : Mixing induced by a finite mass flux plume in a ventilated room CV2— Carlos Haertel; Fredrik Carlsson and Mattias Thunblom (Switzerland): Analysis and simulation of the lobe formation at a gravity-current head CV3 —Bruce R. Sutherland and Kathleen Dohan (Canada): Excitation of large amplitude internal waves from a turbulent mixing region CV4—Julian C. R. Hunt; James W. Rottman (UK): Singular Coriolis effects in stable mountain flows

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CV5—Pierre Carlotti (UK): Spectra of turbulence near the ground CV6—Paul F. Fischer and Gary K. Leaf; Juan M. Restrepo (USA): Forces on spherical particles in oscillatory boundary layers CW—Elasticity (Raymond W. Ogden, UK, chair)

CW1—Michael A. Hayes; Philippe S. Boulanger (Ireland): Unsheared pairs and limiting directions in finite strain CW2—Thomas C. T. Ting (USA): Materials that uncouple antiplane and inplane displacements but not the stresses, and vice versa CW3—Paolo Podio-Guidugli; Paola Nardinocchi (Italy): Angled plates CW4—Pierre Seppecher and Mohamed Camar-Eddine (France): Non-local interactions resulting from the homogenization of an heterogeneous linear elastic medium CW5—Alan S. Wineman (USA): Network scission and healing effects in an elastomeric body due to deformation and high temperatures CW6—Peter Wriggers and Tarek I. Zohdi (Germany): Some aspects of the computational testing of microheterogeneous solids CX—Flow instability and transition (Alexander Solan, Israel, chair) CX1—Bart S. Ng and William H. Reid (USA): Simple asymptotics for the spectra of viscous shear flows CX2—Alison P. Hooper (UK): Energy methods and convergence problems relating to two-fluid flows CX3—Eli Reshotko; Anatoli Tumin (USA): Optimal spatial disturbances in a circular pipe flow CX4—Norman J. Zabusky, Alexei D. Kotelnikov, and Yuriy F. Gulak (USA): Nonlinear evolution and vortex localization in Richtmyer-Meshkov accelerated interfaces CX5—Jason G. Oakley, Bhalchandra P. Puranik, Mark H. Anderson, and Riccardo Bonazza (USA): An investigation of shock-induced interfacial instabilities for strong incident shocks CX6—Katherine Prestridge, Paul M. Rightley, and Robert F. Benjamin; Peter Vorobieff (USA): Validation of an instability growth model using particle image velocimetry measurements DD—Damage and failure of composites—Introductory Lectures (Zvi Hashin, Israel, chair)

DDl —George J. Dvorak (USA): Damage analysis and prevention in composite materials DE—Mechanics of foams and cellular materials—Introductory Lectures (Norman A. Fleck, UK, chair) DE1—Stelios Kyriakides (USA): In-plane crushing of honeycomb DP—Electromagnetic processing of materials—Introductory Lectures (Nagy El-Kaddah, USA, chair) DP1—John S. Walker (USA): Electromagnetic phenomena in crystal growth

DQ—Vehicle system dynamics—Introductory Lectures (J. Karl Hedrick, USA, chair) DQ1—Werner O. Schiehlen; Willi Kortüm and Martin Arnold (Germany): Software tools: From multibody system analysis to vehicle system dynamics DW—Turbulent mixing—Introductory Lectures (Emil J. Hopfinger, France, chair) DW1—Carl I. Wunsch (USA): Overview of oceanographic mixing problems from the microscale to the general circulation DX—Granular flows—Introductory Lectures (Robert P. Behringer, USA, chair) DX1—Michel Y. Louge and James T. Jenkins (USA): Particle segregation in collisional shearing flows ED—Sectional Lecture by Pedro Ponte Castañeda (Zvi Hashin, Israel, chair) ED1—Pedro Ponte Castañeda (USA): Nonlinearity and microstructure evolution in composite

materials

EE—Sectional Lecture by Denis L. Weaire (Norman A. Fleck, UK, chair) EE1—Denis L. Weaire (Ireland): Hard problems with soft materials: The mechanics of foams EP—Sectional Lecture by Hamda Ben Hadid (Nagy El-Kaddah, USA, chair) EPl—Hamda Ben Hadid (France): MHD damped buoyancy-driven flows

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EQ—Sectional Lecture by Timothy J. Gordon (J. Kari Hedrick, USA, chair) EQl—Timothy J. Gordon ( U K ) : Adaptive, nonlinear and learning techniques for the control of vehicle ride dynamics EW—Sectional Lecture by Paul E. Dimotakis (Emil J. Hopfinger, France, chair) EW1—Paul E. Dimotakis (USA): Recent advances in turbulent mixing EX—Sectional Lecture by Isaac Goldhirsch (Robert P. Behringer, USA, chair) EX1 —Isaac Goldhirsch (Israel): Kinetic and continuum descriptions of granular flow FA—Turbulence (Olivier J. Métais, France, chair) FA1—Markus Klein, Amsini Sadiki, and Johannes Janicka (Germany): Direct numerical simulations of plane turbulent jets at moderate Reynolds numbers FA2—Venkatesa I. Vasanta Ram and Maren Schmidts; Gerta Rocklage-Marliani (Germany): Swirling turbulent flow in a pipe: 3-D laser-Doppler velocimetry measurements and comparison with theories FA3—Joël Delville, Fabienne Ricaud, Christophe Picard, Laurent Neau, and Laurent E. Brizzi (France): Large scale structures identification in an axisymmetric shear layer from the near field pressure FA4—Stanislav V. Gordeyev and Flint O. Thomas (USA): Extraction and dynamical modeling of large-scale coherent structure in the planar turbulent jet FA5—Pierre Comte; Emmanuel Briand (France): Formation of coherent structures in spatially-growing boundary layers in large-eddy simulation FB—Fluid mixing (Robert S. Brodkey, USA, chair) FB1—Igor Mezic (USA): Chaotic advection in bounded Navier-Stokes flows FB2—Vered Rom-Kedar; Andrew C. Poje (USA): Universal properties of chaotic transport in the presence of diffusion FB3—Yiannis Ventikos (Switzerland): Spatio-temporal correlations for chaotic advection in aperiodic flows FB4—Jerry P. Gollub, David Rothstein, Emeric Henry, and Wolfgang Losert (USA):

Comparison of chaotic and turbulent transient mixing in two dimensions FB5—Alain Pumir; Misha Chertkov; Boris I. Shraiman (France): Geometry and statistics in Lagrangian dispersion FB6—Haris J. Catrakis, Joshua W. Hearn, Brenda A. McDonald, and Robert D. Thayne (USA): Structure and evolution of fluid interfaces in turbulence

FC—Boundary layers (Yury S. Kachanov, Russia, chair) FC1—Peter W. Duck (UK): On a class of unsteady, non-parallel, three-dimensional disturbances to boundary-layer flows FC2—Stefan Braun and Alfred Kluwick (Austria): Three-dimensional marginal separation of laminar viscous wall jets and boundary layer flows FC3 —Aleksandr V. Obabko and Kevin W. Cassel (USA): Navier–Stokes solutions of thickcore vortex induced unsteady separation FC4—Christopher D. Tomkins and Ronald J. Adrian (USA): Spanwise perturbation of energetic modes in a turbulent boundary layer

FD—Mechanics of foams and cellular materials (Howard A. Stone, USA, chair) FDl—Michael Thies, Carolin Körner, and Robert F. Singer (Germany): Foam formation

and evolution: Numerical simulation with a lattice Boltzmann model FD2—Andrew M. Kraynik; Douglas A. Reinelt (USA): Microrheology of dry soap foams with random structure FD3—Douglas J. Durian, Anthony D. Gopal, Arnaud Saint-Jalmes, and Loic Vanel

(USA): The melting of aqueous foams FD4—Stephan A. Koehler, Sascha Hilgenfeldt, and Howard A. Stone (USA): Foam drainage: Polydispersity and two-dimensional flow FD5—Dmitri L. Vainchtein and Hassan Aref (USA): A phase transition in compressible foam FD6—Bernd Markert and Wolfgang Ehlers (Germany): Finite viscoelastic deformations in fluid-saturated porous solids FE—Damage and f a i l u r e of composites (William A. C u r t i n , USA, chair) FE1—Y. Jack Weitsman (USA): Damage, deformation and failure in randomly reinforced polymeric composites FE2—Peter Gudmundson and Jan-Erik Lundgren (Sweden): Moisture absorption and desorption in composite laminates containing transverse matrix cracks

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FE3—Linda S. Schadler and Chaohui Zhou; S. Leigh Phoenix; Irene J. Beyerlein; Maher S. Amer (USA): Time dependent damage development in graphite/epoxy composites under constant load FE4—Peter Matic, Andrew B. Geltmacher, Kirth E. Simmonds, and Richard K. Everett; Carl T. Dyka (USA): Microtomography and finite element simulations of TiB2 reinforced 1100 aluminum microstructures FE5—David H. Alien (USA): A rate dependent fracture model for predicting multiple damage

during impact in laminated composites FE6—Gyaneshwar P. Tandon; Nicholas J. Pagano (USA): Effect of process-induced damage on the micromechanical response of a multiphase composite

FF—Vehicle systems dynamics (jointly with IAVSD) (Robin S. Sharp, UK, chair) FFl—Hiroshi Yabuno, Takuto Okamoto, and Nobuharu Aoshima (Japan): Stabilization control for the hunting motion of railway wheelset FF2—Oldrich Polach (Switzerland): Influence of locomotive tractive effort on the forces between wheel and rail FF3—Thomas B. Meinders (Germany): The influence of unbalances on the development of polygonalized railway wheels FF4—Hans True; Rolf Asmund (Denmark): The dynamics of a railway freight wagon wheelset with dry friction damping FF5—Yoshihiro Suda, Takefumi Miyamoto, and Norihiko Katoh (Japan): Active controlled rail vehicles for improved curving performance and response to track irregularity FF6—Steven A. Velinsky (USA): Mechanics of wheeled-vehicle based mobile manipulators FG—Plasticity and viscoplasticity (Zenon Mróz, Poland, chair)

FG1—Morton E. Gurtin; Paolo Cermelli (USA): The characterization of geometrically necessary dislocations in finite plasticity FG2—Otto T. Bruhns (Germany): A consistent description of finite elastoplasticity FG3—Keh-Chih Hwang and Han Q. Jiang; Yonggang Huang; Huajian Gao (China, PRC): A finite deformation theory of strain gradient plasticity

FG4—David L. McDowell, George C. Butler, and Robert D. McGinty (USA): An internal state variable model for dislocation substructure formation in polycrystal plasticity FG5—Laurent Langlois and Marcel Berveiller (France): Softening induced by change of dislocation micro structure in metals FG6—Henryk Zorski (Poland): Statistical theory of dislocations in 2D elastic body

FH—Impact and wave propagation (Rodney J. Clifton, USA, chair) FH1—Tsolo Ivanov; Radijanka Savova (Bulgaria): Surface wave propagation FH2—John G. Harris (USA): Elastic surface waves in curved structures FH3—Xiaodong Wang (Canada): High-frequency elastic wave propagation induced by piezoelectric actuators FH4—Lester W. Schmerr Jr. and Matthias Rudolph; Alexander Sedov (USA): Ultrasonic transducer wave field modeling for complex geometries and anisotropic materials

FH5—Paul Fromme and Mahir B. Sayir (Switzerland): On the scattering of flexural waves at a notched hole in a plate FH6—R. Bruce Thompson (USA): Ultrasonic models for the influence of micro structure on flaw detection in aircraft engines

FK—Electromagnetic processing of materials (jointly with HYDROMAG) (John S. Walker, USA, chair) FK1—Duane Johnson; Ranga Narayanan (USA): The effect of magnetic suppression on the convective flow patterns in cylindrical containers FK2—Amnon J. Meir and Paul G. Schmidt (USA): Liquid-metal flow in a cylindrical crucible rotating in a transverse magnetic field FK3—Claude B. Reed; Sergei Molokov (USA): The effect of a uniform magnetic field on a liquid-metal flow in a rivulet

FK4—Koichi Kakimoto (Japan): Heat and mass transfer in Czochralski silicon crystal growth under magnetic fields FK5—Thierry Alboussière; Jean-Paul Garandet; René J. Moreau ( U K ) : Asymptotic magnetohydrodynamic convection and symmetries FK6—René J. Moreau, Alban Potherat, and Karim Messadek; Joel Sommeria (France): Quasi-two-dimensional turbulent shear flows: Modelling versus experiment FL—Granular flows (Stéphane Roux, France, chair)

FL1—John M. N. T. Gray (UK): An exact solution for the mixing of a monodisperse granular

material in a rotating drum

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ICTAM 2000

FL2—Prabhu R. Nott (India): Kinematics of mixing and segregation of granular flows in rotating drums FL3—Kimberly M. Hill; Nitin Jain and Julio M. Ottino (USA): Axial segregation patterns in non-circular tumbling mixers FL4—Daniel M. Mueth, Georges F. Debregeas, Gregory S. Karczmar, Peter J. Eng, Heinrich M. Jaeger, and Sidney R. Nagel (USA): Non-invasive studies of shear bands in dense, three-dimensional granular flow FL5 —Daniel W. Howell and Robert P. Behringer; Christian Veje (USA): Stress fluctuations in slow shear flow FL6—Wolfgang Losert and Jerry P. Gollub (USA): Particle dynamics and shear forces in sheared, air-fluidized granular matter

FO—Computational strategies for multiscale phenomena in mechanics (jointly with IACM) (Erwin Stein, Germany, chair) FO1—J. Tinsley Oden, Gregory Rodin, Kumar Vemaganti, and Yuhong Fu (USA): Adaptive modeling of heterogeneous materials: Theory, fast summation methods and applications FO2—Jacob Fish and Qing Yu (USA): Multiscale models for problems in heterogeneous media FO3 —William A. Goddard, Tahir Cagin, Alejandro Strachan, and Richard P. Muller (USA): First principles multiscale approaches to prediction of mechanical properties FO4—Nicholas J. Pagano and Valeri A. Buryachenko (USA): Multiscale analysis and edge effects for multiply interacting inclusion problems FO5—Christian H. Huet (Switzerland): Integrated micromechanics strategies in random composites bodies with distributed granulometry FO6—Krishna Garikipati; Thomas J. R. Hughes (USA): Embedding micromechanical laws in continuum macromechanics: A variational multiscale approach

FR—Experimental methods in solid mechanics (Isaac M. Daniel, USA, chair) FRl—Michel M. Géradin, Georges Magonette, and Artur V. Pinto (Italy): Large dynamic and pseudo-dynamic testing at the European Laboratory for Structural Assessment FR2—Eiichi Watanabe, Tomoaki Utsunomiya, Kazutoshi Nagata, and Yousuke Tsumura; Yukihide Kajita (Japan): A shaking table test for pounding between girders in elevated bridges and the numerical simulation FR3—Eric Clément, Guillaume Reydellet, and Loic Vanel; Daniel W. Howell, Junfei Geng, and Robert P. Behringer (France): Response functions in granular pilings FR4—Lily D. Poulikakos and Mahir B. Sayir; Manfred N. Parti (Switzerland): In-line nondestructive evaluation of asphalt plug joints FR5—Junji Yoshida, Masato Abe, and Yozo Fujino (Japan): Constitutive law for high damping rubber material and its experimental verification

FS—Mechanics

of porous materials (Alberto Carpinteri, Italy, chair)

FSl—Zhuping Huang; Yi Liu (China, PRC): The macroscopic strain potentials in nonlinear porous materials FS2—Alain Molinari and Sebastien Mercier (France): Micromechanics of porous materials at high strain rates FS3—Pia Redanz; Norman A. Fleck (Denmark): On affine motion of metal powder particles undergoing compaction

FS4—Sankara J. Subramanian and Petros Sofronis (USA): Micromechanical modeling of powder compaction FS5—Ragnar Larsson and Jonas Larsson (Sweden): Nonlinear analysis of porous medium with compressible fluid FS6—Stefan Diebels (Germany): Compressible porous media and the evolution of the volume fractions

FV—Combustion

and flames (Amable Liñán, Spain, chair)

FV1—Francisco J. Higuera (Spain): Thermocapillary flow due to a flame spreading over a liquid fuel FV2—Portonovo S. Ayyaswamy, Ira M. Cohen, and Srinivas S. Sripada (USA): Electric field induced convection effects on flames FV3—John Brindley, John F. Griffiths, and Andy C. Mclntosh (UK): Initiation of combustion by spherical hotspots in reactive solids FV4—Thomas L. Jackson; John D. Buckmaster and Abdelkarim Hegab; Gregory M. Knott ( U S A ) : Random packing of heterogeneous propellants and the flames they support

FV5—Bernard J. Matkowsky; Anatoly P. Aldushin (USA): Steady state combustion wave characteristics

FV6—John D. Buckmaster (USA): The dynamics of edge-flames

Scientific program

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GA—Mechanics of foams and cellular materials (Stelios Kyriakides, USA, chair) GA0l—William E. Warren; Esben Byskov (USA): Three-fold symmetry restrictions on twodimensional micropolar materials GA02—Xiaolin Wang; William J. Stronge (Singapore): Micropolar theory for elastodynamics

of honeycomb GA03—Jörg Hohe and Wilfried Becker (Germany): A refined analytical model for the effective

elasticity tensor of general cellular sandwich cores GA04—Changqing Chen, Tian Jian Lu, and Norman A. Fleck ( U K ) : Stiffness and strength of two dimensional cellular solids containing holes and inclusions GA05—Ashraf Bastawros (USA): Notch effects and associated length scales in the mechanics of cellular metals GA06—Karam Sab; Myriam Laroussi and Amina Alaoui (France): Scale effects in high strain compression of periodic open cell foams GA07—Yu Wang and Alberto M. Cuitiño; Gustavo Gioia (USA): Inhomogeneous deformation in compressed open-cell solid foams GA08—Xavier Chateau; Luc Dormieux (France): The behavior of unsaturated porous media:

A micromechanical approach GA09—Chris G. Poulton, Ross C. McPhedran, and Nicolae A. Nicorovici; Alexander B. Movchan; Yuri A. Antipov (Australia): Spectral problems and phononic band gaps for periodic structures GA10—Rafael J. Mora and Anthony M. Waas; Jacob Pawlicki (USA): Strength scaling of brittle graphitic foams GA11—Tian Jian Lu and Changqing Chen; Guangrui Zhu ( U K ) : The crushing of corrugated

board panels GA12—Martin Schanz; Alex H.-D. Cheng (Germany): Compressional waves in a onedimensional poroelastic column GA13—Hanxing Zhu and Alan H. Windle ( U K ) : The high strain compression of irregular open-cell foams

GB—Boundary layers and waves (Roger H. J. Grimshaw, UK, chair) GB01—Georgiy Korolev ( U K ) : Flow separation and non-uniqueness of the solution of the boundary-layer equations

GB03—David H. Wood and Yuhua Guo (Australia): Measurements in the vicinity of a stagnation point GB04—Ronald Abstiens, Stefan Fühling, and Wolfgang Schröder (Germany): Boundarylayer measurements on the ELAC configuration at Re = 20 × 106 GB05—Rudolph A. King; Kenneth S. Breuer (USA): Oblique transition in a Blasius boundary layer GB06—Xuesong Wu ( U K ) : Receptivity of boundary layers with distributed roughness to vortical disturbances GB07—Christian A. Maresca, Eric C. Berton, Daniel P. Favier, and Christine M. Allain (France): Investigation of boundary layers of moving walls by use of embedded laser-Doppler velocimetry methods GB08—Stefan Herr, Werner Wuerz, Anke Woerner, Ulrich Rist, and Siegfried Wagner; Yury S. Kachanov (Germany): Experimental and numerical investigation of 3D wall roughness acoustic receptivity on an airfoil

GB09—Alexander D. Kosinov (Russia): Flow characteristics measurements in attachment line

boundary layer on a swept cylinder at Mach 2 GB10—Alric P. Rothmayer (USA): A modified Kuramoto-Sivashinsky equation for air driven films GB11—Tim David and Peter G. Walker ( U K ) : Platelet activation and adhesion in general 2D flows GB12—Oleg G. Derzho and Roger H. J. Grimshaw (Australia): Axisymmetric waves and eddies in swirling flows in a tube GB13—Pearu Peterson; Brenny (E.) van Groesen (Estonia): The direct and inverse problem of wave crests

GB14—Janet M. Becker and David Bercovici (USA): Pattern formation on the interface of a two-layer fluid with a bi-viscous lower layer

GC—Geomechanics and geophysical fluid dynamics (Julian C. R. Hunt, UK, chair) GC01—María-Arántzazu Alarcón, John W. Rudnicki, Richard J. Finno, and Amy Rechenmacher (USA): Constitutive modeling of direct measures of strain in simulated fault gouge GC02—Michio Kurashige, I. Taguchi, and Kazuwo Imai (Japan): Integral equations for an arbitrarily-loaded crack in oil reservoirs of poroelastic solid

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ICTAM 2000

GC03—Jean-François Thovert and Farchid Yousefian; Pierre Adler and Samir Bekri (France): Macroscopic properties of heterogeneous porous media GC04—Krzysztof Wilmanski (Germany): Propagation of weak and strong discontinuities in poroelastic materials GC05—Marie-Angele Abellan and Jean-Michel Bergheau; Paul Jouanna (France): Impact of initial conditions on environmental previsions: A case study in unsaturated soils GC06—Xiang Song Li; Yannis F. Dafalias (Hong Kong): Unified modeling of stress-strain behavior of loose and dense sand GC07—Thomas Zwinger and Alfred Kluwick; Peter Sampl (Austria): Dynamics of dry snow avalanches on arbitrarily shaped slopes GC08—Fridtjov Irgens (Norway): Simplified simulation model of snow avalanches and landslides GC09—Chiu-On Ng (Hong Kong): Mass transport following release of contaminated particles into a steady channel flow GC10—Michihisa Tsutahara, Takeshi Kataoka, and Masaya Tanaka (Japan): Point sink flow in a rotating stratified fluid GC11—Takeshi Miyazaki and Tomoyuki Shimonishi (Japan): Stability of a quasigeostrophic ellipsoidal vortex GC12—Rajesh Bhaskaran and Sidney Leibovich (USA): Eulerian and Lagrangian Langmuir circulation patterns GC13—Hideshi Hanazaki; Julian C. R. Hunt (Japan): Linear processes in unsteady stably stratified turbulence with mean shear GC14—Patrick F. Hodnett and Raymond McNamara (Ireland): Zonal influences in a modified Stommel–Arons model of the abyssal ocean circulation

GD—Fracture and crack mechanics (jointly with ICF) (L. Ben Freund, USA, chair) GD01—Mark E. Mear (USA): A weakly-singular, weak-form traction integral equation for analysis of cracks in a half-space GD02—Christian F. Niordson (Denmark): Analysis of steady-state ductile crack growth along a laser weld GD03—Mikhail Perelmuter (Russia): Bridge model for an interface crack GD04—Chunyu Li; George J. Weng; Zhuping Duan (China, PRC): A cylindrical crack in a functionally graded material interlayer under torsional impact GD05—Nadim I. Shbeeb and Wieslaw K. Binienda (USA): Analysis of the driving forces for multiple cracks in an infinite non-homogeneous plate GD06—Masaaki Watanabe (Japan): The angled crack problem revisited GD07—Igor A. Guz and Alexander N. Guz’ (Ukraine): Stability of two different half-planes in compression along interfacial cracks: Analytical solutions GD08—Chien H. Wu and Ming L. Wang (USA): The effect of crack-tip point loads on fracture GD09—Kavi Bhalla and Alan T. Zehnder; Robert Thomas and Lawrence Favro (USA):

Thermal analysis of crack tearing GD10—Adélaïde Feraille and Alain Ehrlacher (France): Behavior of a wet crack submitted to heating up to high temperature GD11—Nicolas Moes and Ted Belytschko (USA): An extended finite element method to model crack growth without remeshing GD12—Alina A. Zalounina and Jens H. Andreasen (Denmark): An edge crack in a coated solid subjected to contact loading GD13—Jens H. Andreasen (Denmark): Straight sided buckling of a coating on a stiff substrate GD14—Liviu Marsavina; Matthew J. Ekman and Andrew D. Nurse (Romania): On the near-tip stress intensity factor mode mixed at bimaterial interface cracks under small-scale

yielding GD15—Vera E. Petrova; Klaus Herrmann (Russia): Thermal cracking of a bimaterial containing an interface crack and microcracks GE—Structural optimization (jointly with ISSMO) (Niels Olhoff, Denmark, chair) GE01—Valeri L. Markine, Amy de Man, and Coenraad Esveld (The Netherlands): Optimal design of ballastless railway track structure GE02—Sergio R. Turteltaub (USA): Optimal distribution of material properties for transient phenomena

GE03—Seil Song, Jong S. Im, and Yung M. Yoo; Jung K. Shin, Kwon H. Lee, and Gyung J. Park (Korea): Automotive door design with the ultra light steel auto body concept using structural optimization GE04—Julian A. Norato, Tyler E. Bruns, and Daniel A. Tortorelli; Kevin J. Hinders; I. Dennis Parsons (USA): Computational tools for architectural design GE05—Alejandro R. Diaz and Brian Feeny (USA): Optimization of two-phase material distributions generated by iterated affine maps

Scientific program

xlix

GE06—Ani P. Velo; Robert P. Lipton (USA): Optimal design of gradient fields with application to problems of heat conduction GE07—Yuanxian Gu, Biaosong Chen, and Hongwu Zhang (China, PRC): Thermalstructural optimization and sensitivity analysis with precise time integration GE08—Ciro A. Soto; John E. Taylor and Jianbin Du; Helder C. Rodrigues (USA): Treatments for simultaneous structural topology and material properties design optimization GE09—Yoon Y. Kim and Gil H. Yoon (Korea): Multi-resolution multi-scale topology optimization in general domains GE10—Thomas Borrvall and Joakim Petersson (Sweden): Topology optimization using regularized intermediate density control GE11—Sándor Kaliszky and János Lógó (Hungary): Layout optimization of elastoplastic structures with plastic deformation and displacement constraints GE12—Niels L. Pedersen (Denmark): On topology optimization of plates with prestress GE13—Thomas Buhl (Denmark): Topology optimization with cost of supports GE14—Chien Jong Shih and Ting Jhang Tseng (Taiwan): Reliability based optimum structural design for robust performance using fuzzy formulation strategy GE15—Gih Keong Lau, Hejun Du, and Mong King Lim (Singapore): Topology optimization of compliant mechanism subjected to harmonic loads GF—Structural vibrations (Irina Goryacheva, Russia, chair) GF01—Shanshin Chen and Daniel A. Tortorelli (USA): Domain decomposition of energy conserving flexible multibody dynamic systems GF02—Peter H. Ruge and Nils Wagner (Germany): Continuous versus discontinuous timefinite-elements for vibration systems with fading memory GF03—Ye Ping Xiong; Jing Tang Xing and W. Geraint Price (China, PRC): A generalized mobility progressive method of power flow analysis for complex coupled dynamic systems GF04—Manfred W. Zehn and Alexander Saitov (Germany): Improvement of finite element models for the dynamics of active composite materials GF05—Carlos E. N. Mazzilli, Mário E. S. Soares, and Odulpho G. P. Baracho Neto (Brazil): Perturbation techniques applied to the evaluation of nonlinear modes GF06—Matti Martikainen, Erno K. Keskinen, and Veli-Matti Jarvenpää; Michel Cotsaftis (Finland): Transient oscillation dynamics of elastic structure due to rolling contact source GF07—Robert G. Parker and Yehong Lin (USA): Parametric instability of axially moving media under multi-frequency tension and speed fluctuations GF08—Vladimir I. Babitsky and Alexander M. Veprik (UK): Synchronization of vibration by collisions in structures and periodic Green’s functions GF09—Igor E. Poloskov (Russia): Analysis of random regimes and computer algebra GF10—Thomas P. Mitchell (USA): The response of an axially translating one-dimensional continuum to stochastic excitation GF12—M. Alaeddin Arpaci and S. Ergun Bozdag (Turkey): Triply coupled vibrations of thin-walled open cross-section beams including rotary inertia effect GF13—Noël Challamel (France): Some limit cycles in drilling mechanics GF14—Xia Lu and Sathya V. Hanagud (USA): Irreversible thermodynamic models for damping GG—Multibody dynamics (Timothy J. Gordon, UK, chair)

GG01—Michel Fayet (France): Dynamics of semi-free systems GG02—Bodo Heimann and Martin Grotjahn (Germany): Newton–Euler equations for multibody systems in minimal dimensional parameter-linear form GG03—Martin Arnold (Germany): Coupled dynamical simulation of coupled mechanical systems GG04—Jens Wittenburg; Ljubomir Lilov (Germany): Decomposition of a screw displacement into three consecutive screw displacements with fixed axes GG05—Brian Moore; Jean-Claude Piedboeuf; Pierre Sicard (Canada): Dynamics identification of mechanisms using a symbolic approach GG06—Hiroaki Yoshimura and Takehiko Kawase (Japan): A method of tearing and interconnecting for multibody dynamics GG07—Yu-Hung Hsu and Kurt S. Anderson (USA): Analytical full-recursive sensitivity analysis for multibody system designs GG08—Viktor Berbyuk; Anders Boström (Ukraine): Optimization problems of controlled multibody systems having spring-damper actuators GG09—Andreas Voile (Germany): The influence of wobbling masses on biomechanical systems GG10—Anatoly A. Khentov (Russia): The principle of strong interaction and the orbital motion of some Saturn’s and Jupiter’s satellites

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ICTAM 2000

GG11—Vigen Arakelian and Marc Dahan; Mike Smith (France): Shaking force and shaking moment balancing of slider-crank mechanisms GG12—Johannes Gerstmayr and Hans Irschik (Austria):

Computational strategies for

vibrating elastoplastic structures with rigid-body degrees of freedom

GH—Computational solid mechanics (jointly with IACM) (J. Tinsley Oden, USA, chair) GH01—Muneo Hori and Tsuyoshi Ichimura (Japan): Macro–micro analysis for computation of earthquake wave propagation in metropolis

GH02—Grzegorz Zboinski and Wieslaw Ostachowicz (Poland): A unified approach to adaptive modeling and analysis of complex structures GH03—Basudeb Bhattacharyya; Subrata Chakraborty (India): Probabilistic sensitivity analysis of structures involving parametric uncertainty GH04—Per Kettil and Nils-Erik Wiberg (Sweden): Adaptive solid modeling and analysis applied to design of bridges GH05—Stanislaw A. Lukasiewicz; Marcin Kaja and Marek Stanuszek (Canada): Filtering and enhancement of the numerical and experimental data at arbitrarily located points GH06—Joaquim B. Cavalcante Neto; Luiz F. Martha (Brazil): A prototype for a system for adaptive analysis in three dimensions GH07—Michael L. Dambach; Alexander Tessler; Donald W. Oplinger (USA): Adaptive mesh refinement strategies for modeling adhesively bonded joints GH09—Gakuji Nagai and Akira Wada; Takahiro Yamada (Japan): 3-D mesoscopic failure simulation of concrete materials by using image-based finite element method GH10—Martin Ammann and Wolfgang Ehlers (Germany): H-adaptive finite element strategies applied to multi-phase problems GH11—Bruno Cochelin; Noureddine Damil; Michel Potier-Ferry (France): Advances in the asymptotic numerical method

GH12—Gang Li and Narayana R. Aluru (USA): Finite cloud meshless method for geometrically nonlinear analysis of micro-electro-mechanical systems GH13—Kent T. Danielson, Shaofan Li, and Wing Kam Liu; R. Aziz Uras (USA): Parallel computational methods for meshfree analysis GH14—Noriko Katsube and Dan Zeng; Wole Soboyejo (USA): Hybrid finite element method for heterogeneous materials and cracks GH15—Zhenhan Yao, Yongqiang Chen, and Xiaoping Zheng (China, PRC): Numerical simulation of failure processes in 3-D heterogeneous brittle material GK—Mixing and multiphase flow (Grétar Tryggvason, USA, chair)

GK01—Fotis Sotiropoulos, Tahirih C. Lackey, and Donald R. Webster (USA): Chaotic and quasi-periodic dynamics in 3D steady vortex breakdown bubbles GK02—Clotilde Chagny-Regardin, Cathy Castelain, and Hassan Peerhossaini (France): The limits of chaotic advection flow regime on convective heat transfer GK03—Patrick D. Anderson, Peter G. M. Kruijt, Oleksey S. Galaktionov, Gerrit W. M. Peters, and Han E. H. Meijer (The Netherlands): A mapping approach for 3D distributive mixing flows

GK04—Gregory P. King; Murray Rudman; Athanasios N. Yannacopoulos; George Rowlands; Igor Mezic (UK): Can effective diffusion coefficients be extracted from symmetry measures? GK05—Keisuke Araki; Katsuhiro Suzuki (Japan): A Riemannian geometrical analysis of relative diffusion of Lagrangian particle pair in turbulence GK06—Andrew W. Cook; Paul E. Dimotakis (USA): Investigation of mixing effects in incompressible Rayleigh–Taylor instability GK07—Xiaolin Li, James Glimm, Andrea Marchese, and Zhiliang Xu ( U S A ) : Computation of 3-d Rayleigh–Taylor mixing rate with randomly perturbed interface

GK08—Yuan-nan Young, Anshu Dubey, and Robert Rosner; Henry M. Tufo (USA): Miscible Rayleigh–Taylor instability: 2D and 3D GK09—Jerzy Baldyga and Wioletta Podgorska (Poland):

Application of the multifractal

model of turbulence to chemical engineering mixing problems GK10—Sami A. Bayyuk, De-Ming Wang, and Samuel A. Lowry (USA): A self-consistent higher-order accurate scheme for the volume-of-fluid methodology for multi-phase flows GK11—Christopher M. Varga and Juan C. Lasheras; Emil J. Hopfinger (USA): On the steady and unsteady breakup of a liquid jet by a coaxial gas stream GK12—Frédéric Risso and Kjetil Ellingsen (France): Dynamics of the path oscillations of a rising ellipsoidal bubble

GK13—Nelya K. Vakhitova, Robert I. Nigmatulin, Iskander Sh. Akhatov, Raisa Kh. Bolotnova, Andrei S. Topolnikov, and Elvira Sh. Nasibullayeva (Russia): Resonant supercompression of gas bubbles (sonoluminescence, sonochemistry and bubble fusion)

Scientific program

li

GL—Fluid-structure interaction (Earl H. Dowell, USA, chair) GL01—Hong-Hui Shi and Motoyuki Itoh; Takuya Takami (Japan): The motion of an underwater blunt body after water entry GL02—Jean-Francis Ravoux; Ali Nadim; Hossein Haj-Hariri (USA): An embedding method for bluff body flows GL04—Alexander I. Zubkov and Anatoly I. Glagolev (Russia): Reduction of aerodynamic drag of bodies of revolution using heat and mass injection in near wake GL05—Morten H. Hansen and Joergen T. Petersen (Denmark): Flutter phenomena in wind turbines GL06—Gregory Hagen and Igor Mezic (USA): Fluid modeling and control of axial compressors GL07—Emmanuel de Langre (France): Absolutely unstable waves in hydroelastic systems GL08—Henrik Møller and Erik Lund (Denmark): Shape sensitivity analysis of strongly coupled fluid–structure interaction problems GL09—Lawrence N. Virgin, Steve T. Trickey, and Earl H. Dowell (USA): Sudden jumps in the oscillatory behavior of a nonlinear aeroelastic system GL10—Alexander Galper and Touvia Miloh (Israel): Rotational flow hydrodynamics GL13—Viorel I. Anghel (Romania): Resonances analysis of hingeless rotor helicopter using the equivalent spring restrained blade model

GM—Plasticity, viscoplasticity, and dynamic plasticity of structures (Stephen R. Reid, UK, chair) GM02—Elhem Ghorbel and Didier Baptiste (France): A generalized yield criterion for unoriented polymers GM03—Douglas J. Bammann (USA): Incorporation of a natural length scale into a crystal plasticity model GM04—Hyszard B. Pecherski; Katarzyna Korbel; Andrzej Korbel (Poland): Finite plastic deformation due to the sequence of slips: Experimental evidence and theoretical model GM05—Franck Ferrer, Thierry Bretheau, and Jérorne Crépin; Alain Barbu (France): Plasticity of zirconium at intermediate temperatures: Effect of a small quantity of sulfur GM06—Wenhui Zhu, Shinji Tanimura, Koji Mimura, and Tsutomu Umeda (Japan): Dynamic response of new sensing block systems and its application to dynamic buckling tests GM07—Michael L. Falk; James S. Langer (USA): A microscopically motivated theory of dynamic plastic deformation in noncrystalline solids GM08—Jilin Yu; Marcilio Alves (China, PRC): On the choice of material model and its parameters in structural impact GM09—Miroslav D. Nestorovic and Nicolas Triantafyllidis; Yves M. Leroy (USA): On the stability of rate-dependent solids with application to the uniaxial plane strain test GM10—Mark R. Liberzon (Russia): Impulse treatment by high-energetic electromagnetic field on materials GM11—K. Lawrence DeVries and Paul R. Borgmeier (USA): Stress distribution in and

failure of standard adhesive joints GM12—Jesus Chapa-Cabrera and Ivar E. Reimanis; Eric D. Steffler (USA): Fracture behavior for cracks perpendicular to the gradient in functionally graded materials GM13—Mark E. Walter (USA): Analysis of the deformation of an aluminide coating system under thermal loading

GN—Continuum mechanics and nonlinear dynamics (Gerard A. Maugin, France, chair) GN01—Peter A. Dashner (USA): Elastic shadow flow and its theoretical implications for inelastic solids GN02—Alain Merlen and Saloua Ben Khélil; Vladimir Preobazhenski; Philippe Pernod (France): Numerical simulation of acoustic waves phase conjugation in active media GN03—Judith E. Skolnik and Mark H. Wähling (Germany): Thermo-mechanical processes in saturated porous solids GN04—Miles B. Rubin; Oleg Yu. Vorobiev, Ilya Lomov, Lewis A. Glenn, and Tarabay Antoun; Alexander Chudnovsky (Israel): Fracture surface energy in a thermomechanical model for bulking of porous viscoplastic material GN05—Valery I. Levitas (USA): Unified approach to phase transitions, fracture and chemical reactions in inelastic materials GN06—Dimitris C. Lagoudas and David H. Allen; Chongho Yoon (USA): A model for predicting fracture toughness and damage evolution in viscoelastic solids GN07—Klaus Hackl (Germany): On induced microstructures occurring in models of finitestrain elastoplasticity

GN08—Kam Tim Chau and Xiao Yang (Hong Kong): Resistance of nonlinear soil to a horizontally vibrating pile

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ICTAM 2000

GN09—Pawel Dluzewski and Grzegorz Maciejewski; Slawomir Kret (Poland): Finite element simulation of residual stress in semiconductor epilayers GN10—Markus Zahn (USA): Nonlinear ferrofluid duct flow profiles in alternating and travelling wave magnetic fields GN11—Philip J. Morrison (USA): An integral transform for the continuous spectrum in fluid and plasma dynamics GN12—Jacob P. Meijaard (The Netherlands): Non-classical bifurcations and chaos in a positioning mechanism GN13—S. John Hogan, Mario Di Bernardo, and Martin E. Homer; Mark I. Feigin (UK): Local analysis of c-bifurcations in n-dimensional piecewise-smooth dynamical systems GN14—Haider N. Arafat and Ali H. Nayfeh (USA): Energy transfer between widely-spaced in-plane bending modes of cantilever beams GR—Elasticity (Michael A. Hayes, Ireland, chair) GR01—Xanthippi Markenscoff (USA): An inverse problem in elasticity: The Eshelby inclusion GR02—George V. Jaiani (Georgia): On a mathematical model of bars with variable rectangular cross-sections GR03—George Mejak (Slovenia): Shape optimization of composite bars in torsion GR05—David P. Mason and Nicolette Roussos (South Africa): Nonlinear superposition principle for radial oscillations of a thin Mooney–Rivlin cylindrical tube GR06—Yi-chao Chen (USA): Dynamic solutions of thermoelastic rods under impact loads GR07—Anthony J. Paris and Sarah E. Zeller (USA): On the mechanics of cord-composite materials GR08—Jian Cao and Nan Song (USA): Effect of tooling design on the variation of final product geometry GR09—Allan X. Zhong (USA): Effects of incompressibility in numerical submodeling of hyperelastic materials GR10—Bing-Zheng Gai and Qing-Cai Chen (China, PRC): BEM analysis of interaction of elastic waves with unilateral interface crack GR11—Youn-Sha Chan; Glaucio H. Paulino; Albert D. Fannjiang (USA): Gradient elasticity theory for a mode I crack in homogeneous and nonhomogeneous materials GR12—Natasha V. Movchan (UK): Effect of interfacial bonding on stress singularity at the vertex of a conical inclusion GR13—Jurgen Jäger (Germany): A generalized Coulomb law for elastic bodies

GS—Plates, shells, and stability of structures (Arthur w. Leissa, USA, chair) GS01—Eric M Mockensturm (USA): On the finite twisting of thick elastic plates GS02—Yu-Hsuan Su and S. Mark Spearing (USA): Large deflection analysis of an annular plate with a rigid boss under axisymmetric loading GS03—Sachiko Nakagoshi and Hirohisa Noguchi (Japan): A new wavelet Galerkin method for analysis of Mindlin plates GS04—Manfred Bischoff; Kai-Uwe Bletzinger (USA): Stabilized discrete shear gap plate and shell elements GS05—Natacha M. Buannic and Patrice M. Cartraud (France): Higher-order asymptotic modeling for heterogeneous periodic plates GS06—Eliza M. Haseganu and Ying Liu (Canada): Equilibrium analysis of cylindrical elastic membranes subjected to conservative pressure loading GS07—Dobroslav D. Ruzic and Ljubisa S. Markovic (Yugoslavia): New formula for the critical external lateral pressure around the cylindrical shell GS08—Guo-Hua Nie (China, PRC): An exact analysis of buckling of imperfect shallow spherical shells on an elastic foundation GS09—Tyler E. Bruns and Daniel A. Tortorelli (USA): Topology optimization of nonlinear elastic structures and compliant mechanisms GS10—Bogdan Bochenek and Jacek Kruzelecki (Poland): Selected problems of optimization for postbuckling behavior GS11—Makoto Ohsaki (Japan): Sensitivity analysis of critical load factor of symmetric systems for minor imperfection GS12—Donald W. Raboud, A. William Lipsett, and M. Gary Faulkner (Canada): An evaluation of the stability of flexible end loaded cantilever beams GS13—Sylvain Drapier, Lionel Léotoing, and Alain Vautrin (France): Influence of scale effects on the buckling of sandwich structures GS14—Rüdiger Schmidt; Liviu Librescu (Germany): Geometrically nonlinear theory of laminated shells weakened by interlaminar bonding imperfections

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GV—Low-Reynolds-number flow, microfluid mechanics, and fluid mechanics of materials processing (Patrick Huerre, France, chair) GV01—Evgeny S. Asmolov (Russia): Unsteady flow past a sphere undergoing unidirectional motion at small Reynolds number GV02—Avinoam Nir and Yaron Rosenstein (Israel): Hindered rotation in a dilute suspension GV03—Alexander Prokunin (Russia): Towards a paradox in the rough-particle motion along a wall in a liquid

GV04—Nadav Liron and Efrath Barta (Israel): Motion of a particle towards and within an orifice of finite length at zero Reynolds number GV05—Xueyan Guo and Antonio Delgado (Germany): Numerical simulation of pure elongational flow around particle-aggregates GV06—Asimina Sierou and John F. Brady (USA): Accelerated Stokesian dynamics simulations of concentrated suspensions GV07—Slavtcho G. Slavtchev and Svetla P. Miladinova; Georgy Lebon; Jean-Claude Legros (Bulgaria): Long wave instabilities of non-uniformly heated falling films GV08—Sahraoui Chaïeb; Gareth H. McKinley (USA): Mixing immiscible fluids: Drainage induced cusp formation GV09—Antonio Castellanos, Heliodoro Gonzalez, Antonio Ramos, and Francisco J. Garcia (Spain): Dynamics of liquid columns subjected to magnetic or electric fields GV10—Andrey V. Kuznetsov (USA): Investigation of diffusion effects in free surface flows

with solidification GV11—Emily S. Nelson; Phillip Colella (USA): Infiltration dynamics of reactive melt infiltration GV12—John B. Szczech and Constantine M. Megaridis; Dan Gamota and Jie Zhang (USA): Manufacture of microelectronic circuitry by drop-on-demand dispensing of nanoparticle suspensions GV13—Carl D. Meinhart and Shannon W. Stone; Steve Wereley (USA): A microfluidicbased nanoscope HA—Convective phenomena (Detlef Lohse, The Netherlands, chair) HA2—Alexander Yu. Gelfgat, Pinhas Z. Bar-Yoseph, and Alexander Solan; Eliezer Kit (Israel): Axisymmetry breaking of natural convection in a cylinder with a parabolic sidewall

temperature HA3—Blas Echebarria and Hermann Riecke (USA): Defect chaos in non-Boussinesq rotating convection HA4—Friedrich H. Busse, Oliver Brausch, Markus Jaletzy, and Werner Pesch (Germany): Hexarolls, knot convection and phase turbulence in the rotating cylindrical annulus HA5—Karen E. Daniels, Brendan B. Plapp, and Eberhard Bodenschatz (USA): Inclined layer convection HA6—Rafael Delgado-Buscalioni and Emilia Crespo del Arco (Spain): Dynamical coupling between shear rolls and a longitudinal thermal wave in a 3D inclined cavity HB—Flow control (Peter W. Carpenter, U K , chair) HB1—Jeffrey D. Crouch, Gregory D. Miller, and Philippe R. Spalart (USA): Active control for breakup of airplane trailing vortices HB2—Markus Schatz, Holger Lübcke, and Ulf Bunge (Germany): Flow simulation around movable flaps under high-lift conditions HB3—Nadine Aubry and Zhihua Chen (USA): Electro-magnetic feedback control of wake flows HB4—Dietmar Rempfer and John L. Lumley (USA): Low-dimensional models for turbulent shear-flow control HB5—Owen R. Tutty; Lubomir Baramov and Eric Rogers ( U K ) : Low dimensional robust control of channel flow HB6—Rama Govindarajan and Balaji T. Ranganathan (India): Control of flow stability in a channel by location of viscosity stratified fluid layer HC—Flow instability and transition (Sidney Leibovich, USA, chair)

HC1—Paul Manneville (France): Modeling of transitional plane couette flow HC2—Bruno Eckhardt, Armin Schmiegel, and Holger Faisst (Germany): Transition to turbulence in plane Couette flow HC3—Alp Akonur and Richard M. Lueptow (USA): Two-plane velocity field in wavy Taylor– Couette flow HC4—Gerd Pflster, Jan Abshagen, and Arne Schulz (Germany): Higher instabilities in Taylor–Couette flow HC5—Hendrik C. Kuhlmann, Stefan Albensoeder, and Hans J. Rath (Germany): 3D-flow instability in generalized lid-driven cavities

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ICTAM 2000

HC6—Alexander Solan, Alexander Yu. Gelfgat, and Pinhas Z. Bar-Yoseph (Israel): Threedimensional instability of flow in a rotating lid-cylinder enclosure HD—Elasticity (Alan S. Wineman, USA, chair) HD1—Christian Licht; Oana Iosifescu and Gérard Michaille (France): Variational limit of a one-dimensional discrete system of material points HD2—Lev Truskinovsky (USA): Is a chain with bi-stable springs elastic? HD3—David M. Haughton (UK): Nonlinear elastic materials which behave differently in tension and compression HD4—Roderic Lakes (USA): Deformation of elastic and viscoelastic chiral materials HD5—Raymond W. Ogden (UK): A pseudo-elastic model for damage in elastic solids under large strain HD6—Millard F. Beatty (USA): The Mullins effect in combined triaxial stretch and simple shear of a block HE—Structural Optimization (jointly with ISSMO) (Sándor Kaliszky, Hungary, chair) HE1—George I. N. Rozvany (Hungary): Latest developments in the exact layout theory of grid-like continua HE2—Michael Ryvkin, Moshe B. Fuchs, and Eyal Moses (Israel): Topologtcal optimization of structures with 2D translational symmetry HE3—Richard J. Balling (USA): A blueprint for simultaneous analysis and optimization of structural topology via multigrid HE4—Tadeusz S. Burczynski, Witold Beluch, Marek Nowakowski, and Piotr Orantek (Poland): Hybrid evolutionary methods in structural optimization HE5—Martin P. Bendsøe; Ole Sigmund (Denmark): Grey-scales in topology optimization: Hashin–Shtrikman bounds and realization by composites HE6—Bing Chung Chen and Noboru Kikuchi (USA): Topology optimization of structures and compliant mechanism with design dependent loads HF—Fatigue (Bruno A. Boley, USA, chair) HF1—Masahiro Jono and Atsushi Sugeta (Japan): Atomic force microscope observation and mechanism of fatigue crack growth HF2—Tung Hua Lin; Naigang G. Liang; Kevin F. F. Wong; Ning J. Teng (USA): Micromechanic analysis of crack initiation and hysteresis loops of aluminum single crystals HF3—Yoshikazu Nakai (Japan): Micro-mechanisms of slip-band growth and crack initiation in fatigue of 70-30 brass HF4—Jian-Ku Shang (USA): Micromechanisms of interfacial fatigue in solder interconnects HF5—Keisuke Tanaka and Yoshiaki Akiniwa (Japan): Microstructural effect on crack closure and propagation thresholds of small fatigue cracks HF6—Robert O. Ritchie and Da Chen (USA): Fatigue of ceramics at elevated temperatures HG—Computational Solid mechanics (jointly with IACM) (Pierre Ladevèze, France, chair)

HG1—Walter Wunderlich and Rudolf Findeiss; Harald Cramer (Germany): Space and time adaptive computation of localized failure in geomechanics HG2—Scott G. Bardenhagen; Jeremiah U. Brackbill; Deborah L. Sulsky (USA): A numerical study of sheared granular material HG3—Osamu Kuwazuru; Nobuhiro Yoshikawa (Japan): Nonlinear finite element for plain woven fabrics HG4—Sohichi Hirose; Can-Yun Wang; Jan D. Achenbach (Japan): Boundary element method for elastic wave scattering by a crack in an anisotropic solid HG5—Tod A. Laursen (USA): New conservative finite element algorithms for dynamic inelastic impact HG6—Erik Lund; Jorn S. Hansen (Denmark): Two-dimensional shape sensitivity analysis: A fixed basis function finite element approach

HH—Contact and friction problems (jointly with IAVSD) (James R. Barber, USA, chair) HH1—Nadia Lapusta; James R. Rice (USA): Instability of dynamic frictional sliding HH2—Michel Raous and Iulian Rosu; João A. C. Martins (France): Instabilities and nonsmooth solutions related to frictional contact HH3—Bernardino M. Chiaia, Mauro Borri-Brunetto, and Alberto Carpinteri (Italy): Contact force transmission at rough interfaces: Numerical and experimental investigation HH4—Pakalapati T. Rajeev and Thomas N. Farris; Irina G. Goryacheva (USA): Wear in partial slip contact HH5—George G. Adams and Mikhail Nosonovsky (USA): Elastic waves induced by the frictional sliding of two elastic half-spaces HH6—Marius Cocu, Elaine Pratt, and Jean-Marc Ricaud (France): Analysis of a class of dynamic viscoelastic contact problems with friction

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HK—Waves (Chiang C. Mei, USA, chair) HK1—Roger H. J. Grimshaw (UK): Models for instability in inviscid fluid flows HK2—Lev Shemer, Haiying Jiao, and Eliezer Kit (Israel): Nonlinear wave groups: Experi-

ments versus simulations based on the Zakharov and Dysthe models HK3—Brenny (E.) van Groesen and Jaap-Harm Westhuis; Renee H. M. Huijsmans (The Netherlands): Unstable evolution of bichromatic wave trains HK5—Alfred Kluwick; Edward A. Cox; Stefan Scheichl (Austria): Negative and nonclassical hydraulic jumps in two layer fluid flow HK6—Gregory P. Chini; Sidney Leibovich (USA): Nonlinear Langmuir circulation–internal wave interactions HL—Granular flows (James T. Jenkins, USA, chair) HL1—Stefan Luding, Mark Laetzel, and Hans J. Herrmann (Germany): Macroscopic material properties from microscopic simulations of a 2D shear-cell HL2—Olivier Pouliquen and Yoel Forterre (France): Friction law for dense granular flows down rough planes HL3—Stephen C. Keast and Michel Y. Louge (USA): On dense granular flows down flat frictional inclines HL4—E. Bruce Pitman (USA): Modeling and simulation of debris flows

masses across 3-D terrain HL6—Howard A. Stone and Johann M. Schleier-Smith (USA): Puddling, heaping, and cracking in vertically vibrated liquid-saturated granular material HO—Dynamic plasticity Of Structures (Victor P. W. Shim, Singapore, chair)

HO1—Li-Lih Wang (China, PRC): Dynamic unloading response of structures at high strain rates HO2—Michael J. Forrestal; Andrew J. Piekutowski (USA): Penetration of aluminum targets by ogive-nose steel projectiles at striking velocities to 3.0 km/s HO3—Stephen R. Reid; Jia Ling Yang; Tongxi Yu (UK): Large deflection elastic-plastic, hardening-softening model for pipe whip HO4—William J. Stronge (UK): Generalized impulse and generalized momentum applied to

multibody impact with an energetic coefficient of restitution HO5—Victor P. W. Shim and Liming Yang (Singapore): Relationship between microstructure and the dynamic plastic response of crushable foam under impact HO6—Fan Song, Xiaolei Wu, and Yilong Bai (China, PRC): Some distinctive microstructures and mechanical behaviors of abalone nacre HR—Control of Structures (Alexander F. Vakakis, USA, chair) HR1—Vekatesh Deshmukh and S. C. Sinha (USA): Control of a class of nonlinear systems with time periodic coefficients HR2—Gabor Stepan and Laszlo Kovacs (Hungary): Dynamics of structures subjected to digital force control HR3—Kurt Schlacher and Andreas Kugi (Austria): Active control of smart structures, a Lagrangian approach HR4—Fabrizio Vestroni; Vincenzo Gattulli (Italy): Non-collocated longitudinal control for oscillating suspended cables

HR5—Alois Steindl (Austria): Stabilizing a flexible dumbbell satellite by tension control HR6—Uwe Jungnickel and Peter Maißer (Germany): Dynamic control of under-actuated systems HS—Rock mechanics and geomechanics (Bernhard A. Schrefler, Italy, chair) HS1—Joseph F. Labuz and Li-Hsien Chen; Luigi Biolzi (USA): Fracture of rock from wedge indentation HS2—Alexei A. Savitski and Emmanuel Detournay; Dmitry I. Garagash (USA): Asymptotic solutions for a penny-shaped fluid-driven fracture HS3—Ioannis Vardoulakis; Maria Stavropoulou; George Exadaktylos (Greece): Nonlinearity of hard crystalline rocks in uniaxial tension–compression loading HS4—Roberto Nova and Riccardo Castellanza (Italy): Modelling weathering effects on the mechanical behavior of soft rocks HS5—Felix A. Darve and Farid Laouafa (France): Instability mechanisms in granular materials

HS6—Sol R. Bodner; Moshe Merzer (Israel): An explanation of a possible magnetic field precursor to the 1989 Loma Prieta earthquake

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ICTAM 2000

HV—Complex and Smart fluids (Eric S. G. Shaqfeh, USA, chair) HV1—Lei Li and Ronald G. Larson (USA): Brownian dynamic simulations of dilute polystyrene and DNA solutions in extension and shear flows HV2—Bamin Khomami and Madan Somasi (USA): Modeling of viscoelastic flows in complex

geometries by Brownian dynamics/finite element techniques HV3—Peter Wapperom and Roland Keunings (Belgium): Numerical simulation of transient

flow of polymer melts with integral models HV4—Roger I. Tanner, Yurun Fan, and Nhan Phan-Thien (Australia): Computations of

chaotic rheological mixing HV5—Daniel J. Klingenberg and Yuri M. Shkel (USA): Magnetorheology and magnetostriction of anisotropic composites of ferromagnetic particles HV6—Andrejs Cebers and Ivars Drikis (Latvia): Magnetic fluid free interface dynamics in Hele-Shaw cells IA—Boundary layers (Hans H. Fernholz, Germany, chair) IA1—Richard E. Hewitt and Peter W. Duck ( U K ) : Non-axisymmetric rotating-disk flows IA2—Jan L. Van Ingen and Dirk M. Passchier (The Netherlands): Experimental investigations of transitional and turbulent flows for airfoil analysis and design IA3—Vladimir B. Zametaev and Marina A. Kravtsova (Russia): 3D interacting flow theory and lateral edge stall IA4—Nickolay V. Semionov, Alexander D. Kosinov, and Viktor V. Levchenko (Russia): Experimental study of disturbances development in a supersonic boundary layer on a swept wing IA5—Anatoly I. Ruban and Imrahim Turkyilmaz (UK): On laminar separation from a corner point of a rigid body contour in transonic flow

IA6—Thierry Maeder, Nikolaus A. Adams, and Leonhard Kleiser (Switzerland): Direct

numerical simulation of supersonic turbulent boundary layers IB—Biological fluid dynamics (Timothy J. Pedley, UK, chair) IB1—Bruce G. Bukiet, Hans R. Chaudhry, and Thomas Findley; Arthur B. Ritter (USA): Residual stresses in oscillating thoracic arteries increase blood flow IB2—Shi-Gui Wu, Li-Min Hu, Ai-Ke Qiao, Yu-Hua Peng, Jia-Quan Wang, and You-Jun

Liu (China, PRC): A numerical approach to nonlinear blood flow in arterial stenosis IB3—Marie Oshima and Ryo Torii; Toshio Kobayashi; Kiyoshi Takagi (Japan): The hemodynamic study of the cerebral artery using numerical simulations based on the CT angiog-

raphy IB4—Kazuo Tanishita, Takaaki Deguchi, and Tetsuya Tsuji; Takako Nishiya and Yasuo Ikeda (Japan): Motion of artificial platelet in the model small artery IB5—Joe S. Hur and Eric S. G. Shaqfeh; Hazen P. Babcock, Douglas E. Smith, and

Steven Chu (USA): Dynamics of dilute and semi-dilute dna solutions in shear flow IB6—Toshiro Ohashi, Shinji Seo, Yasuaki Ishii, and Masaaki Sato (Japan): Intracellular stress distribution and cytoskeletal structure in sheared endothelial cells IC—Turbulence (Marcel R. Lesieur, France, chair) IC1—Elias Balaras, Ugo Piomelli, and Andrea Pascarelli (USA): Turbulent structures in accelerating boundary layers IC2—Alexander P. Kozlov and Nikolay I. Mikheyev; Colin J. Bates; Valery M. Molochnikov (Russia): On some characteristics of wall turbulence in separated flows IC3—Olivier J. Métais and Martin Salinas-Vasquez (France): Large eddy simulations of the turbulent flow in a rectangular duct with asymmetrical wall-heating IC4—Nikolai V. Nikitin; Sergey I. Chernyshenko (Russia): Regeneration mechanism of the near-wall turbulence IC5—Mosa Chaisi and Derek Stretch; James W. Rottman (South Africa): Turbulence and waves at a gas–liquid interface IC6—K. Johan A. Westin and P. Henrik Alfredsson; Alessandro Talamelli (Sweden): Experimental investigation of a correction procedure for hot-wire x-probes in shear flows ID—Fracture and crack mechanics (jointly with ICF) (Walter J. Drugan, USA, chair) IDl—Viggo Tvergaard (Denmark): Resistance curves for mixed mode interface crack growth between dissimilar elastic–plastic solids ID2—Lew V. Nikitin; Vladimir N. Odintsev (Russia): Tensile fracture of brittle solids under compression ID3—Huajian Gao; Yonggang Huang; Farid F. Abraham (USA): Continuum and atomistic analysis of intersonic crack propagation ID4—Dhirendra V. Kubair and Philippe H. Geubelle (USA): Intersonic crack propagation in homogeneous media under shear-dominated mixed-mode loading

Scientific program

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ID5—Leon M. Keer and Igor A. Polonsky (USA): Stress analysis of solids with complex crack patterns based on the Mura formula ID6—Jianxiang Wang and Jing Fang; Bhushan L. Karihaloo (China, PRC): Asymptotic bounds on overall moduli of cracked bodies IE—Structural optimization (jointly with ISSMO) (John E. Taylor, USA, chair) IE1—Daniel A. Tortorelli, Nahil A. Sobh, and Robert B. Haber; Robert W. Hyland Jr. (USA): Quench process modeling and optimization IE2—Vassili V. Toropov and Luis F. Alvarez ( U K ) : Selection of structure of response surface approximation models using genetic programming IE3—Satchi Venkataraman and Raphael T. Haftka; John T. Wang and Chauncey K. Wu (USA): Response surfaces for predicting load redistribution in multi-level structural optimizations IE4—Zenon Mróz; Krzysztof Dems (Poland): Damage identification using natural frequency measurement for varying support and mass parameters IE5—Pauli Pedersen (Denmark): Sensitivity of the optimal shapes to changes in material parameters IE6—Uri Kirsch and Panos Y. Papalambros (USA): Accurate displacement derivatives for structural optimization using approximate reanalysis IF—Plates and shells (jointly with IACM) (Christopher R. Calladine, UK, chair) IF1—Arthur W. Leissa (USA): Some observations on plate and shell analysis IF2—Wojciech Pietraszkiewicz (Poland): The nonlinear theory of thin irregular shell structures in terms of rotations IF3—Timothy J. McDevitt; James G. Simmonds (USA): Reduction of the Sanders–Koiter equations for fully anisotropic circular cylindrical shells IF4—J. N. Reddy; C. M. Wang (USA): Relationships between the solutions of the classical and shear deformation plate theories IF5—J. Mark F. G. Holst; Christopher R. Calladine ( U K ) : Surface-inversion phenomena in thin-walled elastic structures IF6—Herbert A. Mang and Christian Schranz (Austria): Ab initio estimates of stability limits of shells by means of arc-length extrapolation IG—Mechanics of thin films and nanostructures (Jürg Dual, Switzerland, chair) IG1—Terry J. Delph; Ralph J. Jaccodine (USA): The mechanics of silicon oxidation IG2—Johannes Vollmann, Dieter Profunser, and Jürg Dual (Switzerland): Femtosecond ultrasonics for the characterization of thin films and microstructures IG3—Zhigang Suo and Min Huang; Qing Ma and Harry Fujimoto (USA): Thin film cracking and ratcheting caused by temperature cycling IG4—Lei Lian and Nancy R. Sottos (USA): Stress effects in ferroelectric thin films IG5—George C. Johnson, Peter T. Jones, Ming Ting Wu, and Zachary Bell (USA): Testing and analysis of thin film strength using micro-electro-mechanical systems 1G6—Kenneth M. Liechti and Kalyan C. Vajapeyahula; Hyacinth Cabibil and John M. White; Robb M. Winter (USA): Interfacial force microscopy studies of polymer / glass interphases

IH—Impact and wave propagation (Werner Goldsmith, USA, chair) IH1—Alexander M. Samsonov, Galina V. Dreiden, Alexey V. Porubov, and Irina V.

Semenova (Russia): Strain solitons in solids—and how to do them IH2—Rodney J. Clifton and Seung-Yong Yang (USA): Computational modeling of stresswave-induced martensitic phase transformations IH3—Ajit Mal; Jungki Lee (USA): Wave propagation in a locally heterogeneous unbounded solid IH4—Federico J. Sabina and Oscar Valdiviezo; Valery M. Levin (Mexico): Thermoelastic waves in randomly particulate composites IH5—Shiming Zhuang and Guruswaminaidu Ravichandran; Dennis E. Grady (USA): Shock wave propagation in layered heterogeneous media IH6—Pablo D. Zavattieri; Horacio D. Espinosa (USA): An investigation of rate dependent decohesion for capturing damage kinetics in brittle materials IK—Flows in thin films (Yves Couder, France, chair)

IK1—Lou Kondic; Javier Diez (USA): Nonlinear dynamics of thin film flows IK2—Jean-Marc Vanden-Broeck ( U K ) : Air blown solitary waves IK3—Christian Ruyer-Quil and Paul Manneville (France): On the introduction of viscous dispersion in the modeling of film flows down inclined planes IK4—Mahesh Tirumkudulu and Andreas Acrivos (USA): Coating flows within a rotating horizontal cylinder: Lubrication analysis and numerical computations

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ICTAM 2000

IK5—Boris Y. Rubinstein, Alexander A. Golovin, and Leonid M. Pismen (Israel): Effect of van der Waals forces on fingering instability of thermally driven contact lines IK6—Takao Yoshinaga and Takeshi Uchiyama (Japan): Nonlinear wave behavior on a liquid

sheet with a temperature difference between both surfaces IL—Multiphase flows (L. Gary Leal, USA, chair) IL1—S. Balachandar and Prosenjit Bagchi (USA): Steady and unsteady forces on a spherical

particle in nonuniform flows IL2—Simon Crispel; Alain Cartellier (France): Hindering effect and velocity variance in a sedimenting suspension IL3—Warren B. Tauber and Grétar Tryggvason (USA): Direct numerical simulations of primary breakup IL4—Alexander Z. Zinchenko and Robert H. Davis (USA): Shear flow of a highly concentrated emulsion of spherical drops by numerical simulation IL5—Alberto Aliseda, Franek Hainaux, and Juan C. Lasheras; Alain Cartellier (USA): Settling velocity and clustering of particles in homogeneous and isotropic turbulence IL6—Shu Takagi, Kazuyasu Sugiyama, and Yoichiro Matsumoto (Japan): New approach to the modeling of sub-grid scale structures of disperse multiphase flows IO—Stability of Structures (Stephen H. Crandall, USA, chair)

IO1—Alexei A. Mailybaev (Russia): Stability domains of non-conservative systems with small parametric excitation IO2—Paolo S. Valvo and Salvatore S. Ligarò (Italy): Stress-concentration in a partly wrinkled elastic membrane IO3—Theodoro Netto; Stelios Kyriakides (Brazil): Dynamic propagation and flip-flopping of buckles in pipelines

IO4—Anthony N. Kounadis and Charis J. Gantes (Greece): Approximate dynamic buckling loads of nonconservative structural systems via energy considerations IO5—Piergiovanni Marzocca and Liviu Librescu; Walter A. Silva (USA): Nonlinear sta-

bility and response of lifting surfaces via Volterra series IO6—Walter V. Wedig (Germany): Strong stability conditions in elastic structures under stochastic parametric loading IR—Willis Symposium (Pierre M. Suquet, France, chair) IR1—Davide Bigoni; Domenico Capuani (Italy): On decay effects in nonlinear elasticity IR2—Sia Nemat-Nasser (USA): Dislocations-based rate- and temperature-dependent deforma-

tion of FCC, BCC, and HCP metals IR3—Xi Chen and John W. Hutchinson (USA): The influence of foreign object damage on fatigue cracking IR4—John W. Morrissey; James R. Rice (USA): Perturbative simulations of crack front

waves IR5—Hansuk Lee and Ares J. Rosakis; L. Ben Freund; Elizabeth Kolawa (USA): Optical

measurements of curvatures of film-substrate systems in the nonlinear deformations range IR6—Jiang Yu Li and Kaushik Bhattacharya (USA): The macroscopic behavior of ferroelectric polycrystals: Variational bounds and estimations IS—Viscoelasticity and creep (John L. Bassani, USA, chair) IS1—Holm Altenbach, Chunxiao Huang, and Konstantin Naumenko (Germany): Numer-

ical creep-damage predictions in thin-walled structures based on standard and refined approaches IS2—Nobutada Ohno, Takaaki Ando, and Shiro Biwa; Takushi Miyake (Japan): Varia-

tional method for fiber stress profile analysis of unidirectional composites with matrix creep IS3—Petros Sofronis; Prasad B. R. Nimmagadda (USA): Stress redistribution in creeping matrix composite materials: Effect of interface slip and diffusion IS4—S. Leigh Phoenix, Chung-Yuen Hui, and David Shia; Leonid Kogan (USA): Time dependent fiber fragmentation in a composite with matrix creep leading to creep-rupture IS5—Ren Wang and Qi-Feng Yu (China, PRC): Life expectation of fiber reinforced composites IV—Vortex dynamics (Dale I. P u l l i n , USA, chair) IV1—Rene Steijl and H. W. M. Hoeijmakers (The Netherlands): Numerical simulation of cooperative instabilities in a counter-rotating vortex pair IV3—Vyacheslav V. Meleshko and Alexandre A. Gourjii; Andreas Doernbrack, Frank Holzaepfel, Thomas Gerz, and Thomas Hofbauer (Ukraine): Interaction of twodimensional trailing vortex pair with a shear layer IV4—Paul Billant; Jean-Marc Chomaz (France): Zigzag instability and the origin of the pancake turbulence

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IV5—Christophe Eloy, Patrice Le Gal, and Stéphane Le Dizès (France): Multipolar instability of a vortex in a deformed elastic cylinder IV6—Yasuhide Fukumoto (Japan): Three-dimensional motion of a vortex filament with elliptically deformed core JA—Flow instability and transition (Michio Nishioka, Japan, chair)

JA1—Jonathan J. Healey (UK): Wave envelope steepening effects in the Blasius boundary

layer JA2—Masaharu Matsubara; P. Henrik Alfredsson (Japan): Streaky structure in boundary

layer transition induced by free stream turbulence JA3—Donatella Ponziani, Carlo M. Casciola, and Renzo Piva; Francesco Zirilli (Italy): Formation of structures and regeneration mechanisms in by-pass transitional flows JA4—Maxim V. Ustinov (Russia): Streaks production by free-stream nonuniformity interaction with blunt leading edge JA5—Masahito Asai and Akira Fukuoka; Michio Nishioka (Japan): Experimental investigation of the instability of low-speed streak in a boundary layer JA6—Victor V. Kozlov, Genrih G. Grek, Mikhail M. Katasonov, and Valeriy G. Chernoray (Russia): Study of the development of a lambda-structure and its transformations into a turbulent spot

JB—Convective phenomena (S. Balachandar, USA, chair) JB1—Hao-wen Xi; Xiao-jun Li; James D. Gunton (USA): Phase turbulence in stress-free Rayleigh–Bénard convection JB2—Thierry Passot, Dimitri Laveder, Yannick Ponty, and Pierre-Louis Sulem (France): Scaling of the correlation length in rotating convection JB3—Detlef Lohse; Siegfried Grossmann (The Netherlands): Scaling in thermal convection: A unifying view JB4—Jackson R. Herring and Robert M. Kerr (USA): Prandtl dependence of Nusselt in direct numerical simulation JB5—Richard L. Fernandes and Ronald J. Adrian (USA): Large scale structure of turbulent Rayleigh–Bénard convection: Proper orthogonal decomposition analysis

JB6—Axel Günther and Philipp Rudolf von Rohr (Switzerland):

Visualization of fluid

temperature and velocity in Rayleigh–Bénard convection: Mean and second-order quantities JC—Fluid mechanics of materials processing (Wilhelm Schneider, Austria, chair)

JC1—Kenneth J. Craig, Danie J. de Kock, and Jan A. Snyman (South Africa): Continuous casting optimization using CFD and Dynamic-Q JC2—Valod Noshadi and Wilhelm Schneider (Austria): Internal melt flow in horizontal continuous casting JC3—Yuri M. Gelfgat (Latvia): Influence of combined electromagnetic fields on convective transfer processes in conducting fluids JC4—Christian Resagk, Joerg Grabow, and Ulrich Schellenberger (Germany): Optical characterization of interface waves using particle image velocimetry and laser vibrometer techniques JC5—Jean N. Koster and Hongbin Yin (USA): In-situ solidification and flow visualizations with x-rays

JC6—Taketoshi Hibiya, Takeshi Azami, and Shin Nakamura; Kusuhiro Mukai (Japan): Marangoni flow of molten silicon: Role of oxygen at melt surface JD—Elasticity (Paolo Podio-Guidugli, Italy, chair) JD1—Elena F. Grekova (Russia): On the choice of strain tensors for 3D mechanical polar media, shells, and magnetic materials JD2—Andrew N. Norris (USA): Stress invariance in three-dimensional elasticity JD3—Arthur H. England (UK): Finite elastic deformations of a tyre modelled as an ideal fibre-reinforced shell JD4—Thomas J. Pence and Jose Merodio (USA): Equilibrium shocks in a directionally reinforced nonlinearly elastic material JE—Material instabilities (Ahmed Benallal, France, chair)

JE1—Henryk Petryk (Poland): Uniqueness and stability in materials with multiple mechanisms of inelastic deformation JE2—Henrik M. Jensen (Denmark): Model of delamination buckling instability JE3—Tracy J. Vogler, Sheng-Yuan Hsu, and Stelios Kyriakides (USA): On the initiation

of kink bands in fiber composites JE4—Jean-Claude Grandidier; Sylvain Drapier; Michel Potier-Ferry (France): A structural approach of plastic microbuckling in long fibre composites

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ICTAM 2000

JE5—Karine Siruguet and Jean-Baptiste Leblond (France): Influence of inclusions upon void growth and coalescence in porous ductile solids JE6—Kenneth Runesson and Thomas Svedberg (Sweden): An adaptive finite element algorithm for gradient theory of plasticity __ __

JF—Multibody dynamics (Martin Arnold, Germany, chair) JF1—Lorenzo Trainelli and Marco Borri (Italy): Solution of constrained dynamical systems via an embedded projection method JF2—John M. Hansen (Denmark): Synthesis of kinematically driven systems using dynamics JF3—Jean W. Zu and Friedrich P. Rimrott (Canada): The collinearity principle applied to a gyrosatellite JF4—John J. McPhee, Marcus Scherrer, and Pengfei Shi (Canada): A unified modelling approach for electro-mechanical multibody systems JF5—Sotirios Natsiavas, George Vaklavas, and George Verros (Greece): Dynamics and control of vehicles with dual-rate suspension dampers JG—Computational solid mechanics (jointly with IACM) (Peter Wriggers, Germany, chair)

JG1—Tian Yun Wu and Gui Rong Liu (Singapore): A generalization of the differential quadrature method JG2—Peter Betsch and Paul Steinmann (Germany): Time finite elements for nonlinear elastodynamics JG3—Alain G. Combescure and Anthony Gravouil (France): A time-space multi-scale algorithm for transient structural non linear problems JG4—Pierre Ladevèze, David Dureisseix, and Olivier Loiseau (France): A micro–macro computational strategy including homogenization for analysing heterogeneous structures JG5—Wing Kam Liu and Gregory J. Wagner (USA): Bridging scale hierarchical enrichment

JG6—Hannes Schmidt and Simon D. Guest (UK): Symmetry-adapted equilibrium of repetitive structures JH—Impact and wave propagation (Jan D. Achenbach, USA, chair)

JH1—K. T. Ramesh and Yulong Li; Ernest S. C. Chin (USA): Dynamic response of uniaxial continuous fiber-reinforced metal matrix composites JH2—Werner Goldsmith, Gabrielle Cipparrone, and Benjamin C. Bourne; Dennis L. Orphal (USA): Momentum exchange alteration in projectile / target impact JH3—Tongxi Yu; Jia Ling Yang; Stephen R. Reid (Hong Kong): Energy partitioning between two colliding beams JH4—Igor Ye. Telitchev (Russia): Study of burst conditions of thin-walled pressure vessels under hypervelocity impact JH5—Janusz R. Klepaczko and Ahmed Brara (France): Impact tension of concrete by spalling JH6—Min Zhou (USA): An experimental characterization of the impact failure of mortar JK—Waves (Lev A. Ostrovsky, USA, chair)

JK1—Mei C. Shen and Chia-Chin Chang (USA): Forced nonlinear surface waves in a waterfilled circular basin JK2—Chad M. Topaz and Mary Silber; Anne C. Skeldon (USA): Pattern selection of twofrequency forced Faraday waves JK3—Christian Kharif; Regis Pontier (France): Occurrence of new 3D patterns in water wave fields JK4—Evgeny A. Demekhin and Hsueh-Chia Chang (USA): Wave transitions in a falling film JK5—Gérard M. Iooss (France): Gravity and capillary-gravity travelling waves for superposed fluid layers, one being deep JK6—Triantaphyllos R. Akylas and David C. Calvo (USA): Stability of steep gravity– capillary solitary waves in deep water JL—Drops and bubbles (John R. Blake, UK, chair) JL1—Yulii D. Shikhmurzaev (UK): Coalescence and capillary breakup of liquid volumes JL2—Jerzy Blawzdziewicz, Vittorio Cristini, and Michael Loewenberg (USA): Nearcritical behavior of drops in linear flows JL3—Yuriko Renardy and Jie Li (USA): Breakup of a viscous drop under shear JL4—L. Gary Leal, Jong-Wook Ha, and Derek C. Tretheway (USA): Experimental studies of the deformation and breakup of polymeric drops JL5—David I. Bigio, Richard V. Calabrese, and Charles R. Marks (USA): Drop breakup and deformation in sudden onset strong flows JL6—Craig D. Eastwood and Juan C. Lasheras (USA): The breakup time of a droplet in a fully developed turbulent flow

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lxi

JO—Cellular and molecular mechanics (Jüri Engelbrecht, Estonia, chair)

JO1—Sheldon Weinbaum; Jianjun Feng (USA): The mechanics of skiing from red cells to humans JO2—Dennis E. Discher, James Lee, Carol Kwok, and Philippe Carl (USA): From single

molecules to cell membrane mechanics JO3—Michael Elbaum and Hanna Salman (Israel): Dynamics of DNA uptake to the cell nucleus JO4—Christopher R. Calladine (UK): Mechanics of tether formation in liposomes JO5—Anna J. Diaz and Dominique Barthès-Biesel (France): Entrance of a cell into a pore JO6—Alexander A. Spector (USA): Nonlinear electroelastic constitutive relations for the cochlear outer hair cell JR—Willis Symposium (Pedro Ponte Castañeda, USA, chair)

JR1—Norman A. Fleck and Vikram Deshpande (UK): Constitutive models for foams JR2—Renaud Masson; Michel Bornert and André Zaoui; Pierre M. Suquet (France): An affine formulation for the prediction of the overall behavior of nonlinear heterogeneous media JR3—Graeme W. Milton and Sergey K. Serkov (USA): Neutral inclusions and composite assemblages JR4—Walter J. Drugan (USA): A higher-order nonlocal constitutive equation and optimal effective moduli for elastic composites JR5—Pierre M. Suquet, Herve Moulinec, and Jean-Claude Michel (France): A numerical

method for composites with complex micro structure JR6—Nicolas Triantafyllidis (USA): Onset-of-failure surfaces in periodic solids: applications

Theory and

JS—Microgravity mechanics (Vadim I. Polezhaev, Russia, chair) JS1—Constantine M. Megaridis and Kevin Boomsma; Dimos Poulikakos; Vedha Nayagam (USA): Contact angle dynamics of molten solder droplets impacting onto flat

substrates in reduced gravity JS2—Valentin S. Yuferev, Olga N. Budenkova, and Elvira N. Kolesnikova (Russia): Thermal convection under quasi-static component of microgravity field

JS3—Günter Wozniak; R. Balasubramaniam; R. S. Subramanian (Germany): Thermocapillary bubble and drop motion—Experiments versus theory JS4—Alexander L. Yarin; Günter Brenn, Oliver Kastner, and Dirk Rensink; Cameron Tropea (Israel): Evaporation of acoustically levitated droplets of pure liquids and binary liquid mixtures JS5—J. Iwan D. Alexander and Lev A. Slobozhanin (USA): Stability of doubly connected and disconnected free surfaces in a cylinder under microgravity JS6—You-Seop Lee, Hendrik C. Kuhlmann, and Hans J. Rath; Chung-Hwan Chun (Germany): Flow instability in thermocapillary liquid bridges with and without vortex breakdown JV—Topological fluid mechanics (H. Keith Moffatt, UK, chair)

JV1—Pieter G. Bakker and Michael Proot (The Netherlands): Topology of laminar juncture flows JV2—Morten Brøns; Lars K. Voigt and Jens N. Sørensen; Andreas Spohn; Johan Hartnack (Denmark): Two- and three-dimensional streamline topology in the vortex breakdown in a closed cylinder JV3—Konrad Bajer (Poland): Heat flow around a diffusing vortex JV4—Carlo F. Barenghi and David C. Samuels; Renzo L. Ricca (UK): The evolution of vortex knots JV5—Vladimir A. Vladimirov; H. Keith Moffatt; Konstantin I. Iline (Hong Kong): Generalized isovorticity conditions for fluid flows JV6—Robert W. Ghrist; John B. Etnyre (USA): Contact topological techniques for steady

Euler flows KD—Sectional Lecture by Peter A. Monkewitz (Nicholas Rott, USA, chair)

KD1—Peter A. Monkewitz (Switzerland): Flow instabilities and transition in spatially inho-

mogeneous systems KE—Sectional Lecture by Vladimir A. Palmov (Keh-Chih Hwang, China, chair)

KE1—Vladimir A. Palmov (Russia): Stationary waves in elasto–plastic and visco–plastic bodies

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ICTAM 2000

KP—Sectional Lecture by Erwin Stein (Günther R. Kuhn, Germany, chair)

KP1—Erwin Stein (Germany): Error controlled adaptivity for hierarchical models and finiteelement approximations in structural mechanics KQ—Sectional Lecture by Stephen J. Cowley (Roddam Narasimha, India, chair) KQ1—Stephen J. Cowley (UK): Laminar boundary-layer theory: A 20th century paradox? KW—Sectional Lecture by Erik van der Giessen (James R. Rice, USA, chair) KW1—Erik van der Giessen (The Netherlands): Plasticity in the 21st century KX—Sectional Lecture by Charles S. Peskin (Philip J. Holmes, USA, chair) KX1—Charles S. Peskin (USA): Muscle and blood: Fluid dynamics of the heart and its valves LD—Sectional Lecture by Shigeo Kida (Tomomasa Tatsumi, Japan, chair) LD1—Shigeo Kida (Japan): Vortical structure in turbulence LE—Sectional Lecture by R. Narayana lyengar (Jean Salençon, France, chair) LE1—R. Narayana lyengar (India): Probabilistic methods in earthquake engineering LP—Sectional Lecture by Gedeon Dagan (Sol R. Bodner, Israel, chair) LP1—Gedeon Dagan (Israel): Effective, equivalent and apparent properties of heterogeneous media

LQ—Sectional Lecture by Amable Liñán (Moshe Matalon, USA, chair) LQ1—Amable Liñán (Spain): Diffusion controlled combustion LW—Sectional Lecture by Subra Suresh (Johannes Weertman, USA, chair) LW1—Subra Suresh (USA): Nanomechanics and micromechanics of thin films, graded coatings and mechanical/nonmechanical systems LX—Sectional Lecture by Ole Sigmund (Pauli Pedersen, Denmark, chair)

LX1—Ole Sigmund (Denmark): Optimum design of Micro Electro Mechanical Systems (MEMS) MA—Flow instability and transition (Peter A. Monkewitz, Switzerland, chair)

MA1—William D. Thacker; Chester E. Grosch; Thomas B. Gatski (USA): Dynamics of ensemble-averaged linear disturbances in a transitioning boundary-layer MA2—Leonhard Kleiser, Dirk Wilhelm, and Carlos Haertel (Switzerland): A numerical simulation study of the 2D-3D transition in forward-facing step flow MA3—Herbert Steinrück (Austria): Upstream travelling waves in the boundary layer of a horizontal mixed convection flow MA4—Artur Chung-Che Kuo and Heinrich Fiedler (Germany): Weakly nonlinear model for the coherent structures in a curved turbulent mixing layer MA5—Michael G. Olsen; J. Craig Dutton (USA): Stochastic estimation of large structures in incompressible and weakly compressible mixing layers MA6—Jean-Marc Chomaz; Arnaud Couairon; Ivan Delbende (France): Nonlinear global modes in wakes behind bluff bodies MB—Geophysical fluid dynamics (Kolumban Hutter, Germany, chair)

MB1—Andrey N. Salamatin and Dina R. Malikova (Russia): Structural dynamics of ice sheets MB2—Dambaru Raj Baral; Kolumban Hutter (Nepal): Higher order asymptotic theories of large scale motion, temperature and moisture distribution in ice sheets

MB3—Gunter Gerbeth and Frank Stefani; Agris Gailitis, Olgerts Lielausis, and Ernests Platacis (Germany): Magnetic field self-excitation in the Riga dynamo experiment MB4—Sidney Leibovich; Thomas M. Haeusser (USA): Langmuir circulation patterns in a rotating ocean

MB5—Fabrice Veron and W. Kendall Melville (USA): Experiments on the initiation of Langmuir circulations and surface waves MB6—William R. Phillips (USA): Longitudinal vortices in wavy shear flows with a free surface MC—Low-Reynolds-number flow (Antonio Delgado, Germany, chair)

MC1—Jens G. Eggers (Germany): Stability of free surface cusps MC2—Jeremy Teichman, Maria-Isabel Carnasciali, Sahraoui Chaïeb, Gareth H. McKinley, and Lakshminarayanan Mahadevan (USA): Wrinkling of sheared viscous sheets MC3—Taher A. Saif and Chad Sager (USA): Interaction force between two thin solids on the surface of a liquid: Theory and experiment

Scientific program

1xiii

MC4—Dominique Legendre and Jacques Magnaudet (France): The relation between the

forces acting on a spherical drop and those acting on a solid sphere MC5—Ehud Gavze; Michael Shapiro (Israel): Gravity induced drift of ellipsoidal inertial particles in a simple shear flow MC6—Ludwig C. Nitsche and Uwe Schaflinger; Gunther Machu and Walter Meile (Aus-

tria): Coalescence, torus formation and break-up of settling drops: Experiments and computer simulations MD—Mechanics of foams and cellular materials (Gustavo Gioia, USA, chair) MD1—Vikram Deshpande, Norman A. Fleck, and Michael F. Ashby (UK): Effective prop-

erties of a ”cubic” lattice material MD2—Michael K. Neilsen and William M. Scherzinger; Wei-Yang Lu; Andrew M. Kraynik (USA): Mechanics of aluminum honeycomb

MD3—Patrick R. Onck (The Netherlands): Modeling of notch effects in metallic foams MD4—Thomas Daxner, Franz G. Rammerstorfer, and Helmut J. Boehm (Austria): Adap-

tation of the density distribution in weight-efficient metal foam structures MD5—Mark W. Schraad and Francis H. Harlow (USA): A stochastic two-field approach to

modeling the mechanical response of cellular materials ME—Damage and failure of composites (S. Leigh Phoenix, USA, chair) ME1—Alexander E. Bogdanovich (USA): Three-dimensional analysis of inhomogeneous solids

using Bernstein polynomial approximations ME2—Irene J. Beyerlein; S. Leigh Phoenix; Linda S. Schadler (USA): Distributions and size scalings for strength and lifetime of a short-fiber composite

ME3—William A. Curtin (USA): Multiscale models of damage accumulation and failure in fiber-reinforced composites ME4—Zdenek P. Bazant; Drahomir Novak (USA): Energetic probabilistic theory of fracture

scaling of composites ME5—Olivier Allix and Pierre Ladevèze (France): Application of a meso-model of laminates

to low-energy impact ME6—Anthony A. Caiazzo; Francesco Costanzo (USA): Effective constitutive relations for

the structural analysis of components with evolving cracks MF—Structural vibrations (Utz von Wagner, Germany, chair) MF2—James F. Doyle; Elizabeth M. Webster (USA): A dynamic view of static instabilities MF3—Nathan van de Wouw and Dick H. van Campen; Henk Nijmeijer (The Netherlands):

Statistical bilinearization in stochastic nonlinear dynamics MF4—Alexander F. Vakakis; Igor Rozhkov (USA): Energy pumping phenomena in nonlinear

coupled oscillators MF5—Iliya I. Blekhman (Russia): Vibrational mechanics of dry friction and impact systems MF6—Sami F. Masri; Elias B. Kosmatopoulos; Andrew W. Smyth; Anastasios G. Chas-

siakos (USA): Robust neural estimation of internal forces in nonlinear structures under random excitation MG—Plasticity and viscoplasticity (Erik van der Giessen, The Netherlands, chair) MG1—K. Jimmy Hsia; Huajian Gao; Yun-Biao Xin (USA): Effects of dislocation nucleation

on fracture in silicon single crystals MG2—Kazuwo Imai, Satoshi Sugawara, Tsuyoshi Ganbe, and Michio Kurashige; Koji

Sumino (Japan): Dislocation plasticity of semiconductor crystals MG3—Yonggang Huang and Zhenyu Xue; Huajian Gao; William D. Nix (USA): A study

of micro-indentation hardness experiments by strain gradient plasticity MG4—Véronique Favier, Stéphane Berbenni, and Marcel Berveiller; Xavier Lemoine

(France): Coupling between elastic storage and viscoplastic dissipation by a new selfconsistent scheme MG5—Johannes A. W. van Dommelen, W. A. Marcel Brekelmans, and Frank P. T.

Baaijens; David M. Parks and Mary C. Boyce (The Netherlands): A micromechanical

model for the large-strain constitutive behavior of semi-crystalline polymers MG6—Mohammed A. Zikry and W. M. Ashmawi (USA): Microstructurally induced failure evolution and global failure in porous crystalline aggregates MH—Contact and friction problems (jointly with IAVSD) (João A. C. Martins, Portugal, chair) MH1—Joachim Larsson and Bertil Storåkers (Sweden): On inelastic impact and dynamic

hardness MH2—Sinisa Dj. Mesarovic; Norman A. Fleck (USA): Spherical indentation of elastic–plastic

solids

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ICTAM 2000

MH3—Lars-Erik V. Andersson (Sweden): Quasistatic frictional contact problems with finitely many degrees of freedom MH4—Hachmi Ben Dhia and Malek Zarroug; Patrick Massin and Isabelle Vautier (France): Finite elements/points for a hybrid formulation of contact-friction problems in large transformation MH5—Christoph Glocker (Germany): A standard NLCP formulation for rigid body systems with spacial Coulomb friction MH6—James A. Tzitzouris Jr. and Jong-Shi Pang (USA): Dynamic linear complementarity systems MK—Waves (W. Kendall Melville, USA, chair) MK1—Lev A. Ostrovsky (USA): Strongly nonlinear, multisoliton internal waves MK2—Takeshi Kataoka, Michihisa Tsutahara, and Tomonori Akuzawa (Japan): Transverse instability of finite-amplitude internal solitary waves in a weakly stratified fluid MK3—John Grue, Atle Jensen, Per-Olav Rusås, and Johan Kristian Sveen (Norway): On the dynamics of highly nonlinear solitary waves in a stratified fluid MK4—Vladimir E. Zakharov (USA): Weak and solitonic turbulence of gravity waves on the

surface of a finite-depth fluid MK5—Frédéric Dias (France): Wave turbulence in one-dimensional models MK6—Stefan G. Llewellyn Smith; Rupert Ford (USA): Acoustic scattering by vortices in the Born limit ML—Drops and bubbles (Andrea Prosperetti, USA, chair)

ML1—Philip L. Marston, Mark J. Marr-Lyon, and David B. Thiessen (USA): Novel methods for suppressing drop formation from capillary instabilities ML2—Claus-Dieter Ohl; Andrea Prosperetti (The Netherlands): Response of rising bubbles

to a sudden depressurization ML3—Daniel A. Weiss; Alexander L. Yarin; Günter Brenn and Dirk Rensink (Germany): Acoustically levitated drops: Drop oscillation driven by ultrasound modulation ML4—Tatyana P. Lyubimova; Dmitry V. Lyubimov and Sergey V. Shklyaev (Russia): Quasi-equilibrium shape of a drop on oscillating rigid plate ML5—Daniel Attinger, Vincent Butty, Stephan E. Haferl, and Dimos Poulikakos (Switzerland): Numerical modeling of microdroplet impact and solidification with substrate melting ML6—Steve Glod, Dimos Poulikakos, Ziqun Zhao, and George Yadigaroglu (Switzerland): An investigation of microscale explosive vaporization of water on an ultrathin Pt-wire MO—Biological solid mechanics (Lakshminarayanan Mahadevan, USA, chair)

MO1—Jacques M. Huyghe, Menno M. Molenaar, and Marcel W. Wijlaars; Arjan J. H. Frijns (The Netherlands): Mechanics of 3D finite swelling of ionised media MO2—Jüri Engelbrecht and Marko Vendelin; Peter H. M. Bovendeerd and Dick H. van Campen; Theo Arts (Estonia): Contraction of cardiac muscles and ATP consumption MO3—Alexander A. Stein (Russia): Modeling of plant tissue growth on the basis of continuum mechanics MO4—Timothy J. Van Dyke and Anne Hoger (USA): A comparison of two methods for analyzing finite growth of soft biological materials MO5—Alexander Rachev and Miglena Kirilova (Bulgaria): A mathematical model of arterial regeneration over bioresorbable grafts MS—Chaos in fluid and solid mechanics (Bruno Eckhardt, Germany, chair) MS1—Giuseppe Rega; Rocco Alaggio and Francesco Benedettini (Italy): Dimensionality

and reduced-order models for complex dynamics from experimental observations MS2—Frantisek Peterka and Stanislav Cipera; Tadashi Kotera (Czech Republic): Additional impact causes the intermittency chaos of instable subharmonic motions of impact oscillator MS3—Francis C. Moon; Masaharu Kuroda (USA): Complexity measures in large arrays of fluid-elastic oscillators MS4—Timothy J. Burns; Matthew A. Davies (USA): On solid-dynamical chaos in high-speed machining

MS5—Gert van der Heijden ( U K ) : The static deformation of a twisted elastic rod constrained to lie on a cylinder MS6—Philip J. Holmes; John Schmitt (USA): Mechanical models for insect locomotion: Dynamics and stability on the horizontal plane MV—Vortex dynamics (Vyacheslav V. Meleshko, Ukraine, chair)

MV1—Stéphane Le Dizès (France): Two-dimensional breakdown of a vortex subject to a rotating strain f i e l d

Scientific program

lxv

MV2—Henry M. Tufo; Paul F. Fischer (USA): A numerical study of hairpin vortices induced by a hemispherical roughness element MV3—Victor F. Kopiev and Sergey A. Chernyshev (Russia): On possible mechanism of turbulence generation outside the vortex ring core MV4—Helene Politano, Thomas Gomez, and Annick Pouquet; Michele Larcheveque (France): A Lundgren spiral vortex model for compressible turbulence

MV5—Keri A. Aivazis and Dale I. Pullin (USA): Velocity structure functions from the Hill spherical-vortex model for isotropic turbulence MV6—Anthony Leonard (USA): Evolution of localized packets of vorticity and scalar in turbulence NA—Granular flows and complex and smart fluids (Isaac Goldhirsch, Israel, chair) NA01—Frank Spahn and Miodrag Sremcevic (Germany): ”Fingerprints” of the size distribution in planetary rings? NA02—Farzam Zoueshtiagh and Peter J. Thomas (UK): Computational and experimental study of spiral patterns in granular media under a rotating fluid NA03—Robert P. Behringer and Sarath Tennakoon; Guy Metcalfe (USA): The transition to flow in a horizontally shaken system NA04—Milica Medved, Heinrich M. Jaeger, and Sidney R. Nagel (USA): Coupled response modes in horizontally vibrated granular material NA05—Osamu Sano, Akiko Ugawa, and Katsuhiro Suzuki (Japan): Regular and quasicrystal patterns on the vertically vibrated thin granular layer NA06—Clara Salueña and Thorsten Pöschel; Sergei E. Esipov (Germany): From ”solid” to ”fluid”: Time-dependent hydrodynamic analysis of dense granular flows NA07—Tatyana S. Krasnopolskaya; GertJan F. van Heijst and Jan H. Voskamp; Sergei A. Trigger ( U k r a i n e ) : Similarities of patterns in fluid and granular flow inside a horizontally rotating cylinder NA08—Nitin Jain and Julio M. Ottino; Kimberly M. Hill; Alp Akonur and Richard M. Lueptow ( U S A ) : Particle tracking techniques for measuring granular flow NA09—Christine M. Hrenya and Meheboob Alam; Richard Clelland (USA): Observations of inelastic collapse in granular shear flow NA10—Gregory Pacitto, Cyrille Flament, and Jean-Claude Bacri; Mike Widom (France): Rayleigh–Taylor instability at the interface of a ferrofluid NA11—Philippe Corvisier, Cherif Nouar, Rene Devienne, and Michel Lebouché (France): Analysis of the rheological structure evolution for a thixotropic fluid flow in a pipe NA12—Nathanael J. Woo; Eric S. G. Shaqfeh (USA): The rheology of polymer solutions in ultrathin films NA13—Abdulwahab S. Almusallam, Ronald G. Larson, and Michael J. Solomon (USA): Phenomenological model for droplet shape changes and stresses in the flow of immiscible

blends NA14—Stephan E. Haferl, Dimos Poulikakos, and Daniel Attinger; Ziqun Zhao (Switzerland): Fluid dynamics and solidification in the pile-up of picoliter droplets in micromanufacturing NB—Biomechanics (Sheldon Weinbaum, USA, chair) NB01—Jan Rosenzweig and Oliver E. Jensen (UK): Capillary–elastic instabilities and draining flows in buckled lung airways NB02—Hans R. Chaudhry and Bruce G. Bukiet; Arthur B. Ritter (USA): Mathematical analysis of heart reduction surgery NB03—Djenane C. Pamplona; Claudio R. Carvalho (Brazil): Numerical analysis of tissue expansion using finite elements method NB04—Hans I. Weber; Djenane C. Pamplona and Larissa M. Freire (Brazil): A simple experimental technique for the analysis of tissue expansion NB05—Werner Winter (Germany): Trabecular bone behavior and remodeling—based on equations of continuum damage mechanics NB06—Eduard Rohan and Robert Cimrman (Czech Republic): Identification of constitutive parameters for macroscopic modelling of smooth muscle tissues NB07—Jennifer H. Shin and Lakshminarayanan Mahadevan; Paul Matsudaira (USA): Dynamics of an actin spring NB08—Stephen M. Klisch and Anne Hoger (USA): Development of a growth mixture theory for cartilaginous tissues NB09—Henry W. Haslach Jr. (USA): A nonlinear dynamical model for bruit generation by an intracranial saccular aneurysm NB10—Tomasz Lekszycki (Poland): Modeling of bone adaptation process with use of structural optimization methods

NB11—Wendy C. Crone and Kumar Sridharan (USA): Surface modification of NiTi with plasma source ion implantation for biomedical applications

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ICTAM 2000

NB12—Sarah L. Thelen and L. Catherine Brinson (USA): Titanium foam for use in bone implants: Micro structure effects on mechanical properties NB13—Oren Vilnay, Konstantin Yu. Volokh, and Michael Belsky (Israel): Do living cells possess the structure of tensegrity? NC—Transition and turbulence (Gregory p. King, UK, chair) NC01—Luigi de Luca and Giro Caramiello (Italy): On some aspects of transient growth in fluid dynamics stability NC02—Anatoli Tumin and Genxing Han; Israel Wygnanski (Israel): Transition in pipe Poiseuille flow NC04—Alessandro Talamelli and Isabella Gavarini (Italy): Absolute and convective instability of coaxial jets NC05—Jorg Schumacher and Bruno Eckhardt (Germany): Evolution of turbulent spots in a plane shear flow NC06—Osamu Mochizuki, Masaru Kiya, and Hisashi Yamamoto (Japan): Precursor of separation NC07—Valery L. Okulov (Russia): Changes in the helical symmetry of the vorticity field during vortex breakdown into tubes NC08—Katsuhiro Suzuki, Masayuki Hirano, and Osamu Sano (Japan): An experimental study on the reconnection process of vortex tubes NC09—Yasuaki P. Kohama, Yasuhiro Egami, Mitsuyoshi Kawakami, Yoshiyuki Haruta, and Mitsuru Shimagaki (Japan): Transition mechanism, prediction, and control of crossflow field NC10—Yutaka Miyake, Koichi Tsujimoto, and Masaru Nakaji (Japan): Turbulence mixing in a rough-wall channel of two-dimensional square rods NC11—Jie Cui, Virendra C. Patel, and Ching-Long Lin (USA): Large-eddy simulation of

turbulent flow over k-type and d-type rough surfaces NC12—John C. Wells, Yaaufumi Yamamoto, Yuki Yamane, Shinji Egashira, and Hiroji Nakagawa (Japan): Time-resolved particle-image velocimetry in the cross-stream plane of an open channel NC13—Keiji Kishida; Keisuke Araki; Seigo Kishiba; Katsuhiro Suzuki (Japan): Orthonormal divergence-free vector wavelet analysis of nonlinear interactions in turbulence NC14—Honglu Wang, James R. Sonnenmeier, Stephan Gamard, and William K. George (USA): Evaluating DNS simulations of isotropic decaying turbulence using similarity theory ND—Fracture and crack mechanics (jointly with ICF) (Jean-Baptiste Leblond, France, chair)

ND01—Chang-Chun Wu and Zi-Ran Li; Ming Liu (China, PRC): Fracture assessment of piezoelectric material: Bound theorem and numerical implementation ND02—Tov Elperin, Gregory Rudin, and Arkady Kornilov (Israel): Formation of surface microcrack in non-metallic materials using laser thermal shock ND03—Giorgio Novati, Roberta Springhetti, and Antonio Cazzani; Attilio Frangi (Italy): 3D fracture analysis by the symmetric Galerkin boundary element method ND04—Martin L. Dunn and Paul E. W. Labossiere (USA): Fracture initiation at threedimensional bimaterial interface corners ND05—Benjamin A. Gailly; Horacio D. Espinosa (USA): A new model of ceramic based on crack propagation and motion of fragments ND06—Maria Kashtalyan and Costas Soutis ( U K ) : Multi-ply matrix cracking in multidirectional fibre-reinforced composite laminates ND07—Su Hao and Dong Qian; Patrick Klein (USA): A multi-scale damage model ND08—Jürgen Molter, Theo Fett, and Dietrich Munz; Dietrich Munz (Germany): Mixed mode fracture of interface cracks determined by a new fracture test and finite element method ND09—Michael A. Sutton, Fashang Ma, and Xiaomin Deng (USA): Mixed mode I/II cracktip fields ND10—Jerzy T. Pindera; Friedrich W. Hecker; Yuanpeng Zhang (Canada): Advanced experimental mechanics in three-dimensional stress determination ND11—Leon L. Mishnaevsky Jr. and Siegfried Schmauder; Nils Lippmann (Germany): Simulation of fracture in real structures of tool steels as a basis for the steel design NE—Contact and friction problems (jointly with IAVSD) (Anders I. Klarbring, Sweden, chair) NE01—Feodor M. Borodich (UK): Contact problems for fractal punches NE02—Ekaterina E. Pavlovskaya (Russia): Dynamic contact between rough bodies NE04—John M. Golden; G. A. C. Graham (Ireland): The problem of a viscoelastic cylinder rolling on a rigid half-space NE05—Paul M. Rightley, Robert A. Pelak, and James E. Hammerberg (USA): Interfacial dynamics of metal-on-metal sliding at high speeds and loads NE06—Kilian Funk, Andreas Stiegelmeyr, and Friedrich Pfeiffer (Germany): A compact time stepping algorithm for planar friction problems

Scientific program

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Some of the posters presented during the seminar presentation sessions on

29 August and 31 August 2000. NE07—Nikolay A. Lavrov (Russia): Dynamic interaction between elastic bodies having a slender area of contact NE08—Haralambos G. Georgiadis; Louis M. Brock; George Lykotrafitis (Greece): Thermoelastodynamic Green’s functions for problems of rapid sliding contact NE09—Faruk Yigit; Louis G. Hector Jr. (Saudi Arabia): A thermomechanical model of pure metal solidification on a deformable mold with sinusoidal surface NE10—Przemyslaw Zagrodzki (USA): Transient solutions of a thermoelastic contact problem

with fractionally excited instability NE11—Dong Qian, Shaofan Li, and Jian Cao (USA): 3D simulation of manufacturing process

by a meshfree contact algorithm NE12—Paolo Decuzzi; Giuseppe Monno (USA): The effect of disc thickness ratio on hot spotting in multiple-disc clutches and brakes NE13—Stephane Pagano, Karim Ach, Mikael Barboteu, and Pierre Alart (France): Modelling of multi-jointed structures NE14—Sirpa S. Launis and Erno K. Keskinen; Michel Cotsaftis (Finland): Mechanics of wood motion in groundwood manufacturing system NE15—Michelle Schatzman; Laetitia Paoli (France): Numerics for vibroimpact in many

degrees of freedom NF—Damage and failure of composites (Dusan P. Krajcinovic, USA, chair)

NF01—Qing-Jie Zhang, Peng-Cheng Zhai, Li-Sheng Liu, and Run-Zhang Yuan; ShinIchi Moriya and Masayuki Niino (China, PRC): Mechanics of functionally graded materials: Recent and prospective development NF02—Hodrigue Desmorat (France): Quasi-unilateral conditions in anisotropic elasticity NF03—James J. Mason and Carlos Rubio-Gonzalez (USA): Closed form solutions for dynamic stress intensity factors in composites NF06—Paolo M. Mariano, Giuliano Augusti, and Furio L. Stazi (Italy): Finite elements for a multifield model of microcracked bodies NF07—Chi L. Chow and Xianjie Yang (USA): An anisotropic damage model for forming limit diagram under nonproportional loading NF08—Isaac M. Daniel, Emmanuel E. Gdoutos, and Kuang-An Wang (USA): Failure modes in composite sandwich beams under three-point bending NF09—Anthony M. Waas and JungHyun Ahn (USA): Damage initiation and failure of notched composite laminates

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NF10—George Z. Voyiadjis; Babur Deliktas (USA): Bridging length scales from meso to macro through gradient damage NF11—Yufu Liu, Yoshihisa Tanaka, and Chitoshi Masuda (Japan): Experimental and numerical study of transverse cracking in cross-woven fiber-reinforced composites NF12—Thomas A. Godfrey; John N. Rossettos (USA): The onset of tear propagation at large slits in biaxially stressed woven fabrics NF13—Federico M. Sciammarella and Cesar A. Sciammarella; Bartolomeo Trentadue (USA): Effect of interfaces in the mechanical properties of particle reinforced rubber composites NF14—Tungyang Chen (Taiwan): Inclusions with variable interfacial parameter and its effect on the effective conductivity tensor NG—Impact and wave propagation (Anders Boström, Sweden, chair) NG01—Avraham Benatar; Alexander L. Yarin and Daniel Rittel (USA): Experimental

measurement of the dynamic moduli of polymers using an instrumented Kolsky bar NG02—Weinong W. Chen and Fangyun Lu; Danny J. Frew (USA): Challenges and novel techniques in dynamic testing of soft materials NG03—Shinichi Maruyama, Toshihiko Sugiura, Akihiro Inoue, and Masatsugu Yoshizawa (Japan): Effects of a flaw in a specimen on ultrasonic detection by an elec-

tromagnetic acoustic transducer NG04—Daniel Gsell and Jürg Dual (Switzerland): Characterization of material properties in thin structures using a finite difference approach NG05—Robert B. Haber and Lin Yin; Amit Acharya (USA): A space–time discontinuous Galerkin method for elastodynamics using implicit–explicit grids NG06—Tobias Leutenegger and Jürg Dual (Switzerland): Numerical modeling of the influence of cracks on wave propagation in cylindrical structures NG07—Miroslav Premrov (Slovenia): An iterative method for solving harmonic elastodynamics of a halfspace NG08—Lev G. Steinberg (USA): Resonance of self-excited defect bar oscillations NG09—Serge N. Gavrilov (Russia): Passage through the critical velocity by a load moving along a string: Nonlinear approach NG10—Leonid I. Manevitch (Russia): Short wavelength asymptotics in nonlinear dynamics NG11—Mihai V. Predoi and Silviu Livescu (Romania): Complex wave numbers for elastic waves at plane interfaces NG12—Bo Yan and Zhanfang Liu; Xiangwei Zhang (China, PRC): Numerical simulation of two types of body waves and Rayleigh wave in fluid-saturated porous media NG13—Joseph A. Turner and Phanidhar Anugonda (USA): Elastic wave propagation and scattering in random, two-phase materials NG14—François Hild; Christophe Denoual (France): A multi-scale model for the dynamic fragmentation of silicon carbide ceramics NH—Smart materials and structures (Yuji Matsuzaki, Japan, chair) NH01—Alief N. Yahya and Wendy C. Crone (USA): Fabrication and characterization of nanostructured NiTi NH02—Chang-Sun Hong, Chi-Young Ryu, and Chon-Gon Kim (Korea): Experiments for smart composite structures using optical fiber sensors NH03—Fumihiro Ashida; Theodore R. Tauchert (Japan): Control of a distribution of transient thermoelastic displacement in a composite circular disk NH04—Reinaldo Rodríguez-Ramos, Raúl Guinovart, Idania Urrutia, and Julián Bravo;

Federico J. Sabina (Cuba): Overall properties in multiphase composite: Applications NH05—Xisheng Cao and Ulrich Gabbert (Germany): Crack analysis of smart composites with piezoceramics NH06—Hiroshi Asanuma, Haruki Kurihara, and Genji Hakoda (Japan): Fabrication of a new actuator material based on continuous SiC fiber reinforced aluminum composite NH07—Tarak Ben Zineb, Shabnam Arbab Chirani, and Etienne Patoor (France): Behavior modeling of piezoelectric ceramics NH08—Xianwei Zeng and R. K. Nimal D. Rajapakse (Canada): Toughening of conducting cracks due to domain switching NH09—Mitsunori Denda; Yook-Kong Yong (USA): Two-dimensional piezoelectric, timeharmonic dynamic BEM for quartz surface wave resonators NH10—Christophe Niclaeys, Tarak Ben Zineb, Shabnam Arbab Chirani, and Etienne

Patoor (France): Interaction between groups of variants in shape memory alloys NH11—Shabnam Arbab Chirani and Etienne Patoor (France): Martensite transformation criterion in shape memory alloys NH12—George J. Weng; Jackie Li (USA): Martensitic transformation of shape-memory alloys

and domain switch of ferroelectric ceramics

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NH13—Hungyu Tsai and Xinjian Fan (USA): Elastic deformations in shape memory alloy fiber reinforced composites NK—Convective phenomena, fluid mechanics of materials processing, and microgravity mechanics (Simon S. Ostrach, USA, chair) NK01—Pierre Carlès (France): The convective stability of supercritical fluids NK02—Sakir Amiroudine; Andrei B. Kogan and Horst Meyer; Bernard Zappoli (Guadeloupe FWI): Convection onset in a near-critical fluid: Comparison between experiment and numerical simulation NK03—Alexander Roxin and Hermann Riecke (USA): Stability of waves coupled to an advected slow field NK05—Alexander A. Nepomnyashchy and Ilya B. Simanovskii; Leonid M. Braverman (Israel): Phenomenon of anticonvection NK06—Naftali A. Tsitverblit (Israel): Bifurcation from infinity in double-component convection driven by different boundary conditions NK07—Cho Lik Chan, Chuan F. Chen, and Wen-Yau Chen (USA): Salt-finger instability generated by lateral heating of a solute gradient NK08—Robert M. Kerr, Eileen M. Saiki, and William G. Large (USA): Numerical simulation of three-dimensional thermohaline fingering NK09—Haruhiko Kohno and Takahiko Tanahashi (Japan): Three-dimensional numerical simulation of unsteady Marangoni flows using GSMAC finite element method NK10—Gennady F. Putin, Igor A. Babushkin, Gennady P. Bogatyrev, and Alexander F. Glukhov; Sergey V. Avdeev, Alexander I. Ivanov, and Marina M. Maksimova (Russia): Measurement of thermal convection and low-frequency microaccelerations aboard orbital station Mir NK11—Haïk Jamgotchian, Nathalie Bergeon, Dominique Benielli, Philippe Voge, and Bernard Billia (France): Transient convective modes during solidification of succinonitrile0.2 wt % acetone alloy NK12—Valeriya A. Brailovskaya and Ludmila V. Feoktistova; Victor V. Zilberberg (Russia): Numerical modelling of nonstationary forced convection in solution at high rate crystal growth NK13—Lev A. Slobozhanin and J. Iwan D. Alexander (USA): The stability margin for zero-gravity liquid bridges NK14—Vadim I. Polezhaev, Oleg A. Bessonov, and Sergei A. Nikitin (Russia): Microgravity mechanics: A bridge between spacecraft’s accelerations and gravity-dependent systems NL—Computational Solid mechanics (jointly with IACM) (Subrata Mukherjee, USA, chair)

NL01—Elio Sacco; Daniela Addessi; Sonia Marfia (Italy): An elasto–plastic nonlocal damage model NL02—Eric Lorentz (France): Constitutive relations with gradient of internal variables NL03—Thomas Warren; Kevin L. Poormon (USA): Penetration of aluminum targets with spherical-nose steel projectiles at oblique angles NL04—Yasser M. Shabana; Naotake Noda (Japan): Thermal stresses in functionally graded materials taking the fabrication process into consideration NL05—Christof Messner and Gerd Reisner; Ewald Werner (Germany): Autocatalysis in the propagation of martensite bands in a shape memory alloy specimen under tension NL07—Magnus Ekh, Kenneth Runesson, and Lennart Mähler (Sweden): Thermoviscoplasticity for porous solids with application to sintering processes NL08—Xiaomin Deng and Chandrakanth Shet (USA): A finite element analysis of residual stresses and strains in orthogonal metal cutting NL09—Sylvie Aubry, Michael Ortiz, and Eduardo A. Repetto (USA): Computation of microstructure

NL10—Laurens P. Evers, W. A. Marcel Brekelmans, and Frank P. T. Baaijens; David M. Parks (The Netherlands): Meso-scale modeling of the elasto-viscoplastic behavior of polycrystal FCC metals NL11—Schalk Kok, Armand J. Beaudoin, and Daniel A. Tortorelli; Paul J. Maudlin (USA): Parameter estimation and texture augmentation of a polycrystal model through identification studies NL12—Claus Oberste-Brandenburg (Germany): Simulation of phase-transformations in steels using tensorial transformation kinetics NL13—Issam Doghri, Svetoslav Nikolov, Serge Munhoven, and Olivier Pierard (Belgium): Micromechanics of the small-strain behavior of polyethylene NM—Functionally graded materials, porous materials, phase transformations, and thin

films (Glaucio H. Paulino, USA, chair) NM01—Joseph Nadeau (USA): Sensitivity of optimal functionally graded systems to variations

in design parameters

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NM02—Marek-Jerzy Pindera; Jacob Aboudi; Steven M. Arnold (USA): Thermal barrier coating stress management using graded bond coat architectures: A critical higher-order theory analysis NM03—Magnus Jinnestrand; Sören Sjöström (Sweden): Stress state in thermal barrier coatings from 3D finite element simulations NM04—Gregory Rudin and Tov Elperin (Israel): Thermal test of functionally graded materials using laser thermal shock method NM05—Ryan S. Elliott, John A. Shaw, and Nicolas Triantafyllidis (USA): Stability of thermally loaded NiTi perfect crystals NM06—Qing-Ping Sun (Hong Kong): On nucleation and propagation of martensite band in superelastic NiTi memory alloys under tension NM07—Christian Hellmich and Herbert A. Mang; Franz-Josef Ulm (Austria): Thermochemomechanics of phase transitions in shotcrete NM08—Toshio Tsuta, Yajun Yin, and Takeshi Iwamoto (Japan): Ductility and micro damage analyses in rigid plastic porous material with mixed hardening property NM09—Karim Beddiar, Yves Berthaud, and André Dupas (France): Thermo-hydroelectromechanics coupling: Application to electro-osmosis in porous media NM10—Robb M. Winter; Jack E. Houston (USA): Nanomechanical properties of polymeric systems measured by interfacial force microscopy NM11—Horacio D. Espinosa and Maximiliano Fischer (USA): Identification of residual stress state in a radio-frequency MEMS device NM12—J. Gregory Swadener; George M. Pharr (USA): Effects of surface stress state on nanoindentation measurements NM13—Sven Strohband and Reinhold H. Dauskardt (USA): Interface debonding under mixed mode loading in the presence of bridging tractions NM14—Takuya Uehara and Tatsuo Inoue (Japan): Molecular dynamics simulations of nucleation and crystallization processes in multi-layer thin films NN—Vehicle dynamics and control of Structures (Dick H. van Campen, The Netherlands,

chair) NN01—Gregory M. Hulbert and Fan-Chung Tseng (USA): A manifold orthogonal projection method for constrained dynamical systems NN02—Tony Postiau and Jean-Didier Legat; Paul Fisette (Belgium): Generation and parallelization of multibody system equations of motion by a symbolic approach NN03—Vaasilios Valtetsiotis; Robin S. Sharp (Germany): Car driving as an optimal linear preview control problem

NN05—All Haj-Fraj and Friedrich Pfeiffer (Germany): Optimization of gear shift operations in automatic transmissions NN06—Alex F. A. Serrarens, Roell M. van Druten, and Bas G. Vroemen (The Netherlands): Mechanical motor assist applied to a vehicle powertrain NN07—Chih-Yu Kuo, Regina A. G. Graf, Ann P. Dowling, and William R. Graham (UK): On the horn effect of a tyre–road interface NN08—Chijoke O. Mgbokwere (USA): Quenching of a ring gear blank NN09—Jan A. Snyman and Willie J. Smit (South Africa): The optimal design of a planar Stewart platform for prescribed machining tasks NN10—Jens Bormann and Heinz Ulbrich (Germany): Active vibration isolation of a hexapod system (Stewart platform) NN11—Anupam S. Ahlawat and Ananth Ramaswamy (India): Design of an optimal hybrid control system for wind excited buildings NN12—Daniil V. Iourtchenko and Mikhail F. Dimentberg; Alexander S. Bratus’ (USA): Optimal control of random vibrations by bounded stiffness variations NN13—Michael J. Winckler and Irena Otasevic (Germany): Fast and reliable computation of sensitivities for valvetrain design using ODEOPT NN14—Katja Dauster and Friedrich Pfeiffer (Germany): Observation of robotic manipulators in automated assembly NN15—Raphael R. Burgmair and Friedrich Pfeiffer (Germany): Modelling and simulation of a rubber seal assembly NS—Viscoelasticity, Creep, and fatigue (Nobutada Ohno, Japan, chair)

NS01—Marina V. Shitikova and Yuriy A. Rossikhin (Russia): A new method for solving dynamic problems of fractional calculus viscoelasticity NS02—Carl R. Schultheisz; Gregory B. McKenna (USA): Measurement and analysis of torque, axial force and volume change in the NIST torsional dilatometer NS03—Harry H. Hilton and Cristina E. Beldica (USA): Piezoelectric control of deformations and failure probabilities in viscoelastic Timoshenko beams

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NS04—Ting-Qing Yang, Qunli An, and Wenbo Luo (China, PRC): Craze growth at a crack tip in polymers under creep condition NS05—Koichi Goda (Japan): Semi-discretization analysis for shear lag and creep rupture by the break-influence superposition method NS06—Aleksandra M. Vinogradov, Shane Schumacher, and Iakov M. Klebanov (USA): Creep response of polymers under cyclic loading conditions NS07—Christopher H. M. Jenkins and Zhiyu Liu; Isamu Kitahara and Robb M. Winter; Shane Schumacher and Aleksandra M. Vinogradov (USA): Morphology evolution in polymers subjected to vibrocreep NS08—Pranav Shrotriya and Nancy R. Sottos (USA): Influence of fabric architecture on viscoelastic properties of woven composites NS09—Qi Zhu and Philippe H. Geubelle (USA): Process-induced residual stresses and warpage in polymer composites NS10—Bong-Kyu Kim and Sung-Kie Youn (Korea): A constitutive model of rubber under small vibrations superimposed on large static deformation NS11—David A. Hills; Ludwig Limmer (UK): A theoretical and experimental investigation of fretting fatigue of complete contacts NS12—Richard W. Neu, Dana R. Swalla, and John A. Pape (USA): Plastic deformation in fretting fatigue: Experiments and modeling NS14—Thomas Erber; Sidney A. Guralnick (USA): Novel measurements of damage accumulation and the prediction of fatigue failure

NV—Combustion, compressible flow, and computational fluid dynamics (John Brindley, UK, chair) NV01—Moshe Matalon; Philippe Metzener (USA): Premixed flames in closed cylindrical tubes

NV02—Andy C. Mclntosh and Xinshe Yang; John Brindley (UK): The effect of large step pressure drops on strained premixed flames

NV03—Chun W. Choi and Ishwar K. Puri (USA): Flame stretch effects on partially premixed flames

NV04—Dieter E. Bohn and Joachim Lepers (Germany): Simulation of premixed combustion in highly turbulent flows: Comparison of a joint PDF and an eddy break-up model

NV05—Andrzej P. Szumowski; Gerd E. A. Meier (Poland): Airfoil flow instabilities induced by background flow oscillations NV06—Egon Krause; M. Kharitonov and M. D. Brodetsky (Germany): Supersonic lee-side flow on delta wings NV07—Takashi Adachi and Susumu Kobayashi (Japan): On the transition angle from Mach to regular reflection over concave surface of arbitrary shapes NV08—Piotr P, Doerffer, Jaroslaw Kaczynski, Krystyna Namisnik, and Ryszard Szwaba

(Poland): Flow structure at the interaction of oblique and normal shock waves NV09—Gerd E. A. Meier and Markus Raffel (Germany): Optical flow diagnostics by background oriented schlieren methods NV10—Vit Dolejsi (Czech Republic): Anisotropic mesh adaptation method for viscous compressible flows NV11—Jayandran Palaniappan, Robert B. Haber, and Robert D. Moser; Daniel A. Tortorelli (USA): A space–time discontinuous Galerkin procedure for nonlinear conservation laws NV12—Serge M. Prudhomme (USA): A goal-oriented adaptive strategy to enhance local structures in numerical flows NV13—Noboyuki Satofuka and Mitsuru Ishikura (Japan): Large scale numerical simulation of three-dimensional duct flows using lattice Boltzmann method

OA—Microfluid dynamics (Leen van Wijngaarden, The Netherlands, chair) OAl—John R. Melrose (UK): The transmission of force in flowing colloids OA2—Andreas Acrivos and Marco Marchioro (USA): Shear-induced particle diffusivities from numerical simulations OA3—Jason E. Butler and Eric S. G. Shaqfeh (USA): Dynamic simulations of sedimenting, rigid fibers OB—Biological f l u i d dynamics (Kazuo Tanishita, Japan, chair) OB1 —Hao Liu (Japan): A computational fluid dynamic study of the helical flow in aorta OB2—Olivier Boiron, Valérie Deplano, Boris Wilbois, and Robert Pelissier (France): A numerical and experimental study of physiological flows in a curved tube OB3 —Frans N. van de Vosse, Marco M. Stijnen, and Frank P. T. Baaijens (The Netherlands): Computational and experimental investigation of the flow in the left ventricle

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ICTAM 2000

OC—Turbulent m i x i n g (Anthony Leonard, USA, chair) OC1—Robert E. Breidenthal (USA): Super-exponential Rayleigh–Taylor flow OC2—Yasuhiko Sakai; Shinichi Nakajima; Ikuo Nakamura; Hiroyuki Tsunoda (Japan): The statistical analysis of two-particle structural diffusion in homogeneous isotropic turbulence OC3—William M. Pitts; Cecilia D. Richards; Mark S. Levenson (USA): Interactions between large- and small-scale structures in the scalar mixing field of a turbulent jet OD—Boundary layers (Anatoly I. Ruban, UK, chair) OD1—Yury S. Kachanov (Russia): On a universal nonlinear mechanism of breakdown to turbulence in wall bounded shear flows OD2—Scott E. Hommema and Ronald J. Adrian (USA): Structure of wall-eddies at

106

OD3—Jens M. Österlund and Arne V. Johansson; Hassan M. Nagib and Michael H. Hites (Sweden): Spectral characteristics of the overlap region in turbulent boundary layers

OE—Elasticity (Thomas J. Pence, USA, chair) OE1—Alexander N. Guz’ (Ukraine): On foundation of non-destructive method of determination of three-axial stresses in solids OE2—David M. Barnett (USA): One-component surface waves and supersonic non-radiating defects in triclinic linear elastic solids OE3—Ecevit Bilgili, Barry Bernstein, and Hamid Arastoopour (USA): Boundary layer formation in a sheared rubber-like slab subjected to thermal gradients

OF—Damage mechanics (Davide Bigoni, Italy, chair) OF1—Dusan P. Krajcinovic (USA): Statistical damage mechanics OF2—Quanshui Zheng and Danxu Du (China, PRC): An explicit and universal estimate for effective properties of composites and microcracked media OF3—Ron H. J. Peerlings, Marc G. D. Geers, and W. A. Marcel Brekelmans; René de Borst (The Netherlands): A comparative study of nonlocal and gradient-enhanced continuum damage models OG—Mechanics of thin films and nanostructures (Bertil Storåkers, Sweden, chair) OG1—Simon P. A. Gill, Fei Long, and Alan C. F. Cocks (UK): A continuum model for the growth and relaxation of multi-component epitaxial thin film systems OG2—Vijay Shenoy; L. Ben Freund (India): Compositional stability of alloy films OG3—Michael Ortiz, Eduardo A. Repetto, Kaushik Bhattacharya, and Yi-Chung Shu (USA): A continuum model of kinetic roughening and coarsening in thin films OH—Fatigue (Keisuke Tanaka, Japan, chair) OH1—Yukitaka Murakami (Japan): Mechanism of superlong fatigue failure in Nf ³ 107 and disappearance of conventional fatigue limit OH2—Ryuichiro Ebara (Japan): Influence of hydrogen on fatigue strength of structural steels OH3—Kwai S. Chan, Yi-Der Lee, David L. Davidson, and Stephen J. Hudak (USA): Propagation of small cracks under high cycle fretting fatigue

OK—Fluid-Structure interaction (Michael P. Païdoussis, Canada, chair) OK1—Anatol Roshko, Anthony Leonard, and Doug Shiels (USA): Flow induced vibration of a circular cylinder: A unified description OK2—Noel C. Perkins and Wanjun Kim (USA): Coupled slow and fast dynamics of flow excited cables OK3—Morten Huseby and John Grue (Norway): Experiments on higher harmonic wave forces on a vertical cylinder in periodic waves

OL—Drops and bubbles (Carlo Cercignani, Italy, chair) OL1—Grétar Tryggvason and Bernard Bunner (USA): Direct numerical simulations of bubbly flows OL2—Han E. H. Meijer, Frans N. van de Vosse, and Maykel Verschueren (The Netherlands): On the scaling of diffuse interface models for multiphase flows OL3—James P. Ferry; S. Balachandar (USA): A fast Eulerian method for particle-laden flows

OO—Electromagnetic processing of materials (jointly with HYDROMAG) (Peter A. Davidson, UK, chair) OO1—Martin P. Volz and Konstantin Mazuruk (USA): The effect of a rotating magnetic field on flow stability during crystal growth OO2—Hiroyuki Ozoe and Koichi Kakimoto; Masato Akamatsu; Yoo Cheol Won (Japan): Application of various magnetic fields for the melt in a Czochralski crystal growing system

Scientific program

lxxiii

OO3—Jochen Friedrich and Oliver Gräbner; Bernd Fischer, Daniel Vizman, and Georg

Müller (Germany): Influence of magnetic fields on convection in Czochralski melts: Experimental and numerical results OS—Mechanics of phase transformations (jointly with IACM) (Franz-Josef Ulm, USA, chair) OS1—Franz-Josef Ulm; Luc Dormieux and Eric Lemarchand (USA): A micromechanical approach to the modeling of swelling due to alkali–silica reaction OS2—Gustavo Gioia; Alberto M. Cuitiño ( U S A ) : Two-phase particle rearrangement in cohesive powder densification OS3—Christian Lexcellent and Christophe Bouvet; Sylvain Calloch (France): Experimental investigations under multiaxial loadings on Cu–Al–Be shape memory alloy OW—Turbulence (William K. George, USA, chair) OW1—Tomomasa Tatsumi (Japan): The velocity distributions of homogeneous turbulence under the cross-independence closure hypothesis OW2—Benoit M. Chabaud, Bernard Hebral, Sylvain Pietropinto, and Philippe Roche; Bernard Castaing and Yves Ladam (France): Very high Reynolds and Rayleigh numbers experiments in low temperature gaseous helium OW3—Joseph J. Niemela, Ladislav Skrbek, and Russell J. Donnelly; Katepalli R. Sreenivasan (USA): Cryogenic turbulent convection PA—Microfluid dynamics (Leen van Wijngaarden, The Netherlands, chair) PA1—Todd M. Squires; Eric Dufresne and David G. Grier; Michael P. Brenner (USA): Like charge attraction and hydrodynamic interaction PA2—Mohamed Gad-el-Hak (USA): Flow physics in microdevices PA3—Marion N. Volpert, Carl D. Meinhart, Igor Mezic, and Mohammed Dahleh (USA): An actively controlled micromixer PB—Biological fluid dynamics (Kazuo Tanishita, Japan, chair) PB1—Shigeo Wada, Makoto Kojiya, and Takeshi Karino (Japan): Computational study of LDL transport in stenosed arteries based on a true-to-scale anatomical model PB2—Sylvie Lorthois; Francis Cassot and Jean-Pierre Marc-Vergnes; Pierre-Yves Lagree (France): Maximal wall shear stress in carotid stenoses and functionality of the

circle of Willis PB3—Jennifer S. Stroud and Stanley A. Berger (USA): Numerical simulation of blood flow in the severely stenotic carotid artery bifurcation PC—Turbulent mixing (Anthony Leonard, USA, chair) PCI—Daniel I. Meiron; Paul E. Dimotakis, Ronald Henderson, Branko Kosovic, Dale I. Pullin, and Ravi Samtaney (USA): Direct numerical simulation and subgrid-scale modeling of compressible turbulent mixing PC2—Qing-Zeng Feng (China, PRC): Inertial range scaling of passive scalars mixed by turbulence PC3—Amsini Sadiki (Germany): A new approach to modeling of turbulent mass and heat transport based on extended thermodynamics

PD—Granular flows (Prabhu R. Nott, India, chair) PD1—James T. Jenkins and Birgir Ö. Arnarson (USA): A theoretical analysis of a laboratory debris flow PD2—Joe D. Goddard and Anjani Kumar Didwania (USA): Fluid mechanics of vibrated

granular layers PD3—Michael Shapiro, Victor Royzen, Alexander Goldshtein, and Vladislav Dudko (Israel): Vibrofluidization and mixing in vibrated granular layers: From repacking to doubleimpact regimes

PE—Fracture and crack mechanics (jointly with ICF) (K. Jimmy Hsia, USA, chair) PEl—John R. Willis (UK): Dynamic perturbation of a crack propagating in a viscoelastic medium PE2—Alexander Chudnovsky; Boris Nuller; Michael Ryvkin (USA): Problem of crack interaction with bimaterial interface revisited PE3—Alexander B. Movchan ( U K ) : Asymptotic models of multi-structures with cracks PF—Material instabilities (Davide Bigoni, Italy, chair) PF1 —Kristina Nilsson (Sweden): Effects of inertia on dynamic neck formation during highrate tension PF2—Niels Sorensen (Sweden): Unstable neck formation in impact problems PF3—John L. Bassani (USA): Strain localization in ductile crystals

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ICTAM 2000

PG—Plasticity and viscoplasticity (Alan C. F. Cocks, UK, chair) PGl—João A. C. Martins and Fernando M. F. Simoes (Portugal): Flutter instabilities in non-associative elastic–plastic solids PG2—Nicolae D. Cristescu and Oana Cazacu (USA): Compaction and flow of granular materials on chutes PG3—Ann Bettina Richelsen; Erik van der Giessen (Denmark): Size effects of plasticity in thin sheets

PH—Fatigue (Keisuke Tanaka, Japan, chair) PH1—Reinhard Pippan; Franz O. Riemelmoser (Austria): The effect of discrete nature of plasticity on the fatigue crack propagation behavior PH2—Toshihiko Hoshide (Japan): Statistical aspect on simulated failure life of notched components in multiaxial fatigue PH3—Thomas N. Farris and Ganapathy Harish (USA): Coupled thermoelastic evaluation of fretting stresses: Application to life prediction

PK—Fluid–Structure interaction (Michael P. Païdoussis, Canada, chair) PK1—Sergey V. Sorokin (Russia): Nonlinear vibrations of elastic thin-walled structures in heavy fluid loading conditions PK2—Anthony D. Lucey, Gerard J. Cafolla, and Peter W. Carpenter ( U K ) : Evolution of flexible-wall disturbances in a boundary-layer flow PK3—Marco Amabili; Francesco Pellicano; Michael P. Païdoussis (Italy): Nonlinear dynamics and stability of circular cylindrical shells with flow P L — M u l t i p h a s e flows (Carlo Cercignani, Italy, chair) PL1—Dmitri O. Pushkin and Hassan Aref (USA): Self-similarity theory of stationary coagulation PL2—Ehud Yariv; Itzchak Frankel (Israel): Kinetic theory for bubbly flows PL3—Élisabeth Guazzelli, Paul Duru, and Maxime Nicolas; E. John Hinch (France): Constitutive laws in liquid fluidized beds

PO—Electromagnetic processing of materials (jointly with HYDROMAG) (Peter A. Davidson, UK, chair) POl—Toshio Tagawa; René J. Moreau and Guillaume Authié (Japan): Buoyant flow in long vertical enclosures under the presence of a horizontal magnetic field PO2—Valdis Bojarevics and Koulis Pericleous; Janis Freibergs (UK): Nonlinear wave models for aluminum reduction cells PO3—Ulrich Burr; Ulrich Müller (Switzerland): Rayleigh–Bénard convection under the influence of magnetic fields PS—Mechanics of porous materials (Wolfgang Ehlers, Germany, chair) PS1—Anjani Kumar Didwania; Reint de Boer (USA): Capillarity in porous solids—a continuum thermo-mechanical approach PS2—Martti J. Mikkola and Juha Hartikainen (Finland): Finite element solution of soil freezing problem by using bubble functions PS3—Pietro Cornetti, Alberto Carpinteri, and Bernardino M. Chiaia (Italy): Scaling laws in the mechanics of porous and damaged materials

PW—Flow instability and transition (William K. George, USA, chair) PW1—Stephane Leblanc (France): Destabilization of a vortex by acoustic waves PW2—Gilles Bouchet and Eric Climent; Agnès Maurel (France): An experimental investigation of free surface instabilities induced by an upward impinging jet PW3—Franck Plouraboué; E. John Hinch (France): Kelvin–Helmholtz instability in a HeleShaw cell QA—Turbulence (Hassan M. Nagib, USA, chair) QA1—Thomas S. Lundgren (USA): The pdf of the velocity difference between two points in homogeneous isotropic turbulent flow QA2—Susumu Goto and Shigeo Kida (Japan): A turbulence closure theory based on sparseness of nonlinear coupling QA3—Javier Jiménez (Spain): Imperfect multiplicative cascades in turbulence QB—Biological fluid dynamics (John O. Kessler, USA, chair) QB1—Matthias Heil and Joseph P. White (UK): Time-dependent non-axisymmetric instabilities of liquid-lined elastic tubes QB2—Sarah L. Waters; Caterina Guiot ( U K ) : Hæmodynamic aspects of the umbilical circulation

Scientific program

lxxv

QB3—Poul S. Larsen; Hans Ulrik Riisgård (Denmark): Chimney spacing in encrusting bryozoan colonies

QC—Electromagnetic processing of materials (jointly with HYDROMAG) (J. Iwan D.

Alexander, USA, chair) QC1—Nagy El-Kaddah; Thinium Natarajan; Ashish Patel (USA): Theoretical study of the magnetohydrodynamic flow around a sphere in crossed electric and magnetic fields QC2—Behrouz Abedian and Livia M. Racz; Roberts W. Hyers; Gerardo Trapaga (USA): Transition to turbulence in an electromagnetically levitated droplet QC3—Ramesh K. Agarwal (USA): A novel algorithm for computing combined free- and forced

convection magnetohydrodynamic flows QD—Fracture and crack mechanics (jointly with ICF) (John R. Willis, UK, chair) QDl—Leonid I. Slepyan (Israel): Distinctive solutions for fracture and phase transition in lattices QD2—Thomas H. Siegmund (USA): Local versus global failure: Mechanical integrity of thin walled structures QD3—John E. Dolbow; Nicolas Moes (USA): Modeling crack growth and fractional contact

with the extended finite element method QE—Functionally graded materials (Marek-Jerzy Pindera, USA, chair) QE1—Cornelius O. Horgan; Alice M. Chan (USA): Boundary-value problems for functionally graded linearly elastic materials QE2—Glaucio H. Paulino and Zhi-He Jin (USA): Viscoelasticity theory of functionally graded materials QE3—Fazil Erdogan, Serkan Dag, and M. A. Guler (USA): Contact and crack problems in functionally graded materials QF—Control of structures (Ali H. Nayfeh, USA, chair) QF1—Hans Irschik and Uwe Pichler (Austria): Dynamic deformation control of structures

by shaped piezoelectric actuators QF2—Martin Ruskowski and Karl Popp; Hans-Kurt Tönshoff and Richard Kaak (Germany): Modal control of a magnetically guided machine axis QF3 —Jon J. Thomsen; Dmitri M. Tcherniak (Denmark): Is Chelomei’s pendulum a beam? QG—Continuum mechanics (Donald E. Carlson, USA, chair) QG1—Francis J. Rooney, Stephen E. Bechtel; Qi Wang; M. Gregory Forest (USA): Models for thermal expansion: Ill-posedness of constrained theories and constitutive limits QG2—Gérard A. Maugin (France): Universality of the thermomechanics of forces driving singular sets in continuum mechanics QG3—James Casey (USA): On internally constrained thermoplastic continua QH—Smart materials and structures (Richard D. James, USA, chair) QH1—Yuji Matsuzaki (Japan): Thermomechanical studies of shape memory alloys based on interaction energy function QH2—Mark A. ladicola and John A. Shaw (USA): Parametric study of localized thermo-

mechanical behavior in shape memory alloy wire QH3—L. Catherine Brinson; Michele Brocca and Zdenek P. Bazant (USA): Three dimensional constitutive model for shape memory alloys based on microplane model QK—Drops and bubbles (Lev Shemer, Israel, chair) QK1 —Antoine W. G. de Vries and A. Biesheuvel; Leen van Wijngaarden (The Netherlands): Relation between path and wake of free rising and bouncing bubbles QK2—Charles D. Eggleton; Tse Min Tsai; Kathleen J. Stebe (USA): Tip streaming from a drop in the presence of surfactants QK3—Olga M. Lavrenteva, Alexander M. Leshansky, and Avinoam Nir (Israel): Effects of unsteady convective transport on the thermocapillary interaction of drops and bubbles

QL—Computational solid mechanics (jointly with IACM) (Robert B. Haber, USA, chair) QLl—Roman Lackner, Jürgen Macht, Christian Hellmich, and Herbert A. Mang (Austria): Adaptive analysis of shotcrete shells QL2 —Subrata Mukherjee and Mandar K. Chati; Glaucio H. Paulino (USA): Adaptive meshless boundary node methods for three-dimensional problems QL3—Ted Belytschko, Jingxiao Xu, Hao Chen, and Jack Chessa (USA): Meshfree methods and level sets

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ICTAM 2000

QO—Fluid–Structure interaction (John Grue, Norway, chair) QO1—Dean T. Mook and Ali H. Nayfeh; Sergio Preidikman; Benjamin D. Hall (USA): Nonlinear modeling of unsteady aeroelastic behavior QO2—Earl H. Dowell and Deman Tang (USA): Effects of steady angle of attack on nonlinear flutter of a delta wing plate model QO3—Yosuke Watanabe and Nobumasa Sugimoto (Japan): Nonlinear flexural waves on an elastic beam traveling along its axis in an air-filled tube QR—Compressible flow (Jean-Paul Bonnet, France, chair) QR1—Andrew J. Szeri and Hao Lin (USA): Shock formation in the presence of entropy gradients QR2—Francesco Grasso and Sergio Pirozzoli (Italy): Interactions of shock waves with compressible vortex pairs

QR3—Jeff Eldredge, Tim Colonius, and Anthony Leonard (USA): Particle methods for compressible flows QS—Solid mechanics in manufacturing (Tatsuo Inoue, Japan, chair) QS1—Vinay S. Rao; Krishna Garikipati (USA): Mathematical modeling of silicon oxidation QS2—Wojciech K. Nowacki and Nguyen Huu Viem (Poland): Strain localization in dynamic simple shear at high strain rates QS3—Louis G. Hector Jr.; John A. Howarth; Woo-Seung Kim (USA): Mold microgeometry effect on contact pressure evolution in pure metal solidification QV—Complex and smart fluids (Roger I. Tanner, Australia, chair) QV1—Boris Khusid; Andreas Acrivos and Anne D. Dussaud (USA): Particle separation in a flowing suspension subject to high-gradient electric fields QV2—Stephen K. Wilson, Andrew B. Ross, and Brian R. Duffy (UK): Thin-film flow of a viscoplastic material round a large horizontal cylinder QV3—Patrick Bontoux and Isabelle Raspo; Sakir Amiroudine; Bernard Zappoli (France): Rayleigh-Benard instability in a near critical fluid RA—Flow control (Hassan M. Nagib, USA, chair) RAl—Dale I. Pullin (USA): Large-eddy simulation of passive-scalar mixing by the stretchedvortex subgrid stress model RA2—Sergei V. Manuilovich (Russia): Semi-active control of laminar–turbulent transition in three-dimensional boundary layer RA3—John C. Lin and Chung-Sheng Yao; Steven M. Klausmeyer (USA): Micro-vortex generators for flow separation control and high-lift performance enhancement RB—Biological fluid dynamics (John O. Kessler, USA, chair) RB1—Takeshi Sugimoto (Japan): Aeroelastic reason for asymmetry in primary feathers of flying birds RB2—Michel Versluis, Anna von der Heydt, and Detlef Lohse; Barbara Schmitz (The Netherlands): On the sound of snapping shrimp: The collapse of a cavitation bubble RB3—Junji Seki (Japan): Pulse wave propagation in microvessels studied by a dual laserDoppler anemometer microscope RC—Fluid mechanics of materials processing (J. Iwan D. Alexander, USA, chair) RC1—Tomasz A. Kowalewski and Andrzej Cybulski; Janusz Szmyd and Marek Jaszczur (Poland): Experimental and numerical investigations of buoyancy driven instability in a vertical cylinder RC2—Shoichiro Nakamura, Robert S. Brodkey, and Yang Zhao (USA): Investigation of mixing processes by numerical and experimental approaches

RC3—Markus Scholle, Nuri Aksel, and Andreas Wierschem (Germany): Gravity driven film flows in the presence of side walls, edges and wavy bottom profiles RD—Fracture and crack mechanics (jointly with ICF) (John R. Willis, UK, chair) RD2—Roberta Massabò; Brian N. Cox (Italy): Mixed mode delamination and large scale bridging RD3—Erin Iesulauro, Anthony R. Ingraffea, Paul A. Wawrzynek, and Sanjay R. Arwade (USA): Simulation of grain boundary decohesion and crack propagation in aluminum microstructure models RE—Functionally graded materials (Marek-Jerzy Pindera, USA, chair) RE1—Hareesh V. Tippur and Carl E. Rousseau (USA): Evaluation of dynamic crack-tip f i e l d s and fracture parameters in functionally graded materials

Scientific program

lxxvii

RE2—Rowland M. Cannon, Terrence L. Becker, and Robert O. Ritchie (USA): Statistical fracture modeling of functionally graded materials RE3—Antonios E. Giannakopoulos and Subra Suresh (USA): Contact-damage resistant graded surfaces

RF—Vehicle systems dynamics (jointly with IAVSD) (Ali H. Nayfeh, USA, chair) RFl—Daniele Casanova and Robin S. Sharp; Pat Symonds ( U K ) : On minimum time opti-

mization of Formula One cars: The influence of vehicle mass RF2—Philipp Heinzl, Manfred Plöchl, and Peter Lugner (Austria): Different actuation strategies for a yaw moment control of a passenger car RF3—Axel Fritz (Germany): Lateral and longitudinal control of a vehicle convoy

RG—Continuum mechanics (Donald E. Carlson, USA, chair) RG1—Marcelo Epstein (Canada): Material geometry of functionally graded bodies RG2—Paul Steinmann, Dirk Ackermann, and Franz-Josef Barth (Germany): Theory and computation of material forces in geometrically nonlinear fracture mechanics RG3—Samuel Forest (France): Strain localization phenomena as elementary deformation mechanism of different materials classes

RH—Smart materials and structures (Richard D. James, USA, chair) RH1—S. Mark Spearing and Catherine L. Sanders; Michael C. Shaw (USA): The use of piezo-mechanical loading to investigate the thermomechanical fatigue of solder joints RH2—Eric N. Brown and Nancy R. Sottos; Scott R. White and Philippe H. Geubelle; Jeffrey S. Moore (USA): Self-heating composites using embedded microspheres: Fracture toughness of epoxy matrix RH3—Sathya V. Hanagud, Steven Griffin, and Maxime Bayon de Noyer (USA): Active structural control for smart violins and guitars RK—Waves (Lev Shemer, Israel, chair) RK1—Chiang C. Mei and Zhenhua Huang (USA): Wind, waves and currents in shallow seas RK2—W. Kendall Melville, Fabrice Veron, and Christopher White (USA): Coherent structures and turbulence under breaking waves RK3—Alexander B. Ezersky and Vladislav V. Papko; Innocent Mutabazi (Russia): Laboratory modeling of impurity transport in a bottom boundary layer of surface waves

RL—Computational solid mechanics (jointly with IACM) (Robert B. Haber, USA, chair) RL1—David Littlefield, J. Tinsley Oden, and Serge M. Prudhomme (USA): A posteriori error estimation for highly nonlinear impact problems RL2—Tina Liebe, Andreas Menzel, and Paul Steinmann (Germany): Computational aspects of higher gradient softening materials RL3—Natarajan Sukumar and Brian Moran (USA): Natural neighbor Galerkin methods

RO—Fluid–Structure interaction (John Grue, Norway, chair) RO1—W. Geraint Price and Jing Tang Xing (UK): A mixed finite element–finite difference method for nonlinear fluid–solid interaction dynamics RO2—Allan Larsen (Denmark): Experimental and numerical study of fluid–structure interaction of an H-shaped cross section RO3—Jørgen J. Jensen, Preben T. Pedersen, and Peter F. Hansen (Denmark): Hydroelastically induced ship hull vibrations

RR—Compressible flow (Jean-Paul Bonnet, France, chair) RR1—Victor V. Golub, Tatiana V. Bazhenova, and Tatiana A. Bormotova (Russia): Post diffracted shock waves 3-D flow RR2—Max Elena and Joel Deleuze (France): Some turbulence characteristics in a shock wave-boundary layer interaction RR3—Bruno Auvity, Mark B. Huntley, and Alexander J. Smits; Michael R. Etz (USA): Boundary layer control at Mach 8 using helium injection

RS—Solid mechanics in manufacturing (Tatsuo Inoue, Japan, chair) RS1—Friedrich Pfeiffer and Günther Prokop (Germany): Optimization of assembly processes with manipulators RS2—Witold Gutkowski; Krzysztof Dems (Poland): Manufacturing tolerances and multiple loading conditions in structural configuration optimization RS3—Claus Ropers and Eckart Doege (Germany): Finite element simulation of sheet metal forming processes considering the elastic nature of the tools

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ICTAM 2000

RV—Low-Reynolds-number flow (Roger I. Tanner, Australia, chair) RV1—Dudley A. Saville and John R. Glynn Jr. (USA): Electrohydrodynamics in non-

homogeneous fluids—studies in a Hele-Shaw cell RV2—Antoine Sellier (France): Electrophoresis of solid particles in an unbounded electrolyte SA—Flow instability and transition (Robert D. Moser, USA, chair) SA1—Yves Couder; Eric Lajeunesse (France): Tip-splitting instability of Saffman–Taylor fingers SA2—Igor N. Veretennikov; Alexandra E. Indeikina and Hsueh-Chia Chang (USA): Air entrainment at low viscosities SA3—Itai Cohen and Sidney R. Nagel (USA): Snakes eating elephants: The selective withdrawal problem and its uses in coating technology

SB—Biological solid mechanics (Hassan Aref, USA, chair) SB1—Jeffrey E. Bischoff, Ellen M. Arruda, and Karl Grosh (USA): An anisotropic constitutive law for human skin SB2—Albrecht Eiber and Hans-Georg Freitag (Germany): Simulation models of the human middle ear SB3—Lakshminarayanan Mahadevan (USA): Limbless undulatory locomotion on land

SC—Geophysical fluid dynamics (William R. Phillips, USA, chair) SC1—Scott Wunsch (USA): Density layer formation in stably stratified turbulence SC2—James W. Rottman; Derek Stretch; Charles W. Van Atta (USA): Using rapid distortion theory to interpret experiments on stably stratified turbulent shear flows SD—Continuum mechanics (James Casey, USA, chair) SD1—Giancarlo Losi (Italy): A discrete constitutive model for nonlinearly viscoelastic solids with structurally dependent viscosity SD2—Michel E. Jabbour; Kaushik Bhattacharya (USA): A continuum model of growth of thin solid films by chemical vapor deposition SE—Damage mechanics (Jean Lemaitre, France, chair) SE1 — Bhushan L. Karihaloo; Jianxiang Wang (UK): Material instability in the tensile response of short-fibre-reinforced quasi-brittle composites SE2—Milena Vujosevic; Dusan P. Krajcinovic (USA): Cooperative effects in brittle deformation of disordered materials SE3—Raimondo Luciano; John R. Willis (Italy): Non-local effective relations for composites subject to random body forces SF—Structural vibrations (Iliya I. Blekhman, Russia, chair) SF1—Utz von Wagner; Jörg Wauer ( G e r m a n y ) : On the optimization of the vibration in dental tools by compensation masses SF2—Haiyan Hu and Dongping Jin (China, PRC): Dynamics of a traveling cable subject to transverse fluid excitation SF3—Toshihiko Sugiura, Tsuyoshi Tanaka, Takuo Kugiya, and Masatsugu Yoshizawa (Japan): Nonlinear interaction between translation and rotation of a magnet above a highsuperconductor

SG—Mechanics of thin films and nanostructures (Scott R. White, USA, chair) SGl—Hiroshi Kitagawa, Tomotsugu Simokawa, and Akihiro Nakatani (Japan): Deformation mechanism of nano-crystalline material under uniaxial tension/compression SG2—Harley T. Johnson; L. Ben Freund (USA): The influence of strain on confinement in semiconductor quantum dots SG3— Moises E. Smart and Alberto M. Cuitiño; Gustavo Gioia; Antonio DeSimone; Michael Ortiz (USA): The energetics of folding in anisotropically compressed thin-film diaphragms SH—Smart materials and structures (Wendy C. Crone, USA, chair) SH1—Hisaaki Tobushi, Kazuyuki Takata, and Kayo Okumura; Shunichi Hayashi (Japan): Thermomechanical properties of polyurethane-shape memory polymer SH2—Masataka Tokuda, Koichi Sugino, and Tadashi Inaba (Japan): New training of shape memory alloy for smart function by multi-axial loading S H3—Wei Yang and Fei Fang (China, PRC): Domain switching versus fracture of ferroelectrics controlled by poling

Scientific program

1xxix

SK—Chaos in fluid and Solid mechanics (Francis C. Moon, USA, chair) SK1—Jonathan Kobine ( U K ) : Nonlinear sloshing modes in shallow fluid layers with horizontal time-periodic forcing SK2—Ken Kiyono, Tomoo Katsuyama, Takuya Masunaga, and Nobuko Fuchikami (Japan): The dripping faucet as a chaotic dynamical system SK3—Mark A. Stremler; Hassan Aref (USA): Quasi-periodic vortex arrays

SL—Plates and Shells (jointly with IACM) (Herbert A. Mang, Austria, chair) SL1—Walter K. Vonach and Franz G. Rammerstorfer ( A u s t r i a ) : A general solution to the local instability problem of sandwich plates SL2—Khaled W. Shahwan; Anthony M. Waas (USA): Delamination buckling and growth incorporating unilateral contact SL3—Wolfgang A. Wall, Michael Gee, and Ekkehard Ramm; Manfred Bischoff (Germany): Tuning of a 3D shell model in nonlinear statics and dynamics SO—Plasticity and viscoplasticity (Petros Sofronis, USA, chair) SO1—Alan C. F. Cocks; Frederick A. Leckie ( U K ) : The thermal loading of ceramic matrix composites

SO2—Piotr Perzyna; Wojciech Dornowski (Poland): Localized fracture phenomena in thermo-viscoplastic flow processes under cyclic dynamic loadings SO3—Igor K. Senchenkov and Yaroslav A. Zhuk (Ukraine): Thermomechanical coupling effects in the elastic-viscoplastic bodies with internal defect

SR—Acoustics (Ann P. Dowling, UK, chair) SR1—Paul W. Hammerton (UK): Propagation of sonic booms through a stratified atmosphere SR2—Nobumasa Sugimoto (Japan): Mass, momentum and energy transfer by the propagation of the acoustic solitary wave SS—Solid mechanics in manufacturing (Friedrich Pfeiffer, Germany, chair) SS1—Tatsuo Inoue and Hiroyuki Inoue; Fumiaki Ikuta and Takashi Horino (Japan): Simulation of dual frequency induction hardening process of a gear wheel SS2—Alan T. Zehnder and Yogesh Potdar; Xiaomin Deng ( U S A ) : Measurement and sim-

ulation of temperature and strain fields in orthogonal metal cutting SS3—Spandan Maiti and Philippe H. Geubelle ( U S A ) : Simulation of damage mechanisms in high-speed grinding of ceramics

SV—Flows in thin films (Jean-Marc Vanden-Broeck, UK, chair) SV1—Hsueh-Chia Chang, Pavlo V. Takhistov, and Alexandra E. Indeikina (USA): Electrokinetic displacement of air bubbles in microchannels SV2—Peter Vorobieff and Marc S. Ingber; Robert E. Ecke (USA): Two-dimensional cylinder wakes: An experimental and numerical study SV3—Igor L. Kliakhandler (USA): Inverse cascade in film flows

TA—Flow instability and transition (Robert D. Moser, USA, chair) TA1—Jitesh S. Gajjar and Andrew Gibson; Pradeep K. Sen ( U K ) : Instability of compliant pipe flows TA2—James P. Denier; Andrew P. Bassom (Australia): Nonlinear short waves in buoyant boundary layers TB—Multibody dynamics (Hassan Aref, USA, chair) TB1—Andrés Kecskemethy, Christian Lange, and Gerald Grabner (Austria): Robospine:

Simulating cervical vertebrae motion using elementary contact pairs TB2—Harry J. Dankowicz; Jesper Adolfsson, Arne Nordmark, and Petri Piiroinen (USA): Stability analysis of passive, bipedal gait in a three-dimensional environment TB3—Jean H. Heegaard and Matt L. Kaplan (USA): Fast optimal control algorithm for multibody models of the human musculoskeletal system TC—Rock mechanics and geomechanics (loannis Vardoulakis, Greece, chair) TC1—Bernhard A. Schrefler and Lorenzo Sanavia; Paul Steinmann ( I t a l y ) : Geometrical

and material nonlinear analysis of partially saturated porous media TC2—Wolfgang Ehlers (Germany): Dilatant and compressive shear bands in saturated geomaterials TC3—Corey S. O'Hern and Andrea J. Liu; Stephen A. Langer; Sidney R. Nagel (USA): Comparing force distributions and chains in glasses and granular materials

TD—Lighthill Memorial Session (Stanley A. Berger, USA, chair) TD1—John E. Ffowcs Williams ( U K ) : Active flow-avoidance

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ICTAM 2000

TD2—I. David Abrahams; Andrew N. Norris (UK): On the existence of edgewaves on fluid loaded elastic plates TD3—Vladimir N. Shtern and Fazle Hussain (USA): Buoyancy as a driving mechanism for

bipolar jets TE—Damage mechanics (Jean Lemaitre, France, chair) TE1—Beate Lauterbach and Dietmar Gross; Michal Basista (Germany): Influence of microcracks on brittle solids' response under compression TE2—Nicola Alberti; Fabrizio Micari (Italy): Application of damage mechanics approach to predict material formability in deep drawing TE3—Kayleen L. E. Helms and David H. Alien (USA): A model for predicting grain boundary crack growth in viscoplastic polycrystalline materials

TF—Control of Structures (Iliya I. Blekhman, Russia, chair) TF1—Ali H. Nayfeh, Ziyad N. Masoud, and Dean T. Mook (USA): Control of cargo pendulation for ship-mounted cranes TF2—Bala Balachandran and Ying Yong Li (USA): Control of nonlinear crane-load oscillations TG1—Yakov Benveniste; Tungyang Chen (Israel): Torsion of composite bars with imperfect interfaces and neutral inhomogeneities in torsion problems

TG—Damage and failure of composites (Scott R. White, USA, chair) TG2—Fabrizio Greco and Domenico Bruno (Italy): Delamination failure of layered composite plates TG3—Reaz A. Chaudhuri (USA): Delocalization/percolation of inelastic kinkband instability in a thick laminated cylindrical shell

TH—Smart materials and structures (Wendy C. Crone, USA, chair) THl—Prashant K. Purohit and Kaushik Bhattacharya (USA): Dynamics of strings made of phase transforming material TH2—Chih-Kung Lee and Yu-Hsiang Hsu (Taiwan): Theory and experiment of autonomous

phase-gain piezoelectric optimal sensing device TH3—Andrew W. Smyth; Sami F. Masri (USA): Representation and transmission of stochas-

tic loads for nonlinear systems response and control TL—Experimental methods in solid mechanics (James W. Phillips, USA, chair) TLl—Martin Haueis, Rudolf Buser, and Jürg Dual; Claudio Cavalloni and Marco Gnielka (Switzerland): Single crystalline microresonator for force sensing with on-chip vibration excitation and detection

TL2—Markus M. Conrad and Mahir B. Sayir (Switzerland): Detection of flaws in metalceramic compound plates with flexural waves and holography TL3—Arun Shukla and Vikas Srivastava (USA): Strength of adhesive bonded lap joints under high loading rates TV—Flow in porous media (Jean-Marc Vanden-Broeck, UK, chair) TV1—Anthony J. C. Ladd and Rolf Verberg (USA): A lattice-Boltzmann model with sub-grid

scale boundary conditions TV2—Robert A. Beddini and Corwyn E. Low (USA): Hydrodynamic stability of flow in homogeneous porous media

UJ—Closing Lecture by H. Keith Moffatt (Grigory Isaakovich Barenblatt, Russia/USA, chair) UJ1—H. Keith Moffatt (UK): Local and global perspectives in fluid dynamics

NEW PERSPECTIVES ON CRACK AND FAULT DYNAMICS James R. Rice Division of Engineering and Applied Sciences and Department of Earth and Planetary Sciences Harvard University, Cambridge, Mass., USA [email protected] Abstract

1.

Recent observations on the dynamics of crack and fault rupture are described, together with related theory and simulations in the framework of continuum elastodynamics. Topics include configurational instabilities of tensile crack fronts (crack front waves, disordering, sidebranching), the connection between frictional slip laws and modes of rupture propagation in earth faulting, especially conditions for formation of self-healing slip pulses, and the rich faulting and cracking phenomena that result along dissimilar material interfaces due to coupling between slippage and normal stress alteration.

INTRODUCTION

The dynamics of cracking and faulting has seen much recent progress, with implications for structural mechanics, materials physics, tribology, and seismology. In this brief review, the following topics will be discussed: Crack front waves Limiting rupture speeds Mode of rupture on faults: enlarging shear crack or self-healing slip pulse? Interfacial fracture dynamics, slip, and opening To put things in context, we will be using linear isotropic elasticity everywhere except along slip or crack zones. The governing equations of that theory are the equations of motion where or u or a combination are given on the boundary, together with stress-displacement gradient relations 1 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 1–24. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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leading to the Navier equations for the displacement field:

Those equations admit a family of waves. There are primary (or dilational) and secondary (or shear) body waves, with respective speeds and typically with and also Rayleigh surface waves with typical speed As we review recent progress, we shall learn of a newly discovered type of wave that lives along a moving crack front, and also see the significance of a little known generalization of Rayleigh waves for the dynamics of dissimilar materials. Much of the numerical simulation that is reviewed here has been done by a spectral elastodynamic method [l]–[3], which is of limited general utility but is very well tailored to produce good numerical solutions, without grid dispersion, for crack or fault problems that can be con-

sidered as events at the border between two homogeneous elastic half spaces. The border at y = 0, which is the fault or crack plane, undergoes displacement discontinuities

The spectral method treats these problems by writing as a large but finite Fourier sum of terms in form the rupture domain is replicated periodically in x and z. The elastodynamic equations are solved exactly in each half space, for terms of that same space dependence, so that in the end the traction stress components are given as a corresponding Fourier sum of terms in the form Each can then be determined in terms of the in an expression involving a convolution over prior That, effectively, lets us relate the history of the at fast-Fourier-transform sample points along the interface, to the history of the and to any given loading stresses, in such a way that the elastodynamic equations are satisfied in the two half spaces. The system is then closed, so that a definite solution can be computed, by specifying another relation between the and the that is a constitutive relation for crack opening [2, 3] (cohesive model) or frictional slip [1]. Different forms will be noted as we go along.

2.

CRACK FRONT WAVES

Figure 1 shows a tensile crack growing in a 3D solid, along the plane y = 0. Earlier work [4, 5] addressed a simplified version of this problem for a model elastic theory with a single displacement component u,

New perspectives on crack and fault dynamics

3

Figure 1 Crack propagating along a plane in an unbounded elastic solid. Inset shows cohesive stress versus crack opening for non-singular crack model.

satisfying a scalar wave equation. It showed that there was a long lived response to a local perturbation of the crack front, e.g. by the crack passing through a region where the fracture energy was slightly different than elsewhere, although in that scalar model the crack ultimately recovered a perfectly straight front. That led to great interest in solving such problems in the context of actual elasticity. Willis and Movchan [6] produced the corresponding small perturbation solution, although that was difficult to interpret and the fuller implications of their solution were revealed later [7], confirming what had been suggested from spectral numerical simulations [8, 9]: for crack growth in a perfectly elastic solid with a constant fracture energy, perturbation of the crack front leads to a wave that propagates laterally, without attenuation or dispersion, along the moving crack front. The wave is called a crack front wave. Two different fracture formulations have been used for these investigations. The first is a singular crack model, in which one sets on the mathematical cut y = 0, which is the crack surface. That leads to a well known singular field of structure

where r, are polar coordinates at the crack tip and the are universal functions for a given cracking mode. The strength has been normalized in terms of G, which is the energy release rate (energy flow to crack tip singularity, per unit of new crack area), expressed by

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Here, W is the strain energy density, coordinate points in the direction of crack growth, is a circuit that loops around the crack tip at the place of interest, and s is arc length along whose outer normal is n. Owing to the great advances on unsteady crack motion in 2D elasticity by Kostrov [10] and Eshelby [11] for mode III (anti-plane shear), Freund [12] for mode I (tensile cracking), and Fossum and Freund [13] for mode II (in-plane shear), we know that G = g(v(t)) Here g(v) is a universal function of crack speed v for each mode, normalized to and diminishing to at a limiting speed (at least for fully subsonic fracturing), which is for modes I and II, and for mode III. The variable is a complicated and generally untractable functional of the prior history of crack growth and of external loading, but is independent of the instantaneous crack speed v(t). Because of that structure for G, it is possible for cracks to instantaneously change v if the requisite energy which must be absorbed for fracture, changes discontinuously along the fracture path. Also, for a solid loaded by a remotely applied stress, it will generally be the case that increases as the crack lengthens, and increases quadratically with the intensity of the applied stress. Thus, if is bounded, then in a sufficiently large body g(v) will be driven towards 0, which means that v will accelerate towards For the crack growing on a plane in 3D, as in Fig. 1, it has so far been possible to solve for the elastic field [6] only for a crack whose front position x = a(z,t) is linearly perturbed from that is, from a straight front moving at uniform speed An alternative fracture formulation, often more congenial to numerical calculation, explicitly accounts for a gradual decohesion, imposing a weakening relation between stress and displacement-discontinuity as a boundary condition on the potential crack plane. (See the inset diagram of Fig. 1.) Thus, the singularity at the crack tip is smeared into a displacement weakening zone. Its width R scales [14] as, roughly, with being the maximum cohesive strength and the displacement at which cohesion is lost, and with f(v) being universal for a given mode, and with f(0) = 1 and The latter limit poses a challenge for numerical simulation of fracture at speeds very close to When all length scales in problem (crack length, distance of wave travel, etc.), predictions of the displacement-weakening model agree with those of the singular crack model, with identified (Fig. 1) as the area under the cohesive relation [14]. The spectral methodology was used with the cohesive model [8, 9] to study what happens when a crack front, moving as a straight line across y = 0 with uniform rupture speed suddenly encounters a localized

New perspectives on crack and fault dynamics

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“asperity” region for which is modestly higher than its uniform value, say, prevailing elsewhere. The resulting perturbation in the crack front propagation velocity, v(z, t) is shown in Fig. 2. There, is the periodic replication distance in the z direction, as required within the spectral method. The asperity has diameter and has

Figure 2 Numerical simulation results for perturbation of rupture propagation velocity v(z,t) when the crack front, moving at uniform speed encounters a small region of altered fracture energy. Adapted from Ref. [9].

both and were increased by 5% within it. The motivation to examine the response to such a small localized perturbation arose from understanding of the scalar model [5], for which the form of response to such isolated excitation was critical to understanding the response to (small) random excitation. Quite remarkably, when those calculations were done (Fig. 2) for the mode I crack in true elastodynamics, the perturbation of the crack front seemed to propagate as a persistent wave, for as long as it was feasible to do the numerical calculation. This is the crack front wave. Returning to the singular crack model, the existence of the wave was proven, as discussed, by application [7] of the solution [6] for arbitrary, but sufficiently small (linearized) perturbation Writing the critical fracture energy as

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and understanding that its perturbation at the crack front is within a strictly linearized formulation, one must have an expression of the form

where

The function has a simple zero [7, 15] at a certain real value of That proves the existence of the wave, whose speed in the direction parallel to the moving crack front is Figure 3 shows the resulting wave speed measured relative to a place on the fracture surface from which the wave was launched, or

Figure 3 Crack front wave speed, as a function of unperturbed crack speed different values of Possion ratio From Ref. [15].

for

through which it passed; is the speed parallel to the front. The speed is very near to and consistent with the simulation result in Fig. 2. By analyzing the small perturbation problem for the case

New perspectives on crack and fault dynamics

in which

7

it was shown [7] that the wave attenuates if

The existence of the crack front wave leads to rapid disordering in crack propagation through a small random fluctuation about mean of fracture energy on the plane y = 0. Suppose that the variation is statistically stationary and isotropic, with correlation function

and that a sample of that variation is first encountered by the crack at t = 0, with the crack front being straight and moving at unperturbed velocity for t < 0. Within the linearized perturbation formulation, remains the mean velocity after perturbation begins. We consider the random variables A(z, t) and the slope S(z, t) = of the crack front. Then, using the method of Ref. [5] for the scalar case, applied in way outlined in Ref. [9], one derives (for length scales in the correlation function) that

and that

where is a certain function. Thus the variance in crack location grows in direct proportion to distance of propagation into the nonuniform region, whereas the two-point correlation between the positional fluctuations will vanish when has become large enough that there is zero correlation at such distances. This conveys a picture of rapid disordering of the crack front. That is also expressed by the associated result

Note that such second order statistic will not exist if the correlation function has a non-zero slope at separation. Samples of perturbed crack growth histories, within the linearized ideally elastic framework show that strong perturbations cluster in space and time along wave fronts [15]. Many features remain to be understood. These include saturation due to nonlinearities (one must, of course, have damping due to and to material viscoelasticity, and, most especially, to interactions with non-planarity of growth.

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3.

AJ—O PENING L ECTURE: J AMES R. RICE

LIMITING RUPTURE SPEED

The theoretical prediction, according to the singular crack model, for ruptures along a plane has already been mentioned. Under typical remote loading conditions we expect that for modes I and II, and that for mode III, as limiting rupture speeds. However, to summarize observations [16]–[18] for tensile (mode I) cracks, from laboratory tests: The crack speed in brittle amorphous solids (glass, PMMA) has an upper limit of order 0.50–0.60 (i.e. 0.55–0.65 The fracture surface is mirror-smooth only for v < 0.3–0.4 The crack surface roughens severely, and becomes severely oscillatory, at higher (average) speeds. The crack forks at the highest speeds.

There are exceptions for which (or a large fraction of brittle highly anisotropic single crystals (W, mica); and incompletely sintered solids [19]. Up to very recently, the only observations for shear ruptures (modes II and III) have come from inversions of seismograms for slip histories on faults: The range is a commonly inferred range of rupture speeds, although it is not well constrained.

Rarely, bursts of intersonic rupture with inferred [20, 21].

have been

In the case of tensile cracks in brittle materials, interferometric studies [16] of propagating fractures in PMMA sheets showed that the propagation process became quite intermittent as the limit speed was approached. That was evidenced by strong stress waves being radiated from the tip in distinctly separated pulses. (It is actually the mean value of , averaged over a few such pulses, and not necessarily the local itself, which should be thought to have a well defined limit, much less than Later studies [17] clearly tied the strongly intermittent , and intensity of fluctuation of about the mean, to a side branching process. In that, small fractures seem to have emerged out of the walls of the main fracture near to its tip, such that the main crack and one or a pair of such side-branching cracks coexisted for a time, although the branches were ultimately outrun by the main fracture, at least at low mean propagation speeds. It may instead [22] be that the side branched cracks began their life as part, not well aligned with the main crack plane, of a damage cluster of microcracks developing ahead of the main crack tip

New perspectives on crack and fault dynamics

9

(Fig. 4(c)), and grew into the main crack walls. In any event, the density of cracks left as side-branching damage features increases substantially

Figure 4 (a) Yoffe consideration of hoop stress near tip. (b) Eshelby question of when can one fracture, by slowing down, provide enough energy to feed two. (c) Possibility that profuse microcrack nucleation, with some clusters linking with the main fracture, may be a more appropriate mechanism than the side-branching from the tip in (b).

with increase of the loading that drives the fracture. That process also results in the well-known increased roughening of the fracture surface with increasing crack speed. Ultimately, the response of the fracture to further increase in loading is no longer to increase the average , which is the result expected from the theoretical analysis of a crack moving on a plane, but rather to absorb more energy by increasing the density of cracks in the damage cluster formed at the tip of the macroscopic fracture. Thus the energy adsorption rises steeply with crack speed [17]. The deviation of the fracture process from a plane is tied yet more definitively to the intermittency and low limiting crack speeds by studies [19] of weakly sintered plates of PMMA. The resistance to fracturing along the joint was much smaller than for the adjoining material, and that led to smooth propagation of the fracture (no evidence of jaggedness of interferograms by stress wave emission), with reaching 0.92 The attempts to explain low limiting speeds and fracture surface roughening begin with the first paper, by Yoffe [23], in which the elastodynamic equations were solved for a moving crack. That was done for a crack that grew at one end and healed at the other, so that the field depended on x — t and y only (2D case), but sufficed to reveal the structure of the near tip singular field, including the dependence

of the functions above. Yoffe found that (Fig. 4(a)), at fixed small r within the singularity-dominated zone, reached a maximum with

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respect to for when > 0.65 That gives a plausible explanation of limiting speeds and macroscopic branching, but does not explain the side-branching and roughening that sets in at much smaller speeds. Eshelby [24] posed the question of what is the at which a fracture must be moving so that, by slowing down, it can provide enough energy to drive two fractures (Fig. 4(b)). The extreme is to slow down to That question still has no precise answer when the angle (Fig. 4(b)) is non-zero, but Eshelby answered it for mode III—Freund [12] had not yet solved the mode I case—in the limit which reduces to sudden change of speed of a fracture moving along the plane. The condition is g( ) < 1/2 (assuming no dependence of which gave > 0.60 for mode III. Freund addressed the same issue based on his mode I solution [12], again in the limit that, with recalculation [25], leads to > By working out the static version of the problem, which has at least approximate relevance since is near 0, and using approximate considerations of wave travel times to limit the interaction of one branch with the other, Adda-Bedia and Sharon [25] optimize with respect to to estimate that the lowest u for branching is Material response with will reduce that threshold but, at present, it seems unlikely that the inferred onset of roughening at speeds in the range 0.3–0.4 can be explained by a branching instability at a crack tip in a solid that is modeled, otherwise, as a linear elastic continuum. Nevertheless, it is interesting that some discrete numerical models of cracking do give branching at roughly realistic speeds. A transition to zig-zag growth was shown to set in around 0.33 in molecular dynamics simulations [26]. It is not yet clear to what extent the at its onset may depend on details of the force law. Cohesive finite elements [27] or cohesive element interfaces [28] that fully model the separation process allow, within constraints of the mesh, for self-chosen fracture paths, and have shown off-plane fracturing. Xu and Needleman [28] nicely reproduce the observations that when the crack is confined to a weak plane, it accelerates towards whereas for the uniform material, side branches form at 0.45 or at lower speed [29] if statistical variation in cohesive properties is introduced. Recent work [30] suggests that such propensity for low-speed side branching may not be universal for all cohesive finite element models. In particular, models that involve linear behavior until a strength threshold is reached [31], after which there is displacementweakening (much like for the inset in Fig. 1) seem less prone to development of side-branching. The procedure of Ref. [28] allows substantial nonlinear deformation at the element boundaries before achieving their peak cohesive strength. Thus local nonlinear features of the pre-peak

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11

deformation response may be critical to understand the onset of roughening [32]. In contrast to tensile cracks, earthquake ruptures (large scale shear cracks) do seem to approach much more closely to and, as mentioned, bursts of intersonic rupture have been inferred [20, 21]. In fact, theoretical prediction [33, 34] of intersonic rupture from rupture simulations with slip-weakening models (mode II or III versions of the tensile

displacement-weakening model discussed above) preceded the observations. A concentration of stress in a shear wave peak develops ahead of a mode II shear rupture. In a model with a finite strength to get slip started, that allows for the possibility that slip-weakening will initiate ahead of the main crack tip. As part of the linking up with the main slip-weakening zone, that allows a fracture to emerge at a high speed in the range of (which is the unique intersonic speed at which a mode II rupture can propagate in the singular crack model [18]). Such had never been seen in the laboratory until Rosakis et al. [35] demonstrated that for weakly sintered Homalite-100 plates, impact loaded in shear, a mode II fracture propagated intersonically with a speed that fluctuated to high values, near to but approached at greater propagation distances a speed near The importance of the weak channel for the rupture is that typical attempts to form a mode II (or III) rupture

in the laboratory, except under quite high pressure [36], lead to mode I cracking from the rupture tip.

4.

MODE OF RUPTURE ON FAULTS: ENLARGING SHEAR CRACK OR SELF-HEALING SLIP PULSE WHEN THERE IS VELOCITY-WEAKENING FRICTION?

In an influential paper, Heaton [37] argued that the mode of rupture in large earthquakes, as inferred from seismic inversion studies for well recorded events, was such that the duration of slip at a point on the fault was much shorter than the overall duration of the rupture. He argued that a point on the fault begins to slip as the rupture front arrives but that this slipping phase is of short duration and that the fault “heals” (by which is meant that it stops slipping) after a time that is much less than the overall duration of the rupture. That is called the “self-healing” rupture mode. That is to be contrasted with what may be called the “crack-like” mode. For it, a point on the fault plane is assumed to slip for a significant fraction of the overall rupture duration, i.e. beginning when the rupture

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front arrives and continuing until waves generated in the arrest of the front, at a barrier of some sort, carry back signals to stop slipping. The crack mode has been widely observed in numerical simulations of spontaneous rupture. These include cases of prescribed uniform strength drop on the fracture surface in singular crack models [38]–[41], nonsingular slip-weakening models [33, 42, 43, 44], and simple, if grid-size dependent, models with a critical stress failure condition at the rupture front [45]–[47]. Heaton’s paper thus went against a widely accepted view of how rupture occurs. It launched an active body of theoretical research, to understand what does indeed determine the mode of rupture, and what could lead to the self-healing mode, which seems to be supported by observations. There are now a few candidates [48], one to be discussed in this section and one in the next. Here we focus on recent theoretical understanding of a possibility already suggested by Heaton, namely that strong velocity-weakening of friction could allow self-healing. In fact, simulations of rupture with strong velocity weakening sometimes showed crack-like rupture and sometimes self-healing [1, 49, 50], and it has been an important goal to understand what controls. One requirement is that the friction strength increase with time on slipped portions of the fault that are momentarily in stationary contact [1]. Another is that the overall driving stress be low [48] in a way that we analyze here. Velocity-weakening friction on faults is interpreted in the rate and state framework, which includes laboratory-based state evolution features. Those also regularize ill-posed or paradoxical features of models of sliding between two identical elastic solids [51]. Thus, as in Fig. 5, we shall think of the heavy solid line as giving the basic response between shear stress and slip rate V; here we regard normal compressive stress as constant. A full constitutive description, which must be used in numerical simulations, involves strength expressed as where the state variable is —e.g. representing lifetime of current population of asperity contacts, or of current gouge packing— and where (note path of instantaneous change, i.e. change at constant in Fig. 5), and The state variable follows an evolution law, e.g. of form although other forms are sometimes used instead [1, 48]. The characteristic slip distance L for state evolution is typically found to be in the 1–50 range. The evolution law has the property that (= L/V for the law above) in sustained slip at fixed V, so that where and, for the cases of interest here, That is a somewhat complicated formulation, and the state evolution is over in micron range changes of slip, which would be very small

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13

Figure 5 Velocity weakening friction. Interpreted in the rate and state framework, including laboratory-based state evolution features that also regularize ill-posed or paradoxical features in models of sliding between two identical elastic solids. A stress level is also shown, below which crack-like ruptures become impossible.

for applications to tectonic faulting. So the question arises, why use rate- and state-dependent friction instead of a pure rate-dependent friction law, say, The answer is that in addition to attaining

consistency with laboratory evidence and with the microphysical understanding of friction, we eliminate the following [51]: Ill-posedness of the pure rate-dependent formulation when An perturbation of a steady sliding state, with V = then elicits response > 0; see the discussion of a similar issue in the next section. Supersonic propagation of rupture fronts [52] when < An perturbation elicits response V(x,t) —

where r >

for mode II and r >

for mode III

slip. (While supersonic propagation of rupture fronts seems to be

precluded in the rate and state formulation, phase velocities at sufficiently low within the range for which there is unstable response to perturbations, do become supersonic [53]). These shortcomings of the pure rate-dependent model are not widely known, probably because friction studies directed to machine technology

have often focused on sliding rigid blocks rather than deformable elastic continua. The rate and state formulation provides a regularization. Now consider a fault surface, which we treat as the boundary y = 0 between two identical half spaces (Fig. 6). An initial shear stress

on

a constant level too small to cause failure, acts everywhere (the xz plane) except in small nucleation region which will

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Figure 6 Identical elastic half-spaces meeting on a fault plane y = 0; for discussion of crack-like versus self-healing rupture mode.

be overstressed to start the rupture. An important stress level is given by as marked in Fig. 5. It is understood as the highest value that could have if we were to require for all Suppose that As will be seen, that effectively precludes the possibility that rupture could occur on in the form of an indefinitely expanding shear crack [48]. Note that

Important along the way is an elastodynamic conservation theorem [48]

which holds throughout the rupture. This can be derived from relations involving spatial Fourier transforms of elastodynamic fields [2] in the zero wavelength limit. As an aside, it provides an interpretation for the seismic moment release rate,

which has apparently not appeared before in seismology. Let us now assume that with the loading rupture has been locally nucleated and grows on in the form of an indefinitely expanding shear crack. We shall try to develop a contradiction. The integrand everywhere along the rupturing surface except for and for small regions at the rupture front affected by the rate/state regularization, is equal to

where the inequality follows from the above consequence of Thus, letting = region of lying outside the

New perspectives on crack and fault dynamics

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rupture at t, and noting that V = 0 there, we must then have

That seems to be a contradiction: We expect cracks to increase the net force carried outside themselves, or at least to not decrease it. (However,

such non-decrease of force is a proven result, thus far, only for 2D antiplane shear crack solutions [48].) The result does nevertheless suggest that the crack-like rupture mode should not occur if That is consistent with a range of calculations [48, 54] for different stress levels

and forms of slip pulses—when

Crack-like ruptures are not found—only self-healing In the loading regime one

may focus on a dimensionless measure T of the strength of the velocity weakening at a representative slip rate [48]. When T is very small, the rupture surface could reasonably be expected to respond somewhat like

that for a model with constant stress drop, which is in a crack-like mode; that is what is found in simulations.

5.

MATERIAL PROPERTY DISSIMILARITY ACROSS A FAULT PLANE Here we consider two dissimilar solids in sliding frictional contact of

mode II type (Fig. 7). We consider first the simplest case of a constant friction coefficient f, independent of slip rate V or its history. Thus

Figure 7 Frictional sliding of dissimilar elastic solids on one another.

whenever V(x,t) > 0 and normal stress the shear stress The general effect that can drive highly unstable response in this case is that spatially inhomogeneous slip induces a local change in

and such reduction of

allows easier slip.

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Interesting results have been found in the following two types of analyses: Dynamic stability of steady sliding [55]–[59] on the surface between dissimilar elastically deformable materials, loaded with Slip rupture dynamics on faults between dissimilar materials that are loaded below threshold [60]–[65], For the first type of problem, we assume that the two solids are sliding at a uniform rate and are loaded by remote stresses that just meet the sliding conditions, and that the stress state is uniform within each solid. We then consider the response of this system to perturbation, such that the friction law remains valid everywhere along the interface and there is no opening. Solutions to the Navier equations are then sought in the form

For real k, one finds that p is of the form p = – ick, where a and c are real. Most importantly, over a wide range of parameters [56] (values of f, ratios of elastic constants, and densities), it turns out that a > 0 for at least one such solution. That remarkable fact not only implies that the dissimilar material system is modally unstable, but also implies ill-posedness of response to general perturbation. That is, the response to a perturbation that has Fourier strength in that unstable mode is, formally,

which diverges when t > 0 for generic if a > 0. Response in a given mode does not diverge in finite time, but the solution ultimately loses validity because either V < 0 or is predicted. The following has been shown [59] for dissimilar materials: Response is unstable (a > 0) for all f > 0 when a generalized Rayleigh (GR) wave1 exists, and for f > > 0 when a GR wave does not exist. For sufficiently large f, two unstable solutions (i.e. both with a > 0) may exist, with different growth rates a and propagation 1

A GR wave corresponds to a motion with free slip, = 0, but no opening gap at the interface. It exists when there is a real solution c (the wave speed to

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speeds c of different magnitudes and signs (directions), and neutrally stable solutions (a = 0) with supersonic propagation speeds may exist. The second type of problem envisions that the two solids in contact (Fig. 7) are loaded below the friction threshold, and that an event is somehow nucleated (e.g. by local reduction of ) and the focus is on how it propagates into previously locked locations along the interface. Weertman [60] suggested that steady rupture propagation, in a form of what is now called a self-healing pulse, might exist in that circumstance, and such has been confirmed [61]. That solution exists only when the GR wave exists. Rupture propagates at (its possible significance for rupture propagation had been noted earlier [67]), in a unique direction along the interface, which is that of slip in the slower material—i.e. in the +x direction in Fig. 7, for slip as shown. The solution has the remarkable feature that remains unaltered from and that slip occurs because reduces in the slipping region so as to just meet . The speed V is constant within the pulse, and proportional to . Its stability and possibility of emergence from initial conditions remain unclear. Transient dynamic analysis of locally nucleated rupture has also been addressed, first in finite difference simulations [62]–[64]. They did indeed find rupture in the form of a self-healing pulse but results indicated problems of convergence with grid refinement, or with time of rupture propagation, which is now understood to be related to the ill-posedness mentioned above. That ill-posedness was confirmed [65] by using the spectral numerical formulation, as generalized to bimaterials [3]. For a fixed replication distance along strike, the numerical results became progressively more jagged with increase from 256 to 2048 terms in the underlying Fourier series representation of in a way that was consistent with theoretical understanding of the instability. Some possible regularizations of the problem have been discussed [58], e.g. basing the friction law on the average of over a finite patch size around the position of interest. Another approach [59, 65] was based

with k = 1,2 to denote the different materials. GR waves were discovered independently by several investigators [66]–[68]. They satisfy and

and exist only for modestly different materials, typically for

< 1.30–

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on results of oblique shock wave experiments [69, 70]. In those, a shock reflected from the back surface of the target specimen causes an abrupt decrease of normal stress on the sliding interface. As shown schematically in Fig. 8, there is then only a gradual evolution of shear strength (over a few micrometers of slip, or few tenths of microseconds time)

Figure 8 Schematic of graduate evolution of shear strength (over a few micrometers of slip, or few tenths of microseconds time) in oblique shock wave experiments of Refs. [69, 70]. A shock reflected from the back surface of the target specimen causes an abrupt decrease of normal stress.

towards the new level consistent with the altered normal stress. A simple representation of this result is to replace with [59]

assuming where L = const > 0. This regularizes the first group of problems above, i.e. perturbation of steady frictional sliding, as follows [59] (now a and c in p = – ick do depend on k, at least when L/V is of order 1 or larger): When a GR wave exists, the unstable response approaches neutral stability, as (i.e. as wavelength In practical terms, that means stability at length scales

When a GR wave does not exist, there is a critical wavenumber such that response is stable (a < 0) when (or when where scales in proportion to Nothing in those results would change if V/L was replaced by where with at least one > 0, or indeed by any positive factor. The same regularization was then used [65] to address the second class of problems above, systems that are frictionally locked with but for which rupture is nucleated somewhere by local unclamping. With

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the regularization, the problem becomes well posed, in that convergent numerical solutions occur upon refinement of the Fourier basis set. Further, the rupture is in the form of a self-healing pulse propagating at a speed close to In cases for which does not exist, the propagation speed seems to be close to These are all problems of slip rupture along bimaterial interfaces, but there are also challenging problems, with remarkable features seen experimentally and rationalized theoretically, involving combined tensile fracture and slip rupture along bimaterial interfaces. For example, for polymer–metal systems (PMMA–steel [71] and Homalite 100– aluminum [72], respectively), impact loading created fractures that began their life as shear slippage over a millimeter scale at the rupture front, and subsequently opened. Their propagation speed was faster than for the polymer, leading to shock wave structures in high speed photographs of a photoelastic fringe pattern [72]. The essential features of those complex near tip slip and opening features, and the propagation speed, were rationalized in spectral numerical calculations [3], based on a non-singular numerical model with combined displacement weakening in tension and shear.

6.

CONCLUSION

It is hoped this set of short examples conveys some idea of the new discoveries and scientific excitement in understanding the dynamics of rupture. We have seen the discovery of previously unsuspected waves along crack fronts, of reasons why a previously inferred speed limit for fracturing (the Rayleigh speed) is sometimes too high and sometimes too low, of unsuspectedly ill-posed problems in frictional slip dynamics, and of rupture modes, such as the self-healing mode, that were largely unsuspected a relatively short time ago. These are a small sample from a large and vigorously growing body of science.

Acknowledgment The studies discussed were supported by the Office of Naval Research, the U.S. Geological Survey, and NSF through the Southern California Earthquake Center.

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References [1] Perrin, G., J. R. Rice, and G. Zheng. 1995. Self-healing slip pulse on a frictional surface. Journal of the Mechanics and Physics of Solids 43, 1461–1495. [2] Geubelle, P., and J. R. Rice. 1995. A spectral method for 3D elastodynamic fracture problems. Journal of the Mechanics and Physics of Solids 43, 1791– 1824. [3] Breitenfeld, M. S., and P. H. Geubelle. 1998. Numerical analysis of dynamic debonding under 2D in-plane and 3D loading. International Journal of Fracture

93, 13–38. [4] Rice, J. R., Y. Ben-Zion, and K. S. Kim. 1994. Three-dimensional perturbation solution for a dynamic planar crack moving unsteadily in a model elastic solid. Journal of the Mechanics and Physics of Solids 42, 813–843. [5] Perrin, G., and J. R. Rice. 1994. Disordering of a dynamic planar crack front in a model elastic medium of randomly variable toughness. Journal of the Mechanics and Physics of Solids 42, 1047–1064. [6] Willis, J. R., and A. B. Movchan. 1995. Dynamic weight functions for a moving crack—I. Mode I loading. Journal of the Mechanics and Physics of Solids 43, 319–341. [7] Ramanathan, S., and D. Fisher. 1997. Dynamics and instabilities of planar tensile cracks in heterogeneous media. Physical Review Letters 79, 877–880.

[8] Morrissey, J. W., and J. R. Rice. 1996. 3D elastodynamics of cracking through heterogeneous solids: Crack front waves and growth of fluctuations (abstract). EOS Transactions of the American Geophysical Union 77, F485. [9] Morrissey, J. W., and J. R. Rice. 1998. Crack front waves. Journal of the Mechanics and Physics of Solids 46, 467–487.

[10] Kostrov, B. V. 1966. Unsteady propagation of longitudinal shear cracks. Journal of Applied Mathematics and Mechanics 30, 1241–1248 (in Russian). [11] Eshelby, J. D. 1969. The elastic field of a crack extending non-uniformly under general anti-plane loading. Journal of the Mechanics and Physics of Solids 17, 177–199. [12] Freund, L. B. 1972. Crack propagation in an elastic solid subject to general

loading—I, Constant rate of extension, II, Non-uniform rate of extension. Journal of the Mechanics and Physics of Solids 20, 129–152. [13] Fossum, A. F., and L. B. Freund. 1975. Non-uniformly moving shear crack model of a shallow focus earthquake mechanism. Journal of Geophysical Research 80, 3343–3347. [14] Rice, J. R. 1980. The mechanics of earthquake rupture. In Physics of the Earth’s Interior, Proceedings of the International School of Physics ‘Enrico Fermi’ (A. M. Dziewonski and E. Boschi, eds.). Italian Physical Society and North-Holland,

555–649. [15] Morrissey, J. W., and J. R. Rice. 2000. Perturbative simulations of crack front wave. Journal of the Mechanics and Physics of Solids 48, 1229–1251. [16] Ravi-Chandar, K., and W. G. Knauss. 1984. An experimental investigation into dynamic fracture—I, Crack initiation and crack arrest, II, Microstructural aspects, III, Steady state crack propagation and crack branching, IV, On the interaction of stress waves with propagating cracks. International Journal of Fracture 25, 247–262; 26, 65–80, 141–154, 189–200.

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[17] Sharon, E., S. P. Gross, and J. Fineberg. 1995. Local crack branching as a mechanism for instability in dynamic fracture. Physical Review Letters 74, 5096–

[18] [19]

[20] [21]

[22]

[23]

5099. Also, Sharon, E., and J. Fineberg. 1996. Microbranching instability and the dynamic fracture of brittle materials. Physical Review B 54, 7128–7139. Broberg, K. B. 1999. Cracks and Fracture. San Diego, Calif.: Academic Press, 752 pp. Washabaugh, P. D., and W. G. Knauss. 1994. A reconciliation of dynamic crack velocity and Rayleigh wave speed in isotropic brittle solids. International Journal of Fracture 65, 97–114. Archuletta, R. J. 1984. A faulting model for the 1979 Imperial Valley earthquake. Journal of Geophysical Research 89, 4559–4585. Beroza, G. C., and P. Spudich. 1988. Linearized inversion for fault rupture behavior: Application to the 1984 Morgan Hill, California, earthquake. Journal of Geophysical Research 93, 6275–6296. Ravi-Chandar, K., and B. Yang. 1997. On the role of microcracks in the dynamic fracture of brittle materials. Journal of the Mechanics and Physics of Solids 45, 535–563. Yoffe, E. H. 1951. The moving Griffith crack. Philosophical Magazine 42, 739– 750.

[24] Eshelby, J. D. 1970. Energy relations and the energy-momentum tensor in con-

tinuum mechanics. In Inelastic Behavior of Solids (M. F. Kanninen, W. F. Adler, A. R. Rosenfield, and R. I. Jaffe, eds.). New York: McGraw-Hill, 77–115.

[25] Adda-Bedia, M., and E. Sharon. 2000. Private communication. [26] Abraham, F. F., D. Brodbeck, R. A. Rafey, and W. E. Rudge. 1994. Instability dynamics of fracture: A computer simulation investigation. Physical Review Letters 73, 272–275. [27] Johnson, E. 1992. Process region changes for rapidly propagating cracks. International Journal of Fracture 55, 47–63. [28] Xu, X.-P., and A. Needleman. 1994. Numerical simulations of fast crack growth in brittle solids. Journal of the Mechanics and Physics of Solids 42, 1397–1434. [29] Xu, X.-P., A. Needleman, and F. F. Abraham. 1997. Effect of inhomogeneities on dynamic crack growth in an elastic solid. Modeling and Simulation in Materials Science and Engineering 5, 489–516.

[30] Falk, M. L., A. Needleman, and J. R. Rice. 2001. A critical evaluation of dynamic

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fracture simulations using cohesive surfaces. Submitted to 5th European Mechanics of Materials Conference (Delft, 5–9 March 2001). Camacho, G. T., and M. Ortiz. 1996. Computational modelling of impact damage in brittle materials. International Journal of Solids and Structures 33, 2899– 2938. Gao, H. 1996. A theory of local limiting speed in dynamic fracture. Journal of the Mechanics and Physics of Solids 44, 1453–1474. Andrews, D. J. 1976. Rupture velocity of plane strain shear cracks. Journal of Geophysical Research 81, 5679–5687. Burridge, R., G. Conn, and L. B. Freund. 1979. The stability of a rapid mode II shear crack with finite cohesive traction. Journal of Geophysical Research 84, 2210–2222. Rosakis, A. J., O. Samudrala, and D. Coker. 1999. Cracks faster than the shear wave speed. Science 284, 1337–1340.

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[36] Broberg, K. B. 1987. On crack paths. Engineering Fracture Mechanics 28, 663– 679. [37] Heaton, T. H. 1990. Evidence for and implications of self-healing pulses of slip in earthquake rupture. Physics of the Earth and Planetary Interiors 64, 1–20.

[38] Kostrov, B. V. 1964. Self-similar problems of propagation of shear cracks. Journal of Applied Mathematics and Mechanics 28, 1077–1087 (in Russian). [39] Madariaga, R. 1976. Dynamics of an expanding circular fault. Bulletin of the Seismological Society of America 66, 639–666. [40] Freund, L. B. 1979. The mechanics of dynamic shear crack propagation. Journal of Geophysical Research 84, 2199–2209. [41] Day, S. M. 1982. Three-dimensional simulation of spontaneous rupture: The effect of nonuniform prestres. Bulletin of the Seismological Society of America 72, 1889–1902. [42] Ida, Y. 1972. Cohesive force across the tip of a longitudinal-shear crack and Griffith’s specific surface energy. Journal of Geophysical Research 77, 3796–3805. [43] Andrews, D. J. 1985. Dynamic plane-strain shear rupture with a slip-weakening friction law calculated by a boundary integral method. Bulletin of the Seismological Society of America 75, 1–21. [44] Harris, R., and S. M. Day. 1993. Dynamics of fault interactions: Parallel strike-

slip faults. Journal of Geophysical Research 98, 4461–4472. [45] Das, S., and K. Aki. 1977. A numerical study of two-dimensional spontaneous rupture propagation. Geophysical Journal of the Royal Astronomical Society 50, 643–668. [46] Das, S. 1980. A numerical method for determination of source time functions for general three-dimensional rupture propagation. Geophysical Journal of the Royal Astronomical Society 62, 591–604. [47] Das, S. 1985. Application of dynamic shear crack models to the study of the earthquake faulting process. International Journal of Fracture 27, 263–276. [48] Zheng, G., and J. R. Rice. 1998. Conditions under which velocity-weakening

friction allows a self-healing versus cracklike mode of rupture. Bulletin of the Seismological Society of America 88, 1466–1483. [49] Cochard, A. and R. Madariaga. 1996. Complexity of seismicity due to highly rate dependent friction. Journal of Geophysical Research 101, 25321–25336. [50] Beeler, N. M., and T. E. Tullis. 1996. Self-healing slip pulse in dynamic rupture models due to velocity-dependent strength. Bulletin of the Seismological Society of America 86, 1130–1148. [51] Rice, J. R., N. Lapusta, and K. Ranjith. 2001. Rate and state dependent friction and the stability of sliding between elastically deformable solids. Submitted to Journal of the Mechanics and Physics of Solids. [52] Weertman, J. 1969. Dislocation motion on an interface with friction that is dependent on sliding velocity. Journal of Geophysical Research 74, 6617–6622.

[53] Lapusta, N., J. R. Rice, and R. Madariaga. 2000. Research in progress. [54] Nielsen, S. B., and J. M. Carlson. 2000. Rupture pulse characterization: Selfhealing, self-similar, expanding solutions in a continuum model of fault dynamics. Bulletin of the Seismological Society of America, in press. [55] Renardy, M. 1992. Ill-posedness at the boundary for elastic solids sliding under Coulomb friction. Journal of Elasticity 27, 281–287. [56] Adams, G. G. 1995. Self-excited oscillations of two elastic half-spaces sliding with a constant coefficient of friction. Journal of Applied Mechanics 62, 867–872.

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[57] Martins, J. A. C., J. Guimarães, and L. O. Faria. 1995. Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions. Journal of Vibration and Acoustics 117, 445–451. [58] Simões, F. M. F., and J. A. C. Martins. 1998. Instability and ill-posedness in some friction problems. International Journal of Engineering Science 36, 1265–1293.

[59] Ranjith, K., and J. R. Rice. 2001. Slip dynamics at an interface between dissimilar materials. Journal of the Mechanics and Physics of Solids 49, 341–361. [60] Weertman, J. 1980. Unstable slippage across a fault that separates elastic media of different elastic constants. Journal of Geophysical Research 85, 1455–1461. [61] Adams, G. G. 1998. Steady sliding of two elastic half-spaces with friction reduction due to interface stick-slip. Journal of Applied Mechanics 65, 470–475. [62] Andrews, D. J., and Y. Ben-Zion. Wrinkle-like slip pulse on a fault between different materials. Journal of Geophysical Research 102, 553–571. [63] Ben-Zion, Y., and D. J. Andrews. 1998. Properties and implications of dynamic rupture along a material interface. Bulletin of the Seismological Society of America, 88, 1085–1094. [64] Harris, R., and S. M. Day. 1997. Effects of a low velocity zone on a dynamic rupture. Bulletin of the Seismological Society of America 87, 1267–1280. [65] Cochard, A., and J. R. Rice. 2000. Fault rupture between dissimilar materials: Ill-posedness, regularization and slip-pulse response. Journal of Geophysical Research 105, 25891–25907. [66] Weertman, J. 1963. Dislocations moving uniformly on the interface between isotropic media of different elastic properties. Journal of the Mechanics and

Physics of Solids 11, 197–204. [67] Gol’dshtein, R. V. 1967. On surface waves in joined elastic media and their relation to crack propagation along the junction. Prikladnaya Matematika i Mekhanika 31(3), 468–475 (English translation, Journal of Applied Mathematics and Mechanics 31, 496–502). [68] Achenbach, J. D., and H. I. Epstein. 1967. Dynamic interaction of a layer and a half-space. Journal of Engineering Mechanics 5, 27–42.

[69] Prakash, V., and R. J. Clifton. 1992. Pressure–shear plate impact measurement of dynamic friction for high speed machining applications. Proceedings of VII International Congress on Experimental Mechanics. Bethel, Conn.: Society for Experimental Mechanics, 556–564. [70] Prakash, V. 1998. Frictional response of sliding interfaces subjected to time varying normal pressures. Journal of Tribology 120, 97–102. [71] Lambros, J., and A. J. Rosakis. 1995. Development of a dynamic decohesion criterion for subsonic fracture of the interface between two dissimilar materials.

Proceedings of the Royal Society of London A 451, 711–736. [72] Singh, R. P., J. Lambros, A. Shukla, and A. J. Rosakis. 1997. Investigation of the mechanics of intersonic crack propagation along a bimaterial interface using coherent gradient sensing and photoelasticity. Proceedings of the Royal Society

of London A 453, 2649–2667.

24

Dr. Richard M. Christensen (left), one of the ICTAM 2000 Introductory Lecturers, jokes with Prof. L. Ben Freund, Treasurer of IUTAM, during a poster session. ICTAM 2000 featured 19 Introductory Lectures in 6 Mini-Symposia, and 18 Sectional Lectures.

A SURVEY OF AND EVALUATION METHODOLOGY FOR FIBER COMPOSITE MATERIAL FAILURE THEORIES Richard M. Christensen Department of Aeronautics and Astronautics Stanford University, Stanford, Calif., USA [email protected] Abstract

1.

The long-standing problem of characterizing failure for fiber composite materials will be reviewed. Emphasis will be given to the lamina level involving nominally aligned fibers in a matrix phase. However, some consideration will also be given to laminate failure using the lamina form as the basic building block along with the concept of progressive damage. The many different lamina-level theories will be surveyed along with the commitment necessary to produce critical experimental data. Four particular theories will be reviewed and compared in some detail, these being the Tsai–Wu, Hashin, Puck, and Christensen forms. These four theories are reasonably representative of the great variety of different forms with widely different physical effects that can be encountered; also, for comparison, the rudimentary forms of maximum normal stress and maximum normal strain criteria will be given. The controversial problem of how many different individual modes of failure are necessary to describe general failure will receive attention. A specific and detailed methodology for evaluation of all the various theories will be formulated.

INTRODUCTION

Theories of failure for anisotropic materials have been propounded for at least the past forty years. The advent of high strength, highly anisotropic fiber composite materials has accelerated the activity and accentuated the importance of the search. The lack of agreement on a single, best theory has not been for lack of activity. If one includes all forms of theoretical failure characterization, there are probably well over one hundred different theoretical forms, sometimes applicable over widely different conditions. To put some scope and limits on the present considerations, only reasonably comprehensive theories (not individual 25 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 25–40. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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mechanism theories) will be considered, meaning theories applicable to fully three-dimensional states of stress and strain. That the consideration and evaluation of such theories of behavior is a difficult proposition should be self evident. Let it just be said that even in the case of isotropic materials, the theoretical characterization of failure is not a settled issue. The corresponding problem for highly anisotropic materials could reasonably be expected to be much more difficult than that for isotropy. A sampling of some fully three-dimensional theories of failure is given in Table 1. The sources for these theories are given in the list of references. By no means are the theories limited to the case of just 10 adjustable parameters. There are theories with 15 or more parameters. This situation immediately raises the issue of the degree of practicality for theories with large numbers of parameters. It would appear that somewhere in the range of 7–9 parameters is pushing the upper limit. There is a corresponding problem at the other end of the scale. What is the fewest number of parameters that can capture the immensely complicated interactive physical effects that occur at the threshold of failure? Additionally, in developing theories of failure, one must first make the decision of the basis type, namely stress-based or strain-based. By far the more common are the stress-based forms, and they will receive primary emphasis here. Most of the consideration here will be given to the lamina-level form of aligned fiber composites. After examining this level at some length, consideration will be given to the laminate form, which is used in most applications. Nevertheless the emphasis here is upon the lamina level, since it is the basic building block for most composites applications. At the lamina level, the fiber composite material will be idealized as being

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transversely isotropic, and all theories to be considered will be of that type. Since these are macroscopic theories of failure, the micro-scale failure mechanisms could be due to a wide array of processes related to fiber degradation, matrix degradation, interface failure and all manner of interactive complications. When matrix-controlled modes of failure are designated, these will be implicitly taken to include interface-failureprecipitated events. It is important to acknowledge the existence of an ongoing fiber composite failure theory evaluation program. This commendable effort, organized by Hinton and Soden (1998) and Soden, Hinton and Kaddour (1998) was initiated some years ago and is nearing completion. It considers many more theories than just the sampling shown here in Table 1, and some theories included here are not part of their evaluation. The present evaluation examination is not coordinated with the Hinton, Soden, Kaddour study for two reasons. Their evaluation interests and procedures are entirely of a 2-D, plane-stress nature, whereas the present interests are entirely of a three-dimensional nature. Secondly, their aim is to evaluate 2-D lamina-level theories primarily from laminate-level behavior. Theirs is a rather complex undertaking because of the lamina-to-lamina interaction that occurs in a laminate. Even though the ultimate objective must be to predict laminate-level behav-

ior, it is here felt that the most firm ground for evaluating lamina-level theories is from procedures and results obtained at the same level, that of the lamina. Four of the theories of failure behavior shown in Table 1 are selected here for examination. These are the Tsai–Wu, Hashin, Puck, and Christensen theories, all expressed in terms of stresses. These four theories are quite representative of the great variety of different forms with widely different physical effects that can be encountered. Also, for comparison, the rudimentary forms of normal-stress and normal-strain criteria will be considered. All four of the fully 3-D theories are at the quadratic level of representation. After outlining these theories and showing some aspects of the varied behavior, we will formulate a specific evaluation methodology. The experimental commitment needed to effect the evaluation will receive consideration. Finally, some aspects of laminate-level behavior will be considered.

2.

SPECIFIC THEORIES OF FAILURE

The four theories of failure to be considered here are those of Tsai–Wu (1971), Hashin (1980), Puck (Puck and Schürmann (1998), Kopp and Michaeli (1999)) and Christensen (1997, 1998). Table 2 shows some char-

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acteristics of these four theories. All the theories except the Tsai–Wu form are decomposed into matrix-controlled modes of failure or fibercontrolled modes of failure. The Tsai–Wu form is said to be fully interactive, with all effects folded into one overall mode of failure. In the present context the number of modes of failure is defined as the number of distinct branches that are in evidence in the failure envelope in stress space. These branches may or may not intersect, but if they do, the slopes are discontinuous. Prom Table 1 it is seen that the number of modes of failure range from one to a number equal to the number of parameters in the theory. This variation perhaps as much as anything illustrates the great diversity that is encountered when comparing these theories. An even more graphic form of the differences between the theories is as shown in Fig. 1, which are examples of the failure envelopes in a subspace of stress. The matrix-controlled modes of failure are schematically shown in space, where axis 1 is always in the fiber direction. The forms for the various theories are representations taken from related references. They are not specifically calibrated to identical properties, but rather serve only to illustrate the great variety of different behaviors that are possible. A similar comparison between fiber-controlled modes of failure would likely be even more divergent; however such forms are not readily available in the literature. The four individual theories of failure will now be displayed for comparison purposes. For the most part, the terminology followed will be that of the authors.

2.1.

Tsai–Wu criterion

The Tsai–Wu (1971) theory is often called the tensor polynomial theory since that is exactly what it is. The stress invariants for transversely isotropic symmetry are used in a polynomial expansion up to terms of

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Figure 1 Biaxial stress failure examples.

second degree. The following form is then the failure criterion:

where and are the seven parameters that are to be evaluated from data. Five of the parameters are evaluated directly from the relations

where and and and are the uniaxial tensile and compressive failure stress magnitudes in the axial and transverse directions, respectively, and is the axial shear failure stress. The remaining two parameters, and must be evaluated from more complicated stress condition testing, or they must be estimated; the latter procedure is normally followed (Hahn and Kallas 1992). This Tsai–Wu theory is by far the simplest of the four theories; yet it contains all the same threedimensional terms. To the extent allowed by the form of the invariants,

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it has all stress components interacting with each other through (1) and (2).

2.2. Hashin criteria The Hashin (1980) criteria begin with the second-degree polynomial expansion in the invariants. The failure modes are then decomposed into matrix-controlled and fiber-controlled forms, depending upon which stress components act upon the failure planes, these planes being taken parallel and perpendicular to the fiber direction, respectively. Also, interaction parameter in (1) is taken to vanish. Next each mode is further decomposed into tensile-controlled and compressive-controlled forms, introducing 4 additional parameters. Finally, 4 separate assumptions or conditions are imposed, bringing the total parameter count to 6.

Thus, the final forms are composed of 4 modes of failure along with 6 parameters, as follows: Tensile matrix mode,

Compressive matrix mode,

Tensile fiber mode,

Compressive fiber mode,

The 6 parameters have been evaluated from the fiber failure normal and shear stresses in the axial direction,

and the matrix-controlled normal and shear stresses in the transverse direction,

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As seen in (3) and (4), the tensile- and compressive-type matrix modes of failure are differentiated by the sign of the transverse direction mean normal stress.

2.3.

Puck criteria

The Puck criteria have evolved over a period of many years. The works by Puck, his students, and colleagues are here represented by Puck and Schürmann (1998) and Kopp and Michaeli (1999). Whereas the Hashin criterion was somewhat motivated by the Coulomb–Mohr approach for isotropic materials, the Puck procedure goes further along this direction and follows the Coulomb–Mohr procedure quite strictly, at least insofar as matrix-controlled failure modes are concerned. The Puck criteria are the most complicated forms considered here, ultimately resulting in numerical procedures. First consider the matrix-controlled modes of failure involving failure planes parallel to the fiber direction, with the corresponding normal and shear stresses upon the failure plane. A failure criterion as shown in Fig. 2 is taken in terms of the axial shear stress, the transverse shear stress, and the normal stress due to the transverse normal stresses. This

Figure 2 Puck theory failure surface.

procedure introduces 7 parameters. Next, all failure plane orientations are scanned to find the failure plane orientation with the worst combination of normal and shear stresses that produces failure, when compared with the failure criterion of Fig. 2. The end result of this numerical scanning program is the generation and display of failure surfaces in stress space. Seven different modes of failure are identified by this process.

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With regard to the fiber-controlled modes of failure, these are written in plane stress form with fiber-direction strain as

for tension, and a similar expression for compression. The two parameters are the strains to failure, and Symbols and are the fiber-phase Poisson’s ratio and Young’s modulus, respectively. The “magnification” factors are said to be known for different fiber types.

2.4.

Christensen criteria

The procedure followed by Christensen (1997, 1998) starts with starts with the general 7-parameter polynomial expansion of second degree. Micromechanics is used to decompose the polynomial form into 2 separate criteria—matrix- and fiber-controlled. Then independence of both these criteria from failure under hydrostatic pressure is imposed. This independence reduces the parameter count from 7 to 5. The matrix-controlled criterion is given by

where

with and being the transverse normal stress failure levels and the axial shear stress failure level. Condition applies here. The fiber-controlled criterion is given by

where

and where are the fiber-direction failure stresses. This fibercontrolled criterion decomposes into two separate branches, as seen in particular applications.

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3.

33

EVALUATION METHODOLOGY

Probably the most contentious argument that ensues when discussing composite material failure theories is how many different modes of failure should be expected. The sampling of different theories shown in Tables 1 and 2 ranges from 1 mode of failure through 9 modes of failure. This discussion or argument probably does not have a simple, easily established answer. Arguing the merits or demerits of various theories on a conceptual basis seems particularly open-ended and nonproductive. The approach to be sought here avoids this argument by seeking a critical comparison of the theories with experimental data. The question then becomes one of what type of experimental data could be used for such an important purpose. The types of experiment to be used to evaluate the theories is intimately connected with the character of the experiments used to evaluate the determining parameters in each and every theory. An informal survey of experimentalists has indicated a consensus that the maximum number of independent strength-property experiments that is practical under most circumstances is about five. Furthermore, the most practical explicit set of experiments is that used to determine the set of failure properties given by

which are the fiber-direction tensile and compressive strengths, the transverse tensile and compressive strengths, and the axial shear strength, respectively. These are the one-dimensional tests that embody current, standard practice. The method for evaluating the various theories is that they should be evaluated in the immediate, near region of the five data points used to calibrate the theories. If a given theory cannot give a successful prediction of behavior in the neighborhoods of the data points used for the calibration of the theory, then that theory is unlikely to be generally successful in the stress-state regions far removed from the data calibration points. In selecting a criterion to be used for the evaluation, there could be several options, but one particularly attractive one, and the one followed here, is that of the effect of superimposed hydrostatic pressure. For example, if the test used to calibrate a given theory were to give a value of the transverse tensile strength as then the evaluating experiment would involve a second testing procedure to determine by how much the transverse tensile strength is changed by the presence of a superimposed pressure. And, the superimposed pressure should be small in magnitude so that one is probing the near neighborhood of

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the data point. Testing under superimposed pressure is a well developed and well validated technique that has been extensively used with isotropic materials. The technique is particularly appropriate for composite materials, since the common characteristics of and are direct manifestations of the effect of and importance of the mean-normal-stress effect. The pressure-effect forms corresponding to the properties (10) that are to be determined experimentally are

where p is the superimposed pressure. After the experimental database for these is established, any theory calibrated by the properties data (10) would then be used to predict the values for (11). Examples of this procedure will be given shortly. It may be noted that the procedure just stated corresponds to examining the first two terms of a Taylor series. The theory calibration data (10) correspond to the first term; the derivatives in (11)—both theory and measured—correspond to the second term. It should be recognized that this proposed approach would not constitute a comprehensive evaluation, but rather would be the first step in such a direction. An overall evaluation would likely be a graduated process. For example, the possible coupling of the fiber-controlled modes of failure with the axial shear stress is of potential significance and possible controversy. Nevertheless, the present approach could be an important first step through which many or most theories could be eliminated from consideration. It also should be noted that any theory with any number of parameters could be evaluated by this method. If the number of parameters were greater than 5, the extra parameters would need to be evaluated by auxiliary information beyond that of (10), as has always been done in such cases. It is helpful to change the notation slightly before illustrating the method with examples. We write the total stress tensor as

where p is the applied pressure and is the difference between the total stress and the pressure-induced stress. With (12) the derivatives are related by

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Rather than using the total stress derivatives in (11) for the evaluation, we use the derivatives of the stress difference given in (13). Let

The advantage of the notation in (14) is that a value of is indicative of a Mises-like behavior involving independence of mean normal

stress. Three examples will now be given, the first two being quite simple, but still relevant. Two of the oldest failure criteria are those imposed upon normal stress and normal strain. We take these criteria for the fiber direction stress as

The derivatives in (14) are easily shown to be, from (15),

These results do not even differentiate between different composite materials with different properties. Using the data from Parry and Wronski (1982, 1985) for graphite–epoxy composites, one finds to

– 1.1 and

to 0.42. Based upon these rather old data sets,

even the sign of (16) is incorrect, while the scale of (17) appears to be too large. Now consider normal strain in the fiber direction as the failure criterion:

It can be shown that for (18) the derivatives (14) are given by

where is the axial Poisson’s ratio. Compared with the data of Parry and Wronski mentioned above, relation (19) appears to be of the incorrect sign. These two examples show that this evaluation method is highly discriminating. But before any conclusions can be drawn, it would be necessary to have modern, highly reliable data. Now a more comprehensive example will be given to show that the procedure is well posed and entirely practical. Using the failure criteria of Christensen, (8) and (9),

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one can determine the derivatives in (14) in a straightforward manner, giving

Typical data for graphite–epoxy composites are given by

Using data set (22) in the derivatives (21) gives

It is the specific results such as in (23) that are to be compared with modern experimental measurements, permitting the formal evaluation. These examples illustrate the well grounded and basic nature of this program for composite failure-theory evaluation. If an advocate of a particular failure theory should allege that it is too difficult to evaluate the derivatives in (14), then there could be cause to doubt the utility and practicality of the theory under question. The vital need for well characterized failure data will be considered in the last section. These four theories plus the simple normal stress and strain criteria are employed here as samples, albeit significant ones. Any theory could be subjected to this evaluation methodology. It is possible to build up a picture of overall behavior through the combination of individual modes of failure, although these four theories do not illustrate that possibility. Such a procedure has been recommended by Hart-Smith (1993), and any overall theory of behavior so formed could also be evaluated by this method.

Fiber composite materials failure theories

4.

37

LAMINATES Treating the failure of laminates is inherently much more complicated

than treating the failure of a lamina. Nevertheless, there is a widely used approach for the case of laminate failure. This is usually called progressive damage, and it is a natural extension of lamina-level failure theory. The simplest form is that of first-ply failure, but that approach is too conservative for most purposes. The decomposition of moduli-type and failure-type properties into matrix-controlled and fiber-controlled

modes is the motivation for progressive damage. With strain compatibility in adjacent lamina within a laminate, the damage spreads from lamina to lamina. Matrix failure occurs at much lower strains than fibercontrolled effects, so the matrix-controlled degradation usually starts the process. As loading continues, the properties must be degraded selectively or collectively. Then, when fiber-controlled failures occur over a

sufficiently large region, structural failure follows. One could use fracture mechanics to motivate and describe a critical scale of local failure beyond which uncontrolled fracture follows. This properties-degradation process works fairly well; it is semi-empirical, but well established. The transverse cracking at the lamina level can lead to delamination between lamina. Although appealing in its step-by-step approach, the method can become quite complicated, with many pitfalls and traps. The status of the progressive-damage methodology is probably best described by Rohrauer (1999) in a Ph.D. thesis of unusual scope and gravity. He

states: “The quest to ascertain what is happening inside a failed lamina and how it affects the continued existence or catastrophic destruction of

the whole laminate is a continuing one. Few subjects are as confusing and complex. The search for answers has lead to voluminous publications; few if any definitive methods useful to the designer exist as of yet.”

Certainly significant steps have been taken to define the lamina-level damage problem in precise terms that admit extension and generalization. These works include the essentially lamina-level matrix-cracking problem by Hashin (1996) and Nairn and Shu (1994). An alternative to the lamina-to-laminate failure sequencing through progressive damage can be seen. This alternative would be to treat

failure entirely at the laminate level and thereby obtain a complete, self-contained treatment. This approach apparently has not been taken in the past. It has more-or-less been taken as obvious truth that the laminate level is too difficult to approach directly, and that it must be approached incrementally through progressive damage. That view notwithstanding, the full problem should be approached full on; with

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the proper metrics and invariants, there is a reasonable case to be made for optimism.

5.

CONCLUSION

The overall conclusion is that it is possible and practical to compare and evaluate fiber composite material failure theories at the lamina level. This approach would avoid the complications of evaluating lamina-level theories from behavior at the laminate level, which inter-

mixes all the assumptions and idealizations of the lamina-level theory with all the assumptions and idealizations of any particular progressive damage scenario. In reality, two separate and distinct evaluations should be conducted—that for the lamina-level failure theory, and that for the

progressive damage treatment of laminates. The present work has been focused upon the lamina-level evaluation. The specific recommendations are as follows: (i) One should collect the various failure theories that are to be

considered—four of them have been discussed here—and reduce them to the form that predicts the quantitative effect of a superimposed pressure upon the five basic strength characteristics for the aligned fiber lamina form.

(ii) To calibrate and normalize the theories, one should experimentally determine the five basic strength properties for the fiber composite systems of interest, repeat the experimental determination of the five strength properties under the effect of a superimposed pressure, and combine results from (i) and (ii) to evaluate the laminalevel theories. It is of great importance that the experimental investigation receive the highest possible priority. The detailed

evaluation will be no better than the quality and reliability of the data. An enormous effort has been expended on the development of the many different theoretical forms over many years. A commensurate effort should be dedicated to the experimental investigation in order to complete a meaningful evaluation process.

(iii) As a completely separate matter, the progressive damage methodology should be more highly developed; then the various proposed steps and procedures should be evaluated. (iv) As another completely separate but related matter, failure characterization should be formulated directly at the laminate level, and ultimately compared with projections and predictions obtained from lamina-level theory combined with progressive damage.

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Acknowledgments The author is appreciative of many helpful discussions with Dr. S. J. DeTeresa of LLNL. The author is also appreciative of support from the Office of Naval Research, Dr. Y. D. S. Rajapakse, contract monitor. This work was performed under the auspices of the U.S. Department of Energy, University of California Lawrence Livermore National Laboratory, under contract W-7405-Eng-48.

References Boehler, J. P., and M. Delafin. 1979. Failure criteria for unidirectional fiber-reinforced composites under confining pressure. Proceedings of the Euromech Colloquium 115. The Hague: Martinus Nijhoff, 449–470. Christensen, R. M. 1988. Tensor transformations and failure criteria for the analysis of fiber composite materials. Journal of Composite Materials 22, 874–897. Christensen, R. M. 1997. Stress based yield/failure criteria for fiber composites. Inter-

national Journal of Solids and Structures 34, 529–543. Christensen, R. M. 1998. The numbers of elastic properties and failure parameters for fiber composites. Journal of Engineering Materials and Technology 120, 110–113. Cuntze, R. G. 1999. Progressive failure of 3d-stressed laminates: Multiple nonlinearity treated by the failure mode concept. Duracosys Conference Manuscript (from the author). Feng, W. W. 1991. A failure criterion for composite materials. Journal of Composite Materials 25, 88–100. Gosse, J. 1999. Boeing Aerospace. Private communication.

Hahn, H. T., and M. N. Kallas. 1992. Failure criteria for thick composites. Ballistic Research Laboratory Report CR-691, Aberdeen Proving Ground, Md. Hart-Smith, L. J. 1993. Should fibrous composites failure modes be interacted or superimposed? Composites 24, 53–55. Hashin, Z. 1980. Failure criteria for undirectional fiber composites. Journal of Applied Mechanics 31, 223–232. Hashin, Z. 1996. Finite thermoelastic fracture criterion with application to laminate cracking analysis. Journal of the Mechanics and Physics of Solids 44, 1129–1145. Hinton, M. J., and P. D. Soden. 1998. Predicting failure in composite laminates: The background to the exercise. Composites Science and Technology 58, 1001–1010. Kopp, J., and W. Michaeli. 1999. The new action plane related strength criterion in comparison with common strength criteria. Proceedings of the International Conference on Composite Materials 12, Paris. Nairn, J. A., and S. Shu. 1994. Matrix microcracking. In Damage Mechanics of Composites (R. Talreja, ed.). New York: Elsevier Science, 187–243. Parry, T. V., and A. S. Wronski. 1982. Kinking and compressive failure in uniaxially aligned carbon fiber composite tested under superimposed hydrostatic pressure. Journal of Materials Science 17, 893–900. Parry, T. V., and A. S. Wronski. 1985. The effect of hydrostatic pressure on the tensile properties of pultruded CFRP. Journal of Materials Science 20, 2141–2147. Puck, A., and H. Schürmann. 1998. Failure analysis of FRP laminates by means of physically based phenomenological models. Composites Science and Technology

58, 1045–1067. Rohrauer, G. 1999. Ultra-high pressure composite vessels with efficient stress distributions. Ph.D. thesis, Concordia University, Montreal, Canada.

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Soden, P. D., M. J. Hinton, and A. S. Kaddour. 1998. A comparison of the predictive capabilities of current failure theories for composite laminates. Composites Science and Technology 58, 1225–1254. Tsai, S. W., and E. M. Wu. 1971. A general theory of strength for anisotropic materials. Journal of Composite Materials 5, 58–80.

DAMAGE MECHANICS OF METAL-MATRIX COMPOSITES: VARIOUS LEVELS OF APPROACH Jean-Louis Chaboche Département de Mécanique du Solide et de l’Endommanagement (DMSE) Office National d’Etudes et de Researches Aérospatiales (ONERA) Châtillon Cedex, France and Université de Technologie de Troyes [email protected]

Abstract

1.

Constitutive and damage mechanics of composite materials is discussed, with a special attention to unidirectional metal-matrix composites. Macroscopic thermo–elasto–visco–plastic and damage models based on micromechanical approaches are presented that offer various modeling capabilities between a two-phase and a multi-subvolume model. A micromechanics analysis of transverse creep, using periodic homogenization and finite elements demonstrates the role of the fiber–matrix interphase in the creep resistance of the composite. Such a numerical approach can be used to deliver directly the constitutive response at the component level by a multiscale structural analysis using the FE2 imbricated finite-element method.

INTRODUCTION

Composite materials and structures differ from metallic ones in the following aspects: their low weight, and the corresponding design improvement perspective offered by their high specific mechanical properties; their versatility—in many applications, the possibility to optimize the material (volume fraction, sizes, positions, orientations of reinforcements) during the same design stage as the structural component itself; their ability to support such inserted systems as optical fibers, piezo-electric sensors, and actuators, leading to many future “smart structure” capabilities, for instance in active or passive in situ health monitoring systems; 41 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 41–56. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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Figure 1

Schematic of the various scales in composite materials and structures.

their cost, which is often a serious obstacle to their extensive use, as well as the relatively unstable material sources and material manufacturers; their multiscale structure, showing many different successive levels, as illustrated schematically in Fig. 1 for structural composites used in aeronautical applications—this is one of the difficulties in the mechanical analysis associated with the use of composites; metals also have several levels of microstructures, but at lower scales, and with a much more random organization; their damaging processes, during manufacturing and during operation: in metals initiation of any microcrack of some size could have a significant impact on the remaining lifetime, leading often to an accelerated process under load; contrarily, in composite structures the microcracks can appear very early, leading to a certain deterioration of mechanical properties, but without really affecting the component lifetime—damage in that case is one of the mechanisms by which the heterogeneous material accommodates the applied strain.

In this paper the two last aspects mentioned above will be illustrated and discussed in terms of modeling methodologies, as applied to unidirectional metal-matrix composites (MMCs). The specific structural

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applications underlying the developed methods are the bling components (bladed rings) that should replace in some aerojet engines the classical technologies for compressor and turbine discs. The material in the corresponding experimental studies is a SiC/Ti MMC, but many aspects of the methods are still applicable to other classes of metallic composites (at least long fibers). In 2 we present a micromechanics-based set of constitutive and damage equations, exploiting Dvorak’s transformation field analysis (TFA) method (Dvorak 1992, Dvorak and Benveniste 1992), which offers several variants for the damage-growth modeling—in the matrix, in the fiber, and at the fiber–matrix interface. The models are set in such a way that they can be considered and exploited as macroscopic constitutive laws. Section 3 summarizes a specific study of creep behavior of the composite (essentially transverse creep), in which is shown the duality between the matrix elasto–visco–plasticity and the interfacial damage, in order to explain experimental results of creep tests realized under vacuum. Other aspects are treated elsewhere, for instance the component inelastic and damage analysis, based on a true “multiscale analysis”, the FE2 method, using imbricated finite-element models in order to replace the macrolevel constitutive equations directly (Feyel 1999).

2.

DEVELOPMENT OF A SET OF MICROMECHANICS-BASED CONSTITUTIVE EQUATIONS

For the inelastic and damage modeling of composite materials, within a micromechanics based approach, we have various possibilities:

the tools using averaged quantities in each constituent, similar to those employed for random microstructures, with Eshelby-based operators: in this context localization rules are often based on a secant or tangent writing of the local constitutive equations; the more-or-less analytical models based on a mixture of tangent and initial elastic operators, such as the ones developed by Voyiadjis and co-workers (Voyiadjis and Kattan 1993) or by Baptiste (Guo et al. 1997);

the class of model developed by Aboudi and co-workers (Aboudi 1985, Pindera and Aboudi 1988), which defines a geometrically more-or-less sophisticated unit cell, with local fields approximated through a weak compatibility of constituent elasticity equations; the transformation field analysis (TFA) developed by Dvorak (Dvorak 1992, Dvorak and Benveniste 1992), which introduces a

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systematic way of defining concentration and influence tensors, but writing a purely elastic rule, treating plastic strain and thermal expansion as eigenstrains of the same nature.

We decide to concentrate on the last type of approach, because it continues to treat elasto–plasticity (or elasto–visco–plasticity) of constituents through the correct strain partitioning. Moreover, within a single and consistent methodology, it offers various refinement possibilities, from the two-subvolume model (two-phase model) to the multisubvolume model, which eventually degenerates to the finite-element periodic unit cell.

2.1.

Thermo–elasto–visco–plastic model

The model is built up by combining two kinds of equation. The first is the constitutive equation of each constituent:

in which s denotes the subvolume numbering (there are possibly several subvolumes for each phase), Ls is the initial elastic stiffness operator, and the functional will not be defined more precisely—all kinds of local constitutive equations can be used, with internal state variables symbolized here as “ISV”. The thermal expansion can be isotropic or not. The second kind of equation is the localization rule that relates overall to local quantities, taking into account field interactions with eigenstrains. Using TFA, we have for the local strain in the subvolume s:

where is the average strain (in which cr denotes the volume fraction of subvolume r), As and Dsr being respectively the elastic strain concentration tensor and the influence tensor (between subvolume r and subvolume s). Averaging in (1) leads to the overall elastic constitutive equation

where tration tensor by L = =

with B the stress concenand L the overall elastic stiffness defined The reciprocal formulation can also be defined

Damage mechanics of metal-matrix composites

Figure 2

45

Correction with the asymptotic tangent stiffness for various macroscopic

strain rates (volume fraction 35%).

consistently for the stress localization (not needed here). In the case of a two-phase composite system, with one subvolume each, the operators A s , Bs can be obtained from the Eshelby tensor, either through the Mori–Tanaka method (Mori and Tanaka 1973) or through the selfconsistent method. The influence tensors are also easily determined for the two-subvolume model, with exact closed-form relations (Dvorak 1992). When using a more sophisticated model, with several subvolumes per phase, the initial elastic concentration tensors A s , B s and the influence tensors Dsr are determined once, by a prior set of numerical analyses (using for instance periodic homogenization and finite-element method, as indicated in Dvorak et al. 1994 or Pottier 1998). Considering the overly stiff results generally obtained for the twosubvolume model (see Suquet 1997, Zaoui and Masson 1998), we have proposed an asymptotic correction method (Pottier 1998, Chaboche et al. 2000) that modifies the localization rule into

where the 4th-rank tensors Kr are determined by identification of the rate form of (5) with its “tangent” format in which the tangent concentration tensor is determined from the knowledge of the asymptotic tangent stiffness of each subvolume, directly related with

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the linear kinematic hardening model used in the matrix. More details are given in (Chaboche et al. 2000). The improvement of using this correction factor instead of the original TFA is illustrated in Fig. 2 for the transverse traction at three overall strain rates (the matrix is visco–plastic) by comparison with a periodic unit cell finite-element calculation similar to the ones that will be reported in 3. Other comparisons are reported elsewhere (Chaboche et al. 2000).

2.2.

Various damage modeling approaches

In terms of delivering a micromechanics-based overall constitutive equation of composite materials, three approaches could be used, depending on the kind of deterioration that develops:

If microcracks are large enough compared with the heterogeneous microstructure (fiber size) we could proceed in two steps: define first an effective homogeneous behavior of the undamaged composite, then input microcracks in it and deliver the overall behavior of the damaged material via a second micro-to-macro analysis. This is the method implicitly used in laminate composites when introducing transverse cracking as a meso-damage in the ply volume elements (Allix et al. 1990). If microcracks are small enough, in the matrix or in the fiber, compared with the fiber size, we can use a continuum damage mechanics (CDM) methodology in each of these phases, then first modify the initial local constitutive equations through a micromechanicsbased effective stress concept, and second, make the micro–macro averaging of the composite with its damaged constituents. This is for example the method used by Voyiadjis (Voyiadjis and Kattan 1993) for modeling damage in MMCs.

If microcracks and fiber are of the same size, especially when the crack develops at the fiber–matrix interface, none of the above approaches is acceptable (i.e. there is no scale separability). One simplification consists in approximating the presence of an interface crack by a transverse stiffness reduction in the fiber (Guo et al. 1997). It is also the solution adopted in the direct simplified model developed below. A better approximation is sought by generalizing the TFA method, with a large number of subvolumes, including interphase subvolumes. We present below the corresponding model, called the generalized eigenstrain model.

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Figure 3 Direct simplified model and finite-element computation for transverse tensile loading–unloading: case of an elastic matrix.

Direct simplified model (DSM). We consider only two phases— the matrix and the fiber—and assume a CDM approach to introduce a damage ds in each phase (either a scalar or a tensorial variable). Now the constitutive equation is modified as

The visco–plastic strain rate depends on damage, as well as the elastic stiffness . Figure 4 explains the principle of the direct simplified model. Knowing damage ds after integration (i), we define (e) the effective damaged elastic stiffness (not expressed here). Using closedform Eshelby solutions and the Mori–Tanaka scheme, we can define the effective strain concentration tensor . For a two-phase system, we know explicitly the corresponding from and (Dvorak 1992). The localization rule (l), Eqn. (3) (with and instead of and D sr ), delivers from the controlled overall strain E. Then, we obtain successively by the elastic constitutive equation (ce) and by averaging (h). In this model, the localization tensors have to be defined at each time step of a given loading. It is the reason that the model cannot be generalized to a refined discretization (more than two subvolumes). Figure 3 gives an example of comparison between this direct simplified model and a complete analysis of a unit cell by finite-element and periodic homogenization. This is a theoretical example of transverse tension–compression of a unidirectional elastic metal-matrix composite. The matrix is elastic, the fiber is elastic damageable, and damage deactivation is taken

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Figure 4. Flowchart of the two damage models.

into account (closure of microcracks under compression). The comparison appears as excellent in that case, as well as in the case where the visco–plasticity of the matrix is taken into account.

Generalized eigenstrain model (GEM). This model is developed in order to refine the discretization. Subvolumes are used in order to refine the discretization, to describe the interphase behavior (with damage) and the stress redistributions in the matrix. The model uses only the initial (undamaged) concentration and influence tensors of TFA, obtained initially by a number of elastic finite-element calculations. Its key specificity is to transform the elastic changes due to damage into an additional eigenstrain. We replace (6) by

where the damage strain

is given as

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49

Figure 4 indicates the corresponding flowchart of the model. It uses the

following localization equation in place of (5):

We remark that the approach is also able to take into account the variations of the elastic stiffness with temperature (without modifying A s ,

D sr ). Its flexibility and its comparison with experiments appear to be very satisfactory. When applied in the particular case of two subvolumes, the GEM and DSM models deliver exactly the same stress–strain response.

3.

3.1.

MICROMECHANICAL NUMERICAL MODELING OF TRANSVERSE CREEP BEHAVIOR Experimental study

Silicon carbide–titanium MMCs will be used in the future bling technology as the reinforcement to sustain the high hoop stresses generated by centrifugal effects. The essential characteristics (tensile strength, fatigue, creep) have been studied mainly in the axial direction of uni-

directional composites (Ohno et al. 1994). However, transverse creep is a condition that may be of some significance, associated with secondary centrifugal radial stresses generated by external parts and the blades. An experimental study has been performed at ONERA, in order to characterize the creep resistance of the composite under transverse loads (Carrère et al. 2000). The material is a titanium (Ti6242) matrix reinforced with long unidirectional SiC (SCS6) fibers with a volume fraction of 35%. The specimens used for the experiments are taken from an 8-ply plate manufactured by Snecma, using the fiber-foil technique. The behavior of the titanium matrix is elasto–visco–plastic, the fibers are supposed to remain elastic, and the protecting coat around the fibers (to prevent a reaction between the titanium and the SiC) is a brittle zone where the decohesion and damage in the transverse direction of the composite will take place. Creep tests were performed at 500°C, under vacuum to avoid environmental problems, for several loads (Carrère et al. 2000). Results obtained are shown in Fig. 5 for the secondary creep rate, and can be summarized as follows:

Below a critical stress, estimated here around 225 MPa, the creep rate increases gradually and the lifetime is long—there is no rupture even for long creep time (> 1500 h).

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Figure 5 Measured secondary creep rates of SCS6/Ti6242 at 500°C.

For applied stresses over 225 MPa, the creep rate increases drastically, leading to short failure times.

3.2.

Micromechanical analysis

A micromechanical analysis based on the periodic homogenization assumption and the finite-element method is performed in order to explain the experimentally observed behavior. The regular position of the fibers

Figure 6 Composite microstructure, definition of a unit cell, and axial visco–plastic strain field after 1000 hours creep under 270 MPa.

inside the matrix, as shown in Fig. 6, allows us to select a unit cell that defines the representative volume element of the composite.

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51

The constitutive behaviors considered for each constituent, fiber, matrix and interphase, are as follows: The fiber is elastic and transversely isotropic. Its characteristics are given by literature data and independent measurements.

The matrix is elasto–visco–plastic, obeying a classical isotropic cyclic constitutive equation that incorporates two back stresses, one of which obeys the nonlinear kinematic hardening rule, the second being linear. This constitutive equation has been determined previously from tension–compression tests performed on pure Ti6242 over the whole temperature domain (20–870°C) (Baroumes and Vincon 1995). The effects of viscosity observed at high temperature are described with a power-law equation determined from relaxation (Malon 2000) and creep tests (Carrère et al. 2000). The carbon interphase is treated using interface elements that obey Tvergaard’s progressive debonding model (Tvergaard 1990). In fact we use a modified model (Chaboche et al. 1997), which introduces friction effects during the debonding phase, in order to eliminate the complete shear unloading when debonding is complete. The model combines mode I and mode II decohesion, with a unique damage parameter which is the maximum value of the quadratic norm of relative mode I and mode II openings (opening displacement divided by its value at complete debonding). It uses 4 parameters (for each mode a maximum stress and a fracture energy), in addition to the Coulomb friction parameter . The mode II and friction parameters are determined by push-out tests (Guichet 1998) and the complete numerical analysis of this test. The mode I is adjusted from the transverse tensile test. The corresponding simulations allow us to describe fairly well all the important features: onset of decohesion before plasticity takes place, progressive evolution of the unloading stiffness, then appearance of plasticity and irreversible strains (with hysteretic effects under cyclic loads), interfacial closure effects when reverse loadings are performed, etc.

The numerical simulation performed on the unit cell presented in Fig. 6(b) follows as closely as possible the experimental procedure. The main steps of the calculation are summarized as follows: (i) The first step consists in determining the residual stresses induced by the manufacturing process, and applying the load/temperature history undergone by the composite (referred to as “manufacturing” in the following curves).

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(ii) The creep load is imposed, giving the response of the primary state (referred to as “loading”), (iii) Finally, the creep test itself is simulated (secondary stage). The third stage, directly linked to the final failure of the specimen, is not modeled because, in this analysis, the crack initiation and propagation in the matrix are not taken into account. Figure 7 presents simulated results for two loadings: the first one, 200 MPa, below the critical stress, and the second one, 270 MPa, above this stress. The figure shows the evolution of the damage parameter around the interface between the fiber and the matrix, at different out-

Figure 7 Local interfacial damage evolution for creep under 200 and 270 MPa.

put times corresponding to the steps described before. It can be seen that the manufacturing process does not induce significant damage. Once the load is applied, damage starts and increases progressively. The comparison between the damage evolution induced by the two loads shows that for low levels, damage increases slowly (and sometimes saturates) during the creep without breaking the interface, though for higher loads the interface begins to break and this process continues until the complete interface debonds. Additional results are discussed in (Kruch et al. 2000). Therefore, the micromechanical analysis confirms that the matrix controls the global deformation of the composite, but the load carried by the matrix is controlled by the strength of the interface. For high stress levels (near the critical stress), the load is carried totally by the matrix (the interface being completely broken), leading to the failure of the composite when a crack initiates and propagates in the matrix. For lower stress levels the interfacial damage is not very important and the load is carried partly by the fibers and the matrix, leading to long lifetimes. The calculated secondary creep rates underestimate the experimental ones by a signigicant factor. There are two main explanations: (1) the

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interface debonding model was determined at room temperature and applied without change for high temperatures (taking into account only the different thermally induced residual stresses), and (2) the titanium matrix was identified from tests on a standard monolithic material whose microstructure could be different from the one of the foil material used

to manufacture the composite. The anisotropic texture of the matrix is right now analyzed at ONERA using the electron back-scattering diffraction (EBSD) technique. Numerical computations are under way, taking into account the measured textures in the matrix visco–plastic behavior, by using a polycrystalline aggregate model (Pilvin and Cailletaud 1990).

4.

CONCLUSIONS

The considered damage modeling approaches have shown the following capabilities for MMCs or similar composite situations: Micromechanics-based constitutive equations using the TFA approach are able to describe both thermo–elasto–visco–plasticity of the matrix and the damage evolution in each constituent. Two damage approaches—the direct simplified model and the generalized eigenstrain model—with different capabilities in terms of

degrees of freedom, are shown to be consistent with periodic homogenization results. Their application in true components as overall constitutive and damage equations is underway.

The finite-element micromechanics analysis of transverse creep loading conditions is able to predict quite well the experimental results. It assumes periodicity and uses a thermo–elasto– visco–plastic constitutive equation for the matrix and an interface debonding model for the fiber–matrix interphase (determined independently by push-out and transverse tensile tests). Improvements in the quantitative prediction of the secondary creep rate is expected by taking into account the true matrix visco– plastic behavior, including texture effects. Such an analysis is underway, using a polycrystalline aggregate model. A higher-level multiscale analysis of components (not presented) has also been developed (Feyel 1999). The method uses imbricated finite elements in order to replace the overall constitutive equations in a finite-element structural analysis. This method has the objective to deliver all the stress–strain redistributions at the microscale, especially in regions where the two scales do interact, very often the most critical regions of the component.

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References Aboudi, J. 1985. Inelastic behaviour of metal-matrix composites at elevated temperature. International Journal of Plasticity 1(4), 359–372. Allix, O., P. Ladevèze, E. L. Dantec, and E. Vittecoq. 1990. Damage Mechanics for composite laminates under complex loading. In Yielding, Damage and Failure

of Anisotropic Solids (J. Boehler, ed.). London: EGF5, Mechanical Engineering Publications, 551–569. Baroumes, L., and I. Vincon. 1995. Identification du comportement de l’alliage Ti6242. Technical Report contrat Snecma/LMT 762 593F, LMT-Cachan. Carrère, N., S. Kruch, R. Valle, and A. Vassel. 2000. Transverse creep behaviour of a unidirectional SCS-6/Ti-6242 composite. In Composites from Fundamentals to Exploitation, Proceedings of the 9th European Conference on Composite Materials. Brighton. Chaboche, J. L., R. Girard, and A. Schaff. 1997. Numerical analysis of composite systems by using interphase/interface models. Computational Mechanics 20, 3–11. Chaboche, J. L., S. Kruch, J. F. Maire, and T. Pottier. 2000. Towards a micromechanics based inelastic and damage modeling of composites. International Journal of Plasticity, to appear.

Dvorak, G. 1992. Transformation field analysis of inelastic composite materials. Proceedings of the Royal Society of London A 437, 311–327. Dvorak, G., Y. Bahei-El-Din, and A. Wafa. 1994. Implementation of the transformation field analysis for inelastic composite materials. Computational Mechanics 14, 201–228. Dvorak, G., and Y. Benveniste. 1992. On transformation strains and uniform fields in multiphase elastic media. Proceedings of the Royal Society of London A 437,

291–310. Feyel, F. 1999. Multiscale FE2 elastoviscoplastic analysis of composite structures. Computational Materials Science 16, 344–354. Guichet, B. 1998. Identification de la loi de comportement interfaciale d’un composite SiC/Ti. Doctorat d’Université, Ecole Centrale de Lyon.

Guo, G., J. Fitoussi, D. Baptiste, N. Sicot, and C. Wolff. 1997. Modelling of damage behaviour of a short-fiber reinforced composite structure by the finite element

analysis using a micro–macro law. International Journal of Damage Mechanics 6(3), 278–316. Kruch, S., J. L. Chaboche, and N. Carrère. 2000. Micromechanics based creep damage analysis of unidirectional metal matrix composites. In IUTAM Symposium on Creep in Structures, Nagoya, Japan.

Malon, S. 2000. Caractérisation des mécanismes d’endommagement dans les composites à matrice métallique de type SiC/Ti. Doctorat d’Université, ENS Cachan. Mori, T., and K. Tanaka. 1973. Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica et Materialia 21, 597–629. Ohno, N., K. Toyoda, N. Okamoto, T. Miyake, and S. Nishide. 1994. Creep behavior of a unidirectional SCS-6/Ti-15-3 metal matrix composite at 450°C. Journal of Engineering Materials Technology 116, 208–214. Pilvin, P., and G. Cailletaud. 1990. Intergranular and transgranular hardening in

viscoplasticity. In IUTAM Symposium on Creep in Structures (M. Zyczkowski, ed.) (Cracow, Poland). Berlin: Springer-Verlag, 171–178. Pindera, M. J., and J. Aboudi. 1988. Micromechanical analysis of yielding of metal matrix composites. International Journal of Plasticity 4(3), 195–214.

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Pettier, T. 1998. Modélisation multiéchelle du comportement et de l’endommagement

de composites à matrice métallique. Doctorat d’Université, Ecole Nationale des Ponts et Chaussées. Suquet, P. 1997. Effective properties of nonlinear composites. In Continuum Micromechanics (P. Suquet, ed.) (Volume 377 of CISM Lecture Notes). New York: Springer-Verlag, 197–264. Tvergaard, V. 1990. Effect of fibre debonding in a whisker-reinforced metal. Materials Science and Engineering A 125, 203–213. Voyiadjis, G. Z., and P. Kattan. 1993. Damage of fiber reinforced composite materials with micromechanical characterization. International Journal of Solids and Structures 30(20), 2757–2778. Zaoui, A., and R. Masson. 1998. Modelling stress-dependent transformation strains of heterogeneous materials. In IUTAM Symposium on Transformation Problems in Composite and Active Materials (Cairo, Egypt). Dordrecht: Kluwer Academic Publications, 3–15.

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University of Illinois at Urbana-Champaign Department of Theoretical and Applied Mechanics administrative secretary Carol J. Porter (left) assembles submitted papers with assistant Lori C. Melchi and Prof. James W. Phillips for distribution to the various reviewing bodies and the International Papers Committee. Nearly 2000 papers were submitted to ICTAM 2000.

METALLIC FOAMS: STRUCTURE, PROPERTIES, AND APPLICATIONS Lorna J. Gibson Department of Materials Science and Engineering Massachusetts Institute of Technology, Cambridge, Mass., USA [email protected] Abstract

1.

Recently, a number of novel processes have been developed for making metallic foams. Their combination of properties make them attractive in a variety of engineering applications. Their low weight and ability to be formed into complex shapes make them attractive for use in structural sandwich panels. Their capacity to undergo large deformations at almost constant load can be exploited in energy-absorption devices. And the high thermal conductivity combined with the connected porosity in open-cell metallic foams makes them attractive for heat sinks. Here, we summarize the current understanding of the uniaxial behavior of metallic foams and relate their properties to micromechanical models.1

INTRODUCTION

Metallic foams can be made by a number of novel processes. Although aluminum foams are the most common, nickel, copper, zinc and steel foams are also available. Micrographs of four closed-cell and one opencell aluminum foams are shown in Fig. 1. Metallic foams have a combination of properties that make them attractive in a number of engineering applications. In structural sandwich panels, they offer lower weight than conventional honeycomb or stringer-stiffened designs for some, although not all, loading configurations. Recently developed processing techniques allow the manufacture of panels of complex shape with integrally bonded faces at relatively low cost. The capacity of metal foams to undergo large strains (up to about 60–70%) at almost constant stress allows significant energy absorption without generating damaging peak

1 This article first appeared in a more extended form [45], and appears by permission of Annual Review of Material Science.

57 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 57–74. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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stresses, making them ideal for energy-absorption devices. Open-cell metallic foams make excellent heat dissipation devices due to their high thermal conductivity, their high internal surface area, and the connectivity of the voids, which allows a cooling gas to pass through. This article reviews the uniaxial behavior of metallic foams. The results of micromechanical models for the moduli and strength of openand closed-cell foams are presented and compared with data for metallic foams. The effect of imperfections on the uniaxial properties is discussed. A description of further aspects of the mechanical behavior of metallic foams, such as size effects, yield criterion, creep, and fatigue, is available in the book Metal Foams: A Design Guide [1].

2.

MICROSTRUCTURE

Micrographs of a several aluminum foams are shown in Fig. 1. The microstructure is made up of an interconnected network of struts or

Figure 1 Micrographs of metallic foams, reproduced from [2] (with permission from Elsevier Science).

plates, which form the edges and faces of polyhedral cells. The single most important microstructural feature affecting the mechanical properties of foams is the volume fraction of solid, or relative density, the ratio

Metallic foams: Structure and properties

59

of the density of the foam to that of the solid, Metallic foams typically have relative densities in the range of 0.03 to 0.3. Foams are described as open- or closed-celled. Those shown in Fig. l(a)–(d) are all closed-cell, with faces separating the voids of each cell; that shown in Fig. l(e) is open-celled. Micromechanical models for open- and closedcelled foams are described in the next section. The cell size of most metallic foams is in the range of 2–10 mm [2, 3]. Although the mechanical properties of foams are sensitive to the ratio of the cell wall thickness to length, most do not depend on the absolute cell size. The cell shape of metallic foams ranges from equiaxed to ellipsoidal, with the ratios of major to minor axis lengths up to 1.5 [3]. The elongation of the cells corresponds to the rise of gas in the liquid (or mushy) state during processing. Foams are typically stiffer and stronger when loaded parallel to the major principal direction. Finally, the cell wall has an influence on the mechanical behavior of foams. The cell walls of some metallic foams are curved, possibly as a result of partial cell collapse during solidification [2, 3, 4] (Fig. 2). Cell wall curvature reduces the mechanical properties below the values that would be expected if the walls were

Figure 2 Micrograph showing curvature of cell walls in a closed-cell aluminum foam (Alporas), reproduced from [3] (with permission from Elsevier Science).

2

Throughout this paper, a superscipt asterisk refers to a property of the foam, while a

subscript s refers to a property of the solid from which the foam is made.

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planar. The composition of the cell wall also affects the properties of the foam. The additives and foaming agents used to process metallic foams often result in unconventional alloys. For instance, in one process the use of calcium as a thickening agent and titanium hydride as a blowing agent introduces Al–Ca–Ti precipitates into the microstructure [3, 4]. In this case, the yield strength of the cell wall material is measured directly using indentation techniques. Measured values for microstructural features of the aluminum foams shown in Fig. 1 are summarized in Table 1.

3.

MICROMECHANICAL MODELS FOR FOAMS

A schematic stress–strain curve for a foam in uniaxial compression is shown in Fig. 3. Initial linear elasticity is followed by a roughly constant stress plateau, which continues up to large strains beyond which the stress increases sharply. Previous studies of polymer foams have found that the linear elastic response is related to cell edge bending in opencell foams and to the edge bending and face stretching in closed-cell foams [5]–[12]. As the stress increases, the cells begin to collapse at a roughly constant load by elastic buckling, yielding or fracture, depending on the nature of the cell wall material [5, 6, 9, 10, 13, 14, 15, 16, 17, 18, 19]. Once all of the cells have collapsed, further deformation presses opposing cell walls against each other, increasing the stress sharply: this final regime is referred to as densification. In this section we review models for the mechanical behavior of ideal open- and closed-cell foams. The complex cell geometry of foams is difficult to model exactly. Several approaches are described in the literature. The simplest technique is to use dimensional arguments to model

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61

Figure 3 A schematic compressive stress–strain curve for a foam, showing the linear elastic, stress-plateau, and densification regimes.

the mechanisms of deformation and failure [5, 6]. The results give the dependence of the properties on relative density and on cell wall properties but not on cell geometry; the constants relating to cell geometry are found by fitting the model equations to experimental data. A second method is to analyze a repeating unit cell, such as a tetrakaidecahedron, using structural mechanics or finite elements [7, 9, 11, 12, 18, 20]. This method gives the dependence of the properties on the geometry of the unit cell; this can be used as an estimate of the geometrical constants of the foam microstructure. A third approach is to model a spatially periodic arrangement of several random Voronoi cells using finite-element analysis [8]. The advantage of this approach is that it gives the best representation of the cell geometry of foams, although it is computationally intensive. Here we review the use of dimensional arguments to show the way in which the properties depend on the relative density of the foam and the solid cell wall properties. We refer to the results of unit-cell and finite-element models to estimate the magnitude of the constants relating to the cell geometry. In the following section we describe the deformation and failure of metallic foams, modify the models to account for imperfections in their cell structure, and compare the results with data for metallic foams. In open-cell foams, the cell edges initially deform by bending. The Young’s modulus can be calculated from a dimensional analysis of the edge bending deflection. The relative modulus (that of the foam, divided by that of the solid, is proportional to the square of the relative density,

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where is a constant related to the cell geometry. Analysis of a tetrakaidecahedral unit cell with the cell cross section defined by Plateau borders finds = 0.98 [12, 20]. This value is confirmed by data for a wide variety of open-cell polymer foams, which give In shear, the cell edges also respond by bending, so that

with The Poisson’s ratio is simply the ratio of two strains and is independent of relative density; measured values suggest that for many foams it is about 1/3.

Closed-cell foams are more complicated. When the foam is loaded, there is stretching of the planar cell faces in addition to bending of the cell edges, adding a linear density dependence to Eqn. (1):

Finite-element analysis of a closed cell tetrakaidecahedral unit cell and of random, Voronoi closed cells suggests that For low-density foams, the second, face stretching, term dominates so that

The plateau stress for foams made from an elastic–plastic material is reached when the cells begin to collapse plastically. For open-cell foams, cell collapse corresponds to the formation of a mechanism of plastic hinges in the bent cell edges. Dimensional arguments give the plastic collapse stress relative to the yield strength of the solid cell edge material as

where the constant is related to the cell geometry. Data for a wide range of foams suggests that

Closed-cell foams have an additional contribution to their strength: yielding of the stretched cell faces. The stress required for this is linearly proportional to the relative density of the foam, so that

Metallic foams: Structure and properties

63

Finite-element analysis of a tetrakaidecahedral unit cell with flat faces suggests that [21]

For low relative densities, the second term, corresponding to axial yielding of the faces, dominates, so that

The Deshpande and Fleck yield surface suggests that the shear strength is 0.69 times the uniaxial strength [22]:

For open-cell foams, using Eqn. (5), we find

and, for closed-cell foams, using Eqn. (7), we find

In the next section we describe the deformation and failure of metallic foams, modify the models to account for imperfections in their cell structure, and compare the results with data for metallic foams.

4.

DEFORMATION AND FAILURE IN METALLIC FOAMS

Compressive stress–strain curves for several aluminum foams are shown in Fig. 4(a). The shape of the curves is similar to that of the schematic of Fig. 3. The initial responses of an open-cell Duocel foam and a closed-cell Alporas foam are shown in more detail in Fig. 4(b). The Duocel foam is linear elastic: the slopes of the loading and unloading curves are identical. Although the Alporas foam exhibits an initial linear regime at strains of up to about 0.01, the slope of the loading curve is much lower than that of the unloading curve, indicating that there is some plastic deformation even at low strains; similar behavior is observed in the other closed-cell foams of Fig. 4(a). Surface strain

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(a) Behavior up to densification, from [2]

(b) Curves for open-cell Durocel and closed-cell Alporas, showing the slopes of the loading and unloading curves, from [24] Figure 4 Compressive stress–strain curves (reproduced with permission from Elsevier Science).

mapping reveals that the Duocel material deforms relatively homogeneously until the plateau stress is approached, while the Alporas material deforms heterogeneously, yielding locally by deformation banding, at stresses of about half the plateau stress (Fig. 5) [23]. The Alporas material strain hardens up to the peak stress and then softens. Tensile stress–strain curves for three aluminum foams are shown in Fig. 6. As in compression, the loading and unloading moduli of the Duocel ERG foam are the same, while the loading modulus of the Alporas foam is lower than the unloading modulus. The lower loading modulus in the Alporas foam again reflects plastic deformation even within the initial linear region. As the foams yield they strain harden and frac-

Metallic foams: Structure and properties

Figure 5

65

Stress–strain curves and surface strain maps of the incremental principal

strains: (a) Duocel and (b) Alporas. (Reproduced from [23], with permission from Elsevier Science.)

ture at strains of less than 1.5%. The Alcan foam is particularly brittle, fracturing at a strain of about 0.25%. The tensile strength is defined as the maximum stress reached before fracture. A shear stress–strain curve for Alporas aluminum foam is shown in Fig. 7. The shear strength is defined as the peak stress. Data for the unloading Young’s modulus from a number of studies are shown in Fig. 8, along with lines representing Eqns. (1) and (3) for open- and closed-cell foams, respectively [2]–[4], [23]–[29]. The modulus and density data have been normalized by the values for solid aluminum = 70 GPa and The data for the open-cell ERG Duocel foam falls close to the line representing the model for open-cell foams (Eqn. (1)), as expected. Nearly all the data for the closed-cell

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Figure 6 Tensile stress–strain curves for aluminum foams. (Reproduced from [2], with permission from Elsevier Science.)

Figure 7 Shear stress–strain curve for Alporas aluminum foam. (Reproduced from [24], with permission from Elsevier Science.)

foams with relative densities less than 0.3 lie below the line representing the model for closed-cell foams (Eqn. (3)). Most are closer to the line representing the open-cell model (Eqn. (1)), suggesting that bending contibutes significantly to the cell wall deformation. Data for the shear moduli are limited: the only available study is that of Dubbelday [30], who found moduli slightly lower than expected from Eqn. (2), perhaps due to anisotropy in the specimens. Data for the compressive and tensile strengths of aluminum foams are plotted in Figs. 9 and 10, along with lines representing Eqns. (5) and (7) for open- and closed-cell foams, respectively [2]–[4], [23]–[25], [27]–[29], [31]–[37]. The strengths have been normalized by the value for the solid

Metallic foams: Structure and properties

67

Figure 8 Relative Young’s modulus plotted against relative density for aluminum

foams. Beals and Thompson [25] data for loading modulus, Dubbelday [26] and Weber [29] data for vibration test, all other data for unloading modulus. (Reproduced from [45], with permission from Annual Review of Materials Science.)

alloys (Table 2) [3, 4, 38]. Again, the data for the open-cell ERG Duocel foam lie close to the line representing the model for open-cell foams (Eqn. (5)), as expected, while the data for the closed-cell foams lie well below the line representing the model for closed-cell foams (Eqn. (7)). Most are closer to Eqn. (5), suggesting that plastic bending contributes significantly to the failure. The compressive and tensile strengths are, in general, similar in magnitude. The shear strengths of aluminum foams are plotted in Fig. 11, along with lines representing Eqns. (10) and (11) for open- and closed-cell foams, respectively [24, 37]. The shear strength of the closed-cell foams, like the uniaxial strength, lies close to the line representing the open-cell model, suggesting that plastic bending is again significant. Bending deformations can arise in closed-cell metallic foams in several ways. Curvature in the cell faces induces bending from both axial and shear forces. Finite-element analysis of closed-cell foams with curved cell walls suggests that, for the typical measured cell wall curvatures, the modulus is 60% and the strength is 30% of those for planar faces,

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Figure 9 Relative compressive strength plotted against relative density for aluminum foams. Compressive strength is defined in slightly different ways in different stud-

ies; generally it is either the plateau stress or the compressive stress at 20% strain. (Reproduced from [45], with permission from Annual Review of Materials Science.)

accounting for most of the discrepancy between Eqns. (3) and (7) and the data [2, 39, 40]. Irregularities in the cell shape can also induce bending. For instance, observations of the deformations within a closed-cell aluminum foam using computed tomography indicate that the presence of a highly elliptical cell with T-shaped cell wall intersections can induce bending in the neighboring cell walls, initiating the collapse of a deformation band [23]. Fractured cell walls can produce bending [41]. Local variations in relative density within a single specimen also reduce the properties. For instance, Beals and Thompson [25] report variations in relative density through the thickness of their specimens from 0.033 to 0.14—the highest values, measured at the bottom of the specimens, result from drainage during processing.

5.

CONCLUSIONS

Metallic foams have a combination of properties that make them attractive for lightweight structural sandwich panels, energy-absorption devices, and heat-dissipation devices. The recent development of a vari-

Metallic foams: Structure and properties

Figure 10

69

Relative tensile strength plotted against relative density for aluminum

foams. (Reproduced from [45], with permission from Annual Review of Materials Science.)

ety of processes for making metallic foams with improved properties at lower cost has led to increased interest in their use in engineering components. A number of studies have led to an improved understanding of the mechanisms of deformation and failure in metallic foams. Their response to uniaxial, multiaxial, creep, and fatigue loading has recently been characterized for the first time. This improved understanding of the mechanical response of metallic foams, combined with structural optimization design strategies, provides a foundation for their use in engineering applications.

Acknowledgments The author would like to acknowledge the financial support of the ARPA MURI Ultralight Metal Structures Contract Number N00014-96-1-1028. Discussions with a number of collaborators on that project have been insightful. I would particularly like to thank Profs. M. F. Ashby and N. A. Fleck of the Cambridge University Engineering Department, Prof. J. W. Hutchinson of the Division of Engineering and Applied Sciences, Harvard University, Prof. A. G. Evans of the Materials Research Institute, Princeton University, Prof. R. E. Miller of the Department of Mechanical Engineering, University of Saskatchewan, Dr. P. Onck of Micromechanics of Materi-

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als, Delft University of Technology, Dr. E. Andrews, Dr. L. Crews, Dr. A. E. Simone, Mr. G. Gioux, Mr. T. M. McCormack, and Mr. W. Sanders of the Massachusetts

Institute of Technology.

References [1] Ashby, M. F., A. G. Evans, N. A. Fleck, L. J. Gibson, J. W. Hutchinson, and H. N. G. Wadley (eds.). 2000. Metal Foams: A Design Guide. London: Butterworth-Heinemann. [2] Andrews, E. W., W. Sanders, and L. J. Gibson. 1999. Compressive and tensile behaviour of aluminum foams. Materials Science and Engineering A 270(2), 113–124. [3] Simone, A. E., and Gibson, L. J. 1998. Aluminum foams produced by liquid-state processes. Ada Materialia 46(9), 3109–3123. [4] Sugimura, Y., J. Meyer, M. Y. He, H. Bart-Smith, J. Grenestedt, and A. G. Evans. 1997. On the mechanical performance of closed cell Al alloy foams. Acta Materialia 45(12), 5245–5259. [5] Gibson, L. J., and M. F. Ashby. 1982. On the mechanical performance of closed cell Al alloy foams. Proceedings of the Royal Society A 382, 43–59. [6] Gibson, L. J., and M. F. Ashby. 1997. Cellular Solids: Structure and Properties, 2nd ed. Cambridge: Cambridge University Press.

[7] Ko, W. L. 1965. Deformation of foamed elastomers. Journal of Cellular Plastics 1, 45–50.

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Figure 11

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Relative shear strength plotted against relative density for aluminum

foams. (Reproduced from [45], with permission from Annual Review of Materials Science.)

[8] Kraynik, A. M., M. K. Neilsen, D. A. Reinelt, and W. E. Warren. 1999. Foam micromechanics: Structure and rheology of foams, emulsions and cellular solids. In Foams and Emulsions (J. F. Sadoc and N. Rivier, eds.) Dordrecht: Kluwer Academic Publishers, 259–286. [9] Menges, G., and F. Knipschild. 1975. Estimation of mechanical properties for rigid polyurethane foams. Polymer Engineering and Science 15(8), 623–627. [10] 10. Patel, M. R., and I. Finnie. 1970. Structural features and mechanical properties of rigid cellular plastics. Journal of Materials 5(4), 909–932. [11] Warren, W. E., and A. M. Kraynik. 1988. Linear elastic properties of open-cell foams. Journal of Applied Mechanics 55(2), 341–346. [12] Zhu, H. X., J. F. Knott, and N. J. Mills. 1997. Analysis of the elastic properties of open-cell foams with tetrakaidecahedral cells. Journal of the Mechanics and Physics of Solids 45(3), 319–343.

[13] Barma, P., M. B. Rhodes, and R. Salover. 1978. Mechanical properties of particulate-filled polyurethane foams. Journal of Applied Physics 49(10), 4985– 4991. [14] Chan, R., and M. Nakamura. 1969. Mechanical properties of plastic foams. Journal of Cellular Plastics 5(2), 112–118. [15] Christensen, R. M. 1986. Mechanics of low density materials. Journal of the Mechanics and Physics of Solids 34(6), 563–578.

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[16] Gent, A. N., and A. G. Thomas. 1959. Deformation of foamed elastic materials. Journal of Applied Polymer Science 1(1), 107–113. [17] Matonis, V. A. 1964. Elastic behavior of low density rigid foams in structural applications. Society of Plastic Engineers Journal 20(9), 1024–1030. [18] Mills, N. J., and H. X. Zhu. 1999. High strain compression of closed-cell polymer foams. Journal of the Mechanics and Physics of Solids 47(3), 669–695. [19] Zhu, H. X., N. J. Mills, and J. F. Knott. 1997. Analysis of the high strain compression of open-cell foams. Journal of the Mechanics and Physics of Solids 45(11–12), 1875–1904. [20] Warren, W. E., and A. M. Kraynik. 1997. Linear elastic behavior of a low density Kelvin foam with open cells. Journal of Applied Mechanics 64(4), 787–794. [21] Simone, A. E., and L. J. Gibson. 1998. Effects of solid distribution on the stiffness and strength of metallic foams. Acta Materialia 46(6), 2139–2150. [22] Deshpande, V. S., and N. A. Fleck. 2000. Isotropic constitutive models for metallic foams. Journal of the Mechanics and Physics of Solids 48(6), 1253–1283. [23] Bart-Smith, H., A.-F. Bastawros, D. R. Mumm, A. G. Evans, D. J. Sypeck, and H. N. G.Wadley. 1998. Compressive deformation and yielding mechanisms in cellular Al alloys determined using X-ray tomography and surface strain mapping. Acta Materialia 46(10), 3583–3592. [24] Andrews, E. W., G. Gioux, P. Onck, and L. J. Gibson. 2000. Size effects in ductile cellular solids—Part II: Experimental results. International Journal of Mechanical Sciences 43, 701–713. [25] Beals, J. T., and M. S. Thompson. 1997. Density gradient effects on aluminum foam compression behaviour. Journal of Materials Science 32(13), 3595–3600. [26] Dubbelday, P. S. 1992. Poisson’s ratio of foamed aluminum determined by laser Doppler vibrometry. Journal of the Acoustical Society of America 91(3), 1737– 1744. [27] Gradinger, R., F. Simancik, and H. P. Degischer. 1997. Determination of mechanical properties of foamed metals. Proceedings of the International Conference on Welding Technology, Materials and Materials Testing, Fracture Mechanics and Quality Management 2. Vienna University of Technology: Chytra Druck and Verlag GmbH, 701–722. [28] McCullough, K. Y. G., N. A. Fleck, and M. F. Ashby. 1999. Toughness of aluminum alloy foams. Acta Materialia 47(8), 2323–2330. [29] Weber, M., J. Baumeister, J. Banhart, and H.-D. Kunze. 1994. Selected mechanical and physical properties of metal foams. 1994 Powder Metallurgy World Congress—Vol. 1, Editions de Physique, 585–588. [30] Dubbelday, P. S., and K. M. Rittenmyer. 1985. Shear modulus determination of foamed aluminium and elastomers. Proceedings of the 1985 IEEE Ultrasonics Symposium 2, 1052–1055. [31] Banhart, J., J. Baumeister, and M. Weber. 1995. Powder metallurgical technology for the production of metallic foams. Euro Powder Metallurgy ’95 (Light Alloys), 201–208. [32] Banhart, J., and J. Baumeister. 1998. Deformation characteristics of metal foams.

Journal of Materials Science 33(6), 1431–1440. [33] Gioux, G., T. M. McCormack, and L. J. Gibson. 2000. Failure of aluminum foams under multiaxial loads. International Journal of Mechanical Sciences 46(6), 1097–1117.

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[34] Prakash, O., H. Sang, and J. D. Embury. 1995. Structure and properties of Al–

SiC foam Materials Science and Engineering A 199(2), 195–203. [35] Thornton, P. H., and C. L. Magee. 1975. Deformation characteristics of zinc foam. Metallurgical Transactions A 64(9), 1801–1807.

[36] Triantafillou, T. C., J. Zhang, T. L. Shercliff, L. J. Gibson, and M. F. Ashby. 1989. Failure surfaces for cellular materials under multiaxial loads. II. Comparison of models with experiment. International Journal of Mechanical Sciences 31(9), 665–678. [37] von Hagen, H., and W. Bleck. 1998. Compressive, tensile, and shear testing of melt-foamed aluminium. Proceedings of the Materials Research Society (MRS) Symposium 521. Warrendale, Penn.: MRS, 59–64. [38] Davis, J. R. 1993. Properties of cast aluminum alloys. ASM Specialty Handbook: Aluminum and Aluminum Alloys. Materials Park, Ohio: American Society for Metals. [39] Grenestedt, J. L. 1998. Influence of wavy imperfections in cell walls on elastic stiffness of cellular solids. Journal of the Mechanics and Physics of Solids 46(1),

29–50. [40] Simone, A. E., and L. J. Gibson. 1998. Effects of solid distribution on the stiffness and strength of metallic foams. Acta Materialia 46(6), 2139–2150. [41] Chen, C., T. J. Lu, and N. A. Fleck. 1999. Effect of imperfections on the yielding of two-dimensional foams. Journal of the Mechanics and Physics of Solids 47(11),

2235–2272. [42] McCullough, K. Y. G., N. A. Fleck, and M. F. Ashby. 2000. The stress-life fatigue behaviour of aluminium alloy foams. Fatigue and Fracture of Engineering

Materials and Structures 23(3), 199–208. [43] Banhart, J., J. Baumeister, and M. Weber. 1996. Damping properties of aluminium foams. Materials Science and Engineering A 205(1–2), 221–228. [44] Banhart, J., and J. Baumeister. 1998. Production methods for metallic foams. Proceedings of the Materials Research Society (MRS) Symposium 521. Warrendale, Penn.: MRS, 121–132. [45] Gibson, L. J. 2000. Mechanic behavior of metallic foams. Annual Review of Materials Science 30, 191–227.

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To select papers for ICTAM 2000, the International Papers Committee (IPC) met for 5 days in March 2000 on the campus of the University of Illinois at UrbanaChampaign. Gathering for lunch in this photo are (l–r) ICTAM 2000 SecretaryGeneral James W. Phillips; IUTAM Congress Committee secretary Niels Olhoff (Denmark); IPC members Sol R. Bodner (Israel), Alan Needleman (USA), Dick H. van Campen (The Netherlands), Timothy J. Pedley (UK), and L. Gary Leal (USA); and ICTAM 2000 President Hassan Aref.

LINEAR VISCOELASTICITY OF CONCENTRATED EMULSIONS Martin Nemer, Jerzy Blawzdziewicz, and Michael Loewenberg Department of Chemical Engineering, Yale University, New Haven, Conn., USA [email protected]

Abstract

1.

The linear viscoelastic response of an ordered dense emulsion is explored by numerical simulation. At concentrations below maximum packing, the stress relaxation is dominated by a single time scale associated with lubrication, which diverges at maximum packing. For concentrations above maximum packing, the stress relaxation is dominated by fast time scales of the order of the drop relaxation time. A slow time scale appears but does not dominate.

INTRODUCTION

The rheology of emulsions and foams is important for a wide range of applications, including polymer processing, personal care products, food processing, textile finishing, and oil recovery. Emulsions have non-Newtonian rheology [l]–[4], including shear thinning and nonzero normal stress differences, and may exhibit a nonzero yield stress. In small-amplitude oscillatory flows, emulsions exhibit a linear viscoelastic response [5]. Experiments and simulations on the low-frequency elastic behavior have focused on the static elastic modulus because it is easier to define and measure than the yield stress [6]. The static properties of ordered concentrated emulsions and foams have been investigated theoretically [7]–[9]. In three dimensions the equilibrium behavior of small ordered systems has been studied numerically using minimal-energy simulations [10]. Large systems have been explored in two dimensions using minimal-energy simulations [11], and in three dimensions using simplified effective interparticle potentials to model bubble-bubble interactions [12]–[14]. The dynamical properties of foams and compressed emulsions are less understood. Analytical results for the dynamics of thin inextensible films [15] have been incorporated in theoretical models [16] and simulations [13] of the dynamics of two dimensional systems. In three dimen75 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 75–84. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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ET AL.

sions, the effective-interparticle-potential model has been supplemented by simple bubble–bubble frictional dissipation [5, 17]. However, these models cannot describe the effect of detailed drop-scale dynamics on the evolution of the system. In general, the development of theories for the rheology of foams and emulsions will require a detailed understanding of the relevant drop-level dynamics. The focus of this paper is on phenomena that occur on the drop relaxation time scale In this paper we explore the linear viscoelastic response of concentrated ordered two-dimensional emulsions and foams in the frequency regime where the dynamics of drop shape relaxation is important. The response of the system was explored using numerical simulations that incorporate detailed drop-level hydrodynamics.

2.

PROBLEM FORMULATION We consider the linear viscoelastic behavior of a two-dimensional

hexagonal array of drops in which both phases are incompressible Newtonian fluids governed by the Stokes equations. Uniform interfacial tension is assumed. At high volume fractions, the drops are in a state of compression. In response, the drop forms flat films and Plateau borders of curvature higher than that of the undeformed drop. In this work we assume that the Plateau borders contain all the continuous phase liquid, and the thin films have zero thickness; thus the hydrodynamics of the thin films simplifies. Details of the boundary conditions at the contact point are discussed in Appendix A. The magnitude of drop deformation in a flow with shear rate scales with the capillary number Ca = where a is the (volume equivalent) drop radius, and is the continuous phase viscosity. The dynamics of the system also depends on the dispersed-to-continuous phase viscosity ratio drop volume fraction and the external forcing frequency Herein, dimensionless variables are used, where length is scaled with a, time is scaled with drop relaxation time and stress is scaled with This work focuses on linear viscoelastic behavior; thus We consider a hexagonal lattice in which the linear relation between stress and strain is isotropic [18]. Accordingly our system is characterized by a single frequency-dependent linear viscoelastic modulus

Linear viscoelasticity of concentrated emulsions

The drop contribution to the stress, the stress relaxation function

77

can be expressed in terms of

where is the dimensionless shear rate and is the dimensionless effective viscosity of the system in the high-frequency limit. For volume fractions below the critical value (i.e. maximum packing of undeformed drops), the stress relaxation function decays to zero as At the limiting behavior is For a two-dimensional hexagonal lattice, the static elastic modulus has been evaluated [8]. In our simulations, the stress relaxation function is obtained by analyzing the response of the system to a step strain. Accordingly the evolution equation is linearized around the equilibrium state

where is a shape perturbation. In our simulations, the linearized velocity operator A is obtained by solving the Stokes equations using the boundary-integral method [19]. By projecting the resulting normal modes,

onto a step strain perturbation,

the corresponding contribution to the shear stress can be calculated:

The relation between coefficients and is where is the contribution to the stress from the eigenmode The step strain perturbation is affine with the strain only for for determining the step strain perturbation requires a calculation of the hydrodynamic field due to the viscosity contrast, in the absence of surface tension. The results presented in the next section have been obtained for

3.

RESULTS

Selected results for the viscoelastic behavior of a monodisperse hexagonal system of drops are presented in Figs. 1–4. Figure l(a) illustrates

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Figure 1 Stress relaxation function nondimensionalized by mensionalized by drop relaxation time: (a) for method described in

(solid line); (b) for

versus time nondiobtained from the

obtained from direct step

strain simulations—zero-thickness films (solid line), films with finite average thickness (dashed line), static elastic modulus (dotted line).

Figure 2 Relaxation spectra as amplitudes

normalized by

by drop relaxation time, for

versus time scaled

Volume fractions

1.05, 1.03 were obtained from direct step strain simulations.

the evolution of the stress relaxation function volume fraction below critical, where

for dispersed-phase

for a hexagonal array of cylinders. The results indicate that the stress relaxation function decays to zero at long times. The initial decay occurs

on the capillary relaxation time scale however, evolution on a much slower time scale is evident at long times. The relaxation times and amplitudes corresponding to the eigenmode decomposition (5) are plotted in Figs. 2(a,b) for two subcritical values of According to our

Linear viscoelasticity of concentrated emulsions

79

Figure 3 Selected modes for a subcritical and a supercritical volume fraction. (a) Equilibrium drop configuration at (b), (c) normal displacements versus angle defined in (a) (solid lines) for modes with (b) and (c) (d) Equilibrium drop configuration at (e) normal and (f) tangential displacements for mode with Labels a–c in figures (d)–(f) denote contact points. Dashed line in (b) represents analytical result for

results, a small number of modes is sufficient to represent accurately. The dominant mode corresponds to the largest relaxation time associated with the lubrication resistance between closely spaced drops. This slowest time scale diverges as

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Figure 4 Average time scale

ET AL .

given by Eqn. (7) versus

(solid line); aver-

age relaxation time from Eqn. (8) (dotted line). Error bars represent the difference between results obtained from zero-thickness film simulations and finite-thickness film simulations. For results were obtained from simulations with

finite-thickness films.

Figure l(b) shows qualitatively different behavior in the stress relaxation function for At long times, decays to the static elastic modulus In contrast to subcritical volume fractions, the stress relaxation function is dominated by decay on the capillary relaxation time scale a long time scale is evident but with small amplitude. The relaxation times and amplitudes are plotted in Figs. 2(c,d) for two supercritical values of The stress relaxation function obtained from zero-thickness film simulations was compared with the corresponding result for a system with slowly draining films of finite thickness. For average film thickness the results differ by about 1%, as shown in Fig. l(b). Drop-shape perturbations associated with the two slowest modes for a subcritical value of are shown in Fig. 3(b,c). The slowest mode (solid line plotted in Fig. 3(b)) has the largest amplitude according to Fig. 2(b), and results from the lubrication resistance in the near-contact regions between drops. The dashed line shows the limit as

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In this limit, drop shape is associated with point forces acting in the lubrication regions of vanishing extent; between the lubrication regions the drop shape is in quasi-static equilibrium due to the separation of the lubrication time scale and the drop relaxation time where is the minimum gap between drops. As the volume fraction decreases, the dominant mode tends continuously to A mode with the relaxation time is plotted in Fig. 3(c). In this case, the shape of the drop interface in the Plateau border outside the lubrication regions is significantly perturbed from equilibrium. Oscillations are observed in the near-contact region that corresponds to matching of the time scales for the relaxation inside and outside the lubrication regions. For the dominant mode (cf. Fig. 2(c)) is shown in Figs. 3(e,f). The central points of the thin films do not move by symmetry, and the films are only weakly deformed in this region. Oscillations observed near contact point a have diminishing amplitude and wavelength. The result shown in Fig. 3(e) indicates that the film a–a increases length, and films b–b and c–c decrease length. These changes in film lengths occur without significant lubrication resistance; thus the mode evolves on drop-relaxation time Figure 4 displays the average relaxation time

where is the zero-frequency viscosity, and is the drop contribution to the stress immediately after a step strain; for the initial stress The results indicate that for the average relaxation time diverges as This behavior is associated with the lubrication singularity of the zero-frequency viscosity at For Eqns. (7) and (B.8) indicate that

In Fig. 4 this result is represented by the dotted line. The results in Fig. 4 suggest that for does not diverge. A more detailed analysis of this phenomenon is underway.

4.

CONCLUSIONS

In conclusion, we have performed boundary-integral simulations of the linear viscoelastic response of monodisperse ordered two-dimensional emulsions. Our results indicate that for volume fractions below close

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packing of undeformed drops, the evolution is dominated by a single slow mode associated with lubrication resistance. For volume fractions above close packing, the system is dominated by evolution on the drop relaxation time, because thin films between drops can stretch without significant lubrication resistance. There also exists a slow mode that is probably associated with squeezing in the lubrication region where drop interfaces meet, but this mode contributes only weakly to the stress. As the current work progresses, the effects of system parameters, including volume fraction, viscosity ratio, and contact angle, will be explored. The effects of an insoluble surfactant layer will be also be studied.

Appendix A. Zero-thickness films In our model the contact point (where three films meet) is convected by the interfacial velocity. A tangential force balance is maintained at the contact point at all times:

we also assume a continuity relation

for the normal traction jumps Here is the cosine of the contact angle, is the interface curvature, is the unit tangent vector to interface i, and is the unit normal vector. The thin film (subscript 0) has two interfaces and is assumed to have negligible thickness compared with the drop size. Subscripts 1 and 2 denote the interfaces of the Plateau border that connect to the thin film.

Appendix B. Low-frequency viscosity for near-critical volume fractions Here we consider the stress in an ordered emulsion subjected to a linear flow in the limit of low frequency and from below. Under these conditions, the drop contribution to the stress is dominated by the lubrication forces at the near-contact regions centered at positions

where is the relative velocity between drops normal to the interface at and R is the pairwise lubrication resistance between undeformed drops. Thus,

Linear viscoelasticity of concentrated emulsions

where

83

is the drop volume,

E is the dimensionless rate-of-strain tensor, are the lattice vectors. Combining (B.2) and (B.3) gives

is the strain rate, and

where

and m is the dimensionality of the system. For two-dimensional clean drops in near-contact motion with moderate viscosity ratios we find

where is the minimum distance between undeformed drops. For a two-dimensional hexagonal lattice,

Thus, a single transport coefficient

characterizes the rheology, where critical volume fraction.

is the distance from the

References [1] Princen, H. M. 1985. Rheology of foams and highly concentrated emulsions:

II—Experimental study of the yield stress and wall effects for concentrated oilin-water emulsions. Journal of Colloid and Interface Science 105, 150–171.

[2] Princen, H. M., and A. D. Kiss. 1986. Rheology of foams and highly concentrated emulsions: III—Static shear modulus. Journal of Colloid and Interface Science 112, 427–437. [3] Princen, H. M., and A. D. Kiss. 1989. Rheology of foams and highly concentrated emulsions: IV—An experimental study of the shear viscosity and yield stress of concentrated emulsions. Journal of Colloid and Interface Science 128, 176–189.

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ET AL .

[4] Mason, T. G., J. Bibette, and D. A. Weitz. 1996. Yielding and flow of monodisperse emulsions. Journal of Colloid and Interface Science 179, 439–448. [5] Mason, T. G., M. D. Lacasse, G. S. Grest, D. Levine, J. Bibette, and D. A. Weitz. 1997. Osmotic pressure and viscoelastic shear moduli of concentrated emulsions. Physical Review E 56, 3150–3166. [6] Tewari, S., D. Schiemann, D. J. Durian, C. M. Knobler, S. A. Langer, and A. J. Liu. 1999. Statistics of shear-induced rearrangements in a two-dimensional model foam. Physical Review E 60, 4385–4396. [7] Princen, H. M. 1979. Highly concentrated emulsions. Journal of Colloid and Interface Science bf 71, 55–66. [8] Princen, H. M. 1983. Rheology of foams and highly concentrated emulsions. Journal of Colloid and Interface Science 91, 160–175.

[9] Buzza, D. M. A., and M. E. Cates. 1994. Uniaxial elastic modulus of concentrated emulsions. Langmuir 10, 4503–4508. [10] Reinelt, D. A., and A. M. Kraynik. 1996. Simple shearing flow of a dry Kelvin soap foam. Journal of Fluid Mechanics 311, 327–342. [11] Weaire, D., T. L. Fu, and Kermode. 1986. On the shear elastic constant of a

two-dimensional froth. Philosophical Magazine B 54, L39–L43. [12] Lacasse, M. D., G. S. Grest, D. Levine, T. G. Mason, and D. A. Weitz. 1996. Model for the elasticity of compressed emulsions. Physical Review Letters 76,

3448–3451. [13] Okuzono, T., and K. Kawasaki. 1993. Rheology of random foams. Journal of Rheology 37, 571–586. [14] Durian, D. J. 1995. Foam mechanics at the bubble scale. Physical Review Letters

75, 4780–4783. [15] Mysels, K. J., K. Shinoda, and S. Frankel. 1959. Soap Films: A Study of Their Thinning and a Bibliography. Tarrytown, N.Y.: Pergamon Press. [16] Reinelt, D. A., and A. M. Kraynik. 1990. On the shearing flow of foams and concentrated emulsions. Journal of Fluid Mechanics 215, 1235–1253. [17] Durian, D. J. 1997. Bubble-scale model of foam mechanics: Melting, nonlinear behavior, and avalanches. Physical Review E 55, 1739–1751. [18] Frisch, U., B. Hasslacher, and Y. Pomeau. 1986. Lattice-gas automata for the Navier–Stokes equation. Physical Review Letters 56, 1505–1508. [19] Pozrikidis, C. 1992. Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge: Cambridge University Press.

ALUMINUM: APPROACHING THE NEW MILLENNIUM Peter A. Davidson Department of Engineering, University of Cambridge, Cambridge, United Kingdom [email protected]

Abstract

1.

The amount of energy currently used to reduce alumina to aluminum in electrolysis cells is staggering, around 1011 kW·h per annum. Yet much of this energy (almost one half) is lost in the form of I2 R heating of the highly resistive electrolyte. Strenuous efforts have been made to minimize these losses by reducing the volume of electrolyte in the cells. However, the aluminum industry has come up against a fundamental problem: when the depth of the electrolyte is reduced below a critical threshold (around 4–5 cm), the liquids in the cell start to ‘slosh around’ in an uncontrolled fashion. This is an instability, fueled by the intense currents that pass through the liquids. At present, cells operate just above the critical electrolyte depth, but if this depth were reduced from, say, 4.5 cm to 4.0 cm, then the annual savings would exceed £100 million. After a number of false starts, we now have a clear understanding of the physical mechanisms that underpin the instability, and it turns out that these are remarkably simple.

INTRODUCTION

Today, virtually all aluminum is produced by reducing alumina in electrolysis cells. There are two remarkable features of these cells: first, the process has remained virtually unchanged for over a century, and second, the amount of energy consumed by these cells is staggering. Aluminum was first isolated in the 1820s by Oersted and Wöhler. They used chemical, rather than electrical, means. By 1827, Wöhler was producing small quantities of aluminum powder by displacing the metal from its chloride using potassium. Later, in the 1850s, potassium was replaced by sodium, which was cheaper, and aluminum fluoride was substituted for the more volatile chloride. However, those chemical processes were all swept aside by the revolution in electrical technology initiated by Faraday’s discoveries. The electrolytic route was first proposed by Robert Bunsen (he of the ‘burner’ fame) in 1854, but it was not until 1886 that a continuous commercial process was developed. It 85 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 85–98. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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was a young college student from Ohio, Charles Martin Hall, and the Frenchman Paul Héroult, who made the breakthrough. Working independently, they realized that molten cryolite, a mineral composed of fluorine, sodium, and aluminum, readily dissolves alumina and that a current passed through the solution will decompose the alumina, leaving the cryolite almost unchanged. Full commercial production began on

Figure 1

A reduction cell at the turn of the century.

Thanksgiving Day in 1888 in Pittsburgh in a company founded by Hall. An example of an early Hall-Héroult reduction cell is shown in Fig. 1. Remarkably, over a century later, the process is virtually unchanged (Fig. 2).

Figure 2 A modern reduction cell.

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Modern reduction cells consume prodigious amounts of energy. Worldwide, around 18 million tonnes of aluminum are produced annually, and this requires ~ 230 × 109 kW·h per annum (about 2% of all generated electricity). At a notional cost of 3 cents per kW·h, this represents an annual bill of around $10 billion (see, for example, et al. 1999). Yet much of this energy, around 40%, is wasted in the form of I2 R heating of the electrolyte! Needless to say, strenuous efforts have been made to reduce these losses, largely centered around reducing the depth of the electrolyte. (Note the difference in electrolyte depth between Figs. 1 and 2.) However, the aluminum industry is faced with a fundamental problem. When the depth of electrolyte is reduced below some critical threshold, the reduction cell becomes unstable. This instability, which manifests itself as a sloshing of the liquids within the cell, has been the subject of intensive research for over two decades. After several false starts, we have finally reached the point where we understand the cause of the instability, not just in some abstract mathematical sense but in a simple, intuitive, way. It is now clear how to design new cells that are inherently stable, and this is likely to be the next major change in the industry. The potential benefits of such a change would be a rise in productivity (of around 20%) plus a reduction in energy cost from, say, 14 kW·h/kg to around 11 kW·h/kg. At a notional cost of 3 cents per kW·h, this translates to worldwide savings of $1 billion per annum. Given that energy consumption accounts for between 20 and 30% of the total production cost of aluminum, there is scope for substantial savings.

2.

THE STABILITY OF MODERN REDUCTION CELLS

Consider the cell shown in Fig. 2. A large vertical current, perhaps 300 kA, flows downwards from the carbon anode blocks, passing first through the electrolyte (where it reduces the alumina) and then through a liquid aluminum pool before finally being collected at the carbon cathode at the base of the cell. The liquid layers are broad but shallow, perhaps 4m by 10m in plan, yet only a few centimeters deep. (The electrolyte layer might be 4–5 cm thick and the aluminum pool perhaps 20–30 cm deep.) There are two quite distinct efficiencies used to characterize the per-

formance of a cell: the efficiency of electrolysis (usually called the current efficiency) and the energy efficiency. The first is a measure of how many electrons do what they are meant to do, i.e. reduce alumina molecules. This efficiency is very high, typically around 90–95%. The energy effi-

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ciency, on the other hand, which measures the difference between useful and wasted power, is extremely low. Although the aluminum is an excellent conductor, and the carbon a moderate one, the electrolyte (cryolite) is a very poor conductor. Typically, the cell voltage is around 4.5 V, of which only 1.6V is needed for the electrolysis and a staggering 2V is lost in the electrolyte. Thus, around 40% of the input power is lost to Ohmic heating of the thin (4.5 cm) layer of cryolite. There is considerable incentive, therefore, to lower the resistance of the electrolyte layer by reducing its thickness. In fact, worldwide, each millimeter of electrolyte costs around $60 million per year in lost profits! There is a fundamental problem, however. It turns out that unwanted disturbances are readily triggered at the electrolyte–aluminum interface (Fig. 3). In effect, these are interfacial gravity waves, modified by the

Figure 3

The idealized cell geometry.

intense magnetic fields that pervade the cell. When a certain stability threshold is exceeded, these waves can grow, absorbing energy from the ambient electric and magnetic fields. The wave period is measured in minutes, and its growth rate in hours; yet once such a wave takes hold it can disrupt the electrolysis to such an extent that the cell must be withdrawn from operation. These waves represent the primary impediment to increasing both productivity and energy efficiency. In particular, the cell current I must be maintained below a certain critical value in order to ensure stability. Similarly, there is a critical value of the cryolite depth h, below which instability sets in and waves can grow. Most cells operate just on the verge of instability, at a current just below and at an electrolyte depth marginally above To some extent, the mechanism of the instability is clear. Let J0 be the nominal (unperturbed) current density in the cell and j be any

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additional current density arising from movement within the cell. Now consider a tilting of the interface as shown in Fig. 4. Excess current

Figure 4

Long-wavelength disturbances in a cell.

is drawn into the aluminum at points where the thickness of the highly resistive cryolite is reduced, and less current is drawn at points where the electrolyte depth is increased. Since the carbon cathode is much more resistive than the aluminum, these perturbations in vertical current feed into the aluminum pool but do not penetrate the cathode block. The perturbation in current is as shown schematically in Fig. 4. On the left, where excess current is drawn, j points downward. On the right, where there is a deficit in current, the perturbation is negative and so j points upward. Since the perturbed current j must form a closed circuit, and since the fluctuations do not penetrate into the cathode block, the perturbations of vertical current in the electrolyte must short through the aluminum as shown. Now the waves commonly seen in cells have wavelengths comparable with the horizontal cell dimensions. (Higher frequency components are destroyed by friction.) Thus the typical wavelength is much greater than h. There are several consequences of the long-wavelength limit. First, it may be shown that, to leading order in the small quantity the perturbed current density in the cryolite is purely vertical, while that in the aluminum, is horizontal (see Davidson and Lindsay 1998). More importantly, it takes only a minute tilting of the interface to cause a large rush of horizontal current in the aluminum. For example, suppose

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we have a tilt of only 1°, which corresponds to a maximum change in h of around 2% (1 mm). This will induce a 2% change in local resistance (at the crest and valley of the wave) and hence in fluctuation in cryolite current of, say, ~ 2% × 300 kA = 6kA. However, from geometrical considerations, we can show that this leads to a horizontal current of 100 kA! It is this surprisingly large horizontal current that lies at the root of the instability. This perturbation in current density is important because it causes a change in the Lorentz force, F = J × B, which acts on the fluids. That is, as J goes from to + j, and B from to + b, F changes from to F = + × b + j × b. (Here is the ambient magnetic field in the absence of a wave.) In the long-wavelength limit, the dominant contribution to =F– can be shown to be where is the vertical component of the ambient magnetic field (see Davidson and Lindsay 1998). In short, the dominant contribution to the perturbation in force is the interaction of horizontal current in the aluminum with the ambient, vertical magnetic field. This results in a force that acts only on the aluminum and that is directed normal to the plane of Fig. 4. The key question, of course, is whether or not this change in Lorentz force will tend to amplify the initial motion, driving an instability. At first sight it might seem that the answer to this must be ‘no’. Why should there be any interaction between a motion in one plane and a force acting in the perpendicular direction? However, suppose we arrange to have two waves, both of which have the same frequency, and which slosh back and forth in mutually perpendicular planes, say one in the xz plane and the other in the yz plane. (The coordinate system is shown in Fig. 3.) Each wave then produces a force that acts on its companion wave. If we, or rather nature, now choose the relative timing of these waves appropriately, then we have the possibility that the force produced by one wave is in phase with the other wave (and hence does work on that wave) and vice versa. In short, we end up with a kind of resonance. It is this resonance that causes cells to become unstable. There have been many detailed mathematical models of cell stability. These try, in various ways, to capture the instability mechanism described above and to make predictions of and It has to be said, however, that most of these models are rather idealized, simplifying the cell geometry to the point where the quantitative predictions are not very accurate. However, from the perspective of an engineer, this does not matter. The important point is that we now understand the basic mechanisms that cause the waves to amplify, and so we are finally in a position to redesign these cells and engineer the problem away.

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Perhaps the instability mechanisms are best understood by using a simple mechanical analogue. This is due to Davidson and Lindsay (1998) and relies on the fact that, in the long-wavelength limit, the motion in the aluminum is almost purely horizontal (Fig. 4).

3.

A SIMPLE MECHANICAL ANALOGUE FOR THE INSTABILITY

Suppose we replace the liquid aluminum by a thin, rigid, aluminum plate attached to the center of the anode by a light rigid strut. The strut is pivoted at its top and so the plate is free to swing as a compound pendulum about two horizontal axes x and y (see Fig. 5). Let the plate

Figure 5 The compound pendulum shown above contains all the essential physics of the reduction cell instability.

have thickness H, lateral dimensions and density The gap h between the plate and the anode is filled with an electrolyte of negligible inertia and poor electrical conductivity. A uniform current density passes vertically downward into the plate and is tapped off at the center of the plate. Finally, suppose that there is an externally imposed vertical magnetic field Evidently, we have replaced one mechanical system (the cell), which has an infinite number of degrees of freedom, with another that has only two degrees of freedom. However, electrically the two geometries are alike. Moreover, the nature of the motion in the two cases is not dissimilar. In both systems we have movement of the aluminum associated with tilting of the electrolyte–aluminum interface. In a cell this takes the form of a ‘sloshing’ back and forth of the aluminum as the interface tilts first one way and then the other (Fig. 4). Let and be the angles of rotation of the plate measured about the x and y axes. Then it is not difficult to show that, as a result of

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these rotations, the perturbed current in the aluminum plate is

The perturbation in Lorentz force j × can be calculated from these expressions, and the equations of motion for the compound pendulum then follow. Note that a tilting of the plate about the y axis induces both motion and a current flow in the x direction. This then interacts with to create a y-directed force acting on the plate. Thus, movement in one direction, say x, tends to induce a force that drives motion in the perpendicular direction (in this case, y). It turns out that the equations of motion for the pendulum are

where and are the conventional gravitational frequencies of the pendulum. Note the cross-coupling of and If we look for solutions of the form ~ exp(iwt) we find constant amplitude oscillations for small values of and exponentially growing sinusoids (unstable solutions) for large The transition occurs at

Figure 6 Variations of

with

for the compound pendulum.

Figure 6 shows the movement of in the complex plane as is increased. The two natural frequencies and move along the real axis until they meet. At this point, they move off into the complex plane and an instability develops. The important points to note are:

The tendency for instability depends only on the magnitude of and on the natural gravitational frequencies and

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To minimize the danger of an instability it is necessary to keep low and the gravitational frequencies well apart. The closer the natural frequencies are, the lower the threshold value of at which an instability appears. (In the absence of friction, circular and square plates are unstable for vanishingly small values of The system is unstable whenever

When is large the unstable normal mode corresponds to a rotating, tilted plate (see Davidson and Lindsay 1998). Very similar behavior is seen in reduction cells. In particular, the sensitivity of reduction cells to the destabilizing influence of depends on the initial separation of the gravitational frequencies. The closer the gravitational frequencies the lower the stability threshold. Moreover, unstable waves frequently correspond to a rotating, tilted interface (Sele 1977). The physical origin of the instability of the pendulum (and of a cell) is now clear. Tilting the plate in one direction, say produces a horizontal flow of current in the aluminum that interacts with to produce a horizontal force which is perpendicular to the movement of the plate and in phase with This tilting also produces a horizontal velocity, which is out of phase with the force and mutually perpendicular to it. Two such tilting motions in perpendicular directions can reinforce each other, with the force from one doing work on the motion of the other. This is the instability mechanism of the pendulum and essentially the same thing happens in a reduction cell.

4.

CAN WE MODEL THE INSTABILITY?

The instability mechanism in a reduction cell is essentially the same as that of the plate. The main differences are that there is friction in a cell, which tends to stabilize the waves, and that there are many different wave patterns (sloshing modes), each of which may have a different natural frequency. There are many mathematical models of cell stability around. The primary aim of these models is to identify all the possible sloshing modes, and to determine which pair are the most dangerous in the sense that they may interact and go unstable in the way that the pendulum does. Perhaps the best known of these models are due to Sneyd and Wang (1994), Bojarevics and Romerio (1994),

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Urata (1985), Sele (1977), and Davidson and Lindsay (1998). Given the complexity of the dynamics involved, it is remarkable that one can produce a mathematical model at all, and so these models represent a significant achievement. However, they all make a variety of simplifying assumptions, particularly about the cell geometry. For example, the channels at the edges of the cell are usually ignored, and this can influence the accuracy of the predicted gravitational frequencies. Worse still, the background steady-state motion in the fluids is often overlooked in the stability analysis. As a consequence, there is considerable doubt as to the accuracy of the model predictions for particular cells. Nevertheless, the broad trend tends to be captured by these analyses, and perhaps we should be grateful for that. In general, these models tend to end up with a stability criterion of the form (see Davidson and Lindsay 1998)

where is the nominal current density in the cell, some typical measure of the ambient, vertical magnetic field, is the density difference between the two fields, h and H are depths of the two fluid layers, and are the horizontal dimensions of the cell, and F(~, ~)

is some function. The left-hand side is the critical value of at which the cell goes unstable, normalized by the gravitational force The right-hand side includes the cell aspect ratio, because the natural frequencies of the sloshing modes depend critically on the ratio For a given it can be seen that large values of and are destabilizing, as is a small value of h. This is why a high-amperage cell with a thin cryolite layer is prone to instability. The field arises from the bus-bars, which carry current to and from the cell. (The current within the fluids is largely vertical, and so produces only a horizontal magnetic field.) Thus a well designed cell has a bus-bar arrangement that minimizes Another traditional approach to cell design is to choose such as to make F, and hence as large as possible. This is achieved by keeping the natural frequencies of the sloshing modes as far apart as possible through an appropriate choice of aspect ratio. As suggested by (5), this will produce a high critical value of

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A TYPICAL STABILITY MODEL

We shall briefly summarize the stability model of Davidson and Lindsay (1998). We start with conventional shallow-water theory. It is not difficult to show that, to second order in kH, k being wavenumber, the pressure in each layer is hydrostatic. As a consequence, we may apply the conventional shallow-water equation to each layer in turn. This is a two-dimensional equation for the horizontal motion:

Here is the aluminum depth, is the interfacial pressure, and is the horizontal body force in each layer. We now linearize our equation of motion about a base state of zero background motion. Taking we obtain

Although is a two-dimensional velocity field, vertical movement of the interface means that the two-dimensional divergences of and are both non-zero. In fact, it is readily confirmed that

Next, we replace and by the volume fluxes and Also, we may take (to leading order in kH). This is valid because, as we have seen, the current perturbation in the cryolite is an order of magnitude smaller than that in the aluminum. The governing equations become

We now perform a Helmholtz decomposition on q: q= appropriate decomposition is

An

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Evidently, is zero in the electrolyte, while is the same in both layers: We now rewrite (7) and (8) in terms of and eliminate by adding the equations, and use (10) to express in terms of The resulting equation of motion is

where The subscript P on the bracket implies that we take only the irrotational component of the corresponding term. Note that, when the Lorentz force is zero, we recover the conventional equation for interfacial waves in the shallow-water limit. The wavespeed for such waves is We now evaluate and hence using the longwavelength approximation. In the cryolite we have, to leading order in kH, This current passes into the aluminum, and so the boundary conditions on are

It is readily confirmed that the conditions of zero divergence and zero curl, as well as the boundary conditions given above, are satisfied by

Here is the horizontal component of the current density in the aluminum, which satisfies

Comparing equation (14) with (10) we find

This is the key relationship that allows us to express the Lorentz force in terms of the fluid motion and therefore deserves some special attention. The physical basis for (15) is contained in Fig. 4. When the interface tilts, there is a horizontal flow of current from the high to the low side of the interface. Simultaneously, there is a horizontal rush of the aluminum in the opposite direction. It is this coupling that lies at the heart of the instability, and which is expressed by (15).

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We now note that the leading term in the Lorentz force arises from the background component of Substituting for in (12) and introducing

we find, after a little algebra,

This equation is readily solved in any shape of domain and it predicts the unstable growth of waves whenever exceeds a critical value, which depends on the geometry. For rectangular domains the end result is a neutral stability curve that leads to an equation of the form of (6).

6.

CONCLUSIONS

It is over a century since Hall first started aluminum production in Pittsburgh. Reduction cells have been carefully optimized over the years, but basically they are much the same as those envisaged by Hall. If aluminum is to compete for new markets, then its cost of production must fall. Given that energy consumption constitutes a large fraction of this cost, and that 40% of this energy is wasted in I2R heating of the electrolyte, it makes sense to focus our efforts on eliminating this inefficiency. However, if the cryolite losses are to be reduced, then we must somehow neutralize the cell instability. After a number of false starts we now finally understand the mechanisms that underlie the instability and so engineering solutions cannot be far off.

References Bojarevics, V., and M. V. Romerio. 1994. Long wave instability of liquid metal– electrolyte interface in reduction cells. European Journal of Mechanics B 13, 33– 56. Davidson, P. A., W. R. Graham, and H. L. O’Brien. 1999. Control of instabilities in aluminium reduction cells. Light Metals 1999, 327–331.

Davidson, P. A., and R. I. Lindsay. 1998. Stability of interfacial waves in aluminium reduction cells. Journal of Fluid Mechanics 362, 273–295. Moreau, R. J., and D. Ziegler. 1986. Stability of aluminium reduction cells: a new approach. Light Metals 1986, 359–364. Øye, H. A., N. Mason, R. D. Peterson, E. L. Rooy, F. J. Stevens McFadden, R. D. Zabreznik, F. S. Williams, and R. B. Wagstaff. 1999. A history of aluminium production. Journal of Metals 51(2), 29–41. Sele, T. 1977. Instabilities of the metal surface in electrolyte alumina reduction cells. Metallurgy Transactions B 8, 613–618. Sneyd, A. D., and A. Wang. 1994. Instabilities of the metal surface in electrolyte alumina reduction cells. Journal of Fluid Mechanics 263, 343–359.

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Urata, N. 1985. Magnetics and metal pad instability. Light Metals 1985, 581–589.

ELECTROMAGNETIC PHENOMENA IN STEEL CONTINUOUS CASTING Takehiko Toh and Eiichi Takeuchi Technical Development Bureau, Nippon Steel Corporation, Chiba, Japan [email protected] Abstract

1.

The application of magnetohydrodynamics (MHD) in the continuous casting process of steel has now advanced to electromagnetic stirring in the mold and the control of molten steel flow by an in-mold directcurrent (DC) magnetic field brake. These applied MHD technologies are designed to improve further the continuous casting (CC) process capability. They improve the surface quality of cast steel by homogenizing the meniscus temperature, stabilizing the initial solidification, and cleaning the surface layer. They also improve the internal quality of steel slab by preventing the inclusions from penetrating deep into the strand pool and promoting the flotation of argon bubbles. This paper describes the characteristics of the phenomena appearing in these technologies on the basis of MHD. Additionally, a description is given of a technique for controlling the initial solidification of steel in the mold by imposing an alternating-current (AC) magnetic field, which enables the production of a surface-defect-free slab at a higher casting speed.

INTRODUCTION

In the past decade, the continuous casting (CC) process has progressed markedly, has solved many technological problems, and has spread widely as a result. In the meantime, attempts have been made to develop novel casting processes for further innovation in the coming century. Among them, the application of magnetohydrodynamics (MHD) has aroused a great deal of interest as a powerful elementary technology to develop casting processes with higher productivity and better cast steel quality. Nippon Steel started the application of MHD with the development of a strand pool electromagnetic stirrer in the mid-1960s for the purpose of improving the internal quality of the cast steel. The equipment, which has the merit of raising the equiaxed zone ratio, was installed on various continuous casters to prevent center segregation, porosity, and ridging in the late 1970s. The 1980s saw the development of in-mold 99 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 99–112. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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electromagnetic stirring for continuously casting pseudo-rimmed steel. The in-mold electromagnetic stirring technique has proved effective in suppressing the formation of CO bubbles as originally planned, as well as in preventing longitudinal surface cracks and reducing subsurface inclusions. It is now extensively utilized to improve the surface quality of cast steel. In the late 1980s, in-mold electromagnetic braking drew attention as a useful technique for high-speed casting. Nippon Steel also worked on a new type of electromagnetic braking technique that uniformly applies a direct-current (DC) magnetic field across the width of the strand being cast. This resulted not only in preventing the penetration of inclusions deep into the pool, but also in discovering a new function of preventing the mixing in the strand pool. The technique of controlling the initial solidification of continuously cast steel by utilizing electromagnetic pressure has been a subject of ongoing research and development in recent years. In this paper, the details of each application, including the important MHD phenomena, are introduced and are followed by the description of principal results in each plant application.

2.

NUMERICAL SIMULATION METHOD FOR THE INVESTIGATION AND CONTROL OF MHD APPLICATION

The analysis of electromagnetic fields is performed by using simplified scalar or vector potential methods for simple cases, such as round billet CC, or the finite-element method (FEM) for more complicated cases, such as in-mold stirring of a slab caster to obtain the distributions of magnetic field, induced electric current field, and Lorentz force as their product. The basic equations are Maxwell's equations, Ohm’s law, and the Biot–Savart law, which are used as the support equations in the simple treatment of the electromagnetic field. In the case of the electromagnetic brake, a potential equation is used. For the axisymmetric case, a vector potential method, which leads to an integral equation of induced current, is used. Once the distribution of induced electric current is found, the magnetic field and the Lorentz force distribution can be calculated. For the numerical simulation of electromagnetically driven flow and the temperature field, the Navier–Stokes equations, along with the mass conservation and energy conservation equations, are solved by using a finite-difference method, where the Lorentz force is introduced as an external force and Joule heating as source term in the energy conservation equation.

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For the trajectory calculation of particles, which is necessary for the prediction of inclusion behavior in molten steel, a Lagrangian method is used.

3.

IN-MOLD ELECTROMAGNETIC STIRRING BY TRAVELING MAGNETIC FIELD

In-mold electromagnetic stirring is schematically illustrated in Fig. 1. A linear motor is installed in the upper part of the water box in the

Figure 1

Schematic illustration of an in-mold electromagnetic stirrer.

wide face of the mold. The linear motor horizontally stirs the molten steel near the meniscus by generating a traveling magnetic field that covers the full width of the mold [1]. Several new techniques are incorporated to enhance the electromagnetic stirring efficiency. For example, using stainless steel-clad copper plates or light-gage copper plates of low electrical conductivity, and adopting a 10 Hz or lower frequency traveling magnetic field, minimize the attenuation of magnetic flux density. Low-melting-point alloy experimentation and numerical analysis showed that the distance between the linear motor and the meniscus has a great effect on the meniscus velocity and flow pattern.

3.1.

Effect of in-mold electromagnetic stirring on slab quality

In-mold electromagnetic stirring was first introduced to suppress the formation of CO blowholes during casting of pseudo-rimmed steel by

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reducing the concentration of solute elements at the solidification front by the flow [1]. When in-mold electromagnetic stirring was applied to the continuous casting of aluminum-killed steel, it sharply reduced the amount of aluminum oxides within 10mm of the slab surface [2]. As shown in Fig. 2, by applying electromagnetic stirring, the number of inclusions is reduced drastically, even in the unsteady state of the cast

Figure 2 Reduction in subsurface inclusions by in-mold electromagnetic stirring.

(bottom and inter-charge). A longitudinal crack on the slab surface is reduced when the in-mold electromagnetic stirrer causes the molten steel to flow at the front of the solidifying shell in the initial stage of solidification during the continuous casting of medium-carbon steel [2]. The reduction of longitudinal surface cracks is attributable to the washing action of the electromagnetically stirred flow of molten steel. The washing action should level the shell thickness through the promotion of temperature uniformity in the mold pool and reduce the interdendritic segregation.

3.2.

Numerical simulation of in-mold molten steel flow controlled by traveling magnetic field

A magnetic field traveling through a molten steel pool induces an electric current in the molten steel. The interaction of the traveling magnetic field and the induced current produces a Lorentz force, which in turn drives the molten steel. The induced current path is greatly affected by

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the surrounding electric boundary conditions, and the electromagnetic force does not always have a component in the mold width direction alone. The discharge stream from the submerged entry nozzle causes the flow of molten steel in the mold. This flow interferes with the flow driven by the linear motor. As numerical analysis techniques, the FEM and the finite-difference method (FDM) with large-eddy simulation (LES) were used for analyzing the magnetic field and the flow field, respectively. The mold width and thickness were 1,830 and 280 mm, respectively, and the casting speed was 1.3m/min. Figure 3 shows the flow velocity distributions in the mold meniscus position on the transverse section and in the solidifying

Figure 3 Effect of in-mold electromagnetic stirring on the molten steel flow in the mold pool.

shell front position on the longitudinal section without and with in-mold electromagnetic stirring, respectively, when the nozzle port angle is 35°. Without in-mold electromagnetic stirring, the dominant flow is not seen at the solidification front. When electromagnetic stirring is conducted, the molten steel is accelerated to form a relatively uniform rotating flow in the vicinity of the meniscus. The temperature field calculated by the simplified treatment of boundary condition as a Dirichlet condition shows that the stirring increases the uniformity of the temperature field. With regard to the inclusion-removal effect, there are many factors affecting the inclusion behavior, such as flotation, Saffman’s effect, the Magnus effect, and interfacial tension. However, from an order estimation, it can be shown that the flow field itself and Saffman’s effect

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are dominant in the boundary-layer region except within a few dozen micrometers from the solidifying shell. Here, only Saffman’s effect is discussed by using boundary-layer analysis and an estimation of the force acting on a particle. The boundary layer is analyzed by using LES and by introducing fine meshes near the wall. Figure 4 shows the result of numerical calculation. It is noted that with EMS, which generates a flow of about 0.5 m/s inside the mold, the boundary layer becomes thin-

Figure 4

Velocity distribution near the boundary with electromagnetic stirring.

ner and then the Saffman’s lift force becomes higher in the EMS-driven flow than in the one generated by the bulk flow itself.

4.

CONTROL OF MOLTEN STEEL FLOW IN A MOLD BY A LEVEL DC MAGNETIC FIELD

The in-mold electromagnetic braking technique started as a localized MHD application [3] to brake directly the molten steel stream discharged from the submerged entry nozzle. At that time, however, the technique had several problems concerning the stability of the braking effect and the resultant metallurgical benefits. A new technique of stably controlling the molten steel flow in the mold was developed by applying a level DC magnetic field across the mold width; a new function—that of suppressing mixing—was then discovered, in addition to the conventional functions ensuring the slab quality [4, 5, 6, 7].

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Effect of level magnetic field brake on slab quality

Reduction in the number of inclusions. When molten steel was continuously cast on a 1-ton test machine, a level magnetic field (LMF) brake was operated below the mold of the caster, and the metallurgical benefits of this molten steel flow control were investigated. To show the effect of LMF on the penetration depth of injected flow from the submerged entry nozzle, a sulfur tracer wire was fed to the meniscus in the mold. The nozzle discharge stream melted the wire, and the tracer was carried deep into the pool by the downward flow. When a magnetic flux density of 0.55 T was applied, the penetration depth of the tracer was approximately halved, compared with the case in which no electromagnetic braking was applied. This result is similar to the effect of electromagnetic braking in preventing the introduction of inclusions and bubbles into the pool [8]. Minimizing the transition region at ladle change. To study the control of the transition region during a ladle change by imposing the level magnetic field, a medium-carbon steel was cast successively after a low-carbon steel. When the magnetic flux density was 0T, the mixing of the melt proceeded in the transition region mainly by the nozzle discharge stream. As a result, the solute concentration is widely distributed in the cast slab, as shown in Fig. 5. When a 0.55 T magnetic field was applied at the bottom of the mold, the upper part of the pool

Figure 5 Minimizing transition region at ladle change.

was separated by the LMF from the lower pool, thereby suppressing the mixing and minimizing the transition region, as shown in Fig. 5.

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Numerical simulation of in-mold molten steel flow control by level DC magnetic field

When an external DC magnetic field is applied to a moving, electrically conductive fluid, an electric current is induced in the fluid. The current flowing through the fluid establishes a circuit satisfying the law of conservation of electric current, according to the electric boundary condition, and the velocity distribution and magnetic field at right angles induce electromagnetic braking forces on the fluid. The current path formed in this way contains components in the direction opposite to the braking direction. The actual electromagnetic braking phenomenon

is not simple. Hence, substantial understanding of the electromagnetic braking mechanism is indispensable for controlling the flow of molten steel in complex systems, such as the mold of a continuous caster. Concerning the electromagnetic field, a simplified potential equation

is introduced. The electric potential density distribution was obtained and was used to calculate the Lorentz force. For the electrical boundary condition, the solidifying shell drawn at the casting speed was taken into

account. As inlet boundary conditions, a fully developed turbulent flow was given in the upper part of the nozzle tube, and a free jet was formed at the nozzle port. The effect of the free surface was not taken into account.

Figure 6 show the calculated velocity distribution without and with a level magnetic field. Without the magnetic field, the nozzle discharge

Figure 6 Change in velocity distribution in CC mold with LMF.

stream impinges on a narrow face of the pool, and penetrates deep into the pool. On the other hand, with a level DC magnetic field, the nozzle

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stream crossing the DC field is decelerated by an electromagnetic braking force, in which the downward stream at the narrow face is retarded. Figure 7 shows the induced current distributions both in the molten

steel and in the solidifying shell. These three-dimensional current dis-

Figure 7 Electric current distribution in the cross-section of the strand pool with solidifying shell at the level of LMF.

tributions are generated not only by the magnetic and flow fields but also by the complicated electric boundary conditions. Figure 8 shows an example of the calculated results, which demonstrate the suppression

Figure 8 change.

Change of solute distribution at the half width section, 120s after the ladle

of mixing by LMF in the sequential cast of different steel grades. The trajectory calculation for 100 diameter inclusions (Fig. 9) shows the prevention of inclusion movement deep into the strand by the flow. The important factor for setting the amplitude of magnetic field is what kind

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Figure 9 Change in trajectory of inclusions of 100

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without and with LMF.

of defect is to be controlled. For the control of smaller size inclusions, since flotation cannot be expected in plug flow, a choice of proper magnetic field is necessary. In order to control the inclusions and argon bubble behavior by LMF, it is important to know not only the particle behaviors themselves but also the typical phenomena caused by the MHD. The numerical and experimental results showed that the reverse flow is formed around the jet from the submerged entry nozzle. The behavior of argon bubbles is largely affected by this flow.

5.

CONTROL OF INITIAL SOLIDIFICATION BY AN AC MAGNETIC FIELD

The surface of a continuous cast slab is characterized by oscillation marks. The oscillation marks are considered to result from the reciprocal motion of the oscillating mold and the solidifying shell being withdrawn through the layer of mold flux that flows as lubricant between the mold wall and the solidifying shell. These marks are a troublesome obstacle to improving the surface quality of continuously cast steel. In their basic research on the behavior of molten metal in an alternating current (AC) magnetic field, the authors clarified that the free-surface profile of the liquid metal and the static pressure near the meniscus can be controlled by electromagnetic pressure [9]. This phenomenon can be utilized to control the initial solidification of steel being continuously cast. In other words, an electromagnetic force is used to control the meniscus shape and the mold flux channel shape, to relax the dynamic pressure produced in

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the channel by the mold oscillation, and to stabilize the mold lubrication, thereby improving the steel surface quality.

5.1.

Results of plant test

The phenomenon near the initial solidification part and controlling effect of an AC electromagnetic field is illustrated schematically in Fig. 10. A solenoid coil is installed in the mold water-cooling box [10].

Figure 10 Schematic illustration of apparatus for controlling initial solidification of molten steel by electromagnetic force.

Figure 11 shows the relation between imposed magnetic flux density and surface roughness of the cast in the case of electromagnetic casting of stainless steel round billet. The axial component of magnetic flux density at the level of the coil top and on the axis is chosen as the representative value. The bloom surface became smooth and the segregation of nickel, which is typically observed at the bottom of the oscillation mark of the cast, disappeared when the magnetic flux density became more than 0.1 T.

5.2.

Mechanism of initial solidification control

The meniscus shape is calculated from the balance between the electromagnetic force acting on the molten steel pool and the ferrostatic pressure of the molten steel. The velocity distribution of the molten steel in the pool is also obtained. Then, the temperature distribution in the pool is determined by coupled steady-state analysis of heat transfer and

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Relation between imposed magnetic flux density and surface roughness of

fluid flow by the finite-difference method (Fig. 12). By analyzing the flux

flow in the gap between the solidifying shell and mold wall through two-

Figure 12 Distribution of flow velocity and temperature (a) without and (b) with electromagnetic pressure.

dimensional viscous flow field calculations, the dynamic pressure of the

flux channel is evaluated [11]. The static pressure brought about by the electromagnetic pressure generated in the molten steel pool in the mold

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induces the convex shape of meniscus, and the static pressure acting on the solidifying shell is diminished. The amount of heat generated in the molten steel by electromagnetic induction is almost negligible compared with the amount of heat extracted. When compared with the case in which no electromagnetic force is applied, the flux channel thickness is increased, and the dynamic pressure acting on the top of the solidifying shell is decreased; the decrease in the depth of oscillation marks formed

on the bloom surface can be explained in this way. The expansion of the channel thickness also decreases the withdrawal resistance in the mold.

6.

CONCLUSIONS

Examples of the application of MHD in the continuous casting process—the electromagnetic stirring of the strand pool with a traveling magnetic field, the control of molten steel flow by an in-mold DC magnetic field brake, and the initial solidification control by an electromagnetic pinching force—have been introduced, along with results of plant tests. For the interpretation and optimization of these applied MHD processes, numerical simulations have played an important role since the phenomena, which are important to the total control of the

system, are both complex and remarkable. With the aid of numerical simulations, these applied MHD techniques have been designed to improve further the continuous casting process capability. They improve the surface quality of cast steel by homogenizing the meniscus temperature, stabilizing initial solidification, and cleaning the surface layer. They also improve the internal quality of the cast steel by preventing inclusions from penetrating deep into the pool and by promoting the flotation of argon bubbles. The application of a pinching force to control initial solidification will lead to the simultaneous achievement of high productivity and high surface quality in continuous casting of steel.

References [1] Takeuchi, E., H. Fujii, T. Ohhashi, H. Tanno, S. Takao, K. Furugaki, and O. Kitamura. 1983. Continuous casting of pseudo-rimmed steel with in-mold electromagnetic stirring. Tetsu-to-Hagane 69(14), 1615–1622.

[2] Pukuda, J., Y. Ohtani, A. Kiyose, T. Kawase, and K. Tsutsumi. 1998. Improvement of slab quality with in-mold electromagnetic stirrer. Proceedings of the 3rd European Conference on Continuous Casting, 437–445. [3] Nagai, J., K. Suzuki, S. Kojima, and S. Kollberg. 1984. Steel flow control in a high-speed continuous slab caster using an electromagnetic brake. Iron and Steel Engineer 61(5), 41–47.

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[4] Takeuchi, E., H. Harada, H. Tanaka, and H. Kajioka. 1992. Suppression of mixing in the pool of continuous casting strand by DC magnetic field. Magnetohydrodynamics in Process Metallurgy, Minerals, Metals and Materials Society

(TMS), 261–266. [5] Zeze, M., H. Harada, E. Takeuchi, and T. Ishii. 1993. Application of a DC magnetic field for the control of flow in the continuous casting strand, Iron and Steelmaker 20(11), 53–57. [6] Harada, H., E. Takeuchi, M. Zeze, N. Ishii, A. Uehara, and T. Okazaki. 1996.

Control of the molten steel flow in CC with a level magnetic field. Current Advances of Materials and Processes, The Iron and Steel Institute of Japan 9, 205. [7] Ishii, N., N. Konno, T. Okazaki, A. Uehara, E. Takeuchi, H. Harada, T. Kikuchi, and K. Watanabe. 1996. The electromagnetic technique with level DC magnetic field—1, Current Advances of Materials and Processes, The Iron and Steel Institute of Japan 9, 206. [8] Yoneyama, Y., E. Takeuchi, K. Matsuzawa, I. Sawada, Y. Hattori, and Y. Kishida. 1989. Study on the electromagnetic brake of molten steel flow. Seit-

etsu Kenkyu, No. 335, 26–34. [9] Miyoshino, I., E. Takeuchi, T. Saeki, H. Kajioka, H. Yano, and J. Sakane. 1989.

Influence of electromagnetic pressure on the early solidification in a continuous casting mold. The Iron and Steel Institute of Japan International 29(12), 1040– 1047. [10] Toh, T., E. Takeuchi, M. Hojo, H. Kawai, and S. Matsumura. 1997. Electromagnetic control of initial solidification in continuous casting of steel by low frequency alternating magnetic field. The Iron and Steel Institute of Japan International 37 (11), 1112–1119. [11] Takeuchi, E., and J. K. Brimacombe. 1984. The formation of oscillation marks in the continuous casting of steel slabs. Metallurgical Transactions B 15(3), 493–509.

RAIL VEHICLE DYNAMICS FOR THE 21ST CENTURY Ronald J. Anderson Department of Mechanical Engineering, Queen’s University, Kingston, Ontario, Canada [email protected]

John A. Elkins RVD Consulting, Inc., Pueblo, Colorado, USA [email protected]

Barrie V. Brickle Transportation Technology Center, Inc., Pueblo, Colorado, USA [email protected] Abstract

1.

Although railway vehicles were first used in the 18th century, it was only in the 20th century that engineers began to understand their dynamics and were able to write down, and solve, their equations of motion. The single most fundamental property of a railway vehicle is the use of a steel wheel running on a steel rail with all the traction, braking and guidance forces being transmitted through a small contact area between the wheel and the rail. While the inherent guidance provided by a conical wheelset running on rails was known even to George Stephenson in 1821, who described its kinematic oscillation, it is only in the last 40 years that an adequate theory has been available for vehicle suspension designers to use. It is well known that a mechanical system that is subject to non-conservative forces may become dynamically unstable under certain conditions. In a railway vehicle the non-conservative forces arise at the contact point between the wheel and rail due to creepage, and the sustained lateral and yawing oscillation that results in some vehicles is known as hunting. This paper traces the developments in railway dynamics up to the end of the 20th century and discusses the latest developments, with predictions as to what the future might hold.

INTRODUCTION

It is usually the case that predictions made as centuries end are incorrect and not of much use. While this paper has a title that indicates that it addresses the topic of rail vehicle dynamics for a century that is just beginning, it is in fact more of a historical document. The authors feel 113

H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 113–126. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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that this paper provides an opportunity to look at where the field has been in the last 100 years, point out some of the major results during the 20th century, and then provide only a very brief glimpse into the near future. Having taken this approach to writing the paper, it was also felt that, for the readers’ benefit, a significant body of written material on the topic of rail vehicle dynamics should be listed in this paper even if the individual papers there are not all referenced. The result is that a relatively extensive chronological bibliography has been included subject to the constraints on the length of this paper. It is impossible to list all the papers on the subject of rail vehicle dynamics, and the bibliogra-

phy contains only those papers that are important and easily accessible through most technical library systems. The authors are keenly aware of the wealth of literature that is missing due to restricting the list to the English language. With our apologies to those readers from non-English speaking countries, we present the bibliography as a list of core material from which researchers can build a more comprehensive library in all languages by following the leads provided there.

2.

THE LEGACY OF THE CONED WHEELSET

At the beginning of the 20th century, vehicles supported by steel wheels running on steel rails had already been a well-accepted part of the transportation infrastructure worldwide for many years. Rail vehicles had developed rapidly since 1825 when George Stephenson (1781–1848) persuaded the directors of the first public railway between Stockton and Darlington in Northern England to abandon their plans for horse traction and use instead a steam locomotive.

Rigid coned wheelsets were integral to the design of steam trains, as they had been to horse-drawn rail vehicles. This was because of their natural self-centering tendency on tangent track and their ability to negotiate curves by moving outward on the track, thereby developing a differential rolling radius between inner and outer wheels. Even with the low speeds of trains during the early 19th century, Stephenson observed and described a “kinematic oscillation” in 1821. Much later this would be recognized as a “wheelset hunting motion”, which, due to the nonconservative forces acting at the wheel–rail interface, tends to become unstable with increasing speed and increasing conicity. The story of rail vehicle dynamics is a story of two contradictory design goals. Rail vehicles should:

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be designed to travel comfortably at high speeds on straight or “tangent” track, and

be designed to traverse curved track without excessive noise or wear, which arise from misalignment of the wheelsets with the track. The first of these goals requires low conicity to ensure stability. The second requires high conicity to promote large rolling radius differences between the inner and the outer wheel on a curve. As it turns out, the 19th century practice of using coned wheels on rigid wheelsets has carried on with minor modifications throughout the 20th century and the conicity of the wheels has a profound effect on both the stability of running on tangent track and the ability to negotiate curves freely. In most conventional railway vehicles, pairs of wheels are mounted rigidly on a common axle, providing a structure known as a “wheelset”. The wheelset evolved by trial and error during the early 19th century and has changed very little to the present day. It consists of two wheels having a coned or profiled tread with a flange on the inside. The tread cone angle is about 2° while the flange cone angle is about 70°. The wheelset runs on rails that are usually canted inwards at 1 in 20. Figure 1

shows sample wheel and rail profiles in contact.

Figure 1

Wheel and rail profiles.

It is now known that the frequency of Stephenson’s “kinematic oscillation” is proportional to the forward speed of the wheelset and that a free single wheelset is unstable, as shown in Fig. 2. The single wheelset can be stabilized if it is attached elastically to a hypothetical infinite mass that moves along the tangent track at a forward speed V. This is shown schematically in Fig. 3. The longitudinal and lateral primary suspension stiffnesses and respectively, are design variables that can be chosen to achieve some desired level of stability as the wheelset moves along the track. Realistically, the infinite mass concept cannot be achieved and rail–vehicle design tended toward a situation in which a relatively long carbody was supported by single

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Figure 2 The instability of a single wheelset.

Figure 3

A wheelset elastically suspended from an infinite mass.

wheelsets at each end. The carbody and its load approximated the infinite mass, especially for fully loaded cars, but could not totally remove the tendency of the wheelsets to “hunt” at high speed regardless of the suspension stiffnesses employed. In fact, if the stiffnesses were made very large, the entire structure including the two wheelsets and the carbody could still oscillate in an unstable mode, which came to be called “body hunting”. The design trade-off between stability and curving became evident with these early two-axle vehicle designs. Simple geometry (Fig. 4) shows that the carbody, on a curve, must be a chord of that curve since the leading and trailing wheelsets are constrained to remain essentially centered on the track. If the primary suspension is relatively stiff and retains the angular positions of the wheelsets relative to the car-

body, then there must be a misalignment of the wheelsets to the track. This angular misalignment a is called the “angle of attack” and from it results most of the wheel–rail wear and noise generated as a rail vehicle traverses a curve.

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The angle of attack developed by a long carbody.

There are two possible remedies to the curving problem. One is to use softer primary suspensions so that the wheelsets can exercise their natural tendency to align themselves with the rails on the curve. Alternatively, the carbody can be reduced in length so that the angle of attack is reduced through geometry. The first of these possibilities reduces the restraint against instability imposed by the stiff suspension envisioned for the “infinite mass” case (Fig. 3) and can therefore not be exploited to the extent desired without making the vehicle have less than adequate stability. The second option, reducing the carbody length, would seem to imply a reduction in the payload capability of the vehicle. Exploration of this option led to the introduction of small two-axle structures that support the ends of the carbody and are connected to it by another level of

suspension (the “secondary suspension”). These are termed “trucks” or “bogies” and a sample is shown schematically in Fig. 5.

Figure 5 A “truck” or “bogie”.

The bogies are short in comparison with the carbody and are able to swivel relatively freely under it so that they can be better aligned with the track. Geometrically they remain similar to the two-axle carbody

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shown in Fig. 4 and a stiff primary suspension holds the wheelsets parallel to each other, thereby producing angles of attack. Nevertheless, the angles of attack for a carbody fitted with bogies are substantially smaller than those of the same carbody fitted with two wheelsets. Rail vehicles fitted with bogies are found to experience two distinct types of wheelset hunting motion. Figure 6 shows the two modes—one with the wheelsets oscillating out-of-phase with each other (a “bending”

Figure 6 Bending and shear modes.

mode) and the other with the wheelsets oscillating in phase with each other (a “shear” or “lozenging” mode). The displacement in Fig. 6 represents the interaxle shear, which develops as the wheelsets oscillate in phase with each other. In the 1970s it became clear to designers that the bending mode would promote good curving whereas the shear mode would not. Further, it was postulated that vehicles could be designed for the same level of stability with more than one combination of primary stiffnesses. The goal, then, was to design a stable vehicle that was soft in bending and stiff in shear so that it would simultaneously be stable and perform well in curves. It is relatively easy to show that a conventional bogie (Fig. 5) has an effective bending stiffness that can be written as

and an effective shear stiffness

given by

Substituting Eqn. (1) into Eqn. (2) and simplifying yields the expression for as a function of and

Similarly, from Eqn. (1), an expression for be developed:

as a function of

can

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Equations (3) and (4) now provide the ability to specify the actual primary stiffnesses, and in order to achieve the desired bending and shear stiffnesses, and The goal was to choose a combination of and that used high shear stiffness to achieve stability and low bending stiffness to promote good curving. In fact, the designers found that they were not totally free to choose the design parameters because—see the denominator of Eqn. (3)—values of less than required that the stiffness be less than zero, and this was physically impossible. It became common to produce contour plots showing lines of constant critical speed on axes of and Also shown on these plots was the so-called “limit line” above which one could not go without producing a suspension component with a negative stiffness. The design area above the limit line was attractive to many people and the conventional bogie began to change when it was realized that direct interaxle connections could be used to add shear stiffness without affecting the bending stiffness. This led to a generation of bogies variously called “cross-braced” (to represent the direct interaxle connections), “radial” (to indicate that the bending stiffness is small and the wheelsets are expected to align themselves radially on curves to minimize the angles of attack), or “self-steering” (to indicate that the wheelsets are not restrained from allowing their conicity to steer them to a radial position on the curve). These designs remain with us today. Forced-steering bogies were another development that attempted to overcome the trade-off between curving and stability. This is a concept in which a kinematic linkage between the wheelset, the bogie, and the carbody is used to steer the wheelsets in response to the swivel angle of the bogie. The swivel angle is directly related to the curve radius, the length of the carbody, and the length of the bogie. The required steer angle is related to the length of the bogie and the radius of the curve. Using the known lengths, it is a relatively simple matter to derive a relationship between swivel angle and wheelset steer angle that can be realized with a simple linkage. The advantage of such a system is that the linkage, which acts as the longitudinal primary suspension, can be made very rigid to promote stability on tangent track while still guiding the axles to radial positions on the curves. Two terms have been coined for these designs—“forced-steering” for bogies that retain their original primary suspensions and require that the linkage overcome the suspension forces to steer, and “guided-steering” for designs in which the primary suspension has been removed and the linkage acts in its place

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while guiding the wheelsets to radial positions on curves without any resistance.

3.

DEVELOPMENTS IN THE 20TH CENTURY

The development of the ability to write and solve the equations of motion for complex rail vehicles hinges on a few critical developments in

the 20th century. All the authors in the bibliography and many who are not there have made contributions to the field, but some contributions were pivotal to success. These are described in this section.

3.1.

The wheel–rail interface

It was Carter [1, 2, 3, 4]1 who first brought to the field an understanding of the relationship between the creepages and forces acting in the contact patch between the wheel and the rail. Creepage is a function of

the motion of the wheelset, and Carter’s development of a mathematical expression relating the forces acting on the wheelsets to their motion was a fundamental first step in the process of being able to write the

equations of motion for the complete rail vehicle. The development of refined rolling contact theories continued throughout the century with particular achievements being made by Vermeulen and Johnson [16] and, especially, by Kalker [22, 24, 30, 40, 42, 52, 59].

Kalker’s work, while mathematically complex, has produced highly accurate and practical models of wheel–rail forces, which are used almost exclusively in modern simulations of rail vehicle dynamics. Kalker’s models range from linear to fully nonlinear.

3.2.

Linear dynamic models

Wickens [17, 19, 28, 33, 36, 37, 39, 73] is credited with first applying a linear form of the rolling contact theory to a vehicle model. His 1965– 66 paper entitled “The dynamics of railway vehicles on straight track:

Fundamental considerations of lateral stability” [19] is a classic, which provided the impetus for many other authors to develop models of rail vehicle dynamics from which much knowledge was gained. From these linear models came the understanding of vehicle stability and critical speed that is widely in use even today. The introduction of digital computers provided an opportunity for researchers to develop

1 References are arranged by date of publication. Not all entries in the References are cited in the text.

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computer codes capable of calculating the eigenvalues and eigenvectors for rail vehicles and, from them, interpreting the frequencies, damping ratios, and shapes of the dynamic modes of the vehicle. This provided very useful design information that had not previously been available.

3.3.

Multibody dynamics

Once the procedure for including the wheel–rail forces in the equations of motion had been developed and computers were available to solve large numbers of simultaneous equations, the most difficult part of creating dynamic models of new vehicles became the process of deriving the multi-degree-of-freedom equations manually. In many cases, analysts resorted to using only single-bogie or half-body models and accepted the inaccuracy in exchange for gains in efficiency. This all changed with the introduction of multibody dynamics to various fields, including rail vehicle dynamics, in the 1980s and 1990s (see Schiehlen [57]). Analysts were able to exploit these techniques to develop very complex vehicle models, which included nonlinear wheel–rail forces, suspension elements, and geometry. The equations of motion for these complex models can be solved only through numerical integration, and few multibody software packages retain the ability to linearize the equations and produce the eigensolutions that were so useful to designers in the past.

3.4.

Nonlinear dynamic models

Clearly the equations of motion of a rail vehicle are nonlinear and therefore belong to the class of nonlinear dynamical equations that may exhibit chaotic motion. The last two decades of the 20th century saw much effort directed toward these problems in general, with only a few researchers looking at chaotic motions of rail vehicles (see True and Kaas-Petersen [58] and True [65]) but obtaining interesting results. In particular, it has been shown that the critical speed established by linear theory is suspect. More developments in this area are expected.

4.

THE 21ST CENTURY

The ability to analyze and predict the motions of rail vehicles has seen enormous change in the last century. The field developed from one in which vehicle designers had, without a scientific basis, some understanding of the dynamic behavior of the vehicles to one in which we now have the knowledge and tools to make highly accurate predictions. The result of this knowledge base has been the development over time of rail vehicles that are more comfortable, faster, and more efficient. Although the

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basic idea of a vehicle that runs on conical wheelsets rolling on steel rails has not changed, the ability to design suspensions has moved rail vehicle technology ahead in step with all the other technological innovations of the 20th century. Rail vehicle dynamics has now reached the point where a number of general computer programs, such as NUCARS and A’GEM, exist, which include all the latest developments described above and can simulate the dynamic behavior of any type of railroad vehicle, or vehicles, negotiating curves, transitions, or tangent track. These programs have been validated against the results from many tests and benchmarks. As long as an accurate set of vehicle characteristics are input into the programs, they can provide a precise simulation of the dynamic response of the vehicle. The problem of ensuring that accurate vehicle characteristics are used in the computer programs has become a priority for rail vehicle dynamicists. The authors have used a number of different techniques for measuring these characteristics, such as vehicle resonance tests, stiffness tests, and impulse “hammer” tests. It is not sufficient to accept manufacturers’ specification data for components, as these are often found to be different from the actual component characteristics as fitted on the vehicle. As we move to the 21st century, the “mechatronic train” is approaching quickly. Mechatronics exploits a synergy between mechanics and electronics to design, construct, and maintain improved products and processes. The application to rail vehicles is through the use of sensors, actuators, and communication to tune the suspension parameters to achieve the highest possible levels of comfort and safety. It is probable that the operating speeds of even the highest speed trains of today will be significantly increased through this marriage of mechanics and electronics. The development and refinement of mechatronic trains will likely consume the first two decades of the 21st century. Beyond that the authors deem it unwise to predict except to say that some form of the conical wheelset will still be evident, perhaps even in the 22nd century.

References [1] Carter, F. W. 1916. The electric locomotive. Minutes of Proceedings of the Institution of Civil Engineers 201, 221–300. [2] Carter, F. W. 1926. On the action of a locomotive driving wheel. Proceedings of the Royal Society of London A 112, 151–157.

[3] Carter, F. W. 1928. On the stability of running locomotives. Proceedings of the Royal Society of London A 121, 585–611.

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[4] Carter, F. W. 1930. The running of locomotives with reference to their tendency to derail. The Institution of Civil Engineers Selected Engineering Papers 81, 1– 25. [5] Cain, B. S. 1935. Safe operation of high-speed locomotives. Transactions of the American Society of Mechanical Engineers 57, 471–479. [6] Langer, B. F., and J. P. Shamberger. 1935. Lateral oscillations of rail vehicles. Transactions of the American Society of Mechanical Engineers 57, 481–493. [7] Giesl-Gieslingen, A. 1936. Riding qualities of passenger cars. Railway Age 101, 214–216. [8] Davies, R. D. 1938–39. Some experiments on the lateral oscillation of railway vehicles. Journal of the Institution of Civil Engineers 11, 224–288.

[9] Cain, B. S. 1940. Vibration of Rail and Road Vehicles. Toronto: Pitman Publishing Company. [10] Cain, B. S. 1941. Notes on the dynamics of electric locomotives. Transactions of the American Society of Mechanical Engineers 63, A-30–A-36. [11] Newberry, C. W. 1945. A study of the riding and wearing qualities of railway carriage tyres having various profiles. Proceedings of the Institution of Mechanical Engineers 153, 25–40. [12] Davies, R. D., and A. F. Cook. 1948. The motion of a railway axle. Proceedings of the Institution of Mechanical Engineers 158, 426–436.

[13] Johnson, K. L. 1958. The effect of spin upon the rolling motion of an elastic sphere on a plane. Transactions of the American Society of Mechanical Engineers 80, 332–338. [14] Johnson, K. L. 1958. The effect of tangential contact force upon the rolling motion of an elastic sphere on a plane. Transactions of the American Society of Mechanical Engineers 80, 339–346. [15] Katz, E., and A. D. DePater. 1958. Stability of lateral oscillations of a railway vehicle. Applied Scientific Research A 7, 393–407. [16] Vermeulen, P. J., and K. L. Johnson. 1964. Contact of nonspherical elastic bodies transmitting tangential forces. Journal of Applied Mechanics E 31(2), 338–340. [17] Wickens, A. H. 1965. The dynamic stability of railway vehicle wheelsets and bogies having profiled wheels. International Journal of Solids and Structures 1, 319–341. [18] Bishop, R. E. D. Some observations on the linear theory of railway vehicle instability. 1965–66. Interaction Between Vehicle and Track, Proceedings of the Institution of Mechanical Engineers 180(3F), 15–28. [19] Wickens, A. H. 1965–66. The dynamics of railway vehicles on straight track: Fundamental considerations of lateral stability. Interaction Between Vehicle and Track, Proceedings of the Institution of Mechanical Engineers 180(3F), 29–44. [20] Matsudaira, T. 1965–66. Hunting problem of high-speed railway vehicles with

special reference to bogie design for the new Tokaido line. Interaction Between Vehicle and Track, Proceedings of the Institution of Mechanical Engineers 180(3F), 58–72. [21] Gilchrist, A. O., A. E. W. Hobbs, B. L. King, and V. Washby. 1965–66. The

riding of two particular designs of four-wheeled railway vehicle. Interaction

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Between Vehicle and Track, Proceedings of the Institution of Mechanical Engineers 180(3F), 99–119. [22] Kalker, J. J. 1966. Rolling with slip and spin in the presence of dry friction. Wear 9, 20–38. [23] Hobbs, A. E. W. 1967. A survey of creep. Dynamics Section, A.P.T. Group, Research and Development Division, Railway Technical Centre, Derby.

[24] Kalker, J. J. 1968. The tangential force transmitted by two elastic bodies rolling over each other with pure creepage. Wear 11, 421–430. [25] Newland, D. E. October 1968. Steering characteristics of bogies. The Railway Gazette 124(19), 745–750. [26] Matsudaira, T., N. Matsui, S. Arai, and K. Yokose. 1969. Problems on hunting of railway vehicle on test stand. Journal of Engineering for Industry B 91(3), 879–890. [27] Newland, D. E. 1969. Steering a flexible railway truck on curved track. Journal of Engineering for Industry B 91(3), 908–923.

[28] Wickens, A. H. 1969. General aspects of the lateral dynamics of railway vehicles. Journal of Engineering for Industry B 91(3), 869–878.

[29] Boocock, D. 1969. Steady-state motion of railway vehicles on curved track. The Journal of Mechanical Engineering Science 11(6), 556–566. [30] Kalker, J. J. 1970. Transient phenomena in two elastic cylinders rolling over each other with dry friction. Journal of Applied Mechanics E 37(3), 677–688. [31] Law, E. H., and N. K. Cooperrider. 1974. A survey of railway vehicle dynamics research. Journal of Dynamic Systems, Measurement and Control G 96(2), 132– 146. [32] Gilchrist, A. O., and B. V. Brickie. 1976. A re-examination of the proneness to derailment of a railway wheel-set. Journal of Mechanical Engineering Science 18(3), 131–141. [33] Wickens, A. H. 1976. Steering dynamic stability of railway vehicles. Vehicle System Dynamics 5, 15–46. [34] Sheffel, H. January 1976. Self-steering wheelsets will reduce wear and permit higher speeds. Railway Gazette International 132(1), 453–456. [35] Pollard, M. C. 1977. Studies of the dynamics of vehicles with cross-braced bogies. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 5th VSD– 2nd IUTAM Symposium, 518–526. [36] Wickens, A. H. 1977. Static and dynamic stability of a class of three-axle railway vehicles possessing perfect steering. Vehicle System Dynamics 6, 1–19.

[37] Wickens, A. H., and A. O. Gilchrist. July–August 1977. Vehicle dynamics—A practical theory. Railway Engineer 2(4), 26–34. [38] Elkins, J. A., and R. J. Gostling. 1978. A general quasi-static curving theory for railway vehicles. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 5th VSD–2nd IUTAM Symposium, 388–406. [39] Wickens, A. H. 1978. Stability criteria for articulated railway vehicles possessing perfect steering. Vehicle System Dynamics 7, 165–182. [40] Kalker, J. J. 1979. The computation of three-dimensional rolling contact with dry friction. International Journal for Numerical Methods in Engineering 14(7), 1293–1307.

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[41] Pollard, M. G. September 1979. The development of cross-braced freight bogies. Rail International (9), 736–758.

[42] Kalker, J. J. 1979. Survey of wheel-rail rolling contact theory. Vehicle System Dynamics 8, 317–358. [43] Sweet, L. M., J. A. Sivak, and W. F. Putman. 1979. Nonlinear wheelset forces in flange contact. Journal of Dynamic Systems, Measurement and Control 101(3), 247–255. [44] Scheffel, H. August 1979. The influence of the suspension on the hunting stability

of railway vehicles. Rail International (8), 662–696. [45] Horak, D., C. E. Bell, and J. K. Hedrick. 1981. A comparison of the stability and curving performance of radial and conventional rail vehicle trucks. Journal of Dynamic Systems, Measurement and Control 103(3), 191–200. [46] Bell, C. E., and J. K. Hedrick. 1981. Forced steering of rail vehicles: Stability and curving mechanics. Vehicle System Dynamics 10, 357–386.

[47] Newland, D. E. 1982. On the time-dependent spin creep of a railway wheel. Journal of Mechanical Engineering Science 24(2), 55–64. [48] Scheffel, H. 1982. The dynamic stability of two railway wheelsets coupled to each other in the lateral plane by elastic and viscous constraints. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 7th IAVSD Symposium, 385–400. [49] Cooperrider, N. K., and E. H. Law. 1982. The nonlinear dynamics of rail vehicles in curve entry and negotiation. The Dynamics of Vehicles on Roads and on

Tracks, Proceedings of the 7th IAVSD Symposium. 357–370.

[50] Horak, D., and D. N. Wormley. 1982. Nonlinear stability and tracking of rail passenger trucks. Journal of Dynamic Systems, Measurement and Control 104(3), 256–263. [51] Elkins, J. A., and B. M. Eickhoff. 1982. Advances in nonlinear wheel/rail force prediction methods and their validation. Journal of Dynamic Systems, Measurement and Control 104(2), 133–142.

[52] Kalker, J. J. 1982. A fast algorithm for the simplified theory of rolling contact. Vehicle System Dynamics 11, 1–13.

[53] Elkins, J. A., and R. A. Allen. 1982. Verification of a transit vehicle’s curving behavior and projected wheel/rail wear performance. Journal of Dynamic Systems Measurement and Control 104(3), 247–255.

[54] Johnson, K. L. 1982. One hundred years of Hertz contact. Proceedings of the Institution of Mechanical Engineers 196, 363–378. [55] Clark, R. A., B. M. Eickhoff, and G. A. Hunt. 1982. Prediction of the dynamic response of vehicles to lateral track irregularities. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 7th IAVSD Symposium, 535–548. [56] Goodall, R. M., and W. Kortüm.

1983.

Active controls in ground

transportation—A review of the state-of-the-art and future potential. Vehicle System Dynamics 12, 225–257. [57] Schiehlen, W. O. 1984. Modelling of complex vehicle systems. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 8th IAVSD Symposium, 548–563.

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[58] True, H., and C. Kaas-Petersen. 1984. A bifurcation analysis of nonlinear oscillations in railway vehicles. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 8th IAVSD Symposium, 655–665. [59] Kalker, J. J. 1984. A simplified theory for non-Hertzian contact. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 8th IAVSD Symposium, 295–302. [60] Johnson, K. L. 1987. Contact Mechanics. Cambridge: Cambridge University Press. [61] Evans, J. R. 1987. An investigation into the interaction between vehicle ride and tilt systems. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 10th IAVSD Symposium, 101–112. [62] Anderson, R. J., and C. Fortin. 1987. Low conicity instabilities in forced-steering railway vehicles. The Dynamics of Vehicles on Roads and on Railway Tracks, Proceedings of the 10th IAVSD Symposium, 17–28. [63] Smith, R. E., and R. J. Anderson. 1988. Characteristics of guided-steering railway trucks. Vehicle System Dynamics 17, 1–36.

[64] DePater, A. D. 1988. The geometrical contact between track and wheelset. Vehicle System Dynamics 17, 127-140. [65] True, H. 1989. Chaotic motion of railway vehicles. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 11th IAVSD Symposium, 578–587. [66] Prederich, F. 1989. Dynamics of a bogie with independent wheels. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 11th IAVSD Symposium, 217–232. [67] Elkins, J. A. 1989. The performance of three-piece trucks equipped with independently rotating wheels. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 11th IAVSD Symposium, 203–216. [68] Elkins, J. A. 1991. Prediction of wheel/rail interaction: The state-of-the-art. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 12th IAVSD Symposium, 1–27. [69] Sheffel, H., R. D. Fröhling, and P. S. Heyns. 1993. Curving and stability analysis of self-steering bogies having a variable yaw constraint. The Dynamics of Vehicles on Roads and on Tracks, Proceedings of the 13th IAVSD Symposium, 425–436. [70] Scheffel, H. 1995. Unconventional bogie designs—their practical basis and historical background. Vehicle System Dynamics 24, 497–524. [71] Harris, N. R., F. Schmid, and R. A. Smith. 1998. Introduction: Theory of tilting train behaviour. Journal of Rail and Rapid Transit 212(F1), 1–6. [72] Gilchrist, A. O. 1998. The long road to the solution of the railway hunting and curving problems. Proceedings of the Institution of Mechanical Engineers F 212, 219–226. [73] Wickens, A. H. 1998. The dynamics of railway vehicles—from Stephenson to Carter. Proceedings of the Institution of Mechanical Engineers F 212, 209–217.

FUNDAMENTALS OF THE LATERAL DYNAMICS OF ROAD VEHICLES Robin S. Sharp School of Mechanical Engineering, Cranfield University, United Kingdom [email protected] Abstract

1.

Road vehicle steering control is discussed as a man–machine problem. A conceptual framework for thinking about vehicle handling qualities issues is put forward. Fixed control and free control ideas are introduced. Rolling contact between tire and ground is central to understanding of the engineering of the vehicle and a section is devoted to an explanation of it. The steady turning behavior of road vehicles is extremely significant. The interactions of suspensions and chassis have a strong influence on the system qualities and these are highlighted. The full dynamics are then described by reference firstly to the well-known yaw–sideslip linear model and then further by generalization of the results to include nonlinear behavior of the simple car and of more complex, higher-order systems representing vehicles with much design detail. Linearization about a trim equilibrium cornering condition, and the intimate relationship between steady turning behavior and static stability properties, are important topics. Vehicle design alternatives are exposed and the use of the tire longitudinal force capability, via electronic chassis controls, to avoid compromise are mentioned. The paper concludes with a brief description of contemporary vehicle modeling and simulation issues.

INTRODUCTION

The paper is a tutorial review of some of the fundamental issues in the steering behavior of road vehicles. Since the controller in the road vehicle system is a human driver or rider, the problem involves both man and machine and interactions between them. Some discussion of driving is included, leading to ideas on the behavioral qualities necessary in a vehicle. Important aspects of the vehicle engineering are then considered, mainly from the viewpoint of two-track vehicles. These include the external forces acting through the tire–road friction interfaces. Also included 127 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 127–146. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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are the suspension kinematic and compliant chain comprising the linkages between tires and main structure. The suspension and steering links interact with the tire forces by influencing the presentation of the tires to the road surface and the distribution of the total weight of the vehicle between the wheels. The interactions are demonstrable at the first level by considering in detail the steady-state turning behavior. Dynamic properties are revealed via solutions of low-order models of simplified vehicles. Stability in particular is of interest, since the main vehicle classes have characteristic modes of instability that must be understood and guarded against in design. The extension of simple models to more fully represent real vehicles is considered. The main influences come from the additional freedoms afforded by the suspension system and steering system details. Computer-aided model building becomes almost essential for the construction of detailed models. Numerical and symbolic/numerical schemes are mentioned.

2.

MAN, MACHINE, AND INTERACTIONS BETWEEN THEM

The lateral behavior of road vehicles involves strong interactions between man and machine. Contexts vary widely over vehicle type and driver skill level. The human-factors part of the problem is much more complex than the machine part and, correspondingly, it is much less understood. However, it is possible to put the vehicle engineering into a particular context by considering first some human factors issues. Driving skill is acquired by learning, involving repeated small adjustments to mental processes, based on trial and observation. Learned mental processes can be stored and recalled when needed. The capacity to learn is a universal human property. Precisely what is learned is personal and not completely definable. It depends on the machine properties. The main lateral control mechanism on a road vehicle is the steering system. The steering control input can be thought of as a torque, involving so-called free control, or a rotational displacement, involving fixed control. Elements of both free and fixed control are present in any real steering task. The person is free to choose that combination which suits him/her best for the conditions. If the operating mode selected is principally fixed control, the person will provide a local closed-loop steering displacement controller, such that whatever torques are needed to create

the desired steering wheel angle will be provided. Throttle and brake controls will often have some modest influence on lateral behavior. Such influences imply the need for coordination of lateral and longitudinal

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controls, as opposed to completely decoupled actions. In any case, the driver is automatically exposed to both the control device displacement and to its feel. The throttle control is normally arranged to move significantly but not visibly, with spring loading to the closed condition; the brake control moves through a non-executive distance against a spring restraint and then becomes primarily a force control device, moving little once the brakes are properly applied; while the steering wheel moves substantially, is easily observable in doing so, and requires significant driver effort in order to make it move. Clearly, there are man–machine interaction consequences from these different operating modes. Typically, a driver can operate as a feedback controller, subject to the following restrictions. Firstly, he/she must be able to observe any departure from a desired condition. Secondly, there must be an effective control action that will correct the observed error. Thirdly, there must be enough time available for the control to be effective. The comparative slowness of human beings in observing, processing, decision making, and actuation makes the last quite a severe restriction. It implies that rapidly divergent systems and oscillatorily unstable systems with vibration frequencies greater than about 3 Hz will not be controllable by human subjects by active intervention.

2.1.

Steering control

Steering control involves preview of the road ahead. Without sight of the road ahead, safe travel at any reasonable speed is not feasible. Linear optimal preview control theory can be used to find how the steering wheel angle can be derived from road preview information to enable precise path following (Valtetsiotis 1999). The structure of the timeinvariant optimal path following control is shown in Fig. 1. The linear theory can be extended into a nonlinear, force-saturation framework to deal effectively even with limit conditions (Sharp et al. 2000). In providing the necessary steering control, the driver is required to produce appropriate torques and naturally has knowledge of those torques. The steering torque is needed to overcome frictional and hysteretic system losses, to accelerate the massive parts and, in particular, to work against the self-steering effects possessed by a normal system, a large contribution to which comes from the tire–ground interaction forces at the front road wheels. Such torques are modified by torqueassistance systems (power steering), which need to be designed not to spoil the driver’s access to knowledge of the tire–road interface conditions. In practice, these are bound to change substantially from time

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Figure 1 Time invariant optimal steering control structure for previewed path tracking (Sharp et al. 2000).

to time and it is imperative for at least the more adventurous driver to be well informed of them, so that the forward plan can be made with proper recognition of the circumstances.

2.2.

Requirements of the vehicle

The vehicle needs to be steerable and preferably it will be stable. If it is not itself stable, it will rely on the driver to stabilize the system, which may or may not be possible, as discussed above. Ideally, the driver’s observations of the path ahead will be from a stable platform to minimize the complexity of the view, implying the need for minimum pitch response to road roughness input. Minimum response to aerodynamic disturbance is also an ideal, since the driver will need to regulate against the effects of such response (Macadam et al. 1990), adding to the workload. Underdamped oscillatory modes will, in general, be more demanding in terms of driver control, than well damped ones. Decoupled control–response relationships are simpler than cross-coupled ones, as above. Time lags in the controlled system, between control input and significant response, should be very short compared with the corresponding human lags (the latter of the order of 0.3 s) and there should not be several conflicting responses all requiring to be controlled via a single control input.

Learning is most effective when systems behave most consistently. Road vehicle operating conditions are bound to vary according to speed, operating circumstances (loading or wear, for example), and weather. Small changes in conditions should not make large changes to behavior.

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Changes in the weather especially can make radical differences to the contact between tire and ground, dramatically altering the main external force production system. Complete consistency of vehicle behavior is not possible, without designing for worst case conditions, which is not reasonable. The best that can be done is that changes in behavior

are progressive and slow and that adequate warning signals are available to the driver that changes are occurring. In this connection, the steering system is vital. With a good system, it is possible for the continuously observed relationship between steering torque and steering angle, together with potentially confirmatory information in the vehicle responses, to keep the driver more or less fully informed of prevailing tire–road friction levels, even though only mild maneuvering is being done (Sharp 2000).

3.

TIRE SHEAR FORCE GENERATION

In the contact between a pneumatic tire and the road, the road can be considered rigid, with all necessary deformations taking place in the tire. As the tire rolls along, the rubber in the finite contact length continually leaves the contact patch at the back and is replaced by “new” rubber at the front. The simplest view of the shear force generation process comes from the isotropic brush model (Pacejka and Sharp 1991). In this model, the longitudinal and lateral compliances in the tire’s structure are confined to the individually acting tread rubber blocks. The carcass deforms radially to allow conformity of the blocks in the contact region with the flat road surface, and the blocks (or bristles) may also deform laterally and/or longitudinally, with isotropic stiffness, depending on the running conditions. The simplest running condition is free rolling. In this case, each rubber block in turn comes into contact with the ground, adheres to the ground while it is in the contact region, and leaves again. In steady free rolling, the travel direction is along the intersection of the wheel plane and the ground plane and the spin velocity is equal to the forward speed divided by the effective rolling radius (Fig. 2). For a greater spin velocity, the tread bristles need to bend forwards after entering the contact region in order to grip the ground, where they first touched down. The tread base moves backwards relative to the ground. With infinite friction available, increasing forward bending of the bristles would occur through the contact length to the rearmost point. The distribution of the normal loading through the contact length is usually considered to be independent of the running conditions and is parabolic in shape. This implies that the normal pressure between bristle and ground falls continuously to zero at the rear of the contact

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Figure 2 Representation of rolling contact with continuous renewal of contact elements.

patch. With only finite friction available, there must come a point where the elastic forces on the bristle tip due to its bending distortion are too strong for the shear force by friction at the ground contact. Then the bristle tip will slide across the road to relieve the bending strain and the shear force. The greater is the spin velocity, the earlier in the contact region will a bristle start to slide across the road. The front part of the region is characterized by adhesion, while the rear involves sliding (Fig. 3).

Figure 3 A brush model tire in straight running, spinning faster than for free rolling. Only infinite friction will prevent sliding at the rear of the contact region. Then the shear force intensity will be proportional to the distance from the front edge of contact.

The distortion of the bristles depends on the ratio of the tread base velocity, to the modulus of the wheel center velocity, For this special quantity, the term “slip” is reserved. At a certain value of the

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slip, the sliding region will extend over the whole of the contact—the bristle tips will start to slide across the road as soon as they enter the contact region—and the tire then generates the maximum shear force of which it is capable, for the load at which it is operating. This is on the basis of a finite friction coefficient independent of contact pressure and sliding speed, both of which are simplifications of reality. An important consequence of this mechanism is that the gradation from free rolling to full sliding is continuous. The shear force is a continuous function of the slip, being proportional to it for small slips. In the brush model, no inertial effects are considered and the wheel velocity does not matter in an absolute sense. The force–slip relationship is depicted, for various contact lengths and loads, in Fig. 4.

Figure 4 Isotropic brush model driving force as a function of slip for different contact lengths.

The isotropy of the brush model implies similar shear forces when the spin velocity is the free rolling velocity (no longitudinal slip) and the slip is purely lateral slip. In this event, the bristle distortion measure is the (tangent of the) angle between the wheel-center velocity vector and the intersection line common to the wheel plane and the ground plane (Fig. 5). For pure lateral slip with moderate slip angles, Fig. 5 shows that the center of action of the side force developed is behind the geometric center of the contact length. The tire generates an aligning moment. For small slip angles, the ratio of the aligning moment to the side force, called the pneumatic trail, is 1/6 of the contact length. As the lateral slip increases and sliding extends further forward in the contact region,

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Figure 5 Bristle distortions for pure lateral slip with no sliding. Successive positions of the tread base structure at times zero, and are shown in plan view. Bristle distortion is proportional to distance back from the first point of contact; tan(slip) =

the pneumatic trail decreases, reaching zero when sliding extends over the whole contact patch and the tire is generating its peak side force (Fig. 6).

Figure 6 Side force and aligning moment for isotropic brush model tire in pure sideslip for a particular load.

The compliance of real tires, which has been represented as belonging entirely to the bristles of the brush model, resides substantially in the tire carcass. The structure is very different longitudinally from laterally, so that the real tire is non-isotropic. The pure slip properties are qualitatively similar but quantitatively different, and the rather simple combined slip properties of the isotropic brush are substantially more complex for real tires. The simplest way of thinking about the carcass flexibilities is to imagine that the circumferential ring of the tire is rigid, taking the place of the undeformable base structure of the brush, and that it is connected to the wheel hub by an elastic restraint system allowing it all six degrees of freedom, relative to the hub. Such a tire has the same pure longitudinal properties in steady state as the corresponding brush. When it is generating longitudinal force, the carcass is simply wound up such that the carcass torsion balances the moment of the shear

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force about the wheel center. However, when sideslipping, the carcass will twist in yaw, by virtue of the aligning moment developed, and the slip angle will be shared between the carcass and the bristles. In the brush tire, of course, the bristles see all the sideslip, not just a part of it, so the carcass twist compliance acts to reduce the side force for a given slip angle roughly in proportion to the aligning moment generated. The carcass deflections also cause the contact friction forces to be displaced relative to the wheel center, so that an overturning moment is developed and a longitudinal force may contribute, via a lateral offset, to the aligning moment. The geometry of the rolling and slipping tire in contact with even flat terrain becomes quite complex if wheel camber is included, so that fundamental tire models become very complex before they are accurate. Rubber-to-road friction is variable also, according to contact pressure and sliding speed, with rules that are difficult to know with any precision. Empirical representation of laboratory measured steady-state force and moment data is the norm in simulation (Pacejka and Besselink 1996). The flexible carcass of the tire causes the force and moment system to lag the motions giving rise to them, under transient conditions. For example, if the straight running, free rolling tire is given a step steer input, the carcass/bristle structure is twisted initially in yaw and no

sideforce will come from that. On rolling forwards though, the bristle tip-ground forces will alter the distortion of the structure until it gets into its steady sideslipping state. The tire carcass relaxes into this state depending on the distance traveled. The relevant tire parameter is called the relaxation length. Again, for reasonable speeds, tire mass effects are not important and the transient response of the tire approximates to that of a first-order lag. The time constant of the lag is the relaxation length divided by the forward speed.

4.

AERODYNAMIC FORCES

Under typical road vehicle conditions, there is little flow dependency on Reynolds number, and compressibility effects are negligible. Forces and moments are roughly proportional to relative velocity squared, so that they become significant to the vehicle dynamics only at higher speeds of travel. For a vehicle running straight in still air, symmetry indicates no side force, rolling moment, or yawing moment. Lift may be a problem, especially at the rear (see below) and downforce is deliberately generated with inverted wings and vehicle-hull–ground interactions in many classes of competition car. The effects are to alter the tire loadings and thereby to modify the shear forces developed by the tires. In

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crosswinds, side forces and rolling and yawing moments will, in general, occur. They will influence the directional motion of the vehicle and oblige the driver to regulate against the disturbance. The typical problem is that the lateral center of pressure is well forwards in the vehicle and the yawing moment causes the vehicle to veer to leeward. The influence is particularly strong with rear-heavy cars. This is a smallperturbation problem and the tire forces are simple. The location of the center of action of the tire forces, the so-called neutral steer point, influences the steady-state yaw rate response to a crosswind excitation, which is a first-order measure of the quality of this aspect of the vehicle (MacAdam et al. 1990).

5.

SUSPENSIONS AND STEADY TURNING VEHICLE ATTITUDES Lateral suspension geometry can be thought of on any one of about

four different levels. At the lowest level, the track change property for

the static, symmetric car is captured in a parameter called the rollcenter height. At the second level, the wheel lateral, camber, and steer motions can be considered geared to the bump motion by constant gear ratios or suspension derivatives (Hales 1964). At the third level, the displacements can be considered analytic functions of bump displacement, determined by some off-line geometric analysis (Mousseau et al. 1992), while at the fourth level, the mechanical details can be included in a multibody analysis, with mechanism and vehicle-dynamics problems being solved concurrently (Orlandea et al. 1977). When a car maneuvers, its attitude is influenced by its suspension geometry, while its geometry is in turn influenced by the attitude. Understanding this interaction problem is not easy and, to simplify the mechanics, it is common to treat the car-cornering attitude problem with a “level one” suspension description. For a single-end car, the simplified problem is illustrated in Fig. 7. The car body is treated as transmitting a torque from one end to the other, with each end then considered to operate independently. By taking moments for the body about the roll center and then for the whole system about a ground point, the roll angle and the load transfer can be found and it can be established how these things depend on the suspension geometry. If the unsprung mass is neglected in comparison with the sprung mass, if the front and rear ends of the vehicle are joined together by a torsionally rigid chassis, and if small

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Figure 7 Equilibrium of single end, simplified car in steady turning.

angles are assumed, it can be shown that

where The impression is gained from these results that raising the roll center is an effective way of restricting the body roll angle and consequently the road wheel camber angles. It may not be appreciated that the system is constrained to disallow vertical movement of the roll center and it fails to represent in any way the geometric changes that are consequent on the body attitude changing. A fuller treatment will reveal the dangers in believing too literally in the results of a simplified analysis. In fact, high roll centers encourage jacking in cornering and the resulting geometry changes may have very strong consequences (Sharp and Segel 1983). Hales (1964) showed how to analyze cornering equilibrium accurately by virtual work, but this work has not been fully developed and the method has been surpassed by the advent of multibody mechanics computer packages (Schiehlen 1990). Notwithstanding that the problem can be solved straightforwardly with modern numerical methods, to the

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author’s knowledge there is still no definitive material in the literature on the suspension–attitude interaction problem. An illustration of how the suspension geometry and the vehicle body attitude interact in cornering is given in Fig. 8. The figure was generated via a multibody model and a simple animation of the behavior in cornering with slowly increasing

Figure 8 Illustration of suspension geometry and vehicle body attitude interactions in cornering at increasing lateral accelerations.

lateral acceleration. The lateral tire forces are shared between the inside and outside tires in proportion to the loads carried by them.

6.

LATERAL DYNAMICS

The simplest useful model of the lateral dynamics of a car is a yaw– sideslip model (Whitcomb and Milliken 1957, Wong 1993, Dixon 1996). The car is represented as a single rigid body moving on a flat plane and having a rigid steering linkage. The input is a steer angle, implying fixed control. Inertial effects from steering the front wheels are ignored but the tire forces due to steering are described as linearly dependent on the lateral slip. Usually, the braking and drive systems distribute torques equally to left and right sides, and the steering and turning geometries are such that there are hardly any differences between right and left sides except for the wheel loads. If the influences of load transfer on the tire forces of a pair of wheels belonging to one axle are ignored, which is approximately the case for mild maneuvering, it is as if the two wheels of an axle were joined together at the vehicle center plane. Capitalizing on the absence of car-position-based forces in the problem, one can describe the motions conventionally by longitudinal, lateral, and yaw velocities, as seen from the car (Fig. 9). These are sufficient to describe tire slip angles and car accelerations and to allow the derivation of the equations of motion. In the absence of the kind of differential longitudinal tire forces used in traction control and active stability control systems (Abe 1998), the equation for the longitudinal translation

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Figure 9 Plan view of longitudinal, lateral, yaw car model.

is only weakly coupled to the lateral and yaw equations, and the latter two describe the handling problem at this level. In linear form, they are (Dixon 1996)

Solving the equations of motion for steady turning reveals that the curvature response to steer angle ratio is given by

the character of which depends on whether K is positive or negative (Fig. 10). The sign of K depends on the relationship between and representing the respective moments of the axle forces about the car mass center for unit slip angle. The “understeering” car, with K positive, requires more steer angle for a given turn as the speed increases. In the “oversteering” case—when the front axle is “too strong” for the rear axle —the curvature response goes to infinity at a certain speed, called the critical speed (Fig. 10). The understeer level, for K > 0, can be characterized by the characteristic speed which is that speed for which the curvature response is half its low-speed limit value. This value is 1/l, being determined by simple geometry. Above the critical speed for the oversteering car, the curvature response is negative, corresponding to the “opposite lock” case most commonly

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Figure 10 Curvature responses of understeering, neutralsteering and oversteering cars, from linear theory, in steady state.

observed in competition motoring on low-friction surfaces. It should be clear, however, that too close a parallel between real behavior involving close approach to friction limits and the results of the linear theory should not be drawn. The characteristic equation of the yaw–sideslip car is of the second order form and the analogy with a mass–spring–damper system indicates the basic car to have a normalized directional stiffness

which is always positive for the understeering car but which becomes zero for the oversteering car at the critical speed. Zero directional stiffness corresponds to neutral stability. Above the critical speed, the oversteering car is divergently unstable. The infinite steady-state response at the critical speed and the neutral stability at the same condition are intimately related. They both depend on the directional stiffness becoming zero. The stiffness S appears in the denominator of the steady-state response relations and is also the zero-order term in the characteristic equation, determining the static stability of the system (Sharp 2000). The damping factor of the simple car is always positive. At low speeds it is near unity and it remains close to unity for neutral steering for all speeds. The understeerer, for high speeds, has a reduced damping fac-

tor and its yaw response may show a significant resonance at the yaw– sideslip natural frequency. The bandwidth of the car is typically less

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than that of the driver (3 Hz say). It is extended by high axle-cornering stiffnesses, small mass, and small yaw inertia. To get a proper understanding of the basic fixed control car, it is

necessary to extend the ideas from the linear yaw–sideslip model first into the nonlinear, tire force saturating, region. It is fundamental that the tire forces will saturate and that the process towards saturation is progressive. Correspondingly, the car behavior changes progressively. Altering only the tire force description from linear to nonlinear in the simple model, one can linearize for small perturbations from any steadyturn equilibrium (trim) condition. The small perturbation equations are the same as the original linear ones, except that the tire cornering stiffnesses are replaced by local stiffnesses, representing the tangents to the

tire side-force–slip-angle curves at the trim state operating point (Hales

Figure 11 Normalized axle force to sideslip relationships, illustrating a nonlinear vehicle understeering or oversteering depending on the trim, from which small perturbations are considered. At a particular trim, the relative slopes determine the understeer.

1970, Sharp 1973). The trim states that the car adopts as the lateral acceleration changes from zero to the limit possible can be represented as in Fig. 11, showing each normalized axle side force as a function of the axle sideslip angle. This is effectively the cornering compliance concept

of Bundorf and Leffert (1976). If the rear axle limits, in the sense of Fig. 11, at lower slip than the front, the steady-turn performance limit of the car is set by the rear

axle. The front axle will have spare force capability at the limit, when the car will be statically unstable. Conversely, if the front axle limits first, the limit behavior is stable but the car is not steerable. To use the

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full force capability in a trim state, the car needs to be neutral at the limit. Figure 11 need not be interpreted in the context of a yaw-sideslip car with nonlinear tire forces. Rather it can be thought of in the more general context of a real vehicle with many extra degrees of freedom and design details. A connection can be made with Fig. 8. As the trim changes, the body attitude alters, the suspension geometry changes, compliant bushes deflect, etc. Even in this quite general context, it remains the case that the steady-turn behavior is related to the stability as before, both being dependent on the static stability coefficient in the characteristic equation—the normalized directional stiffness S (Sharp 2000). Whereas the simple model contains few contributions to S, a complex model will have many, from suspension and steering elastokinematics, body attitude changes, tire properties, load transfers, etc. The relationship between lateral acceleration and steering wheel angle for a car at a fixed speed can be illustrated in a handling diagram (Fig. 12). Again, it is the incremental changes in steer input and lateral

Figure 12 Steady state handling diagram for a particular speed, showing the lateral acceleration as a function of steer angle for three types of car. One is always neutralsteering, while the others change with lateral acceleration level.

acceleration from one trim to a neighboring one that relate to understeering, oversteering, and static stability via a local linearization of the cornering problem. In terms of the vehicle chassis design, static stability can be guaranteed by ensuring that the front axle will limit first in any steady turn. Since the operating conditions vary widely, over vehicle loading, road friction level, tire type and condition, etc., this design strategy may be very conservative and will imply under-utilization of the rear tire forces

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and a relatively narrow performance envelope for the car. On the other hand, designing for a more neutral limit behavior will introduce the risk of the vehicle becoming statically unstable at its limits and relying on the driver to maintain stability of the man–machine system. The major limitations of drivers in this context are twofold. Firstly, they are slow in sensing, processing, decision making, and actuation, with reaction times to expected events of about 0.3–0.5 s. Secondly, they need to learn what to do for advantage by repeated trials and observations. This is the province of electronically controlled vehicle dynamics intervention systems. Sensors can be used to detect events and conditions and their outputs can be used to actuate engine, transmission, and braking controls, such that the above designer’s dilemma can be avoided. Differential longitudinal tire forces, as between left and right sides, can be employed to introduce yawing moments designed to prevent the car from spinning (Abe 1998). In the design and realization of such intervention systems, continuity is an important property, in view of the man–machine nature of the total system. In principle, accurate detailed predictions of the motions of a car in response to a prescribed set of control inputs can be made. Handbuilt models have largely been superseded by computer-built models (Sharp 1994, 1998), some of which are available freely or commercially

(http://www.trucksim.com . ). The difficult areas in relation to complex vehicle modeling are parametric data and interpretation of results in terms of quality. Tire and aerodynamic force systems can be effectively measured only with special facilities (http: //www.MTS. com) .. Special installations are also advantageous for the determination of suspension and steering elasto-kinematics(http://www.abd.uk.com and http: //www. MIRA. co. uk) and inertial properties (Heydinger et al. 1995, Sharp 1997). Some complex and elaborate studies have simple objectives. Understanding the circumstances under which rollover or spinning occurs, establishing the “maximum challenge” set of control inputs (Ma and Peng 1998), or solving a minimum-time maneuver problem (Casanova et al. 2000a, 2000b) are not so difficult in terms of interpretation. There are many other circumstances, however, when interpretation is both difficult and crucial. In the author’s view, the more complex quality judgments are best attempted through reference to the basis set out in 2.

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CONCLUSION

In the above discussion, a perspective on contemporary road vehicle handling dynamics has been offered. Space restrictions have mitigated against a more complete coverage.

References Abe, M. 1998. Vehicle dynamics and control for improving handling and active safety: from 4WS to DYC. Multibody Dynamics: New Techniques and Applications (Institution of Mechanical Engineers, 1998–13). London: Professional Engineering Publishers, 229–248.

Bundorf, R. T., and R. L. Leffert. 1976. The cornering compliance concept for the description of vehicle directional control properties. Society of Automotive Engineers 760713. Casanova, D., R. S. Sharp, and P. Symonds. 2000a. Minimum time manoeuvring: The significance of yaw inertia. Vehicle System Dynamics 34(2), 77–115.

Casanova, D., R. S. Sharp, and P. Symonds. 2000b. On minimum time optimisation of formula one cars: The influence of vehicle mass. Proceedings of Symposium on

Advanced Vehicle Control (AVEC’2000) (Ann Arbor, Mich.), 585–592. Dixon, John C. 1996. Tires, Suspension and Handling, 2nd ed. Warrendale, Penn.: Society of Automotive Engineers.

Hales, F. D. 1964–65. A theoretical analysis of the lateral properties of suspension systems. Proceedings of the Institution of Mechanical Engineers 2A 2 and 3, 73–91.

Hales, F. D. 1969–70. Vehicle handling qualities. Proceedings of the Institution of Mechanical Engineers 2A 12, 233–244. Heydinger, G. J., N. J. Durisek, D. A. Coovert, D. A. Guenther, and S. J. Novak. 1995. The design of a vehicle inertia measurement facility. Society of Automotive Engineers 950309. Ma, W., and H. Peng. 1998. Worst case evaluation methodology—Examples on truck rollover/jackknifing and active yaw control systems. Proceedings of Symposium on Advanced Vehicle Control (AVEC’98), Society of Automotive Engineers of Japan, 299–305. MacAdam, C. C., M. W. Sayers, J. D. Pointer, and M. Gleason. 1990. Crosswind sensitivity of passenger cars and the influence of chassis and aerodynamic properties on driver preferences. Vehicle System Dynamics 19(4), 201–236. Mousseau, C. W., M. W. Sayers, and D. J. Fagan. 1992. Symbolic quasi-static and dynamic analyses of automobile models. In The Dynamics of Vehicles on Roads and on Tracks (G. Sauvage, ed.). Lisse: Swets and Zeitlinger, 446–459. Orlandea, N., M. A. Chace, and D. A. Calahan. 1977. A sparsity oriented approach to the dynamic analysis and design of mechanical systems—Parts I and II. Journal of Engineering for Industry 99, 773–784.

Pacejka, H. B., and R. S. Sharp. 1991. Shear force development by pneumatic tyres in steady state conditions: A review of modelling aspects Vehicle System Dynamics 20(3–4), 121–176. Pacejka, H. B., and I. J. M. Besselink. 1996. Magic Formula tyre model with transient properties In Tyre Models for Vehicle Dynamic Analysis (F. Boehm and H.-P.

Willumeit, eds.) (Supplement to Vehicle System Dynamics 27). Lisse: Swets and Zeitlinger, 234–249.

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Schiehlen, W. (ed.). 1990. Multibody Systems Handbook. Berlin: Springer-Verlag.

Sharp, R. S. 1973. Relationship between the steady-handling characteristics of automobiles and their stability. Journal of Mechanical Engineering Science 15(5), 326– 328. Sharp, R. S., and L. Segel. 1983. Digital simulation of rollover of a military vehicle. Proceedings of the Institution of Mechanical Engineers/Motor Industry Research

Association Conference on Road Vehicle Handling. London: Mechanical Engineering Publications, 23–30. Sharp, R. S. 1994. The application of multibody computer codes to road vehicle dynamics modelling problems. Proceedings of the Institution of Mechanical Engineers D1 208, Journal of Automobile Engineering, 55–61. Sharp, R. S. 1997. The measurement of mass and inertial properties of vehicles and components. In Automotive Vehicle Technologies 1997–7. London: Mechanical Engineering Publications, 209–217.

Sharp, R. S. 1998. Multibody dynamics applications in vehicle engineering. In Multibody Dynamics: New Techniques and Applications, Institution of Mechanical Engineers 1998–13. London: Professional Engineering Publishers, 215–228. Sharp, R. S., D. Casanova, and P. Symonds, 2000. A mathematical model for car steering, with design, tuning and performance results. Vehicle System Dynamics

33(5), 289–326. Sharp, R. S. 2000. Some contemporary problems in road vehicle dynamics. Proceedings of the Institution of Mechanical Engineering C1 214, Journal of Mechanical Engineering Science, 137–148. Valtetsiotis, V. 1999. A discrete time optimal preview control driver model. M. Sc.

thesis, Cranfield University School of Mechanical Engineering. Whitcomb, D. W., and W. F. Milliken. 1957. Design implications of a general theory

of automobile stability and control. Proceedings of the Institution of Mechanical Engineers (a.d.), 367–391. Wong, J. Y. 1993. Theory of Ground Vehicles, 2nd ed. New York: John Wiley and

Sons.

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More than 1400 scientists, engineers, and students from 51 countries participated in ICTAM 2000.

The UIUC Office of Continuing Education, directed by Dr.

James C. Onderdonk, coordinated meeting registration and tours, and assisted with hotel arrangements.

A MECHATRONICS APPROACH TO ADVANCED VEHICLE CONTROL DESIGN Masato Abe Department of System Design Engineering Kanagawa Institute of Technology, Kanagawa, Japan

J. Karl Hedrick Department of Mechanical Engineering University of California, Berkeley, Calif., USA Abstract

1.

This paper presents a “mechatronic” approach to design vehicle chassis controls as well as active suspensions using preview information. The term mechatronics is used to denote a combined mechanical and electronic design to achieve improved vehicle safety and performance. In the chassis control area, two alternative approaches are analyzed to provide compensation for vehicle lateral instability due to nonlinear tire characteristics. First, a direct yaw moment control (DYC) approach that is based upon sensing side-slip and yaw rate and directly controlling the longitudinal tire forces is analyzed. Next, a four-wheel steering (4WS) approach to control the tire lateral forces is analyzed and compared with the DYC method. Simulations are provided to support the conclusion that the DYC approach provides superior performance. In the second half of the paper an active suspension system is analyzed that has access to road preview information. A controller design method called model predictive control (MPC) is developed to provide a real-time constrained optimization procedure, which optimizes the vertical ride quality subject to inequality constraints on the suspension deflection.

INTRODUCTION

The general purpose of chassis control is to control a vehicle’s lateral, vertical, and longitudinal motions in order to improve its handling performance, ride comfort, and traction/braking performance. The first part of this paper focuses on vehicle chassis control, primarily to improve vehicle handling performance and safety. One way to improve handling performance is through chassis control using four-wheel steering (4WS). Four-wheel steering control depends upon the tire lateral forces, which are proportional to the steer angle 147 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 147–164. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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in a range where the lateral acceleration is small. In this range, the control law can rather easily be introduced by using a two-degree-offreedom linear vehicle model. However, at higher lateral acceleration, the lateral force is not necessarily proportional to the steer angle due to its saturation as a function of side-slip angle. The lateral force also becomes strongly dependent upon the tire vertical load and longitudinal force as well, so that the control law becomes rather sensitive to the vehicle motion and its environmental conditions. A direct yaw moment control (DYC), which controls vehicle motion by a yaw moment that is actively generated by the intentional distribution of the tire longitudinal forces, is becoming one of the most promising means of chassis control. One of the major advantages of this control method is that the tire longitudinal force has no feedback from the vehicle lateral motion as long as it is within the limit of the tire capacity due to the vertical load. Therefore, a precise yaw moment (required to control the vehicle lateral motion) can be developed, and the vehicle motion with the control becomes more robust to fluctuations in the vehicle running condition and its environment. It is obvious that any chassis control aimed at improving vehicle handling performance must rely on the tire lateral force, and that the tire longitudinal force can also be used for the chassis control. These two forces each rely upon the vertical load and are interdependent. This relationship makes the vehicle dynamic characteristics very complex, and a comprehensive study is required to develop more effective chassis control strategies. The second part of this paper discusses the use of preview information to improve the ride quality of off-road vehicles. It has long been known that an active suspension system that had access to upcoming road disturbances can significantly improve the vehicle’s ride quality. This paper presents an interesting control methodology called model predictive control (MPC), which uses real-time, on-line constrained optimization to optimize ride quality subject to such constraints as available suspension clearances.

A mechatronic approach to advanced vehicle control design Table 1

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Nomenclature

INDIVIDUAL VEHICLE CHASSIS CONTROL Chassis control to compensate for loss of stability

The basic nature of the lateral dynamics is described by the following equations of motion for a two-degree-of-freedom vehicle plane model:

where m is the vehicle mass, etc., as noted in Table 1. Equation (1) can be rewritten for small perturbation around an equilibrium point as follows:

According to the stability theorem of nonlinear systems, the roots of the characteristic equation with respect to the singular point determine stability. The characteristic equation of the system described by Eqn. (2) is with

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where represent the slopes of the tire lateral forces versus side-slip angle at the equilibrium point considered. These coefficients represent the equivalent cornering power at the front and rear, respectively. For the motion to be stable, A and B must be positive. For A < 0, the instability condition

arises. It is interesting to find from this equation that the nonlinear vehicle motion tends to unstable motion predominantly by a decrease of the equivalent cornering power of the rear tire with an increase of the side-slip angle—the nonlinear tire characteristics. The equation of motion (2) yields the following description for the ratio of steady-state side-slip gain to yaw rate gain:

This result shows that if the equivalent cornering stiffness of the rear tire is reduced due to low friction coefficient between tire and road surface, large slip angle, load transfer from rear to front during braking, etc., then the negative value of the side-slip angle to generate a definite yaw rate increases. This observation suggests that adopting a control to achieve a definite yaw rate yields a larger side-slip response when the vehicle is subjected to the deterioration of rear tire characteristics. On the other hand, if the side-slip motion is controlled, the yaw rate response gain is automatically restricted with the deterioration of rear tire characteristics. This is a reason why, from the stability point of view, it is better to adopt the side-slip control for the chassis control rather than to adopt the yaw rate control directly. Equation (6) also shows that the side-slip angle needed to maintain a definite yaw rate is determined by the rear tire and not by the front tire. Therefore, in order to control the vehicle side-slip angle by active wheel steering, a rear wheel steer has to be used, as in 4WS. Thus, deterioration of control ability due to the load transfer during braking is inevitable, and the vehicle motion is likely to become unstable even though it is due to the side-slip control, not the yaw rate control. The above discussion may lead to the view that side-slip control by DYC is more effective than other controls in preventing the vehicle from falling into unstable motion due to nonlinear tire characteristics. This

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view will be corroborated through experimental tests at the proving ground, as shown below.

3.

CONTROL STRATEGY TO STABILIZE

VEHICLE MOTION In order to compensate for loss of stability due to nonlinear tire characteristics, we propose a model-following control to follow the response of a linear two-degree-of-freedom vehicle plane plant model with constant speed. Then there will be no control requirement so long as the vehicle motion remains within a linear characteristic region. The model responses of side-slip angle and yaw rate are described as follows:

where P, Q, are the response parameters of the side-slip angle determined by the vehicle parameters as follows:

The response error of the vehicle from the model response caused primarily by the tire nonlinear characteristics and acceleration or braking would require the control to compensate the error and stabilize the vehicle motion.

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Side-slip control by DYC

A sliding control is adopted for the model-following control. The model response of side-slip angle, Eqn. (7), is rewritten in the form of a sliding surface for the sliding control as follows:

The sliding condition is and equations for the vehicle motion are

From these equations, the control yaw moment for the DYC is derived as:

where (i = 1,..., 7) are expressed in terms of P, Q, and k. In order to calculate the lateral forces, their slopes, and the equivalent

cornering stiffness in the above equation, we use the tire model described by Eqn. (20) below.

3.2.

Yaw rate control by DYC

The model response of yaw rate, Eqn. (8), is rewritten in the form of a sliding surface for the sliding control as follows:

Using Eqn. (13) instead of Eqn. (9), we can derive the control yaw moment for the yaw rate model-following control in the same manner as the control yaw moment for the side-slip control:

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Side-slip control by 4WS

Putting equal to zero in Eqn. (11), one can solve for Eqns. (9), (10), and (11):

by using

where (i = 1,..., 6) are expressed in terms of P, Q, and k. This is the required lateral force of the rear axles to be produced by rear wheel steer in order to follow the model response of the side-slip angle. Once the lateral force is given, the rear tire side-slip angle can be obtained by inverse use of the tire model described by Eqn. (3) and then the rear wheel steer angle is found to be

If the required lateral force is greater than the saturated value in the tire model, then the rear wheel steer angle is adjusted so that the tire side-slip angle remains at the saturated point by satisfying the equation

3.4.

Estimation of side-slip angle

It is essential to know the side-slip angle to be used in the control. The authors have already proposed an estimate of the vehicle side-slip angle by a model observer for use in the control. The side-slip motion is described by the equation

and side-slip angle is defined as

These two equations are valid no matter how the vehicle motion changes with acceleration or braking. In Eqn. (18), and are given by the following simple tire model:

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depending on whether or respectively. In Eqn. (20), the subscript T corresponds to f and r, representing front and rear, respectively; and the cornering stiffness is regarded as a function of the vertical load By on-line real-time integration of Eqn. (18), the side-slip angle defined by Eqn. (19) can be obtained.

3.5.

Experimental validation

Validation of side-slip angle estimation. Before studying the effects of the control, the estimated side-slip angle was compared with the real side-slip angle measured by an optical side-slip sensor. The configuration for the side-slip angle estimation is shown in Fig. 1 and the results are shown in Fig. 2. It is found that the side-slip angle can be estimated precisely even under such severe running conditions as evasive lane change maneuvering with quick and large steering angle,

turning with near-limit lateral acceleration with large step steer input, and brake-in-a-turn with steering correction. The experimental results of the side-slip estimation on a low-friction road prove that the estimation method introduced is effective during running on a low-friction road as well. Some estimation errors are recognized during maneuvering with near-limit lateral acceleration when a high friction coefficient is used in the on-board-tire model; nevertheless, no catastrophic failure can be seen in the estimation.

Vehicle and control configurations. In order to compare the effects of DYC and 4WS, as well as to compare the effects of the sideslip control and the yaw rate control, a small-size passenger car on the market, equipped with the automatic torque transfer system and the rear wheel steering system for DYC and 4WS, respectively, was used as the experimental test vehicle. Figure 3 shows the vehicle configuration with side-slip and yaw rate controls by DYC. The vehicle configuration with side-slip control by 4WS is shown in Fig. 4. Effects of the controls. The experimental results are shown in Fig. 5, in which the effects of the controls during a double lane change are

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Estimation of side-slip angle.

compared. The vehicle response without control seems almost unstable. On the other hand, the side-slip controls, as well as the yaw rate control, stabilize the vehicle motion and it is found that the effect of the sideslip control by DYC is almost the same as that of the control by 4WS.

However, the vehicle response with the yaw rate control shows larger side-slip angle to follow the reference yaw rate. The experimental results of the single-lane-change test with braking are shown in Fig. 6. The vehicle motion without control seems com-

pletely unstable with large side-slip angle. It is found that the side-slip control by DYC can successfully stabilize the driver–vehicle system; however, the vehicle motions even with the side-slip control by 4WS or the yaw rate control by DYC are almost unstable, showing large side-slip angle. For the side-slip control by 4WS, the instability is likely due to the excessive load transfer from rear to front during braking, which dete-

riorates the rear tire characteristics; the control cannot keep the vehicle stable. In order to follow the model yaw rate strictly, the vehicle with the yaw rate control shows an excessively large side-slip angle, which causes the unstable motion.

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Figure 2 Experimental results of side-slip angle estimation.

4.

ACTIVE SUSPENSIONS WITH PREVIEW INFORMATION

The benefits of active suspensions have been known for several decades. An active suspension has the ability to add power as well as to dissipate power (as is done by passive suspensions). Active suspensions have not achieved commercial success due to the fact that the cost/benefit ratio is not yet low enough to warrant their wide-scale introduction into commercial passenger vehicles. The use of preview information with active suspensions has also been studied and its benefits are clearly understood. Due to the natural frequencies associated with most ground vehicles, it can be shown that road preview information less than or equal to 0.3 seconds provides significant ride quality improvement over fully active suspensions with no preview information. A recent study at UC–Berkeley on the use of active suspensions combined with a preview sensor for application on an off-road military vehicle is described in this paper. Figure 7 shows a schematic of an HMMVW

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Figure 3 Vehicle side-slip and yaw rate controls by DYC.

with active suspensions added at each wheel. Drivers of off-road vehicles

generally feel capable of increasing their speed until all the suspension clearances are used; once metal-to-metal contact has been reached due to the combination of road roughness and vehicle speed, the driver needs to reduce speed in order to maintain control of the vehicle. In this paper a control methodology called model predictive control (MPC) is applied to design a broad-bandwidth active suspension system that has access to upcoming road disturbances from a sensor located at the front of the vehicle. Model predictive control solves a constrained optimization problem in “real time” that optimizes a quadratic performance index subject to constraints on the suspension deflections at the four corners of the vehicle.

4.1.

Vehicle model

Figure 8 shows a quarter car model with an active force element in parallel with the passive suspension. Figure 9 shows a full car model

with four active elements at each wheel. If one considers the force of each

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Figure 4 Vehicle with side-slip control by 4WS.

actuator to be the controlled variable, the linear equations of motion for the quarter car model can be expressed as

If we let

then the discretized versions of Eqns. (21) become

where u(k) represents the control force at time k and v(k) represents the road profile disturbance at time k. The MPC method seeks to minimize

A mechatronic approach to advanced vehicle control design

Figure 5 Vehicle responses during double lane change.

a quadratic performance index

159

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Figure 6 Vehicle responses during single lane change with braking.

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Figure 7 HMMVW–Active suspension schematic.

Figure 8 Quarter car model.

where the index P represents the total preview horizon and N represents the number of time steps in the future at which we will choose the control

variables u(k). The suspension travel constraints can be expressed as

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Figure 9 Full car model.

The constraint over a time horizon

can express as

The minimization problem can be expressed in vector form as

with constraints where

The MPC method uses a predictor to forecast the output over a receding time horizon. It determines the control variables u(k) by minimizing a

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quadratic cost function of the future output and control inputs subject to constraints on the suspension deflection magnitudes. The algorithm implements only the first control variable calculated and then starts the optimization over at each step. The output predictor equation is given

by The predictor for the constrained suspension travel is given by

where

are the future inputs over the constrained horizon:

Using the output predictor equation, one can express the cost function

as

Using the output predictor equation, one can express the suspension constraints as

Thus the final quadratic programming problem can be expressed as

subject to

The MPC algorithm described above was implemented at the Berkeley Active Suspension laboratory. It was determined from the experimental results that the hydraulic actuator dynamics needed to be considered to achieve accurate force tracking results. A simple nonlinear dynamic model was used to design an inner loop. The following equation for the hydraulic actuator was used:

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where u is the spool valve displacement and is the by-pass spool valve displacement, which was kept fixed. A simple first-order lag model was used to model the dynamics between the spool valve displacement and the input voltage v: The sliding control method was used to design a fast inner loop to track the computed MPC desired force command. In the experimental work the MPC force command was computed every 10ms and the inner force tracking loop was updated every 1 ms. The experimental laboratory results show that significant acceleration reductions can be achieved by active suspensions using road profile preview information. Experiments have been conducted in both a laboratory situation as well as on an HMMWV as shown in Fig. 7. The HMMWV field tests are still ongoing but preliminary results are consistent with the laboratory results.

5.

CONCLUSION

It is conclusively found that, compared with the other controls, the side-slip control by DYC has a prominent effect on compensation of vehicle lateral dynamics during severe maneuvering for a loss of stability due to nonlinear tire characteristics. Active suspensions can be designed to provide significant ride quality improvements in a variety of road situations by the use of preview information and such advanced control methodologies as MPC, which can take into consideration such hard operating constraints as available suspension clearances.

References Abe, M., Y. Kano, Y. Shibahata, and Y. Furukawa. 1999. Improvement of vehicle handling safety with vehicle side-slip control by direct yaw moment. Proceedings

of 16th IAVSD Symposium, 665–679. Abe, M. 1999. Vehicle dynamics and control for improving handling and active safety: from four-wheel steering to direct yaw moment control. Proceedings of Institution of Mechanical Engineers K 213, 87–101. Abe, M., Y. Kano, K. Suzuki, Y. Shibahata, and Y. Furukawa. 2000. An experimental validation of side-slip control to compensate vehicle lateral dynamics for loss of stability due to nonlinear tire characteristics. Proceedings of 5th International Symposium for Advanced Vehicle Control, 179–186. Sharp, R. S., and S. A. Hassan. 1986. The relative performance capabilities of passive, active, and semi-active car suspension systems. Proceedings of the Institution of Mechanical Engineers D 203(3), 219–228. Alleyne, A., and J. K. Hedrick. 1995. Nonlinear adaptive control of active suspensions. IEEE Transactions on Control Systems Technology 3(1), 94–102.

MIXING: KINETICS AND GEOMETRY Emmanuel Villermaux Institut de Recherche sur les Phénomènes Hors Équilibre Université de Provence, Aix–Marseille 1, France [email protected]

Abstract

1.

We review the different facets of the phenomenon of mixing, including its geometrical, temporal, and structural aspects. Then we suggest that a complex mixture can be viewed as the superposition of independent sources. Kinetics and geometry are shown to be closely linked to each other when following the transient mixing of an isolated scalar source in a turbulent flow. The composition law between multiple interacting sources is established experimentally, therefore allowing one to reconstruct any scalar field from well defined elementary contributions.

WHY MIXING?

Mixing1 is a subject that suffers from the Bourgeois Gentilhomme complex. Like Monsieur Jourdain in Molière’s play (1670), scientists, engineers, and indeed all of us often “do mixing without even knowing it” (just think about yourself trying to prepare mayonnaise … ) . As an operation, it consists simply in putting together two or more initially segregated constituents and stirring, in order to attain uniformity, or a new product, or the complete disappearance of one of the constituents, etc.; mixing thus is indeed at the crossroads of many different areas of science. One often needs to mix elements in order to make a new product, a homogeneous blend, or to make possible a chemical reaction or an efficient combustion. One also needs to understand how nature mixes or has mixed to gain information on, e.g., the size of a pollutant spot in a valley, the rate of destruction of ozone in the atmosphere, or the dynamics of the earth’s mantle. More than a fascinating subject in itself, mixing is thus ubiquitous and a key process in many complex man-made or natural operations. 1 The text in this section is inspired from the foreword of Mixing: Chaos and Turbulence, Chaté, H., E. Villermaux, and J. M. Chomaz (eds.), Kluwer Academic/Plenum Publishers, New York, 1999.

165 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 165–180. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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It is no doubt because of its universality that mixing, as noted with regret by J. Ottino (1989), “… does not enjoy the reputation of being a very scientific subject …” as is frequently the fate of interdisciplinary topics. The dendritic nature of the subject matter is revealed by the different angles from which the various scientific communities attack the problem, according to their needs. These approaches can be roughly grouped into three naturally overlapping categories, each of them of interest to different schools of thought: Geometry Kinetics Structures Beyond the tools and attitudes developed by different scientific groups regarding the problem of mixing, one might appreciate that mixing, as suggested by common sense, is the operation by which a system evolves from one state of simplicity (the initial segregation) to another state of simplicity (the complete uniformity). Between these two extremes, complex patterns emerge and die. Questions then naturally arise: how can the geometry of complex patterns be characterized; what is the clock, the time-scale of the process; and what are the structures involved in the flow?

1.1.

Geometry

Very early on, emphasis was placed on the geometry of the mixing zone in the combustion context. Indeed the involved, multiscale geometry of the interface that separates two streams being mixed is not only a spectacular facet of the process, but also sometimes at the core of the physical problem. Exothermic reactions in gases, or reactions with a fast kinetics in liquids confine the reaction zone to a thin region of space and the flame appears as a sharp interface for most scales in the flow. The total extent of the flame area dictates the propagation speed of the front in premixed reactants, as noted by Damköhler (1940), and the net combustion rate in diffusion flames (see the celebrated experiments of Hawthorne et al. (1949)). Knowing how and why the front is distorted by the underlying motions is thus crucial for predicting the flame extent. The multiscale structure of the contour of a scalar blob immersed in a disordered flow was recognized by Welander (1955), who suggested how a connection could be made between the internal structure of the underlying motions and the complexity of the blob shape. Welander even

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made reference to the Koch curve, long before fractals were popularized in this and other contexts by Mandelbrot (1975). The program, still very active today, for investigating the geometry of the scalar support, or the iso-concentration surfaces in flows was certainly already contained in Reynolds (1894). Reynolds was probably even more ambitious, suggesting that watching the dynamics of “coloured bands” in a flow was a route towards understanding the driving motions.

1.2.

Kinetics

Mixing is, in the strict sense, a transient process from initial segregation to ultimate homogeneity. This calls for the understanding of the kinetics and timescales of the process. Studies on dispersion address the problem of the growth rate of the radius of a tracer blob in a prescribed displacement field. In addition to pure molecular diffusion in the medium at rest, fluid motion usually enhances dispersion (Taylor 1921, 1953) and alters not only the diffusion law, by a renormalization of the diffusion coefficient, but also the structure of the law itself. Due to persistent ballistic movements and ever larger jumps in turbulent flows, dispersion laws exhibit, in the absence of traps or slow recirculating motions, a faster-than-linear growth of the mean-squared radius of the blob (Prandtl 1925, Richardson 1926). The presence of bypasses or dead-ends in complex geometries alter, in continuous flow systems (a river, a valley through which wind blows, an open chemical reactor), the residence time distribution of a tracer introduced at the inlet of the system. The novel features of the distribution are spikes at short times if a short circuit is present, and/or long tails caused by traps and slow motions in confined cavities. This was first described by Danckwerts (1953). The kinetics of mixing does not only refer to dispersion. Dispersion may result solely from a spatial reorganization of the quantity to be mixed, with no interpenetration with the substrate at the molecular level. Mixing, as opposed to stirring, actually means homogenization at the smallest, i.e. diffusive, scales; and the appropriate quantity to define and measure the mixing time, namely the “intensity of segregation,” was introduced by Danckwerts (1952). The mixing time (for instance in a tank stirred with an impeller, familiar in the chemical industry) is found to be solely determined by the large-scale features of the flow (integral scale, root-mean-square velocity; see, e.g., Nagata (1975)), regardless of the intimate structure of turbulence, provided the Reynolds number is

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large enough (typically larger than in a wide variety of flows. This absence of a role for turbulence and its structure in the characteristic time of the inhomogeneities decay is an important point, which has been somewhat overlooked. The intimate structure of the local rearrangements in the flow does, however, play a role when chemical reactions with a nonlinear kinetics, or consecutive–concurrent chemical reactions occur within the mixture (Danckwerts 1953, Epstein 1990).

1.3.

Structures

Disordered flows develop a broad hierarchy of scales of motion that convect and distort the scalar field. Notwithstanding the fact that mixing is, like the decay of the energy-containing eddies, a transient phenomenon, the problem of the interaction of the scalar field with the underlying turbulent flow has focused on hypothetical “stationary conditions.” This approach parallels the Kolmogorov quasi-equilibrium picture of turbulence (Corrsin 1951, Oboukhov 1949, Batchelor 1959, Batchelor et al. 1959) and it is in this assumed limit that the timescales of the stirring motions that distort the scalar field are all shorter than the global mixing time (i.e. the variance of the scalar fluctuations is stationary as seen on the timescale of these motions), allowing the possibility of resorting to cascade arguments, and spectral analysis. It is customary, in turbulence, to oppose the statistical approaches, the cascade representation being one of those, to the approaches focusing on “structures” more or less coherent or permanent in the flow. This opposition is somewhat artificial. Modern statistical models of turbulence rely in fact on a description of the flow as a hierarchical structure in the real space, possibly fractal, the structure of the hierarchy inducing the statistics. The cascade picture, notably, is thus in a sense a “structural approach.” In this cascade picture, the mixing time is the time needed for the scalar to travel, by a process of successive reductions of scale imposed by the hierarchy of pre-existing scales of motion (eddies) in the flow, from its initial size to the dissipation scale, prescribed by the Reynolds and Schmidt numbers (Corrsin 1964). This description presents many shortcomings, the first one being its inconsistency with the observed fact that the mixing time depends linearly on the initial size of the blob to be mixed, instead of on its size raised to the power 2/3. If it is clear that mixing macroscopic objects implies a reduction of their transverse size and a multiplication of ever smaller scales for diffusion to act efficiently, there is no direct proof that a scalar blob should follow the

“Kolmogorov cascade” step by step to be ultimately erased by molecular

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Figure 1 Start-up vortex. A fast chemical reaction between the ring and the ambient fluid traces the spiraling interface between the two media.

diffusion. Here is a sign that, in turbulence, even many basic properties escape our precise understanding; the future should therefore be rich in forthcoming works (see the program reviewed by Shraiman and Siggia 2000). The will to reduce a complicated problem such as turbulence to a set of elementary objects containing, presumably, all of the desired information, is in keeping with a long tradition in physics and fluid mechanics. Coherent structures, filaments, worms, and sheets are frequentlyinvoked paradigms. In the context of mixing, the discovery of largescale vortical structures sustained by shear flows has prompted a surge of activity (Brown and Roshko 1974). Originating essentially from a Kelvin–Helmhotz type of instability, these intense, long-lived structures are serious candidates to compete with the cascade representation since they directly couple injection scales with dissipation scales, eliminating intermediary steps. Their existence alone is not, however, sufficient to ensure a rapid mixing. When formed by a given velocity difference across a shear layer, their size (equivalent to the Reynolds number) has to grow beyond a critical value to allow the onset of the fine-scales activity, thereby hastening the uniformization within the layer. This is the so-called “mixing transition” (Breidenthal 1981). Although its ubiquitous character is striking and goes far beyond the context of mixing layers, as recently reviewed by Dimotakis (2000), little is known in detail about what occurs during and after this transition. It is, nevertheless, a known fact that the deformation tensor in turbulent flows bears, in the mean, two directions of stretching and one direction of compression as first analyzed by Betchov (1956). This property

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Figure 2

Cuts in the longitudinal and transverse directions of the flow of a coaxial

jet showing the rolling-up structures resulting from the shear instability between the streams (from Villermaux and Rehab 2000).

leads to an increase in the length and area of material lines and surfaces (Batchelor and Townsend 1956). The stretched sheet is thus likely to be the elementary brick of mixing, whether turbulent or not (Ranz 1979,

Ottino 1989).

1.4.

Towards a new paradigm

Restricting our field of sight to the particular limit of mixing in homo-

geneous fluids, and keeping the Pandora’s box of mixing in complex, non-Newtonian fluids, multiphase flows, granular flows, etc., carefully sealed, we may wonder what the minimal set of ingredients should be to reach a satisfactory description of the above three facets of the mixing phenomenon. Let us examine mixing patterns obtained in different flow conditions, going from laminar (Fig. 1) to strongly turbulent (Fig. 3), via transitional shear instabilities (Fig. 2).

Figure 1 shows a start-up vortex (a kind of a smoke ring), several turns after its formation. The fluid constitutive of the ring develops a chemical reaction with the ambient medium in which it is rolled up. This is a fast chemical reaction and the reaction diffusion zone, very thin in

this case, traces the spiraling interface between the two media. The ring is a non-turbulent object. It directly couples its own injection scale with the diffusive scale via the spiraling motion and the accretion of sheets of ambient fluid at each turn. However, the ring presents many scales, even a continuous spectrum of scales, due to the continuous increase of

the striation thickness from the core of the spiral to its edge. This is a fractal whose covering dimension is close to 3/2. This is a multiscale, dissipative object; but where is the cascade?

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Figure 3 Blob of dye deposed in a large scale sustained turbulent flow converted into disjointed sheets which dilute in the surrounding medium (from Villermaux and Innocenti 1999).

Figure 2 represents cuts in the longitudinal and transverse directions of the flow of a coaxial jet. The inner jet is slower than the outer annular jet, and the pictures show the structures that result from the instability of the shear between the streams. One recognizes in the longitudinal structures formed from the shear instability the rolling-up vortices similar to those in Fig. 1. But the coherence of these structures gets lost and the resulting mixture finally looks like Fig. 3. Figure 3 illustrates what happened to a small compact blob of scalar deposed in a large-scale sustained turbulent flow. The blob has been progressively converted into disjointed sheets that dilute in the surrounding turbulent medium. Except for loose, diffuse sheets of dye, no obvious structures can be distinguished. The striking observation is nevertheless the broad fluctuations in the transverse size of the sheets, and in the concentration level they carry. Although captured at the same instant of time in the snapshot of Fig. 3, those sheets have not all experienced the same history. Some are still thick and dark, while others are already so thin that they almost fade away in the diluting medium. What is the paradigm, the elementary structure of mixing in disordered flows? The coherent rolling-up vortex, the cascade, the stretched sheet? It seems that none of these caricatures is fully satisfactory and could give way to a new paradigm that would account for the parallel, distributed histories of cumulated stretchings experienced by the material elements in the course of time. This new paradigm could reconcile the three facets of mixing by unifying geometrical, temporal, and structural aspects of the process: its intrinsic transient nature, its evolutive geometry as a set of particular structures, namely sheets with distributed histories.

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Figure 4 A snapshot of the scalar distribution in the region 8 < x/d < 12 downstream of the injection point, with d = 1 cm and Re = = 6000.

2. 2.1.

FROM ELEMENTARY SOURCES TO COMPLEX MIXTURES The single source

Experiments (Villermaux et al. 1998, Villermaux and Innocenti 1999) have been conducted aiming at following an initially smooth and compact blob of dye released in a turbulent flow along its transient evolution, from the initial segregation, towards uniformity. The scalar to be mixed is injected continuously in the far field and on the axis of a turbulent jet via a small tube whose diameter d is smaller than the local integral scale L (typically The injection point behaves neither as a source nor as a sink of momentum, in the mean, and the properties of the flow (stirring scale L, r.m.s. velocity are constant during the uniformization period of the scalar. These experiments involve three types of scalars: temperature in air temperature in water (Sc = 7), and the concentration of disodium fluorescein in water (Sc = 2000), thus allowing an investigation of the quantitative role of the intrinsic diffusive molecular properties of the scalar being mixed on the process.

Mixing time. The blob is progressively converted into a set of stretched sheets, possibly coalescing as they fade away (Fig. 4), and the scalar concentration fluctuations PDF, where denotes the injection concentration, exhibits rapidly a self-preserving shape, whose

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Figure 5 Fluctuations PDFs, normalized by the initial concentration (temperature), recorded for three Schmidt numbers 20 diameters d downstream from the injection point. The PDFs exhibit an exponential decay of the form Insert: The argument of the exponentials at different times ut/d and three Schmidt numbers. Sc = 2000, Re = 6000 and 12000, d/L = 0.05, 0.1, 0.6. Re

= 6000, d/L = 0.05, 0.1, 0.16. 0.7, d/L = 0.08, Re = 45000.

Sc = 0.7, d/L = 0.08, Re = 23000;

tail is an exponential with an argument increasing linearly in time t as

with the mixing time

being found to be

as shown in Fig. 5. The factor f(Sc) is a slowly increasing function of the Schmidt number. A fit consistent with the data is f(Sc) ln(Sc), although a weak power-law dependence of the form is not inconsistent as well. The use of three different injection diameters indicates that the mixing time of a blob of size d scales like ln(5Sc), as opposed to according to the vision proposed by Corrsin (1964) and Oboukhov (1949). The spectrum of the mixture is also found to decay like in the inertial range of scales, and for scales smaller than the injection scale d, as opposed to Moreover, the mixing time

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Figure 6 Left: Contours of the level sets of the scalar field of Fig. 4, for four different concentration thresholds levels. From left to right and top to bottom, 0.3, 0.4 and 0.5. Right: Corresponding box-counting relationships. Continuous lines fit by Eqn. (5).

has been found to be smaller than the cascade time We have suggested (Villermaux et al. 2000) that the process of mixing does not follow the sequential route expected from cascade arguments, but is on the contrary “bypassed” by a strong and constant stretching rate acting at the injection scale. Transient geometry. The changes in the blob morphology in the course of its dilution reflect closely the uniformization process itself. This is best illustrated by following the transient shape of a particular isoconcentration contour Figure 6 shows four contours corresponding to four different concentration levels Consider a blob of dye initially smooth and segregated. The number of segments, or square boxes of linear size r needed to cover its contour at a threshold level is on a two-dimensional cut such as those of Figs. 3 or 4. If the blob is immersed in a prescribed displacement field whose stationary velocity increments give rise to a scale-dependent stretching rate then the number of segments constitutive of the contour at time t is

where the amplification factor expresses the net increase in the contour corrugations length, all the more rapid that is intense, and where the exponential factor reflects the disappearance of the scalar by molecular diffusion according to the global temporal evolution of the concentration histogram (Eqn. (1)).

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The maximal rate of shear, giving the separation rate of two material

points initially close to each other, is of the order of as suggested by the scale dependence of the mixing time in Eqn. (2). Considering two material points belonging, for instance, to the same isoconcentration contour, initially separated by a distance r not necessarily infinitely small, the rate at which their separation distance increases is smaller than Velocity gradients in the flow tend to vanish, in the mean, for separation distances r larger than the scale of the mean gradient support, namely the injection size d in the present case. A possible model for the elongation rate for any scale r is

The number of boxes are represented, for the corresponding threshold levels by (see Fig. 6)

according to (3) and (4). Accounting for the initial smoothness of the contour, namely the covering relationship (5) is a combination of the trivial 1 / r factor, times a corrective factor, increasing in magnitude with time, and whose weight depends on scale: it is, at a given instant of time, a decreasing function of scale, expressing the

fact that small scales have, in proportion, more contributed to the corrugation of the contour than larger scales, precisely because shearing motions are less efficient at large scales than at smaller ones (Eqn. (4)). The covering relationship (5) thus exhibits a curvature, whose direct

consequence is the scale dependence of the fractal dimension of the contour This fact has been recognized in a number of related instances (see Villermaux and Innocenti 1999 and references therein) where the local (in scale) fractal dimension is found to increase from 1 at small scale, to larger values, possibly reaching 2, characteristic of space-filling objects in two dimensions. The above scenario provides a mechanism for this continuous transition, its origin lying in the close interplay between kinetics and geometry.

2.2.

Multiple sources

A complex mixture in the real world is obviously the result of the

superposition of many different sources. If we think of a hot jet discharging in a quiescent cold environment, the temperature fluctuations on its centerline at a given distance from the nozzle are certainly likely

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Figure 7 Two nearby sources discharging in a turbulent medium in which they mix.

to be the result of cumulated contributions from different parcels of cold fluid entrained at the boundary of the jet in the course of its development. The “signature” of one scalar source presented in the previous section is likely to be altered if another source is present in its vicinity (as shown in Fig. 7), and the interaction between those two has to be considered. If and are the bare PDFs of each of the two sources disposed in the vicinity of each other, the compound PDF is shown in Fig. 8. As long as the plumes emanating from each of the sources do not interfere, the sources develop in an anticorrelated manner: the concentration measurement point is either in one plume, or in the other (see also Warhaft 1984). Then, as soon as the plumes merge, it is observed experimentally that is very close to the convolution of and The concentration C in the resulting mixture is the sum of the concentration from source 1 and of concentration from source 2, and being chosen independently in each of the original distributions provided that

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Figure 8 Composition of sources of heat in water (Sc = 7). Solid line: experiment with two sources. Dashed line: reconstruction of the scalar field from and under the anticorrelated rule. Circles: reconstruction of the scalar field by convolution.

This result is extended to the turbulent jet problem (Duplat and Villermaux 2000), for which it is shown that the scalar fluctuation PDF on the centerline is the result of well defined elementary sources whose size is given by the local width of the jet. These findings, bridging the kinetics and statistical aspects of the phenomenon of mixing, also suggest that a complex mixture can be understood as the superposition of well defined elementary contributions, whose spatial distribution reflects the entrainment properties of the flow.

References Batchelor, G. K. 1959. Small-scale variation of convected quantities like temperature in a turbulent fluid—Part 1. General discussion and the case of small conductivity. Journal of Fluid Mechanics 5, 113–133. Batchelor, G. K., I. D. Howells, and A. A. Townsend. 1959. Small-scale variation of convected quantities like temperature in a turbulent fluid—Part 2. The case of large conductivity. Journal of Fluid Mechanics 5, 134–139.

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Batchelor, G. K., and A. A. Townsend. 1956. Turbulent diffusion. In Surveys in Mechanics, Batchelor, G. K., and R. M. Davis (eds.). Cambridge: Cambridge University Press, 352–399. Betchov, R. 1956. An inequality concerning the production of vorticity in isotropic turbulence. Journal of Fluid Mechanics 1, 497–504. Breidenthal, R. 1981. Structure in turbulent mixing layers and wakes using a chemical reaction. Journal of Fluid Mechanics 109, 1–24. Brown, G. L., and A. Roshko. 1974. On density effects and large scale structures in turbulent mixing layers. Journal of Fluid Mechanics 64(4), 775–815. Corrsin, S. 1951. On the spectrum of isotropic temperature fluctuations in an isotropic turbulence. Journal of Applied Physics 22, 469–473. Corrsin, S. 1964. The isotropic turbulent mixer: Part II. Arbitrary Schmidt number. American Institute of Chemical Engineers Journal 10(6), 870–877. Damkohler, D. 1940. Der Einfluss der Turbulenz auf die Flammen-geschwindigkeit in Gasgemischen. Zeitschrift fur Elektrochemie 46(11), 601–652. Danckwerts, P. V. 1952. The definition and measurement of some characteristic mixtures. Applied Scientific Research A 3, 279–296. Danckwerts, P. V. 1953. Continuous flow systems. Distribution of residence times. Chemical Engineering Science 2, 1–13. Dimotakis, P. E. 2000. The mixing transition in turbulent flows. Journal of Fluid Mechanics 409, 69–98. Duplat, J., and E. Villermaux. 2000. In preparation. Epstein, I. R. 1990. Shaken, stirred—but not mixed. Nature 346, 16–17. Hawthorne, W. R., D. S. Wendell, and H. C. Hottel. 1949. Mixing and combustion in turbulent gas jets. In Third Symposium on Combustion and Flame and Explosion Phenomena. Baltimore: Williams and Wilkins, 266–288. Mandelbrot, B. B. 1975. On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars. Journal of Fluid Mechanics 72(2), 401–416. Nagata, S. 1975. Mixing, Principles and Applications. New York: John Wiley & Sons. Oboukhov, A. M. 1949. Structure of the temperature field in a turbulent flow. Izvestiia Academii Nauk USSR, Seriia Geograficheskaia i Geofizicheskaia 13, 58–69.

Ottino, J. M. 1989. The Kinematics of Mixing: Stretching, Chaos, and Transport. Cambridge: Cambridge University Press. Prandtl, L. 1925. Uber die ausgebildete Turbulenz. Zeitschrift für angewandte Mathematik und Mechanik 5, 136–139. Ranz, W. E. 1979. Application of a stretch model to mixing, diffusion and reaction in laminar and turbulent flows. American Institute of Chemical Engineers Journal 25(1), 41–47. Reynolds, O. 1894. Study of fluid motion by means of coloured bands. Nature 50, 161–164. Richardson, L. F. 1926. Atmospheric diffusion shown on a distance-neighbour graph. Proceedings of the Royal Society of London A 110, 709–737.

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ICTAM 2000 early registrants enjoy conversation and a spectacular view of Chicago's skyline from the 9th floor of the Chicago Marriott Downtown hotel during the Get-Together Party on Sunday, 27 August 2000. In the background is Chicago's landmark Hancock Building.

STATISTICAL APPROACH TO THE MECHANICAL BEHAVIOR OF GRANULAR MEDIA Stéphane Roux Unité Mixte de Recherche CNRS/Saint-Gobain, “Surface du Verre et Interfaces” Aubervilliers Cedex, France [email protected]

Farhang Radjaï Laboratoire de Mécanique et de Génie Civil, Université de Montpellier II Montpellier Cedex, France [email protected] Abstract

1.

We discuss the quasistatic rheology of ideal granular media consisting of rigid discs interacting via the Coulomb law of friction and perfectly inelastic collisions. The macroscopic description of the rheology of quasistatic deformation of such media is rigid–plastic with hardening laws parameterized with internal variables that have to characterize the geometry of the assembly. A phenomenological approach is proposed along these lines. An outline of a microscopic–macroscopic derivation of the required characteristics is presented. Finally, we list some possible effects of fluctuations that may limit the precise quantitative success of this approach.

INTRODUCTION

The quasi-static behavior of granular materials is already a mature field in which a number of elasto–plastic models reproduce very accurately the available experimental tests. They allow us to design civil engineering structures with confidence. However, this description is essentially based on extensions of the elasto–plasticity of other materials rather than a microscopic modeling of a granular assembly (Mróz 1998). On the other hand, by focusing on the details of particle interactions, considerable progress has been made in recent years through various algorithms that allow us to describe different regimes of granular flows. (See e.g. Bardet 1998, Kishino 1999 for recent reviews.) The discrete 181 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 181–196. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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modeling of granular media allows nowadays very accurate simulation of very stiff particles up to particles for a significant cumulative strain. In cases where the accurate modeling of the contact and of the particle properties is less stringent, more than one million grains can be taken into account. The interest of this approach is that it allows us also to follow the response of a granular assembly subjected to uniform stress or strain. In this context, the use of bi- or tri-periodic boundary conditions (in two or three dimensions, respectively) is an important achievement allowing us to reduce significantly the role of walls. We are thus now in a suitable position to answer questions pertaining to the characterization of the geometry of the packing (and hence eventually to answer such unsolved issues as the orientation of localization bands), to address the role of fluctuations, and to measure the effective stress carried by a specific granulometric class in a packing (relevant for fragmentation)— issues that are clearly out of reach within today’s continuum modeling. With this motivation in mind, one is naturally invited to have a fresh look at the continuum modeling and to progress in the direction of introducing geometric information in the macroscopic modeling, even if such an approach will inevitably lead first to a deterioration in the accuracy of the macroscopic modeling. The hope is that, after some time and effort, one may achieve a more satisfactory description in terms of connections to microscopic reality and still with a fair account for experimental tests.

2.

A MODEL SYSTEM

In the following, we will focus on the simple model of a granular assembly of rigid discs in two dimensions. These particles interact only via a hard-core potential (without adhesion) and the Coulomb friction law with a single coefficient of friction. When numerical simulations are used, a slight polydispersity is introduced in order to avoid the crystallization of the system that is a specific feature of two-dimensional systems. However, in our theoretical description we ignore the polydispersity and the particles will be characterized by a single average radius R.

2.1.

General remarks concerning the macroscopic modeling

Since the particles are considered as rigid and with no adhesion, the macroscopic behavior has to be rigid–plastic. Therefore, the domain of admissible stresses where the medium is rigid, with its boundary where plastic flow may take place, has to be specified. The absence of a stress scale imposes moreover that is a cone in stress space.

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The Coulomb friction law itself can be seen as a rigid–plastic behavior. Within this framework, the Coulomb friction does not obey the normality rule, and hence macroscopic modeling has to be a non-associated plastic behavior. Therefore, the direction of the plastic strain rate has to be specified independently from The plastic flow rule, as well as the yield stress, has to be defined as a function of the internal variables that characterize the state of the medium. When considered at the microscopic scale, it is obvious that this state is defined only through its geometry, i.e. the spatial organization of particles with respect to each other, and hence it is natural to require that internal variables have a simple geometrical interpretation. Although this statement seems quite obvious, it readily disqualifies a number of macroscopic descriptions used at present. Stated simply, the fundamental difference between a microscopic modeling and a macroscopic one lies in the number of internal variables. Going from the micro- to the macro-scale, we wish to preserve only a few of them, enough to characterize precisely the geometrical state of the assembly and not too many, so as to be able to have an efficient formulation and a limited number of parameters to adjust through fitting rheological responses or microscopic considerations. As usual, the choice of pertinent state variables is the crucial issue, which often results from a compromise between accuracy and simplicity. In order to highlight this compromise, we proceed in three steps. First, we apply the above stated constraints to the formulation of a (trivial) rigid–plastic description with no internal variable. Then, we go one step further and incorporate a single scalar internal variable. Finally, we add the fabric to the description. We will see that the incorporation of more and more variables leads to a progressively richer description. We will sketch the outline of a systematic procedure. Coming back to the macroscopic description, in order to arrive at a complete formulation we have to specify three points: (1) a yield stress parameterized by the internal variables, (2) a plastic flow rule parameterized by the internal variables, and (3) a “hardening” law that describes the incremental evolution of internal variables with plastic strain.

3.

NO INTERNAL VARIABLE

The most elementary choice is to assume that no internal variable is necessary. In this case, there is obviously no need for a “hardening law”. The medium has to be assumed isotropic—otherwise the anisotropy parameters would constitute internal variables. In two dimensions, the requirement of objectivity implies that the domain is a function of

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stress invariants. The easiest case refers to the principal stresses, i.e. a two-dimensional description of the stress space. Then, is a cone whose axis has to lie along the isotropic pressure direction. Thus, a single parameter is required, which quantifies the relative magnitude of the deviatoric stress with respect to the average stress (trace). This single parameter can be rephrased in terms of the classical friction angle in the Mohr–Coulomb framework. For similar reasons, the plastic flow rule consists of defining the relative amount of dilation/contraction with respect to the deviatoric part of the plastic strain rate. Since the macroscopic description should hold for an arbitrarily large cumulative strain, the only physically admissible choice is to require that the plastic strain be isochoric, or This corresponds to the fact that the internal state of the granular medium is assumed to be unique, and thus it has a well-defined packing fraction that is independent of its prior deformation history. Moreover, along the plane where the Mohr–Coulomb criterion is reached, the axial strain has to be zero. This defines uniquely the relative orientations of the principal axes of the strain rate and stress. The above considerations are sufficient to determine fully the macroscopic bahavior of the medium in two dimensions. In particular, this shows that a single scalar parameter is needed—the Mohr–Coulomb friction angle. It is obvious that the resulting description is very crude. Nevertheless, in cases where the cumulative shear strain is large (e.g. surface flow in avalanches), this level of description may prove quite sufficient. The unique state of the medium corresponds to the “critical state” of soil mechanics reached after a sufficiently large deformation (Schofield and Wroth 1968).

4.

ONE INTERNAL VARIABLE

Let us now try to add some information about the geometrical state of the medium. The first and most obvious property pertaining to the geometry of a granular packing is the packing fraction c, defined in two dimensions as the fraction of surface covered by the particles. A number of equivalent variants, such as void ratio, density of particle centers, global density, and porosity may be used instead. We can proceed as in the previous case but using c as an internal variable for the yield surface and for the plastic flow direction. The yield surface is still parameterized by a friction angle, but now the latter is a continuous (increasing) function Concerning the direction of the incremental plastic strain, dilation and contraction are now admissible.

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The ratio of the spherical part to the deviatoric part of the plastic strainrate tensor is denned to be the dilation angle satisfying

where are the two eigenvalues of In the absence of internal variables we argued that = 0. Following the same argument, one infers that there should exist a specific packing fraction such that

This packing fraction characterizes then the critical state. Now a hardening law has to be specified. It should describe the way c evolves under an increase of plastic strain. In the present case, this evolution law is dictated by the very definition of our internal variable, namely

This description provides already a more detailed description of the transient stages leading to the critical state. One also recovers the previous model if only large strains are considered. The price to pay for this more accurate description is that we have to specify two functions and There we have different routes at our disposal: either we consider a purely phenomenological approach and thus we try to identify these functions from simple tests (Schofield and Wroth 1968), or we may try to relate these functions to a microscopic description of the packing. Eventually, we could also combine both approaches by using part of the information from the micro-scale and identify only a few parameters from experiments. Along the latter direction, one could for instance use Taylor’s hypothesis (1948) in order to relate the two functions and together, and measure then only one of them experimentally. We will come back to this issue after the following section devoted to a richer description of the geometrical state of the medium. In order to simplify the analysis and the number of parameters to be introduced, we may focus on the neighborhood of the critical state and expand the functions of interest in Taylor series. This leads to a simple three-parameter description, based on and (Roux and Radjaï 1999). An interesting point to make is that this approach naturally leads to the occurrence of localization in a dense granular medium. We will however not enter this issue, which would require lengthy developments.

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FABRIC AS STATE VARIABLES

The previous modeling predicts that once the system has reached its critical state, it remains in this state for all directions of shearing. This is not what is observed experimentally. In particular, if the shear is simply reversed, one typically observes a long transient deformation where the system evolves towards a new critical state. This discrepancy clearly indicates that a scalar internal variable such as c (which is unable to encode a specific anisotropy of the medium) is insufficient for the description of the state of the medium. From symmetry arguments, the most elementary object that can characterize such information is a second-order tensor. The latter has to be built from statistical information describing pairs of particles. Moreover, assuming that the required information is local, one should focus on particles in contact. These remarks point to the internal variables pertaining to the distribution of contact normals. Let us consider the probability distribution of contact normals . This is the probability that a given particle has a contact along the direction parameterized by the polar angle of the contact normal n. The function is -periodic and it can be Fourier expanded as (Rothenburg and Bathurst 1989)

Truncation of the Fourier expansion after the second term provides the most salient features of the texture of the medium. In a totally equivalent fashion, one could simply construct the classical fabric tensor where the brackets denote averaging over all particles in a representative element of volume, and is the dyadic (tensor) product (Satake 1982). While the fabric tensor is often normalized with respect to the total number of contacts, here we choose to normalize it by the number of particles. In other words, in our case A = where z is the coordination number. Assuming that the latter is simply related to the packing fraction, we see that the three descriptions considered above can be seen as retaining more and more terms (0, 1, or 2) in the Fourier expansion of p. Let us now characterize all the required information to obtain a complete mechanical description of the behavior using A, B, and as internal variables:

Yield stress: Since the medium has some anisotropy, a single friction angle is no longer appropriate. However, in order to keep close to the concepts used within the previous cases, we consider all potential slip planes characterized by a polar angle of the

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normal to the slip plane. The largest ratio of the tangent to the normal stress that can be supported by this plane naturally defines a friction angle that is a function of and the above three internal variables. Galilean invariance implies that only the difference is meaningful. We thus have to specify the function

Plastic strain rate: Similarly, along each potential slip plane we can characterize the orientation of the relative velocity of two points aligned perpendicular to the slip plane. This defines a dilation angle which as before depends on and the internal variables, i.e. Hardening rule: A hardening rule here means that and are functions of their current values and the value of the incremental plastic strain as well as the rotation i.e. the antisymmetric part of velocity gradient, because of the induced anisotropic texture of the medium. This hardening law should account for the advection of contacts in the plastic flow, as well as the creation and opening of contacts. A last constraint to be taken into account comes from the quasistatic nature of the loading we are interested in. This implies that the rheology should be rate independent, i.e. time as such should not play a role in these equations. Thus (as well as all other such rates) should depend on through a positively homogeneous function of degree 1. As in the previous case, there are now different routes to follow according to different strategies. Again, one possible route is a purely phenomenological approach with the perspective of identifying all the required information from tests. Some basic principles should still be used to constrain the hardening rule. However, due to the crowding of internal variables, this task is quite challenging. A second route is to circumvent first the form of the possible dependencies by means of expansions around special values of the internal variables, such as the isotropic case B = 0 or, alternatively, around values that would correspond to the “critical states”. We use the latter in plural form because the critical state may well depend, for instance, on the relative rotation rate with respect to the deviatoric strain rate, i.e. the critical state under pure shear may differ from the critical state under simple shear. To be more precise, we may assume that, if the deviatoric part of the fabric tensor is reasonably small, may be expanded as

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so that the determination of the full function of three parameters is reduced to four scalar parameters for i = 0 , . . . , 3. Another possible strategy is to resort to the microscopic world in order to identify the above-listed unknown functions. We now discuss some elements along this direction.

6.

MICRO–MACRO TRANSITION

At the macroscopic level, we deal with stress and strain as continuous fields. However, at the particle level, we have a discrete set of forces transmitted by the interparticle contacts and velocities of particles. The connection between these two levels of description has been the subject of intensive work. A clear review of different proposed relations has been presented by Bardet (1998). Let us first recall very briefly a few essential results.

6.1.

From discrete to continuous

The first key point is to define the average stress over any domain, which encompasses an arbitrary number of particles. This can be done consistently down to the scale of one single particle. Let us label each contact around one particle by an index i, and denote the unit normal vector by and the force transmitted at the contact by The stress that characterizes the particle is (Cundall and Strack 1979)

where S is an area associated with the particle from a Voronoï construction from particle centers. This expression exploits the fact that the particles are at rest and no torque is transmitted at the contacts. We have assumed that the particles are circular with a radius R. We see that the transition from forces to stress requires the definition of a representative environment of a particle where each neighbor is specified,

i.e. a “node” of the contact network. The analogous treatment of the displacement field is somewhat more subtle. The main point is that the object in which we are interested at the macroscopic scale is the displacement field particle center. Whether a particle rotates or not will affect the velocity of a material element in the grain, but not the overall displacement. Thus, the strategy usually adopted in this respect is to construct a continuous field from an interpolation of the velocity field that matches exactly the particle velocity at the particle center.

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The standard technique is again to resort to a tesselation of space with polygons whose vortices are the particle centers, and whose edges connect the contacting particles. Once the problem is reduced to the estimation of the average strain inside an elementary polygon whose boundary velocity is prescribed, the solution is standard (Kruyt 1996). As in the case of the force–stress relation, here we need a representative elementary structure—a polygon formed by contiguous particles—which will be referred to as a “cell” in the sequel.

6.2.

Environments: Nodes and cells

A major difficulty is that, as discussed above, the macroscopic description of the geometrical state of a granular medium is based on the fabric represented by (including a more or less severe truncation). However, in order to construct the two elementary tensors, we have to deal with nodes or cells and this requires richer information. For instance, a node is characterized by its coordination number z and the orientations of its contact normals. Thus, we need the probability distributions The corresponding issue is trivially written in similar terms for the cells. In order to construct these representative environments, we have to propose an “educated guess” for these multicontact distributions. An easy solution is to assume the most “disordered” situation, i.e.

In our case, this solution is obviously wrong since a contact in the direction with a given particle impedes other contacts to be established in a direction such that hence

Here, the strategy that we propose is still to resort to a similar “most disordered” situation, provided these steric constraints are taken into account. Operationally, this translates into the maximization of an entropy functional under constraints. The entropy S is classically defined as

and S is maximized over the set of functions that (1) fulfill the steric hindrance conditions, Eqn. (11), and (2) whose partial summation over all but one angle gives back the known The second constraint

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is imposed through Lagrange multipliers. Let us introduce a periodic function such that = 0 for and H = 1 otherwise. It can be shown that takes the form

The unknown function g is then determined from an implicit equation resulting from the resummation condition (2), and whose solution can be obtained from a simple iterative scheme. Note that without steric hindrance, the simple solution is recovered. We note that not all values of A and B in the truncated form of p admit a solution. The condition is that

which simply states that no more than one contact can be found in any sector of opening angle Using the simple truncated form Eqn. (5), we obtain

which together with the condition (no negative probability) gives the domain of physically accessible values of the fabric.

7.

YIELD STRESS AND PLASTIC STRAIN DIRECTIONS

The previous section proposed an operational way of generating representative nodes (and using a similar construction, cells) with acceptable statistics, i.e. consistent with the known information concerning fabric. Thus, we have now a key that will allow us to compute the yield stress. From the previous discussion, we have to compute the maximum allowed deviatoric stress for all orientations of the principal axes of stress with respect to that of the fabric. We propose a Monte-Carlo procedure, which consists in generating a large collection of node configurations with the computed statistics. Then, for fixed orientation and trace of the stress tensor we compute the maximum value of its deviatoric component that can be obtained from the contact forces with the following constraints: (1) the forces are balanced, (2) they yield the imposed stress when using Eq. (9), and (3) each force fulfills Signorini (no traction) and Coulomb conditions. A simple average over the configurations gives an estimate of the sought yield surface.

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A similar procedure can be designed for the strain. Cells are generated, and for each configuration the minimal dilation angle for a principal strain orientation is computed. This is done by computing an admissible set of particle center velocities that is consistent with the average strain, and with the steric exclusion conditions. An average over configurations gives the sought dilation angle.

8.

HARDENING RULE

We still have to address the final point, which is the hardening rule, i.e. the evolution of the fabric, in terms of or fabric tensors as a function of strain. The two effects to take into account are (1) the advection of contacts by the plastic flow and (2) the induction (gain or loss) of contacts (Roux and Radjaï 1999). We can write the time evolution of as a balance equation:

where the divergence operator is simply J is the contact “current”, and I the induction term. The mean velocity field with respect to a particle center in polar coordinates is written For contacting particles, where r = 2R, the mean velocities are written The expression of the contact current is thus simply written as

The Signorini condition requires that Galilean invariance dictates that the effect of a rotation rate should come into play only in the tangential component, as and We note that current can also be split into two contributions, one part from the rotation, and one from the pure deformation term. Extracting the rotation term from the current, we can rewrite the left-hand side of the balance equation as

where the total time derivative includes the rotation term. With this in mind, we can now safely ignore the rotation effect. The velocity field for non-contacting particles can be deduced simply from a mean-field assumption since no direct steric constraints are at play. Thus u and are directly related to The induction term I consists of a creation of contacts, and an opening term with The creation of contacts involves

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the non-contacting normal velocity when the latter is negative, and the probability that a given particle lies in the region sufficiently close to reach the reference particle. This involves the packing fraction, through the areal density of centers The creation term is written

where [ … ] _ denotes the negative part of the velocity. The contact

opening term is Up to now, we have specified all terms except the contacting velocities They are obviously related to through a geometric function that is positively homogeneous of degree 1, because of the rate independence. This could be computed together with the direction of the plastic strain rate, because at this stage we generate a representative set of local configurations where we have access to the relative particle velocities. Alternatively, we may consider two arbitrarily remote particles along

a given direction, which can be reached through a path of contacting particles. When the particles are sufficiently far from each other, their relative velocity is equal to the macroscopically determined one. However, it is also equal to the sum of the relative contacting particles along the path. This constrains to be equal to when averaged over an interval of angles that allows for the existence of a path. This

allows the deficit in negative U values to be compensated by a larger

value of V at a different angle. The difference between and i.e. the deviation from the mean field, may be interpreted as some kind of diffusive contribution to the current due to the steric hindrance transmitted through particles outside our “shell” description. This section has been written directly in terms of the entire distribution and not its truncated Fourier expansion. In order to express the hardening equations in terms of A, B, and it suffices to consider weak formulations of the latter by multiplying Eqn. (16) by 1,

and

and by integrating over all angles

Then, a term-by-term

identification gives a direct transcription corresponding to the reduced description of the fabric.

Let us finally mention an additional important point. We pointed

out that the fabric is restricted by geometrical constraints (see e.g. Eqn. (15)) at the particle level. However, these constraints do not appear in the present hardening rules, so they might be violated if one proceeds along the proposed route. A consistent way to circumvent this difficulty is to require that the fabric always lies in the admissible domain. This

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will in turn generate an admissible set of plastic strain increments. The latter may thus be used to define the plastic flow rule. This would constitute an alternative way to end up with a consistent rigid–plastic formulation, and it would bypass the stage that consists in generating representative “cells”. In other words, it would exploit different ways of relating the relative velocities of contacting particles to the mean strain field, through paths rather than elementary cells.

9.

EFFECTS OF FLUCTUATIONS

In this section, we briefly discuss potential limitations to the scheme proposed above, as inferred from numerical observations. Most of these difficulties result from fluctuations either in time or in space that have received in the past a much more limited attention than average behaviors. Indeed, the above scheme is based at each time step on rest configurations. However, when running numerical simulations we observe that from time to time the system reaches unstable points where a dynamical rearrangement of particles has to take place. In this dynamical phase, inelastic collisions dissipate the potential energy drop (i.e. the work of the loading forces), due to the reorganization of the assembly. The cumulative effect of these restructuring events may have paradoxical effects. In particular they can produce an effective mean dissipation that is “Coulomb”-like even for ideal frictionless particles. Indeed, we can imagine that averaging the instantaneous dilation rate during a long steady shear strain (once the steady state has been reached) over all rest configurations, we may obtain a positive value that is exactly counterbalanced by the compression taking place during these unstable events. The dynamics of restructuring being much faster than any external loading (quasistatic condition), the (time averaged) dissipation appears to be rate independent. Moreover, the dissipation at each event is equal to the external work accumulated prior to the event. Therefore, the dissipation is simply proportional to the external loading. Both these features characterize a solid “Coulomb” friction. It is amusing to note that a similar explanation based on elastic asperity interactions is at the heart of microscopic physical modeling of friction (Caroli 1996, Tanguy 1998). Trying to quantify the above effect in a more realistic case, we quantified the relative part of dissipation due to inelastic collisions, as compared with the total dissipation, i.e. including contact friction (for an interparticle coefficient of friction = 0.5). The

resulting ratio was close to 30% due to restructuring events. This effect being absent from the proposed description, the effective macroscopic friction angle is at best off by this amount if a global dissipation balance

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is performed. In contrast, an analysis of experimental results based on instantaneous values of the stress may be in much closer agreement. Fluctuations may be important for basically two reasons. The first one is their amplitude relative to the mean, and the second is the existence of long-range correlations, which may be present either in time or space and which may affect very significantly the validity of the proposed approach. In order to investigate the latter, we studied the trajectory of particles in a simple shear test. The analysis of the fluctuating part of the displacement and of the distance between two neighboring particles reveals the existence of long-range temporal fluctuations that give rise to an anomalous diffusion regime. Similarly, the instantaneous strain field also displays large scale inhomogeneity, which was unexpected. These observations, which are not taken into account in the approach we presented, may potentially invalidate a quantitative agreement between theory and numerical or experimental tests.

10.

CONCLUSIONS

We presented a systematic approach to progressively enrich a description of the mechanical behavior of a granular medium from both macroscopic and microscopic (particle level) points of view. This constitutes a template, and a number of these elementary steps have still to be validated in particular from detailed numerical simulations. We proposed an original scheme based on entropy maximization that allows us to generate representative environments around particles—an essential step for determination of the yield stress or the plastic strain rate direction. We also proposed a simple “shell model” description of the velocity field around a particle, which allows us to obtain an evolution equation for the fabric. Finally, we pointed out some potential difficulties of this kind of approach, where salient aspects of fluctuations, not included in the theoretical approach, may affect the accuracy of the description. Nevertheless, the global framework, i.e. the form of the equations, should be unaffected. A macroscopic description of these fluctuations might thus appear a necessary step for a complete description, and that is, in our view, a major challenge for the future.

Acknowledgments The authors wish to thank R. Behringer, J. C. Charmet, M. Jean, J. J. Moreau, and H. Troadec for valuable discussions on the subject covered in this chapter.

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References Bardet, J. P. 1998. Introduction to computational granular mechanics. In Behaviour of Granular Media (B. Cambou, ed.). Vienna: Springer, 99–169. Caroli, C., and P. Nozieres. 1996. Dry friction as a hysteretic elastic response. In Physics of Sliding Friction (B. N. J. Persson and E. Tosatti, eds.). Dordrecht: Kluwer Academic Publishers, 27–49. Cundall, P. A., and O. D. L. Strack. 1979. The distinct element method as a tool for research in granular media. National Science Foundation Report 76-20711, University of Minnesota, Minneapolis, Minn. Kishino Y., and C. Thornton. 1999. Discrete element approaches. In Mechanics of granular materials (M. Oda and K. Iwashita, eds.). Rotterdam: Balkema, 147–223. Kruyt, N. P., and L. Rothenburg. 1996. Micromechanical definition of the strain tensor for granular materials. Journal of Applied Mechanics 118, 706–711. Mróz, Z. 1998. Elastoplastic and viscoplastic constitutive models for granular materials. In Behaviour of Granular Media (B. Cambou, ed.). Vienna: Springer, 269–337. Rothenburg, L., and R. J. Bathurst. 1989. Analytical study of induced anisotropy in idealized granular materials. Geotechnique 39, 601–614. Roux, S., and F. Radjaï. 1999. Texture-dependent rigid-plastic behaviour. In Physics of Dry Granular Media (H. J. Herrmann, J. P. Hovi, and S. Luding, eds.). Dordrecht: Kluwer Academic Publishers, 229–235. Satake, M. 1982. Fabric tensor in granular materials. Proceedings of the IUTAM Symposium on Deformation and Failure of Granular Materials (P. A. Vermeer and H. J. Luger, eds.). Balkema: Delft, 63–68. Schofield, A. N., and C. P. Wroth. 1968. Critical State Soil Mechanics. New York: McGraw-Hill.

Tanguy, A. 1998. Un modèle de frottement solide sec par multistabilité de milieux elastiques. Ph.D. dissertation, University of Paris VII, France. Taylor, D. W. 1948. Fundamental of Soil Mechanics. New York: John Wiley and Sons.

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Ukranian participants Tatyana S. Krasnopolskaya and her husband, Vyacheslav V. Meleshko, receive ICTAM 2000 “stress relief” apples from session aide Gareth Block as they enter the Opening Ceremony. The instructions on the apple read: “Archimedes, float! Galileo, drop! Newton, throw! all others, squeeze!”

DAMAGE ANALYSIS AND PREVENTION IN COMPOSITE MATERIALS George J. Dvorak Rensselaer Polytechnic Institute, Troy, New York, USA [email protected] Abstract

1.

A new model for incremental analysis of distributed damage evolution in heterogeneous solids is developed with the Transformation Field Analysis. Stress changes caused by local debonding under increasing overall loads are described by a selected model of imperfectly bonded inhomogeneity and represented by equivalent eigenstrains that act together with the applied loads and prescribed local transformation strains on an undamaged aggregate. Interaction between the still bonded and partially debonded phases at any damage state is described by transformation influence functions. Damage rates are derived from the local fields, in terms of a prescribed probability distribution of interface strength and local energy released by debonding. An incremental procedure is outlined that predicts the extent of damage and its effect on local and overall response under variable loads. Also, potential applications of fiber prestress in damage control and prevention in laminated structures are illustrated by two examples. One shows how prestress release can expand the tensile damage-free region of symmetric laminates. Another derives the fiber prestress magnitudes needed to eliminate free edge stress concentrations caused in any laminated plate by cooling from the processing temperature.

INTRODUCTION

A significant effort has been devoted in recent years to modeling of interface decohesion in heterogeneous solids, which is responsible for most of the distributed damage preceding failure of composite materials and polycrystals. In most models, a single cylindrical or spherical inhomogeneity is bonded to the surrounding matrix by an imperfect interface or interphase. Elastic or inelastic interlayers or spring-type interfaces, and also sliding and frictional interfaces in contact, have been used in analytical studies of the stress and deformation fields in partially debonded inhomogeneities. Moreover, both numerical unit cell models and analytical tools have been developed that incorporate separation laws for imperfect interfaces consistent with fracture mechanics 197

H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 197–210. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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concepts. While such models are essential building blocks of micromechanical damage evolution models, their utilization has been rather limited. Available studies tend to predict only uniform or periodic damage distribution in the representative volume, which do not correspond to observations. The next chapter of this paper briefly outlines a new analytical framework for modeling of distributed damage evolution that can incorporate different local debonding models, together with extensive information on local material properties. The damage process is assumed to start in a few weakly bonded interfaces at a certain initial level of applied load, and proceed under increasing load or time by affecting ever larger internal surface areas and volume fractions. Interaction between both the still bonded and partially or completely debonded inhomogeneities at each loading step is described using certain transformation influence functions. To this end, the local average stress changes in the debonded phase are simulated by equivalent transformation strains applied to an otherwise undamaged aggregate. Local thermal and variable transformation strains are also included in the applied load set. Local stresses and strains, overall deformations and compliance changes are found in terms of applied loads and damage density, and then updated under increasing load according to a prescribed probability distribution of local debonds, and an energy criterion for debonding of each individual volume. The third chapter describes some recent results on using selected magnitudes of fiber prestress, applied prior to and released after matrix consolidation, to create residual stress fields that help prevent damage initiation in laminates. In particular, fiber prestress is shown to change the position of the initial damage envelopes in the overall loading space, allowing for higher tensile stresses to be applied in a damage-free region. Another example illustrates how prestress release can be used to cancel free edge stresses caused in laminates by cooling from processing temperature.

2.

2.1.

DAMAGE EVOLUTION IN A TWO-PHASE COMPOSITE SYSTEM Local stresses in current damage state

A representative volume V of a composite aggregate contains a statistically homogeneous distribution of dispersed particles or fibers of stiffness initially bonded to a matrix of stiffness Incremental loading is applied by a superposition of surface tractions on 5 of V, derived from a uniform overall stress with thermal and other transformation strains (r = 1, 2) in the phases. Under increasing loads,

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starting at some of the interfaces between the matrix and dispersed reinforcements may undergo partial or complete debonding. At any stage k of the damage process, the volume V contains a constant matrix volume while the total volume of the reinforcements is divided into fully bonded (r = b) and debonded (r = d) parts; the respective volume fractions are related by After partial debonding, certain interface tractions are transmitted by friction or by interlocking asperities; hence an average stress is supported by each debonded particle or fiber. If the overall loading path is reversed, parts of the debonded interfaces may reconnect under local compression. Therefore, in general, some or all components depend on the loading set except under complete debonding, when In what follows, debonding-induced changes in local stresses, local and overall strains are simulated by introducing equivalent local eigenstrains in the r = d subvolumes of the undamaged composite system. Since the eigenstrain currently present in the bonded reinforcements must be present in the entire volume the total eigenstrain in is and will be decomposed again in below. At any stage of the process, the components of the current load set are regarded as independent loads acting on a fully bonded aggregate. All local eigenstrains are assumed to be uniform, or to represent certain averages of the actual local fields. Also, all reinforcement particles or fibers are assumed to have the same shape and orientation, but each can have a different “diameter” or size. Local stress averages caused in a heterogeneous medium by a uniform overall stress and a piecewise uniform distribution of local transformation strains are written in the form (Dvorak 1990)

where are Hill’s (1965) mechanical stress concentration factors of the undamaged system, which depend only on the local and overall stiffnesses. The local stresses satisfy the requirement hence and Since there is and The are transformation stress concentration factor tensors that evaluate the residual stresses contributed by the equivalent eigenstress in the transformed and debonded phase volumes to stresses in all phases r. Phase definition is expanded here to mean homogeneous subvolumes of the same

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stiffness in a fixed overall space, subjected to the same average eigenstrain. Thus the debonded and still bonded reinforcements are regarded as different phases of the same stiffness loaded by different eigenstrains and Evaluation of the must take into consideration all three phases in (2); this is reflected by adopting the two equivalent forms valid for multiphase systems (Dvorak and Benveniste 1992)

where r, s = 1, b, d, the local and overall compliances and also and but

M =

Both and are evaluated in a perfectly bonded system, where the overall compliance M has been estimated by a selected averaging method. The stress concentration factor where is the compliance of the cavity containing in a large volume of a certain homogeneous comparison medium S is the Eshelby tensor of a transformed homogeneous inclusion in According to the above assumption, all reinforcements have the same shape and orientation; hence only a single S or are needed. The choice of defines the averaging method used in evaluation of M, and Preferred here is the choice which implies the Mori–Tanaka method. The S and are then evaluated in the matrix material and M can be found using the expression (M + = Other choices of are possible, providing that the estimate of M remains within the Hashin–Shtrikman bounds. In any case, selecting as a certain function of only is required to render the stress field independent of in the limiting case of porous medium, when and Since the choice of affects only the evaluation of S and , any such choice can be easily implemented in what follows. The local stress averages can now be found as functions the current total load set and the debonded phase volume fraction Substituting from (4) into an expanded form of (2) and using the notation (Hill 1965)

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together with algebra

201

provides after some

Since one can readily verify the requirement As expected, the stresses depend only on the differences between the local eigenstrains in the reinforcement and matrix; application of a uniform eigenstrain in the entire volume of any solid has no effect on its stress state. Also, it turns out that

At the current state of debonding the is an unknown quantity that needs to be found from additional information. This is usually provided by a local decohesion model that relates the local stress field in the partially debonded inhomogeneity to that in the surrounding matrix. A suitable general form connects the averages of the two fields

by means of a partial stress concentration factor It should provide an interface for implementation of many different models that simulate interface decohesion modes. For example, those employing an elastic interphase (Christensen and Lo 1979, Chen et al. 1990) or a spring-type interface (Hashin 1991a,b) provide fairly simple forms of that may change with values of the interface parameters, but do not depend on the debond angle or area. A more realistic class of models that evaluate the extent of debonding either by numerical or analytical methods (Achenbach and Zhu 1989, Hutchinson and Jensen 1990, Karihaloo and Vishvanathan 1985, Gao 1995), and rely on specific models of interface decohesion (Needleman 1987, Tvergaard 1990) would yield dependent also on as would models that incorporate interface sliding and friction (Steif and Dollar 1988, Furuhashi, et al. 1992). Implementation of the latter class may rely on a separate numerical routine for evaluation and updating of Using (8) in yields the equivalent eigenstrain that, when applied within together with on S of V, and within and respectively, creates the local stresses (6) and (8), where

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Substituting this into (6) readily yields the local stress in terms of and eliminating this with (7) and (8), in favor of the prescribed quantities

and current value of

Again,

finally gives

As one would expect, in contrast to (6), the

introduction of in (8) has rendered the fields dependent on the damage parameters and even under purely mechanical loading. The former property is retained in the case of complete debonding, where and

2.2.

Local and overall strains in current damage state

The relevant expressions evaluating the local and overall strain averages are

where, according to the Levin formula and (5), the overall eigenstrain is

Noting that

it follows from (9)

that the overall strain (12) can then be reduced to the form where the damage-induced addition to the overall elastic compliance of the damaged aggregate is found as

and the overall eigenstrain induced by the applied local eigenstrains

becomes

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Since both Q and are diagonally symmetric, is also diagonally symmetric in the case of complete debonding, where and when The latter symmetry has been established for the Hashin (1991) spring layer model of an imperfect interface, applied to an isotropic inclusion or an isotropic and transversely isotropic fiber, each embedded in an isotropic matrix. Other debonding models can be used, providing that they yield the said diagonal symmetry of Note that for the overall strain (14) and eigenstrain (16) both revert to their respective values in the undamaged state, where and Finally, we recall that where the is the actual equivalent eigenstrain representing the effect of damage by local debonding on the average local stresses and strains and on the overall strain in the composite aggregate. A similar decomposition of small strains can be introduced in the matrix, where the total eigenstrain may consist of thermal and inelastic components. Since the local stresses (10) and strains (12) depend only on the total transformation strains in the respective volumes, such decompositions do not alter the above results. Of course, a somewhat different form of the governing equations would result if inelastic strains or other local eigenstrains were denned as functions of the local stress history (Chaboche et al. 2000).

2.3.

Energy changes

Progressive debonding of the reinforcements from the matrix under either changing or constant overall stress and local eigenstrains applied to the representative volume involves release of potential energy. For the damage process to proceed, the amount of energy released by partial or complete debonding of each particle or fiber must be equal or exceed the work required for interfacial separation. In the context of the loading and damage process described in the change in total potential energy or Gibbs free energy of the representative volume V is caused by changes in both the total load set and local debonded volume during transition from the current state k to the next state k + 1. This can be recorded as

with the understanding that is applied in the current debonded volume and in and that and are applied in and respectively. However, the energy change that can be attributed to debonding alone depends only

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on changes in the damage-related variables under a constant total load set, selected as This can be written in the form

where the

in the second term is now applied only in

Specific evaluation utilizes the expression

where

is the local elastic strain field and the surface integral is

The local stress and strain components in the first integral (19) are taken from (10) and (11), and the overall strain in the second integral from (14), with values of the respective parameters indicated in (18). The actual volume of the reinforcing particles or fibers that experience debonding in the transition from current state k to the next state k + 1, is reflected in the evaluation of the energy change by substituting in the fields (10), (11) that enter the integrals (18)–(20).

2.4.

Damage evolution

Two distinct conditions need to be satisfied for interfacial decohesion to proceed. Local stress at the interface of a bonded fiber or particle has to reach a certain minimum level needed to nucleate an interfacial crack. For this crack to propagate, the rate of energy release must exceed the interfacial toughness. Both complete and partial debonding with asperity contacts require potential energy release equal or greater than the total surface energy dissipated by interface decohesion. Specific forms of the two criteria depend on the material system considered and on the debonding model employed in evaluation of (8). For example, a stress-based debonding criterion that estimates the volume fraction of debonded reinforcements as a function of the local stresses can be selected as a Weibull distribution

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where, depending on the available resolution of the local fields, can be a function of the tension or shear stress maxima or invariants,

such as the isotropic tension component. The

is the value of

at

initial debonding, which may be detected experimentally. The p and q

are scale and shape parameters; no damage occurs but it becomes widespread

= 0) for

when the second term in (21)

tends to zero. In principle, the in (21) can be found as a function of using (13) and (14), from unloading compliances measured in an experiment under incrementally increasing overall stress. In the decohesion models that estimate the extent of local debonding

as a function of applied stress, the energy criterion is typically satisfied during the entire separation process. In the imperfect interface models that estimate only the average stress in the debonding inhomogeneity, a simple criterion can be written as

where

taken from (18)–(20) is an estimate of the total energy

released by partial or complete decohesion of the surface area one or more particles or fibers in the same loading step, and

of is the

interface toughness. Under constant overall stress and typical reinforcement volume, the energy released by debonding of each particle or fiber increases as the

growing number of debonds reduces overall stiffness. Also, the average stress in the stiffer and still bonded reinforcements increases with increasing porosity of the matrix. Therefore, in agreement with observations, the present damage process model suggests acceleration of the

debond rate with increasing volume fraction of debonded reinforcements. An iterative solution procedure under overall stress control starts with (i) selecting trial values of the next load increments and (ii) These are added to the current values at step k, and the updated totals substituted into (10), which yield estimates of the

updated local stress averages, (iii) A new value of is then found from (21) and the increment found by subtracting The is rounded to correspond to debonding an integer number of reinforcement volumes, (iv) The energy criterion (22) is examined for compliance,

and the increments (i) are adjusted as needed for debonding of a preselected number of reinforcements, (v) The debonding model may yield an updated matrix for the new value of in (8). (vi) Outcomes of steps (iii) to (v), substituted into (10), yield updated local stress aver-

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ages, (vii) Steps (iii) to (vi) are repeated until selected error criteria are satisfied by the updated local stress and magnitudes. Under overall strain control, one selects the increments of and and finds from (14)–(16). The updated local stress and Cd values are then found by retracing (ii)–(v). Next, the strains (13)–(16) are updated and adjusted as required by the selected Again, these iterations continue until selected error criteria are satisfied, as in (vii).

3.

DAMAGE CONTROL BY FIBER PRESTRESS

Damage in fiber composites can be regarded, in part, as a constrained fracture or failure process in the matrix, impeded by undamaged fibers that bridge the prospective cracks and thus limit their extension. Another part of the process involves tensile or compressive fiber failure that can be contained to some extent by the surrounding matrix or plies. In both cases, damage can be prevented or retarded by redistribution of internal stresses induced by fiber prestress, applied prior to and released after matrix consolidation. Prestressing of structural components for improvement of internal stress distributions and overall deflections is widely used in concrete structures, and to a lesser degree in steel structures. However, prestressing of fibers in composite plies and laminates, which offers a similar method for stress redistribution, has been apparently used only in reduction of fiber waviness for enhancement of ply compressive strength. Since the forces needed to apply significant stresses to fiber tows are relatively small, within the capacity of conventional equipment used in filament winding or fiber placement, fiber prestress could be used in many composite structures for damage retardation or control. Examples of several such applications that have been recently identified are briefly described below. In all cases studied, this method of damage control by fiber prestress relies on creating fixed residual stress states that keep local ply stresses within allowable ranges in composite structures exposed to prescribed thermomechanical loading programs. One application of fiber prestress in damage control has been developed for laminated plates, where release of prestress after matrix consolidation is equivalent to application of fixed compressive stresses in the fiber direction of each ply (Dvorak and Suvorov 2000). Analysis of these problems involves application of available laminated plate theories, with the released prestress acting as fixed load, in superposition with applied thermomechanical loads. An example of the prestress effect on estimated onset of damage is shown in Fig. 1. A 9-layer

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S-glass/epoxy laminate is loaded by biaxial normal stresses and with the 0° fibers aligned in the x direction. Initial damage envelopes are constructed from interior branches that reflect ply strength magnitudes estimated by the critical stress criterion on the basis of experimentally measured longitudinal and transverse tensile or compressive and shear strengths of the individual plies. Application of different fiber prestress magnitudes to all plies causes mostly rigid body translation of the envelope in the direction of the prestress force resultant, although different ply strength branches may assume the interior position at different levels of prestress. The outcome is reminiscent of the shakedown effect on positions of loading surfaces of laminates with

kinematically hardening matrices. In a more recent study, we have developed a procedure for evaluation of fiber prestress distributions in individual plies that reduce to prescribed strength ranges the free edge stress concentrations caused in homogenized plies by thermal changes and variable mechanical loads. Also, the procedure accommodates the average stresses in the laminate interior within translated initial damage envelopes. The useful fiber prestress magnitudes are typically limited by the requirement that ply com-

Figure 1 Effect of fiber prestress applied in all plies, and cooling by and position of initial damage envelopes for the 9-ply laminate.

–150°C, on shape S-glass/epoxy

pressive strengths are not exceeded after mechanical unloading, which

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implies that the damage envelopes must contain the origin of overall stress coordinates (Fig. 1). These values are much lower than typical fiber strengths, hence the effect of prestress on fiber stresses is relatively small, especially in systems with compliant matrices. As an example, consider a laminated plate of any layup that is subjected to a uniform thermal change applied during cooling from the fabrication temperature. In-plane moduli and CTEs of all plies need to be identical, and as is the case in most composites, the expansion coefficient of the matrix is assumed to be larger than that of the fiber; hence during cooling a free ply contracts more in the transverse direction. This allows application of fiber prestress that cancels the thermal stress concentrations at the free edges of the laminate, and thus creates a uniform stress and strain field in each ply. To evaluate the required prestress magnitude, the plies are separated, and each ply is subjected to the uniform, stress-free thermal strain. In general, these strains are different in the plane of each ply, so the laminate cannot be reassembled; however, the plies remain plane and parallel to each other. Next, suppose that a certain fiber prestress has been released; this is a product of the fiber volume fraction and tensile stress applied to

the fibers in the ith ply prior to matrix consolidation; the local coordinate system in each homogenized ply is selected such that the fibers are aligned with the direction in the plane of the ply. The prestress magnitude is selected such that after its release, and in superposition with the thermal strain, the elastic strains create an isotropic in-plane strain

where the are ply compliances and the linear coefficients of thermal expansion. The required prestress value in each ply is thus found as

and the uniform strain in all plies after cooling and prestress release as

Note that

the assumption is that > 0 and < 0. Since all plies are deformed by the same isotropic in-plane strain, and are free of tractions, the laminate can be reassembled. The strain (25) is preserved through laminate thickness. Note that neither this strain nor the prestress (24) depends on ply orientation or laminate layup. Of course, local concentrations at the free fiber ends are not eliminated by the prestress.

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CONCLUSIONS

Space limitations prevent description of specific applications of the damage evolution model, of its extension to inelastic composites, and of implications for stability of the damage process. These and other subjects will be described elsewhere. Many other fiber prestress applications in preventing initiation of both distributed damage and free edge debonding also await exposure in a different setting. However, the examples shown should provide some insight into the two research trends that open the way both to a better understanding of distributed damage evolution in heterogeneous systems, and to active control and prevention of damage in laminated composite structures.

Acknowledgments This work supported by a grant from the Army Research Office; Dr. Mohammed Zikry served as program monitor. The author appreciates contributions by Mr.

Alexander P. Suvorov and Mr. Jian Zhang, who also kindly produced the manuscript file.

References Achenbach, J. D., and H. Zhu. 1989. Effect of interfacial zone on mechanical behavior and failure of fiber-reinforced composites. Journal of the Mechanics and Physics of Solids 37, 381–393. Chaboche, J. L., S. Kruch, J. F. Maire, and T. Pottier. 2000. Towards a micromechanics based inelastic and damage modeling of composites. International Journal of Plasticity, to appear. Chen, T., G. J. Dvorak, and Y. Benveniste. 1990. Stress fields in composites reinforced by coated cylindrically orthotropic fibers. Mechanics of Materials 9, 17–32. Christensen, R. N., and K. H. Lo. 1979. Solutions for effective properties of composite materials. Journal of the Mechanics and Physics of Solids 27, 315–330. Dvorak, G. J. 1990. On uniform fields in heterogeneous media. Proceedings of the Royal Society of London A 431, 89–110. Dvorak, G. J., and Y. Benveniste. 1992. On transformation strains and uniform fields

in multiphase elastic media. Proceedings of the Royal Society of London A 437, 291–310. Dvorak, G. J., and A. P. Suvorov. 2000. Effect of fiber prestress on residual stresses and onset of damage in symmetric laminates. Composite Science and Technology 60, 1129–1139. Furuhashi, R., J. H. Huang, and T. Mura. 1992. Sliding inclusions and inhomogeneities with frictional interfaces. Journal of Applied Mechanics 59, 783–788.

Gao, Z. 1995. A circular inclusion with imperfect interface: Eshelby’s tensor and related problems. Journal of Applied Mechanics 62, 860–866. Hashin, Z. 1991a. The spherical inclusion with imperfect interface. Journal of Applied Mechanics 58, 444–449. Hashin, Z. 1991b. Thermoelastic properties of particulate composites with imperfect interface. Journal of the Mechanics and Physics of Solids 39, 745–762.

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Hill, R. 1965. Continuum micromechanics of elastoplastic polycrystals. Journal of the Mechanics and Physics of Solids 13, 89–101. Hutchinson, J. W., and H. M. Jensen. 1990. Models of fiber debonding and pullout in brittle composites with friction. Mechanics of Materials 9, 139–163. Karihaloo, B. L., and K. Vishvanathan. 1985. Elastic field of a partially debonded elliptical inhomogeneity in an elastic matrix. Journal of Applied Mechanics 52, 835–840. Needleman, A. 1987. A continuum model for void nucleation by inclusion debonding. Journal of Applied Mechanics 54, 525–531. Steif, P. S., and A. Dollar. 1988. Longitudinal shearing of a weakly bonded fiber composite. Journal of Applied Mechanics 55, 618–623. Tvergaard, V. 1990. Micromechanical modeling of fiber debonding in a metal reinforced by short fibers. In Inelastic Deformation of Composite Materials, G. J. Dvorak (ed.). Berlin: Springer, 99–111.

ELECTROMAGNETIC PHENOMENA IN CRYSTAL GROWTH John S. Walker Department of Mechanical and Industrial Engineering University of Illinois at Urbana-Champaign, USA [email protected]

Abstract

1.

Integrated circuits and optical devices are produced on wafers sliced from single crystals that are grown from a body of liquid semiconductor or melt. The crystal must have few defects, such as dislocations, and must have uniform distributions of dopants, which are added to the melt to give the crystal the desired electrical or optical properties. Since molten semiconductors have large electrical conductivities, magnetic fields can be used to eliminate hydrodynamic instabilities in the melt and to tailor the convective dopant transport. For silicon, controlling the convective transport of oxygen is important, and a particular nonuniform axisymmetric magnetic field is optimal for this purpose. For compound semiconductors, such as gallium-arsenide, eliminating instabilities in the buoyant convection is necessary for the growth of large crystals with few defects.

INTRODUCTION

Computer processors and memory chips, high-speed optical-fiber communications, lasers and light-emitting diodes, wireless communications, electronic controls for automotive systems, and many other new technologies begin with wafers or substrates sliced from single crystals of semiconductors. These crystals are grown from a body of liquid or melt, and the crystal properties needed for each application depend on the melt motion during crystal growth. Dramatic advances in these new technologies will require radical improvements in the quality of semiconductor crystals through melt-motion control. Most molten semiconductors behave as liquid metals: their electrical and thermal conductivities are large and their viscosities are small. With the large electrical conductivity, (1) a steady magnetic field can be used to eliminate deleterious hydrodynamic instabilities and to control the steady laminar melt motion, or (2) a periodic magnetic field can be used to drive a beneficial melt motion.

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While some effects of the melt motion are common to all semiconductors, other effects are important only for certain materials. There are three types of semiconductor crystals: single-material, compound and solid-solution. A single-material semiconductor crystal consists of an element with four valence electrons, such as silicon or germanium. Silicon is currently the basis of computer processors and memory chips. Single crystals with diameters of 30 cm and lengths of 1 m are currently produced commercially by the Czochralski process. Each crystal yields hundreds of substrates, and many integrated circuits are produced on each substrate. Dopants with three valence electrons, such as boron, may be added to the melt in order to create a missing electron or hole at each location of a dopant atom in the crystal. An electron can jump to this hole, so these dopants are called acceptors and produce a p-type semiconductor. Alternately dopants with five valence electrons, such as phosphorus, may be added to the melt to create an extra electron at the location of each dopant atom in the crystal. Such dopants are called donors and produce an n-type semiconductor. With adjacent pieces of n-type and p-type, a small voltage can cause the donor electrons to jump to the holes or back. If the substrate is n-type, sections of the surface are converted to p-type by various processes to create transistors. Other sections of the surface are coated with metal to carry electrical signals between the transistors or are converted to SiO2, which serves as an electrical insulator. Currently a silicon-based integrated circuit contains about 8 million transistors, while the next generation will contain about 240 million on the same area. With smaller transistors, the time for signals to travel between transistors is reduced, making the computer faster. Optical applications, such as optical-fiber data transmission, optical data storage and retrival, other applications of lasers, light-emitting diodes and photodetectors, are based on compound or solid-solution semiconductor crystals because silicon lacks the optoelectronic properties required for these applications. Compound semiconductors involve equal amounts of elements with three and five valence electrons (IIIV’s), such as gallium-arsenide (GaAs) and indium-phosphide (InP), or of elements with two and six valence electrons (II-VI’s), such as mercurytelluride (HgTe) or cadmium-telluride (CdTe). The elements in compound semiconductors maintain their molecular bond in the liquid state as long as the temperature is below the dissociation temperature, which is a different function of pressure for each material combination. Therefore compound semiconductors grow like single-material semiconductors. On the other hand, solid-solution crystals are grown from liquid mixtures of unassociated elements or molecules. Examples include

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95% silicon and 5% germanium (SiGe), 95% germanium and 5% silicon (GeSi), and 80% mercury-telluride and 20% cadmium-telluride (HgCdTe or Hg0.8Cd0.2Te). Since the elements are not bound in the melt, the melt motion must be controlled so that the convective mass transport during crystal growth produces an axially and radially uniform distribution of both species in the crystal. Compound and solid-solution crystals also provide the substrates for the integrated circuits in cell phones and other wireless communication devices, and for high-temperature applications. For example, on automotive engines, silicon-based sensors and control circuits require expensive cooling, while GaAs-based devices operate well at high temperature and require no cooling. It is much more difficult to produce compound and solid-solution crystals with uniform dopant and species distributions, and with very few dislocations or other crystal defects than it is for silicon crystals. The poorer quality of these crystals places severe limits on many important technologies. Ten years ago, GaAs was expected to replace silicon in the fastest supercomputers because an integrated circuit on a GaAs substrate would perform at roughly ten times the speed of the same circuit on a silicon substrate. However, the same integrated circuit cannot be produced on both materials because of nonuniformities and defects in current GaAs crystals. At about the time silicon-based integrated circuits reach 240 million transistors, GaAs-based integrated circuits may reach 1.5 million transistors. With the shorter distance between transistors, silicon-based computer chips remain faster than GaAs-based ones. However, silicon cannot be used for the technologies which are currently revolutionizing communications, so that there is a great need to improve the quality of compound and solid-solution crystals. Without flow control, hydrodynamic instabilities lead to unsteady melt motions in virtually all crystal-growth processes. In the Czochralski growth of 30 cm diameter silicon crystals, the initial melt volume is very large and the temperature difference in the melt is 50–100 K, so that the uncontrolled buoyant convection is turbulent. For the smaller melt volumes involved in the growth of compound and solid-solution crystals, the uncontrolled melt motion is generally not turbulent, but it is cer-

tainly unsteady and involves enough different frequencies that the effects of melt-motion oscillations appear to be rather chaotic. The convective heat transfer associated with unsteady melt motions produces fluctuations in the heat flux from the melt to the crystal-growth interface, and these heat-flux fluctuations produce temporal oscillations in the solidification rate. Oscillations in the solidification rate: (1) are a major cause of dislocations in the crystal (Kuroda et al. 1984) and (2) produce spatial oscillations of the dopant or species concentration in the crystal

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(Garandet 1993, Wilson 1980). Most dopants are rejected into the melt during solidification, and the amount of excess dopant in the melt adjacent to the growth interface depends on the balance between the rate of rejection and the rate of diffusion through the melt. The part of the crystal grown during an increase in the heat flux from the melt has a lower dopant concentration because the rejection rate decreases with the solidification rate, so that diffusion gains in the balance and carries more rejected dopant away from the interface. The part of the crystal grown during a decrease in the heat flux from the melt has a higher dopant concentration because the rejection rate increases with the solidification rate, so that rejection gains in the balance and less rejected dopant can escape by diffusion before being solidified. When a section of the crystal is etched, the spatial oscillations in dopant or species concentration appear as lighter and darker bands, so they are called striations. Both the wavelength and amplitude in these oscillations often appear to be rather chaotic, due to the superposition of several frequencies in the periodic laminar melt motion. Without melt-motion control, a local maximum dopant concentration may be twice the local minimum, while an average distance between adjacent maxima on an etched section is generally a few micrometers. When there were only a few thousand transistors in an integrated circuit, each transistor spanned many striations, so that its function was determined by a local average and striations were not a major problem. Now each transistor is so small that it may

fall on a maximum or minimum in the local striation, and this can drastically alter its performance, which depends on the density of donors or acceptors. When a steady magnetic field is applied during crystal growth, any motion of the electrically conducting melt across the magnetic field produces an electric current, which interacts with the magnetic field to produce an electromagnetic (EM) body force opposing the melt motion. A sufficiently strong steady magnetic field can eliminate hydrodynamic instabilities and can thus eliminate the striations produced by unsteady melt motions. In addition a properly tailored steady magnetic field may control the convective mass transport in the melt in order to produce axially and radially uniform dopant and species distributions in the crystal. While a steady magnetic field damps the melt motion, a periodic magnetic field may produce an EM body force that drives a melt motion. With a cylindrical melt, a rotating magnetic field (RMF) produces an azimuthal EM body force, while a travelling magnetic field (TMF) produces a meridional EM body force with radial and axial components. In the next two sections we discuss some applications of steady and periodic magnetic fields.

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STEADY MAGNETIC FIELDS

When the Sony Corporation obtained a patent on the Czochralski growth of silicon with a steady magnetic field in 1980, there was a general hope that any strong steady magnetic field would produce wonderful crystals with no striations and very low dislocation densities. Experiments revealed that the magnetic damping that eliminates the hydrodynamic instabilities may also produce undesirable effects in the crystal. There are two types of deleterious effects of a steady magnetic field. First a magnetic field may produce a deviation from axisymmetry in the melt motion, temperature, and dopant distribution, or it may amplify the effects of an intrinsic deviation from axisymmetry. Since the crystal is rotated about its vertical centerline, a steady nonaxisymmetric temperature distribution leads to temporal oscillations in the solidification rate at a point on the rotating crystal-growth interface, and thus leads to striations for a steady melt motion. Since one or two striations are produced with each revolution of the crystal, the spacing between light and dark bands in an etched section of the crystal is very uniform, and these striations are called rotational striations. The second type of deleterious effect from a steady magnetic field is that it may damp a beneficial part of the melt motion more that it damps a deleterious part, and thus leads to radial or axial variations in the dopant or species concentration in the crystals, called radial or axial macrosegregation, respectively. For examples of both types of deleterious effects and the tailoring of the steady magnetic field to achieve the beneficial elimination of hydrodynamic instabilities without any negative effects, we focus on the Czochralski (CZ) growth of silicon crystals. In the CZ process, the melt is contained in an open cylindrical crucible. Crystal growth begins when a single-crystal seed touches the center of the melt’s free surface. The diameter is increased from the seed diameter to the desired crystal diameter, and then the crystal is moved vertically upward so that the diameter remains constant and so that the growth-interface remains close to the horizontal plane of the free surface. The unique aspect of the CZ growth of silicon crystals is oxygen contamination because the melt at more than 1700 K dissolves the quartz (SiO2) crucible. To achieve acceptably low oxygen concentrations in the crystal, 95–98% of the oxygen from the crucible must evaporate as SiO from the free surface. The buoyant convection involves upward flow near the vertical crucible wall, radially inward flow beneath the free surface, downward flow beneath the growth interface, and radially outward flow near the crucible bottom. The melt accumulates oxygen as it travels outward along the crucible bottom and upward along the vertical cru-

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cible wall. Since the diffusion coefficient for oxygen in molten silicon is very small, namely 10–8 m2/s, the oxygen-rich melt must flow very close

to the free surface in order to lose enough oxygen to keep the crystal concentration low. The thermocapillary convection due to the increase in the surface tension of the free surface from a minimum at the hot vertical crucible wall to a maximum at the colder periphery of the crystal-growth

interface is very important because it pulls the radially inward streamlines very close to the free surface. If the crystal is not rotated about its vertical centerline, there is much more oxygen at the center of the crys-

tal than at its periphery where oxygen concentrations are low because of evaporation from the free surface. Therefore the crystal is rotated at 5–25 rpm so that the axial variation of the azimuthal velocity overwhelms the radially inward buoyant convection and produces a radially outward melt motion over the entire crystal-growth interface. The associated convective mass transport leads to radially uniform oxygen and dopant distributions in the crystal. The first experiments and models for the CZ growth of silicon with a steady magnetic field focused on uniform fields that were either vertical (axial) or horizontal (transverse). Since the EM body force opposes the velocity components perpendicular to the magnetic field but not the

component parallel to the field, a uniform axial magnetic field provides more damping of radial velocities than of axial ones. Since the beneficial thermocapillary convection and radially outward flow due to crystal

rotation are primarily perpendicular to an axial magnetic field, they are strongly damped, while the upward buoyant convection near the vertical

crucible wall is not damped as much and continues to convect oxygen from the crucible to the melt and crystal.

Therefore silicon crystals

grown by the CZ process with an uniform axial magnetic field have no striations, but they have average oxygen concentrations that are several times the maximum acceptable level and they have extreme radial macrosegregation with much higher oxygen and dopant concentrations near the center than at the periphery (Ravishankar et al. 1990). Hurle and Series (1994) estimated that a crystal rotation rate of 700 rpm would be required to produce a radially uniform oxygen distribution in a silicon crystal grown in an axial magnetic field with a magnetic flux density B = 0.2 T, while 30 rpm is roughly the maximum practical rotation rate.

A uniform transverse magnetic field produces less damping of the ther-

mocapillary convection and of the radially outward flow due to crystal rotation and more damping of the upward buoyant convection near the vertical crucible wall. Hence oxygen levels in the crystal are very low and can be adjusted by varying the crucible rotation rate. Unfortunately

any steady uniform transverse magnetic field that is strong enough to

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eliminate hydrodynamic instabilities also produces large deviations from axisymmetry in the melt motion and hence in the temperature, dopant, and oxygen through nonaxisymmetric heat and mass transport. Thus crystals grown with uniform transverse magnetic fields have severe rotational striations (Ravishankar et al. 1990). A certain nonuniform axisymmetric magnetic field called a cusped field was first proposed independently by Series (1989) and by Hirata and Hoshikawa (1989). This field is produced by two identical solenoids placed symmetrically above and below the horizontal plane of the free surface and crystal-growth interface. The two solenoids carry electrical current in opposite azimuthal directions so that their axial magnetic fields cancel at the plane of the free surface and growth interface. Since the local magnetic field is radial, it provides only weak damping of the beneficial thermocapillary convection and radially outward flow due to crystal rotation. The fringing magnetic field lines in the melt involve significant normal components at both the bottom and vertical wall of the crucible, so that the local buoyant convection is strongly damped, which reduces the transport of oxygen from the crucible to the melt and crystal. A cusped field leads to low oxygen concentrations in the crystal, radially uniform oxygen and dopant distributions, and no striations. As the diameter of a commercial silicon crystal increased from 8 cm in 1985 to the current 30 cm, steady magnetic fields became progressively more important for the elimination of turbulence and all other unsteadiness in the melt motion during the CZ growth of silicon. Compound III-V semiconductor crystals are also grown with the CZ process. Two differences from silicon are that: (1) the sizes of the crystal and crucible are much smaller since the diameter and length of a large InP crystal are both 10 cm, and (2) there is a layer of liquid boron oxide in the annular region between the periphery of the crystal and the vertical crucible wall above the melt. This liquid encapsulant prevents the evaporation of the volatile component, i.e., arsenic in GaAs and phosphorus in InP. The melt does not dissolve the quartz crucible so that there is no contamination. The primary concerns are (1) dislocations and other crystal defects and (2) the uniformity of the dopant distribution in the crystal. All of the dopant in the melt before the start of crystal growth ends up in the crystal or in the small amount of melt left in the crucible at the end of the process. Compared with silicon, compound semiconductor crystals are much more prone to the development of dislocations and other defects, such as twinning. Twinning occurs when a part of the crystal begins to grow with a different crystallographic orientation, i.e., one crystal twins into two crystals. For GaAs and InP and other III-V crystals grown by the liquid encapsulated Czochralski

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(LEC) process, most experiments with magnetic fields have focused on the dramatic reduction in dislocation densities, the eliminating of twinning, and the elimination of unsteady melt motions with a uniform axial magnetic field with B = 0.2–0.5 T. Bliss et al. (1993) have developed a process to grow large InP crystals by a slight modification of the LEC process with a uniform axial magnetic field. The crystals have very low and radially uniform dislocation densities and no twinning. Unfortunately many GaAs and InP crystals grown by the LEC process with an axial magnetic field have rotational striations. While a uniform axial magnetic field does not produce any deviations from axisymmetry, it can amplify the deviations from axisymmetry in the melt motion and dopant distribution due to deviations from axisymmetry in the heat flux from the external heater. Many heaters have a gap at one azimuthal position where the leads are connected and there is less heat flux at this location. With induction heating, a slight eccentricity between the induction coil and the graphite susceptor holding the crucible can lead to a large deviation from axisymmetry in the heat flux into the melt. Without a magnetic field, crucible rotation at a few rpm is considered sufficient to eliminate the adverse effects of nonaxisymmetric heaters. For the purposes of explanation we split the three-dimensional buoyant convection in the melt into an axisymmetric convection, i.e., the azimuthal average, and a nonaxisymmetric convection, i.e., the threedimensional convection minus its azimuthal average. A uniform axial magnetic field produces strong damping of the axisymmetric convection because the radial components of this motion produce azimuthal electric currents, which in turn produce large EM body forces opposing the radial motion. The nonaxisymmetric convection can involve upward and downward flows near the hotter and colder azimuthal locations, respectively, and large azimuthal velocities near the vertical crucible wall from the hot to the cold azimuthal positions in the upper part of the crucible and from cold to hot in the lower part. There is very little EM damping of this nonaxisymmetric convection because the upward and downward flows are parallel to the magnetic field, and the electrically insulating vertical crucible wall blocks the radial electric currents that would produce the EM body force opposing the large azimuthal velocities near this wall. Therefore, with a uniform axial magnetic field, a small deviation from axisymmetry in the heat flux to the melt can lead to a large deviation from axisymmetry in the magnetically damped melt motion, leading in turn to rotational striations through nonaxisymmetric dopant transport. It appears that these rotational striations can be eliminated by insuring that the heat input is axisymmetric.

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Most solid-solution crystals are grown by the vertical Bridgman process, which uses a three-zone furnace with a vertical centerline. The top hot zone has a temperature above the solidification temperature, and the bottom cold zone has a temperature well below the solidification temperature. The middle zone is often thermally insulated so that it carries an axial heat flux from the hot zone to the cold zone. The two species and any dopant are contained in a sealed cylindrical ampoule which is initially moved along the furnace centerline into the hot zone until all the semiconductor material is melted except the single-crystal seed at the bottom of the ampoule. Then the ampoule is moved slowly downward through the middle adiabatic zone and into the cold zone, so that the crystal grows upward through the ampoule from the seed at the bottom. A conical insert guides the increase in diameter from that of the seed to that of the ampoule. During solidification, the heavier of the two species is generally rejected into the melt, i.e., Ge in GeSi or SiGe and HgTe in HgCdTe. Since the difference between the thermal conductivities of the melt and crystal leads to a crystal-growth interface that is concave into the crystal, the heavier rejected species sinks to the lower center of the growth interface. Since the solidification temperature decreases as the concentration of the heavier species increases, the growth interface becomes more concave into the crystal with a higher solidification temperature at the periphery than at the center. The increase in the slope of the growth interface accelerates the sinking of the rejected heavier species. As a result, almost all solid-solution crystals grown without a magnetic field have severe radial macrosegregation with much more of the heavier species at the center and much more of the lighter species at the periphery. Watring and Lehoczky (1996) and Becla et al. (1992) grew solid-solution crystals by the vertical Bridgman process with very strong axial magnetic fields with B = 4–5 T. Such a strong field provides enough EM damping of the radially inward melt motion with the higher concentration of the heavier species that the rejected species is solidified before there is significant radial migration. There was far less radial macrosegregation in these crystals than in crystals grown by the

same process without a magnetic field. While good solid-solution crystals are harder to produce than good compound semiconductor crystals, the potential advantages of solid-solution crystals are great. Each compound semiconductor has a limit on how much its properties can be changed by varying dopant concentrations, while the properties of solidsolution crystals can be varied much more by changing the percentages of the two species. The application of strong steady magnetic fields during the vertical Bridgman growth of solid-solution crystals appears to be the key to achieving both radially and axially uniform compositions and

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to reducing the number of dislocations, but far more research is needed to realize the potential benefits of magnetic damping for solid-solution crystals.

3.

PERIODIC MAGNETIC FIELDS

A rotating magnetic field (RMF) is produced by connecting the successive phases of a multiphase AC power source to a number of inductors at equally spaced azimuthal positions around a crystal-growth process. An RMF is a transverse magnetic field with an essentially fixed spatial pattern that rotates about the centerline of the process with an angular velocity which depends on the frequency of the AC power source and the ratio of the number of inductors to the number of phases in the power source. Davidson and colleagues (1987, 1992, 1995, 1999) have defined the fundamental phenomena in the melt motions driven by an RMF. The characteristic ratio of the induced magnetic field produced by the electric currents in the melt to the applied magnetic field produced by the external inductors is the shielding parameter where and are the magnetic permeability and electrical conductivity of the melt, while R is the melt radius, i.e., the inside radius of the crucible or ampoule. For a 50 or 60 Hz power source, the radius of a vertical Bridgman ampoule is small enough that is small and the induced magnetic field can be neglected. On the other hand, for the CZ growth of 30 cm diameter silicon crystals, the crucible radius is large, so that is not small, and the induced magnetic field is important. Two other important parameters are the magnetic Taylor number, and the interaction parameter, where and are the density and dynamic viscosity of the melt, while B is the characteristic magnetic flux density of the RMF. For all current applications of RMFs in crystal-growth processes, the magnetic fields are very weak, B = 1–10 mT, the values of N are extremely small, and the values of are large. An important combination of these two parameters is which is small for all current crystal-growth applications. The first consequence of the assumption that 1 is that the induced electric field due to the melt motion across the instantaneous magnetic field is negligible in Ohm’s law. As a result, the electric current, electric field, and magnetic field are independent of the melt motion, and their solutions determine the EM body force on the melt. Therefore the melt motion is not a magnetohydrodynamic flow with an intrinsic coupling between the electric current and the velocity. Instead it is an ordinary hydrodynamic flow that is driven in part by a known body force. The EM body force consists of a steady axisymmetric

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body force, plus a force that varies as where t is time and is the azimuthal coordinate. The second consequence of the assumption that is that the temporally and azimuthally periodic part of the EM body force can be neglected, so that the only important part of the EM body force is steady and axisymmetric. For the only non-zero component of this force is in the azimuthal direction. Since the magnetic flux density in an RMF is very small, the RMF has no direct effect on hydrodynamic instabilities. Since an RMF creates an azimuthal flow in the melt, there is a stabilizing effect through the axial variations of the centripetal acceleration created by the convection of angular momentum due to the radially inward and outward flows in the buoyant or thermocapillary convection. Experiments have indicated that this may increase the frequency of the periodic melt motions, so that the distance between striations and the amplitude of the dopant concentration variations in the striations both become smaller, which is certainly beneficial (Dold and Benz 1999). If an RMF is applied and the crystal is not rotated, the axial variation of the centripetal acceleration drives a radially inward melt motion near the crystal-growth interface, which is generally undesirable. This is easily corrected by rotating the crystal in one azimuthal direction and applying the RMF in the opposite azimuthal direction. For the vertical Bridgman process, this involves rotating the ampoule about its vertical centerline. An RMF for drives an azimuthal melt motion, and the axial variations of the centripetal acceleration drive the radial and axial velocities. For which is almost always true for crystal-growth applications, the azimuthal velocity is nearly independent of the axial coordinate except inside Ekman or Bödewadt layers, so that meridional motion due to the RMF becomes relatively weak. A traveling magnetic field (TMF) is also a periodic magnetic field, but it is created by connecting the successive phases of the multiphase AC power source to inductors at successively higher or lower axial locations outside the ampoule or crucible. The assumptions for a TMF are very similar to those for an RMF, but now the steady axisymmetric part of the EM body force is concentrated near the periphery of the melt and is axially upward or downward. Therefore a TMF drives a meridional motion directly and does not create any azimuthal melt motion. The use of TMFs in crystalgrowth processes has only recently been proposed.

4.

CONCLUSIONS

Since molten semiconductors are good electrical conductors, a magnetic field can be used to improve the quality of semiconductor crys-

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tals through electromagnetic control of the melt motion during crystal growth or by driving a beneficial melt motion. Different uses involve

strengths from 0.001 T to 5 T, steady and periodic fields, and uniform and nonuniform fields. The spatial variation of the strength and direction of a magnetic field can be tailored in many different ways through the placement of coils, ferromagnetic materials, and shields. More experimental and modeling studies are needed in order to further optimize the benefits of magnetic fields in various crystal-growth processes.

Acknowledgment The author’s research is supported by the National Aeronautics and Space Administration under Grant NAG8-1453.

References Becla, P., J. C. Han, and S. Motakef. 1992. Application of strong vertical magnetic fields to growth of II-VI pseudo-binary alloys: HgMnTe. Journal of Crystal Growth

121, 394–398. Bliss, D. F . , R. M. Hilton, and J. A. Adamski. 1993. MLEK crystal growth of large diameter (100) indium phosphide. Journal of Crystal Growth 128, 451-445. Davidson, P. A., and J. C. R. Hunt. 1987. Swirling recirculating flow in a liquid-metal column generated by a rotating magnetic field. Journal of Fluid Mechanics 185,

67–106. Davidson, P. A. 1992. Swirling flow in an axisymmetric cavity of arbitrary profile,

driven by a rotating magnetic field. Journal of Fluid Mechanics 245, 669–699. Davidson, P. A., D. J. Short, and D. Kinnear. 1995. The role of Ekman pumping in confined electromagnetically driven flows. European Journal of Mechanics

B/Fluids 14, 795–821. Davidson, P. A., D. Kinnear, R. J. Lingwood, D. J. Short, and X. He. 1999. The role of Ekman pumping and the dominance of swirl in confined flows driven by Lorentz forces. European Journal of Mechanics B/Fluids 18, 693–711. Dold, P., and K. W. Benz. 1999. Rotating magnetic fields: Fluid flow and crystal growth applications. Progress in Crystal Growth and Characterization of Materials 38, 7–38. Garandet, J. P. 1993. Microsegregation in crystal growth from a melt: An analytical approach. Journal of Crystal Growth 131, 431–438. Hirata, H., and K. Hoshikawa. 1989. Silicon crystal growth in a cusp magnetic field. Journal of Crystal Growth 96, 747–755. Hurle, D. T. J., and R. W. Series. 1994. Use of a magnetic field in melt growth. In Handbook of Crystal Growth 2, Bulk Crystal Growth. Amsterdam: North-Holland, 259–285. Kuroda, E., H. Kozuka, and Y. Takano. 1984. The effect of temperature oscillations at the growth interface on crystal perfection. Journal of Crystal Growth 68, 613–623. Ravishankar, P. S., T. T. Braggins, and R. N. Thomas. 1990. Impurities in commercialscale magnetic Czochralski silicon: Axial versus transverse magnetic fields. Journal of Crystal Growth 104, 617–628.

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Series, R. W. 1989. Effect of a shaped magnetic field on Czochralski silicon growth. Journal of Crystal Growth 97, 92–98. Watring, D. A., and S. L. Lehoczky. 1996. Magnetohydrodynamic damping of convection during vertical Bridgman–Stockbarger growth of HgCdTe. Journal of Crystal

Growth 167, 478–487. Wilson, L. O. 1980. The effect of fluctuating growth rates on segregation in crystals grown from the melt, I. No backmelting, II. Backmelting. Journal of Crystal Growth 48, 435–458.

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Between sessions, ICTAM 2000 participants mingle while enjoying coffee and pastries served by Chicago Marriott Downtown wait staff. The Congress filled three

floors of the conference hotel.

SOFTWARE TOOLS: FROM MULTIBODY SYSTEM ANALYSIS TO VEHICLE SYSTEM DYNAMICS Willi Kortüm DLR German Aerospace Center, Institute of Aeroelasticity, Wessling, Germany [email protected]

Werner O. Schiehlen Institute B of Mechanics, University of Stuttgart, Stuttgart, Germany [email protected]

Martin Arnold DLR German Aerospace Center, Institute of Aeroelasticity, Wessling, Germany [email protected] Abstract

1.

After successful application to spacecraft and appropriate theoretical examinations, multibody system (MBS) approaches, their formalisms, and software became of interest to the vehicle system dynamicists both for rail and road vehicles. This introductory paper sketches a few important milestones in the MBS general development related to vehicle system dynamics. While at the beginning the absence of system-specific force laws was the major stumbling block, later on the numerical methods and the efficiency of the formalisms became of prime focus. More recently, the transition from system analysis to system design and optimization, as well as the integration into multidisciplinary computer aided engineering (CAE) of vehicle systems, was and still is a challenge.

MULTIBODY DYNAMICS—THE BEGINNING

The origin of modern multibody dynamics is closely related to the beginning of the space age in the ’50s and ’60s of the 20th century. The development of such space vehicles as spacecraft and orbiting satellites was essentially based on a computer-aided dynamic analysis, in particular for the rotational behavior, see e.g. [1]. To this end multibody systems were reconsidered from a more algorithmic point of view that took into account the increasing efficiency of computers. 225 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 225–238. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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Since then, the term multibody system (MBS) has been used more precisely for a system of a finite number of rigid or flexible bodies and their interconnections. It is supposed that the mass of the system is concentrated in the bodies that are connected by such idealized massless elements as joints, springs, dampers, and actuators [2]. The state of an MBS is described by a finite-dimensional vector q(t) of position coordinates and the corresponding velocities v(t). Based on principles of classical mechanics, such as d’Alembert’s principle, the equations of motion are derived in a systematic way [3]. For a single body these equations may be written down straightforwardly, but for complex MBS the derivation becomes technically very complicated and is most conveniently performed by computers using

multibody formalisms. These formalisms are algorithms to generate the equations of motion for a user-specified MBS in a form that allows an efficient numerical solution. The multibody system approach, multibody formalisms, and the derived software packages became a powerful basis for the kinematic and dynamic analysis of mechanisms. Shortly after the first applications in space technology and beyond its prime academic appeal, multibody dynamics was found to be useful also in vehicle system dynamics both for rail (including magnetically levitated) and road vehicles. This introductory paper sketches a few important milestones in the development of multibody dynamics in general, as well as their particular evolution in the vehicle system dynamics area. In the final part of the paper some new challenges are summarized, which result from the concurrent engineering approach to vehicle system dynamics software.

2.

MULTIBODY SYSTEMS AND VEHICLE SYSTEM DYNAMICS

Essential contributions to multibody dynamics and to the efficient numerical solution of the equations of motion were made during and after the late 1980s—see 3. Nevertheless, as described by two of the authors in a first survey paper in 1985 [4], the MBS approach had already been used successfully in vehicle system dynamics much earlier. All these early approaches were adapted to the restricted computing power in the early 1980s; they were not designed to handle fully nonlinear spatial MBS models with loop-closing constraints that represent today’s stateof-the-art. A typical representative of these early attempts is the software package MEDYNA developed at DLR [5]. It is based on kinematically linearized equations of motion, a restriction that was motivated by the applica-

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tions to rail-guided vehicle system dynamics that includes both rail and magnetically levitated vehicles. Ten years later this type of software reached a deadlock because of the inherently linearized kinematics, but in the beginning this restriction was very helpful since the implementation was substantially simpler compared with a fully nonlinear formalism and furthermore the handling of closed loops in the MBS was strongly simplified. Despite its shortcomings, MEDYNA met a number of nontrivial needs that are typical of applications in vehicle system dynamics: flexible bodies were considered on the basis of a modal approach; the package offered reasonable system specific library elements for force laws and joints (wheel–rail interaction, magnetic levitation force models, tire models); and there was a basic interface to other simulation packages, such as ACSL. Furthermore, MEDYNA offered from the early beginning not only solvers for (nonlinear) time integration but also linear system analysis methods in the time and frequency domain that are especially important for applications in vehicle system dynamics. MEDYNA and other early commercial MBS packages are the starting point of an ongoing development that made MBS software one of the basic analysis tools of modern vehicle system dynamics. This application

to real-life vehicle system dynamics is a complex issue that has been addressed in a number of surveys and reviews, which were necessarily supported by extensive benchmarking activities—see e.g. [6, 7]. There are basic requirements on general-purpose MBS software, such as MBS modelling elements, typical modes of analysis, robust and efficient solvers, software aspects, and user-friendly interfaces. Nevertheless, the application to industrial vehicle dynamic simulation definitely requires a number of additional qualities and functionalities, such as:

a) Additional modeling requirements as, for instance, the consideration of elastic deformations in ride/comfort analysis, tire and wheel–rail contact models, and such other important nonlinear effects as friction, nonlinear and discontinuous force laws, b) Powerful solvers to deal with the high complexity of the models

characterized by many degrees of freedom, highly nonlinear phenomena, stiff equations with or without discontinuities, and control systems that may include digitally sampled data, c) Complex geometries of track and rail profiles (including switches), as well as deterministic, stochastic, and measured irregularities,

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d) The reliability of simulation results that is determined by the status of verification and validation of the models used and of the MBS software itself.

New challenges resulted from the introduction of electronics and control in vehicle systems. Control circuits had to be included into the simulation software, and engineers had to be enabled to work in an MBS simulation environment as well as in a control system analysis and design environment using computer-aided control system design

(CACSD) [8, 9, 10].

3.

ADVANCES IN MULTIBODY SYSTEM THEORY

In parallel with the progress in vehicle system dynamics and its associated simulation software, the basic problems of multibody systems as a topic in applied and computational mechanics attracted and motivated quite a number of research-oriented mechanical engineers as well as applied mathematicians. Also specific problems from such other fields

as robotics, biomechanics, and mechanisms stimulated research groups in MBS. The results of these early efforts and the state of the MBS software were first discussed in a 1977 IUTAM Symposium in Munich [11] and then during the 1985 lUTAM/IFToMM Symposium in Udine [12]. Finally, in 1990 the known MBS software packages of the 1980s were put together in the Multibody System Handbook [13] with contributions from 17 research groups. The topics discussed within the MBS community ranged from the choice of coordinates (absolute, relative, mixed) and the basic mechanical principles in the derivation of the equations of motion (Newton– Euler, d’Alembert, Jourdain), to the most efficient numerical solution strategies (O(N) formalisms, explicit vs. residual formalisms), to software aspects (analytical vs. numerical representation of the equations of motion, user interfaces, standardization of input data). Many important issues in these areas were examined in the research program “Dynamics of Multibody Systems” of the German Research Council (DFG) [14]. A key problem of multibody dynamics is the handling of joints that connect bodies in the MBS and restrict their relative motion. These joints define the topology of the MBS that forms the backbone of all multibody formalisms. The topology is represented by a graph with N knots representing the N bodies of the MBS. The branches of the graph represent the joints in the MBS: two knots are connected by a branch if and only if the corresponding bodies are connected by a joint.

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If this graph has tree structure (tree-structured MBS), then the translational and rotational degrees of freedom of all joints define a natural set of generalized coordinates q that allows the analytical elimination of all constraints such that the equations of motion take the general form of ordinary differential equations (ODEs)

with the symmetric positive-definite inertia matrix M and the vector f of generalized applied and gyroscopic forces [3]. If Cartesian coordinates q are used instead, then the equations result in differential–algebraic equations (DAEs)

The

constraints generate the constraint forces with (t, q) and Lagrangian multipliers At first sight (2) has a higher dimension and a more complicated structure than (1) but the topology of the MBS implies a characteristic sparsity structure in (2) that may be exploited in the numerical solution

[15]. If the graph of the MBS contains loops (MBS with closed loops), then the generalized coordinates have to be evaluated using complex kinematics while the Cartesian coordinates are subject only to additional constraints—see e.g. [16, 17]. By exploiting the MBS topology, the right-hand sides of (1) and (2) may be evaluated with O(N) arithmetic operations (O(N) formalisms) in contrast to a complexity of O for a standard matrix inversion, which is a characteristic of early multibody formalisms. The numerical solution of DAEs (2) was an unsolved problem in the early days of multibody dynamics, so the explicit or implicit transformation to ODEs according to (1) found much interest in the MBS community. Nowadays the time integration methods for DAEs (2) are as highly developed as the ones for ODEs (1) [18, 19, 20]. Note that these robust DAE integrators require also the evaluation of hidden constraints, such as

so the multibody formalisms have to be extended accordingly. Time integration is the most challenging numerical problem in multibody dynamics; but also linearization, linear system analysis in time and frequency domains, as well as kinematic analysis, static analysis, and inverse dynamic analysis, are of importance [19].

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Besides these algorithmic and numerical improvements, multibody dynamics itself extended its classical approach of rigid body systems to flexible MBS with elastic deformation of bodies by using a modal representation [21, 22]. Furthermore the classical concept of equality constraints in (2) was extended to unilateral constraints, which allow a very natural formulation of contact conditions [23].

4.

FROM SYSTEM ANALYSIS TO SYSTEM DESIGN, FROM MECHANICAL MULTIBODY SYSTEMS TO MECHATRONICS

Traditional vehicle dynamics and MBS simulation concentrated on the analysis and the evaluation of given vehicle systems, perhaps including parameter variation and sensitivity studies of purely mechanical systems. More recently, much work has been done to overcome these limitations, grasping the full system with its multidomain (multiphysical) properties and supporting the optimization of system parameters with respect to performance criteria [24] (simulation-based design or design by simulation). Optimization of MBS models has been studied for more than a decade [25], but even today it is not yet available with all standard industrial MBS packages. A fully automated numerical multi-objective parameter optimization of industrial MBS models is provided by SIMPACK, the general purpose MBS software being developed at DLR [26]. With the advent of mechatronics thinking in vehicle systems—see [27] for a survey of current trends and future possibilities—the request for a higher integration of mechanical and control/electronics software packages became more demanding [28]. A key feature of modern MBS software is therefore an interface to the leading CACSD packages MATLAB/Simulink and Matrixx/System Build. Figure 1 illustrates the advanced interface SIMAT [29] between SIMPACK and MATLAB/Simulink, that provides a simple export of system matrices A, B, C, D, a bidirectional function-call interface, code export and code import facilities, and an advanced co-simulation interface including an option for parallel computing using inter-process communication. Originally, these interfaces were developed for and tailored to the coupling of MBS and CACSD software, but they have been extended to the more general integration of mechanical system properties and other physical domains. Simulator coupling [30] is the method of choice for simulating the interaction of mechanical and hydraulic or pneumatic sys-

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Figure 1 Interface between SIMPACK (MBS) and MATLAB/Simulink (CACSD).

tem components. Typical examples are the interfaces of SIMPACK to AMEsim [31] and DSHplus [32]. Furthermore, DLR is currently developing fluid–structure–MBS interfaces to connect aeroelastic and MBS

models [33]. Last but not least, the importance of computer-aided design (CAD) for vehicle system design in general, and in relation to MBS software in particular, should be mentioned. Computer-aided design not only provides high quality graphical output in 3D, but also—even more important—comprises all geometrical data and such other data as material properties, masses, and inertias. Therefore CAD may be considered as the parent database for such other computations as CFD, FE, and MBS analysis. As a consequence, connections between CAD and MBS have been developed [34]. Industrial MBS packages provide advanced interfaces that consider both views: the MBS view to derive MBS data from the CAD source as well as the CAD engineer’s view, which allows MBS functions to be called from the CAD user interface [35].

5.

CURRENT AND CONCURRENT PROBLEM AREAS—A CRITICAL REVIEW AND OPEN PROBLEMS

Industrial MBS software, such as ADAMS, SIMPACK, and DADS, provides powerful tools for vehicle system dynamics. Nevertheless, from time to time, they leave the frustrated user unsatisfied, both in ordinary

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and high-end applications. Therefore, a critical assessment of current problems may be helpful. Fundamental problems result from the fact that by 1985 the basic conceptions were already established for all the MBS packages that are today extensively used in industry. Later developments in multibody dynamics and MBS numerics had to be added afterwards, and these additions proved to be extremely difficult and tedious to manage if their development had not been foreseen in the conception phase. Other problems result from the rapidly increasing demands and increasing complexity of real-life applications in vehicle system dynamics. MBS formalisms and time integration methods. Today it is well known that a numerically stable and robust time integration of DAE (2) requires the evaluation of hidden constraints, such as (3) [18]. In the early days of multibody dynamics [15], nobody knew about that and the multibody formalisms were not prepared to evaluate (3) during time integration. A later redesign to include the necessary information proved to be nontrivial because of the large set of already existing library elements and user routines. For similar reasons, modern non-stiff time integration methods for (2) are not yet available in industrial MBS packages since they require the separate evaluation of M, f and [36] but the formalisms evaluate M, f and simultaneously. Efficiency. There are mainly two sources for efficiency and low simulation times in MBS software. In the widely used MBS packages

the time integration methods have been adapted to the special structure of (1) and (2) and the equations of motion are evaluated efficiently by O(N) formalisms. For MBS with small displacements, further improvements may be achieved by linearizing the kinematics. For nonlinear problems the O(N) formalism may be extended by a pre-processor that eliminates for a fixed MBS topology all arithmetic operations with vanishing terms. See [37] for a comprehensive overview of new and adapted algorithms in SIMPACK, which is today known among the commercial packages for its high efficiency. Further improvements are necessary to handle MBS models with a

few hundred degrees of freedom, which are more and more often used in industry. Flexible MBS.

Elastic bodies in MBS models may cause major difficulties or even wrong simulation results with MBS packages that

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were originally designed for rigid bodies. Only a few industrial packages have taken into account the elastic deformation from the outset. Typical examples are DADS and SIMPACK, which use a modal approach with eigenmodes as well as static modes. Second-order terms have to be included because of geometrical stiffening effects [21, 22, 37]. Eigenmodes and static modes are obtained in a separate pre-processing step using industrial FE packages and interfaces between FEM and MBS. More recent developments are dealing with the inverse connection from MBS to FE analysis, called LOADS in SIMPACK. The results of MBS simulation (states, forces) are transferred to the FE packages to perform a load analysis including such post-processing evaluations as life prediction [38]. Semi-symbolic equations of motion may be also derived using NEWEUL and FE codes [39].

Open software structures. Generally speaking, earlier software packages were designed as closed systems that compute for given input data the simulation results, without any additional output. By using numerical MBS formalisms to evaluate the equations of motion, there was no need to generate these equations explicitly in source code form. Later it was recognized that open software structures, including code export, facilitate interfaces to other CAE packages and real-time applications. Symbolic MBS formalisms have been developed to generate

FORTRAN or C statements representing the equations of motion. Code export for MBS models has the inherent problem that the MBS topology has to be exploited for an efficient numerical solution. This may be done by using such symbolic formalisms as NEWEUL [40, 41] or the Symbolic Code Module of SIMPACK [37], which eliminate all constraints resulting from tree-spanning joints during the generation of symbolic code. Other packages supplement their exported symbolic code by special sparse-matrix solvers, but this approach is less straightforward and is nontrivial to handle. With the increasing importance of hardware-in-the-loop simulations, there is an increasing interest in efficient symbolic code running in real time in a digital signal processor environment [42]. The robust numerical solution of DAEs (2) in real time is currently a field of active research, since standard DAE solvers use iterative methods to satisfy the constraints 0 = (t, q) but real-time simulation has to be based on noniterative algorithms. Underestimating the mechanical problems. Visualization and graphical user interfaces are definitely important features, since they support users in their daily work and allow a presentation of simula-

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tion results in a form that is also easily understandable for non-experts. But some packages overemphasize the “show” to impress potential customers and seem to neglect the difficulties in the modeling and solution of complex problems. A typical example is the wheel–rail problem with its intrinsic and complex contact geometry and creep-force laws—and the resulting numerical problems. The difficulties caused by the strongly coupled problems of modeling and numerical solution are discussed in [43], the background for the Wheel-Rail Module of SIMPACK, including multiple contact problems and switching maneuvers of railroads. Automotive. The automotive industry is one of the main customers of MBS software. To create accurate and reliable vehicle models quickly and easily, the packages offer special parameterized substructures for such model components as wheel suspensions, tires, and engines. Furthermore the pre- and post-processing is supported by specialized model setup and evaluation tools that allow easy access to standardized driving maneuvers and other typical analysis methods [44]. Without such extensions, an MBS package is of very restricted use for engineers in the automotive industry, and further developments are necessary to meet the demands of these users even better. Open problems and future developments. The solution of the above-mentioned problems, such as robustness, efficiency, and handling of MBS packages, is of prime importance for industrial vehicle system dynamics. But there are also a number of more basic ongoing developments that will definitely have a strong impact on these analysis tools. Multibody system software has been used in industry for more than 15 years. There is a huge knowledge of how to set up MBS models so that they may be analyzed efficiently, how to handle error messages and problems, and how to use these tools most efficiently. This expert knowledge has to be built in. Solver-independent pre- and post-processors, such as MotionView for applications in the automotive industry, make the user independent of the selected MBS package. To support this strategy, standards have to be defined and implemented for all basic MBS modeling elements and data structures. More and more often, engineers use pre-defined substructures to set up complex MBS models. Further improvements in numerical solvers are necessary to guarantee nevertheless acceptable simulation times. The analysis of nonlinear elastic structures with large rigid-body motion is a topic of very active academic research that may result in a much

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closer connection between structural dynamics and multibody dynamics in the near future—see e.g. [45, 46, 47]. Today it is an open question if there will be software that provides the full functionality of FEM as well as MBS software in one and the same industrial package. Simulator coupling is a promising approach for incorporating MBS simulations in a general CAE environment. Standard coupling strategies may, however, reduce the efficiency of time-integration methods; they may even become unstable. Improved numerical coupling strategies and extended co-simulation interfaces are a topic of actual research—see

e.g. [30, 48]. A natural extension of coupled software tools in simulation is a coupling to optimize a complex technical system with respect to some performance criteria, i.e. multidisciplinary design optimization [49, 50].

6.

SUMMARY

The developments in multibody dynamics and vehicle system dynamics have been intertwined for more than 20 years. Today, industrial MBS packages are powerful and important analysis and design tools in vehicle dynamics that are well prepared for the demands of concurrent engineering processes in industry. Further improvement of the software tools requires fundamental research, which is provided by the multibody dynamics community.

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geschlossenen kinematischen Schleifen. Fortschritt-Berichte VDI Reihe 11, Nr. 167. Düsseldorf: VDI–Verlag. [17] Blajer, W., W. Schiehlen, and W. Schirm. 1994. A projective criterion to the coordinate partitioning method for multibody dynamics. Archive of Applied Mechanics 64, 86–98.

[18] Hairer, E., and G. Wanner. 1996. Solving Ordinary Differential Equations—II. Stiff and Differential–Algebraic Problems, 2nd ed. Berlin: Springer–Verlag.

[19] Eich-Soellner, E., and C. Führer. 1998. Numerical Methods in Multibody Dynamics. Stuttgart: Teubner–Verlag.

[20] von Schwerin, R. 1999. MultiBody System SIMulation—Numerical Methods, Algorithms, and Software. Berlin: Springer. [21] Wallrapp, O. 1991. Linearized flexible multibody dynamics including geometric stiffening effects. Mechanics of Structures and Machines 19, 385–409.

[22] Schwertassek, R., and O. Wallrapp. 1999. Dynamik flexibler Mehrkörpersysteme. Braunschweig: Vieweg.

[23] Pfeiffer, F., and Ch. Glocker. 1996. Multibody Dynamics with Unilateral Contacts. New York: John Wiley and Sons.

[24] Wentscher, H., and W. Kortüm. 1996. Multibody model-based multiobjective parameter optimization of aircraft landing gears. Proceedings of the

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[38] Dietz, S. 1999. Vibration and Fatigue Analysis of Vehicle Systems using Component Modes. Fortschritt-Berichte VDI Reihe 12, Nr. 401. Düsseldorf: VDI– Verlag.

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differential–algebraic systems. BIT Numerical Mathematics 41, 1–25. [49] Eschenauer, H., J. Koski, and A. Osyczka, eds. 1990. Multicriteria Design Optimization. Berlin: Springer–Verlag. [50] Schiehlen, W. 1997. Multibody system dynamics: Roots and perspectives. Multibody System Dynamics 1, 149–188.

GRANULAR SEGREGATION IN COLLISIONAL SHEARING FLOWS Michel Y. Louge, James T. Jenkins, Haitao Xu, and Birgir Ö . Arnarson Cornell University, Ithaca, New York, USA [email protected] Abstract

1.

This paper considers segregation in collisional granular shearing flows from the experimental, computational, and theoretical standpoints. We focus on a phenomenon of segregation where the separation of grains by size or mass is driven by spatial gradients in the fluctuation energy of the grains. We report experiments carried out in microgravity with a shear cell shaped as a race track and containing a mixture of two types of spherical grains. In those experiments, a gradient of fluctuation energy was produced between an inner moving boundary driving collisions among the grains and an outer, more dissipative boundary at the periphery of the cell. The grain segregation and the velocity statistics were captured by a rapid video camera and analyzed using computer-vision software. We briefly outline a kinetic theory and simulations for these flows. We compare the corresponding profiles of granular concentration, mean velocity, and fluctuation energy with the experimental results.

INTRODUCTION

The size segregation of flowing or shaken grains is a commonly observed phenomenon in industrial processes and in nature. In a gravitational field, flows are generally dense and dominated by enduring contacts; their segregation mechanisms are diverse and complex (Savage and Lun 1988, Haff and Werner 1986, Rosato et al. 1986, 1987). In reduced gravity, collisional flows become possible and the segregation is driven mainly by spatial gradients in the energy of granular velocity fluctuations. In steady, fully-developed flows, the balance of momentum exchanged in collisional interaction among different species of grains requires that gradients of particle fluctuation energy be balanced by concentration gradients, thus giving rise to segregation (Jenkins and Mancini 1989). 239 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 239–252. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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To observe collisional segregation driven by a fluctuation energy gradient in the direction perpendicular to the flow, we constructed a shear cell in the form of a race track in which the segregation of a sheared binary mixture of two different types of sphere is maintained by the motion of the inner boundary. We conducted experiments on NASA’s KC-135 microgravity aircraft. To analyze the resulting images, we developed appropriate vision software. We then compared the profiles of volume fraction and mean and fluctuation velocities with computer simulations of the apparatus. The link between the computer simulations and the physical experiments was achieved by measuring the parameters that characterize individual impacts (Foerster et al. 1994). To interpret the results of the simulations and experiments, we solved equations governing the spatial variation of the mixture fluctuation energy and velocity. For simplicity, we employed the approximate treatment for the mixture segregation and the transport coefficients of Arnarson and Jenkins (2000), and carried out the averaging of Jenkins and Arnarson (2000) to capture the effects of side walls. We begin with a brief outline of the theory. Then, we summarize the principle of the simulations, describe the experiments, compare the results, and discuss their significance.

2.

SKETCH OF THE THEORY

Because physical experiments inevitably involve boundaries, their interpretation must be conducted with a theory involving these. We briefly outline such a theory for a steady, fully-developed, rectilinear, unidirectional flow in a rectangular cross section bounded by two bumpy boundaries and two flat, frictional walls. In the absence of an interstitial gas, the steady, fully-developed momentum balance in the flow direction is

where is the granular stress tensor. The directions x, y, and z are shown in Fig. 1. The origin is located midway between the flat side walls at an ordinate above the centers of the stationary boundary bumps, where is the sum of the two species radii and d is the bump diameter. Because the kinetic theory follows the center of interior spheres, its domain occupies a narrower range than the physical experiment, namely and With bumps of uniform size, and

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Figure 1 Sketch of the bottom half of the cell. Its straight section (I) has length L, width Y between centers of cylindrical boundary bumps and depth Z between the two flat side walls. Bumps are affixed to both boundaries, including the curved regions (II) and (IV). The upper straight section (III) and the other halves of regions (II) and (IV) are omitted for clarity. Fully developed flows are simulated using a periodic boundary condition in the observation region with to scale.

Dimensions are not

The equation governing spatial variations of the mixture fluctuation energy is

where is the volumetric rate of collisional dissipation and energy flux. The mixture velocity u along the flow is given by

where and velocity of species is given by

is the

are the material density, volume fraction, and mean respectively. Similarly, the mixture temperature T

where is (2/3) the average fluctuation kinetic energy of species (Jenkins and Mancini 1989), is its number density, is its mass, and The shear stress is related to the velocity gradient through the viscosity In fully-developed flow,

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Similarly, the flux is given by

where is the coefficient of thermal conduction. In this work, we adopt the simplified forms of the transport coefficients calculated by Arnarson and Jenkins (2000) for slight differences and in the radii and masses of the two species,

and

where n is the mixture number density, is the sum of masses, and G, J, and M are functions of the mixture volume fraction

In Eqns. (7) and (8), is the number fraction of species A and the overbar denotes its average value in the cross section. For nearly elastic spheres, the rate of dissipation per unit volume is

where, following Zhang (1993), we incorporate the dissipation due to a modest friction in collision between grains using an effective restitution coefficient on the order of

Finally, the mixture pressure is

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Our experience with the simulations is that the flat side walls play a relatively minor role in this flow. Thus, rather than carrying out a numerical integration of Eqns. (1) and (2), we follow Jenkins and Arnarson (2000) by first integrating these along z. For simplicity, we assume that the mixture kinetic energy and species volume fractions are uniform in that direction, that the average along z of the square of the shear stress is equal to the square of its average, and that the shear stress is proportional to z and vanishes at the centerline by symmetry. In that case, Eqns. (1) and (2) reduce to

and

where In these expressions, brackets denote the averaging along z; they are omitted for those variables which, from our assumptions, are constant along that direction. The superscripts + and – represent quantities at the flat side walls at and respectively. Jenkins (1992) and Jenkins and Louge (1997) derived expressions for, respectively, the granular stress and the granular flux of fluctuation kinetic energy. If all spheres are sliding at the side wall,

and

where is the Coulomb friction coefficient of spheres impacting the walls and is a known function of volume fraction and wall impact parameters. When the spheres roll rather than slide, the stresses and fluxes are functions of the mean relative velocity of the contact point of the grains with the wall, which we evaluate in terms of the mean center of mass velocity assuming that the granular spin equals half the granular vorticity. At the bumpy boundaries, we employ the conditions derived by Jenkins, Myagchilov, and Xu (2000), who refined the conditions calculated by Richman and Chou (1988) for smooth bumps by adding nonlinear corrections in the slip velocity that remain accurate when the latter becomes large. We incorporate frictional interactions on the bumps in the simple way proposed by Jenkins and Arnarson (2000). Then, the

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boundary conditions for the temperature are found from a balance of fluctuation energy at the stationary wall,

and at the moving wall,

where and are complicated functions of the parameters of impact with the bumps, the stress ratio and the wall bumpiness

which is an average measure of the penetration of a sphere of either species in bumps of spacing s between adjacent cylinder edges. Similarly, the velocity boundary conditions are derived from a momentum balance near the stationary boundary,

and the boundary moving with speed U,

where and are complicated functions of bump impact parameters, wall bumpiness and relative velocity. To capture the segregation, we adopt the one-dimensional form of the simplified transport equation for proposed by Arnarson and Jenkins (2000). In the absence of gravity,

where

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and

Here, we determine lateral profiles of T, and by solving Eqns. (15), (16), and (24) numerically subject to conditions (19), (20), (22), and (23), and for given values of the mean volume fraction and mean number fraction

3.

SIMULATIONS

The simulations follow the dynamics of an ensemble of two species of spheres interacting with the boundaries and among themselves through individual impacts. They are based on the algorithm described by Hopkins and Louge (1991). In that algorithm, collisions occur when a sphere overlaps slightly with another sphere or with the wall. The algorithm adjusts its time step periodically to ensure that the mean overlap is kept below a negligible tolerance. In addition, a search grid is superimposed on the flow domain to permit fast identification of near neighbors. Because this method makes it superfluous to maintain a list of future impacts, its computing time is merely proportional to the number of spheres and, consequently, it can simulate the entire shear cell on a relatively small workstation. Profiles of solid volume fraction, velocity, and temperature for the two grain species are measured by dividing the flow domain into a number of averaging slices (Fig. 1). The average value of an intrinsic grain property in a slice is calculated by considering a number of instantaneous realizations of the flow, by summing all contributions from spheres passing through the slice over all realizations, and by dividing the result by and This center-averaging is consistent with the theory outlined earlier. In the absence of gravity, all grain velocities scale with the velocity U of the boundary. In addition, we make the fluctuation energy dimensionless with and Dimensionless quantities are denoted by a dagger (†). Because any rigid boundary tends to order spheres in its neighborhood, the transverse profiles of solid volume fraction exhibit spatial oscillations near the bumpy inner and outer walls with wavelength on the order of a sphere diameter. Because the kinetic theory ignores these fluctuations, their presence can hinder comparisons of the measured segregation with the corresponding theoretical predictions. In grain mixtures, we alleviate this difficulty by focusing on the relative number fraction rather than the volume fractions of each individual species. For example,

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for species A,

Values of above unity indicate a local surplus of species A spheres. As long as the radii of the two species are not greatly different, the relative number fractions incorporate ratios of volume fractions that nearly oscillate in phase and, consequently, their own spatial fluctuations are considerably reduced. In the simulations, impacts are characterized in terms of three coefficients (Walton 1988). The first is the coefficient of normal restitution e. The second is the coefficient of friction for sliding impacts. The last is the coefficient of tangential restitution for impacts that do not involve sliding. For collisions between two free spheres or between a sphere and one of the fiat side walls, we measured these parameters with the facility described by Foerster et al. (1994); for collisions with a cylindrical boundary bump, we adopted the method of Lorenz et al. (1997). Table 1 summarizes the impact parameters.

4.

EXPERIMENTS

The apparatus is shaped as a race track. Its straight sections have dimensions L = 419 mm, Y = 29 mm and Z = 40 mm; they permit granular segregation in a rectilinear shearing flow without significant body forces. The grains are recycled through circular regions of a similar cross section and an inner radius of 62 mm. The moving boundary consists of a chain to which closely spaced hemi-cylindrical stainless steel rods with d = 3.2mm are affixed. The chain is driven by a direct-current motor connected to one of the sprockets through a timing belt.

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The surface of the outer stationary boundary is made bumpy with similar hemi-cylinders. In the curved regions, these cylinders are closely

spaced (s = 0). In the upstream half of the straight regions, the separation gradually increases to s = 1.6mm, and it remains at this value in the downstream half where observations were carried out. The texture of the stationary boundary is meant to maintain the grains as agitated as possible in the curved sections, while providing a smooth transition to a steep temperature profile in the neighborhood of the observation window.

Flat side walls provide lateral containment and allow the flow to be viewed through glass windows. A digital Kodak EktaPro R0 camera images a 25 mm wide region of the cell with abscissa from the upstream sprocket axis in the range where = 292 mm and

326 mm. The spheres exhibit excellent sphericity and narrow size distribution. Their finish allows the vision software to track their movement

with accuracy. Louge et al. (2000) describe the computer image analysis. A typical image is shown in Fig. 2. As long as the displacement between two consecutive frames remains less than a sphere radius, this method

Figure 2 A typical image for the conditions of Test I. Circles and lines are superimposed to indicate the location and trajectory of detected spheres. The moving boundary can be discerned at the bottom of the picture.

permits unambiguous tracking of the moving spheres. Adequate lighting is an important condition for success of the computer vision. The field of view is illuminated with two optical-fiber light guides. The shutter speed of the camera, the opening of the lens, and the illumination level

are tuned by trial and error until grains are reliably detected.

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The vision algorithm typically detects grains located within a sphere diameter from the window. The two species are then distinguished according to their distinct gray scale or size. Each frame is subdivided into ten horizontal strips of constant width spanning the entire length of the image. By incrementing the observed sphere cross-sectional area intersecting each strip, the vision algorithm calculates the fraction of the strip surface occupied by each species within the field of view and

estimates the corresponding relative number fraction by substituting this fraction for in Eqn. (28). The simulations showed that transverse profiles of produced by this method closely represent the state of segregation in the interior. Sphere velocities are then calculated from the positions of each sphere center in two consecutive images and the corresponding center-averaging statistics are incremented as in the simulations. They are made dimensionless with the chain speed directly measured from the image sequence. Table 2 summarizes experimental conditions for the flight tests in the KC-135 microgravity aircraft. A typical KC-135 trajectory included a gravitational pull of order followed by a parabolic flight yielding 20 s of reduced gravity. We began each test by starting the motor at the onset of the parabola. Peak-to-peak gravity fluctuations remained typically below The simulations helped us prescribe mixtures for which these fluctuations had minimal effects on the flow, and provided estimates of the dimensionless time required for each experiment to re-establish steady segregation after canceling that level of residual acceleration. Based on those estimates, we waited a time after the onset of reduced gravity to acquire images with the camera. Electrostatic charging of the dielectric flow spheres was mitigated by maintaining high relative humidity in the apparatus. With this precaution, no evidence of such charging could be observed.

5.

RESULTS AND DISCUSSION

The opacity of the grain assembly and the flow development in the straight section make it challenging to compare experiments and theory

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directly. While the latter predicts average quantities in the interior of a fully-developed flow, the former observe these quantities through side walls in a developing flow. Although flow variables vary little in the z direction, thus making lateral observations representative of the state in the interior, the flow development remains a more serious impediment to direct comparisons between experiments and theory (Fig. 3).

Figure 3 Development of the cross-sectional averaged solid volume fraction along the cell for the conditions of Test I.

In contrast, the simulations can be matched separately with experiments or theory. For comparisons with experiments, we simulate the flow in the entire cell while reproducing the method of imaging, as follows. Because the vision software detects spheres within a depth of focus

on the order of

the simulations calculate observable quantities only

from spheres centered within that distance from the window. To evaluate the corresponding statistics, the simulations ignore their knowledge of sphere velocities. Instead, they collect the locations (x, y) of visible

sphere centers from virtual images generated at the frame rate of the camera. From these, they infer two components of the center velocity

and proceed to calculate strip statistics in the same manner as in the experiments. As Figs. 4 and 5 illustrate with Test II, the simulations reproduce well the fluctuation velocities and segregation observed in the experiments.

Louge et al. (2000) reported similar results with Test I. This agreement demonstrates the utility of the simulations as an equal partner to theory and physical experiment. In this event, fewer physical experiments need be done, and the simulations can be used to evaluate the accuracy of

the theoretical modeling.

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Transverse profiles of the dimensionless fluctuation velocities for the con-

ditions of Test II. The symbols and lines are experimental measurements and predictions of the simulations, respectively. The open symbols and thin lines refer to the x direction, while the closed symbols and thick lines refer to the y direction. The squares and circles represent small and large acrylic spheres, respectively.

Figure 5 Transverse profiles of the relative number fraction for the conditions of Test II. For symbols and lines, see Fig. 4.

Such an evaluation is shown in Fig. 6. Here, after simulating the entire cell, we extracted an observation region in the range to which we imposed the periodic boundary condition sketched in Fig. 1. The spheres contained in this region thus experienced a fully developed flow, which could be matched with results of the theory. Note that, because the periodic simulations were carried out with an actual sample of the developing flow, their species volume fractions were not necessarily equal to those in the entire cell. As this figure illustrates, the theory captures segregation better at low solid volume fraction. Our conjecture is that higher particle number densities introduce geometric obstacles that the continuum kinetic theory ignores.

Acknowledgments This research was sponsored by NASA’s Microgravity Science and Applications

Division under contracts NCC3-468 and NAG3-2112.

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Figure 6 Transverse profiles of dimensionless mixture mean velocity, mixture fluctuation velocity, and relative number fraction for the conditions of Test I (left) and II (right). Triangles and lines represent simulations and theoretical predictions, respectively. In the bottom graphs, circles, diamonds, and squares are simulations results for large acrylic, ceramic, and small acrylic spheres, respectively. In these fullydeveloped flows, Test I has volume fractions = 12.0% and = 27.5%, and Test II has = 25.0% and = 4.3%.

The authors are indebted to Anthony Reeves for designing the computer vision algorithm and for producing the corresponding velocity statistics; to Gregory Aloe,

Roshanak Hakimzadeh, Jeffrey Larko, and Elaina McCartney for participating in the KC-135 flights; and to the engineering team at the NASA–Glenn Research Center: Joe Balombin, Christopher Gallo, Frank Gati, Roshanak Hakimzadeh, Jeffrey Larko, Pamela Mellor, Emily Nelson, Enrique Ramé, Leon Rasberry, John Yaniec and Gary Wroten; and to the KC-135 flight crews and ground support team.

References Arnarson, B.Ö., and J. T. Jenkins. 2000. Binary mixtures of inelastic spheres: simplified constitutive theory. In preparation.

Foerster, S., M. Y. Louge, H. Chang, and K. Allia. 1994. Measurements of the collision properties of small spheres. Physics of Fluids 6, 1108–1115.

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Haff, P. K., and B. T. Werner. 1986. Computer simulation of the mechanical sorting of grains. Powder Technology 48, 239–245. Hopkins, M. A., and M. Y. Louge. 1991. Inelastic microstructure in rapid granular flows of smooth disks. Physics of Fluids A 3, 47–57. Jenkins, J. T. 1992. Boundary conditions for rapid granular flows: flat, frictional walls. Journal of Applied Mechanics 59, 120–127. Jenkins, J. T., and B. Ö. Arnarson. 2000. The influence of side walls on a collisional shearing flow. In preparation.

Jenkins, J. T., and M. Louge. 1997. On the flux of fluctuation energy in a collisional grain flow at a flat, frictional wall. Physics of Fluids 9, 2835–2840. Jenkins, J. T., and F. Manciní. 1989. Kinetic theory for binary mixtures of smooth, nearly elastic spheres. Physics of Fluids A 1, 2050–2057. Jenkins, J. T., S. V. Myagchilov, and H. Xu. 2000. Nonlinear boundary conditions for collisional grain flows. In preparation.

Lorenz, A., C. Tuozzolo, and M. Y. Louge. 1997. Measurements of impact properties of small, nearly spherical particles. Experimental Mechanics 37, 292–298. Louge M. Y., J. T. Jenkins, A. Reeves, and S. Keast. 2000. Microgravity segregation in collisional granular shearing flows. In Segregation in Granular Flows (A. D. Rosato, ed.). Dortrecht: Kluver Academic Publishers, in press. Richman, M. W., and C. S. Chou. 1988. Boundary effects on granular shear flows of smooth disks. Zeitschrift für angewandte Mechanik und Physik 39, 885–901. Rosato, A., K. J. Strandburg, F. Prinz, and R. H. Swendsen. 1986. Monte Carlo simulation of particulate matter segregation. Powder Technology 49, 59–69. Rosato, A., K. J. Strandburg, F. Prinz, and R. H. Swendsen. 1987. Why the Brazil nuts are on top: Size segregation of particulate matter by shaking. Physical Review Letters 58, 1038–1040.

Savage, S. B., and C. K. K. Lun. 1988. Particle size segregation in inclined chute flow of dry cohesionless granular solids. Journal of Fluid Mechanics 189, 311–335. Walton, O. R. 1988. Granular solids flow project. Quarterly report, January–March 1988, UCID-20297-88-1, Lawrence Livermore National Laboratory. Zhang, C. 1993. Kinetic theory for rapid granular flows. Ph.D. thesis, Cornell University.

NONLINEAR COMPOSITES AND MICROSTRUCTURE EVOLUTION Pedro Ponte Castañeda Department of Mechanical Engineering and Applied Mechanics University of Pennsylvania, Philadelphia, Penn., USA [email protected]

Pierre Suquet Laboratoire de Mécanique et d’Acoustique, CNRS, Marseille, France [email protected]

Abstract

1.

Recently developed methods for estimating the effective behavior of nonlinear composites are reviewed. The methods follow from variational principles expressing the effective behavior of the given nonlinear composites in terms of the behavior of suitably chosen “linear comparison” composites. These methods allow the use of classical bounds and estimates (e.g. Hashin–Shtrikman, effective medium approximations) for linear materials to generate corresponding information for nonlinear ones. Comparisons are made with numerical simulations for metalmatrix composites, showing that the new methods are significantly more accurate than earlier ones, especially at high nonlinearity and heterogeneity contrast. The methods can be extended to incorporate evolution of the microstructure and its influence on the effective response under finite-strain conditions. An application to a forming process involving a porous metal is considered for illustrative purposes.

INTRODUCTION

In the context of linear elasticity, rigorous and reliable methods have been available for quite some time to estimate the effective or overall behavior of heterogeneous materials. These so-called homogenization methods include the variational methods of Hashin and Shtrikman (1962) and Beran (1965), both of which are particularly well suited to composites with particulate random microstructures. There is also the self-consistent approximation, which is known to be fairly accurate for polycrystals and other materials with granular microstructures. For a review, see e.g. Willis (1981). 253 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 253–274. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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For heterogeneous materials with nonlinear (e.g. plastic, viscoplastic) properties, rigorous methods have not been available until fairly recently, even though efforts along these lines have been going on for a long time, particularly in the context of ductile polycrystals. Following an extension of the Hashin–Shtrikman (HS) variational principles by Willis (1983), the first bounds of the HS type for nonlinear composites were derived by Talbot and Willis (1985). A more general approach consisting in the use of an optimally chosen “linear comparison composite” was proposed by Ponte Castañeda (1991), and, independently for the special case of power-law materials, by Suquet (1993). This approach not only is capable of delivering bounds of the HS type for nonlinear composites, but also can be used to generate bounds and estimates of other types, by making use of the corresponding bounds and estimates for the linear comparison composite. More recently, Ponte Castañeda (1996a) proposed an alternative approach making use of a more sophisticated linear comparison composite, which while not yielding bounds, appears to give more accurate results. In particular, this method gives the only general homogenization estimates to date capable of reproducing exactly to second order in the contrast the asymptotic expansions of Suquet and Ponte Castañeda (1993). While the idea of using linear composites to estimate the effective behavior of nonlinear ones is quite old, the key feature in these novel linear comparison methods is the use of rigorous variational principles to determine the best possible choice of the linear comparison composite of a given type. Within the context of the “secant” approximation, first used by Chu and Hashin (1971), it follows from the work of Suquet (1995) (and, independently, Hu (1996)) that the optimal choice is that made in the variational method of Ponte Castañeda (1991), i.e. the secant moduli of the phases evaluated at the second moments of the relevant fields in the phases, and not at the phase averages, or first moments, as had been done previously in the classical secant approaches. Alternatively, within the context of “tangent”-type approximations, first used by Hill (1965) in his popular incremental method, the work of Ponte Castañeda and Willis (1999) shows that the optimal choice is, in some restricted sense, that made in the second-order procedure of Ponte Castañeda (1996a), i.e. the tangent moduli of the phases evaluated at the phase averages of the relevant fields in a more general thermoelastic comparison composite. In this short review paper, we attempt to summarize these recent developments, focusing on applications and on comparisons with recent numerical simulations (Moulinec and Suquet (1998), Michel et al. (1999), Michel et al. (2000)). For more detailed reviews, concentrating on the

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more theoretical aspects, the reader is referred to Ponte Castañeda and Suquet (1998) and Willis (2000).

2.

EFFECTIVE BEHAVIOR

The assumption is made that the material is composed of N different phases, which are distributed randomly in a specimen occupying a volume at a length scale that is much smaller than the size of the specimen and the scale of variation of the loading conditions. The constitutive behavior of the nonlinear phases will be characterized by convex energy functions (r = 1 , . . . , N), such that the local stress–strain relation (Fig. l(a)) is defined by

where the function is equal to 1 if the position vector x is inside phase (i.e. and zero otherwise. The relations (1) can be used to describe several constitutive models, including deformation theory of plasticity, in which case and

Figure 1 Constitutive relation.

are identified with the infinitesimal strain and stress, respectively. The relation applies equally well to viscoplastic materials, in which case the associated deformations are finite and and are associated with the Eulerian strain rate and Cauchy stress, respectively. A commonly used

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form for

is the isotropic, incompressible power-law form

where m is the rate-sensitivity parameter, such that is the flow stress, is a reference strain rate, and is the equivalent von Mises strain or strain rate. Making use of the symbols and to denote volume averages over the composite and over phase respectively, we determine the effective behavior of the composite by the effective energy function

where the scalars denote the volume fractions of the given phases and denotes the set of kinematically admissible strains such that there is with in Thus, physically corresponds to the energy stored in the composite when subjected to an affine displacement on the boundary with prescribed average strain It can be shown (Hill (1963)) that the average stress is then related to the average strain via

3.

HOMOGENIZATION VIA LINEAR COMPARISON COMPOSITES

In this section, a brief introduction is given to the “linear comparison composite” methods. A linear composite is introduced with potential

where the are symmetric, positive definite, constant tensors. Following Ponte Castañeda (1996b), we make use of the basic result that

where the assumption has been made that the functions (or functionals) f and are bounded below within the set A.

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Noting that under the hypotheses associated with expression (2) the function is bounded below, and applying the result (6) to the function we find that

and therefore that

where is the effective potential associated with the linear elastic comparison composite with local potential given by (5):

Here is the effective modulus of the linear comparison composite. Next, relaxing the constraint in the second term on the right of expression (8), we arrive at

Observing that this result must hold for any choice of the tensors and bringing the infimum over inside of the averages, we find that

It is noted that if the phases of the nonlinear composite are isotropic, the optimal choice of the comparison moduli is also isotropic with bulk and shear moduli and respectively. Expression (11) is the bound first proposed by Ponte Castañeda (1991) in the context of nonlinear composites with general isotropic phases. A generalization for polycrystals was given by deBotton and Ponte

Castañeda (1995). When it is used in conjunction with the upper bound of Hashin and Shtrikman (1963) for the bounds of Talbot and Willis (1985) are recovered. However, there are pathological cases (not including the standard models of plasticity) for which the bounds generated by the Talbot–Willis procedure are superior (see Willis (1992); Talbot and Willis (1992)). On the other hand, the result (11) can be used together with any other bound (e.g. three-point bounds) or estimate (e.g. the

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self-consistent estimates) for linear composites to generate corresponding estimates for nonlinear composites. For example, Ponte Castañeda (1992) used this procedure to compute three-point bounds. In addition, for the special case of power-law composites, an alternative, but equivalent, form has been given by Suquet (1993) using Hölder-type inequalities. The result is given by

The effective constitutive relation for the nonlinear composite is obtained by making use of expression (11) for in expression (4) and enforcing the optimality condition in the variables to obtain

This means that the effective constitutive relation for the nonlinear composite is precisely the same as that of the linear comparison composite, where the local properties of the linear comparison composite are determined as the solution of the procedure (11) for the optimized variables Clearly, since these variables depend on the applied strain the above relation is nonlinear in this variable, as expected. It is emphasized, however, that the microstructure of the linear comparison composite is identical to that of the nonlinear composite. Finally, it is noted that the optimal choice of the variables can be given an interpretation in terms of the second moment of the strain field in the linear comparison composite. Full details about this derivation can be found in Suquet (1995), Suquet (1997), Ponte Castañeda and Suquet (1998); therefore, we limit ourselves here to simple heuristic arguments when the phases are isotropic. The condition for the optimal choice of the variables in the inner infimum problem in (11) is given by

which physically means that should be chosen to be the secant modulus tensor of nonlinear phase (Fig. l(b)). Assuming stationarity

with respect to we generate the condition for the optimal choice of these variables by substituting the expression (9) for in (11) to obtain

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In other words, the modulus of phase r in the linear comparison composite is the nonlinear secant modulus evaluated at some “effective” strain defined through relation (15), where the right-hand side is the second-order moment of the strain field over phase r of the linear comparison composite. It is emphasized that the expression (15) may not be satisfied in general. However, Suquet (1995) has remarked that, for the case of isotropic phases, this tensorial relation is not needed and that the optimal linear comparison composite can be defined using the second moment of the von Mises strain The more general case was considered in Ponte Castañeda and Suquet (1998) by making use of a suitable extension of the above ideas. The important point to retain here is the following. As mentioned earlier, many attempts have been made to estimate the effective behavior of nonlinear composites in terms of the effective behavior of linear comparison composites—in particular, making use of secant-type approximations. However, because the strain field in the composite is highly nonuniform, it is not obvious what strain to use in the evaluation of the secant moduli for the phases of the linear comparison composite. In the past, ad hoc prescriptions have been tried, mostly making use of the average, or first moment, of the strain field in the phases of the composite. Expression (11) shows that the best choice—within the context of a rigorous variational principle—is the second moment of the strain field over the phases. As will be seen in the next section, this approach gives much better results than the classical schemes. It should be noted that Buryachenko and Lipanov (1989) were apparently to be the first to use second-moment type quantities in the context of their “multi-particle effective field scheme.” However, as already noted in the Introduction, the estimates (11) are not able to recover the perturbation estimates of Suquet and Ponte Castañeda (1993) for weakly inhomogeneous nonlinear materials, which are exact to second order in the heterogeneity contrast. This is in contrast with the Hashin–Shtrikman and self-consistent estimates for linear composites, which are known to be exact to second-order in the contrast. A homogenization method that has the capability to recover small-contrast results exactly to second order in the contrast was introduced recently by Ponte Castañeda (1996a). A full derivation of this result cannot be given here, but the result can be stated succinctly as

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where is the average of the strain field over phase r in a linear thermoelastic composite with phase r defined by the constitutive relation

where the modulus tensor

is defined by

It was shown by Ponte Castañeda and Willis (1999) that this “secondorder” procedure follows from suitable approximations in the context of a rigorous variational principle. However, the principle is only stationary and therefore leads only to stationary estimates and not to bounds. It is further noted that this procedure leads naturally to the choice of the tangent modulus tensors (Fig. 1(b)) for the linear comparison composite. But these should be evaluated at the phase averages of the strain—not at the second moments. As will be seen later, although the second-order estimates are not bounds, they are complementary to the bounds discussed earlier and appear to be more accurate. Other recent developments in the field of nonlinear composites, which will not be reviewed here for lack of space, include the computation of the “hard” bounds by Talbot and Willis (1997), who made use of suitably chosen nonlinear comparison composites; bounds on the strain fields (as opposed to the energies) by Milton and Serkov (2000); an affine procedure due to Masson et al. (2000) that is closely related to the second-order method, but that also works for elastic–viscoplastic composites; new self-consistent estimates for polycrystals (Bornert and Ponte Castañeda (1998), Nebozhyn et al. (1999) and Gilormini et al. (2001)) showing dramatic improvements over the classical estimates; and applications of the second-order method in finite elasticity (Ponte Castañeda and Tiberio (2000)), where the relevant potentials are nonconvex, leading to the failure of other methods.

4.

SAMPLE RESULTS

Sample results are presented here for various special cases, including particle-reinforced composites and porous metals. For simplicity, all constituents are assumed to be governed by relation (2) with the same exponent m, but with different flow stresses The microstructures are assumed to be statistically isotropic, so that the effective potentials of the nonlinear composite are isotropic functions of the (traceless)

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strain thus depending only on its second and third invariants. They can therefore be written in the form

where the effective flow stress

is seen to be a function of the plastic

phase angle which in turn is related to the two invariants of through the relation cos = 4 det Note that the extreme values of the variable correspond to uniaxial tension and simple shear Some of the results presented here are for two-dimensional composites with transverse isotropy subjected to plane-strain conditions, in which case a result analogous to (19) is generated, but where the equivalent strain has been suitably redefined. The corresponding effective flow stresses are constant (independent of in this case. In the figures and discussion below, the following abbreviations will be used for simplicity: Hashin–Shtrikman (HS), self-consistent (SC), upper bound (UB), and lower bound (LB). Similarly, the following terminology will be used: “Variational” refers to the relation (11) of Ponte Castañeda (1991), or, equivalently, for power-law materials, relation (12) of Suquet (1993). “Second-order” refers to relation (16) of Ponte Castañeda (1996a). “Incremental” refers to the procedure first proposed by Hill (1965), in the form developed by Hutchinson (1976) for power-law viscous materials. “Secant” refers to the classical secant procedure, used by various authors, starting with Chu and Hashin (1971). “Tangent” refers to the procedure first proposed by Molinari, Canova, and Ahzi (1987) in its full anisotropic form (no further approximations). “Voigt” and “Reuss” will be used to denote the classical microstructureindependent upper (uniform strain) and lower (uniform stress) bounds. Finally, FEM and FFT will be used to denote the the results of the finiteelement method and fast-Fourier-transform simulations of Moulinec and Suquet (1998), Michel et al. (1999), and Michel et al. (2000).

4.1.

Particle-reinforced composites

In Fig. 2, the variational HS UB and the second-order HS estimate are compared against the classical Voigt UB and Reuss LB, the incremental, secant, and tangent procedures, and the FEM simulations of Michel et al. (1999) for two-phase, isotropic, rigid ideally plastic composites with 15% concentration of the harder phase, subjected to uniaxial tension. The effective flow stress is thus plotted as a function of the flow stress ratio of the two phases. It is emphasized that the FEM simulations are for composites with periodic microstructures and cylindrical unit cells, while all the HS estimates correspond to random microstructures with

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Figure 2 Rigidly reinforced, ideally plastic composites with statistically isotropic microstructures: Influence of the heterogeneity contrast.

overall isotropy. Therefore, the HS and FEM estimates are not expected to be in good agreement for general values of the volume fraction of the reinforcing phase. (In particular, the FEM predictions will not lead to exactly isotropic behavior.) However, for small enough volume fractions, the HS estimates would be expected to be in good agreement with the FEM simulations and hence the comparisons shown the figure at 15% concentration are considered to be meaningful. The main observation in this figure is that the second-order HS esti-

mate gives the best overall agreement with the FEM simulations, even at large contrast (within 1%). The variational HS result, which is an upper bound for all other HS estimates, is satisfied by the second-order and tangent estimates, but not by the secant and incremental estimates. However, the tangent prediction in this limiting case agrees exactly with the Reuss lower bound, which is believed to be too soft. In conclusion, for this extreme nonlinearity, essentially all the classical schemes break down and we are left with the variational bound and second-order estimates, which are complementary to each other and can be used to estimate the effective behavior of the composite fairly accurately.

4.2.

Fiber-reinforced composites

Figure 3 (a) depicts a comparison between yield surfaces computed with the variational procedure using the HS lower bounds to estimate

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Figure 3 Effective flow surfaces and deformation maps for fiber-reinforced composites.

the properties of the linear comparison composite (continuous lines) and FFT simulations (various symbols) for generalized plane strain loading of a fiber-reinforced, ideally plastic composite. The fibers, which are aligned in the out-of-plane direction, are taken to be 2, 5, and 10 times stronger than the matrix. The horizontal axis corresponds to inplane shear loading of the form and the vertical axis to out-of-plane axial loading. It can be seen that the agreement between the analytical predictions and the FFT simulations is quite good, at least in qualitative terms. The variational procedure is able to capture the “bimodal” character of the yield surfaces, which has also been observed experimentally (Dvorak and Bahei-El-Din (1987)). Figure 3(b) presents strain intensity maps for transverse shear loading of the plastic composite. Note that the microstructure in the

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FFT simulations is taken to be periodic, with 64 fibers thrown at random in the unit cell. The shear bands (white lines) that develop through matrix channels between the hard fibers (black circles) are the source of the relatively “weak” behavior in this mode of deformation. For comparison purposes, results are also given in Figs. 3(c) and (d), for the same mode of deformation in linear elastic composites with the same microstructure and shear moduli ratios of 6 and 100, respectively. It is interesting to note that while strain intensification is also observed in the regions of close proximity between the fibers, shear bands going from one end of the specimen to the other end are not observed. It may then be surprising that the variational procedure, which uses the solution of a linear elastic problem to estimate the effective behavior of the nonlinear problem, does so well, in particular, capturing the strongly nonlinear “flat” sector on the yield surface. However, the important thing to remember here is that the variational principle is designed to select the optimal properties of the linear comparison composite to model as accurately as possible the effective behavior of the nonlinear composite, which is characterized in this case by the effective yield surfaces.

4.3.

Effect of the third invariant: Particle-reinforced composites

In this section, the effect of the third invariant of the loading, as measured by the plastic angle on the effective flow stress of two-phase, ideally plastic composites with overall isotropy is considered. Thus, in Fig. 4(a), a comparison is given between the variational and second-order procedures, used in conjunction with the HS lower bounds for the relevant linear comparison problem, and the FFT simulations carried out by Moulinec and Suquet (unpublished) for “quasi-random” microstructures, as defined by the unit cell with 15% reinforcement shown in Figure 4(b). The FFT simulations show dependence on as expected from general considerations. This dependence is also captured by the second-order procedure quite well near the axisymmetric end and less well near the pure shear limit where it tends to the flow stress of the matrix. The variational HS estimate is independent of but on the other hand provides an upper bound for the effective flow stress. Figures 4(c) and (d) provide deformation maps in the FFT simulations for the two extreme cases, axisymmetric tension and pure shear, respectively. As can be seen in the transverse diagonal planes highlighted in

the figures, the deformation is distributed more uniformly in the matrix for axisymmetric tension, while it tends to become localized on shear bands passing through the matrix for pure shear, thus elucidating the

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Figure 4 Rigidly reinforced, ideally plastic composites with statistically isotropic microstructures: Influence of the third invariant of the applied stress.

physical source of the dependence on the third invariant for strongly nonlinear composites.

4.4.

Cellular microstructures

In this section, two-phase composites with transversely isotropic cellular microstructures, subjected to plane strain loading, are considered. Comparisons are shown in Figs. 5(a) and (b) between the predictions

of the second-order procedure, using the SC estimates for the linear

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comparison composite, and FFT simulations for composites with ran-

Figure 5 Two-dimensional composites with cellular microstructures: Effect of volume fraction and nonlinearity.

dom distributions of hexagons defined by unit cells of the type shown in Fig. 5(c) (25 configurations were used; the circles denote the aver-

age value, and the error bars the maximum and minimum values). It is known from earlier work in the context of linear elastic composites that these microstructures are well approximated by the self-consistent model, at least at finite heterogeneity contrast. It is thus seen in Fig. 5(a)

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that the second-order SC estimates predict with fair accuracy the dependence on volume fraction at this moderate level of nonlinearity. (The results are not as good as the nonlinearity increases for intermediate values of the volume fraction.) Similarly, Fig. 5(b) shows relatively good agreement even at infinite contrast and high values of the nonlinearity (m tending to zero), for a volume fraction of 25% of the rigid phase. (Also shown in this figure are the corresponding second-order

HS estimates, which are compared with periodic FFT results for a perfect lattice consisting of a hexagonal distribution of rigid hexagons in the matrix material.) Finally, Fig. 5(d) shows the deformation maps for the configuration defined by Fig. 5(c) with an ideally plastic matrix

and reinforcement phase 5 times as strong. (The soft phase is shown in black in Fig. 5(c).) Note, once again, that shear bands develop passing through channels of the weaker phase.

4.5.

Microstructure evolution in porous materials

In forming processes involving porous metals, for example, the strains are large enough to cause the microstructure to evolve as a function of

the deformation. An isotropic porous plasticity model that has been shown to work extremely well under nearly hydrostatic loading conditions was proposed by Gurson (1977). However, for processes involving lower triaxiality conditions, such as rolling and extrusion, the material

is expected to develop anisotropy, even if its initial state is isotropic. An anisotropic model, which is based on the use of the HS variational estimates of Ponte Castañeda (1991) and Michel and Suquet (1992) for porous media, was proposed by Ponte Castañeda and Zaidman (1994) and Kailasam et al. (2000) to account for the evolution of pore shape and orientation. In Fig. 6, a comparison is made between the predictions of the aniso-

tropic model of Ponte Castañeda and Zaidman (1994) and finite-strain FEM simulations for a periodic composite with axisymmetric unit cell, as depicted in the figure. In addition, the predictions of an extension of the isotropic Gurson model, due to et al. (2000), for a powerlaw viscoplastic matrix material are also shown. The initial porosity is the strain-rate sensitivity is m = 0.2, and uniaxial tension is applied with triaxiality X = 1/3. It can be seen that the predictions of the anisotropic model—in contrast with those of the isotropic model—

are at least in qualitative agreement with the FEM simulations at this level of triaxiality. In particular, the anisotropic model captures not only the evolution of the average shape of the voids but, in addition,

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Figure 6 Porous materials subjected to uniaxial tension.

also predicts the gentler increases in the porosity f (which is found to saturate at large enough strain ), as well as the overall hardening of the porous material (positive slope in the uniaxial stress–strain relation, as opposed to the negative slope predicted by the isotropic model). Analogous observations have been made for other low-triaxiality situations, including uniaxial compression, where the anisotropic model is found to predict full densification at around 70% strain, which is in much better agreement with numerical and experimental evidence than the value predicted by the Gurson-type models (around 250% strain). On the other hand, it should be emphasized that the Gurson-type models give the most accurate predictions for high triaxialities. Finally, in Fig. 7, contour plots are given (Kailasam et al. (2000)) for the distribution of the porosity predicted by the anisotropic and Gurson models, as well as for the pore aspect ratios predicted by the anisotropic model, in a disk compaction experiment (Parteder et al.

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Figure 7 Compaction of a tapered disk with a height reduction of 37.5%.

269

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(1999)). The initial porosity of the tapered axisymmetric disk (only one quarter of the cross-section of which is shown) is = 15%, and the macroscopic height reduction is 37.5%. The main observation in the context of this figure is that the porosity reduction predicted by the anisotropic model (Fig. 7(a)) is considerably larger than that predicted by the Gurson model (Fig. 7(b)). More specifically, the size of the region where the porosity is less than 1% (inner dark region) is much larger for the anisotropic model than for the Gurson model. Also, in this region, the anisotropic model predicts that the pores have become nearly flat (i.e. they have become cracks) and are aligned with the horizontal direction (Fig. 7(c) and (d)). For more details on these simulations, the reader is referred to Kailasam et al. (2000).

5.

CONCLUDING REMARKS In this review, two nonlinear homogenization methods have been

briefly described, the main ingredient in both being the use of linear com-

parison composites in the context of suitably designed variational principles. Their predictions for various types of composite material have been compared with the corresponding predictions of the classical schemes, as well as with numerical simulations. It is found that the “second-order” method, which makes use of the tangent moduli evaluated at the average strain in the phases, usually leads to the most accurate estimates, while the less accurate “variational” method, which makes use of secant moduli evaluated at the second moments of the strain, gives rigorous bounds. Thus, in some sense, they provide complementary information. The main advantage of these methods is their relative simplicity and computational efficiency. Explicit (or nearly so) expressions are available for most of the cases considered here, including rigidly reinforced composites and porous materials—see Ponte Castañeda and Suquet (1998) for the most comprehensive set of results. These results compare favorably with FEM and FFT simulations, which, while very accurate for the specific configurations chosen, are much more computationally intensive. It is important to emphasize that, by their very nature, these homogenization methods cannot be more accurate than the estimates that are used as input in the computation of the effective behavior of the relevant linear comparison composites. Thus far, reasonable accuracy has been achieved for composites that are modeled well by the Hashin–Shtrikman and self-consistent approximations. More accurate estimates, incorporating higher-order statistics, are likely to be needed for other types of microstructure and for improved performance. Concerning the nonlinear schemes themselves, improvements are still needed to be able to handle

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the most extreme circumstances, usually involving very highly nonlinear behavior.

Acknowledgments PPC acknowledges the support of the NSF through grants DMS-99-71958 and CMS-99-72234. We are grateful to J. C. Michel and H. Moulinec for the FFT results shown in Figs. 3 to 5, which are taken from the joint works of PS with them. The financial support of NSF (grant INT-97-26521) and CNRS through the joint project “Micromechanics of nonlinear composites and polycrystals” is also acknowledged.

References Beran, M. 1965. Use of the variational approach to determine bounds for the effective permittivity of random media. Nuovo Cimento 38, 771–782.

Bornert, M., and P. Ponte Castañeda. 1998. Second-order estimates of the selfconsistent type for viscoplastic polycrystals. Proceedings of the Royal Society of London A 356, 3035–3045. Buryachenko, V., and A. M. Lipanov. 1989. Prediction of nonlinear flow parameters for multicomponent mixtures. Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki 4, 63–68. Chu, T., and Hashin, Z. 1971. Plastic behavior of composites and porous media under isotropic stress. International Journal of Engineering Science 9, 971–994. deBotton, G., and P. Ponte Castañeda. 1995. Variational estimates for the creep behavior of polycrystals. Proceedings of the Royal Society of London A 448, 421– 442. Dvorak, G., and Y. Bahei-El-Din. 1987. A bimodal plasticity theory of fibrous com-

posite materials. Acta Mechanica 69, 219–241. M., J. C. Michel, and P. Suquet. 2000. A micromechanical approach of damage in viscoplastic materials by evolution in size, shape and distribution of voids. Computer Methods in Applied Mechanics and Engineering 183, 223–246. Gilormini, P., M. Nebozhyn, and P. Ponte Castañeda. 2000. Accurate estimates for the creep behavior of hexagonal polycrystals. Acta Materialia 49, 329–337. Gurson, A. 1977. Continuum theory of ductile rupture by void nucleation and growth: Part I—Yield criteria and flow rules. Journal of Engineering Materials and Technology 99, 1–15. Hashin, Z., and S. Shtrikman. 1962. On some variational principles in anisotropic and nonhomogeneous elasticity. Journal of the Mechanics and Physics of Solids 10, 335–342. Hashin, Z., and S. Shtrikman. 1963. A variational approach to the theory of the elastic behavior of multiphase materials. Journal of the Mechanics and Physics of Solids 11, 127–140. Hill, R. 1963. Elastic properties of reinforced solids: Some theoretical principles. Journal of the Mechanics and Physics of Solids 11, 357–372. Hill, R. 1965. Continuum micro-mechanics of elastoplastic polycrystals. Journal of the Mechanics and Physics of Solids 13, 89–101. Hu, G. 1996. A method of plasticity for general aligned spheroidal voids or fiberreinforced composites. International Journal of Plasticity 12, 439–449.

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Hutchinson, J. W. 1976. Bounds and self-consistent estimates for creep of polycrystalline materials. Proceedings of the Royal Society of London A 348, 101–127.

Kailasam, M., N. Aravas, and P. Ponte Castañeda. 2000. Constitutive models for porous metals with developing anisotropy and applications to deformation processing. Computer Modeling in Engineering and Sciences 1, 105–118. Masson, R., M. Bornert, P. Suquet, and A. Zaoui. 2000. An affine formulation for nonlinear composites and polycrystals. Journal of the Mechanics and Physics of

Solids 48, 1203–1227. Michel, J. C., and P. Suquet. 1992. The constitutive law of nonlinear viscous and porous materials. Journal of the Mechanics and Physics of Solids 40, 783–812.

Michel, J. C., H. Moulinec, and P. Suquet 1999. Effective properties of composite materials with periodic microstructure: A computational approach. Computer Methods in Applied Mechanics and Engineering 172, 109–143. Michel, J. C., H. Moulinec, and P. Suquet. 2000. A computational method based on augmented Lagrangians and fast Fourier transforms for composites with high contrast. Computer Modeling in Engineering and Sciences 1, 79–88. Milton, G., and S. K. Serkov. 2000. Bounding the current in nonlinear conducting composites. Journal of the Mechanics and Physics of Solids 48, 1295–1324. Molinari, A., G. R. Canova, and S. Ahzi. 1987. A self-consistent approach of the large deformation polycrystal viscoplasticity. Acta Metallurgica Materialia 35, 2983– 2994. Moulinec, H., and P. Suquet. 1998. A numerical method for computing the overall response of nonlinear composites with complex microstructure. Computer Methods in Applied Mechanics and Engineering 157, 69–94. Nebozhyn, M. V., P. Gilormini, and P. Ponte Castañeda. 1999. Variational selfconsistent estimates for viscoplastic polycrystals with highly anisotropic grains. Comptes Rendus de l’Académie des Sciences, Paris IIB 328, 11–17. Parteder, E., H. Riedel, and R. Kopp. 1999. Densification of sintered molybdenum during hot upsetting experiments and modeling. Materials Science and Engineering A 264, 17–25. Ponte Castañeda, P. 1991. The effective mechanical properties of nonlinear isotropic composites. Journal of the Mechanics and Physics of Solids 39, 45–71. Ponte Castañeda, P. 1992. New variational principles in plasticity and their application to composite materials. Journal of the Mechanics and Physics of Solids 40, 1757–1788. Ponte Castañeda, P. 1996a. Exact second-order estimates for the effective mechanical properties of nonlinear composites. Journal of the Mechanics and Physics of Solids 44, 827–862. Ponte Castañeda, P. 1996b. Variational methods for estimating the effective behavior of nonlinear composite materials. In Continuum Models and Discrete Systems (CMDS 8) (K. Z. Markov, ed.). Singapore: World Scientific, 268–279. Ponte Castañeda, P., and P. Suquet. 1998. Nonlinear composites. Advances in Applied Mechanics 34, 171–302. Ponte Castañeda, P., and E. Tiberio. 2000. A second-order homogenization method in finite elasticity and applications to black-filled elastomers. Journal of the Mechanics and Physics of Solids 48, 1389–1411. Ponte Castañeda, P., and J. R. Willis. 1999. Variational second-order estimates for nonlinear composites. Proceedings of the Royal Society of London A 455, 1799–

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Ponte Castañeda, P., and M. Zaidman. 1994. Constitutive models for porous materials with evolving microstructure. Journal of the Mechanics and Physics of Solids 42, 1459–1497. Suquet, P. 1993. Overall potentials and extremal surfaces of power-law or ideally plastic materials. Journal of the Mechanics and Physics of Solids 41, 981–1002.

Suquet, P. 1995. Overall properties of nonlinear composites: A modified secant moduli theory and its link with Ponte Castañeda’s nonlinear variational procedure. Comptes Rendus de l’Académie des Sciences, Paris IIB 320, 563–571. Suquet, P. 1997. Effective properties of nonlinear composites. In Continuum Micromechanics (P. Suquet, ed.), CISM Lecture Notes 377. New York: Springer, 197–264. Suquet, P., and P. Ponte Castañeda. 1993. Small-contrast perturbation expansions for the effective properties of nonlinear composites. Comptes Rendus de l’Académie des Sciences, Paris II 317, 1515–1522. Talbot, D. R. S., and J. R. Willis. 1985. Variational principles for inhomogeneous nonlinear media. IMA Journal of Applied Mathematics 35, 39–54.

Talbot, D. R. S., and J. R. Willis. 1992. Some simple explicit bounds for the overall behavior of nonlinear composites. International Journal of Solids and Structures 29, 1981–1987. Talbot, D. R. S., and J. R. Willis, 1997. Bounds of third order for the overall response of nonlinear composites. Journal of the Mechanics and Physics of Solids 45, 87– 111. Willis, J. R. 1981. Variational and related methods for the overall properties of composites. Advances in Applied Mechanics 21, 1–78. Willis, J. R. 1983. The overall response of composite materials. Journal of Applied Mechanics 50, 1202–1209. Willis, J. R. 1992. On methods for bounding the overall properties of nonlinear composites: correction and addition. Journal of the Mechanics and Physics of Solids 40, 441–445. Willis, J. R. 2000. The overall response of nonlinear composite media. European Journal of Mechanics A/Solids 19, S165–S184.

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The technical sessions were kept running smoothly by these dedicated session aides, most of whom were graduate students from Midwestern schools—front row (l–r): Zhiqun Deng (UIUC*), Gareth Block (UIUC), Carmen Hernandez (Northwestern), Zhengyu Zhang (UIUC), Susan Olson (Notre Dame), Almila Guvenc (UIC†); back row (l–r): Mihajlo Golubovic (UIC), Michael J. Rodgers (Northwestern), Russell Todres (UIUC), Michael R. Kessler (UIUC), John Aldrin (Northwestern), Igor Kouznetsov (UIUC), Amine Benzerga (Brown), Prof.

Hassan Aref, and Ilker S. Bayer (UIC). Not pictured is Brett E. Soltz (UIUC). *University of Illinois at Urbana-Champaign

†University of Illinois at Chicago

HARD PROBLEMS WITH SOFT MATERIALS: THE MECHANICS OF FOAMS Denis L. Weaire and Stefan Hutzler Physics Department, Trinity College, Dublin, Ireland [email protected], [email protected] Abstract

1.

The static properties of foams with low liquid fraction are now well understood. We review current topics of foam research, which include the search for appropriate boundary conditions for drainage models, and dynamic effects, such as convective bubble motion. The formation of metal foams poses interesting problems with regard to the solidification of draining metal during the cooling process.

NEW FRONTIERS IN FOAM RESEARCH

Many aspects of the behavior of foams (both liquid and solid) have come to be well understood in the last twenty years [1]. The lowestorder description of their properties is virtually complete. Nevertheless the remaining terra incognita is large, varied, and essential to many practical applications [2]. In the case of liquid foam (Fig. 1), a new frontier is approached as the foam becomes very wet, or is subjected to rapid deformation. The problems presented to theory and experiment are not merely those of mapping out quantitative corrections to established models, but include new phenomena, such as convection and avalanches. Our intention here is not to dwell too long on what is by now accepted as common knowledge. Rather we shall concentrate on sketching the boundaries of the unknown. Of the four key aspects of the physics of liquid foams, indicated in Fig. 2, our focus will be concentrated on drainage and rheology. From the present perspective these are closely interlinked fields of inquiry.

2.

STATIC EQUILIBRIUM

The local laws of equilibrium of a foam together with their consequences have been well understood (with some exceptions) for a long time. A recent flurry of interest in problems that relate to static equi275 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 275–288. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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Figure 1

Foam as seen through the camera lens of art photographer Michael Boran.

Figure 2

The physics of liquid foams includes at least four key phenomena, which

may occur in combination. The physicist’s approach is to separate these for study in isolation.

librium has been provoked not so much by fresh insights as by fresh computing power, and the software to exploit it. In three dimensions, Ken Brakke’s Surface Evolver [3] gives an accurate representation of static structures, within the usual idealized model. Figure 3 shows such a computed element of a foam structure. It may also be useful in the more demanding context of dynamic effects, at least at the outset.

3.

FOAM DRAINAGE

Whenever a foam is freshly created, perhaps by shaking up a liquid, and is then left to come into equilibrium under gravity, much of the liquid

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Figure 3

277

Junction of four Plateau borders.

will be seen to drain away. It does so primarily through the network of Plateau borders, which are the liquid-filled interstices at the edges of the polyhedral bubbles (Fig. 3). Provided that the foam is stable, this is almost all that happens over the first few minutes. The “wet” foam

Figure 4

Sketch of the equilibrium density profile of a liquid foam.

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is said to become “dry”, but not uniformly so. There is an equilibrium profile of density or liquid fraction, such that capillarity balances gravity at all points. The sketch of Fig. 4 makes use of static two-dimensional computer simulations, for purposes of illustration. The equilibrium density profile was described by Henry Princen [4, 5], one of the modern pioneers of the subject. Another was Robert Lemlich [6, 7, 8], who understood the essentials of the drainage process, but failed to arrive at a palatable version of the theory, or to stumble on the “solitary wave” phenomenon described below. More recently our own group, together with Guy Verbist, has discovered, rediscovered, and rendered more coherent various components of the study of drainage [9, 10]. The resulting theory neatly describes drainage under various circumstances, including the usual case of “free drainage” and that in which liquid is continually added, which we call “forced drainage”. When drainage is initiated, the downward progress of the wet region takes the form of a solitary wave. This means that the partial differential equation (the drainage equation) for the vertical profile of the liquid fraction has a solution in which the profile is of constant shape, moving steadily downward. There are many attractive features of this experiment and theory, not least of which is the existence of an analytic solution for the solitary wave. Once established, forced drainage creates a column of essentially uniform liquid fraction The corresponding volume flow rate of liquid varies as

for steady flow. This might be regarded as the primary law of foam drainage within such a theory. It is to be contrasted with electrical conduction, for which current is proportional to for a given applied voltage. Equation (1) rests on very strong assumptions (particularly that of Poiseuille flow) and approximations that might seem to limit its validity to relatively dry foams. Nevertheless, it has been repeatedly found to agree with data from detergent foams, over quite a large range, as have the more exotic solutions of the drainage equation. Recently a body of fresh experimental results was presented by the group of Howard Stone, which deviated significantly from Eqn. (1) [11]. The new data were more consistent with

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which can be justified by the assumption that flow is not of Poiseuille character in the Plateau borders, but closer to plug flow, so that most of the dissipation takes place in the hitherto neglected junctions. In due course the apparent discrepancy has been resolved in an amusing manner: the two groups used leading brands of dishwasher detergent (Fairy Liquid and Dawn) from their respective sides of the Atlantic, and it would seem that their surface rheology is different. This comes down to the magnitude of the surface viscosity, which must be sufficiently high to establish Poiseuille flow.

4.

A COMPREHENSIVE THEORY?

In order to put such conclusions on a firmer basis, we need to capture many effects in a more comprehensive theory of drainage. It is a formidable task, even if one is prepared to adopt a semi-empirical approach. We require a description of the bulk flow through the borders and the junctions, coupling this with the surface flow. Various additional complications may also be necessary, including coupled motion of the adjoining films, the distortion by fluid pressure of the shape, and even the arrangement of junctions and borders. All this is called for without invoking any more exotic effects from physical chemistry (Marangoni effect, etc.) [12], and at every multidisciplinary conference we are told by the physical chemists that these are important. All this might seem too daunting, were it not for the challenge of new qualitative effects that are observed at high drainage rates (and liquid fractions), such as the convective instability described in the following section. These are not just numerical corrections to the basic theory. We have taken only tentative first steps towards reliable calculations in an augmented theory. Specifically, we have calculated the correction due to the inclusion of junctions that maintain the shape dictated by static equilibrium, while keeping Poiseuille flow. The challenges involved in further progress are taken up again in

5.

CONVECTIVE INSTABILITY

Convective instability at high rates of forced drainage, as sketched in Fig. 5, was first reported by our group [13], and confirmed by Douglas Durian and co-workers [14], with some additional features. It is found that, whenever forced drainage is impressed on a vertical column, uniform steady drainage becomes unstable when the liquid fraction exceeds about one half its maximum possible value for a foam (about 0.35, at which point the bubbles come apart). The instability

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Figure 5

Forced drainage at large flow rates leads to the establishment of a convective

bubble motion.

takes the form of a slow convective circulation of bubbles, the speed of which increases as the flow rate is further increased. That this instability awaits a satisfactory explanation is a good indication of the limited state of our success so far—it does not extend to combining drainage and rheology (or bubble motion), as seems required. Cilliers et al. have made a start on this in the context of foam flotation [15].

6.

FOAM RHEOLOGY

How does a foam respond to stress? Attempts to answer this have had to cope with awkward experiments and difficult theory, and there remains a good deal of uncertainty about anything beyond the most basic aspects of foam rheology [16]. Let us begin with those basic properties. A foam remains solid under low stress (Fig. 6), so it has bulk and shear elastic moduli. The bulk modulus is rarely significant, and is almost entirely attributable to the constituent gas. While surface tension or energy makes only a tiny contribution to the bulk modulus, it is entirely responsible for the shear rigidity, so that the shear modulus G is proportional to the surface tension On dimensional grounds we

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Figure 6

281

Sketch of the stress–strain relation for a liquid foam, subjected to increasing

strain. Shear modulus and yield stress depend strongly on the liquid fraction of the foam.

might expect

where c is a numerical constant (depending on and d is the mean diameter of bubbles. Such an equation, with the constant c being largely independent of the details of structure, is well supported by experiment and simulations. Indeed there is an old argument due to Stamenovic, which provides an excellent estimate for c for the dry foam, by considering a single junction and its adjoining Plateau borders, in the spirit of a mean field theory. There is a strong dependence of the shear modulus on the liquid fraction. For the usual case of a disordered foam, the shear modulus goes continuously to zero as the limit of stability is reached. Exactly how it does so is a tricky problem, not yet quite resolved. Of course, the elastic regime is quite limited: plasticity sets in at higher stresses and beyond a certain yield stress, the foam yields continuously, conforming to its designation as a “soft solid” (Fig. 6). At the heart of this plasticity are topological rearrangements, in which the local configuration of Plateau borders or bubbles is changed. At the dry limit, Plateau’s rules of equilibrium tell us to expect these sudden changes, whenever more than four Plateau borders come together in a single junction. Extensive simulations in both 2D and 3D have given us a good feeling for the role of these topological changes. Two kinds of simulation should

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be distinguished. In the first of these, the surface area of a large sample is minimized at all times, so that the effect of stress is simulated quasistatically [17, 18]. Such accurate simulations are very demanding in computer resources, at least in the case of 3D [19]. Accordingly they are usefully complemented by less accurate models, in which the contact of bubbles is roughly represented by simple forces. Using these, much larger samples can be used, and viscous effects at high strain rates can be included, at least in a heuristic style [20]. Again, the justification for the pursuit of such a task should be seen to lie in the promise of new phenomena. Indeed it became evident at an early stage of the simulations, that the conventional picture of the yielding process in terms of individual uncorrelated topological events was wrong, at least for wet foams. Instead they suffer large avalanches of these events [21]. However, no coherent physical picture of these avalanches has yet emerged (and not even a coherent definition), as the scale of the rearrangements seems to be strongly dependent on the simulation method [20, 22]. Experiments by Abd el Kader and Earnshaw [23] on two-dimensional bubble rafts under strain showed that bubble rearrangements may indeed occur in the form of correlated clusters. Whether the clusters can be system-wide if the foam is wet enough is still unanswered.

7.

FOAM DYNAMICS AT THE CELL LEVEL

In both drainage and rheology we are eventually drawn to detailed consideration of foam dynamics at the cell level. While this is part of the frontier of today’s research, it should not be thought that it has not been addressed before. In the case of the drainage process, Leonard and Lemlich [6] already recognized the role of surface viscosity and the need for a model that described the coupling of bulk and surface motions. Indeed they framed just such a model, and arrived at a criterion for the neglect of surface motion, which has been the prevailing assumption since that time. The criterion is

Here is the bulk viscosity, is the surface viscosity, and is the radius of curvature of a Plateau border. Note that where d is the average bubble diameter and is the liquid fraction. Since this criterion has been cited without any experimental verification (as far as we are aware) it is worth examining its basis.

The model used by Leonard and Lemlich [6] for this purpose uses only a single Plateau border. Surface flow velocity is set equal to zero where

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the adjoining films are attached. This condition entails surface shear if the surface is mobile. This shear is resisted by surface viscosity and driven by the forces associated with bulk shear. From today’s perspective, following the line of Koehler et al. [11], this physical model seems questionable. One would rather expect the junctions to play an important role, and the films to be compliant, that is, exerting no significant longitudinal force on the borders and moving freely with them. Such a revised model will lead to a different criterion for the neglect of surface viscosity, and experimental tests need to be made, by analyzing drainage experiments for various bubble sizes and liquid fraction. In the case of rheology, the history of ideas on local dynamics was reviewed by Kraynik in 1988 [25], and it would hardly be unfair to say that little has changed, on the fundamental level. A more comprehensive review has recently been prepared by H. M. Princen [26]. Again, we are confronted by reasonable but largely unconfirmed models, and again must hope for more systematic experiments. The models address the following question. What additional forces, beyond the quasi-static description, are induced by the local distortion of foam structure? Khan and Armstrong [27] made an estimate of the viscous resistance to the stretching of films, with no direct reference to

Plateau borders. However, in 1987 Schwartz and Princen introduced a different model, in which the pulling of films out of Plateau borders is assumed to be the dominant effect [28]. This model has since been used extensively in simulations by Kraynik and collaborators [19].

8.

SOLID FOAM FABRICATION

Many solid foams result simply from the freezing of a liquid foam, so that they have the same structure. As stiff, insulating, structural, energy-absorbing or cushioning materials, they have many familiar uses. While industrial processes are highly sophisticated, the underlying science is not yet well developed. Among those who have pioneered the science are Gibson and Ashby [24] and Kraynik [25]. In the formation of a solid foam, the processes shown in Fig. 7 are interlinked. It is generally expanded by blowing agents and solidified (by freezing or chemical reaction) before drainage can create inhomogeneity and collapse. Drainage is of particular concern for metallic foams, a relatively new class of materials presently under active assessment. In Fig. 8 some results of a simulation of metallic foam formation by our own group are presented [29].

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Figure 7 In one process for formation of a metal foam the metal is expanded by a blowing agent at its melting point and must be solidified before drainage leads to collapse.

Figure 8 Drainage occurs during the solidification process of a metal foam, resulting in a peculiar density profile [29].

Again, further progress may depend on better models of local dynamics.

9.

SOLID FOAM MECHANICS

A classic book on solid foams is Cellular Solids—Structure and Properties, now in its second edition [24]. A distinction must be made between closed-cell solid foams, where the cell faces of the parent liquid foam have been left intact, and open-cell foams, where they have been removed by some mechanism. For example, closed-cell foams find applications in heat insulation. The qualitative behavior of solid foams under an applied load is strongly dependent on the type of solid material. If it is elastomeric,

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the deformation is entirely elastic, that is, the foam recovers its initial shape once the stress is removed. A ductile material will result in some unrecoverable deformation, while foams made of a brittle material will simply fracture at some point of deformation. Analytic calculations performed on simple space-filling structures have now given way to finite-element computations of Surface Evolver generated foam structures, for both open- and closed-cell foams [19].

10.

CONCLUSIONS

Let us delineate the challenges that arise out of the work described in the previous sections (Fig. 9). (a) Present theories using approximations appropriate to dry foams need to be extended to wet foams (Fig. 2).

(b) Dynamic effects need to be added to static and quasi-static models. (c) We do not yet have realistic formulations that simulate simultaneous drainage and flow, wherever they are strongly coupled.

(d) It is likely that (b) and (c) require extensive studies of local dynamics, in which the roles of Plateau borders, films, and junctions are

all recognized.

Figure 9 State of the art of foam physics, frontiers (a) and (b). While static effects in dry foams are well understood, dynamic effects in wet foams evade an understanding. Question marks hint at areas that have in parts been tackled successfully but are still full of unresolved issues.

All these theoretical objectives will be greatly furthered by more and better experiments. In the face of continuing uncertainty about much of the underlying physics and chemistry, theory may diverge from reality, without systematic comparisons with data. Some readers may recognize within the stated desiderata certain problems already addressed in the literature—indeed we have cited references

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from the mid-1980s, to quite advanced ideas. But in reviewing such contributions, one is constantly reminded of the story of the blind men and the elephant .... Beyond the most elementary level, a comprehensive picture is needed, rather than isolated contributions. It may require another decade to accomplish this. In the meantime, foam scientists can enjoy the satisfaction of certain agreed basic principles, a luxury not to be found in most other branches of the study of soft matter. Our colleagues who devote their time to granular matter cannot even agree on the fundamental equations of their subject!

Acknowledgments This research was supported by Enterprise Ireland, Shell Amsterdam, and the European Space Agency, through a Topical Team and a contract under the Prodex program. Thanks are due to S. J. Cox for providing various diagrams.

References [1] Weaire, D., and S. Hutzler. 1999. The Physics of Foams. Oxford: Oxford University Press. [2] Zitha, P. L. J. (ed.). 2000. Proceedings of EuroFoam 2000, the EuroConference on Foams, Emulsions and Applications. Delft: Kluwer. [3] Brakke, K. 1992. The Surface Evolver. Experimental Mathematics 1, 141–165. [4] Princen, H. M. 1986. Osmotic pressure of foams and highly concentrated emulsions—I. Theoretical considerations. Langmuir 2, 519–524. [5] Princen, H. M., and A. D. Kiss. 1987. Osmotic pressure of foams and highly concentrated emulsions—II. Determination from the variation in volume fraction with height in an equilibrated column. Langmuir 3, 36–41. [6] Leonard, R. A., and R. Lemlich. 1965. A study of interstitial liquid flow in foam—Part I. Theoretical model and application to foam fractionation. American Institute of Chemical Engineers Journal 11, 18–25. [7] Leonard, R. A., and R. Lemlich. 1965. A study of interstitial liquid flow in foam—Part II. Experimental verification and observations. American Institute of Chemical Engineers Journal 11, 25–29. [8] Shih, F. S., and R. Lemlich. 1967. A study of interstitial liquid flow in foam— Part III. Test of theory. American Institute of Chemical Engineers Journal 13, 751–754. [9] Weaire, D., S. Hutzler, G. Verbist, and E. A. J. F. Peters. 1997. A review of foam

drainage. Advances in Chemical Physics 102, 315–374. [10] Cox, S. J., D. Weaire, S. Hutzler, J. Murphy, R. Phelan, and G. Verbist. 2000. Applications and generalizations of the foam drainage equation. Proceedings of the Royal Society of London A 456, 2441–2464. [11] Koehler, S. A., S. Hilgenfeldt, and H. A. Stone. 1999. Liquid flow through aqueous foams: The node-dominated drainage equation. Physical Review Letters 82, 4232–4235. [12] Exerowa, D., and P. M. Kruglyakov. 1998. Foam and Foam Films. Amsterdam: Elsevier.

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[13] Hutzler, S., D. Weaire, and R. Crawford. 1998. Convective instability in foam drainage. Europhysics Letters 41, 461–465. [14] Vera, M. U., A, Saint-Jalmes, and D. J. Durian. 2000. Instabilities in a liquid–

fluidized bed of gas bubbles. Physical Review Letters 84, 3001–3004. [15] Neethling, S., and J. J. Cilliers. 1999. Visualization and drainage of coalescing,

flowing foams. In Foams and Films (D. Weaire and J. Banhart, eds.). Bremen: MIT–Verlag. [16] Weaire, D., and M. A. Fortes. 1994. Stress and strain in liquid foams. Advances in Physics 43, 685–738. [17] Bolton, F., and D. Weaire. 1991. The effects of Plateau borders in the twodimensional soap froth—I. Decoration lemma and diffusion theorems. Philosophical Magazine B 63, 795–809. [18] Bolton, F., and D. Weaire. 1992. The effects of Plateau borders in the twodimensional soap froth—II. General simulation and analysis of rigidity loss transition. Philosophical Magazine B 65, 473–487. [19] Kraynik, A. M., M. K. Neilsen, D. A. Reinelt, and W. E. Warren. 1999. Foam micromechanics. In Foams and Emulsions (J. F. Sadoc and N. Rivier, eds.). Dordrecht: Kluwer Academic Publishers. [20] Durian, D. J. 1997. Bubble-scale model of foam mechanics: Melting, nonlinear behaviour, and avalanches. Physical Review E 55, 1739–1751. [21] Hutzler, S., D. Weaire, and F. Bolton. 1995. The effects of Plateau borders in the two-dimensional soap froth—III. Further results. Philosophical Magazine B 71, 277–289. [22] Jiang, Y., P. J. Swart, A. Saxena, M. Asipauskas, and J. A. Glazier. 1999. Hysteresis and avalanches in two-dimensional foam rheology simulations. Physical Review E 59 5819–5832. [23] Abd el Kader, A., and J. C. Earnshaw. 1999. Shear-induced changes in twodimensional foam. Physical Review Letters 82, 2610–2613. [24] Gibson, L. J., and M. F. Ashby. 1997. Cellular Solids—Structure and Properties, 2nd ed. Cambridge: Cambridge University Press. [25] Kraynik, A. M. 1988. Foam flows. Annual Review of Fluid Mechanics 20, 325– 357. [26] Princen, H. M. 2000. To be published.

[27] Khan, S. A., and R. C. Armstrong. 1986. Rheology of foams: I. Theory for dry foams. Journal of Non-Newtonian Fluid Mechanics 20, 1–22. [28] Schwartz, L. W., and H. M. Princen. 1987. A theory of extensional viscosity for flowing foams and concentrated emulsions. Journal of Colloid Interface Science 118, 201–211. [29] Cox, S. J., G. Bradley, and D. Weaire. 2000. Modelling metallic foam formation: The competition between heat transfer and drainage. Submitted.

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Tony Carwardine, Group Marketing Manager for John Wiley & Sons, displays his company’s materials while ICTAM 2000 Sectional Lecturer Hamda BenHadid examines a book. In all, eight exhibitors hosted booths, and others provided promotional materials for participants’ registration packets. (See pp. 561–562.)

MAGNETOHYDRODYNAMIC DAMPED BUOYANCY-DRIVEN CONVECTION Hamda BenHadid Laboratoire de Mécanique des Fluides et d’Acoustique, UMR-CNRS 5509 Université Lyon1/Ecole Centrale de Lyon, France [email protected]

Abstract

1.

We are particularly interested in the flows occurring in metallic liquids confined in cylindrical cavities and subjected to a temperature gradient and a constant or rotating magnetic field. We review previous experimental, numerical, and theoretical works about this topic and summarize the basic phenomena and the scaling laws for such characteristic quantities as velocities and boundary layers. The main features of the action of a constant magnetic field are analyzed, considering the effects of its orientation, intensity, and spatial uniformity on the nature and structure of convective flows. For the combined buoyancy and a rotating magnetic field-driven flow, the stability diagram for the onset of the oscillatory flow is given for a moderate aspect ratio Az = 4, and a representative value of the Prandtl number Pr = 0.026.

INTRODUCTION

In materials processing facilities, constant or rotating magnetic fields are used for a contactless control of fluid flow and thus mass transport during the processing. Both static and rotating magnetic fields can lead to a suppression of the temperature fluctuations. However, the two fields act on the fluid in different ways: for a static field the strength of the Lorentz forces (body forces) depends on the vigor of the convective flow and diminishes as the intensity of the convective flow decreases, whereas the rotating magnetic field can induce a flow even in the melt initially at rest. In the last decades the use of magnetic fields in the crystal-growth process from the melt was limited mainly to stationary magnetic fields, owing to their damping effects on the melt motion and thus the reduction of temperature fluctuations. The strength of the applied stationary field is generally in the range of several hundred millitesla (Utech and Fleming 1966, Series and Hurle 1991, Hurle 1993), but rotating fields of 289 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 289–306. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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a few millitesla are sufficient to affect the melt motion considerably and eliminate time-dependent flows (Dold and Benz 1995, 1997). Experimental observations of Dold and Benz (1995) indicate that forced convection in circular cylindrical cavities generated by a rotating magnetic field produces very complicated flows. They also show that above a critical imposed magnetic field or rotation speed, the flow fields exhibit several complicated oscillatory modes. This timedependent oscillatory convection is not only of inherent phenomenological significance but also of prime practical interest to materials processing. For example, in crystal growth, time-dependent oscillatory convection implies unavoidable crystal–liquid interface oscillations that influence solute segregation in the crystal. Thus, understanding the nature and origin of the oscillatory forced convection becomes necessary for applying an efficiently rotating magnetic field to control the fluid motion. In this paper, the influence of static or rotating magnetic fields on buoyancy-driven convection is investigated. We present results for two different geometries and several physical situations. At first we consider the influence of an externally imposed, static magnetic field on natural convection and investigate the flow in electrically insulating parallelepipedic or cylindrical cavities that are heated and cooled at the opposite side walls. Then we investigate the influence of a rotating magnetic field on the flow in circular cylindrical cavities. Natural convection is driven by a horizontal temperature gradient while forced convection is driven by spatially uniform, transverse, rotating magnetic field. The dynamical behavior of the forced convection is analyzed and its influence on the regime and structure of the convective natural flow is presented.

2.

MATHEMATICAL MODEL

Consider a container of length L and height H (or diameter D for a circular cylinder), containing an electrically conducting metallic liquid. The temperature of the right and left side walls are uniformly maintained at and respectively. The container lateral walls are thermally insulating. The material properties of the fluid—kinematic viscosity thermal diffusivity density thermal expansion coefficient magnetic permeability and electrical conductivity —are assumed to be constant. The container is subjected to an external magnetic field acting on the melt. Neglecting dissipation and Joule heating in the energy equation and using the Boussinesq approximation, we obtain

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the following system of nonlinear partial differential equations:

where t is the time, v = (u,v,w) is the velocity, p is the pressure, T is the temperature, and B is the magnetic induction. Since the magnetic Reynolds number (where V is the characteristic velocity) is very small with regard to unity, the unsteady induced field b is negligible (see e.g. Moreau 1990). In typical melt zones of semiconductor crystal growth, does not exceed Thus we can assume that the magnetic induction B is that of the applied field The Lorentz force is given by where the current density J is determined by Ohm’s law:

Conservation of electric current density gives

These equations are subject to no-slip conditions at all the walls. Because of the electrically insulating walls, the normal component of the electric current vanishes at the wall. Thus the boundary condition for the electric currents is where n denotes the normal to the boundary. For an externally imposed, stationary magnetic field, the induced electric field can be expressed as the gradient of an electric potential and thus, applying Ohm’s law to the conservation of the electric current density, one obtains

with a boundary condition With such a condition, the value of the potential is not unique and needs thus to be fixed at one point of the cavity. Note that the dimensionless form of the system (1)– (6) contains three dimensionless quantities: the Prandtl number Pr = which represents a physical property of the melt; the Hartmann number Ha = which represents the square root of the ratio of electrodynamic to viscous forces; and the Grashof number Gr = which represents the ratio of buoyant energy release to viscous energy dissipation.

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For an externally applied rotating magnetic field, the magnetic field may be described by

where is the magnetic field amplitude, denotes the frequency of the applied current used to generate the magnetic field, and t is the time. We assume that the well-known skin effect is neglected and thus the skin depth We also assume that the frequency is much larger than any typical velocity in the melt; e.g. The electric and magnetic fields are coupled through the Maxwell equations

Note that for a rotating magnetic field the specific governing parameters are the rotating Reynolds number and the magnetic Taylor number

3. 3.1.

DISCUSSION Stationary magnetic field

Theoretical approaches. The control of fluid flows with the help of external magnetic fields seems to be promising. Therefore, current research has intensified in the field of magnetohydrodynamic (MHD) convection, and several new results have been obtained in the last decade. In this context we refer to papers of Garandet et al. (1992) and Alboussière et al. (1993, 1996) for a theoretical approach. These authors proposed an analytical model based on the assumption of a parallel core flow, derived expressions for both the velocity and the induced current density in the presence of a horizontal transverse magnetic field, and predicted that for a rectangular cavity, these should vary as and respectively. They also showed that the analytical model is in good agreement with the numerical results over a limited parameter range, provided that the cavity has an aspect ratio larger than 3. The influence of the cross-section shape on the magnetic damping in the case of long horizontal cavities has been studied analytically by Alboussière et al. (1996). In the case of electrically insulating boundaries, the nature of the symmetry is found to govern the magnitude and structure of the damped velocity. In fact, with electrically insulated walls, the magnetically damped convection velocity varies as when the cross section has a horizontal plane of symmetry, and for nonsymmetrical shapes. When the walls are perfectly conducting, the damped veloc-

ity always varies as

The stability of the parallel core flow has

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been investigated by Bojarevics (1995), who assumed the existence of a two-dimensional solution in the rectangular cross section. Numerical simulations. Ozoe and Okada (1989) reported the results of a three-dimensional numerical investigation of the directional effect of a steady magnetic field on side wall convection in a cube of molten silicon. They found that the longitudinal orientation of the magnetic field is most effective for suppressing convection, whereas the transverse direction is least effective. The results of an experimental study of the flow of molten gallium in a cube are reported in Ozoe and Okada (1992), where they confirmed qualitatively their earlier numerical finding. BenHadid and Henry (1996, 1997) carried out numerical calculations of three-dimensional flows of mercury (Pr = 0.026) in both cylindrical and rectangular cavities of aspect ratio 4, with different orientations of the magnetic field. They found good agreement with the analytical estimates of Garandet et al. (1992) and Alboussière et al. (1993, 1996) for the damping of the velocity field. For the rectangular cavity, the investigation included free-surface effects. Interesting changes in the flow structure are reported and these appear to be closely linked to the distribution of the induced electric currents and their interaction with the applied magnetic field. Three-dimensional numerical simulations of melt convection carried out by Baumgartl and Müller (1992) in a cylindrical geometry subjected to a constant magnetic field show that only the models that include the electric potential equation agree well with the experimental results. The numerical results of Baumgartl et al. (1993) show that for flow fluctuations with typical amplitudes of about 4% of the mean values are suppressed, and the resulting steady flow has essentially the same features as the unsteady one. Möbner and Müller (1999) investigated three-dimensional natural convection driven by a horizontal or vertical temperature gradient under a stationary magnetic field.

Experimental approches. The pioneering experimental work on this problem was carried out by Utech and Fleming (1966) and Hurle, Jakeman, and Johnson (1974). They observed thermal oscillations in the flow and showed that a magnetic field applied orthogonally to the main convective flow can be used to damp the time dependence. Juel et al. (1999) investigated the effect of a horizontal transverse magnetic field on temperature fluctuations in a horizontal Bridgman configuration of aspect ratio four. Their combined experimental and numerical investigations show that the base flow state in a rectangular box filled with liquid gallium is modified into a new two-dimensional configuration. A value

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of Ha = 64 was sufficient to damp the initial temperature fluctuations caused by buoyancy-driven convection. Based on the vertical temperature difference in the center of the cavity, the comparisons between the experimental and numerical results are very good. In agreement with a numerical simulation, the dependence of the vertical temperature difference and similarly the maximum horizontal and vertical velocities, which vary approximately as for Ha > 100, have been experimentally validated. Recent experiments by Davoust et al. (1999) demonstrate the damping of thermogravitational flow of mercury in a horizontal circular cylindrical cavity of aspect ratio 10 by means of a uniform vertical magnetic field. The structure of the steady flow is investigated at large values of Hartmann number up to 235, in the light of current theoretical predictions, and good qualitative agreement is found. A study of the time-dependent convection flow is also performed, which shows in particular that damping is found to occur for small values of the Hartmann number between 1 and 10. Rayleigh–Bénard convection. The three-dimensional numerical simulations performed by Touihri et al. (1999) show that the flow in a vertical circular cylindrical cavity heated from below and subjected to a vertical or horizontal magnetic field keeps the same symmetries as that without a magnetic field. It is also noted that similar convective modes (m = 0, m = 1, and m = 2, where m is the azimuthal wave number) occur and are generally stabilized by the magnetic field. However, they are not equally stabilized. For an aspect ratio Az = 0.5 and for sufficiently large values of the Hartmann number, the mode m = 2 becomes the critical mode in place of m = 0. The horizontal magnetic field breaks some symmetries of the flow. The axisymmetric mode disappears, giving an asymmetric mode m = 02 (which is the combination of the m = 0 and m = 2 modes), whereas the asymmetric modes (m = 1 and m = 2), which were invariant by azimuthal rotation without a magnetic field, now have two possible orientations—either parallel or perpendicular to the applied magnetic field. These five modes are stabilized differently— weakly for those having the axis of the rolls parallel to the direction of the applied magnetic field, and strongly if the axis is perpendicular. The nonlinear evolution of the convection beyond its onset is also modified by the horizontal magnetic field. In fact, the secondary bifurcation, found in the pure thermal case for Az = 0.5, becomes an imperfect bifurcation consisting of two disconnected branches. Thermoelectromagnetic convection. Another potential use of magnetic fields is to create thermoelectromagnetic convection (TEMC).

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This convection is based on the fact that in systems where the solid– melt interface is not isothermal, the temperature gradient produces a Seebeck electromotive force where S is the thermoelectric power of the material. If the two gradients and are not parallel, thermoelectric currents are generated in the fluid. In the presence of an applied magnetic field, these induced electric currents produce a driving thermoelectric body force, which is able to generate flow in the melt. Lehmann et al. (1998a,b) used this possibility to modify the interdendritic convection in directional solidification by adjusting TEMC through the control of the applied uniform magnetic field. Direct numerical simulations of the effect of TEMC on Bridgman-growth semiconductors was carried out by Yesilyurt et al. (1999, 2000). Theoretical studies on the subject were done by Gelfgat (1994). Nonuniform magnetic field. The sensitivity of magnetohydrodynamic convection to a slight nonuniformity of the magnetic field (where b is the magnetic field perturbation) was studied experimentally and analytically by Neubrand (1995). The asymptotic approach confirms the experimental finding in the sense that, for electrically insulating walls, the existence of a slight magnetic field nonuniformity can entail a velocity disturbance of the order

Ha. Thus, it

is clear that for Bridgman-growth systems, there is a more severe relation than expected between efficiency and the magnet size and design. Following the experiments of Neubrand et al. (1995), BenHadid et al. (1996) carried out numerical calculations in a cylindrical cavity subjected to a magnetic field with cusp-shape nonuniformity equal to 10% of the uniform magnetic field. Their results agree rather well with the experimental finding of Neubrand (1995) and show that the MHD flow inside a horizontal cylinder is strongly changed by the nonuniformity in structure as well as in intensity.

Measure of diffusion coefficient.

Based on recent progress in the

knowledge of the damping effect of a uniform constant magnetic field on convective flow, an analytical model for mass transport by molecular diffusion and unsteady solute buoyancy-driven convection under magnetic field was developed by Maclean and Alboussière (2000) in continuation of the previous analytical study of Garandet et al. (1995). Experimental approaches based on the shear-cell technique (see Praizey 1989) were

performed recently by Botton et al. (2000) in order to measure under normal gravity the solute diffusivity in liquid metals and semiconductors for Sn–In (l%at) and Sn–Bi (0.5%at) couples. A vertical magnetic field (0.3 to 0.75 T) was used to damp out the convection that develops within

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a horizontal capillary tube subjected to a longitudinal temperature gradient. Starting from an initial one-dimensional step difference in composition, three phases in the diffusion process were identified and assessed by analytical models. During the first phase, where solutal convection dominates, the apparent diffusion coefficient depends on the duration of the experiment and decreases as In the last phase, thermal convection prevails and the scaling law for the apparent diffusion coefficient is of the form For the intermediate phase, the developed onedimensional semi-integral analytical model provides a good description of the experimental concentration profile evolution. Prom the measure of the apparent diffusion coefficient, the extrapolation to infinite magnetic field value gives the molecular diffusivity value. Note that the use of a magnetic field to measure the diffusion coefficient appears to be a promising method, as it gives results in good agreement with previous microgravity experiments on the same metallic alloys. However, it needs a careful control of the experimental conditions and a strong magnetic field of the order of 2 T or more.

Influence of static magnetic field parameters on the flow.

It

can be generally stated that an increase in the strength of the applied magnetic field leads to several fundamental changes in the properties of thermal convection. The convective circulation progressively loses its intensity, and when Ha reaches a certain critical value, which is found to depend on the direction (longitudinal, vertical, or transverse) of the applied magnetic field, the decrease of the flow intensity takes on an asymptotic form with important changes in the structure of the flow circulation. In order to characterize the effects of a constant magnetic field in three-dimensional parallelepidedic geometry, we plot in Fig. 1 the velocity vectors in the transverse vertical plane at Az/2, and in Fig. 2 the velocity vectors in the longitudinal horizontal plane at H/2. From these figures it appears that the flow structure may be separated into three regions: the core flow; Hartman layers (thickness which develop along walls that are not aligned with the applied magnetic field; and the parallel layers (thickness appearing along walls parallel to the applied magnetic field. Note that viscous effects are confined to these layers, whereas the remainder of the flow, namely the core flow, may be considered as inviscid. When the magnetic field is applied longitudinally, the main effect is observed on the vertical velocity (Fig. 2(a)), which is perpendicular to the direction of the applied field. A strong decrease is obtained in the core region, whereas the largest velocities appear in two (parallel) layers near the lateral walls. The characteristic decrease of the flow

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Figure 1 Horizontal velocity vectors in the transverse vertical (y, z) plane at Az/2

for three orientations of the applied magnetic field. The parameters are Gr = = 0.026, Ha = 100, and Az = 4.

Pr

Figure 2 Vertical velocity vectors in the horizontal (x, z) plane at y = 0.5, for three orientations of the applied magnetic field. The parameters are Gr = Pr = 0.026, Ha = 100, and Az = 4.

with increasing Ha corresponds to an asymptotic variation for the maxima of the absolute values of the longitudinal and vertical velocities. When the magnetic field is applied vertically, the horizontal velocity profile is linear with respect to the vertical coordinate y in the core region. The profile is given by and is connected to the lower and upper rigid boundaries by the classical Hartmann exponential

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profiles. The decrease of the intensity of the flow reveals an asymptotic behavior for large Ha values The velocities vary as and whereas in the parallel layers the velocity follows an law. For the transverse case the flow becomes two-dimensional except for a quick decrease of the velocities near the lateral walls where the noslip condition holds (Figs. l(c) and 2(c)). In this case, as has been shown by Alboussière (1994), the horizontal velocity profile is given by . The decrease of the intensity of the flow with increasing Ha affects more symmetrically the values of and and follows roughly an law for The velocity distributions are easily explained by analyzing the paths of the electric currents and the resulting Lorentz forces. The electric current and the related electric potential fields are presented in Figs. 3 and 4, respectively. As indicated by Eqn. (4), the induced electric cur-

Figure 3 The electric current projection in the transverse vertical (y, z) plane at Az/2, for three orientations of the applied magnetic field. The parameters are Gr = Pr = 0.026, Ha = 100, and Az = 4.

rents have two distinct sources, due to the melt motion through the magnetic field lines, and the electric field part, which appears to allow conservation of electric currents in the melt. The resulting electric currents from E produce two effects: on the one hand they allow the closing of the electric currents along the side walls and on the other hand they compete with the current produced by mainly in the vicinity of the corners. In general, the resulting total electric currents form loops in the sense dictated by the main values of and give rise to Lorentz forces that counteract the buoyancy-induced flow. Therefore, a significant braking effect on the convection is generated in regions where the electric current vectors are perpendicular to the applied magnetic field, whereas a weak braking effect is obtained in regions where the electric current is no longer perpendicular—in par-

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Figure 4 The electric potential field in the transverse vertical (x, z) plane at y = 0.5, for three orientations of the applied magnetic field. The parameters are Gr = Pr = 0.026, Ha = 100, and Az = 4.

ticular near the side walls, where consequently layers of overvelocities develop. It is interesting to note that for the longitudinal case, rather than generating long longitudinal current loops, conservation of the current still occurs mainly in cross-sectional planes with the creation of two superposed counter-rotating loops, with opposite senses of circulation in the right and left parts of the cavity (Fig. 3(c)). Detailed information concerning the scaling laws and the subtle actions of the electric currents in other situations can be found in papers of Alboussière et al. (1993) and BenHadid and Henry (1996, 1997). The damping of the oscillatory flow by a magnetic field is analyzed with respect to the amplitude of the temperature signals that are relevant to crystal growth. This was performed for Gr = 1.5 x Pr = 0.026, and Az = 3 by progressively increasing the Hartmann number up to stabilization. Note that for Ha = 0 the transition from timeindependent to time-dependent flow occurs around under the form of a Hopf bifurcation. It corresponds spatially to a breaking of the longitudinal and center-point symmetries, but keeps the symmetry that concerns the transverse direction. In Fig. 5 we give the time signal for the dimensionless temperature at the middle of the cavity (x = 0, y = 0 and z = Az/2) for the longitudinal direction of the applied magnetic field. From the results, the stabilization of the flow is obtained for and for vertical, transverse, and longitudinal magnetic field orientation, respectively. Thus, the vertical direction is the most efficient orientation for the applied magnetic field to produce a rapid damping on the flow. The least effective one is the longitudinal orientation. In general, as Ha is increased, the time signals become gradually more simple, going through a doubleperiod signal before giving a purely periodic signal that persists with

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Figure 5 Time series of the temperature at (x = 0, y = 0, and z = Az/2) for the longitudinal orientation of the applied magnetic field at Gr = 1.5 × 105 and increasing values of Ha.

a decreasing intensity up to stabilization. We found in addition that the corresponding power spectra show the progressive disappearance of

the low-frequency peaks, whereas the fundamental frequency is maintained up to the stabilization. This finding raises the question of whether the damping of oscillatory flow is due to the modification of the averaged base flow, or to a direct interaction between the magnetic field and the three-dimensional time-dependent flow. However, the original symmetry properties of the steady flow are not restored. Moreover, if Ha is now decreased, a periodic flow is found again as soon as Ha has a value below the critical one. This result does not give any evidence of hysteresis phenomena.

3.2.

Rotating magnetic field (RMF)

Recently, there has been a growing interest in the use of a rotating magnetic field (RMF) to control transport phenomena in the melt and thereby improve the quality of the grown crystals. There have been a number of papers devoted to this subject. Among the earliest significant contributions, Davidson and Hunt (1987) conducted a combined theoretical and experimental study and derived some of the basic scaling

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relationships relating the flow velocity to RMF. More recently, Gelfgat (1994) provided a good review on the possibility of using various types of RMF and combinations of these to control the hydrodynamics and heat/mass transport in the crystal melt. The theoretical works of Moffatt (1965), Richarson (1974), and Witkowski et al. (1998, 1999) provide a framework to study RMF in a liquid-metal column related to growth systems. The solutions obtained in these results established the parametric dependence of the flow field induced by RMF on the controlling nondimensional parameters. Kaiser and Benz (1998) performed a numerical stability analysis for the flow induced by transverse RMF and investigated the effect of the aspect ratio for finite circular cylinders. For the infinite cylinder, their numerical simulation stability results are in good agreement with those of Richardson (1974) and show that instability sets in with Taylor vortex structure. Friedrich et al. (1999) studied by experiments as well as by numerical simulations the interaction of buoyancy and RMF-driven convection in the classical Rayleigh–Bénard configuration. For a cavity of aspect ratio 2, they give

evidence of the existence of thermal waves with wave number k = 2, traveling azimuthally in the same direction as the rotation direction of the RMF. They also conclude that the small-scale temperature oscillations are caused by Taylor instabilities. Barz et al. (1997) in a model cavity with an aspect ratio 2, filled with InGaSn and under the conditions that found experimentally the ratio of the velocity of the secondary flow to that of the primary flow to be 1/5. They also found that, for magnetic Taylor number the boundary-layer regime prevails and the azimuthal velocity follows asymptotically a When buoyancy forces act on the melt, the structure of the flow depends on the relative strength of the forced and natural convection. The azimuthal flow does not influence the heat and mass transport directly; it sustains a secondary recirculating cell, which may interact with the prevailing buoyancy-driven flows or promote mixing of an otherwise quiescent melt. The experiment of Dold and Benz (1995) shows that for a circular cylindrical cavity of aspect ratio 2, filled with liquid gallium and subjected to a vertical temperature gradient, buoyancydriven convection is time-dependent for a Rayleigh number Ra = Magnetic fields in the range of 1 mT and rotation frequency of 50 Hz are sufficient to dominate the buoyancy convection completely and to reduce the amplitude of the buoyancy-caused temperature fluctuations by more than one order of magnitude. At the same time, the fluctuation frequency increases with the field strength.

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Influence of RMF parameters on the flow. In this part we give some numerical results for the stability thresholds of RMF-driven motions of isothermal fluid for a cylindrical geometry of aspect ratio Az = 8. The stability thresholds of buoyancy and RMF-driven motions are also considered. In general for RMF flows, exceeding a critical rotating Reynolds number leads to the generation of Taylor vortices in the middle of the cavity. For fixed Ha, it is interesting to note that the onset of Taylor vortex structure is found to depend essentially on the aspect ratio and on the rotation speed, i.e. From the results, which are in agreement with previous publications, it is clear that, due to the stabilizing influence of the bounding walls, the critical decreases with increasing aspect ratio. In Fig. 6 it is also clear that the number of Taylor vortices increases with the aspect ratio; we observe one pair of

Figure 6 Velocity vectors projected onto the longitudinal horizontal plane for four aspect ratios: Az = 2, 3, 5, and 8. The flow is induced by a rotating magnetic field at Ha = 1.

vortices for Az = 2, whereas eight pairs of vortices exist for Az = 8. At the end regions, due to the existence of the Eckman cell, the last vortices do not develop fully. In addition, we note that the Hartmann number has a significant stabilizing effect and produces a significant increase of the value of the critical magnetic Taylor number for the transition from time-independent to time-dependent flow. Starting from a situation where the buoyancy-driven convection is time-dependent (Gr = 1.5 × 105, Pr = 0.026 and Az = 3), we keep the Hartmann number constant at Ha = 1, and increase stepwise.

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The influence of is investigated from zero to the limiting value corresponding to the transition from time-dependent to time-independent flow. The results show that RMF can be used to damp significantly the temperature fluctuations caused by time-dependent buoyancy-driven convection. The stability diagram in space given in Fig. 7 shows clearly two distinct behaviors in the evolution of the critical Grashof

Figure 7 Stability diagram in the plane for combined buoyancy–rotating magnetic field-driven flow for Az = 3 and Pr = 0.026.

number, When the RMF is considered, increases and reaches a maximum. In this first stage, which illustrates the stabilizing effect of the RMF, the critical Grashof number evolves as Above a certain limit of a stabilizing effect of RMF is no longer valid, and a further increase in the leads to a rapid decrease of the value of which follows an law. Therefore, at moderate Gr numbers, a region of exists where steady state prevails. This region becomes narrower with increasing Grashof number. From the flow patterns given in Fig. 7, we note that in the first stage the flow patterns possess the characteristics of the buoyancy-driven flow, whereas in the second stage significant changes in the flow patterns occur and the flow becomes progressively dominated by RMF convection.

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4.

SUMMARY

We now summarize the basic action of a static or rotating magnetic field on convection. In the case of static magnetic field, for high Ha number the characteristic quantities undergo an asymptotic behavior and significant modifications of the flow structure occur. These modifications are governed mainly by the electric-current circulation. The results show also that a static field can be used to damp significantly the temperature fluctuations caused by time-dependent buoyancy-driven convection. The vertical orientation (parallel to the gravity vector) of the applied magnetic field is the most efficient orientation to damp the flow intensity; the longitudinal orientation is the least efficient. The recent investigations concerning RMF show that RMF also can be used to damp fluctuations induced by buoyancy-driven convection. For Ha = 1, the stabilization of the flow is obtained by RMF when its rotation speed exceeds some threshold value, which is found to depend on the vigor of the buoyancy convection. But this stabilization by RMF of the

initially time-dependent buoyancy-driven melt flow is found to occur in a

narrow range of the values, as destabilization by RMF occurs soon. It can be generally stated that the characteristics of time-dependent melt flows due to RMF at high frequencies and low amplitudes, are less detrimental to real crystal growth processes than the time-dependent flow due to buoyancy convection with typical high-amplitude and low-frequency fluctuations. Thus the interesting features are the relative intensity of the two flow modes—forced or buoyancy-driven flow—in order to know which flow mode dominates and determines the heat and mass transport in the melt.

Acknowledgments The author wishes to thank Prof. R. Moreau from MADYLAM, Dr. D. Henry from LMFA Lyon, and Dr. J. P. Garandet from CEN Grenoble for helpful discussions. This work is supported by the Centre National des Etudes Spatiales. The calculations were carried out on IBM-SP and SGI computers with the support of the CINES.

References Alboussière, T., J. P. Garandet, and R. Moreau. 1993. Buoyancy driven convection with a uniform magnetic field. Part 1. Asymptotic analysis. Journal of Fluid Mechanics 253, 545–563. Alboussière, T., J. P. Garandet, and R. Moreau. 1996. Asymptotic analysis and symmetry in MHD convection. Physics of Fluids 8, 2215–2226. Alboussière, T., A. C. Neubran, J. P. Garandet, and R. Moreau. 1997. Segregation during horizontal Bridgman growth under an axial magnetic field. Journal of Crys-

tal Growth 181(1–2), 133–144.

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Barz, R. U., G. Gerbeth, U. Wunderwald, E. Buhrig, and Y. M. Gelfgat. 1997. Modelling of the isothermal melt flow due to rotating magnetic fields. Journal of Crystal Growth 180, 410–421. Baumgartl, J., and G. Müller. 1992. Calculation of the effects of magnetic field damping on fluid flow. Comparison of magnetohydrodynamic models of different complexity. Proceedings of the 8th European Symposium on Materials and Fluid Sciences in Microgravity (ESA-SP-333), 161–164. BenHadid, H., and D. Henry. 1996. Numerical simulation of convective threedimensional flows in a horizontal cylinder under the action of a constant magnetic field. Journal of Crystal Growth 166, 436–445. BenHadid, H., and D. Henry. 1998. Numerical simulation of convection in the horizontal Bridgman configuration under the action of a constant magnetic field. Part II. Three-dimensional flow. Journal of Fluid Mechanics 333, 57–83. Bojarevics, V. 1994. Buoyancy driven flow and its stability in a horizontal rectangular channel with an arbitrary transversal magnetic field. Proceedings of the Second International Conference on Energy Transfer in Magnetohydrodynamic Flows. Assois, France, 1 (PAMIR). Botton, V., P. Lehmann, R. Moreau, and R. Haettel. 2000. Measurement of solute diffusivities. Part 2. Experimental measurements in a convection-controlled shear cell. Interest of a uniform magnetic field. International Journal of Heat and Mass Transfer, in press. Chandrasekhar, S. 1961. Hydrodynamic and Hydromagnetic Stability. Mineola, N.Y.: Dover Publications. Davidson, P. A., and J. C. R. Hunt. 1987. Swirling recirculating flow in a liquid metal column generated by rotating magnetic field. Journal of Fluid Mechanics 185, 67–106. Davoust, L. 1996. Convection naturelle MHD dans une cavité horizontale élancée. Ph.D. thesis, Institut National Polytechnique de Grenoble, France. Davoust, L., M. D. Cowley, R. Moreau, and R. Bolcato. 1999. Buoyancy-driven con-

vection with a uniform magnetic field. Part2. Experimental investigation. Journal of Fluid Mechanics 400, 59–90. Dold, P., and W. Benz. 1995. Convective temperature fluctuations in liquid gallium in dependence on static and rotating magnetic fields. Crystal Research and Technology 30, 1135–1145. Dold, P., and W. Benz. 1997. Modification of fluid flow and heat transport in vertical Bridgman configurations by rotating magnetic fields. Crystal Research and Technology 32, 51–60.

Friedrich, J., Y. S. Lee, B. Fischer, C. Kupfer, D. Vizman, and G. Müller. 1999. Experimental and numerical study of Rayleigh–Bénard convection affected by a rotating magnetic field. Physics of Fluids 11(4), 853–881. Garandet, J. P., C. Barat, and T. Duffard. 1995. The effect of natural convection in mass transport measurement in dilute liquid alloys. International Journal of Heat

and Mass Transfer 38(12), 2169–2174. Gelfgat, Y. M. 1994. Electromagnetic field application in the process of single crystal growth under microgravity. Proceedings of the 45th Congress of the International Astronautical Federation. Jerusalem, Israel. Oxford: Pergamon. Hurle, D. T. J., E. Jakeman, and C. P. Johnson. 1974. Convective temperature oscillations in molten gallium. Journal of Fluid Mechanics 64, 565–576. Juel, A., T. Mullin, H. BenHadid, and D. Henry. 1998. Magnetohydrodynamic convection in molten gallium. Journal of Fluid Mechanics 378, 97–104.

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Kaiser, Th., and K. W. Benz. 1998. Taylor vortex instabilities induced by a rotating magnetic field: A numerical approach. Physics of Fluids 10, 1104–1110. Lehmann, P., R. Moreau, D. Camel, and R. Bolcato. 1998a. Modification of interdendritic convection in directional solidification by a uniform magnetic field. Acta Materialia 46(11), 4067–4079. Lehmann, P., R. Moreau, D. Camel, and R. Bolcato. 1998b. A simple analysis of the effect of convection on the structure of the mushy zone in the case of horizontal Bridgman solidification—Comparison with experimental results. Journal of Crystal Growth 183(4), 690–704. Maclean, D. J., and T. Alboussière. 2000. Measurement of solute diffusivities. Part 1. Analysis of coupled solute buoyancy-driven convection and mass transport. International Journal of Heat and Mass Transfer 44(9), 1639–1648. Moffatt, H. K. 1965. On fluid flow induced by a rotating magnetic field. Journal of Fluid Mechanics 22, 521–528. Moreau, R. 1990. Magnetohydrodynamics. Dordrecht: Kluwer. Mössner, R., and U. Müller. 1999. A numerical investigation of three-dimensional magnetoconvection in rectangular cavities. International Journal of Heat and Mass Transfer 42, 1111–1121. Neubrand, A. C. 1995. Convection naturelle et ségrégation en solidification Bridgman sous champ magnétique. Ph.D. thesis, Institut National Polytechnique de Grenoble, France. Neubrand, A. C., G. P. Garandet, R. Moraeu, and T. Alboussiére. 1995. Influence of a slight nonuniformity of the magnetic field on MHD convection. Magnetohydrodynamics 31(1–2), 1–17. Ozoe, H., and K. Okada. 1989. The effect of direction of the external magnetic field on the three-dimensional natural convection in a cubical enclosure. International Journal of Heat and Mass Transfer 32, 1939–1954. Ozoe, H., and K. Okada. 1992. Experimental heat transfer rates of natural convection of molten gallium suppressed under an external magnetic field in either the x, y, or z direction. Journal of Heat Transfer 114, 107–114. Praizey, J. P. 1989. Benefits of microgravity for measuring coefficient transport in metallic alloys. International Journal of Heat and Mass Transfer 32, 2395–2401. Richardson, A. T. 1974. On the stability of a magnetically driven rotating fluid flow. Journal of Fluid Mechanics 63, 593–605. Series, R. W., and D. T. J. Hurle. 1991. The use of magnetic fields in semiconductor crystal growth. Journal of Crystal Growth 113, 305–328. Utech, H. P., and M. C. Fleming. 1966. Elimination of solute banding in indium antimonide crystal growth. Journal of Applied Physics 37, 2021–2023. Witkowski, L. M., and P. Marty. 1998. Effect of a rotating field of arbitrary frequency on a liquid metal column. European Journal of Mechanics B/Fluids 17(2), 239– 254. Witkowski, L. M., J. S. Walker, and P. Marty. 1999. Nonaxisymmetric flow in a finitelength cylinder with a rotating magnetic field. Physics of Fluids 11(7), 1821–1826. Yesilyurt, S., L. Vujisic, S. Motakef, F. R. Szofran, and M. P. Volz. 1999. A numerical investigation of the effect of thermoelectromagnetic convection (TEMC) on the Bridgman growth of . Journal of Crystal Growth 207(4), 278–291. Yesilyurt, S., L. Vujisic, S. Motakef, F. R. Szofran, and A. Croll. 2000. The influence of thermoelectromagnetic convection (TEMC) on the Bridgman growth of semiconductors. Journal of Crystal Growth 211(1–4), 360–364.

ADAPTIVE, NONLINEAR, AND LEARNING TECHNIQUES FOR THE CONTROL OF VEHICLE RIDE DYNAMICS Timothy J. Gordon Department of Aeronautical and Automotive Engineering Loughborough University, United Kingdom [email protected]

Abstract

1.

The ride dynamics of road vehicles is concerned with the control of whole-body vibration, to provide comfort and vibration isolation for occupants and transported goods. Ride isolation from road unevenness is conventionally achieved through the pneumatic tire, coupled with a spring and damper in the suspension; however substantial benefits can be derived from active computer control of the suspension system. Active ride control also benefits from on-line adaptation, and similar advantage can be derived via nonlinear feedback control. This paper reviews the fundamental issues and considers the potential for future vehicles, against a background of increasing total system complexity and interaction, as well as the continuing need for robust, safe, and faulttolerant operation. Consideration is also given to the use of “intelligent” control systems that adapt and learn in real-time on the vehicle.

INTRODUCTION

The tire and suspension of a road vehicle provides an interface between the vehicle structure and the road surface, to transmit forces for lateral and longitudinal handling control—braking, acceleration and cornering— and isolate road surface irregularities for vertical ride control. This paper is concerned with the control of ride dynamics, wherein the principal degrees of freedom are body-bounce, pitch, and roll, as well as the relative motion of the wheels to the body. Assuming a rigid vehicle structure, this corresponds to seven principal degrees of freedom for a four-wheeled passenger vehicle. The ride control problem is to minimize accelerations in the three body degrees of freedom, as well as control body attitude (angular deflections) with respect to roll and pitch. This is to be achieved for a large range of road conditions and vehicle speeds, and working within the limited deflections available for both the suspen307 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 307–326. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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sion workspace and the tire structure. A further limitation derives from the safety requirement that sufficient vertical tire loads are maintained, in order to provide in-plane forces for cornering and braking. Active suspension control is achieved through some form of mechanical actuation—typically a servo-hydraulic unit incorporating an electronically controlled spool valve to regulate hydraulic pressure, and hence applied force or torque. The details of such actuators will not be considered here; and while there are significant practical issues relating to actuator technology, reliability and failure effects, cost, and power consumption, it will be sufficient here to regard the suspension actuator as a sub-system that can deliver ‘force on demand’. Also required is a sensor set and a state observer (or equivalent signal processing) to reconstruct real-time dynamic states from measured outputs. Again, we shall assume this has been done, and the details will be skipped (see however [1]). Although this paper has a specific focus on ride control, many of the issues covered apply to a much wider class of dynamic control problems. The general context includes adaptive vs. nonlinear control, effectiveness of feed-forward information, intelligent and learning control, as well as issues of dynamic system integration. In the next section, a review is undertaken of the main issues for automotive ride control. Section 3 considers some aspects of control system adaptation, which are taken further in Section 4 in the form of on-line reinforcement learning. Optimal nonlinear control techniques are then described in Section 5, and the concluding section includes an outline of future research challenges.

2.

OPTIMAL RIDE CONTROL

The quarter-car suspension model, shown in Fig. 1, has been used widely for fundamental investigations into ride control [2, 3, 4]. Though it contains just two of the seven ride degrees of freedom, it represents much of the basic dynamics, and has proved particularly suitable for the evaluation of fundamental control concepts. Vehicle parameters have been based on a medium-sized passenger car, and typical values may be found in the cited references. For the standard passive system, the active force is zero, and there is relatively limited scope for suspension tuning, at least within the context of linear feedback and the quarter-car model. With the introduction of computer controlled actuation, there is considerable scope for delivering additional forces that might improve suspension performance in some way. The most direct way to optimize the ride performance of the suspension is to use linear feedback of suspension states, via such optimal

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Figure 1 Quarter-car suspension model.

design methods as linear quadratic regulator (LQR) and Here we focus on the simpler LQR method, where the time integral or statistical expectation of a quadratic cost function is to be minimized, and it can be rigorously shown that the closed-loop system gives minimum cost response to sudden events and Gaussian white noise inputs. The cost function is commonly taken to be of the form

where T(t) is the dynamic vertical tire load, S(t) is the suspension workspace deflection, and A(t) is the vertical sprung mass (body) acceleration. The positive weighting parameters are adjustable to suit the particular vehicle parameters and specific design requirements. Since it is only the ratio between these parameters that is significant, we may set without any loss of generality. Parameters and may be regarded as Lagrange multipliers, used to impose constraints on the peak or RMS values of T and S, under design conditions. In that case, the LQR optimal controller provides a rigorous minimum RMS value for A(t), or in other words gives a “best comfort” optimal controller. Note that when Gaussian white noise is considered as an input for controller design, it is more for analytical convenience than for real-

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world accuracy. However, the approximation is not unreasonable when the model equations are expressed so that the dynamic input is the vertical velocity v(t) of the road–tire interface. The effectiveness of active control is indicated in Fig. 2, where frequency response gains are shown for both active and passive suspension systems. In each case v(t) is the input, and T(t), S(t), and A(t) are the three outputs. Active 1 has been tuned against the passive sys-

Figure 2 Active and passive frequency response gains.

tem, to give identical peak tire load and suspension deflections under design conditions, here prescribed as unit velocity initial conditions for

the unsprung and sprung masses respectively. For comparison, Active 2 was allowed 25% more tire load variation and suspension deflection under design conditions. From the lower plot, both active systems enjoy substantially improved ride isolation, compared with Passive, with the majority of improvement being shown near the “body bounce” resonance frequency at approximately 1 Hz. Active 2 has the lowest body acceleration gain across the whole frequency range, except for a single frequency, which coincides with the ideal “wheel-hop” resonance frequency for free vibration of the unsprung mass on its tire spring. This is a particular example of an “invariant point”, one of a small number that constrain dynamic

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responses in the quarter-car system; all active and passive suspension system variants have coincident gains at such specific frequencies [6]. The frequency responses in the upper two plots, for the “constraint” variables, show that Active 1 is always at least comparable with Passive, with the exception of increased suspension gain at low frequencies, a feature that is described further below. Active 2 on the other hand allows greatly increased wheel motion at the wheel-hop resonance, giving rise to increased dynamic tire loads and suspension variations at this frequency. However, it is fundamental to what follows to note that this is not always detrimental to system performance; provided the excitation amplitude at these frequencies is sufficiently low, Active 2 can usefully improve ride comfort isolation compared with Passive and Active 1. While the active suspension is an interesting concept for dynamic control, it may not always be considered feasible, due to practical considerations of weight, packaging, cost, and power consumption. In this case, an interesting alternative is the semi-active suspension, which is designed to modulate the dissipation of energy from the suspension, but without any external source of mechanical power [7]. One way this can be achieved is via an electronically controlled spool valve within an otherwise standard hydraulic damper; provided the valve has sufficiently fast transient response, and the range of damping rates is sufficiently large, the semi-active system performance can theoretically approach that of the active suspension. This is demonstrated in Fig. 3, where three road bumps generate significant disturbance for the passive system, while active and semi-active both filter the input approximately equally. The semi-active model is very much idealized (though so is the active system), but the conclusion that a well designed semi-active system can produce significant ride benefits is certainly valid. An interesting problem for active suspensions arises through the use of “skyhook damping”. In the LQR active control, all available suspension states are fed back into the actuator, including the “absolute” vertical body velocity—i.e. the velocity of the sprung mass relative to the original rest frame. In fact, without such skyhook damping, the ride control system reverts to something very similar to a standard passive suspension, and therefore its use is particularly significant. The problem alluded to occurs whenever the active suspension vehicle ascends a road of constant incline; this induces a steady-state velocity that feeds back a non-zero signal to the actuator. This results in a steady-state suspension offset, where the suspension spring force (including any suspension deflection feedback gain in the active control law) cancels the erroneous skyhook damping force. The problem is also apparent from the nonzero suspension gain at 0 Hz shown in Fig. 2. It can be “solved” in a number

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Figure 3 Comparison of passive, active and semi-active systems.

of ad hoc ways, but is fundamentally due to an incomplete description of the road surface geometry in the underlying optimal design. The incompleteness relates to low frequencies generally, not just to steady-state, and a systematic solution to this problem requires a subtle modification to the control optimization [8].

3.

ADAPTATION OF RIDE CONTROL FEEDBACK It is apparent from Fig. 2 that there is a compromise, or trade-off, between tire and suspension deflections on the one hand, and ride comfort on the other. When the road input amplitude is large, there is danger of exceeding available suspension workspace limits, or of losing contact between the tire and road surface; in this case ride comfort must be sacrificed for the suspension, and hence the vehicle, to operate safely. At lower amplitudes, where such limits are not an issue, it is desirable to ‘soften’ the active suspension and improve ride comfort. To achieve this, some form of on-line adaptation is necessary. Conceptually, the simplest approach is “gain scheduling”; a set of LQR gains are designed off-line to accommodate the various expected road inputs types (varying with amplitude and frequency content), and as different conditions

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Figure 4

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Control of vertical tire load for severe events.

are experienced, the vehicle response is used to estimate ‘best-fit’ design conditions, and hence adapt controller gains [9]. Table 1

Test road profile component events (at 15 m/s)

Figures 4 and 5 show suspension responses to an aggressive test road profile that consists of a series of raised sinusoidal bumps and linear inclines—one large one up, followed by four shorter ones down—see Table 1. The road speed is a moderate 15 m/s, but gives rise to severe

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Figure 5 Control of suspension workspace.

vertical inputs, as can be seen from the Passive responses in the following plots. Figure 4 shows vertical tire loads for Passive, Active 1, and Active 2, as well as a further Adaptive system. These tire loads now include the static component, and should therefore remain well above zero for safe handling control. Clearly this is far from the case for Passive; and while the Active systems are greatly improved, the Adaptive system provides a much better and consistent minimum load across the various surface changes. In Fig. 5, unconstrained suspension travel is shown, even though a real vehicle is always subject to workspace limits. For a medium-sized passenger vehicle, workspace of around 0.1 m would be typical, or even

generous, and it is clear that the non-adaptive systems all exceed this limit on occasion. In reality, a suspension will include bump rubbers to reduce the worst effects of metal-on-metal contact, but it is reasonable to expect an active suspension to be designed to control such events as far as possible, and it clear that the adaptive system is superior in this aspect. An important point here is that an adaptive suspension of this type combines the above authority over tire load and workspace, with high levels of ride comfort wherever this is possible. Figure 6 shows the corresponding vertical body accelerations, where Passive gives relatively

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Figure 6 Vertical body accelerations.

poor ride comfort whenever the road generates significant amounts of body bounce—on the initial set of bumps, and on the later ramp events— though between t = 6 s and 15 s, where the input frequencies are higher, the passive ride is comparatively reasonable. Overall, the ride comfort predicted for Active 1 is much improved over Passive; on the other hand, Active 2 is ‘too good to be true’, because the system design has sacrificed suspension and tire load control for the apparent improvement in ride comfort. For example, for 15 s < t < 15.5 s, the ramp induces very little body acceleration, but there is very large suspension compression, as seen in Fig. 5. A more reasonable comparison can be made between Adaptive and Active 1, where it is clear that Adaptive gives equal or lower accelerations, except where the need to meet tire or suspension constraints leads to higher suspension forces, and hence higher body accelerations.

In terms of real-time application, there are a number of remaining issues. The adaptation implemented here is based on a full knowledge of the road profile, where in reality the system must adapt as a function of vehicle response. For example, the sudden events occurring between 19 s and 23 s on the above plots, which cause poor tire load control for the other systems, each require an adaptive ‘switch to hard’ response to occur within around 20 ms. A second issue relates to the integration with

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handling dynamic control, which is important at low lateral accelerations on smooth roads; a very ‘soft’ suspension provides poor control of body motion, and also poor feedback to the driver. Both of these aspects are taken up later.

4.

LEARNING CONTROL

An alternative approach to control system design and adaptation is via reinforcement learning. Such an approach uses a computer system to act on the dynamic controller, and adapt the operation of the controller to reinforce desirable closed-loop performance. As with traditional adap-

Figure 7 System architecture for the CARLA learning process.

tive control, reinforcement learning must operate on a slower timescale than the underlying system dynamics, and normally the timescale is very much slower. The simplest such approach is non-associative reinforcement learning, where the learning system treats the controller and vehicle as a ‘black box’, and applies actions by setting control parameters. Recent work on the application of reinforcement learning to ride control has used stochastic learning automata in both simulation and realtime practical implementation [10, 11, 12]. An effective approach has been to employ a ‘team’ of Continuous Action Reinforcement Learning Automata (CARLA) [13]. Each CARLA defines an action based on probability densities, and in this case each action corresponds to defining a control system parameter, such as a feedback gain (Fig. 7).

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At each iteration n, the ith CARLA has an associated probability density where is the corresponding control parameter with a pre-defined range Using a pseudo-random number generator with uniformly distributed variate, is selected and used to define according to the formula

The full set of control parameters are then implemented, in real time or in simulation, to control the vehicle dynamics for a pre-assigned period of time, and then a cost function, possibly similar to that of Eqn. (1), is used to assess performance cost J(n) and define a reward signal A reward–inaction reinforcement algorithm alters the set of probability density functions whenever ‘improved’ performance is achieved, and no change is made otherwise. The reward and probability update equations are

where and are respectively the median and minimum cost values measured during the previous m samples, is a scale factor, defined to normalize to unit area within and is a symmetric Gaussian neighborhood function, which ‘diffuses’ the reinforcement of probability over neighboring values of the controller gain

The CARLA approach has been applied to a variety of simulation and real-world control problems. In the simulation of a fully active suspension control system, the method reproduces optimal LQR gains with minimal errors. However, being much less limited in its underlying mathematical assumptions, the CARLA approach can successfully optimize controller gains for more general conditions than are possible with LQR—for example, the learning system can directly use real or realistic road surface data in place of Gaussian white noise [12]. Figure 8 shows a typical probability density function, created during real vehicle learning of a semi-active controller; in this case the learning took place using stationary random disturbances from a four-poster road simulator rig. In the figure, the sharp peak created at around = 1000

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Figure 8 Probability density convergence for semi-active ride control.

corresponds to a preferred suspension spring stiffness gain of around 50% of the existing passive value on the vehicle. Thus improved control results from the passive suspension force cancelling 50% of the spring force whenever energy considerations make this possible. Furthermore, this result does not depend on any modeling assumptions, since it was ‘derived’ from the actual vehicle test; the

approach thus provides a valuable diagnostic tool, as well as providing an optimal controller. It is worth noting that learning on real roads is a much more challenging exercise, due to the nonstationary nature of the input, and while

some progress has been made to achieve satisfactory learning and adaptation, further work remains to be done.

5.

NONLINEAR OPTIMAL CONTROL

There are many situations where nonlinear control techniques may be preferred over linear methods, though there are often serious technical difficulties associated with achieving—or even defining—the ‘best’ nonlinear control for any given problem. Particular cases are when the underlying system model is taken to be nonlinear [14, 15] or when the control objectives naturally lead to nonlinearity [16, 17]. For ride control, the former category includes the application of semi-active or other force-limited actuation, while the latter includes the imposition of fixed

limits on suspension workspace and absolute criteria for vertical tire load control. The use of gain scheduling and other forms of adaptive control is essentially a heuristic approach, used in the absence of more powerful nonlinear control design methods.

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Nonlinear optimal control (NOC) is a well-known concept that is under-used, mainly because implementation is often impractical. The results described below are obtained using the famous Pontryagin maximum principle, via numerical techniques outlined in [15, 17]; the approach is computationally intensive and is not suitable for real-time application. And while neural networks offer new opportunities for real-time application of NOC, the underlying ‘curse of dimensionality’ remains—for anything other than low-order systems, large amounts of information must be encoded into the neural network, and excessive training times may result. However, NOC offers interesting opportunities for investigating the theoretical limits of a control system, and hence provides a benchmark for more practical control schemes; here we shall use this approach to assess the active suspension results obtained above. For demonstration, we modify the control objectives, but retain an underlying linear description of the suspension. The dynamic cost for suspension workspace is modified to impose rigid limits at with a = 100 mm in simulation. A term

is added to the cost function (1)—with Active 2 parameters—and clearly

as To improve the tire load control of Active 2, a further term is also added that becomes significant only for large dynamic tire load variations: The resulting suspension and tire components of the cost function (including quadratic terms from Eqn. (1)) are shown in Fig. 9. The resulting dynamic responses on the test road surface are shown in Fig. 10, where the nonlinear results are referred to as NOC, and again compar-

isons are made using Active 1 as a reference. Overall, as expected, the NOC suspension workspace is rigorously maintained within the available limits. There is a single event at t = 15.5 s, at the end of the upward ramp, where NOC tire load is controlled less effectively, but it is easy to understand why. Active 1 exceeds the suspension constraint at this point, in an effort to hold the tire on the road surface; this is simply not possible, and the NOC results reflect this. Elsewhere (e.g. 6 s < t < 10 s) NOC maintains somewhat better tire load control, and where constraints are unimportant (e.g. 10 s < t < 15 s) the body acceleration for NOC is also improved. Thus NOC shares all of the benefits of the previous Adaptive control, but without the associated difficulties of detection and inference.

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Figure 9 Modified cost functions for tire load and suspension workspace.

An interesting result arises from the ability of nonlinear optimal control to exploit a wide range of cost function modifications. The reader may have noticed an anomaly, that the above cost functions—whether quadratic or more general—have been chosen symmetrical with respect to both tire load T and suspension deflection S. While this is certainly sensible for S, it is clear that only reductions in tire load cause concern for handling control, and therefore in place of Eqn. (1) we might substitute the following:

Simulations show that, provided the value of is doubled from the previous value (to reflect the fact that it is used only ‘half of the time’) the dynamic responses are virtually identical to the previous linear system. Far more significant is the effect of using previewed (or ‘look-ahead’) information from the road surface. Although this has already been

included in a very simple manner for the Adaptive system, there are great potential benefits when such information is available as a dynamic input to the controller [4, 15, 18, 19]. Again both linear and nonlinear approaches are available, though linear methods are not valid for

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Figure 10

321

Comparison of NOC with linear Active 1 responses.

semi-active suspension, or other conditions where the underlying model is nonlinear, or where as above the control objectives force the nonlinearity. Figure 11 compares the previous NOC results with the case where accurate one-second preview is available. Here the improvement in body acceleration is impressive and overwhelming—the active suspension makes effective use of available suspension workspace and tire load variations, without losing control, and to effect excellent ride vibration isolation. Of course there are questions over the degradation of such results due to modeling errors, sensor type, and accuracy, and over signal processing requirements for preview-based control; but the fact remains that ride isolation is very sensitive to the addition of such information, and even imperfect preview is potentially effective. For example, in [18] a linear analysis shows that rear suspension response may be significantly improved by feeding forward the dynamic responses from the front wheels.

6.

CONCLUSIONS

The above work has addressed a number of fundamental issues concerned with the ride dynamics of road vehicles. The literature on the subject is vast, and the references given here provide only a sampled

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Figure 11

Inclusion of preview information for NOC.

introduction. Many of the issues are well known, though the results presented here on nonlinear optimal control are new. In this short paper it is not possible to go very far into the issues that lie beyond the quartercar concept, and indeed much of the literature is also restricted to this simple model. However, in conclusion is seems appropriate to consider current and future research issues involving ride dynamics, and these very definitely go beyond such a limited perspective, and involve interactions with handling, safety, and driver information systems—in other words, are concerned with ‘dynamic systems integration’. In its simplest practical form, systems integration means increasing the use of common hardware, sensors, communication, and data analysis within the vehicle environment, on the basis that duplication is wasteful. A more fundamental aspect of dynamic systems integration is to balance design objectives for various systems, so that interactions and compromises (e.g. between ride and handling) are properly addressed at the design stage. This is essentially the application of multivariable control to a large and complex system—the vehicle—so that multiple objectives can be optimized in a systematic and simultaneous fashion. Unfortunately, a global system approach to vehicle dynamics control is not necessarily practical or even desirable. A basic problem is that vehicle manufacturers and systems suppliers are often not the same com-

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panies, so that complete sharing of design knowledge—especially control algorithms—simply does not occur. Control algorithms are often embedded as ‘black boxes’, and a systems supplier is more likely to ‘tune the system to the vehicle’ rather than share details of embedded algorithms

with the vehicle manufacturer or other systems suppliers. Although this is ‘only’ a commercial problem, it is a very real one, and is associated with a more fundamental problem of vehicle–system combinatorics; as new vehicles and systems are developed, between various different companies, each offering several different options to customers, the number of distinct vehicle variants grows factorially, and it becomes totally

unreasonable to design separate software for every such combination. Also, as vehicles and their control systems become increasingly complex, there is less scope for providing underlying mathematical models for the total system behavior; and where such models do exist, they are

limited by the need for linearity in most existing multivariable control methods. Thus the expected trend for dynamic systems integration on road vehicles is for more intensive real-time computation and on-line optimization, more intelligence and learning within the vehicle systems, more parallelism and modularity in design, and for the integration (including ‘trade-offs’ and compromises) to actually take place dynamically on

the vehicle. Some small progress in this direction has been reported by the author [20, 22] but it is clear that new techniques are required. The work presented here suggests that a minimum requirement for such

dynamic integration is that it should include scope for nonlinear behavior, and make use of adaptation and learning in the real-world operating environment. A further key requirement, that potentially comes ‘for free’, is that of robust and fault-tolerant operation; dynamic integration and optimization in real time can also accommodate dynamic re-optimization once localized failures are detected [20, 21]. Finally there must be suitable interfaces with the driver and passengers—so that individual preferences and driving styles are taken into account in the dynamic integration process. These future challenges are immense, but also immensely exciting.

References [1] Best, M. C. 1995. On the modelling requirements for the practical implementation of advanced vehicle suspension control, Ph.D. Thesis, Loughborough University. [2] Thompson, A. G. 1970. Design of active suspensions. Proceedings of the Insti-

tution of Mechanical Engineers (Part I) 185, 553–563.

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[3] Thompson, A. G. 1976. Active suspensions with optimal linear state feedback. Vehicle System Dynamics 5, 187–203. [4] Hac, A. 1992. Optimal linear preview control of active vehicle suspension. Vehicle System Dynamics 21, 167–195. [5] Yamashita, M., K. Fujimori, C. Uhlik, R. Kawatani, and H. Kimura. 1990. control of an automotive active suspension. Proceedings of the 29th Conference on Decision and Control, 2244–2250. [6] Hedrik, J. K., and T. Butsuen. 1988. Invariant properties of automotive suspensions. Proceedings of the Institution of Mechanical Engineers International Conference on Advanced Suspensions, London. [7] Margolis, D. L. 1982. Semi-active heave and pitch control for ground vehicles. Vehicle System Dynamics 11, 31–42.

[8] Fairgrieve, A., and T. J. Gordon. 2000. On-line estimation of local road gradient

for improved steady-state suspension deflection control. Vehicle System Dynamics 33 (Supplement—Dynamics of Vehicles on Roads and Tracks), 590–603.

[9] ElBeheiry, E. M., and D. C. Karnopp. 1996. Optimal control of vehicle random vibration with constrained suspension deflection. Journal of Sound and Vibration 189, 547–564. [10] Gordon, T. J., C. Marsh, and Q. H. Wu. 1993. Stochastic optimal control of active vehicle suspensions using learning automata. Proceedings of the Institution of Mechanical Engineers—Part I (Journal of Systems and Control Engineering) 207, 143–152.

[11] Marsh, C., T. J. Gordon, and Q. H. Wu. 1995. The application of learning automata to controller design in slow-active automotive suspensions. Vehicle System Dynamics 24, 597–616. [12] Frost, G. P., T. J. Gordon, M. N. Howell, and Q. H. Wu. 1996. Moderated reinforcement learning of active and semi-active vehicle suspension control laws. Proceedings of the Institution of Mechanical Engineers—Part I (Journal of Systems and Control Engineering) 210, 249–257. [13] Howell, M. N., G. P. Frost, T. J. Gordon, and Q. H. Wu. 1997. Continuous action reinforcement learning applied to vehicle suspension control. Mechatronics 7, 263–276. [14] Ono, E., S. Hosoe, H. D. Tuan, and Y. Hayashi. 1996. Nonlinear control of active suspension. Vehicle System Dynamics 25 (Supplement—Dynamics of Vehicles on Roads and Tracks), 489–401. [15] Gordon, T. J., and R. S. Sharp. 1998. On improving the performance of automotive semi-active suspension systems through road preview. Journal of Sound and Vibration 217, 163–182. [16] Gordon, T. J., C. Marsh, and M. G. Milsted. 1991. A comparison of adaptive LQG and nonlinear controllers for vehicle suspension systems. Vehicle System Dynamics 20, 321–340.

[17] Gordon, T. J., and M. C. Best. 1994. Dynamic optimization of nonlinear semiactive suspension controllers. Proceedings of the Institution of Electrical Engineers (IEE) 1994 International Conference “Control 94” (IEE Publication no. 389), Warwick, U.K., 332–337.

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[18] Sharp, R. S., and D. A. Wilson. 1990. On control laws for vehicle suspensions accounting for input correlations. Vehicle System Dynamics 19, 353–363. [19] Louam, N., D. A. Wilson, and R. S. Sharp. Optimization and performance enhancement of active suspensions for automobiles under preview of the road. Vehicle System Dynamics 21, 39–63. [20] Gordon, T. J. 1995. An integrated strategy for the control of complex mechanical systems based on sub-system optimality criteria. Proceedings of the IUTAM Symposium on Optimization of Mechanical Systems (Stuttgart, 1995) (D. Bestle and W. Schiehlen, eds.). Dordrecht: Kluwer, 97–104. [21] Gordon, T. J. 1996. An integrated strategy for the control of a full vehicle active suspension system. Vehicle System Dynamics 25 (Supplement—Dynamics of Vehicles on Roads and Tracks), 229–242. [22] Frost, G. P., M. N. Howell, T. J. Gordon, and Q. H. Wu. 1996. Dynamic vehicle roll control using reinforcement learning. Proceedings of the 1996 United Kingdom Automatic Control Council International Conference on CONTROL 96 (Exeter, U.K.), 1107–1118.

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ICTAM 2000 participants line up at one of several carving tables at the Welcome

Reception on Monday evening, 28 August 2000. University of Illinois president James J. Stukel hosted the event.

RECENT ADVANCES IN TURBULENT MIXING Paul E. Dimotakis Graduate Aeronautical Laboratories California Institute of Technology, Pasadena, Calif., USA [email protected]

Abstract

1.

Experimental, modeling, and direct-numerical-simulation (DNS) studies have advanced our understanding of turbulence and mixing. The mixing transition that occurs across a broad spectrum of flows at outerscale Reynolds numbers in the vicinity of or Taylor Reynolds numbers of will be discussed. Inflow/initial-condition effects in high-Re shear layers will also be discussed. Comparisons between DNS studies of strained diffusion-flame regions and experiments in chemically reacting shear layers provide some new insight in the overall chemical-product formation. DNS studies of the Rayleigh–Taylor instability (RTI) between miscible fluids, with identical boundary conditions and different initial perturbations, reveal an initial-growth regime dominated by diffusion, with a subsequent nonlinear growth that depends on the details of the initial perturbations for as long as the simulations were run. Mixing in RTI flow is found even more sensitively dependent on initial conditions. The discussion concludes with some general comments on high-Re turbulence.

INTRODUCTION

Turbulent mixing dynamics span the full range of flow scales, from the outer spatial scales that define the extent of the turbulent region, or perhaps some internal large scale (such as the mesh in grid turbulence) of the flow, to molecular-diffusion scales where all mixing occurs. Outerscale structure and dynamics dominate the spatio-temporal growth of the turbulent region and the attendant entrainment of fluid, or fluids, that are eventually mixed, as well as the mixed-fluid composition. Intermediate- and small-scale eddies and dynamics are responsible for stirring the entrained fluid(s) and the generation of the high surfaceto-volume ratio of interfacial surface, across which (molecular) mixing occurs. 327 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 327–344. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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The inner (small) scales of the flow are set by viscosity, with the ratio of outer to inner (viscous) scales, scaled by the outer-flow Reynolds number, as proposed by Kolmogorov [1, 2]:

where is the (mixed-fluid) density, the characteristic (outer-scale) velocity that drives the flow, the outer spatial scale, across which is applied, and the (dynamic) viscosity. Diffusion scales can be comparable with, or smaller than, the viscous scales, depending on the fluid Schmidt number Sc. In particular [3],

for gases; for water). For turbulent flows, such as the flow behind a grid, for which an outer-flow Reynolds number is not appropriate, scaling can be referenced to the Taylor Reynolds number

where is the root-mean-square velocity fluctuation and the Taylor microscale [4, 5]. For flows for which both can be defined, one finds empirically that and therefore

Proportionality constants in (1), (2), and (4) are found to be of order unity. Following Kolmogorov (1941) [1], turbulence theories implicitly assume that in high-Re flows, for which inner and outer spatial scales are well separated (1), small-scale dynamics, when scaled by inner-flow parameters, i.e. the mean kinetic-energy dissipation rate and the kinematic viscosity v, may be regarded as universal [4]. While subsequent proposals by Kolmogorov and Oboukhov have refined this assumption by treating the dissipation rate as a stochastic variable [6]–[8], the consequent modifications to spectral measures, for example, are slight and the premise of small-scale universality is held as a good approximation [9]. As (diffusive) mixing is a small-scale phenomenon, small-scale dynamics must be correctly modeled to capture turbulent mixing. Conversely,

experimental mixing measures may be used to interrogate small-scale behavior. Some recent progress in our understanding of turbulence and turbulent mixing will be discussed below.

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2.

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THE MIXING TRANSITION IN TURBULENT FLOWS

Turbulent flows exhibit a conspicuous transition at outer-scale Reynolds numbers (1) given by

The latter, in terms of a Taylor Reynolds number (3), can be used where an outer-scale Reynolds number is not appropriate. These are necessary but not sufficient criteria for this transition; laminar flows can exist at higher Re yet [10]. This transition was first noted in gas-phase shear layers [11, 12]. For zero-pressure-gradient shear layers, the velocity difference, const and owing to a linearly increasing thickness of the turbulent region, i.e. For this flow, a rather abrupt transition to a more disordered, enhanced-mixing state was noted (Fig. 1), at

Figure 1 Side-view schlieren image of a gas-phase shear layer over a densitymatched mixture of Ar and He). Note abrupt transition in the early part of the flow

([11], Fig. 2f).

Similar behavior was found in liquid-phase shear layers [13], demonstrating that this is not a Schmidt number (2) dependent phenomenon (Fig. 2). At least for high-Sc flow (water), fluid within the shear-layer extent is mostly pure high- and low-speed fluid prior to the mixing transition, with very little mixed. While not as conspicuous for gas-phase shear layers, the mixed-fluid fraction, X (x,t), i.e. the fluid for which for some small increases as for both gas- and liquid-phase shear layers [10]. Moderate-Re shear layers can remain quasi-2-D, forming near-cylindrical interfaces between the entrained fluids (e.g. Fig. 2(a)). Beyond the mixing transition, however, the flow becomes decidedly 3-D. It was unclear at the time whether this was a Re-dependent transition, or a transition in flow dimensionality.

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Laser-induced fluorescence streak images of the scalar field in a liquid-phase

shear layer. Color assignment denotes high-speed-fluid mole fraction ([13], Figs. 7(b) and 9).

Turbulent jets are three dimensional, even at relatively low Reynolds numbers. Experiments in liquid-phase jets also indicate a transition at, This is shown in Fig. 3, which depicts the jet-fluid concentration in the plane of symmetry of liquid-phase turbulent jets [14]. Unmixed reservoir fluid (black) penetrates throughout the turbulent region in the lower-Re jet ((a), This is not the case in the higher-Re case Measurements in chemically reacting, gas-phase jets indicate that, for jets, the transition is completed at i.e. roughly twice as high as for shear layers.1 The length scale for a jet should be the (local) radius and not the diameter—shear is applied across the former—offering a possible explanation for the difference in transition Reynolds numbers between shear layers and jets [10]. That jets also exhibit this transition indicates that this is not a 2-D to 3-D transition. A similar transition occurs in liquid-phase transverse jets in a uniform crossflow (Fig. 4) [17]. Figure 5 depicts the probability density functions (PDFs) of far-field jet-fluid concentration of jets discharging in (a) a quiescent reservoir is pure jet-fluid concentration) [18] and in (b) the far-field of trans1

As opposed to shear layers,

in jets. The whole flow is either below, or above,

the mixing transition. Compare Fig. 1 and Fig. 3; this is a Reynolds number effect.

Recent advances in turbulent mixing

Figure 3

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Jet-fluid concentration in the plane of symmetry of round turbulent jets.

([14], Figs. 5 and 9).

Figure 4

Transverse jets in cross-flow:

Jet-fluid concentration in

the plane of symmetry. Intensity compensated for downstream concentration decay

verse jets [17]; Reynolds numbers are based on jet-nozzle parameters in both cases. Both flows are liquid phase with all PDFs

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Figure 5 Jet-fluid concentration PDFs: (a) Jets in a quiescent reservoir with lines of increasing solidity denoting increasing and ([18], Fig. 8); (b) Transverse jets with increasing PDF at associated with increasing and

normalized to differential area fraction, i.e. [19]

where A(c) is the area portion where the concentration is less than PDF normalization is a subtle issue, with differences between PDFs normalized in time, i.e. relative frequency, vs. space, i.e. line/area/volume fraction. (Space limitations do not permit its discussion here.) The differences are interesting. Jets in a quiescent reservoir exhibit a preferred composition at low Re, whereas transverse jets in a cross flow develop a preferred concentration that develops as Re increases. This can probably be attributed to the difference in entrainment vs. stirring/mixing rates between the two flows and indicates that the shape and Re-dependence of scalar PDFs may not be treated as possessing universal behavior. In addition to flows discussed above, transitions in the qualitative behavior of turbulence and the consequent mixing have been documented in a variety of other flows, spanning both low- and high-Sc fluids, freeshear and wall-bounded flows, uniform-density and stratified flows, as well as incompressible and (weakly) compressible flows, all occurring at values of Re, or as indicated in (5) [10]. The wide variety of flows that exhibit this transition argues for reasons not associated with the large-scale structure, which varies considerably across the spectrum of flows. Its apparent universality suggests a quasiuniversal cause. It was proposed that the reason lies in the requirement for an inviscid range of eddies in the flow. This translates into a separation between the Liepmann–Taylor scale, and the (inner) viscous

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scale,

333

(1), estimated as

where is the Kolmogorov scale. The former represents the thickness of an internal layer formed as the result of a plunging motion across The factor of 5 is for a Blasius boundary layer but is also appropriate for free shear layers at near-unity velocity ratios. The latter represents the inner scale where viscous effects begin to act, as diagnosed by where power-law spectra deviate for a near-5/3 exponent—this occurs at As the scale range between and can be spanned by viscous action and is the inner viscous scale, inertial size eddies require, at a minimum,

in accord with observation [10].

3.

HIGH-Re SHEAR LAYERS: EFFECTS OF INITIAL/INFLOW CONDITIONS

It is a tenet of turbulence theory that, for high-enough Reynolds numbers, sufficiently far from inflow boundaries, and for times long after the initial conditions are established, the flow, at least in a statistical sense, is characterized by local flow parameters [4]. In free-shear layers and some other flows, however, inflow, or “initial” conditions in a convective frame, are known to exert a significant influence on such far-field properties as growth rate, even at high Re [20]. The question is whether such an influence can be felt at the inner scales of the flow, where mixing occurs, far enough downstream and at high enough Re. Experiments in gas-phase shear layers employed undisturbed or tripped initial boundary layers. The Reynolds numbers were high, with measurements at The Reynolds number for the high-speed boundary layer at the splitter plate trailing edge, for which tripping transitions a laminar boundary layer to turbulence. The experiments employed low-heat-release chemical reactions between hydrogen and nitric oxide, in the high-speed stream, and fluorine in the low-speed stream. They were in the fast-kinetic (high Damköhler number) regime, where chemical-product formation is limited by the mixing rate. Schlieren visualization covered the whole flow, with local measurements of temperature rise (product formation) at in particular, further than has been stated as required for self-similar behavior [21]–[24] by some margin [20].

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Tripping changed large-scale attributes, increasing the visual growth rate, and decreasing spanwise coherence throughout. Figure 6 summarizes the mixing results and plots the temperature rise profile, normalized by the adiabatic temperature rise, i.e.

where is the freestream temperature and is the adiabatic flame temperature, for each reactant blend. The kinetic rate for the slowest reactant blend at a stoichiometric mixture ratio in the untripped-flow case (a) was reduced further by lowering the reactant

Figure 6 Normalized temperature-rise data ([20], Figs. 3 and 6). Diamonds:

Triangles:

Asterisks:

reduced chemical-kinetic rate.

concentrations. The resulting normalized temperature rise (asterisks) is within experimental reproducibility of the kinetically faster, but otherwise identical run (diamonds), confirming that all runs in this series were in the fast-kinetic (diffusion-/mixing-limited) regime [20]. Tripping had a major effect on mixing and mixed-fluid composition, as evidenced in the data for the “flip” experiments in the limit of high and low stoichiometric mixture ratios, for the two freestreams. The width of the mixed-fluid region decreased with tripping, as opposed to the visual thickness, which increased. More importantly, the location of peak chemical production does not shift, for the untripped initial boundary layers, as the stoichiometry changes from lean to rich, indicating a “non-marching” PDF, i.e. one characterized by a distribution of mixed-

fluid compositions that is nearly the same across the shear layer (e.g. [25], Fig. 11, and [13], Figs. 10 and 20). Tripping increases and shifts peak production towards the lean freestream, as changes, as expected for a “marching PDF”, i.e. one whose peak composition changes across

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the shear layer. Tripping and the resulting changes in shear-layer structure permeate to the smallest scales of the flow, altering the mixing environment [20]. In this flow, the Reynolds number and kinetic-energy dissipation, did not change with tripping. While it may not be conceptually difficult to attribute the effects of tripping to entrainment/stirring/mixing environment changes imposed by changes in largescale structure and geometry, this is not the classical view. The documented changes in mixing behavior, even at high Reynolds numbers, indicate that small-scale behavior may not be regarded as universal, in the conventional sense.

4.

HIGH-Re SHEAR LAYERS: MIXING

Entrained fluid mixes, residing over a spectrum of scales, and is subjected to a broad range of strain-rates. In the model proposed by Broadwell and Breidenthal [26], further refined by Broadwell and Mungal [27], mixed fluid is treated as residing in either a homogeneously mixed state, at a composition set by entrainment, or in strained diffusion layers, subject to a strain rate associated with the Taylor microscale, e.g. as modeled by a Kolmogorov cascade. In an alternative model by Dimotakis [28], mixed fluid is treated as residing over the spectrum of outer-to-inner scales and subject to strain rates associated with each scale size, again as modeled by a Kolmogorov cascade. In modeling nonpremixed, chemically reacting flows, it is of some interest to associate the amount of mixed fluid that contributes to chemical-product formation, as a function of the strain rate to which it is subjected. Chemical production in thin, strained, diffusion zones, can be described in terms of “flamelet” models [29], if the amount in each strain-rate state is known. The highest strain rates in turbulent free-shear flows are associated with the smallest (viscous) scales, i.e. (assuming Kolmogorov scaling),

The prefactor is for shear layers at such as the ones described in for which We then have for the expected strain-rate values in such high-Re shear layers,

The higher values are the more relevant; a surface-to-volume weighting of strained diffusion interfaces will favor smaller scales [28, 30]. Non-premixed, opposed-jet diffusion flames were simulated numerically (by DNS) with the same reactant/diluent blends used in shearlayer experiments. The simulations were for a jet of diluent,

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against a jet of diluent, for both rich and lean reactants, for blends. Chemical production for the blends simulated is not limited by chemical kinetics, at least to an experimentally discernible extent (see discussion above, Fig. 6(a)). The peak normalized adiabatic temperature rise in the chemical-reaction zone (9), i.e. was calculated as a function of the imposed strain rate, For the reactants investigated, which are both hypergolic (require no ignition source) and kinetically fast, the results indicate that, for strain rates above or so (Fig. 7), the expected contribution from internal strained diffusion layers is negligible [31]. Considering the

Figure 7 Calculated peak scaled temperature rise in non-premixed strained flame region for

and

(diluted) reactants (Ref. [31], Fig. 1).

strain-rate range for such flows (11), we conclude that, at least for such flows, for which experimentally the contribution from internal strained diffusion layers is negligible. Substantial reactant leakage must occur in internal strained layers, setting an environment for chemical production best described as occurring in small-scale, (partially) premixed regions. Considering the Lagrangian time for the mixing/reaction sequence, the time delay for this leakage to be completed, is relatively small and overall chemical production is negligibly affected by it.

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5.

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RAYLEIGH–TAYLOR FLOW: EFFECTS

OF INITIAL CONDITIONS Rayleigh–Taylor instability (RTI) occurs whenever fluids of different density are subjected to acceleration in a direction opposite that of the density gradient [32]–[35]. These flows eventually become turbulent, as secondary, Kelvin–Helmholtz instabilities (KHI) develop and grow. If the fluids are miscible, species diffusion and mixing, which reduce density differences and hence forcing, can play a dynamic role.

Models of the Rayleigh–Taylor instability mixing zone have it growing quadratically in time, following an initial, linear-stability evolution [36]– [38]. For constant acceleration, g, the vertical penetration, of the light fluid into the heavy fluid (dubbed “bubbles”), and of the advancing heavy into light fluid (“spikes”), are modeled as

where and s denote “bubbles” and “spikes”, with and model constants, and the Atwood number, with and the densities of the (pure) heavy and light fluids, respectively. The extent h(t) of the RTI mixing zone is then given by

where

Empirical estimates are in the range Such growth models implicitly assume that the

density difference across the advancing fronts remains constant and that the flow structure remains self-similar [40]. Direct numerical simulation of the Navier–Stokes equations, aug-

mented by a species-conservation equation (binary, Fickian diffusion) was performed for this flow, on a grid with matched fluid viscosities. Initial error-function density (mole-fraction) profiles were perturbed by displacing the intermediate isosurface, nominally

at by an amount in units of the initial profile scaling length [40]. Three simulations, Cases A, B, and C, were performed, each with a different initial perturbation spectrum, but otherwise identical in every other respect. The initial

spectra are plotted

in Fig. 8, along with the linear-stability exponential-growth coefficient, (k) [34]. High-k perturbations are initially damped by viscosity and diffusion. The perturbation energy is the same for all cases:

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Spectra of initial interfacial perturbations (14) for Cases A, B, and C.

The exponential growth coefficient from viscous and diffusive linear stability theory (LST) [34] is also plotted. Left axis: Perturbation spectra scale. Right axis: Lineargrowth rate coefficient, with units of inverse time [40].

The evolution for Case C is depicted in Fig. 9. Pure heavy fluid is red. pure light fluid is blue, and the intermediate

Figure 9 Time evolution of intermediate heavy fluid is red, pure light fluid

isosurface (green) for Case C. Pure is blue [40].

isosurface (1:1 molar ratio) is green. The initial

isotropic

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perturbations are barely discernible on the interface. By mushroom-like structures have formed on the upper side of the isosurface is the transverse extent of the RTI cell). By mushrooms have merged into larger structures and a long, thin protuberance (“spike”) can be seen penetrating deep into the light fluid [40]. Figure 10 depicts the extent, h(t), of the RTI mixing zone (13), along with indicatrices for and growth. An initial, diffusion-dominated

Figure 10 Mixing-zone extent,

vs.

for the three cases [40].

regime is the same for all cases, i.e. independent of the initial perturbations, and in good agreement with analytical predictions for purely diffusive growth. A faster-growth stage occurs at a different time for each of the three cases, reflecting differences in initial modal seeding. Case C growth is well represented as quadratic. Growth for Cases A and B is different. Even greater differences between the three cases are found in mixing, as measured by the amount of product that would be formed from a stoichiometric chemical reaction between the two fluids being mixed. These differences persist as far as the simulations could be run, to [40]. While final Reynolds numbers attained are below the anticipated mixing transition, the flow evolved over a significant multiple of the initial transverse extent, which is greater than that for the high-Re shear layers, i.e. described above.

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SOME COMMENTS ON HIGH-Re TURBULENCE The equations for the turbulent flow, at least for incompressible flow are the (scaled) Navier–Stokes equations,

which depend on one parameter, the Reynolds number, Re. Of course, solutions also depend on initial and boundary conditions; we cannot

envisage The Book of Turbulence, simply indexed by Re. If initial conditions were eventually “forgotten”, one could conceive of a volume for turbulent shear layers—one for boundary layers, one for pipes, etc.—a finite project, at least as regards the classical set of flows. The potential geometric complexity of boundary conditions, however, as encountered in internal flows in many engineering devices, for example, quickly elevates this to an unmanageable project. To make matters worse, experimental and computational evidence indicates that the influence of initial conditions can persist, not only as discussed above for shear layers and Rayleigh–Taylor flows, but also in other flows, including grid turbulence [42], which has been treated as the universal flow par excellence. Once the flow has been launched into a particular dynamic state, or strange attractor, there is no “entropic spring” that will return it to some universal, dynamic-equilibrium state; the prerequisites that render this the expected behavior in statistical mechanics and thermodynamics, for example, are not at work here. Conversely, the evidence indicates that Re effects on the dynamics and mixing become substantially weaker as Re increases and, in particular, for i.e. beyond the mixing transition. This is illustrated in the image sequence in Fig. 11, which includes (a) a buoyant flame at the pair of jet images (b, c) at and 104, respectively, and (d) a rocket exhaust at an estimated The first three differ considerably, even though separated from each other by a factor of, roughly, four in Re. In contrast, the difference between the last two is not great, even though their Reynolds numbers are separated by four orders of magnitude. Consequently, experimental/DNS results at should be viewed with care, with extrapolation to the higher Reynolds numbers typically of interest of dubious validity. Figure 11 illustrates the caveat.

7.

CONCLUSIONS

Laboratory and DNS experiments indicate that turbulence can transition to a fully developed state, characterized by enhanced mixing, at

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Figure. 11

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Turbulent-jet flows at increasing Re. [*Photo (d) from Los Angeles (Menlo

Park, Calif.: Sunset Lane, 1968), 246–247.]

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or At least for flows away from solid boundaries, the Re-dependence of the dynamics for is found to be weak. This suggests that DNS at just about within reach today for many flows, may serve as a surrogate for simulations of higher-Re flows, at least for free-shear flows. There can be no question that, if not DNS, then large-eddy simulation (LES) and/or sub-grid scale (SGS) simulations will serve as the solution to the “turbulence problem” in the long run. At this time, however, LES/SGS approaches to mixing and chemically reacting flows have had only limited success. To my knowledge, only one recent SGS scalar-mixing model appears to have captured elements of the mixing transition [43], indicating decreasing scalar variance, until and essentially constant beyond that. The effects of variable density and compressibility, Schmidt number, and differential or multicomponent diffusion are not in the offing in SGS mixing models at this time, however. In closing, and considering that this is part of the ICTAM 2000 Proceedings, it is worth noting that of all the classical physics problems bequeathed as such by the 19th century to the 20th, turbulence is the only one we are handing the baton to the 21st. I would like to hope that some lecturer at the transition of the next century will not be making a similar observation.

Acknowledgments I would like to acknowledge the extensive and expert support and contributions by D. Lang in the Caltech experiments reviewed, in all matters regarding digital imaging, electronic, computing, and networking, as well as the JPL imaging team, led by M. Wadsworth. Discussions and contributions by the many collaborators cited, as well as R. D. Henderson, S. V. Lombeyda, D. I. Meiron, and J. M. Patton of Caltech, and P. L. Miller and T. A. Peyser of the Lawrence Livermore National Laboratory (LLNL) are also gratefully acknowledged. The recent Caltech work on mixing reviewed was sponsored by the Air Force Office of Scientific Research Grants F49620–94–1–0353 and F49620–98–1–0052, and the DOE/Caltech ASCI/ASAP subcontract B341492. The collaborative work with LLNL was performed under U.S. Department of Energy contract W–7405–Eng–48.

References [1] Kolmogorov, A. N. 1941. Local structure of turbulence in an incompressible viscous fluid at very high Reynolds numbers. Akademiia Nauk SSSR Doklady 30, 301–305. [2] Kolmogorov, A. N. 1941. Dissipation of energy in locally isotropic turbulence. Akademiia Nauk SSSR Doklady 32, 19–21. [3] Batchelor, G. K. 1959. Small-scale variation of convected quantities like temperature in turbulent fluid. Part I—General discussion and the case of small conductivity. Journal of Fluid Mechanics 5, 113–133.

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[4] Batchelor, G. K. 1953. The Theory of Homogeneous Turbulence. Cambridge, U.K.: Cambridge University Press. [5] Tennekes, H., and J. L. Lumley. 1972. A First Course in Turbulence. Cambridge, Mass.: MIT Press. [6] Kolmogorov, A. N. 1962. A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. Journal of Fluid Mechanics 13, 82–85. [7] Oboukhov, A. M. 1962. Some specific features of atmospheric turbulence. Journal of Fluid Mechanics 13, 77–81. [8] Monin, A. S., A. M. Yaglom, and J. Lumley (eds.). 1975. Statistical Fluid Mechanics, Vol. 2: Mechanics of Turbulence. Cambridge, Mass.: MIT Press. [9] Hinze, J. O. 1975. Turbulence, 2nd ed. New York: McGraw-Hill. [10] Dimotakis, P. E. 2000. The mixing transition in turbulence. Journal of Fluid Mechanics 409, 69–97. [11] Konrad, J. H. 1976. An experimental investigation of mixing in two-dimensional turbulent shear flows with applications to diffusion-limited chemical reactions. Ph.D. thesis, California Institute of Technology. [12] Bernal, L. P., R. E. Breidenthal, G. L. Brown, J. H. Konrad, and A. Roshko. 1979. On the development of three-dimensional small scales in turbulent mixing layers. Proceedings of the 2nd International Symposium on Turbulent Shear Flows. New York: Springer-Verlag, 305–313. [13] Koochesfahani, M. M., and P. E. Dimotakis. 1986. Mixing and chemical reactions in a turbulent liquid mixing layer. Journal of Fluid Mechanics 170, 83–112. [14] Dimotakis, P. E., R. C. Miake-Lye, and D. A. Papantoniou. 1983. Structure and dynamics of round turbulent jets. Physics of Fluids 26, 3185–3192. [15] Gilbrech, R. J. 1991. An experimental investigation of chemically-reacting, gasphase turbulent jets. Ph.D. thesis, California Institute of Technology. [16] Gilbrech, R. J., and P. E. Dimotakis. 1992. Product formation in chemicallyreacting turbulent jets. AIAA 30th Aerospace Sciences Meeting, Paper 92–0581. [17] Shan, J. W. 2001. Mixing and isosurface geometry in turbulent transverse jets. Ph.D. thesis, California Institute of Technology.

[18] Catrakis, H. J., and P. E. Dimotakis. 1996. Mixing in turbulent jets: Scalar measures and isosurface geometry. Journal of Fluid Mechanics 317, 369–406. [19] Dimotakis, P. E., and H. J. Catrakis. 1996. Turbulence, fractals, and mixing. NATO Advanced Studies Institute series on Mixing: Chaos and Turbulence. Report FM97–1, Graduate Aeronautical Laboratory, California Institute of Technology. [20] Slessor, M. D., C. L. Bond, and P. E. Dimotakis. 1998. Turbulent shear-layer mixing at high Reynolds numbers: Effects of inflow conditions. Journal of Fluid Mechanics 376, 115–138. [21] Bradshaw, P. 1966. The effect of initial conditions on the development of a free shear layer. Journal of Fluid Mechanics 26(2), 225–236.

[22] Dimotakis, P. E., and G. L. Brown. 1976. The mixing layer at high Reynolds number: Large-structure dynamics and entrainment. Journal of Fluid Mechanics 78, 535–560 + 2 plates.

[23] Ho, C.-M., and P. Huerre. 1984. Perturbed free shear layers. Annual Review of Fluid Mechanics 16, 365–424. [24] Karasso, P. S., and M. G. Mungal. 1996. Scalar mixing and reaction in plane liquid shear layers. Journal of Fluid Mechanics 323, 23–63.

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[25] Mungal, M. G., and P. E. Dimotakis. 1984. Mixing and combustion with low heat release in a turbulent mixing layer. Journal of Fluid Mechanics 148, 349–382. [26] Broadwell, J. E., and R. E. Breidenthal. 1982. A simple model of mixing and chemical reaction in a turbulent shear layer. Journal of Fluid Mechanics 125, 397–410. [27] Broadwell, J. E., and M. G. Mungal. 1991. Large-scale structures and molecular mixing. Physics of Fluids A 3(5), Part 2, 1193–1206. [28] Dimotakis, P. E. 1987. Turbulent shear layer mixing with fast chemical reactions. In Turbulent Reactive Flows (R. Borghi and S. N. B. Murthy, eds.), Lecture notes in Engineering 40. New York: Springer-Verlag, 417–485. [29] Peters, N. 1984. Laminar diffusion flamelet models in non-premixed turbulent combustion. Progress in Energy and Combustion Science 10, 319–339. [30] Dimotakis, P. E. 1991. Turbulent free shear layer mixing and combustion. Chapter 5 in High Speed Flight Propulsion Systems. Progress in Astronautics and Aeronautics 137, 265–340. [31] Egolfopoulos, F. N., P. E. Dimotakis, and C. L. Bond. 1996. On strained flames with hypergolic reactants: The H2/NO/F2 system in high-speed, supersonic and subsonic mixing-layer combustion. Proceedings of the 26th International Combustion Symposium, 2885–2893. [32] Rayleigh, Lord. 1883. Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proceedings of the London Mathematical Society 14, 170–177. [33] Taylor, G. I. 1950. The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. Proceedings of the Royal Society of London A 201, 192–196. [34] Duff, R. E., F. H. Harlow, and C. W. Hirt. 1962. Effects of diffusion on interface instability between gases. Physics of Fluids 5, 417–425. [35] Sharp, D. H. 1984. An overview of Rayleigh–Taylor instability. Physica D 12, 3–18. [36] Annuchina, N. N., Yu. A. Kucherenko, V. E. Neuvazhaev, V. N. Ogibina, L. I. Shibarshov, and V. G. Yakovlev. 1978. Turbulent mixing at an accelerating interface between liquids of different densities. Izvestiia Akademiia Nauk SSSR, Mekhanika Zhidkosti i Gaza 6, 157–160. [37] Read, K. I. 1984. Experimental investigation of turbulent mixing by Rayleigh– Taylor instability. Physica D 12, 45–58. [38] Youngs, D. L. 1984. Numerical simulation of turbulent mixing by Rayleigh– Taylor instability. Physica D 12, 32–44. [39] Dimonte, G., and M. Schneider. 2000. Density ratio dependence of Rayleigh– Taylor mixing for sustained and impulsive acceleration histories. Physics of Fluids 12, 304–321. [40] Cook, A. W., and P. E. Dimotakis. 2000. Transition stages of Rayleigh–Taylor instability between miscible fluids. Report No. UCRL–JC–139044, Lawrence Livermore National Laboratory. Journal of Fluid Mechanics (submitted). [41] Dimotakis, P. E. 1997. Non-premixed hydrocarbon flame. Nonlinearity 7, 1–2. [42] George, W. 2000. Decay of isotropic turbulence. Physics of Hydrodynamic

Turbulence (31 Jan–30 June, 2000, University of California, Santa Barbara), http://online.itp.ucsb.edu/online/ hydrot00/george/. [43] Pullin, D. I. 2000. A vortex-based model for the subgrid flux of a passive scalar. Physics of Fluids 12, 2311–2319.

KINETIC AND CONTINUUM DESCRIPTIONS OF GRANULAR FLOWS Isaac Goldhirsch Department of Fluid Mechanics and Heat Transfer, Tel-Aviv University, Tel-Aviv, Israel [email protected] Abstract

1.

Unlike the basic units of molecular systems, the elementary constituents of granular matter (the grains) experience dissipative interactions. This fact is the root cause of many of the difficulties encountered in the study of granular materials and among its major consequences are the existence of unique states and instabilities (e.g. collapse and clustering) as well as the multistable/metastable nature of most granular states. Another central consequence is the inherent lack of scale separation. The latter is responsible e.g. for the prominent normal stress differences (anisotropic pressures), long-range correlations, scale-dependent stress (and other) fields and, in general, the rheological nature of these materials. Constitutive relations and boundary conditions for dilute and near-elastic rapid granular flows have been derived from the pertinent Boltzmann equation with extensions to moderate densities obtained by employing the Enskog–Boltzmann equation or (systematically, via) response theory. Unlike the rapid flows, the dense, static and quasistatic regimes have not been treated in a systematic fashion heretofore. Preliminary results on elasticity in the static regime are presented.

INTRODUCTION

There are at least three major reasons to study granular materials: they are abundant and of great importance on Earth (sand, gravel, soil) and in space (Saturn’s and other planets’ rings, interstellar dust), they are of central importance in a large number of industries (chemical, construction, food) and in the environment (pollutants, snow avalanches, mud and rock slides), and they exhibit numerous unusual and unexpected phenomena whose understanding poses exciting challenges [1]. The storage, transport, and, in general, handling of granular matter encounters serious difficulties [2], which range from undesired clogging, through unwanted mixing or segregation, to the structural collapse of silos. Granular matter is often described as ‘unpredictable’, ‘irreproducible’ or ‘erratic’. These and other adjectives used to characterize 345 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 345–358.

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granular materials are a clear sign that much is still lacking in our understanding of these materials. A variety of approaches have been proposed and applied to the description of the statics and dynamics of granular matter, ranging from tables (e.g. Jenike tables [3]) for engineering purposes, through phenomenological/empirical constitutive relations [4], to equations of motion that are based on fundamental considerations and systematic derivations. The latter are usually limited to the fully fluidized regime of granular matter, otherwise known as “rapid granular flow” [5], and they are restricted to the near-elastic regime as explained below. This paper has two major goals. The first is to identify some of the root causes for the difficulties encountered in the study of granular materials. The second is to report some recent progress in the theoretical description of these materials.

2.

SOME CONSEQUENCES OF DISSIPATIVE INTERACTIONS

When granular materials are strongly forced (e.g. by vibration or shear) they can become fully fluidized, the grains interacting by (inelastic) collisions. This ‘rapid granular flow’ is strongly reminiscent of the classical model of a gas. The analogy led to the definition of granular temperature [6] as a measure of the velocity fluctuations in ‘granular gases’, and later to the development of kinetic, Boltzmann-based theories [7]–[14] for these ‘gases’. The other extreme, i.e. the static state of granular media, is clearly reminiscent of the solid state of matter; as typical granular solids are not in lattice configurations, the molecular analogue would be amorphous or disordered solids. These observations suggest that methods borrowed from statistical mechanics (such as the Boltzmann equation mentioned above) should be useful for the description of granular systems. However, in importing these methods to the study of granular materials one should exercise extreme care as the dissipative nature of grain interactions (inelastic collisions, friction) is the source of significant differences between ‘molecular’ and ‘granular’ materials. Practically all states of granular matter are metastable. For instance, the ground state of a sand pile is one in which all grains reside on the ground; common piles owe their very existence to the frictional forces among the grains. Notice also that static granular piles cannot spontaneously rearrange themselves; thus their configurations are history dependent. Granular gases are different from classical gases not only in the typical dimensions of the constituents. For instance, a granular gas possesses

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no equilibrium state at finite temperature, as any initial state must lose energy to inelasticity (i.e. to internal degrees of freedom of the grains); hence its granular temperature must decay to zero (the only possible equilibrium state). The only way one can prevent this energy loss is by external replenishment, but then the gas is in a non-equilibrium state. The absence of equilibrium is the source of some difficulties in the development of a perturbative (Chapman–Enskog [15]) approach to the kinetics of granular gases, as this state is usually the zeroth order in the Chapman–Enskog expansion. Typical grains are (by definition) macroscopic. Hence grain/container size ratios are always far larger than the corresponding ratio in molecular systems. This ‘practical’ aspect of the lack of scale separation in granular systems is not a fundamental property of these systems. It turns out that granular systems lack scale separation in a fundamental way, i.e. even in the ideal ‘thermodynamic’ limit. This fact has far reaching consequences [16].

2.1.

Lack of scale separation

Granular systems possess correlations whose typical scales are comparable with the scale of the system itself; as such they may be coined mesoscopic [16]. For instance, static granular systems may possess arches that span the entire size of the system (or a significant part thereof). Recent experiments [17] revealed the existence of ‘stress chains’ or ‘force chains’ in static and quasistatic granular systems, which are fractallike strings of particle contacts that span distances comparable with the size of the system and along which the ‘large’ forces (beyond a cutoff) ‘propagate’. Interestingly, rapid granular flows lack scale separation as well [16]. Consider the case of a linearly sheared monodisperse granular system in which the particle interactions are collisions characterized by a fixed coefficient of normal restitution e. Let the macroscopic velocity field V be given by (a well known solution of the pertinent equations of motion), where is the shear rate, y is the spanwise coordinate and is a unit vector in the streamwise direction. Both phenomenological [18, 19] (even dimensional) and kinetic theoretical calculations [10]–[12] show that the granular temperature, T (defined as the mean square fluctuating velocity), in this system approaches a steady state value given by

where C is a volume fraction dependent prefactor, whose value at low volume fractions can be shown to equal approximately 0.6 in two dimen-

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sions [10, 11] and 3 in three dimensions [12], is the ‘degree of inelasticity’ defined by and is the mean free path of the particles. The change of the macroscopic velocity over a distance of a mean free path, in the y direction, is given by A shear rate can be considered small if is small with respect to the thermal speed Here, following i.e. the shear rate is not small unless the system is nearly elastic. (Notice that for , for example, This result alone implies that the Chapman–Enskog expansion of the constitutive relations (which is truly a gradient expansion, formally an expansion in the Knudsen number K; in the discussed case, must be carried out beyond the Navier–Stokes order (linear in K), the lowest next orders being the Burnett [10, 11, 12, 20] and super-Burnett orders. This is why the normal stress differences, a Burnett order effect, are prominent in granular gases. As the Burnett (and super-Burnett) equations are ill-posed [21, 22], resummation techniques [22, 23] may have to be used to tame the ill-posedness. Another problem associated with the gradient expansion is that higher orders in the gradients may be non-analytic [24], indicating non-locality. (A second argument for non-locality is presented below.) Still another consequence of the above findings is that granular flows are typically supersonic [16]. This can be understood on the basis of the following argument: consider two particles moving (approximately) in the same direction; as they collide their relative velocity decreases but their total momentum remains unchanged, which amounts to a decrease in the fluctuating component of their velocities, i.e. the granular temperature. Moreover, the same mechanism is responsible for the emergence of correlated strings of particles—an effect which is also known [25] in molecular systems at high shear rates; here it is generic—hence to a violation of the hypothesis of molecular chaos [26]. The mean free time equals the ratio of the mean free path and the thermal speed: Clearly, is the microscopic time scale characterizing the system at hand and is the macroscopic time scale characterizing it; hence the ratio is a measure of the degree of temporal scale separation. Employing (1), one obtains an O(1) quantity except in the near-elastic case. Thus typically the considered system lacks temporal scale separation irrespective of its size or the size of the grains. Consequently, one cannot a priori employ the assumption of “fast local equilibration” and/or use local equilibrium as a zeroth order distribution function unless the system is nearly elastic. The latter condition severely limits the applicability of the hydrodynamic description. For instance, consider the application of these equations to a stability study. As expected (and is well known [19, 27]) some of the eigenvalues of the granular stability problem (includ-

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ing those corresponding to instabilities) must be of the order of the only “input” inverse time scale (in the absence or irrelevance of gravity), i.e. Since, as explained above, one obtains instabilities whose characteristic times are comparable with the mean free time. Thus one encounters a paradoxical situation in which the time constants corresponding to the development of instabilities (the inverse values of the corresponding growth rates) predicted on the basis of the hydrodynamic equations may be shorter than the time scales that the hydrodynamic

equations themselves resolve. Another conclusion from this observation is that one cannot distinguish between microscopic and macroscopic fluctuations; for instance, pressure fluctuations which are merely results of single particles impacting on a wall are measurable on macroscopic time scales and appear as ‘intermittent events’ [28]. A similar phenomenon has been reported in the dense case [29]. The free path of a particle, i.e. the distance it moves in between consecutive collisions, depends on its absolute velocity; hence it is not Galilean invariant. It follows that the mean free path is not a Galilean invariant entity either (one can define a Galilean invariant mean free path in a co-moving, Lagrangian frame; this entity is denoted above by Denote the instantaneous velocity of a particle by v. Let where V denotes the macroscopic velocity at the particle’s position and is the fluctuating part of its velocity. It follows (assuming statistical independence of and V) that the average (denoted by an overbar) of is hence the typical velocity u of a particle is or in the specific case considered, Multiplying u by the mean free time (as mentioned, a Galilean invariant entity), one obtains the mean free path At values of y at which the speed is subsonic (following the above considerations this happens when one can neglect with respect to T, in which case However, when , in particular when the thermal speed is far smaller than the average speed (i.e. the flow is supersonic) and in this

case, i.e. the ‘true’ mean free path is (much) larger than the equilibrium mean free path. Moreover it is of macroscopic dimensions, being an O(1) quantity times This implies that the considered system has long-range correlations, unless is small enough for a given system size, a fact that may invalidate the hydrodynamic equations, unless additional fields are used [16].

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Scale dependence of stress

In the realm of molecular fluids (when they are not under very strong thermal or velocity gradients) there is a range, or plateau, of scales that are larger than the mean free path and far smaller than the scales characterizing macroscopic gradients, and that can be used to define “scale independent” densities (e.g. mass density, momentum density, energy density or temperature) and fluxes (e.g. stresses, heat fluxes). Such

plateaus are virtually non-existent in systems in which scale separation is weak, and therefore these entities are scale dependent. By way of example, the “eddy viscosity” in turbulent flows is a scale-dependent (or resolution-dependent) quantity, since in the inertial range of turbulence there is no scale separation. There is a plenitude of “rheological materials” in which the lack of scale separation is associated with scale dependence of stresses and other fields. The scale-dependent entity discussed below [28] is the stress tensor. For simplicity we shall mostly discuss the kinetic part of the stress tensor,

which dominates at low volume fractions. The kinetic theoretical expression for this tensor is where is the ensemble average of A, v' is the fluctuating part of the velocity, and is the mass density. It can be shown [28, 30, 31] that the stress field can be defined for single realizations (hence one does not need to invoke the notion of an ensemble; the latter may not be known) in such a way that the standard continuum equation of motion, holds. The general theory of [28] allows for a variety of coarse graining functions (or averaging weights). Here we choose to study the simplest version, which also corresponds to standard practice in computer simulations. Let the macroscopic velocity field V(r, t) be defined as the center of mass velocity, at time t, in (say) a cube of side w of which r is the geometrical center. The fluctuating velocity of a particle i residing in this cube at time t is defined as The stress tensor at point r at time t is defined as where denotes the average over all particles in the cube. Define next to be the fluctuation of the velocity of a particle with respect to the average velocity at its instantaneous position and let be the difference between the average velocity at and the coarse graining center r. Clearly The above decomposition yields two contributions to the kinetic stress tensor: the first is a ‘standard’ kinetic contribution, and the second is a contribution which is proportional to the square of the velocity gradient tensor and which stems from the fact that the variation of the macroscopic velocity in the coarse graining cell

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is a contribution to the velocity fluctuations. The latter contribution is negligibly small in molecular systems as changes of velocity over typical coarse graining scales (of the order of the mean free path) are small. In contrast, the second contribution can be dominant in the realm of granular flows. In the case of a linear shear flow, the contribution of the second term to the normal stress is proportional to whereas the kinetic contribution (in the dilute case) is The ratio of these contributions, is typically a very small quantity in molecular gases; in contrast, in granular gases this ratio is proportional (following (1)) to a large quantity for The above (and other, related) results have been corroborated by simulations [28].

2.3.

Collapse and clustering

Clustering [19, 32] and collapse [33] are effects that are specific to granular systems; they are results of the inelastic nature of these systems and have no equivalent in atomic gases.

Collapse. The essence of the collapse phenomenon [33] can perhaps be explained as follows. Consider a hard sphere bouncing off a floor. Assume that when the sphere hits the floor with velocity v, it bounces off with velocity ev, where is the coefficient of restitution. An elementary exercise shows that if the sphere is dropped from rest at an initial height it reaches, upon rebouncing n times, a maximal height of Let denote the time that elapses between consecutive maximal height positions of the particle. It is easy to show that Since the sum of is finite it follows that an infinite number of collisions occurs in a finite time. A similar effect occurs in many-body granular systems where the “floor” is replaced by a collection of particles and where the relative velocities of the colliding particles become very small, leading (via a theoretically infinite number of collisions) to the emergence of strings of particles whose relative velocities vanish [33]. Clustering. Granular materials have a tendency to form dense clusters even when the ‘initial condition’ is homogeneous. Unlike the collapse phenomenon, clustering is a hydrodynamic effect (but it may precede collapse [9]). An intuitive explanation [19] of the clustering phenomenon proceeds as follows. Consider a region in a granular gas in which the density is increased due to a fluctuation, without a change in the granular temperature T. In this region the frequency of collisions is larger than in neighboring, less dense regions. Since the collisions are inelastic, the granular temperature in the dense region decreases faster than in the neighboring

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regions, causing the pressure in this region to be lower than that in the neighboring regions. The resulting pressure gradient leads to a migration of particles into the dense region, thus further increasing its density and decreasing its pressure. Thus, once commenced, a density fluctuation leads to the formation of a dense cluster, provided the (hydrodynamic) mechanisms that may disperse the agglomeration of particles are slower than the clustering process. A more quantitative analysis of the clustering phenomenon requires a detailed study of the hydrodynamic equations for granular gases. Here, a simplified derivation is presented. Consider a monodisperse collection of spheres of radius a whose collisions are characterized by a fixed coefficient of normal restitution e. The pressure in this dilute granular gas is given by where is the (mass) density. Let m be the mass of a particle and n be the number density. Recall the standard kinetic (or mean free path theory) formula for the mean free path, where is the total collision cross section of two particles. It follows that i.e. the pres-

sure is inversely proportional to the number density n, in accord with the above physical considerations. Numerical simulations of unforced as well

as of sheared granular gases [19, 27] corroborate this result. An analysis [19] of the equation for the momentum density, reveals that the clustering time is shorter the shorter the scale of the fluctuation, an intuitively plausible result (it takes less time for mass to move a shorter distance). On the other hand, an inspection of the equation of motion of the temperature field reveals that the above neglect of the diffusive terms in the hydrodynamic equations is justified only when Hence the fastest, and consequently dominant, cluster formation process occurs at i.e. the typical cluster separation distance is When is larger than the system size, one does not expect clustering to occur. Instead, one expects a shear mode and an unstable density fluctuation mode of the smallest wavenumber allowed

by the system’s geometry [19]. Consider a system under time-dependent shear [27]. As energy is pumped into this system at the boundaries, it heats up near the boundaries, increasing the pressure there; the result is the creation of dense clusters (or plugs) ‘far’ from the boundaries. Other states of sheared flows are possible as well. In general, the clustering mechanism is responsible for many of the hysteretic/multistable properties of rapid granular flows.

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It is easy to modify the Boltzmann equation to account for inelastic collisions [8, 10, 11, 12]. Pioneering studies [7] of granular kinetics employed the Enskog equations of motion for the moments of the velocity and obtained closures by assuming specific forms of the distribution function. Recent progress includes the development of a systematic approach to the problem, which is based on the Chapman–Enskog expansion [10, 11, 12, 14]. Further developments include the use of response theory [34] to extend the validity of the Boltzmann constitutive relations to moderately dense granular gases. The rational approaches to the granular kinetic problem have led to the discovery of novel terms in the constitutive relations; there are also some quantitative differences between the phenomenological results and those obtained systematically. Another discovery is that the normal stress differences are truly Burnett effects, which are prominent in the granular case because of the lack of scale separation. As mentioned above, these recent studies have revealed limitations of the constitutive relations, in particular the fact that they are restricted to the near-elastic case.

3.1.

Developing a Chapman—Enskog expansion

Following Eqn. (1) it is convenient to consider the mathematical limit in which both the shear rate (more generally, the Knudsen number) and the degree of inelasticity tend to zero in such a way that the temperature is kept fixed (i.e. In this (elastic, unforced) limit the system evolves toward a state of equilibrium. It can be shown that this limit is not singular. Consequently, one can scale the shear rate by define an expansion (of the distribution function) in powers of and employ this expansion to solve the pertinent Boltzmann equation in powers of this small parameter. This expansion is limited to steady states alone [10, 11]. In order to obtain constitutive relations that are not restricted to steady state situations, one can define a double expansion in powers of the Knudsen number K and the degree of inelasticity [12]. As expected, this expansion reduces to that described above in the steady case. The leading order (elastic) viscous contribution to the stress tensor is O ( K ) (the Navier–Stokes order) and the leading inelastic correction is Since in the steady state, this correction is also hence one needs to calculate the super-Burnett, i.e. for consistency. Some of the conclusions stemming from this finding have been stated above. Explicit constitutive relations can be found in [12] for three dimensions and in [10, 11] for two. It is appropriate to mention the

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existence of a competing method [14] for deriving constitutive relations for rapid granular flows; the latter is based on an expansion around the ‘decaying’ unforced, homogenous granular system.

3.2.

Boundary conditions

Except for the study of (idealized) infinite systems, the constitutive relations are useless without corresponding boundary conditions [35]. Even in the realm of molecular systems, there have been no systematic derivations of boundary conditions at gas–solid interfaces heretofore. The basic problem encountered in attempts to derive such boundary conditions is the lack of a small parameter. However, it turns out that one can derive an expansion of the boundary conditions in which the zeroth order corresponds to the Maxwell condition (in which it is assumed that the distribution function of the particles colliding with the boundary is Maxwellian) and the nth order corresponds to a particle that has collided n times (in between collisions with the boundary); hence the incoming distribution function is affected by the presence of the boundary [36]. While this expansion does not have a formal small parameter, it does converge as (due to the spectrum of the linearized Boltzmann operator) it takes only a few collisions per particle to achieve local equilibration. The latter assumption restricts the theory to near-elastic systems, much like the restriction on the Boltzmann-based constitutive relations. One of the results obtained using this formulation is that for a boundary orthogonal to the z direction that is characterized by a degree of inelasticity the boundary condition for the z component of the velocity in the presence of a thermal gradient in the z direction is to lowest order in the collisions. This boundary condition is quite surprising as it implies a violation of mass conservation in steady states when The resolution of this “paradox” is found by noting that in a steady sheared state, the order (which is the order of the above expression for and the order are the same; hence a super-Burnett correction should be added to the above expression for to obtain the vanishing of at the boundary in a steady state.

3.3.

Moderately dense gases

The Enskog–Boltzmann equation, in which a phenomenological, density-dependent collision kernel represents the effect of finite density, has been useful for deriving constitutive relations for moderately dense molecular and granular gases alike. This equation does not contain information on dynamic correlations (in particular repeated collisions, also

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known as ring events). A general approach to this problem is offered by the (nonlinear) response theory, originally developed for molecular systems. A direct generalization of this theory to granular systems is inapplicable as the theory requires energy conservation to hold. However, a variant of the response method, proposed by Ronis [37], can be modified for the derivation of constitutive relations for granular gases [34]. For sake of this derivation it is useful to measure time in units of the number of collisions accumulated (on average) by a particle and consider the (unstable) homogeneous cooling state as the zeroth order in a gradient expansion. (In this case, unlike that in which the Boltzmann equation is employed, there is no a priori restriction on the density.) Among the results obtained by employing this formulation are Green–Kubo formulae for the dissipative coefficients, which, similarly to the molecular case, assume the form of time integrals of correlations of dissipative fluxes. A contribution to the energy sink term (due to inelasticity) in the equation of motion for the granular temperature, which is proportional to the divergence of the velocity field, is one of the new finite-density terms obtained by employing the response formulation.

4.

ELASTICITY OF GRANULAR SYSTEMS

Dense granular systems in three dimensions and polydisperse granular systems in two and three dimensions are disordered. The derivation of constitutive relations for these systems requires the identification of relevant characterizations of their configurations. A sufficient characterization is not known at present [1]. Surprisingly, even the elastic state of amorphous solids or granular systems is not well characterized or understood. (The only elastic theories truly derived from microscopic descriptions pertain to lattice configurations or very specific non-lattice ones.) As a matter of fact, even the definition of such a basic entity as strain is under debate in the literature [38]. It is the goal of this short section to shed some light on the latter problem [39]. A common method for determining strain in a granular (or random) system is to fit the displacements of the particles to a linear or piecewise linear function of their coordinates, the ‘prefactor’ being identified as the strain. The mean field methods that are defined by this procedure (e.g. [38]) are based on the (implicit or explicit) assumption that ‘to lowest order’ a particle’s displacement is determined by the local strain field at its position. This assumption obviously breaks down when strain fluctuations are significant or when (another implicit assumption) the number density of the particles is significantly non-constant. It is claimed here that continuum mechanics actually teaches us how to

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resolve this problem in a manner that is consistent with general principles [39]. Indeed, the displacement field is defined as the Lagrangian integral of the (macroscopic) velocity field. The latter can be expressed as where n is the number density (assuming here a monodisperse system) and is a coarse graining (averaging) function. This definition is compatible with the continuum mechanics equations of motion. Upon integrating this expression from time 0 to time t and assuming, for sake of simplicity, that the system is unstrained at one obtains, following a straightforward integration by parts, that the displacement field u(r, t) is given, up to a quadratic error in the strain, by where are the displacements of the particles. This formula has been corroborated by extensive numerical simulations. One of the findings of the simulations

(and the theory) is that the more random the system is the smaller the range of strain values for which linear elasticity holds [39].

5.

CONCLUSION

Granular gases, as well solids, are generically mesoscopic and require rheological descriptions except in the case of near-elastic granular gases. The lack of scale separation, which follows from the dissipative nature of the grain interactions seems to be one of the most important factors underlying the difficulties encountered in developing theories of granular matter. The lack of scale separation also offers certain advantages, e.g. one is in a unique situation to ‘see’ the interior of a shock wave, as the latter may have a macroscopic width. Granular materials possess states and instabilities specific to them; in addition they amplify properties of molecular systems (such as normal stress differences) from nearly negligible values to O(1), thus serving (among other things) as a macroscopic laboratory for the study of molecular systems (provided the differences between the two classes of systems are understood). Dense granular matter bears some similarities to amorphous solids. The properties of the latter are not well understood and thus the study of dense granular materials forces one to consider fundamental questions that are not limited to this field alone.

Acknowledgments This work has been partially supported by the National Science Foundation, the Department of Energy, the United States–Israel Binational Science Foundation (BSF) and the Israel Science Foundation (ISF).

References [1] —. 1999. The entire issue of Chaos 9(3) is devoted to granular matter.

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[2] Knowlton, T. M., J. W. Carson, G. E. Klitzing, and W.-C. Yang. April 1994. Particle technology: The importance of storage, transfer and collection. Chemical Engineering Progress 44, 44–54. [3] Jenike, A. W. 1964. Storage and flow of solids. Bulletin of the University of Utah Engineering Experiment Station 123. [4] Johnson, P. C., P. Nott, and R. Jackson. 1990. Frictional–collisional equations of motion of particulate flows and their application to chutes. Journal of Fluid Mechanics 210, 501–535. [5] Campbell, C. S. 1990. Rapid granular flows. Annual Reviews of Fluid Mechanics 22, 57–92, and references therein. [6] Ogawa, S., A. Unemura, and N. Oshima. 1980. On the equations of motion of fully fluidized granular materials. Zeitschrift für Mechanik und Physik 31, 482– 493, and references therein. [7] For pioneering works in this field, see [5] and references therein, and Lun, C. K. K. 1991. Kinetic theory for granular flow of dense, slightly inelastic, slightly rough spheres. Journal of Fluid Mechanics 223, 539–559, and references therein. [8] Goldshtein, A., and M. Shapiro. 1995. Mechanics of collisional motion of granular materials, Part I: General hydrodynamic equations. Journal of Fluid Mechanics 282, 75–114. [9] Sela, N., and I. Goldhirsch. 1995. Hydrodynamics of a one-dimensional granular medium. Physics of Fluids 7(3), 507–525. [10] Goldhirsch, I., N. Sela, and S. H. Noskowicz. 1996. Kinetic theoretical study of a simply sheared granular gas—to Burnett order. Physics of Fluids 8(9), 2337– 2353.

[11] Goldhirsch, I., and N. Sela. 1996. Origin of normal stress differences in rapid granular flows. Physical Review E 54(4), 4458–4461 (1996). [12] Sela, N., and I. Goldhirsch. 1998. Hydrodynamic equations for rapid flows of smooth inelastic spheres, to Burnett order. Journal of Fluid Mechanics 361, 41–74, and references therein. [13] Brey, J. J., F. Moreno, and J. W. Dufty. 1996. Model kinetic equations for low density granular flow. Physical Review E 54(1), 445–456. [14] Brey, J. J., J. W. Dufty, C. S. Kim, and A. Santos. 1998. Hydrodynamics for granular flow at low density. Physical Review E 58, 4638–4653.

[15] Chapman, S., and T. G. Cowling. 1970. The Mathematical Theory of Nonuniform Gases. Cambridge: Cambridge University Press. [16] Tan, M.-L., and I. Goldhirsch. 1998. Rapid granular flows as mesoscopic systems. Physical Review Letters 81(14), 3022–3025. [17] Jaeger, H. M., S. R. Nagel, and R. P. Behringer. 1996. Granular solids, liquids and gases. Reviews of Modern Physics 68(4), 1259–1273, and references therein. [18] Haff, P. K. 1983. Grain flow as a fluid mechanical phenomenon. Journal of Fluid Mechanics 134, 401–430. [19] Goldhirsch, I., and G. Zanetti. 1993. Clustering instability in dissipative gases. Physical Review Letters 70, 1619–1622. See also Goldhirsch, I., M.-L. Tan, and G. Zanetti. 1993. A molecular dynamical study of granular fluids I: The unforced granular gas in two dimensions. Journal of Scientific Computation 8(1), 1–40. [20] Burnett, D. 1935. The distribution of molecular velocities and the mean motion in a nonuniform gas. Proceedings of the London Mathematical Society 40, 382– 435.

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[21] See, for example, Bobylev, A. V. 1984. Exact solutions of the nonlinear Boltzmann equation and the theory of Maxwell gas relaxation. Theoretical and Mathematical Physics 60(2) 280–310. See also Gorban, A. N., and I. V. Karlin. 1992. Structure and approximations of the Chapman–Enskog expansion for linearized Grad equations. Transport Theory and Statistical Physics 21, 101–117. [22] Rosenau, P. 1989. Extending hydrodynamics via the regularization of the Chapman–Enskog expansion. Physical Review A 40, 7193–7196. [23] Slemrod, M. 1999. Constitutive relations for monoatomic gases based on a generalized rational approximation to the sum of the Chapman–Enskog expansions. Archive for Rational Mechanics and Analysis 146, 73–90. See also Jin, S., and M. Slemrod. Regularization of the Burnett equations for fast granular flows via relaxation. 2000. Preprint. [24] Ernst, M. H., and J. R. Dorfman. 1976. Nonanalytic dispersion relations for classical fluids II: The general fluid. Journal of Statistical Physics 12(4), 311– 357. [25] Lutsko, J. F. 1996. Molecular chaos, pair correlations and shear induced ordering

of hard spheres. Physical Review Letters 77, 2225–2228. [26] Luding, S., M. Miiller, and S. McNamara. 1998. The validity of molecular chaos in granular flows. World Congress on Particle Technology, Brighton. See also

Soto, R., unpublished. [27] Tan, M.-L., and I. Goldhirsch. 1997. Intercluster interactions in rapid granular

shear flows. Physics of Fluids 9(4), 856–869, and references therein. [28] Glasser, B. J., and I. Goldhirsch. 1999. Scale dependence, correlations and fluctuations of stresses in rapid granular flows. Preprint. [29] Miller, B., C. O’Hern, and R. P. Behringer. 1996. Stress fluctuations for continuously sheared granular materials. Physical Review Letters 77, 3110–3113. [30] Babic, M. 1997. Average balance equations for granular materials. International Journal of Engineering Science 35, 523–548. [31] Murdoch, A. I. 1998. On effecting averages and changes of scale via weighting functions. Archives of Mechanics 50, 531–539, and references therein. [32] Hopkins, M. A., and M. Y. Louge. 1991. Inelastic microstructure in rapid granular flows of smooth disks. Physics of Fluids A 3(1), 47–57.

[33] McNamara, S., and W. R. Young. 1992. Inelastic collapse and clumping in a one-dimensional granular medium. 1992. Physics of Fluids A 4, 496–504. Recent work is cited in Kadanoff, L. P. 1999. Built upon sand: Theoretical ideas inspired by granular flows. Reviews of Modern Physics 71(1), 435–444. [34] Goldhirsch, I., and T. P. C. van Noije. 2000. Green–Kubo relations for granular fluids. Physical Review E 61(3), 3241–3244. [35] For previous work on boundary conditions for granular gases, see Jenkins, J. T., and E. Askari. 1991. Boundary conditions for rapid granular flows. Journal of Fluid Mechanics 223, 497–508, and references therein. [36] Goldhirsch, I. 1999. Scales and kinetics of granular flows. Chaos 9(3), 659–672. [37] Ronis, D. 1979. Statistical mechanics of systems nonlinearly displaced from equilibrium I. Physica 99A, 403–434. [38] See, for example, Liao, C.-L., T.-P. Chang, D.-H. Young, and C. S. Chang. 1997. Stress–strain relationship for granular materials based on the hypothesis of best fit. International Journal of Solids and Structures 34, 4087–4100. [39] Goldenberg, C., and I. Goldhirsch. 2000. Elasticity of microscopically inhomogeneous systems. Preprint.

STATIONARY WAVES IN ELASTO-PLASTIC AND VISCO-ELASTO-PLASTIC BODIES Vladimir A. Palmov St. Petersburg State Technical University, St. Petersburg, Russia [email protected]

Abstract

1.

Attention is paid to the analysis of longitudinal vibration of a semiinfinite rod loaded by a harmonic force at one end. A rod with different constitutive equations is considered, and vibration decay with distance from the loaded end is sought. In order to overcome problems caused by nonlinearity of the constitutive equations, the method of harmonic linearization is used. As a result simple analytical solutions are obtained and analyzed. If the material of a rod is rigid-plastic and has linear kinematical hardening, the vibration occupies only finite domain near the loaded end. The same is true for an elasto-plastic material with a bilinear diagram, while elastic waves are traveling in the remaining part of the rod. In the case of so-called amplitude-dependent internal friction, vibration occupies the entire rod and possesses a saturation property. This property implies that there exists a limit of attainable level of vibration, which depends on the material properties, the distance from the loaded end, and frequency. One special sort of continuous media with complex structure is formulated in a latter section. It is shown that dynamical structure, rather than input power and constitutive equations of components, influences the wave propagation.

FORMULATION OF THE PROBLEM

We consider the semi-infinite rod longitudinal vibration is given by

The governing equation for

where Q denotes a tensile axial force in the cross section with coordinate x, u is axial displacement, and m is the mass per unit length. The dot and prime indicate differentiation with respect to time t and spatial coordinate x, respectively. The axial strain of the rod is

359 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 359–372. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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The constitutive equations for the rod material is assumed to be given by the equation

where is the visco-elasto-plastic functional. We formulate boundary conditions at the cross section can take one of the forms

They

or

At the Sommerfeld condition should be satisfied: it is possible to have waves traveling only in the x direction and not coming back. We intend to consider different constitutive equations (3) and to analyze the peculiarity of traveling and standing waves in a semi-infinite rod.

2.

WAVES IN VISCO-PLASTIC MATERIAL

For determinacy we use the generalized Maxwell rheological model. Acting force Q and strain are connected by the integral functional

where

is the so-called relaxation function. A steady-state vibration of the rod is sought in the form

where Re denotes real part of the complex function and const. Substituting of Eqn. (8) in Eqn. (6) and then in Eqn. (1) leads to the equation for the determination of the unknown quantity

Only that root of this equation having the form

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should be used; only in this case will we have a wave traveling in the x direction:

For pure elastic material of the rod we should put for all relaxation times T except infinite ones. In this case the right hand side of Eqn. (9) becomes real-valued. So, and Eqn. (11) gives a wave that travels in the x-direction without any spatial decay. The simplest analysis of this section is well known. But we pay attention to this analysis for the convenience of comparison with future results.

3.

WAVES IN RIGID-PLASTIC MATERIAL WITH LINEAR HARDENING Let the constitutive equation of the rod material have the form

where c and H are constants. One can easily recognize the deformation law of a rigid-plastic material with linear hardening and the Bauschinger effect being taken into account. Equations (1), (2) and (12) exhibit a nonlinear problem. The method of harmonic linearization (see Ziegler 1995, Palmov 1998) will be applied to solve this problem. This method is one of the simplest and most effective methods of nonlinear vibration theory and is actively used in the theory of automatic control, theory of vibration protection, etc. An essential feature of the method of harmonic linearization is its simplicity and clearness of physical results obtained. In this paper this method is applied to the analysis of wave propagation in a rod whose material is governed by a nonlinear equation. Using the method of harmonic linearization, the strain ε is assumed to be a nearly harmonic process in each cross-section:

where a(x) is the amplitude and is the phase. Now we replace the nonlinear function (12) by the linearized version

Substituting Eqn. (14) in equations (2) and (1), and equating sine and cosine coefficients, we get the system of equations

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Although we have not succeeded in constructing a general solution with four integration constants, we have obtained a solution that has two integration constants and satisfies the Sommerfeld conditions. This solution has the form

Direct substitution into (15) and (16) shows that the constants have the values

and

The constants and are still arbitrary and must be determined from the boundary condition at Solution (17) is valid only for those values of x for which i.e.

For

it should be taken that

A direct substitution of the latter solution into the system of equations (15) and (16) shows that this system is satisfied. Besides, the solutions (17) and (16) meet. Inserting the obtained expressions for amplitude and phase into (17), we get the following expression for the deformation:

Let Eqn. (4) be the boundary condition at Satisfying it by means of a linearized expression for force Q, we obtain the following values for the integration constants:

It follows from this equation that the constructed solution makes sense only if the amplitude of the force in the cross section at is larger then From a physical perspective, this means that force must exceed the yield stress H in law (12). If the force is smaller than then no motion in the rod occurs.

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Displacement in this problem is obtained by integration of the equation for deformation (21), to give

It is seen that in the region near the driven end the displacement is a superposition of running and standing waves. For there is no wave motion. The region of wave motion is obtained by substitution of Eqs. (22) and (18) into (19) and is given by

The lower the material hardening and the higher the frequency of vibration, the smaller is the zone of motion. Further, decreasing the yield stress H or increasing the amplitude enlarges the region of motion. In addition, as Hence, in a rod made of a rigid-plastic material without hardening, the vibration localizes in the cross section where the source of vibration is located.

4.

WAVES IN ELASTO-PLASTIC MATERIAL WITH LINEAR HARDENING

Let the rod material be an elasto-plastic material with a linear hardening. For developed plastic deformation the rod behavior is governed by the following system of equations (see Palmov 1998):

Here denotes the plastic strain. The method of harmonic linearization can be applied directly to Eqn. (25). Following this method, we put

where b and are the amplitude and phase of the plastic strain, respectively. Next, the nonlinear function in (25) is approximated by a linear one

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Inserting this approximation into (25) we find the following expressions for and Q in terms of

Differentiating the equation of dynamics (1) with respect to x and using eqs. (2) and (28), we obtain an equation for the plastic strain:

Substituting the expression for due to (26) into (25) and combining coefficients in sine and cosine, we arrive at following system of two differential equations for the amplitude and phase of plastic strain:

This system of equation has a solution where and are integration constants, and and are easily obtained by substitution of (32) into the system of equations (30) and (31). These are given by

Solution (32) makes sense only in that part of the rod where the amplitude b is positive, i.e. only for x satisfying the inequality

For

one must take

Direct substitution of solution (35) into the system of equations (30) and (31) shows that these equations are satisfied. In addition, the solutions

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(32) for and (35) for meet. Indeed, at the amplitude and phase of the plastic strain are continuous. By virtue of (28) the total strain and force in the rod are continuous as well. Therefore, the plastic strain is equal to zero for The solution constructed satisfies the Sommerfeld radiation condition at infinity, as as Let us determine the integration constants and find the main variables of the problem, namely the total strain and the displacement u. A convenient way to do this is to transform to a complex exponential form in Eqn. (26): which is allowed since Eqs. (28) are linear in It goes without saying that only the real parts of (36) and others have a physical meaning. Inserting (36) into (28) yields the equations

Using the first equation in (37) and the boundary condition in (4) at we easily obtain the amplitude and phase of the plastic strain in the plastic zone:

The solution obtained is seen to be meaningful only if the amplitude of the external force satisfies the inequality

This inequality is a condition of initiation of plastic deformations in the rod. This condition is an approximate one and differs from the exact condition only in an inessential factor in As seen from the second Eqn. (37) the strain amplitude is

or, accounting for the exact expression for b,

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It follows from these equations that the strain amplitude decreases from its maximum value at to the value and then remains constant and is equal to maximal elastic strain. Let us determine the displacement u. To this end, we insert expressions for b and from (32) and (41) into (37), and integrate over x taking into account the continuity of u at and the radiation condition as The result is

The parameters that appear in this equation have been determined above. As seen from (42) a wave of constant amplitude is outgoing at infinity for For the rod vibration is a superposition of propagating and standing waves, where amplitude of the running wave decreases with increasing distance from the source of vibration. Let us draw some conclusions. The cross-section is a border between a part of the rod with plastic deformation and part with pure elastic deformation. An expression for is obtained by substitution of the expressions for (38) and and (33) into (34):

Thus, the higher the material hardening is and the lower the mass per unit length and vibration frequency are, the larger the zone of plastic strain is.

5.

WAVES IN ELASTO-PLASTIC MATERIAL WITH POWER-LAW HARDENING

In this section we make use of the generalized Prandtl model. In Russian literature this model is called sometimes as Ishlinsky model (Palmov 1998). The constitutive equations are as follows:

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where C is longitudinal rigidity, is the plastic strain for a Prandtl element with dimensionless plastic limit h, and p(h) is the probability distribution of plastic limits h. For small h we assume

In this case we have a simple expression for the loading function:

It is widely used in the theory of internal and structural friction. Considering the behavior of material under harmonic deformation in time,

we again exploit harmonic linearization:

Substituting this approximate representation in Eqn. (44) we get

Since Eqns. (44) are linear with respect to and we can use the general complex exponential representations for the variables

where A, a, and are amplitudes, and and are phases. After long-term calculations we can get the simple formula

where

In what follows, is referred to as the complex rigidity. As follows from (52), the complex rigidity does not depend on the deformation frequency for any density p(h). It is easy to see that depends on the amplitude of deformation a. These properties of the model correspond to the laws of internal friction observed by most metals and structural materials. In this sense, the model considered here successfully describes internal friction under intensive stresses.

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Differentiating of Eqns. (1) with respect to x and using Eqns. (2) and (51) renders the equation

Using this equation and first Eqn. (50), we get equations for amplitude B and phase of traction force:

where For example, for the power function p(h) given by (45), we have

Equations (54) and (55) are identical with the equations for dynamics of a point mass in a plane non-central field. The established analogy is useful from two points of view. Firstly, it gives an idea of the character of the difficulties one faces when integrating the system (54), (55). Secondly, it allows the exact and approximate methods developed in the dynamics of a point mass—in particular, in the dynamics of satellites and spacecraft—to be applied. We apply one of these methods: we neglect radial acceleration. In our problem it means that we should neglect the second derivative in Eqn. (54). This gives

Substituting the expression of complex rigidity given by (52) into Eqn. (58) and retaining only terms of the first order in effects of plastic deformation yields

For the power function (45) this equation takes the form

Integrating this equation we obtain

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It is seen that vibration occupies the whole rod, decaying from the value at to zero. The following peculiarity of the vibration field is of interest. For large values of x we can neglect the value of unity in the parentheses of (61) to obtain

The strain amplitude at the loaded end is absent in this equation. Hence it indicates an upper limit of vibration at any point. This limit does not depend on the power of the vibration source, but depends on the properties of material, distance x, and frequency. In particular, it means that amplitude of the excitation cannot be reliably determined by measurements from the far field of elasto-plastic vibrations.

6.

WAVES IN VISCO-ELASTO-PLASTIC MATERIAL

Let us consider now a more complicated rheological model. The model consists of the generalized Prandtl model and generalized Maxwell model, connected in parallel. Repeating the derivation of the previous section we find that the strain

amplitude a(x) decreases and is governed by the following ordinary differential equation: where

Here describes the energy dissipation due to viscosity in the generalized Maxwell part of the model. The integral of Eqn. (63) is given by

As seen from Eqn. (65), vibration occupies the whole rod, decaying from the value at to zero. The vibration distribution has the same qualitative feature as in the previous example. Indeed, an increase of the strain amplitude leads to an increase in the amplitude a at any x. However, this increase at a given x has a limit, which is formally obtained from Eqn. (65) as

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Hence the following inequality holds:

which is similar to inequality (62). Again this indicates the existence of an upper limit of vibration that does not depend on the power of the source of vibration, but is a function of frequency, the rod parameters, and the distance from the source. The effect of vibration saturation observed in the two last examples may be explained as follows. As the power of the source of vibration increases, the density of vibrational energy increases in the cross section near the loaded end. The energy dissipation in this zone increases even more intensively. This results in an absorption of vibration in a region adjoining the loaded end, while the vibrational energy in remote cross sections increases negligibly.

7.

WAVES IN MEDIUM WITH COMPLEX STRUCTURE

We postulate a carrier medium and assume that it is governed by the equation of an elastic rod. A set consisting of an infinite number of noninteracting oscillators with continuously distributed eigenfrequencies is attached to each cross section of the carrier rod (Palmov 1998). The equations of dynamics of this rod with complex structure are as follows:

where u is the displacement of a rod cross section, is the displacement of the oscillator mass with eigenfrequency k, E is Young’s modulus of the carrier rod, and is the density of the carrier rod. The quality of is equal to mass of the oscillators having eigenfrequencies in the interval per unit volume. Thus the total mass density of the oscillators attached to the carrier structure is given by

The rigidity of an oscillator’s suspension is denoted in Eqn. (69) by C(k), which is given by

The term is introduced in Eqn. (69) in order to take into account the energy dissipation in the oscillator suspension. Account of

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damping will be achieved firstly by a visco-elastic model, and then will be generalized by an elasto-plastic or visco-elasto-plastic one. The term with is assumed to be small for any oscillator motion in both cases. Lastly we note that K, Q, exhibit external forces applied to the carrier rod and to the attached oscillators. We formulate the boundary condition at the cross section in the form For we formulated the Sommerfeld condition. If K, Q, equal to zero, we are seeking a wave solution in the form

where

are

is governed by the equation

in which We should select only the root of this equation that has the form

Substitution of this representation in Eqn. (73) renders the simple formula for the displacement of carrier rod

Spatial decay of this wave is determined by the factor We intend to prove that the vibration decay does not depend primarily on the oscillator damping and that it remains finite even for vanishingly small damping. To this end, it is sufficient to show that the expression for remains complex when Using the Sokhotsky–Plemelj formula we obtain the following limit in Eqn. (74) as

As the imaginary part in (78) does not vanish, the spatial decay factor has a finite value indeed, and is determined by the dependence of the oscillator masses m(k) on their eigenfrequency k! This effect is typical only for a model that accounts for the complex structure of the medium, i.e. existence of the suspended oscillators. From the physical point of

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view the effect can be explained by the fact that suspended oscillators act as dynamic absorbers. At first glance, this conclusion seems to be paradoxical. However, a careful analysis shows that the energy dissipation has a finite value even for small damping of the oscillators, which is explained by large amplitudes of vibration for the resonating oscillators. The theory presented here may be used for the analysis of vibration in such complex dynamical systems as vessels, spacecraft, and submarines.

8.

SUMMARY

The laws of vibration decay with distance from the source of vibration were investigated in this paper. The character of the decay turned out to be essentially dependent on the rheological properties of rod material. For a purely elastic rod, spatial decay is absent. The more difference of material law from pure elastic, the more spatial decay there is. There exists, however, another important factor that influences the decay character, namely the dynamical structure of the medium. We considered one of the simplest media of this sort. It was an elastic carrier rod with suspended oscillators. The medium under consideration

possesses the following interesting peculiarity: Though the medium is “built” from high-quality elements, such as an ideally elastic carrier rod and slightly damped oscillators, the coefficient of spatial decay has a finite value. The latter model can be successfully used for the description and analysis of vibration of complex mechanical systems.

Acknowledgments This work made possible in part by a grant from the Russian Foundation for Basic Research (N 98-01-01073).

References Ziegler, F. 1995. Mechanics of Solids and Fluids. New York, Vienna: Springer-Verlag. Palmov, V. 1998. Vibrations of Elasto-Plastic Bodies. Berlin, Heidelberg, New York: Springer-Verlag.

HIERARCHICAL MODEL- AND DISCRETIZATION-ERROR ESTIMATION OF ELASTO–PLASTIC STRUCTURES Erwin Stein, Stephan Ohnimus, and Marcus Rüter Institute of Structural and Computational Mechanics (IBNM) University of Hannover, Hannover, Germany [email protected], [email protected], [email protected]

Abstract

1.

A posteriori error estimators of the discretization and model errors are presented for finite-element solutions of small-strain elasto–plasticity and large-strain elasticity problems. The described posterior equilibrium method (PEM) also yields anisotropic error indicators that are more efficient for thin-walled structures than isotropic ones. The model error in a sequence of hierarchical mathematical models and related dimensions can be as important as the discretization error. Therefore, integrated error-controlled adaptivity of finite-element approximations and of models in relevant subdomains, such as thickness jumps, is a challenging task for complex engineering structures. The posterior equilibrium method yields an effective estimation of discretization errors and model errors in general. Based on this approach, an upper bound of the error in a local quantity of interest can be obtained by solving an auxiliary local Neumann problem for the error of the dual problem. Some illustrative examples show the numerical and physical features of the methodologies.

INTRODUCTION

An important task for engineers is the reliable prediction of local and global mechanical behavior of structures under various loading conditions, including damage and failure. This prediction needs adequately simplified physical structures and systems and related mathematical models, based on experimental evidence. Engineering models result from physical idealizations and related mathematical homogenizations on different scales of consideration (from nano- to macro-scales), following the rule “as simple as possible and as complicated as necessary”. For a certain class of problems we define a mathematical master model, e.g. 3D elasto–plasticity, from which a sequence of more simplified or reduced models can be introduced stepwise in a hierarchical or parallel way, determined by further kinematical and/or physical hypotheses (e.g. 373 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 373–388. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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the 2D elastic Reissner–Mindlin plate theory). The analytical solutions of such a nested sequence of models should provide consistent transitions by asymptotic analysis, e.g. concerning the types of operators and singularities depending on constitutive equations, topology, geometry and loading. The global discretization error of the approximated solution of a specific model is estimated by locally computed error estimators of local Neumann type. This estimator was introduced by Bank and Weiser [1], proven by Ainsworth and Oden [2], and extended by Stein and Ohnimus [3, 4]. Since we do want convergence not only with respect to a specific simplified model but for the hierarchically expanded master model, model adaptivity has to be applied in certain subdomains integrated with solution adaptivity for each related model. The goal of hierarchical model- and solution-error analysis therefore is the automatic detection of these subdomains [3, 4]. However, these estimators are based on a global energy norm or norm control of the error. To estimate the error in a local quantity of interest, local error measures based on duality techniques (see Johnson [5] and Becker and Rannacher [6]) were systematically developed in recent years for a wide range of mostly linear problems, e.g. in linear elasticity by Rannacher and Suttmeier [7], Cirak and Ramm [8] and Ohnimus et al. [9], here also with local Neumann problems. In this paper we present model-error estimators as well as implicit a posteriori error estimators of local Neumann type for both the initialboundary-value problem of elasto–plasticity and the boundary-value problem of finite elasticity.

2.

2.1.

VARIATIONAL FORMULATION AND A POSTERIORI ERROR ANALYSIS IN LINEAR ELASTICITY

The weak form and its discretization

Let us begin with a brief outline of the linear elasticity problem and its discretization with the finite-element method. The elastic body occupies a bounded open set with piecewise smooth and Lipschitz continuous boundary such that and where and are the Dirichlet boundary and the Neumann boundary, respectively. If the elastic body is in equilibrium, the displacement field fulfills the equilibrium condition in and the boundary conditions u = 0 on and on where are prescribed body forces, are prescribed surface tractions, is the unit outward normal, is the symmetric

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stress tensor, and is the constitutive equation. Here, C is the symmetric and positive definite elasticity tensor, and is the strain tensor. The variational problem now reads: find a solution such that

where

denotes the Sobolev space and and are bilinear and linear forms, respectively,

given by

In the finite-element method, the Galerkin method for finite subdomains, we subdivide the domain into elements, i.e. with boundary and introduce finite dimensional subspaces Hence, we search for such that which means that we restrict the variational calculus in its action. Replacing by in (1) and subtracting (3) from (1) result in the Galerkin orthogonality for all where we introduced the solution error From (3) it follows with Céa’s lemma that the Galerkin method is optimal in the energy norm Furthermore, subtracting the Galerkin orthogonality condition from yields the main equation of error analysis Note that only by introducing the further test function it is possible to get the upper bound estimate for the interpolation constant of the important Babuška–Miller upper and lower estimate—see Babuška and Miller [10].

2.2.

Equivalent representations of residual and equilibrated error estimators

Upon introducing the local bilinear and linear forms and respectively, on the local space the restrictions of a and l to an element such that

by

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we may present three equivalent forms for the estimated error for all and as follows:

where the residuals

and

are given by

and in respectively, and equilibrated tractions fulfilling Cauchy’s lemma, i.e. and the Neumann boundary condition Further requirements are discussed in 3.1.

3.

are on on

DIRECTIONAL ERROR ESTIMATORS WITH EQUILIBRATED BOUNDARY TRACTIONS, CALLED POSTERIOR EQUILIBRIUM METHOD (PEM)

The implicit “posterior equilibrium method” (PEM) for quantitative local computations of directional (anisotropic) global or local as well as model-error estimates is based on an equivalent problem for direct or explicit residual-error estimation and yields sharper upper bounds and thus much better effectivity indices than the direct “residual error estimation method” (REM), especially for locking problems [3].

3.1.

Introduction of residual error estimators with equilibrated tractions

Let us now consider the following local Neumann problem derived from (6c): find the approximated error such that

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where is an enriched finite-dimensional test space by at least two further hierarchical test modes. For the solvability of (8) we have to ensure that the equilibration condition

is fulfilled, where is the space of rigid-body motions. Hence, a simple averaging ansatz of the form

is possible with linear polynomials

This results in

We remark that the are not really consistent to the original local finiteelement problem with the property if Hence, anisotropic error estimation might not be reliable. In order to get around this problem, we introduce an extended space for the equilibration instead of This leads to a local Neumann problem if Integration by parts yields the necessary identity (see also (7))

The consistency condition

is fulfilled, such that the global equivalence of the weak form

holds. A condition for the equivalence with the Babuška–Miller residual error estimator is that with C > 0 holds. Thus, we choose g such that which is the same regularization condition as

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The determination of is clearly not unique. Hence, different approaches exist. The basic idea of improving the stresses by a post process goes back to Bufler and Stein [11] and Stein and Ahmad [12]. Ladevèze and Leguillon [13] were the first who used this concept for a posteriori error estimation. Different strategies are discussed e.g. in Ainsworth and Oden [14]. Here, we follow Stein and Ohnimus [3], who use the energy-equivalent nodal forces to determine the equilibrated tractions by least-squares regularizations on patches instead of averaging the neighboring tractions. The main part of the error analysis is performed only once on unit elements and then transformed energy equivalently onto real elements. Additional elementwise variational problems are solved for each element in order to get improved displacements from the new tractions using an enriched space The improved solution follows from

but only if is reduced by the rigid-body motions. The locally computed equilibrium discretization error follows from the energy norm of the difference

3.2.

Model adaptivity based on equilibrated boundary tractions (PEM)

The concept of model adaptivity is similar to the one of solution adaptivity, but uses an additional hierarchically expanded model 2, e.g. the expansion from a 2½ D-plate to a 3D-continuum theory. Our goal is to get the global error between the considered model 1 and the expanded model 2 with solutions and respectively. Obviously, the model error should not be larger than the solution error. In conjunction with the model expansion we also need dimensional adaptivity of the ansatz functions, especially in the thickness direction (denoted by such that we realize an integrated solution–dimension–model adaptivity, necessary in boundary layers, thickness jumps, and concentrated loads. We want to compute the dimensionally expanded solution of the new model 2 with the enhanced bilinear form for all

where denotes the dimensionally expanded test space. We remark that the dimensions of and the solution of the reduced model 1, i.e. have to be equal as well as the dimensions of the related strains.

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The equilibrium model error estimator in the energy norm follows from the difference of the improved stresses (from PEM) obtained by

the hierarchically expanded model, by the prolongation from

and the transformed stresses to obtained by the

investigated model as follows:

Note that the using of instead of is mathematically inconsistent and would generally result in an overestimation of the error due to deformation constraints, i.e. locking effects. In order

to suppress influences on the model error caused by large discretization errors, the model is adapted only if the global model error is larger than the global discretization error. Furthermore, the model error is activated only in certain subdomains after reaching a prescribed tolerance of the

discretization (solution) error. The following model-error estimators are possible: The estimator of the model displacement error, i.e. which is not reasonable due to the discussed thicknesslocking effects. The error estimator using stresses of the current and the expanded model, i.e. Here, the locking phenomena do not appear. This can be shown by a mixed variational formulation. A relative-error estimator, taking into account the neglected deformation modes of the reduced model, i.e. This estimator is useful for the expansion of 2D-plate and shell models. For a pure displacement method, an energy-oriented estimator

can be used, too. But in most investigated cases, this indicator tends to zero.

3.3.

Error estimators for Prandtl–Reuss elasto–plasticity by PEM

We show next how to derive a posteriori error estimators for perfect elasto–plasticity (Prandtl–Reuss elasto–plasticity) based on PEM. Therefore, let us first briefly outline the basic theory of elasto–plasticity. The von Mises yield condition at the process time is given by with yield stress The associated

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flow rule is given by with the plastic multiplier and the flow direction The Kuhn–Tucker conditions are and To satisfy these conditions, we introduce the additive split of the strain rates into elastic and plastic parts by with

Differentiation of the specific strain energy with respect to the elastic strains results in the stresses Further, we assume that the time integration is sufficiently exact (quasi exact). Hence, we get with the analytical Prandtl–Reuss tangent n(t). The elasto–plastic problem is now solved approximately with the finite-element method and yields the approximations and In order to estimate the error in the equilibrium and the plastic strains, we follow a two-step strategy. First, we assume that the flow direction and the plastic strain field are exact. Thus, we have only to control the error of the equilibrium for all Hence, we get the residual formulated in rate equations for the error

with the test space on and the definition of the solution space with the important property For the semi-discrete approximation method, we introduce the spatial finite solution and test subspaces and respectively. Thus, we have The residual rates have to be equilibrated in the same way as we have seen above, i.e.

with the equilibrium condition for all In order to determine the equilibrated tractions, one has to solve for each element separately the following problem for all

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since the equilibrated tractions have to represent the correct tractions that solve the local discrete variational problem. Furthermore, the consistency condition has to be satisfied for the rates of the tractions for all For this elliptic problem, we can prove that we get an upper bound error estimator for the current tangential space as follows:

where we defined the total energy norm by

For further details we refer to Stein and Ohnimus [4]. Note that this estimator holds also for the pure (linear) elastic case. We further remark that we estimate the error at time T. If we want to get an error estimator in time that also includes the accumulation error, we have to use a different norm, namely

An upper bound for the error measured in this norm is carried out in the same way as the upper bound (22), and is explained in [4]. From the asymptotical relation, our estimator will lead to an upper error estimator since we describe the plastic domain more and more accurately, such that the error estimator is asymptotically exact. In the second step we consider an error indicator for the plastic strains. Thus we test the local solution e* of the error with respect to the current elasto–plastic tangent. The motivation is that we have to control whether the stress rates and the strain rates are orthogonal to the flow direction Therefore, we introduce the plastic strain error indicator in the energy related norm as

The exact solution of the local variational problem

yields an upper error bound. Since an exact solution is not possible for realistic engineering problems, we solve the local problem in an expanded

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approximation space with additional hierarchically expanded polynomial test and solution shape functions. The polynomial order of the current test space is p, and the local extension is k, such that we get the local polynomial order p + k with at least k = 2. (We tested until k = 5.)

We then define the isotropic error estimator

by

The

corresponding effectivity index is

3.4.

An error estimator in finite elasticity

In what follows, we present an error estimator for the geometrically

as well as materially nonlinear problem of finite elasticity. Therefore, we first sketch below the weak form of the boundary-value problem of finite elasticity. Material points in the reference configuration are mapped to material points in the current configuration by the deformation mapping Let with denoting the deformation gradient; then the mapping is orientation-

preserving if the Jacobian Since it proves convenient to formulate the problem in the current configuration (i.e. an Eulerian formulation), we introduce the left Cauchy–Green tensor as a strain measure. From the second law of thermodynamics we thus obtain the Kirchhoff stresses by where is the specific strain-energy function of an isotropic homogeneous body. For the sake of simplicity, we make the choice of a neo-Hooke material given by with positive Lamé constants and This leads to the Kirchhoff stresses where 1 denotes the second-order unity tensor. Let us now consider the total potential energy of the elastic body

with

Clearly, the minimizer

fulfills the variational formulation

where the semi-linear form which is linear only with respect to its second argument, and the linear form as an element of the dual space are now given by

with

Since this is a nonlinear problem, we apply Newton’s

method and solve within a recursive procedure the linearized (around

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variational problem of finding increments

where the functional weak form of the residual, and

given by

383

such that

is the is a symmetric bilinear

form given by the second variation of the energy functional (27), that is,

with the spatial elastic moduli

Here, I

denotes the fourth-order unity tensor.

Let us now assume that we iterated a finite-element solution of the corresponding discrete analogue to (30) with sufficient accuracy. A linearization of the variational problem (28) around the finite-element solution thus leads to the error representation formula

where we assumed that the error is small. The weak residual R takes now the form for all Replacing by

leads to the nonlinear formulation of the Galerkin orthogonality R(v) = for all Furthermore, the tangent form aT induces a norm, the tangential-energy norm which is the appropriate norm to measure the error e. Analogously to the local forms (5), we define the restriction of to an element given by

Recalling (32), we may now formulate the corresponding local Neumann

problem as follows: find a solution

where the weak residual

such that

contains now the nonlinear finite-element solution—cf. (8). We can now show that the solutions of the local Neumann problems (34) yield an upper bound of the error given by

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However, the local Neumann problems (34) may become negative definite long before a global stability point is reached. Possibilities to get around this problem are discussed in Rüter et al. [15]. Brink and Stein derive in [16] a similar estimator for the stress error. Nonlinear anisotropic problems are discussed in Rüter and Stein [17].

3.5.

A local a posteriori error estimator in finite elasticity

Following the pioneering works of Johnson [5], Eriksson et al. [18], and Becker and Rannacher [6], we show next how to estimate the error in a local quantity of interest, such as local displacements or stresses. Therefore, let us first introduce the dual problem: find a solution such that

where is an arbitrary bounded linear functional that defines the local quantity of interest within an adequate local support. Note that, since the associated linear operator to the bilinear form is self-adjoint, only the right-hand side of the primal problem (30) changes. Substituting in (36), we obtain (Betti’s theorem)

Next, we subtract the Galerkin orthogonality from (37) and apply the Cauchy–Schwarz inequality. We thus arrive at the estimator

The norms can now be estimated by the solutions of the corresponding local Neumann problems, using improved boundary tractions for both the primal and the dual problem. Note that this nonlinear error estimator also includes the linear elasticity problem as analyzed by Ohnimus et al. [9]. Another approach of local error estimators yielding upper and lower error bounds was suggested by Prudhomme and Oden [19].

4.

NUMERICAL EXAMPLES

In the first example (Fig. 1), the automatic error-controlled expansion from a 2D clamped square plate in bending to a 3D-continuum theory is shown as well as the dimensional and model adaptivity. The second example treats the elastic–perfectly-plastic tension of a plate with a hole (only a quarter is shown) in the plane-strain state. The system data and results are given in Fig. 2.

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Figure 1 Refined meshes and isoplots of and for coupled model and solution adaptivity of an elastic square plate (length l = 10 m, thickness t = 0.01 m) with a central hole (radius R = 0.01 m) in bending and constant transverse loading. System data: E = 206900 MPa, and anisotropic polynomial order results for mesh 10.

The last example, depicted in Fig. 3, shows the results of a finite elastic problem, here a plate clamped on the left and subjected to a tension force on the opposite side.

References [1] Bank, R. E., and A. Weiser. 1985. Some a posteriori error estimators for elliptic partial differential equations. Mathematics of Computation 44, 283–301.

[2] Ainsworth, M., and J. T. Oden. 1993. A unified approach to a posteriori error estimation based on element residual methods. Numerische Mathematik 65, 23– 50.

[3] Stein, E., and S. Ohnimus. 1997. Coupled model- and solution-adaptivity in the finite-element method. Computer Methods in Applied Mechanics and Engineering 150, 327–350. [4] Stein, E., and S. Ohnimus. 1999. Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems. Computer Methods in Applied Mechanics and Engineering 176, 363–385.

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Figure 2 Elasto–plasticity: Square plate (l = 200 mm, t = 1 mm) with a hole (R = 10mm) in the plane-strain state. System with E = 206900 MPa, 95% of critical load, elements. [5] Johnson, C. 1993. A new paradigm for adaptive finite element methods. Proceedings of The Mathematics of Finite Elements and Applications (MAFELAP) 93. New York: Wiley. [6] Becker, R., and R. Rannacher. 1996. A feed-back approach to error control in finite element methods: Basic analysis and examples. East-West Journal of Numerical Mathematics 4, 237–264.

[7] Rannacher, R., and F.-T. Suttmeier. 1997. A feed-back approach to error control in finite element methods: Application to linear elasticity. Computational Mechanics 19, 434–446. [8] Cirak, F., and E. Ramm. 1998. A posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem. Computer Methods in Applied Mechanics and Engineering 156, 351–362.

[9] Ohnimus, S., E. Stein, and E. Walhorn. 2000. Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems. International Journal of Numerical Methods in Engineering, submitted. [10] Babuška, I., and A. Miller. 1987. A feedback finite element method with a posteriori error estimation: Part I. The finite element method and some basic proper-

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Figure 3 Finite elasticity: Tension test, neo-Hooke material, plane strain, elements; upper half of the specimen is discretized due to symmetry conditions.

ties of the a posteriori error estimator. Computer Methods in Applied Mechanics and Engineering 61, 1–40. [11] Bufler, H., and E. Stein. 1970. Zur Plattenberechnung mittels Finiter Elemente.

Ingenieur-Archiv 39, 248–260. [12] Stein, E., and R. Ahmad. 1977. An equilibrium method for stress calculation using finite element methods in solid- and structural-mechanics. Computer Methods in Applied Mechanics and Engineering 10, 175–198. [13] Ladevèze, P., and D. Leguillon. 1983. Error estimate procedure in the finite

element method and applications. SIAM Journal of Numerical Analysis 20, 485–509. [14] Ainsworth, M., and J. T. Oden. 1997. A posteriori error estimation in finite-

element analysis. Computer Methods in Applied Mechanics and Engineering 142, 1–88. [15] Rüter, M., S. Ohnimus, and E. Stein. 2001. On global and local a posteriori error estimation in finite elasticity. In preparation. [16] Brink, U., and E. Stein. 1998. A posteriori error estimation in large-strain elasticity using equilibrated local Neumann problems. Computer Methods in Applied Mechanics and Engineering 161, 77–101.

[17] Rüter, M., and E. Stein. 2000. Analysis, finite element computation and error estimation in transversely isotropic nearly incompressible finite elasticity. Computer Methods in Applied Mechanics and Engineering 190, 519–541. [18] Eriksson, K., D. Estep, P. Hansbo, and C. Johnson. 1995. Introduction to adaptive methods for differential equations. Acta Numerica, 106–158. [19] Prudhomme, S., and J. T. Oden. 1999. On goal-oriented error estimation for local elliptic problems: Application to the control of pointwise errors. Computer Methods in Applied Mechanics and Engineering 176, 313–331.

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At the Welcome Reception, the Grady Johnson Orchestra provided Congress participants with a sample of the jazz and blues for which Chicago is famous.

LAMINAR BOUNDARY-LAYER THEORY: A 20TH CENTURY PARADOX? Stephen J. Cowley Department of Applied Mathematics and Theoretical Physics University of Cambridge, Cambridge, United Kingdom [email protected]

Abstract

1.

Boundary-layer theory is crucial in understanding why certain phenomena occur. We start by reviewing steady and unsteady separation from the viewpoint of classical non-interactive boundary-layer theory. Next, interactive boundary-layer theory is introduced in the context of unsteady separation. This discussion leads onto a consideration of large-Reynolds-number asymptotic instability theory. We emphasize that a key aspect of boundary-layer theory is the development of singularities in solutions of the governing equations. This feature, when combined with the pervasiveness of instabilities, often forces smaller and smaller scales to be considered. Such a cascade of scales can limit the quantitative usefulness of solutions. We also note that classical boundary-layer theory may not always be the large-Reynolds-number limit of the Navier–Stokes equations, because of the possible amplification of short-scale modes, which are initially exponentially small, by a Rayleigh instability mechanism.

INTRODUCTION

Sectional lecturers were invited ‘to weave in a bit more retrospective and/or prospective material [than normal] given the particular [millennium] year of the Congress’. This invitation is reflected in the current, possibly idiosyncratic, article. For alternative viewpoints the reader is referred to Stewartson (1981), Smith (1982), Cowley and Wu (1994), Goldstein (1995), and Sychev et al. (1998). We begin with a deconstruction of the components of the title. Boundary-layer theory. Prandtl (1904) proposed that viscous effects would be confined to thin shear layers adjacent to boundaries in the case of the ‘motion of fluids with very little viscosity’, i.e. in the case of flows for which the characteristic Reynolds number, Re, is large. In a more general sense we will use ‘boundary-layer theory’ (BLT) to refer to any 389 H. Aref and J.W. Philips (eds.), Mechanics for a New Millennium, 389–412.

© 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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large-Reynolds-number, asymptotic theory in which there are thin shear layers (whether or not there are boundaries). 20th century. Prandtl (1904) published his seminal paper on the foundations of boundary-layer theory at the start of the 20th century, while ICTAM 2000 was held at the end of the same century. Laminar. Like Prandtl (1904) we will be concerned with laminar, rather than turbulent, flows. Flows that are in the process of laminarturbulent transition will be viewed as unstable laminar flows. A paradox. Experimental flows at large Reynolds numbers are turbulent; yet useful comparisons with laminar-flow experiments at moderately large Reynolds numbers can sometimes be made with largeReynolds-number asymptotic theories. We view as a paradox this seemingly contradictory result, i.e. that useful comparisons with laminar flow can be made with expansions made about Reynolds numbers when flows are almost invariably turbulent. ?. The question this paper will discuss is whether the final ‘?’ is needed in the title. A subjective conclusion is given at the end.

2. 2.1.

CLASSICAL BOUNDARY LAYERS Formulation

We consider incompressible flow of a fluid with constant density and dynamic viscosity past a body with typical length We assume that a typical velocity scale is and that the Reynolds number is given by

For simplicity we will, for the most part, consider two-dimensional incompressible flows, although many of our statements can be generalized to three-dimensional flows and/or compressible flows.

The key approximations. Boundary-layer theory applies to flows where there are extensive inviscid regions separated by thin shear layers, say, of typical width For one such shear layer take local dimensional Cartesian coordinates and along and across the shear layer respectively. Denote the corresponding velocity components by and respectively, pressure by and time by On the basis of scaling arguments (e.g. Rosenhead 1936) it then follows that

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Further, it can also be deduced that the key approximations in classical BLT are that the pressure is constant across the shear layer, i.e.

and that streamwise diffusion is negligible, i.e. if (·) represents any variable, then The former approximation is more significant dynamically.

The governing equations.

and taking the limit the Navier–Stokes equations:

Using the transformations

we can deduce the BLT equations from

For flow past a rigid body the appropriate boundary conditions are

where U(x,t) is the inviscid slip velocity past the body. Further, from (6) evaluated at the edge of the boundary layer,

We define the ‘viscous blowing’ velocity out of the boundary layer to be

The velocity indicates the strength of blowing, or suction, at the edge of the boundary layer induced by viscous effects. It is a good diagnostic for dynamically significant effects within the boundary layer—much better than, say, the wall shear which can remain regular while becomes unbounded.

2.2.

Steady flows

Steady flow past an aligned flat plate: A success. Probably the most famous solution to (6) and (7) is that of Blasius (1908) for flow past an aligned flat plate. A comparison between this similarity solution and Wortmann’s visualization of that flow (Van Dyke 1982) demonstrates that BLT seems to work in this case—at least for where

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Steady flow past a circular cylinder: A failure. On the assumption that far from the boundary the velocity field is irrotational and inviscid to leading order, the inviscid slip velocity for steady flow past a circular cylinder is given by U = 2 sin x. Terrill (1960) showed numerically that the solution to (6)–(8) with this slip velocity terminates in a Goldstein (1948) singularity at with

for some constant k. This singularity occurs at the point where reversed flow is about to develop, i.e.

A serious problem. The occurrence of a singularity often indicates that there is a significant development in the flow physics, e.g. the formation of small-scale structure. In such circumstances new physics can usually be included in the model by introducing an asymptotic scaling

close to

thus enabling a solution to be found for However, Stewartson (1970) showed that, in general, there is no inner rescaling that ‘smoothes out’ the Goldstein singularity. (Smith and Daniels (1981) discuss an exception.) As a result, no BLT solution exists for Moreover this result implies that both the BLT solution for and the inviscid solution far from the cylinder, are incorrect. Boundary-layer theory does not always work.

What has gone wrong? The short answer is that the assumption that the flow far from the wall is irrotational is incorrect. Experimentally it is observed that, other than at very small Reynolds numbers,

there is a rotational eddy behind the cylinder that is at least as large as the cylinder; moreover the flow is steady and symmetric only for (For larger Reynolds numbers the flow is unsteady and asymmetric.) Further, is not an asymptotically large Reynolds number! Steady symmetric solutions for obtained numerically, by specifically excluding the possibility of unsteadiness and asymmetry, suggest that the asymptotic regime for steady symmetric solutions is reached only for (Fornberg 1985), i.e. at Reynolds numbers far larger than those at which the steady flow is stable. One way forward is to study unsteady flows on the basis that these are what are observed experimentally—see Another is to seek the asymptotic form of the steady symmetric solution (even if it is experimentally unobservable) in the hope that the solution will shed light on the failure of classical BLT (although the true justification for studying

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the problem may be closer to that of mountaineers for climbing Everest, i.e. it’s hard and it's there).

Steady symmetric flow past a circular cylinder: An answer. An asymptotic solution for steady symmetric separated flow past a bluff body at large Reynolds number must contain at least two important ingredients. First there must be a local solution at the point of separation of the boundary layer from the body surface. Second, an asymptotic model of the global wake is needed. Local separation is described by Sychev’s (1972) ‘triple-deck’ analysis (see also Smith 1977). This analysis is based on the premise that, at the point of separation on a smooth surface, the pressure gradient is In the context of Kirchhoff free-streamline theory this means that the appropriate free-streamline solution satisfies the Brillouin–Villat condition to leading order (see e.g. Sychev et al. (1998) for a discussion). There have been a number of attempts to fit the above local description of separation into a consistent large-Reynolds-number asymptotic global solution for the flow past a circular cylinder. Building on the work of others, Chernyshenko (1988) proposed an asymptotic structure based on the special Sadovskii (1971) vortex where there is no velocity

jump at the edge of the vortex. While this structure, in which both the length and the width of the wake are O(Re) in magnitude, may not be unique, it overcomes the technical shortcomings, especially as regards wake reattachment, of other proposals (see e.g. Chernyshenko 1998).

2.3.

Unsteady separation

While the large-Reynolds-number asymptotic solution for steady, symmetric laminar flow past a bluff body is not experimentally realizable,

this is not the case for fast starting flow past a smooth bluff body. In particular, when the unsteady term in (6) is much larger than the nonlinear term, and is balanced by the diffusive term (so that the boundary layers are very thin with Hence when each point of the boundary layer looks locally like Rayleigh’s solution for starting motion over a flat plate, and hence separation does not take place at sufficiently early times (see e.g. Goldstein and Rosenhead 1936). Impulsively started flow past a circular cylinder. There have been a number of visualizations of this flow, e.g. Prandtl (1932) and Coutanceau and Bouard (1977). Prandtl’s (1932) film is particularly instructive as regards where separation of the boundary layer starts.

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For many years research into unsteady separation focused on the rear stagnation point (see e.g. Robins and Howarth 1972), since it is there that reversed flow first sets in. However, reversed flow is not the same as separation/breakaway of the boundary layer from the body surface— there is plenty of reversed flow in Stokes’s solution for flow over an oscillating plate, and yet this unsteady boundary layer remains attached to the plate for all Reynolds numbers. In contrast, Prandtl (1932) focused attention on a region approximately from the front stagnation point, because it is approximately there that the boundary layer clearly first separates from the body surface. It is arguable that conventional wisdom, i.e. that the rear stagnation point was the place to look, delayed an understanding of unsteady separation by fifty years. Unsteady separation: Physics. On the rear-side of an impulsively moved cylinder, fluid particles are decelerating. These particles will tend to be squashed in the streamwise direction, with a compensating expansion in the direction normal to the boundary. In Navier–Stokes (NS) or Euler flows it is not possible to squash a particle to zero thickness in one direction and an infinite length in another because the rapid stretching of the fluid particles leads to the generation of a pressure gradient that inhibits the stretching. However, in classical BLT, (see (7)), and hence no pressure gradient can be induced in the direction normal to the wall. Unsteady separation occurs when a fluid particle is squashed to zero thickness in the direction parallel to the wall, thus ejecting the fluid above it out of the boundary layer (van Dommelen 1981).

Unsteady separation: Mathematics. Since unsteady separation is connected with the deformation of a particle, it is natural to seek a mathematical description in terms of Lagrangian coordinates, say (Shen 1978, van Dommelen and Shen 1980). Then with and the momentum equation (6) becomes

while kinematics and mass conservation yield

Van Dommelen and Shen (1980) made the key observation that (12) depends only on x and u, and hence that (12) and the first equation in (13) can be solved independently of the equations governing y and v.

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The solution for y can then be obtained from the third equation of (13), while the solution for v can be deduced subsequently from the second equation of (13). For a given the Jacobian mass conservation relation in (13) is a hyperbolic equation for

with a unit source term. If at some time, say the solution evolves so that (indicating that a particle has been squashed to zero thickness in the x direction), then ‘shock’ singularities can form in y, and hence v (van Dommelen and Shen 1980).

Impulsively started circular cylinder: Results. Numerical calculations show that a ‘shock’ singularity develops at a time and a position The position where the unsteady singularity forms, is not the position where the Goldstein singularity forms, i.e. As the unsteady singularity results in a rapid thickening of the boundary layer over a streamwise distance where is the center of the singularity structure, while the displacement thickness and blowing velocity vary like and respectively—see Fig. 1 for a schematic.

Figure 1

Schematic of a separating boundary layer

Three-dimensional unsteady separation. The above analysis can be extended to describe three-dimensional separation (e.g. van Dommelen and Cowley 1990). When there are no symmetries, the boundary layer still thickens like and the singularity is quasi-two-dimensional with a slower variation in a direction orthogonal to the more rapid variation. However, if there is a plane or axis of symmetry, then the thickening of the boundary layer is more rapid than

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KQ1—S ECTIONAL L ECTURE : S TEPHEN J. C OWLEY

A singularity . . . surely not?

Conventional wisdom is that finite-time singularities do not spontaneously develop in solutions to the NS equations. (Of course conventional wisdom may prove to be wrong—as illustrated by the aforementioned fifty years of misguided study of unsteady separation.) Assuming that conventional wisdom is correct, the BLT singularity must be an artifact of the BLT approximation. Hence, at times very close to at least one of the terms that are usually asymptotically smaller than those included in the BLT equations, must grow to be so large that it cannot be neglected at leading order; e.g. in order to stop the extension of the squashed particle in the y direction, we might anticipate that the approximation will need to be refined. An upper deck. For it follows from (5) and the above scaling for that the dimensional blowing velocity has magnitude

This blowing velocity causes a perturbation to the inviscid flow in a region just above the boundary layer. This perturbation is both inviscid and irrotational; hence it is governed by Laplace’s equation. The perturbation extends over a region sufficient for a pressure gradient normal to the wall to be felt (and so reduce the normal velocity to zero). Since the Laplacian is a ‘smooth operator’ and the extent of the variation in the direction is the extent of the variation in the direction is also From the continuity equation it follows that the perturbation velocity in the streamwise direction is of the same order of magnitude as the blowing velocity (14). Similarly it follows from the linearized version of the time-dependent Bernoulli equation that the dimensional pressure perturbation is Hence the dimensionless pressure-gradient perturbation has a magnitude

This induced perturbation pressure gradient can have a feedback effect on the boundary-layer flow when it is as large as the acceleration, within the boundary layer. This condition occurs when At such times a new asymptotic problem needs to be formulated involving four distinct asymptotic regions in the y direction (Elliott et al. 1983). For this ‘quadruple-deck’ analysis to be valid, i.e. for the four asymptotic regions to be dist