From Waves in Complex Systems to Dynamics of Generalized Continua: Tributes to Professor Yih-Hsing Pao on His 80th Birthday

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From Waves in Complex Systems Dynamics of Generalized Continua 0

Tributes to Professor Yih-Hsing Pao on His 80th Birthday

Y|jWorld Scientific

From Waves in Complex Systems to Dynamics of Generalized Continua

Tributes to Professor Yih-Hsing Pao on His 80th Birthday

8090.9789814340717-tp.indd 1

5/4/11 9:29 AM

From Waves in Complex Systems to Dynamics of Generalized Continua Tributes to Professor Yih-Hsing Pao on His 80th Birthday

Editors

Kolumban Hutter (Federal Institute of Technology, Zurich, Switzerland) Tsung-Tsong Wu (National Taiwan University, Taiwan) Yi-Chung Shu (National Taiwan University, Taiwan) World Scientific NEW JERSEY

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LONDON

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SINGAPORE

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BEIJING

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SHANGHAI

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HONG KONG

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TA I P E I

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CHENNAI

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Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

FROM WAVES IN COMPLEX SYSTEMS TO DYNAMICS OF GENERALIZED CONTINUA Tributes to Professor Yih-Hsing Pao on His 80th Birthday Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN-13 978-981-4340-71-7 ISBN-10 981-4340-71-5

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2/25/2011, 9:50 AM

PREFACE

This book has been assembled from contributions dedicated to Professor Yih-Hsing Pao on the occasion of His 80th birthday, held at the International Symposium on Engineering Mechanics on 21/22 May 2010 and organized by the Institute of Applied Mechanics, National Taiwan University, Taipei. It concentrates on reviews and new developments in the fields of waves in complex systems and dynamics of generalized continua, the two classes of subjects, embracing the more than 50 years of active research to which Professor Pao dedicated his scientific efforts in the field of Applied Mechanics. His scientific achievements and his leading role played as a University teacher and internationally as an educator and researcher are reviewed in the preliminary items of this book in the Laudatio. The fifteen contributions, which are collected in this book are divided into four groups typified as -

Waves, Numerical Methods and Time Series Analyses, Continuum Mechanical Theories, and Wind Energy Techniques.

Waves are analyzed in the context of Lamb waves in phononic band gap structures and as early time responses in trusses and frames, using the reverberation-ray matrix technique. Wave theory is also applied to surface wave nonlinearities for fatigued steel and as acousto-elastic Lamb waves in implications to structural health monitoring. Time series analyses are treated by an adaptive data analysis method via a novel so-called Hilbert-Huang Transformation (HHT), which is particularly apt in the characterization of non-stationary statistical processes in a great variety of data through measured digital quantities. v

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Preface

In the subject of numerical mathematics, computational fluid dynamics is illustrated with the so-called Unified Coordinates (UC); these are generalizations of the Euler-Lagrange Algorithms (ELA), by which initial boundary value problems can be more effectively solved than with the classical ELAs, since the UCs are optimally and automatically generating boundary adjusted meshes. Moreover, in the fields of seismology and exploration geophysics, imperfectly partitioned ambient wave fields allow retrieval of the Green’s function and, thus, yield exact determination of travel times, attenuations, specific intensities and scattering strengths in complex wave fields. A contribution, which applies optimization techniques, shows their use in structural elements, which minimize their size on the basis of periodic micro-structural elements, and inverse techniques are applied to problems in optical sciences. Methods of advanced continuum-thermo-mechanics are applied to such complex systems as lava flows, in which the aggregation states arise as hot fluids, temperate solid-fluid mixtures at the solidification temperature and as cold solids separated by phase change surfaces and nourished by mass transports across these surfaces, as ternary bio-fluid mixtures and as acoustic and electromagnetic meta-materials, and in electro-magneto-mechanical coupled fields with various postulations of electro-magnetic interaction forces, couples and stress tensors. Wind energy is, finally, tackled by studying, how nonlinear vibrowind energy generation competes with its rotational counterpart and how off-shore wind energy can be produced below the costs of coal electricity. The articles either review a topic or present new developments; but both appeal to experts as well as researchers and practitioners alike. The editors express their heartfelt thanks to the distinguished international team of contributors whose scientific efforts unite to form this book. They also express thanks to Professor C. C Chang, chairperson of the Institute of Applied Mechanics at National Taiwan University and his staff, in particular Chi-Wei Liu for organizing the conference and to Fong Kiaw Wong at World Scientific Publishing Company for the assistance in editing this book. Zurich, Switzerland and Taipei, Taiwan, 22. May 2010 K. Hutter, T. T. Wu and Y. C. Shu

CONTENTS

Preface

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Contributors

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Group Photo

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Laudatio for Professor Yih-Hsing Pao on the Occasion of the International Symposium on Engineering Mechanics 2010, on the Occasion of His 80th Birthday (21/22 May 2010)

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Waves 1. Lamb Waves in Phononic Band Gap Structures (by T. T. Wu)

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2. On Generalization of the Phase Relations in the Method of Reverberation-Ray Matrix (by W. Q. Chen)

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3. Surface-Wave Nonlinearity Measured with EMAT for Fatigued Steels (by M. Hirao)

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4. Acoustoelastic Lamb Waves and Implications for Structural Health Monitoring (by J. E. Michaels)

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5. Source Synthesis for Inverse Problems in Wave Propagation (by W. W. Symes)

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Numerical Mathematics / Time Series Analysis 6. An Introduction to an Adaptive Data Analysis Method (by N. E. Huang)

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7. Computational Fluid Dynamics Based on the Unified Coordinates — A Brief Review (by W. H. Hui)

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Contents

8. Towards Green’s Function Retrieval from Imperfectly Partitioned Ambient Wave Fields: Travel Times, Attenuations, Specific Intensities, and Scattering Strengths (by R. L. Weaver)

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9. Study on Two Scale Design Optimization for Structure and Material with Periodic Microstructure (by G. D. Cheng)

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Continuum Mechanical Theories 10. A Continuum Formulation of Lava Flows - From Fluid Ejection to Solid Deposition (by K. Hutter)

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11. Rigorous Mechanics and Elegant Mathematics on the Formulation of Constitutive Laws for Complex Materials: An Example from Biomechanics (by V. Mow)

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12. Professor Pao’s Influence on Research in Coupled Field Problems, Chirality and Acoustic and Electromagnetic Metamaterials and their Applications (by V. V. Varadan)

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13. Transient Response of an Elastic Half Space by a Moving Concentrated Torque (by C.-S. Yeh)

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14. Magnetic Force Model for Magnetizable Elastic Body in the Magnetic Field (by Z. J. Zheng)

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Wind Energy 15. Principles of Nonlinear Vibro-Wind Energy Conversion (by F. Moon)

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CONTRIBUTORS

W. Q. Chen Zhejiang University, China [email protected]

V. C. Mow Columbia University, U.S.A [email protected]

G. D. Cheng Dalian University of Technology, China [email protected]

W. W. Symes Rice University, U.S.A [email protected]

M. Hirao The University of Tokyo, Japan [email protected]

V. V. Varadan University of Arkansas, U.S.A [email protected]

N. E. Huang National Central University, Taiwan [email protected]

T. T. Wu National Taiwan University, Taiwan [email protected]

K. Hutter Laboratory of Hydraulics, Hydrology and Glaciology, Federal Institute of Technology, Zurich, Switzerland [email protected]

R. L. Weaver University of Illinois at Urbana-Champaign U.S.A [email protected] C. S. Yeh National Taiwan University, Taiwan [email protected]

W. H. Hui Hong Kong University of Science and Technology, Hong Kong [email protected]

X. J. Zheng Lanzhou University, China [email protected]

J. E. Michaels Georgia Institute of Technology, U.S.A. [email protected] F. Moon Cornell University, U.S.A [email protected] ix


Conference participants at the entrance of the Institute of Applied Mechanics, National Taiwan University, Taipei, with the jubilee, Professor Y.-H. Pao (seated in the middle, front row)


LAUDATIO FOR PROFESSOR YIH-HSING PAO ON THE OCCASION OF THE INTERNATIONAL SYMPOSIUM ON ENGINEERING MECHANICS 2010, ON THE OCCASION OF HIS 80TH BIRTHDAY (21/22 May 2010)

Kolumban Hutter Retired Professor of Mechanics, Darmstadt University of Technology, Germany NOW: c/o Laboratory of Hydraulics, Hydrology and Glaciology Swiss Federal Institute of Technology, Zurich, Switzerland E-mail: [email protected]

Dear Professor Pao, dear Mrs Pao, Respected representatives of the National Taiwan University (NTU) and other Universities, of governmental and scientific agencies of Taiwan and abroad, Dear Academicians, dear scientists and Professors from Academies and Universities worldwide, Dear guests and friends of Professor Pao We have come together here to celebrate Professor Pao on the occasion of his 80th birthday – it was on 19th January 2010 – and to look back and contemplate about him as a personality, a scientist and simply as a human being. We have also come together to pay tribute to a superior, a colleague and a friend with whom some of us have had extensive work interactions, scientific discussions, disagreements, long working sessions in seminar rooms, intellectually, always as equals, but hard and subject oriented and not person oriented, focused on every detail until xiii

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Laudatio for Professor Y.-H. Pao

intellectual agreement was reached. When experiencing this for the first time in 1969 in seminars together with the late Steve Thau, this was for me, the young Swiss student a unique experience for which I first had to take breath to understand that objective scientific arguments could be separated from professional seniority or superiority. At that time, this was not at all so in Swiss academic circles, but I learned quickly! This, however, does not mean that a clear superior-subordinate relationship would not have been respected. I addressed him as ‘Professor Pao’ and he responded as ‘Yes Kolumban!’, and this is essentially still so now. However, the creation of absolute unquestioned equality of scientific arguing was my first experience of an attitude of Professor Pao, which I had never experienced before and am still sometimes missing in Europe today. There is a further early incidence in my academic education which demonstrated Professor Pao’s generosity. During the first summer break as a student in the Department of Theoretical and Applied Mechanics at Cornell University, I had a research assistantship with the duty to develop finite difference software on elastic waves in a layered half plane as a subject of my Master of Science thesis. This work dragged on in the following semester with no success. In that fall semester 1969 Professors Pao and Rand had organized a reading seminar on Cole’s new book on ‘Perturbation Theory’. One chapter dealt with pre-stressed cables with small bending rigidity. We all know the result: Bending effects are strong in the vicinity of built-in supports due to the small flexural stiffness associated with the highest derivative arising in the differential equation, a problem of matched asymptotics. This problem hit me immediately with high heart beating. A year or so before in Switzerland, I was involved with the construction of a concrete bridge with box cross section and skew supports. We had computed the torsional response by use of the Saint Venant theory, but measurements of the stress distribution in the supports of the realized bridge revealed large deviations from the computed twisting moment along the bridge axis, whilst agreement distant from the supports was good. My heart beating in the seminar room at Cornell’s Thurston Hall was caused by my immediate recognition that warping torsion due to the cantilever pedestrian paths on each side of the box cross section of the bridge was


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ignored; it corresponded to the bending effects of the pre-stressed cable in Cole’s book, whilst Saint Venant torsion was the analogue of the tension-string effects in the cable. I went home that Friday in fall 1969, worked day and night through the weekend and presented Professor Pao a drafted manuscript on Monday, asking – but being very scared – whether he would accept this as a basis for a Master’s Thesis. He did; the Master’s Thesis is from 1970 and the work became my first paper in the Int. J. Solids and Structures in 1971. I was scared because I had failed with the layered media problem and weigh Professor Pao’s decision at that time as generosity, because he measured the success and forgot the failing precursor. My experience in Switzerland years before was the opposite. I have above illustrated objective professional judgment and generosity as two outstanding features which Professor Pao employs in reaching optimal scientific advancement. Endurance in pursuing a chosen approach in reaching intended goals and vision for the longer range development into the future of the field of Applied Mechanics are yet other ones. He also believes that first rank achievements, such as the foundation of the Institute of Applied Mechanics at National Taiwan University (NTU), which he founded single-handed and which is based on his long range vision and firm belief: Namely, that Applied Mechanics has important developments to offer to future technology both in education and industrial development of Taiwan on a global scale. He did this contrary to today’s army of deans and presidents of Engineering Schools and Universities who closed Departments of Theoretical and Applied Mechanics or Departments of General Engineering Sciences. Such educational units disappeared by and large on the entire Globe. In Germany, the Department of Mechanics at Darmstadt University and the Department of Physical Engineering Sciences at the Technical University in Berlin have been closed. The professors were distributed to more professional engineering departments, and at their retirement were not or have been differently replaced by representatives of professional specialties. In the United Kingdom the Department of Theoretical and Applied Mechanics at Nottingham has been closed, as have the Departments of Theoretical and Applied Mechanics at Cornell University and the University of Illinois at

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Urbana-Champaign in the US. In Darmstadt, four Professors now teach statics, dynamics and strength of materials to about 3000 students. (When I was there, we were first 12 and then 8 professors.) There is hardly any time left for specialized courses to form accordingly specialized pupils. Professor Pao’s ‘endurance and vision’ during the foundation period of the Institute of Applied Mechanics at NTU have been very efficient supports for the creation of the institute, and the Institute of Applied Mechanics has already and may further prove to be very valuable for forefront activity not only in the industrial development of Taiwan. What are Professor Pao’s research achievements and how did he pursue research to get where he is now? There seems to be one main theme. This is ‘Waves in complex continuous systems’. This theme is embellished with variations on ‘Dynamics of generalized continua’. Waves became already the main focus when our jubilee started as a Ph. D. student under the supervision of Professor Raymond D. Mindlin, the American born Russian professor of civil engineering at Columbia University, who is most famous for his work on the derivation of the dynamical plate equations by employing a variational formulation (the Principle of Weighted Residuals) and using Cauchy series expansion of the displacement field in the direction perpendicular to the plate reference plane. He is not only known for his applied work on vibrations of elastic plates, but has also done fundamental work in early Cosserat theories, in vibrations of piezoelectric plates, in which polarization gradients were introduced (for the first time?) to describe deformation-induced birefringence. The graduate student Yih-Hsing Pao was exposed to this environment of fundamental applied physics, in which plates were two-dimensional continua with Cosserat structure, rather than just elements of structural engineering, which could also be loaded by electromagnetic forces, which were subjected to electrical and magnetic dipoles, and when transparent and deformable, could change the polarization of the light. The point I want to make is, that the young engineer Y.-H. Pao wrote his first paper on ‘Dispersion of Flexural Waves in an Elastic, Circular


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Cylinder’ with his advisor, Professor Mindlin, on a classical subject of Applied Dynamics, being immersed in a non-classical work environment. He was exposed to a wealth of other subjects of Applied Physics, where expert knowledge of engineers could be used with tremendous advantage. It is quite natural to look at Professor Pao’s early work in the field of scattering of elastic waves from this point of view. Flexural waves in circular cylinders are the beam or shaft analogue of plates, and circular cylinders are bounded by the simplest quadratic surface. Scattering of elastic waves by spherical and elliptical obstacles and parabolic cylinders involves more complex quadratic surfaces and makes a fantastic playground to study the mathematics of spherical harmonics, Mathieu and parabolic cylinder functions. This was work done with C. C. Mow (1963) and S. A. Thau (1966/67). Also in the sixties, work with K. F. Graff on the influence of couple-stresses on wave reflection and scattering by a spherical cavity (1967) shows Mindlin’s tangible influence on dynamics of generalized continua. However, elastodynamics with waves, vibrations, focusing on scattering, diffraction, acoustoelasticity, residual stresses, layered media, generalized ray theory, hyperbolic heat conduction equation, ultrasonic waves and non-destructive testing of materials continued to be Professor Pao’s main research activities. Collaborators through the years on these subjects were, among others, R. Gayewski, W. Sachse, V. K. and V. Varadan, F. Ziegler, U. Gamer, T. T. Wu, M. Hirao and Y. S. Wang. Most of these subjects were initiated and formulated in their elements already in the late sixties and early seventies. I participated in 1970-1972 with Professors W. Sachse and J. T. Jenkins in long work-seminars on ultrasonic waves and nondestructive testing. In the summer of 1971 Professors Jenkins, Pao and Sachse and two graduate students, D. K. Banerjee and myself also wrestled with the formulation of a thermodynamic theory of solids close to absolute zero. A hyperbolic heat equation was a necessity, and it turned out that a linearized theory for the coldness – the inverse of the absolute temperature (not the temperature itself) – was the optimal variable for a linearized hyperbolic heat conduction equation. The work merged into D. K. Banerjee’s Ph. D dissertation and two papers on thermoelastic waves in dielectric and anisotropic solids. (As a side

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remark: Dr. I. Luca from Bucharest, Romania who was 2.5 years at Academia Sinica from 2007 to 2009 and worked there on debris flow, continued that work in the 80s as her Ph. D. dissertation under the supervision of Prof Soos in the Mathematics Department of the University of Bucharest). Waves in plates became the center of activity again in the late seventies and eighties in work with R. L. Weaver and W. J. Parzygnat as waves emerging from point sources and resonating in non-linear vibrations, as propagations of elastic pulses and as inverse source problems with F. Santosa, W. W. Symes, A. N. Ceranoglu and J. E. Michaels. In my opinion, Professor Pao’s activity truly deserving the qualification ‘waves in complex systems’ is work on waves in trusses and frames, done in the late nineties to present. The problem of steady vibration of trusses and frames as linear elastic systems is today a standard linear eigenvalue problem. Much more difficult is the analysis of wave propagation in the transient regime (the early time behavior). This research led to the method of reverberation-ray matrices, an alternative to the method of transfer matrices, in which a wave arriving at a nodal point of an element of the truss or frame, is distributed among the elements at the nodal point and then propagated in each element to the neighboring nodal points. Work on this subject has been done with D. C. Keh, S. M. Howard, X. Y. Su, J. Y. Tian and W.Q. Chen and is presently on an advanced level to reach commercial exploitation. Back to the sixties and early seventies, when a brilliant student, F. C. Moon, performed experiments on buckling of magnetoelastic cantilever plates! The early work of F. C. Moon and Professor Pao was also on waves, but the paper ‘Magnetoelastic Buckling of a Thin Plate’ (1968) was the trigger of a whole surge of Ph. D. dissertations and research articles, and this work is still under way. A fascination of the problem was that electromagnetic forces could generate a quasistatic mechanical instability; the disappointment was that the buckling load predicted by the theory was far off from its measured counterpart in the experiment. The second paper by F. C. Moon and Professor Pao on the subject ‘Vibration and Dynamic Instability of a Beam-Plate in a Transverse Magnetic Field’ did not fully rectify the discrepancy. A thorough study of the foundations of electro-magneto-mechanics was needed, a theory of


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Electrodynamics of Deformable Continua with the flexibility that an arbitrary class of constitutive behavior could be postulated and accordingly reduced to be in conformity with the second law of thermodynamics. A beautiful book by the MIT Professors P. Penfield and H. A. Haus ‘Electrodynamics of Moving Media’ (1967) led us to the complexities of the electric and magnetic force postulates. They were not the same, depending upon, whether magnetization was constructed by a dipole of magnetic monopoles or an electric current in a closed loop. Another beautiful book by W.F. Brown Jr. ‘Magnetolelastic Interactions’ (1966) showed us that the separation of electromagnetic short and long range effects could not be established in a unique way. This result says that the electromagnetic body forces (and couples) are not unique and therefore neither are the stresses. This also means that the definition of a magnetoelastic material is subtle. Moreover, since in an elastic material the stress tensor is given by the derivative of the Helmholtz free energy with respect to the strain tensor, two different formulations of magnetoelasticity have only a chance of furnishing the same solution for observables (e.g., the displacement distribution), if the free energies in the two formulations are related to one another in the correct manner. This discussion shows that the answer to the question whether a certain electromagnetic force postulate is correct, is at last a problem of thermodynamics. Moreover, its discussion in isolation without corresponding stress postulates is futile. The papers by Pao and Yeh ‘A Linear Theory of Soft Ferromagnetic Elastic Solids’ (1973) and of Pao and Hutter ‘Electrodynamics for Moving Elastic Solids and Viscous Fluids’ (1974) are concerned with a proper embedding of the thermomechanical and electrodynamic equations in an entropy principle plus a systematic derivation of magnetic body forces and body couples from a dipole model. This latter topic was subsequently deepened by Professor Pao in his famous 1978-article ‘Electromagnetic Forces in Deformable Continua’, which appeared in Mechanics Today. This work has been continued in the last 32 years by many researchers; the final answer is still not given. The problem with the magnetic force models is reviewed in this book by X. Zheng and K. Jin, however, without touching the question of thermodynamic consistency. It is my belief that disagreements between theoretical results and experimental findings

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which still exist (see X. Zheng & K. Jin, this volume) need a thermodynamic setting to be resolved. The book by K. Hutter, A.A.F. van de Ven ‘Field Matter Interaction in Thermoelastic Solids’ (1978), in the second edition by K. Hutter, A.A.F. van de Ven and A. Ursescu as ‘Field Matter Interaction in Thermoelastic Solids and Viscous Fluids’, Springer Verlag (2006) provides a hint. Let us finish this review with yet another hobby horse of Professor Pao’s scientific activities: Variational Principles of Thermomechanics of Continuous Systems. [Professor Pao has long fought to get his work with L. S. Wang published, now, as I am told, in press in the Int. J. of Engr. Sci.] Publishing novel ideas in this topic is almost certainly bound to be condemned to fail. Why is this so? In my opinion, it probably goes back to several centuries of competing relations between the French and the Britons. On an intellectual level the Britons (and today also the Americans) take forces as the fundamental entities to rein the physical world. Geometry is simply the space in which forces live. The French take work or better the power of working as the fundamental entity. For them, the two elements ‘force’ and ‘trajectory’, which I now identify with ‘geometry’, are equal partners. Geometry reins our physical world as much as do forces. This is d’Alembert’s Principle: force and geometry are combined as equal partners to define the work which reins the world. Note, after all, the French beheaded the king and founded their democracy. The Britons still have a Queen, and a further King certainly in less than 20 years. Professor Pao might want to be a king, but his teacher, Professor Mindlin, showed him, how two-dimensional plate equations are obtained from three-dimensional theory. The vehicle to do this is the Principle of Weighted Residuals, essentially nothing else than the power of working, equality between forces (the momentum equations) and geometry (variations of the dual geometric field). It is a fairly trivial step to extend this to other fields, e.g. the energy equation (~force) and variation of temperature (~geometry) and eventually any other quantity with dual deformation field (angular momentum, angular velocity, see the article of P. Germain in SIAM J. Appl. Math., 1973). This approach is simple and so natural that many scientists treat it as intellectually almost empty. There is no room to make a big fuss about


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such a natural thing, and this is taken by referees to claim that nothing is new in such a formulation and to reject such papers. The mechanics people in the western world (modulo the French) are “Britons” and will likely reject papers on such variational principles, claiming there is nothing to them. Perhaps you should formulate your variational principles in a setting of general relativity. (Beam theories in a covariant setting of the theory of general relativity do exist!) I am joking now, but if you do so, you might have a chance to get a theoretical physicist as a referee. Physicists know very well since Einstein that the curvature of the space is made by the mass distribution which determines the forces. Here, geometry and forces are true brother and sister. I nevertheless believe – and I now return to Professor Pao’s activities on the Principle of Virtual Power – that this is useful as an alternative to the classical balance law approach. However, success is more likely, if it is applied in the context of generalized continua. The role of the ‘forces’ is then played by the evolution equations of the subgrid field variables and the ‘geometry’ is described by dual fields, which need to be invented together with the mathematical properties of the operations which generate a scalar field, representing the ‘power of working’. It is obvious that from a viewpoint of inventing field equations including constitutive relations and boundary (jump) condition, it may be easier to guess scalars, rather than tensors. The apparent triviality of the method is then hidden behind the complexity of the physics. Mention should also be made about Professor Pao’s handling of his blindness, due to retinitis pigmentosa, the tunnel vision which started in the eighties and progressed to complete blindness in the late nineties. Apart from his negligence to learn Braillie, it is absolutely amazing and deserves our highest respect and admiration, how he keeps his spirit and intellectual level. This allows him not only to follow forefront research but also to inspire and take part in actual research activity. I have never seen him presenting a talk at a conference since complete blindness, but I am told that it is a well organized event to see how he guides the audience in a GPS manner through his densely filled transparencies, made by one of his aids. Of course, all of this is only possible, because the faculty and staff members of the Institute of Applied Mechanics at the National Taiwan University provide admirable support to an extent

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impossible in the USA and in Europe. They deserve my admiration and thank for the continued service which they offer. I suppose that similar support is also provided at Zhejiang University in Hangzhu, China. This is a laudatio on Professor Pao, which is rather personal. I do not apologize for it as my relation to him is personal. [I have, however, attached a formal curriculum vitae below]. I shall offer apologies to all whose work has not adequately been mentioned and who feel not to have received the proper credit. I had to be selective. The most important thing is to state that Professor Pao is a very unusual personality, generous, with usually objective judgment; his relations to others and to his and his coauthors work show endurance; he does not give up a topic readily but sticks with it almost indefinitely. He has given me a lot for which I am tremendously thankful and without which I would not be where I am now. I am sure, many of you who experienced support from him think alike. Let us wish him good health and many more years of active research in the years to come. Please join me in a standing ovation for him. Biography (From homepage, Department of Mechanical Engineering, Hong Kong University of Science and Technology, with changes and extensions) Professor Yih-Hsing Pao was born in 1930 at Nanjing, China, and grew up during the Second World War in the Chongqing area; he went to the National Second High School for five years on a government scholarship. He attended the National Chao-Tung University (1947-49) and National Taiwan University (1950-52), receiving the Bachelor degree in Civil Engineering. In 1953, he was awarded a teaching assistantship to do graduate studies at Renssalear Polytechnic Institute, Troy, New York, USA, receiving the degree M. S. in Mechanics (1955). He continued the graduate study at Columbia University in the city of New York (1955-58), receiving the Ph. D. degree in Applied Mechanics (1959). Since 1958, he has been a faculty member at Cornell University, rising from the rank of Assistant Professor, Associate Professor, Professor, Joseph C. Ford Professor of Theoretical and Applied Mechanics, to J. C. Ford Professor Emeritus (2000-present). He served also as


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Chairman of the Department of Theoretical and Applied Mechanics, Cornell University, from 1974 to 1980. By an Intergovernmental Loan Agreement, he visited the National Taiwan University, on leave of absence from Cornell, and founded the Institute of Applied Mechanics at NTU, and served as Professor and Director twice (1984-86, 1989-94). By then, his longtime illness of Retinitis had deteriorated to almost complete blindness. He resigned from the directorship in 1994 and retired from NTU in 1998 as Professor Emeritus. While at Cornell, he had also held visiting assignments at Princeton University and Stanford University in the USA; Technische Hochschule Darmstadt (now Darmstadt University of Technology); Hong Kong University of Science and Technology in China between one and nine months each. In 2003, he was appointed by Zhejiang University, Hangzhou, China, as a professor of the College of Civil Engineering and Architecture with tenure. At Cornell he had taught undergraduate engineering courses in Mechanics and Applied Mathematics in large classes and graduate courses in Theoretical and Applied Mechanics (TAM) every year, advised several hundreds of graduate students majoring in TAM, and specially supervised 17 Ph.D. students and 8 post doctoral fellows. Altogether there were 25 Ph.D. Fellows, after leaving Cornell, 16 of them remain teaching at well-known universities in America, Germany, Australia, Turkey, Austria, Japan, Mainland of China, and Taiwan and so on. From 1984 to1995 his work at NTU was mainly administration. After 1995, he resumed teaching and research in blindness through administrative assistants and graduate students doing reading and dictation in Chinese and English. In addition to teaching and research he was active in several professional societies, and was elected a fellow of the American Society of Mechanical Engineers, and the American Academy of Mechanics. He was the General Chairman and Scientific Committee Chairman of the ninth United States National Congress of Applied Mechanics held at Cornell University in 1982. At Taiwan, he was elected President of the Chinese Society of Theoretical and Applied Mechanics (1992-1995). In 1985, he was elected a member of the US National Academy of Engineering, the highest honor that can be bestowed on an engineer by the US government. In 1986, he was elected an Academician by

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Academia Sinica (founded in 1928 at Nanjing, China), a lifetime honor in China. He received the senior US Scientist Award from the Humboldt Foundation, Germany, and visiting Darmstadt University of Technology in the spring 1989. In 1995, he was awarded an Honorary Doctoral Degree by the National Chiao-Tung University (Hsin-Chu). In 2003, he was one of the three recipients of the Presidential Science Prize of Taiwan. From 1960 to present he has authored and co-authored 118 technique papers as listed in the appendix. The first publication no. 6 is a monograph co-authored with Dr. Henry C. C. Mow which summaries research work on “Scattering-Diffraction of Elastic Waves and Dynamic Stress Concentration” done by Pao with his graduate students at Cornell and by Mow with his associates. The remaining papers numbered Nos. 1-118 may be grouped in five categories: 1. Elastodynamics (Dynamic theory of elasticity) and Mechanical Vibrations of Elastic Solid; 2. Interactions of Elastodynamics with Physical Acoustics, Heat Pulse, Geophysics (Seismic waves and Under-Water Acoustics), Ultrasonic Waves and Non-destructive Testing of Materials; 3. Field-Theory of MagnetoElectro Mechanics and Magneto Elastic Stability; 4. Elastodynamic Theory of Structures; 5. Variational Principles of Thermomechanics of a Continuous Medium, Unifying Theory of Continuum Mechanics with Thermodynamically Irreversible Constitutive Laws for the Medium. In category 4 research works were mainly done at NTU and ZJU after 1999, in category 5 research works on variational principles were done after 2000 with Profs. L. S. Wang and G. Q. Chen at NTU. The list of publications include more than ten comprehensive review articles (marked with *). The one paper ‘Magnetic forces in deformable continua’ in Mechanics Today, (Pergamon Press, 1978), was translated into a Chinese monograph Scientific Press (Beijing, 1996). Another article, ‘Generalized Ray Theory and Transient Responses of Layered Elastic Solids’ in Physical Acoustics (Academic Press, 1977) was chosen by IUTAM as one of the landmark papers in Mechanics of the 20th century. Acknowledgements: I thank Profs. T. T. Wu and C. C. Chang for reading an earlier version of this text and suggesting improvements.


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List of Publications – Professor Y.-H. Pao 1. Y.-H. Pao and R. D. Mindlin, 1960, “Dispersion of Flexural Waves in an Elastic, Circular Cylinder”, Journal of Applied Mechanics, v.27, pp.513-520. 2. M. Onoe and Y.-H. Pao, 1961, “Edge Mode of Thin Rectangular Plate of Barium Titanate”, Journal of the Acoustical Society of America, v.33, no.11, pp.1628. 3. Y.-H. Pao, 1962, “The Dispersion of Flexural Waves in an Elastic, Circular Cylinder — Part 2”, Journal of Applied Mechanics, v.29, pp.61-64. 4. Y.-H. Pao, 1962, “Dynamical Stress Concentration in an Elastic Plate”, Journal of Applied Mechanics, v.29, pp.299-305. 5. Y.-H. Pao and C. C. Mow, 1962, “Dynamic Stress Concentration in an Elastic Plate with Rigid Circular Inclusion”, Proceedings of the Fourth U.S. National Congress of Applied Mechanics, pp.335-345. 6. Y.-H. Pao and C. C. Mow, 1963, “Scattering of Plane Compressional Waves by a Spherical Obstacle”, Journal of Applied Physics, v.34, no.3, pp.493-499. 7. C. C. Chao and Y.-H. Pao, 1964, “On the Flexural Motions of Plates at the Cut-Off Frequency”, Journal of Applied Mechanics, v.31, pp.22-24. 8. Y.-H. Pao and C. C. Chao, 1964, “Diffractions of Flexural Waves by a Cavity in an Elastic Plate”, AIAA Journal, v.2, no.11, pp.2004-2010. 9. M. A. Medick and Y.-H. Pao, 1965, “Extensional Vibrations of Thin Rectangular Plates”, Journal of the Acoustical Society of America, v.37, no.1, pp.59-65. 10. Y.-H. Pao and C. Kowal, 1965, “A Laboratory for Teaching Mechanical Vibration”, Journal of Engineering Education, v.56, no.3, pp.96-100. 11. S. A. Thau and Y.-H. Pao, 1966, “Diffractions of Horizontal Shear Waves by a Parabolic Cylinder and Dynamic Stress Concentrations”, Journal of Applied Mechanics, v.33, pp.785-792. 12. F. C. Moon and Y.-H. Pao, 1967, “Interactions of Point Defects and Elastic Inclusions”, Journal of Applied Physics, v.38, no.2, pp.595-601. 13. S. A. Thau and Y.-H. Pao, 1967, “Stress-Intensification Near a Semi-Infinite Rigid-Smooth Strip Due to Diffraction of Elastic Waves”, Journal of Applied Mechanics, v.34, pp.119-126. 14. F. C. Moon and Y.-H. Pao, 1967, “The Influence of the Curvature of Spherical Waves on Dynamic Stress Concentration”, Journal of Applied Mechanics, v.34, pp.373-379. 15. S. A. Thau and Y.-H. Pao, 1967, “A Perturbation Method for Boundary Value Problems in Dynamic Elasticity”, Quarterly of Applied Mathematics, v.25, no.3, pp.243-260. 16. K. F. Graff and Y.-H. Pao, 1967, “The Effects of Couple-Stresses on the Propagation and Reflection of Plane Waves in an Elastic Half-Space”, Journal of Sound and Vibration, v.6, no.2, pp.217-229. 17. S. A. Thau and Y.-H. Pao, 1967, “Wave Function Expansions and Perturbation Method for the Diffraction of Elastic Waves by a Parabolic Cylinder”, Journal of Applied Mechanics, v.34, pp.915-920.

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18. K. F. Graff and Y.-H. Pao, 1967, “The Effects of Couple-Stresses on the Scattering of Plane Waves by a Spherical Cavity”, Proceedings of the 10th Mid-Western Conference of Mechanics, pp.721-739, Fort Collins, Colorado, 1967. 19. F. C. Moon and Y.-H. Pao, 1968, “Magnetoelastic Buckling of a Thin Plate”, Journal of Applied Mechanics, v.35, pp.53-58. 20. F. C. Moon and Y.-H. Pao, 1969, “Vibration and Dynamic Instability of a BeamPlate in a Transverse Magnetic Field”, Journal of Applied Mechanics, v.36, pp.92-100. 21. S. A. Thau and Y.-H. Pao, 1970, “On the Derivation of Point Source Responses from Line Source Solutions”, International Journal of Engineering Science, v.8, pp.207-218. a. S. A. Thau and Y.-H. Pao, 1973, “Comments on Two- and Three-Dimensional Wave Propagation”, Bulletin of the Seismological Society of America, v.63, no.1, pp.329. 22. Y.-H. Pao and S. A. Thau, 1970, “A Perturbation Method for Boundary-Value Problems in Dynamic Elasticity, Part II”, Quarterly of Applied Mathematics, v.28, no.2, pp.191-204. 23. K. Hutter and Y.-H. Pao, 1971, “Regular and Singular Perturbation Solutions for Bending and Torsion of Beams”, International Journal of Solids and Structures, v.7, pp.1523-1537. 24. Y.-H. Pao, 1972, “Some Recent Developments in Elastic Waves in Solids”, Experimental Mechanics, v.12, pp.83-89. (Invited Review) 25. G. C. Kung and Y.-H. Pao, 1972, “Nonlinear Flexural Vibrations of a Clamped Circular Plate”, Journal of Applied Mechanics, v.39, pp.1050-1054. 26. Y.-H. Pao and C. S. Yeh, 1973, “A Linear Theory for Soft Ferromagnetic Elastic Solids”, International Journal of Engineering Science, v.11, pp.415-436. 27. Y.-H. Pao and D. K. Banerjee, 1973, “Thermal Pulses in Dielectric Crystals”, Letters in Applied and Engineering Sciences, v.1, no.1, pp.33-41. 28. M. A. Oien and Y.-H. Pao, 1973, “Scattering of Compressional Waves by a Rigid Spheroidal Inclusion”, Journal of Applied Mechanics, v.40, pp.1073-1077. 29. Y.-H. Pao and K. Hutter, 1973, “Magnetoelastic Waves in Soft Ferromagnetic Solids”, (Proceedings of the 10th Annual Conference of the Society of Engineering Science, North Carolina State University, Raleigh), Recent Advances in Engineering Science, v.6, pp.83-91. (Invited Lecture) 30. K. Hutter and Y.-H. Pao, 1974, “A Dynamic Theory for Magnetizable Elastic Solids with Thermal and Electrical Conduction”, Journal of Elasticity, v.4, no.2, pp.89-114. 31. *Y.-H. Pao and R. K. Kaul, 1974, “Waves and Vibrations in Isotropic and Anisotropic Plates”, in R. D. Mindlin and Applied Mechanics, pp.149-195, edited by G. Herrmann, Pergamon Press, New York, 1974. 32. Y.-H. Pao and W. Sachse, 1974, “Interpretation of Time Records and Power Spectra of Scattered Ultrasonic Pulses in Solids”, Journal of the Acoustical Society of America, v.56, no.5, pp.1478-1486.


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33. D. K. Banerjee and Y.-H. Pao, 1974, “Thermoelastic Waves in Anisotropic Solids”, Journal of the Acoustical Society of America, v.56, no.5, pp.1444-1454. 34. W. J. Parzygnat and Y.-H. Pao, 1975, “Resonance Phenomena in Non-Linear Systems Governed by the Duffing Equation”, Letters in Applied and Engineering Sciences, v.3, no.2, pp.109-118. 35. Y.-H. Pao and K. Hutter, 1975, “Electrodynamics for Moving Elastic Solids and Viscous Fluids”, Proceedings of the Institute of Electrical and Electronics Engineers, v.63, no.7, pp.1011-1021. 36. U. Gamer and Y.-H. Pao, 1975, “Diffraction of a Plane Harmonic SH Wave by Semi-Cylindrical Layers”, Archives of Mechanics (Archiwum Mechaniki Stosowanej), v.27, no.1, pp.133-139. 37. Y.-H. Pao, 1976, “Electromagnetic Forces in Deformable Moving Media”, Letters in Applied and Engineering Sciences, v.4, no.1, pp.75-83. (Invited Lecture) 38. Y.-H. Pao and C. C. Mow, 1976, “Theory of Normal Modes and Ultrasonic Spectral Analysis of the Scattering of Waves in Solids”, Journal of the Acoustical Society of America, v.59, no.5, pp.1046-1056. 39. Y.-H. Pao and V. Varatharajulu, 1976, “Huygens’ Principle, Radiation Conditions, and Integral Formulas for the Scattering of Elastic Waves”, Journal of the Acoustical Society of America, v.59, no.6, pp.1361-1371. 40. V. Varatharajulu and Y.-H. Pao, 1976, “Scattering Matrix for Elastic Waves, I. Theory”, Journal of the Acoustical Society of America, v.60, no.3, pp.556-566. 41. Y.-H. Pao, 1976, “Spectral Analysis of Diffracted Elastic Pulses and GeneralizedRay Theory for Mechanical Wave Guides”, Elastic Waves, pp.398-412. (Report of the Workshop on “Application of Elastic Waves in Electrical Devices, NonDestructive Testing and Seismology”, to National Science Foundation, edited by J. D. Achenbach, Y. H. Pao and H. F. Tiersten). 42. U. Gamer and Y.-H. Pao, 1977, “Interaction of Shear Wall with Elastic Foundation under the Excitation of SH Waves”, Engineering Transactions, Polish Academy of Science, v.25, no.3, pp.447-461. 43. *Y.-H. Pao and R. R. Gajewski, 1977, “The Generalized Ray Theory and Transient Responses of Layered Elastic Solids”, in Physics Acoustics, v.13, pp.183-265, edited by W. P. Mason and R. N. Thurston, Academic Press, New York, 1977. 44. P. Chen and Y.-H. Pao, 1977, “The Diffraction of Sound Pulses by a Circular Cylinder”, Journal of Mathematical Physics, v.18, no.12, pp.2397-2406. 45. *Y.-H. Pao, 1978, “Electromagnetic Forces in Deformable Continua”, in Mechanics Today, v.4, pp.209-305, edited by S. Nemat-Nasser, Pergamon Press, Oxford, 1978. (Invited Review) 46. Y.-H. Pao and A. N. Ceranoglu, 1978, “Determination of Transient Response of a Thick-Walled Spherical Shell by the Ray Theory”, Journal of Applied Mechanics, v.45, pp.114-122. 47. V. K. Varadan, V. V. Varadan and Y.-H. Pao, 1978, “Multiple Scattering of Elastic Waves by Cylinders of Arbitrary Cross Section, I. SH Waves”, Journal of the Acoustical Society of America, v.63, no.5, pp.1310-1319.

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48. Y.-H. Pao, 1978, “Theory of Acoustic Emission”, Elastic Waves and NonDestructive Testing of Materials, AMD, v.29, pp.107-129, edited by Y. H. Pao, ASME, New York. 49. Y.-H. Pao, 1978, “The Transition Matrix for the Scattering of Acoustic Waves and for Elastic Waves”, Modern Problems in Elastic Wave Propagation, pp.123-144, edited by J. Miklowitz and J. D. Achenbach, John Wiley & Sons, 1978. 50. Y.-H. Pao, 1978, “Betti’s Identity and Transition Matrix for Elastic Waves”, Journal of the Acoustical Society of America, v.64, no.1, pp.302-310. 51. Y.-H. Pao and D. K. Banerjee, 1978, “A Theory of Anisotropic Thermoelasticity at Low Reference Temperature”, Journal of Thermal Stresses, v.1, pp.99-112. 52. W. J. Parzygnat and Y.-H. Pao, 1978, “Resonance Phenomena in the Nonlinear Vibration of Plates Governed by Duffing’s Equation”, International Journal of Engineering Science, v.16, pp.999-1017. 53. W. Sachse and Y.-H. Pao, 1978, “On the Determination of Phase and Group Velocities of Dispersive Waves in Solids”, Journal of Applied Physics, v.49, no.8, pp.4320-4327. 54. Y.-H. Pao, R. R. Gajewski and A. N. Ceranoglu, 1979, “Acoustic Emission and Transient Waves in an Elastic Plate”, Journal of the Acoustical Society of America, v.65, no.1, pp.96-105. 55. R. L. Weaver and Y.-H. Pao, 1979, “Application of the Transition Matrix to a Ribbon-Shaped Scatterer”, Journal of the Acoustical Society of America, v.66, no.4, pp.1199-1206. 56. R. L. Weaver and Y.-H. Pao, 1980, “Multiple Scattering of Waves in Irregularly Laminated Composites”, Journal of Applied Mechanics, v.47, pp.833-840. 57. Y.-H. Pao, 1980, “The Sound of Stress — Acoustic Emission and Elastic Waves”, Cornell Engineering Quarterly, v.14, pp.2-11. 58. Y.-H. Pao, 1980, “Mathematical Theories of the Diffraction of Elastic Waves”, Ultrasonic Materials Characterization, pp.457-473, (National Bureau of Standards Special Publication No. 596), U.S. Government Printing Office, Washington D. C., 1980. 59. Y.-H. Pao, 1980, “Theory of Acoustic Emission”, Transactions of the Twenty-Sixth Conference of Army Mathematicians, (ARO Report 81-1), pp.389-396, U.S. Army Research Office, Research Triangle Park, NC. 60. A. N. Ceranoglu and Y.-H. Pao, 1981, “Propagation of Elastic Pulses and Acoustic Emission in a Plate, Part I: Theory; Part II: Epicentral Responses; Part III: General Responses”, Journal of Applied Mechanics, v.48, pp.125-132, 133-138 & 139-147. 61. S. Sancar and Y.-H. Pao, 1981, “Spectral Analysis of Elastic Pulses Backscattered from Two Cylindrical Cavities in a Solid, Part I”, Journal of the Acoustical Society of America, v.69, no.6, pp.1591-1596 (Part II by S. Sancar and W. Sachse, Journal of the Acoustical Society of America, vol.69, no.6, pp.1597-1609, 1981). 62. R. L. Weaver and Y.-H. Pao, 1981, “Dispersion Relations for Linear Wave Propagation in Homogeneous and Inhomogeneous Media”, Journal of Mathematical Physics, v.22, no.9, pp.1909-1918.


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63. W. Sachse and Y.-H. Pao, 1981, “Signals in The Far-Field of an Acoustic Emission Source”, Ultrasonics International 1981 Conference Proceedings, pp.116-122, IPC Science and Technology Press, Ltd., Guildford, Surrey, UK. 64. Y.-H. Pao, 1982, “Applied Mechanics and Physics”, (in Chinese), Transactions, Society of Theoretical and Applied Mechanics of China, no.18, pp.1-8. 65. Y.-H. Pao and F. Ziegler, 1982, “Transient SH-Waves in a Wedge-Shaped Layer”, Geophysical Journal of the Royal Astronomical Society, v.71, pp.57-77. 66. R. L. Weaver and Y.-H. Pao, 1982, “Axisymmetric Elastic Waves Excited by a Point Source in a Plate”, Journal of Applied Mechanics, v.49, pp.821-836. 67. R. L. Weaver and Y.-H. Pao, 1982, “Spectra of Transient Waves in Elastic Plates”, Journal of the Acoustical Society of America, v.72, no.6, pp.1933-1941. 68. W. Sachse and Y.-H. Pao, 1982, “Locating and Characterizing Sources of Acoustic Emission”, Proceedings of the Fifth International Conference on Nondestructive Evaluation in The Nuclear Industry, pp.326-331, San Diego, California, 1982. 69. Y.-H. Pao, 1982, “Transient AE Waves in Elastic Plates”, Progress in Acoustic Emission, pp.181-197, Proceedings of the 6th International AE Emission Symposium, October 31-November 3, 1982, edited by M. Onoe, K. Yamaguchi, and T. Kishi, Japanese Society for Non-Destructive Inspection, Susono, Japan. 70. *Y.-H. Pao, 1983, “Elastic Waves in Solids”, Journal of Applied Mechanics (50th Anniversary Volume), v.50, pp.1152-1164, Transactions of the ASME. 71. *Y.-H. Pao, W. Sachse and H. Fukuoka, 1983, “Acoustoelasticity and Ultrasonic Measurements of Residual Stresses”, Physical Acoustics, v.17, pp.61-143, edited by W. P. Mason and R. Thurston, Academic Press, New York, 1983. 72. Y.-H. Pao, G. C. C. Ku and F. Ziegler, 1983, “Application of the Theory of Generalized Rays to Diffractions of Transient Waves by a Cylinder”, Wave Motion, v.5, pp.385-398. 73. T. M. Proctor, F. R. Breckenridge and Y.-H. Pao, 1983, “Transient Waves in an Elastic Plate: Theory and Experiment Compared”, Journal of the Acoustical Society of America, v.74, no.6, pp.1905-1907. 74. J. E. Michaels and Y.-H. Pao, 1984, “Deconvolution of Source Time Functions of the Moment Density Tensor”, in Review of Progress in Quantitative Nondestructive Evaluation, v.3, pp.707-715, edited by D. O. Thompson and D. E. Chimenti, Plenum Press, New York, 1984. 75. Von U. Gamer and Y.-H. Pao, 1984, “Grundlagen der akustoelastischen Spannungsmessung”, Osterreichische Ingenieur und Architekten-Zeitschrift, pp.51-54. 76. Y.-H. Pao, F. Ziegler and P. L. Chen, 1984, “Analysis of Transient Waves in Layered Media with Dipping Structure”, Proceedings of the Eighth World Conference on Earthquake Engineering, pp.703-710, San Francisco, California, 1984. 77. Y.-H. Pao, F. Ziegler and P. L. Chen, 1984, “Transient Waves in Layered Media with Dipping Structure”, presented at the 1984 Pressure Vessels and Piping

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78.

79.

80. 81. 82. 83.

84.

85.

86. 87.

88. 89.

90.

91.

Laudatio for Professor Y.-H. Pao Conference and Exhibition, pp.49-59, San Antonio, Texas, June 17-21, 1984, edited by S. K. Datta. F. Ziegler and Y.-H. Pao, 1984, “A Note on Transient Elastic Cylindrical Waves in a Dipping Layer on Top of a Half-Space”, Osterreichishce Akademie der Wissenschaften, B. 193, Heft 8-10, pp.501-512. J. E. Michaels and Y.-H. Pao, 1984, “Characterization of AE Sources from Experiments with Oblique Forces”, in Progress in Acoustic Emission II, pp.189-195, edited by M. Onoe, K. Yamaguchi and H. Takahashi, Japanese Society for Non-destructive Inspection, 1984. F. Ziegler and Y.-H. Pao, 1984, “Transient Elastic Waves in a Wedge-Shaped Layer”, Acta Mechanica, v.52, pp.133-163. F. Ziegler and Y.-H. Pao, 1985, “Theory of Generalized Rays for SH-Waves in Dipping Layers”, Wave Motion, v.7, no.1, pp.1-24. Y.-H. Pao and U. Gamer, 1985, “Acoustoelastic Waves in Orthotropic Media”, Journal of the Acoustical Society of America, v.77, no.3, pp.806-812. M. Hirao and Y.-H. Pao, 1985, “Dependence of Acoustoelastic Birefringence on Plastic Strains in a Beam”, Journal of the Acoustical Society of America, v.77, no.5, pp.1659-1664. J. E. Michaels and Y.-H. Pao, 1985, “The Inverse Source Problem for an Oblique Force on an Elastic Plate”, Journal of the Acoustical Society of America, v.77, no.6, pp.2005-2011. F. Ziegler, Y.-H. Pao and Y. S. Wang, 1985, “Generalized Ray-Integral Representation of Transient SH-Waves in a Multiple Layered Half-Space with Dipping Structure”, Acta Mechanica, v.56, pp.1-15. F. Ziegler, Y.-H. Pao and Y. S. Wang, 1985, “Transient SH Waves in Dipping Layers: The Buried Line-Source Problem”, Journal of Geophysics, v.57, pp.23-32. Y.-H. Pao, C. S. Yeh and Francis C. Moon, 1985, “Dynamic Stresses in a Buried Cylinder due to Ground Shock”, Proceedings of the Trilateral Seminar-Workshop on Lifeline Earthquake Engineering, pp.297-308, Taipei, Taiwan, November, 1985. J. E. Michaels and Y.-H. Pao, 1986, “Determination of Dynamic Forces from Wave Motion Measurements”, Journal of Applied Mechanics, v.53, pp.61-68. T. Ohira and Y.-H. Pao, 1986, “Microcrack Initiation and Acoustic Emission during Fracture Toughness Tests of A533B Steel”, Metallurgical Transactions A, v.17A, pp.843-852. F. Santosa and Y.-H. Pao, 1986, “Accuracy of a Lax-Wendroff Scheme for the Wave Equation”, Journal of the Acoustical Society of America, v.80, no.5, pp.1429-1437. Y. K. Yeh and Y.-H. Pao, 1987, “On the Transition Matrix for Acoustic Waves Scattered by a Multilayered Inclusion”, Journal of the Acoustical Society of America, v.81, no.6, pp.1683-1687.


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92. Y.-H. Pao, 1987, “Theory of Acoustoelasticity and Acoustoplasticity”, in Solid Mechanics Research for Quantitative Non-destructive Evaluation, pp.257-273, edited by J. D. Achenbach and Y. Rajapakse, Martinus Nijhoff Publishers, Dordrecht, the Netherlands, 1987. 93. F. Santosa and Y.-H. Pao, 1989, “Transient Axially Asymmetric Response of an Elastic Plate”, Wave Motion, v.11, pp.271-295. 94. T. Ohira and Y.-H. Pao, 1989, “Quantitative Characterization of Microcracking in A5033B Steel by Acoustic Emission”, Metallurgical Transactions A, v.20A, pp.1105-1114. 95. Y.-H. Pao, F. Ziegler and Y. S. Wang, 1989, “Acoustic Waves Generarated by a Point Source in a Sloping Fluid Layer”, Journal of the Acoustical Society of America, v.85, no.4, pp.1414-1426. 96. K.-C. Weng and Y.-H. Pao, 1989, “ ”, Doctoral Dissertation, National Taiwan University. 97. Y.-H. Pao, T. T. Wu and U. Gamer, 1991, “Acoustoelastic Birefringences in Plastically Deformed Solids: Part I – Theory”, Journal of Applied Mechanics, v.58, pp.11-17., T. T. Wu, M. Hirao and Y.-H. Pao, 1991, “Acoustoelastic Birefringences in Plastically Deformed Solids: Part II-Experiment”, Journal of Applied Mechanics, v.58, pp.18-23. 98. P. Haupt, Y.-H. Pao and K. Hutter, 1992, “Theory of Incremental Motion in a Body with Initial Elasto-Plastic Deformation”, Journal of Elasticity, v.28, pp.193-221. 99. X. Y. Su and Y.-H. Pao, 1992, “Ray-Normal Mode and Hybrid Analysis of Transient Waves in a Finite Beam”, Journal of Sound and Vibration, v.151, no.2, pp.351-368. 100. M. Muller, Y.-H. Pao and W. Hauger, 1993, “A Dynamic Model for a Timoshenko Beam in an Elastic-Plastic State”, Archive of Applied Mechanics, v.63, pp.301-312. 101. Y.-H. Pao and D. C. Keh, 1996, “Moment-Distribution Method and ReDistribution Matrix Analysis of Frame Structures”, The Chinese Journal of Mechanics, v.12, no.1, pp.157-165. 102. *Y.-H. Pao, 1998, “Applied Mechanics in Science and Engineering”, Applied Mechanics Reviews, v.51, no.2, pp.141-153. 103. S. M. Howard and Y.-H. Pao, 1998, “Analysis and Experiments on Stress Waves in Planar Trusses”, Journal of Engineering Mechanics, ASCE, v.37, pp.884-891. 104. Y.-H. Pao and G. S. Lee, 1999, “Analyses of Acoustic Waves in Shallow Water Over A Sloping Bottom — A Review”, Theoretical and Computational Acoustics’97, p.323-328, edited by Y.-C. Teng, E.-C. Shang, Y.-H Pao, M. H. Schultz and A. D. Pierce, World Scientific, Singapore (presented in the 3rd International Conference on Theoretical and Computational Acoustics, 1997). 105. Y.-H. Pao, D. C. Keh and S. M. Howard, 1999, “Dynamic Response and Wave Propagation in Plane Trusses and Frames”, AIAA Journal, v.37, no.5, pp.594-603.

以超音波測定二維殘留應力

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106. Y.-H. Pao, X. Y. Su and J. Y. Tian, 2000, “Reverberation Matrix Method for Propagation of Sound in a Multilayered Liquid”, Journal of Sound and Vibration, v.230, no.4, pp.743-760. 107. P. Borejko, C. F Chen and Y.-H. Pao, 2001, “Application of the Method of Generalized Rays to Acoustic Waves in a Liquid Wedge over Elastic Bottom”, Journal of Computational Acoustics, v.9, no.1, pp.41-68. 108. X. Y. Su, J. Y. Tian and Y.-H. Pao, 2002, “Application of the Reverberation-Ray Matrix to the Propagation of Elastic Waves in a Layered Solid”, International Journal of Solids and Structures, v.39, pp.5447-5463. 109. L. S. Wang and Y.-H. Pao, 2003, “Jourdain’s Variational Equation and Appell’s Equation of Motion for Nonholonomic Dynamical Systems”, American Journal of Physics, v.71, no.1, pp.72-82. 110. Y.-H. Pao and G. Sun, 2003, “Dynamic Bending Strains in Planar Trusses with Pinned or Rigid Joints”, Journal of Engineering Mechanics, ASCE, v.129, no.3, pp.324-332. 111. J. F. Chen and Y.-H. Pao, 2003, “Effects of Causality and Joint Conditions on Method of Reverberation-Ray Matrix”, AIAA Journal, v.41, no.6, pp.1138-1142. 112. Y.-H. Pao et. al, “Forward and Inverse Problems of Elastic Waves in Environmental Engineering, Part I: A General Review” by Y.-H. Pao, G. S. Lee and Y. H. Liu, 2003; “Part II: Inverse Medium Problems of Layered Media” by Y.-H. Pao, Y. H. Liu and T. T. Wu, 2003, Enviromental Vibration: Prediction, Monitoring and Evaluation, ed. Chen Yumin and Takemiya Hirokazu, pp.3-23 & pp.24-37, Proceedings of the International Seminar on Environmental Vibration, Zhejiang University, Hangzhou, China, 16-18 October 2003. 113. Y.-H. Pao, 2003, “A Unified Variational Principle of Thermo-mechanics”; K. C. Chen, L. S. Wang and Y.-H. Pao, 2003, “A Variational Principle for the Dynamics of Rigid Bodies”, Proceedings of the Cross-Strait Workshop on Dynamics, Control, and Variational Principles in Mechanics, Beijing, China, October 2003, pp.1-5 & pp.70-74. 114. Y.-H. Pao, W.-Q. Chen and X.-Y. Su, 2007, “The Reverberation-ray Matrix and Transfer Matrix Analyses of Unidirectional Wave Motion”, Wave Motion, Vol.44, Iss.6, pp.419-438. (Special Issue Guest Ed. Y.H. Pao, Selected papers presented at the International Symposium on Mechanical Waves in Solids, Zhejiang University, Hangzhou, China, May, 2006) 115. Y.-H. Pao and W.-Q. Chen, 2009, “Elastodynamic theory of framed structures and reverberation-ray matrix analysis”, Acta Mechanica, 204/1,2 pp.61-79. 116. F. X. Miao, Guojun Sun and Y. H. Pao, 2009, “Vibration Mode Analysis of Frames by the Method of Reverberation Ray Matrix”, ASME Journal of Vibration and Acoustics, Vol.131. 117. Y.-H. Pao, G.-H. Nie and D.-C. Keh, “Dynamic Response and Wave Propagation in Three-dimensional Framed Structures”. (submitted for publication, 2010)

CHAPTER 1 LAMB WAVES IN PHONONIC BAND GAP STRUCTURES

Tsung-Tsong Wu1, Jin-Chen Hsu2 and Jia-Hong Sun1 1

Institute of Applied Mechanics, National Taiwan University 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan E-mail: [email protected] 2

Department of Mechanical Engineering National Yunlin University of Science and Technology 123, Sec. 3, University Road, Douliou, Yunlin 64002, Taiwan Phononic crystals (PCs) which consist of periodically arranged media have, in the past two decades, attracted intensive studies due to their renewed properties of band gaps and maneuverable band structures. Recently, Lamb waves in thin plates with PC structures started to receive increasing attention for their potential applications in filters, resonators, and waveguides. This paper presents a review of recent works conducted by the authors and co-workers on this topic. Both theoretical and experimental studies of Lamb waves in twodimensional (2D) PC plate structures are covered. On the theoretical side, analyses of Lamb waves in 2D phononic plates using the plane wave expansion (PWE) method, finite-difference time-domain (FDTD) method, and finite-element (FE) method are addressed. These methods were applied to determine the frequency ranges of band gaps of Lamb waves and characteristics of the propagating and localized eigenmodes that can exist in the PC plate structures. The theoretical analyses demonstrated the effects of PC based waveguides and resonant cavities. We discuss the influences of geometrical parameters on the guiding and resonance efficiency and the frequencies of waveguide and cavity modes. On the experimental side, we discuss band gaps of Lamb waves in a surface stubbed phononic plate made of aluminum which were demonstrated using laser ultrasonics. Furthermore, we present design and fabrication of a silicon based two-port Lamb wave resonator which utilizes PC reflective gratings to form the cavity. The measured results 1

2

T.-T. Wu, J.-C. Hsu and J.-H. Sun

showed significant improvement of the insertion losses and quality factors of the resonators when the PCs were applied.

1. Introduction Over the past two decades, propagation of acoustic waves in periodic structures comprised of multi-components has received much attention because of renewed physical properties and potential applications of the periodic structures in a variety of fields, such as noise and vibration isolation, frequency filters in wireless communication, super lens design, etc. These periodic structures, called phononic crystals (PCs),32,55 give rise to forbidden gaps of acoustic waves which are analogous to the band gaps of electromagnetic waves in photonic crystals. Major mechanisms leading to the forbidden gaps are known as Bragg scattering and localized resonances.37 The former opens up the Bragg gap at the Brillouin-zone (BZ) boundaries, and the band-gap frequency corresponds to the wavelength in the order of the structural period, i.e. the lattice constant, and relates to the lattice symmetry. On the other hand, localized resonance creates the resonant gap dictated by the frequency of resonance associated with scattering units and depends less on the lattice symmetry, orderliness, and periodicity of the structure. In the literature, theoretical calculations of bulk acoustic waves (BAWs) in two-dimensional (2D) periodic composites were reported using the plane wave expansion (PWE) method,13,30---33,35,37,55,56,67,70---73 multiple scattering theory,26,36,38,46 and finite-difference time-domain method (FDTD).12,28,61 On the other hand, experimental evidence for the existence of complete acoustic band gaps (BAW modes) have also been reported.12,16,42,50,62,64,66 Among these, existence of band gap, transmission properties, and local resonance of 2D phononic structures in the low frequency range were demonstrated. A comprehensive review of bulk waves in PCs can be found in a review article by Sigalas et al.54 Investigations into the surface acoustic wave (SAW) properties of PCs in which periodic modulation occurs on the traction-free surface did not take place until the late 90’s.20,34,59,60,63,69,75,76 Theoretical studies of surface waves on a square and hexagonal superlattice consisting of cubic (AlAs/GaAs) and isotropic (Al/polymer) materials were reported59,60 and,

Lamb Waves in Phononic Band Gap Structures

3

a couple of years later, extended to the case of general anisotropy by Wu et al.76 Subsequently, the case of SAW modes in the 2D piezoelectric PC was reported.20,34,75 In a recent article,69 the concept of the localization factor was introduced to study the Rayleigh wave propagation and localization in disordered piezoelectric PCs and the authors found that the larger the randomness degree the stronger the degree of wave localization. Since the late 90’s, experimental studies of SAW in millimeter-scale 2D PCs have demonstrated the existence of band gaps, attenuation, localization, and anisotropic transmission of surface states for sonic propagation in finite periodic systems with frequency in the range of a few MHz or lower.4,39,63,68,81 Nevertheless, study in the micrometerscale PCs which may find applications in radio frequency (RF) communications or MEMS devices had not been started until the middle of this decade. In an effort toward the integration of PC and SAW frequency filters or oscillators, Wu et al.79 utilized the Silicon micromachining to fabricate an air/silicon square lattice PC with layered slanted finger interdigital transducers (SFIT) attached. The transmission of surface acoustic waves through 6 layers of a PC has more than 30 dB attenuation, as was observed in the band gap between 183 to 215 MHz. In a subsequent paper, Benchabane et al.1 demonstrated experimentally the existence of a complete SAW band gap in a 2D square lattice piezoelectric PC etched in lithium niobate. The surface acoustic wave was generated by a normal interdigital transducer (IDT) and a complete band gap extending from 203 to 226 MHz was demonstrated experimentally. Scattering and propagation of surface acoustic waves in PCs revealed by optical methods were given by Kokkonen et al.29 and Profunser et al.49 Recently, a design that combines two-port SAW devices and PCs acting as reflective gratings was demonstrated.78 The design consists of a layered ZnO/Si SAW device and a square lattice PC composed of cylindrical holes on silicon. By only using 15-layer PC cylinders, experimental insertion loss shows a 7 dB improvement at the central frequency of 212 MHz. As compared with the conventional metallic reflective gratings, the major advantage of using PC is that the size of the CMOS compatible layered SAW filter can be reduced significantly.

4


Apart from the BAW and SAW modes, propagation of Lamb waves in phononic plate structures has rapidly received increasing interests. It has been known that excitation of high-frequency Lamb waves is achievable by using interdigital transducers (IDTs) deposited on a thin piezoelectric plate25 as that of SAWs on a piezoelectric half space and has been important in a variety of resonators and sensing applications. Theoretical studies on the propagation of Lamb waves in plates with periodic inclusions which are arranged transversely to the propagation direction have not been investigated extensively until recently. For onedimensional (1D) periodic8,17 or quasi-periodic11,53 composite thin plates, the stop bands and transmission properties were investigated and discussed. Dependences of the widths and starting frequencies of band gaps on the plate thickness, filling fraction, and lattice spacing were discussed. As a result, the difference between the band gaps of the bulk and Lamb waves in quasi-periodic composites was revealed. For the 2D case, propagation of Lamb waves in phononic plates with either non-piezoelectric or piezoelectric constituents have also been investigated5---7,21–23,27 utilizing the PWE or modified PWE methods. It was found that, similar to the cases of the bulk and surface modes, directional as well as complete band gaps also existed in phononic plate structures. In order to reduce the computation time of the band structure, a formulation which is based on Mindlin’s plate theory and the PWE method was proposed.21 In addition to the theoretical prediction, the existence of band gaps in phononic plates was also demonstrated through the laser ultrasonics technique.5 In recent years, band structures of phononic plates with three-dimensional (3D) inclusions and/or defects were studied and discussed in the literature using the multiple scattering method,10,51 FDTD method or modified PWE methods.57,58,65,80 Results of these studies showed that with the introduction of defects in phononic plates, wave disturbances can be localized or propagated along the defects and worked as a point resonance cavity or waveguides. In addition to flat phononic plates which have two flat surfaces, complete band gaps and wave guiding in PC plates with periodic stubbed surfaces have been demonstrated both numerically47,48,74,77 and experimentally.74,77 In particular, the results showed that by introducing the periodic stubbed cylinders on the surface


5

of a homogeneous thin plate, a low-frequency gap which is similar to the case of locally resonant structures can be formed.48 On the experimental side, there have been studies, even though not many, on the demonstrations of the band gap properties of phononic plates with lattice constants on the millimeter and micron scales. For the millimeter case (with operating frequencies at a couple of MHz or lower), the propagation of acoustic waves in phononic flat plates or plates with periodic stubbed surfaces was examined experimentally using laser ultrasonics; very good agreements were found with corresponding numerical predictions.3,18,45,77 In addition, demonstration of the wave guiding and frequency selection capability was also conducted.74 Similar to the integration of SAW devices with phononic MEMS structures,1,78,79 the physical realization of Lamb wave phononic MEMS devices had not been started until recently. The first demonstration of the complete band gap of Lamb waves in a phononic slab was given by Olsson et al.44 The phononic slab was realized by including tungsten scatters in a SiO2 matrix. Wide band aluminum nitride piezoelectric couplers were adopted for interrogating the devices with center frequency around 67 MHz. Results showed a 30 dB acoustic rejection with bandwidths exceeding 25% of the midgap. In their work, single and multimode acoustic waveguides have also been investigated by defecting the acoustic crystals through removal of a subset of the tungsten scatters. The other demonstration of a complete band gap of a 2D phononic slab formed by embedding cylindrical air holes in a silicon plate was given by Mohammadi et al.41 The PC structure was made by etching a hexagonal array of air holes through a free standing plate of silicon. A pair of IDT was utilized to measure the transmission of elastic waves through eight layers of the hexagonal lattice PC in the ΓK direction. Their results showed more than 30 dB attenuation was observed at 134 MHz with a band gap to midgap ratio of 23%. In a subsequent paper,40 by etching a hexagonal array of holes in a 15 µm thick slab of silicon, high-Q PC resonators were fabricated using a CMOS compatible process. Results showed that the complete phononic band gap of the PC structure supports resonant modes with quality factors of more than 6000 at 126 MHz. Recent developments in the area of micro-fabricated PCs can be found in a recent review article.43

6


In this article, our recent investigations of Lamb waves in 2D phononic band gaps structures are reviewed and presented, including theoretical methods, numerical calculations, and experimental results, with a demonstrated application of PCs to Lamb wave resonators. This paper is organized as follows. The formulations of the PWE formalism and numerical methods, including the FDTD and finite-element (FE) methods, for Lamb waves in the PC plate structures are summarized in Sec. 2. Section 3 reviews the calculated results of Lamb wave propagation and related phenomena, including the band gaps, in several PC plate structures, and the utilization of the properties of phononic band gaps to construct phononic waveguides and resonant cavities for efficient guiding and confinement of acoustic energy. In Sec. 4, experimental demonstrations of phononic band gaps and a potential application to Lamb wave resonators, based on the simulated results shown in Sec. 3 are given. Finally, Sec. 5 summarizes this presentation. 2. Formulation and Numerical Methods In this section, the theoretical formulations and numerical methods to analyze the propagation of Lamb waves in 2D PC plate structures are summarized. In Sec. 2.1, the typical PWE methods based on Fourier series expansions of the full 3D wave equations,75,76 as well as the Mindlin plate theory based PWE method are introduced. These can effectively reduce the computation time when analyzing the lower-order Lamb modes.21 In Sec. 2.2, the FDTD method is briefly explained.19,57 In the method, in order to tackle wave propagation in PC plate structures, Bloch periodic boundary conditions and absorbing layers with the FDTD algorithm were developed. Moreover, the construction of parallel computation with a personal computer cluster (PC Cluster) that can accelerate the numerical computing of the FDTD method was introduced. Finally, FE analysis employing the solver COMSOL MULTIPHYSICS with Bloch periodic boundary conditions applied to the FE models is introduced in Sec. 2.3.


7

2.1. Plane Wave Expansion Method 21,75,76 2.1.1. Wave Equation and Eigen-Solutions Consider a periodic elastic structure composed of a 2D periodic array of material A in the x1-x2 plane embedded in a background material B. The governing field equations of wave propagation in solids can be expressed as

ρ (r ′)uɺɺj (r , t ) = ∂ i cijkl (r ′)∂ l uk (r, t )  ,

(2.1.1)

where r=(x1, x2, x3)=(r , x3) is the position vector, uj (r, t) is the displacement vector, and ρ (r ′) and cijkl (r ′) are position-dependent mass density and elastic stiffness, respectively. Due to the spatial periodicity in the x1-x2 plane, the material properties can be expanded in Fourier series with respect to the 2D reciprocal lattice vectors, G = (G1 , G2 ) , as '

cijkl (r′) = ∑ eiG⋅r ′ cGijkl ,

(2.1.2)

G

ρ (r ′) = ∑ eiG ⋅r′ ρG ,

(2.1.3)

G

where cGijkl and ρG are the associated Fourier coefficients which can be obtained from

cGijkl = Ac−1 ∫ cijkl (r ′)e − iG ⋅r ′ d 2 r ′ ,

(2.1.4)

ρG = Ac−1 ∫ ρ (r ′)e −iG⋅r ′ d 2 r ′ .

(2.1.5)

Ac

Ac

In the above integrals, Ac is the enclosed area of a primitive unit cell of the 2D periodic structure on the x1-x2 plane. On utilizing the Bloch theorem and expanding the displacement vector uj in Fourier series, we have u j (r , t ) = ∑ AGj eik3 x3 ei (G +k )⋅r ′−iωt , ( j = 1,2,3)

(2.1.6)

G

where k=(k1, k2) is the Bloch wave vector, ω is the circular frequency, k3 is the wave number along the x3-direction, and A jG, ( j=1,2,3), are the amplitudes of the displacement vector. Inserting Eqs. (2.1.2), (2.1.3), and (2.1.6) in Eq. (2.1.1), the following expression can be obtain

8   PG11,G ′  2  21  k3  PG ,G ′   P 31   G ,G ′

T.-T. Wu, J.-C. Hsu and J.-H. Sun PG12,G ′ PG22,G ′ PG32,G′

 Q11 PG13,G ′  G ,G ′   PG23,G ′  + k3  QG21,G ′  31 PG33,G′   Q G ,G ′

Q12 G ,G ′ QG22,G ′ QG32,G′

  R11 Q13 G ,G ′ G ,G ′   QG23,G ′  +  R G21,G ′ QG33,G′   R 31 G ,G ′

R12 G ,G ′ R G22,G ′ R 32 G ,G ′

   A1G ,G′  R13 G ,G ′    R G23,G ′    A G2 ,G′  = 0   3  R 33 G ,G′    A G ,G′ 

(2.1.7) ij G ,G ′

ij G ,G ′

ij G ,G ′

, Q , and R , (i, j=1, 2, 3), are where the submatrices P ijkl functions of k, G, ω, cG and ρG . When the summations in Eqs. (2.1.2), (2.1.3), and (2.1.6) are truncated up to include n reciprocal-lattice vectors G, the matrices PGij ,G′ , QijG ,G′ , and R ijG ,G′ are reduced to n × n matrices, and the expressions of the matrix elements are given by ij 2 α ij β PG,G ′ = ω ρ G − G ′δ ij − (Gα + kα )(Gβ′ + k β )cG − G ′ ,

(2.1.8)

ij α ij 3 3ij β QG,G ′ = − (Gα + kα )cG − G ′ − (Gβ′ + k β )cG − G ′ ,

(2.1.9)

ij 3ij 3 RG,G ′ = −cG − G ′ ,

(α , β = 1, 2).

(2.1.10)

Eq. (2.1.7) can be rewritten in the form of a generalized eigenvalue problem with respect to k3 as

( Pk

2 3

+ Qk3 + R ) U = 0 .

(2.1.11)

Eq. (2.1.11) can be transformed into a standard eigenvalue problem to solve for the eigenvalues k3 and suitably normalized eigenvectors U. By introducing V=k3 U, Eq. (2.1.11) can be rewritten as

I  U   0 U  −1   = k3   . −1  − P R − P Q  V  V

(2.1.12)

2.1.2. Lamb Wave in Phononic Crystals Consider a PC of finite thickness to be a phononic plate with thickness denoted by h. The x1-x2 plane of the coordinate system rests in the middle plane of the plate, and the x3-axis is normal to it and directed upward. When the matrices P, Q, and R in Eq. (2.1.11) are reduced to 3n×3n matrices by choosing n reciprocal lattice vectors, solving Eq. (2.1.12) with an arbitrarily guessed frequency ω yields 6n eigenvalues k3(m) ( m = 1 − 6n) . The eigenvalues and the associated eigenvectors can be used to build the displacement field of Lamb waves in the plate. The general solution can be written as

9

Lamb Waves in Phononic Band Gap Structures 6n

( m)

u j ( r, t ) = ∑∑ X m aGj ( m) ⋅ eik3

x3

⋅ ei ( G + k )⋅r ′ − iω t ,

(2.1.13)

G m =1

where Xm, ( m = 1 − 6n) , are undetermined weighting coefficients. The two parallel surfaces of the plate at x3 = ± h / 2 are assumed to be stress free; so the boundary conditions give

τ i3

x3 =± h 2

= ci 3kl ∂ l uk

x3 =± h 2

= 0.

(2.1.14)

Substituting Eq. (2.1.13) in Eq. (2.1.14) leads to a system of 6n homogeneous linear equations for Xm,  L(1,1)G  1  L(2,)G   ⋮  L(1)  6,G

L(1,G) 2

L(2,)G ⋮ 2

L(6,)G 2

⋯ L(1,G)   X 1     6n ⋯ L(2,G)   X 2  ⋅ = L⋅X = 0,  ⋯ ⋮   ⋮    6n ⋯ L(6,G)   X 6 n  6n

(2.1.15)

where L is a 6n × 6n matrix. L(bm,G) , ( b = 1 − 6 ) , are n × 1 column matrices and their elements L(im,G) are (m)

m L(i ,mG) = ∑ ( Gα′ + kα ) ( cGi 3−kαG ′ aGk (′ m ) ) + k3( ) ( cGi 3−kG3 ′ aGk (′m ) )  eik3

h 2

,

(2.1.16)

G′

(m)

m L(i +m3,) G = ∑ ( Gα′ + kα ) ( cGi 3−kαG ′ aGk (′ m ) ) + k3( ) ( cGi 3−kG3 ′ aGk (′ m ) )  e− ik3

h 2

.

(2.1.17)

G′

Accordingly, the eigenfrequencies of the Lamb waves can be correctly chosen from those guessed frequencies if following condition

det ( L ) = 0 ,

(2.1.18)

is satisfied. Eq. (2.1.18) is the dispersion relation for Lamb waves propagating in 2D periodic composite plates. 2.1.3. Mindlin Theory Based Method Other than the full 3D wave equations, the Mindlin plate theory can be utilized as an alternative.21 According to the assumptions of the Mindlin plate theory, the components of the displacement field uj, ( j=1, 2, 3), for waves propagating in a plate are expanded in power series of the coordinate variable x3:

10

T.-T. Wu, J.-C. Hsu and J.-H. Sun ∞

u j ( r ′, x3 , t ) = ∑ x3n u (j

n)

( r′, t ) = u (j0 ) ( r′, t ) + x3u (j1) ( r′, t ) + …

(2.1.19)

n =0

where uj(n) is called the nth order component of the displacement field. Retaining the zeroth - and first-order terms in Eq. (2.1.19), the equations of motion can eventually be expanded as

ρ uɺɺ1( 0) =

∂  1 (0) (0) (0) g11u1,1 + g12 u2,2 + κ 2 g14 u3,2 + u2( )   ∂x1 

(

)

(2.1.20a)

∂  1 ( 0) (0) (0)  , κ1c56 u3,1 + + u1( ) + c66 u2,1 + u1,2  ∂x2 

(

ρ uɺɺ2( 0) =

)

(

)

∂  1 (0) (0) (0)  κ1c56 u3,1 + u1( ) + c66 u2,1 + u1,2   ∂x1

(

)

(

)

(2.1.20b)

∂  1 ( 0) ( 0) ( 0) g12 u1,1 + + g 22 u2,2 + κ 2 g 24 u3,2 + u2( )  ,   ∂x2

(

ρ uɺɺ3( 0) =

)

∂  2 1 ( 0) ( 0) ( 0)  κ1 c55 u3,1 + u1( ) + κ1c56 u2,1 + u1,2  ∂x1 

(

)

(

)

(2.1.20c)

∂  1 (0) (0) ( 0) + + κ 2 g 24u2,2 + κ 22 g 44 u3,2 + u2( )  , κ 2 g14u1,1  ∂x2 

(

ρ h3 12

1 uɺɺ1( ) =

3 h3 ∂  (1) (1)  h ∂  (1) (1)  u1,1 γ γ 66 u2,1 + γ 12u2,2 + + u1,2 11     12 ∂x1 12 ∂x2

(

− h κ c 

2 1 55

ρ h3 12

1 uɺɺ2( ) =

)

(u

( 0) 3,1

(1)

+ u1

) + κ c (u 1 66

)

(0)

2,1

)

3 h3 ∂  (1) (1)  h ∂  (1)  γ 66 u2,1 γ u (1) + γ 22u2,2 + u1,2 +   12 ∂x2  12 1,1 12 ∂x1 

(

(2.1.20d)

+ u1,2  , 

(0)

)

(

(2.1.20e)

)

− h κ 2 g14u1,1 + κ 2 g 24u2,2 + κ g 44 u3,2 + u2  .   (0)

(0)

2 2

(0)

(1)

In the above expressions, Voigt’s notation has been used to replace the fourth order cijkl (i,j,k,l = 1,2,3) by the second order cIJ , ( I , J = 1 − 6) . gIJ and γIJ are the modified elastic stiffnesses given by

g IJ = cIJ − cI 3c3 J c33 ,

(2.1.21)


γ IJ =

cof sIJ sIJ

,

sIJ

s11 = s21 2 s16

s12 s22 2 s26

2s16 2 s26 , 4 s66

11

(2.1.22)

sIJ being the elastic compliances, and κ1 and κ 3 being the correction factors given by

κ1 =

κ 3 = κ1

c33 + c44 −

π 12

,

( c33 − c44 ) 2 g 44

(2.1.23) 2

2 + 4c44

.

(2.1.24)

By Bloch’s theorem, the zeroth- and first-order components of the displacement field have the forms ⌣ u (j0) ( r ′, t ) = ∑ AGj ⋅ ei ( G + k )⋅r ′ − iω t , j = 1, 2, 3, G (2.1.25) ⌣ α i (G + k )⋅r ′ − iω t (1) , α = 1, 2. uα ( r ′, t ) = ∑ BG ⋅ e G

Expansion of those material properties in Eq. (2.1.20) in Fourier series as Eq. (2.1.3) and then substitution of Eq. (2.1.25) into Eq. (2.1.20) lead to a system of equations in the matrix form, ⌣ ⌣  A1G ′   A1G ′  ⌣   ⌣2   M11   A G2 ′  ⋯ M15 G ,G ′ G ,G ′ A ⌣ G′     ⌣3   A3  = 0 . ⋅ = ⋅ ⋮ ⋱ ⋮ (2.1.26) M A    ⌣ G′   ⌣ 1G ′  1  M 51 ′ ⋯ M 55 ′   B   BG ′  G ,G   G ,G ⌣ G2 ′  B  ⌣2   G′   BG ′  While the summations of Eq. (2.1.25) are truncated to include just n reciprocal lattice vectors in practice, the matrix M in Eq. (2.1.26) is reduced to a 5n × 5n matrix. The explicit expressions of the matrix components can be found in Ref. [21]. The eigenfrequencies of the plate modes in the 2D phononic plate can be correctly chosen by examining the following condition:

det ( M ) = 0 .

(2.1.27)

12


2.2. Finite-Difference Time-Domain Method 2.2.1. Three-Dimensional Difference Equations In a linear elastic material, the equations of motion and constitutive law can be expressed as

ρ uɺɺi = ∂ jτ ij + ρ fi ,

(2.2.1)

τ ij = cijkl ε kl ,

(2.2.2)

where τij is the stress tensor, fi is the body force, and εkl is the strain tensor. Equations (2.2.1) and (2.2.2) describe the motion of an infinitesimal element of an anisotropic material in general. These equations are applicable to the inhomogeneous structure of phononic crystals by arranging the density and elastic constants periodically in space. Further, with staggered grids, the differential equations (2.2.1) and (2.2.2) are transferred into difference equations based on Taylor series expansion to develop the 3D heterogeneous finite-difference formulation.19,57 2.2.2. Boundary Conditions and Perfect Matching Layers The Bloch theorem was introduced to treat the periodic boundary conditions (PBCs) for a unit cell of the PC. According to the theorem, the displacement and stress fields of the PCs can be expressed by multiplying of a plane wave and a periodic function as

ui (r, t ) = eik ⋅rU i (r, t )

(2.2.3)

τ ij (r, t ) = eik ⋅rTij (r, t )

(2.2.4)

where Ui(r, t) and Tij(r, t) are the periodic functions satisfying

U i (r + a, t ) = U i (r, t )

(2.2.5)

Tij (r + a, t ) = Tij (r, t )

(2.2.6)


13

with a lattice translation vector a. Combining Eqs. (2.2.3) and (2.2.4) with Eqs. (2.2.5) and (2.2.6), respectively, the PBCs of displacement ui and stress τij can be written as

ui (r + a, t ) = eik ⋅a ui (r, t )

(2.2.7)

τ ij (r + a, t ) = eik ⋅aτ ij (r, t )

(2.2.8)

With the PBCs, 2D PCs can be analyzed by calculating the unit cell. In analyses of the dispersions, a small disturbance at a random position in the unit cell is set as the initial condition. Thus, all possible wave modes can be transported inside the considered unit cell, and the displacement is recorded and Fourier transformed. Then, the eigenfrequencies with a given wave vector k are indicated by selecting the resonance peaks of the Fourier transformed spectrum, and this allows finding the information about possible types of the waves. On the other hand, Berenger2 introduced the concept of perfect matching layer (PML) to reduce the electromagnetic wave reflection from a boundary, and then, PML was also developed for the elastic wave propagation.9 The PML program was adopted to serve as the nonreflecting boundary condition in analyzing PCs. A stretched coordinate is defined with a complex variable, ei = ai + i

Ωi

ω

(2.2.9)

to derive the code for the PML region. In Eq. (2.2.9), the real part ai is a scale factor, and Ωi ω is the imaginary part with attenuation factor Ωi and circular frequency ω . Then, the differential operation in the stretched coordinate is defined and applied to the equations of motion and constitutive law. After introducing the plane wave solutions into the equations, the numerical attenuation is achieved via the factor Ωi . In addition, the non-reflecting condition at the interface between the PML region and the inner space is obtained by setting corresponding material constants and the unity scale factor. In the elastodynamic equations for the stretched coordinate, displacement and stress components are taken with their spatial partial differential operations in all directions; thus the variables are split into three components to realize the difference

14


Fig. 1. The hardware architecture of the PC Cluster system.19

equations. Then, the actual values are obtained from summation of split components. Finally, the PML equations can be shown to have the form, ρui j + ρ Ω j uɺi j = τ ij , j , (2.2.10)

τɺij m + Ω mτ ij m = cijkl

uɺk ,lδ ml + uɺl , k δ mk

. (2.2.11) 2 In the above equations, ui/j and τij/m are the split displacement and split stress components, which satisfy ui=ui/1+ui/2 +ui/3 and τij=τij/1+τij/2+τij/3. After transforming Eqs. (2.2.10) and (2.2.11) into difference formulae, the PML is arranged outside the space boundaries as a buffer zone with a matched acoustic impedance to suppress the reflection. A numerical attenuation occurs as waves decay rapidly inside the region. With the absorbing layers, the reflection could be reduced to less than 1%, and PML was used in calculations of the dispersion relations and transmission. 2.2.3. Parallel Computing Because of the high-speed computation and low cost of the PC Cluster system, the complex research of PCs can be achieved efficiently by saving CPU computation time. As shown in Fig. 1, the PC Cluster


15

Fig. 2. Parallel model-working queue.19

consists of several personal computers with different operating systems and different hardware architectures. The computers are connected with each other by external network devices, such as network cards based on Ethernet. Thus, the computers in the system can use the messages passing around to communicate with each other. Therefore, the whole computer system can be regarded as a complete parallel machine. The parallel model of the PC Cluster is SPMD (single program multiple data). In SPMD, the parallelism is controlled by the programs, not the CPU instructions. The controller of the system sends the program to every computational node involved in the calculation, and commands every computer to do the task. In the parallel computing, MPICH,14 a highperformance portable implementation of the standard MPI, was adopted to parallel the FDTD program. MPICH offers many kinds of parallel ways in the library so that the programmer can easily pass the message from one of the computers to the others by simply calling the function offered from the library. The FDTD calculation speed was accelerated by paralleling the program. There were several parallel methods in the calculation, such as the method of dividing the time or space and the method of dividing the wave vectors. When calculating the dispersion relations of acoustic

16


waves in PCs, the method of dividing the wave vectors was used. Dividing the calculation by time or space requires a large number of data exchanges. On the other hand, when calculating different wave vectors around the irreducible part of the first Brillouin Zone (BZ), there are much fewer messages passing in the PC Cluster system so that the computation can achieve higher performance. There is no message passing, data exchanging, and synchronization between the calculations with different wave vectors. In this parallel method, as Fig. 2 shows, the working queue is created to store all the wave vectors of the whole computation in the server before calculation. Then, the client sends the request to the server for the jobs asked for in the calculation. If the working queue in the server is not empty, the new job will be sent to the client. After receiving the reply from the server, the client can do the calculation with the specific wave vector. Then, the client can iterate the job requesting process until the working queue is empty. This completes the whole computation. Using the parallel method of wave vector dividing, the whole computation can save more calculation time by increasing numbers of CPUs. Because the numbers of CPUs in calculations are increased, jobs dispatched to each node are also reduced. Moreover, in the method, there is no message passing between the clients and fewer data exchanging between the client and the server. Thus, the total waiting time is spent mostly in the calculation and just a little in the waiting for message passing. Therefore, the parallel method of wave vector dividing is very suitable for use in the FDTD study of the dispersion relations in PCs. 2.3. Finite Element Analysis

In the FE analysis, the commercial FE package and solver COMSOL MULTIPHYSICS was used to carry out the numerical calculations. For the eigenfrequency and eigenmode calculations of Lamb waves, a FE structural model is constructed for the unit cell of the considered phononic structure and then meshed. Assuming a time-harmonic solution, the wave equation results in an eigenvalue problem, which yields the eigenvalues and the eigenfrequencies. According to the Bloch’s theorem,


17

the displacement fields obey the following periodic condition on the boundaries of the unit cell, u j ( x1 + ma1 , x2 + na2 , x3 , t ) = u j ( x1 , x2 , x3 , t ) exp ( ik1ma1 + ik2 na2 ) ,

(2.3.1)

where a1 and a2 are the pitches along the x1- and x2-direction of the periodic array, respectively. m and n are integers. The above conditions were applied on the perimeter of the unit cell, and the top and bottom surfaces of the unit cell plate in the model were set to be free. In the calculations, the COMSOL connected to MATLAB Script can be used to iteratively vary the structural dimensions and/or boundary conditions. 3. Phononic Band Gaps, Waveguides, and Cavities

In this section, numerical results conducted by using the methods described in Sec. 2 are reviewed to explain the behavior of Lamb waves in 2D phononic plates. To start with, Sec. 3.1 focuses mainly on discussions of the phenomenon of phononic band gaps. Their existence was determined by the dispersion relations in a frequency range where no eigenmode is allowed and by the transmitting spectra where the power transmission is extremely low. Apart from finding the phononic band gaps, the eigenmodes in the dispersion curves are also discussed. The eigenmodes of phononic crystals are useful for understanding the band-gap formation, wave behavior in the structures, and designing the phononic band gap materials for further applications. In Sec. 3.2 and Sec. 3.3, we discuss the utilization of phononic plates with complete band gaps (i.e., omni-directional band gaps that can forbid wave propagation in any direction) to construct the waveguides and cavities for Lamb waves, respectively. 3.1. Phononic Band Gaps of Lamb Waves

As the 2D phononic structure has a finite thickness in the x3 direction, the structure becomes a plate, and the wave energy is totally confined between the two traction free surfaces of the plate and developed into the Lamb waves. The following two examples illustrate the calculated results

18

T.-T. Wu, J.-C. Hsu and J.-H. Sun Table 1. Elastic Stiffness (1010 N/m2) and Mass Density (kg/m3) Material

Class

c11

c11

c11

c11

c11

ρ

ZnO Si Au Steel Al Epoxy

6mm m3m m3m isotropic isotropic isotropic

20.97 16.56 19.25 28.02 11.1 0.76

12.11 6.39 16.30

10.51

21.09

4.25 7.95 4.24 8.29 2.5 0.159

5676 2329 19300 7900 2695 1180

of the band structures of Lamb waves in 2D Au/epoxy and steel/epoxy phononic plates using the PWE and FDTD methods, respectively. In the first case, the result of the full 3D PWE method was compared with that of the Mindlin-theory-based PWE (abbreviated as MPWE in the following) method.21 They are in good agreement. The MPWE method also exhibits an improvement in reducing the computation time. In the second case, the FDTD method calculated not only the band structure to show the complete band gaps but also the transmission spectra to obtain a close approach to the experiment. The eigenmodes in the periodic plate structure are also discussed. In the third case, another form of periodic plate structure is discussed, where the plate has periodic stubs on one of the plate surface instead of the periodic fillers. The idea to create such a structure was to produce a periodic variation in geometries rather than in the material properties. Calculated result showed that the structure can generate not only complete band gaps but also some highly resonant modes. The FE method was applied to analyze the structures. The material properties used in the following calculations are listed in Table 1. 3.1.1. Au/Epoxy Phononic Plate In the system, crystalline gold (Au) serves as the filling material, and epoxy as the host material. The band structure of the Au/epoxy plate with thickness h=0.25a, where a is the lattice constant, was first calculated using the full 3D PWE method, and then, compared with that using the MPWE method. The Au circular cylinders of radius r are arranged as a square lattice embedded in the epoxy host as depicted in Fig. 3(a). The

19

Lamb Waves in Phononic Band Gap Structures 5

x1

x2

Normalized Frequency (ωa/ Ct )

k2 M Γ

X

k1

Unit cell

a r

x1

(a)

x3

Full 3D Mindlin

4 3 2 1

h

(b)

0

−

Γ

−X

−

M

−

Γ

Fig. 3. (a) Geometries of the phononic plate. (b) Comparison of the band structures using the full 3D PWE method and the MPWE method.21

filling fraction is F=πr2/a2=0.283. Figure 3(b) displays the calculated dispersion curves along the boundary of the irreducible first BZ, ΓXMΓ, in the wave vector space. The solid lines represent the results obtained by using the full 3D PWE method, and the dots denote the results calculated by using the MPWE method. Due to the large computation time required for the full 3D PWE method, the number of reciprocal lattice vectors G is restricted to 81. The CPU times to calculate the band structure along the boundary of the irreducible part of the first BZ (40500 grid points are included) are about 250 h for the full 3D PWE method and only 2 h for the MPWE with 81 reciprocal lattice vectors on a personal computer equipped with Intel Pentium4 CPU of 2.80 GHz and 512 MB memory. In the figure, it can be observed that the two methods result in good agreement of numerical results. The maximum difference is about 1.6% at the eighth band in the calculated interval of normalized frequency. The above analysis shows that the MPWE method exhibits satisfactory accuracy and affords much less computation time. The results suggest that the MPWE method can serve as a quick and good predicting tool in the design of a phononic plate in the lower frequency bands. Moreover, good convergence of the numerical results should be obtained by implementing the calculations with a large number of reciprocal lattice vectors when a phononic plate consists of materials with a large contrast. 81 reciprocal lattice vectors used in the calculations of Fig. 3(b) do not provide satisfactory convergence. The full 3D PWE

20


(a)

4 Normalized Frequency (ωa/ Ct)

Normalized Frequency ( ωa/ C t)

5 4 3 2 1 0

− Γ

− X

−

M

−

Γ

(b)

3

2

1

0

−

Γ

− X

−

M

−

Γ

Fig. 4. Band structures of Lamb waves in Au/epoxy phononic plates of plate thickness h=0.25a (a) and 0.175a (b), calculated by using the MPWE method with 441 reciprocal lattice vectors.21

method applied to the phononic plate problem, however, is not practical for computation time consideration when the number of the reciprocal lattice vectors is large. Therefore, with the MPWE method, 441 reciprocal lattice vectors were used to re-calculate the dispersion relations of the Au/epoxy example, and the result is shown in Fig. 4(a). In the figure, the complete frequency band gap exists between the sixth and seventh frequency bands extending in the normalized frequency from 2.62 to 3.03. The ratio of gap width to midgap frequency, therefore, is 14.6%. In the phononic plate, the band structure could be quite different from that of an infinite phononic crystal for bulk waves because the waves confined in the finite thickness plate result in an acute dispersion effect in the low frequency region by supporting the flexural and thicknessshear vibrations. Therefore, the ratio h/a can be another influential parameter on opening the complete band gap and band shifting in the band structure. Figure 4(b) shows the band structure corresponding to the phononic plate with a smaller thickness h=0.175a; other parameters employed in the calculations were remained unchanged. In Fig. 4(b) another complete band gap in the thinner phononic plate is found, and the ratio of gap width to midgap frequency is 9.3%. Comparing Fig. 4(b) with Fig. 4(a), the complete band gap between the sixth and seventh frequency bands is closed, and another complete band gap is opened between the fifth and sixth frequency bands, and in the lower frequency


21

50

∆ω/ωm (%)

40 30 20 10 0

0

0.1

0.2

0.3

0.4

0.5

0.6

h/ a

Fig. 5. Variation of the band gap width in the Au/epoxy plate as a function of the plate thickness.21

range, by tuning down the value of the plate thickness. In others words, for Au/epoxy phononic plates, wider and higher complete frequency band gap is obtained in a thicker phononic plate, and lower complete band gap can be created in a thin phononic plate. Figure 5 displays the thickness dependence of the complete band gap width in the Au/epoxy phononic plate. In the thin plate region ( h a ≤ 0.20 ), a complete band gap exists between the fifth and sixth frequency bands sustains a short thickness/lattice spacing range, h / a ≅ 0.075 , and closes down when the thickness h ≤ 0.125a or h ≥ 0.20a . The local maximum band gap width takes place at h=0.175a. In the thicker plate region h a ≥ 0.20 , the complete band gaps appear between the sixth and seventh frequency bands and lie in the relatively higher frequency regions; the band gap width increases progressively with the increase of the thickness when h ≥ 0.20a . It is known that the formation of wide band gaps for BAWs propagating in infinite PCs originates from the interaction between rigid-body resonances of individual fillers and waves propagating in an 52,82 effective homogeneous medium, and the coalescence with Bragg gaps. Correspondingly, in the phononic plate, hybridization of the rigidbody resonance of the individual circular Au plates with the propagation in the effective homogeneous medium corresponding to the periodic

22


plate results in a complete band gap of Lamb waves to exist. Change in the thickness of the plate dramatically changes the scattering properties of the individual circular plates and thus their resonances; therefore, the band structure of the phononic plate is sensitive to the variation of the thickness. Eventually, the eigenfrequencies of the resonance states can be shifted by tuning the plate thickness to create complete band gaps between different frequency bands as shown in Fig. 4. 3.1.2. Steel/Epoxy Phononic Plate57 In this example, a square lattice phononic plate consisting of steel cylinders embedded in an epoxy matrix is analyzed; its usage to construct the phononic waveguides will be shown in Sec. 3.2.1. To calculate the eigenmodes of acoustic waves in the phononic plate, a unit cell was defined as displayed in Fig. 6(a). The cross section on the x1-x2 plane is 1a×1a. a is divided into 48 grids in the calculations. The lattice constant a=8 mm, and the radius r of the steel cylinder is 3 mm. A time step interval of 10 ns in the FDTD calculation was chosen, sufficiently small to satisfy the numerical stability condition. In this case, F=0.442, and h=0.25a. The dispersion relations were calculated and are shown in Fig. 6(b). In the figure, two noticeable complete band gaps appear from 89 to 101 kHz and 125 to 162 kHz, respectively. In the phononic plate,

250

Phononic-Crystal Plate, Cylinder:Steel / Base:Epoxy, r:a:h=18:48:12, sq. k2

M

π a

π a

200

x1

Frequency (kHz)

x2 x3 h

Γ

H

150

k1

The Second complete band gap

G

100 F

a

The First complete band gap

E

50

(a)

X

(b)

D

C B A

0

-40

-20

0 Γ

Transmission(dB)

X

M

Γ

Reduced Wave Vector

Fig. 6. (a) Setup of the unit cell of the phononic plate in using the FDTD method. (b) Band structure and transmission of Lamb waves in a steel/epoxy phononic plate.57


23

the eigenmodes are coupled, and they are identified as flexural (antisymmetric), longitudinal (symmetric), and transverse (shear horizontal) waves such as Lamb waves in classical plates. In order to investigate the wave modes in the dispersion relation, the transmission of the waves and the polarization of the specific modes were analyzed. First, to launch a wide frequency wave packet along the ΓX direction, x2 and x3 polarized line sources were defined on the plate surface. The acoustic waves propagating through a ten-layered phononic plate and a homogeneous plate without steel cylinders were recorded and compared with each other to calculate the transmission. The results are shown in the left part of Fig. 6(b). When the wave is generated from the x3 polarized source, the gaps appear at 72---116, 125---172, and 176---183 kHz, shown by the solid line. The dotted line presents that, when the wave is generated from the x2-polarized source, the pass bands are between 73---87 kHz and the range below 43 kHz. Compared to the dispersion curves along the ΓX direction, the two transmission distributions show a consistent result with the band structure, except a deaf band that can be observed on the top of the second complete band gap. The displacement distribution inside the unit cell was calculated to display the eigenmodes. The calculation setup was the same as in Fig. 6(a) but the initial condition has been replaced by a monochromatic wave source. The displacement distributions of the eigenmodes which belong to the first seven bands along the ΓX direction are shown in Fig. 7. The displacement distributions of the modes A and B are shown in Figs. 7(a) and 7(b). These two modes belong to the lowest band and its folded band with the same wave vector. In these two cases, the displacement fields are almost invariant along the x2 axis. The 3D vector plots show the displacement distribution inside a unit cell. Additional 2D figures are also plotted for clearer understanding. The 2D figures show the polarizations along the plane which is parallel to the x1-x3 plane passing through the center of the steel cylinder. Obviously, the displacement components u1 and u3 dominate the behavior of these modes, which are basically corresponding to the lowest flexural mode in a plate. The next eigenmode, mode C, belongs to the second band in the band structure. Its displacement field on the plate at x3=h/2 is shown in Fig. 7(c). The mode has a primary polarization in the x2 direction. This is the lowest mode of

24


horizontally polarized shear wave of which the distribution does not vary appreciably along the thickness. Mode D has a higher phase velocity than the previous two modes. The distribution of the displacements and the polarization on the plane x3=h/2 of the plate is shown in Fig. 7(d). The polarization in this case is mainly along the x1 direction, and the field does not change much along the plate thickness. This means it is the lowest longitudinal mode. For mode E, the plotted slice of the displacement field is parallel to the x2-x3 plane and passes through the center of the steel cylinder, as shown in Fig. 7(e). The polarization remains at the x2-x3 plane but the magnitude also varies with the x1 direction. Basically, this mode contains u3 components, but they are antisymmetric with respect to the x1-x3 plane passing through the center of the unit cell. Thus, this band cannot be excited by the x3-polarization line source, as shown in the transmission of Fig. 6(b). The result of mode F is shown in Fig. 7(f), and this band determines the bottom edge of the first complete band gap. The displacement distribution shows a circular polarization pattern on the plate at x3=h/2, and this band is excitable by the x2-polarized source. The next mode, mode G, is shown in Fig. 7(g) and exists between the two complete band gaps. The displacement fields of two slices a-a' and b-b' are plotted, respectively, in order to show the polarization clearly. Basically, the displacement field shows the property as a flexural mode in a plate, similar to the case shown in Fig. 7(b), but the field varies significantly along the x2 direction. Finally, an example of the deaf band observed in the transmission is mode H shown in Fig. 7(h). The displacement field is antisymmetric with respect to the x1-x3 plane passing through the central line of the unit cell. The x3 component is antisymmetric and the x2 component shows two opposite directions across the unit cell. Thus, neither the x2 polarization nor the x3 polarization line sources on the plate surface can excite this wave, and this band was identified as a deaf band. Dispersion relations of the phononic plates with the same filling fraction but different thicknesses varied from 0.125a to 1.5a and showed that the bands of longitudinal waves (mode D) and transverse waves (modes C and F) appear at similar ranges. That is, the plate thickness does not alter the gaps of these two wave types. However, the flexural modes (modes A, B, E, G, and H) in the plates are affected by the thicknesses. In a thinner plate, the lowest

25


flexural mode shifts toward the lower frequency range. Thus, the folded bands above the gaps of longitudinal and transverse waves shift downward and narrow the gap width. For a thicker plate, the cutoff frequency of higher order flexural modes is lowered, and then, the higher order modes appear at the range of the band gap. Therefore, the phononic plates usually have narrower complete band gaps than those of the 2D phononic crystal of the same filling fraction. With the complete band gaps, the steel/epoxy phononic plate based waveguide is illustrated in the next subsection.

x1 x2

x1 x2

x3

x3 x2

x1

x3

x3

(e)

(a) x1 x2

x1

x1 x2

x3

x3

x2

x1 x3

(f)

(b) x1

x1 x2

x1

x1

x2

x3

x2

x3

x3

a b

a-a'

a' b'

(c)

(g)

b-b'

x1

x1

x1 x2

x3

x2

a'

x2

x3 b'

x2

x3

a a-a' b

(d)

(h) b-b'

Fig. 7. Vibrations of the eigenmodes in the band structure of the steel/epoxy phononic plate labeled on Fig. 6(b).57

26


3.1.3. Surface Stubbed Phononic Plate77 Different from the two previous cases, a thin plate with a periodic stubbed surface is now considered instead of periodic inclusions. The geometry detail is described as follows. On one side of a thin aluminum plate (plate thickness h1=1 mm), periodic stubbed aluminum cylinders of heights h2 are arranged in a square lattice with lattice constant a=10 mm. The diameter of the cylindrical stubs is 7 mm (i.e., F=0.385, h1/a=0.1). Numerical results of the FE analysis are shown in Fig. 8 to observe the influence of the stub height. Figure 8 shows the dispersion relations of the thin plate with stubs of different heights for waves propagating along the ΓX direction. Upon comparison with the dispersion relation of a homogeneous thin plate [Fig. 8(a)], formation of the special coupling modes can be identified. For example, in Fig. 8(b) of the stub height h2=0.25h1, the purple line marked with S0+A0 means that the mode is transformed from the S0 mode and A0 modes shown in Fig. 8(a). Similarly, the yellow line marked with A0+S0+A1 in Fig. 8(b) indicates that the mode is transformed from the A0, S0, and A1 modes. The evolution of the coupling modes can be identified by comparing Figs. 8(a)–8(c).

(a)

(b)

(c)

250

Frequency (kHz)

200 S0

A1+A0+S0+A0

A1+A0+S0+A0

150

A1+S0+A1

A1+S0+A1

100

A1 A0+S0+A1

A0+S0+A1

T0

50

Γ

0

S0+A0 A0 X

Γ

S0+A0

X

Γ

X

Fig. 8. Dispersion relations along the ΓX direction of stubbed phononic plate with different stub heights h2. (a) Flat plate, (b) h2=0.25h1, and (c) h2=0.5h1. The base plate 77 thickness h1=1 mm, the radius of the stubs is 7 mm, and the lattice constant a=10 mm.


27

250

Frequency (kHz)

200 150

114 – 143 kHz 100 50 0

Reduced Wave Vector (ka/π )

Fig. 9. Band structure of the stubbed phononic plate with h2=10h1.77

If the stub height is gradually increased, some resonances are formed, which result in slower wave velocity and flatter bands near the boundary of the first BZ. It was found that, as the stub height approaches about triple the plate thickness, i.e., h2=3h1, a narrow complete band gap ranging from 167.5 to 171 kHz forms. Moreover, when h2=9h1, the complete band gap is largest, with a range from 119 to 157.5 kHz and a relative band gap width equal to 27.8%. When the stub height is ten times the plate thickness, i.e., h2=10h1, the complete band gap ranges from 119 to 143 kHz (blue region), and three partial band gaps (green region) also appeared. The full band structure of the case h2=10h1 is shown in Fig. 9. 3.2. Phononic Waveguides

Because of the existence of complete band gaps, Lamb wave propagation is forbidden in the frequency ranges with any polarization and propagation direction in the phononic plate structures. The effect allows one to build highly efficient waveguides by utilizing the band gap materials. This subsection describes energy confinement effects for Lamb waves propagating in some defect-contained (line defects) phononic structures with complete band gaps, i.e., phononic waveguides.

28


To understand clearly the wave propagation and energy distributions in phononic waveguides, the dispersion relations and displacement fields of defect modes in the waveguides, and their characteristics related to the structures are discussed. 3.2.1. Waveguides of Lamb Waves in Steel/Epoxy Phononic Plate Guided Lamb waves in the phononic plates were analyzed with the FDTD method.57 Phononic waveguides were formed based on the complete band gaps of Lamb waves. In the steel/epoxy phononic plate presented in Sec. 3.1.2, there exist complete band gaps in the 89---101 and 125---162 kHz windows. Thus, such plates can be used to create waveguides. In the FDTD method, the supercell used to analyze the dispersion of Lamb waves in the waveguide is shown in Fig. 10(a). The supercell contains a defect area and extra ten unit cells. Using the supercell and the PBCs, the eigenmodes (defect modes) can be excited by incident wideband wave sources. The frequencies of the defect modes can be obtained by selecting the local maximum peaks from the Fourier transformed spectra, following the procedure for a unit cell in Sec. 3.1.2.

Phononic-Crystal Plate Waveguide, Cylinder:Steel / Base:Epoxy, w=10 mm

190 180 170

Frequency (kHz)

160

w

150

140 kHz

140 130 120 110 100

95 kHz

90

Supercell

80 70

(a)

(b)

60 _

Γ

Reduced Wave Vector

_ X

Fig. 10. (a) Top view of the phononic waveguide for Lamb waves and the supercell. (b) Band structure of the defect modes in the waveguide of width w=10 mm.57

29


First, consider the waveguide with a width w=10 mm [see Fig. 10(a)]. The dispersion of the waveguide is shown in Fig. 10(b). The figure is focused on the frequency range of 60---190 kHz to observe defect modes in the complete band gaps. The boundaries of complete band gaps are marked with horizontal solid lines. In the waveguide, there are numerous extended modes outside the range of the complete band gaps but these modes are not concerned and the regions are presented in gray. In Fig. 10(b), 13 recognizable defect bands appear in the complete band gaps. The defect bands could be active or deaf. In general, numerous defect bands in the same frequency range support the acoustic waves to propagate inside the waveguide. The x3-polarized monochromatic line source of frequencies equal to 95 and 140 kHz injected from the lefthand inlet of the waveguide showed that wave propagation can be well confined within the scatter-free area. However, observing Fig. 10(b), the monochromatic wave sources of 95 and 140 kHz can excite multiple defect modes inside the waveguide. Thus, the amplitude distribution of the guided waves will be complicated. A waveguide with single mode or only a few defect modes is much preferred for some further applications. The 2D phononic waveguide of narrower width can allow fewer defect modes. The dispersion relations of the defect modes in a narrow waveguide with w=6 mm is shown in

Phononic-Crystal Plate Waveguide, Cylinder:Steel / Base:Epoxy, w=6 mm

190 180 170

Frequency (kHz)

160 150 140

(b)

130 120 110

x1

100

A

x2

90

x3

x1

80 70

x3

60 _

(a)

Γ

Reduced Wave Vector

_ X

(c)

Fig. 11. (a) Band structure of Lamb waves in the steel/epoxy phononic waveguides of width w=6 mm. (b) Amplitude distribution of the 95 kHz Lamb wave [point A in (a)]. (c) Displacement field of eigenmode A in (a).57

30

T.-T. Wu, J.-C. Hsu and J.-H. Sun Phononic-Crystal Plate Waveguide, Cylinder:Steel / Base:Epoxy, w=4 mm

190 180 170

Frequency (kHz)

160 150 140 130 120 110 100 90 80 70 60 _

Γ

Reduced Wave Vector

_ X

Fig. 12. Band structure of Lamb waves in the steel/epoxy phononic waveguide of width w=4 mm.57

Fig. 11(a). Compared to Fig. 10(b), the number of defect bands in the complete band gaps is reduced from 13 to 10. In the first complete band gap, 89---101 kHz, there are only two defect bands. Figure 11(b) shows the amplitude distribution on the surface at x3=0 of a 95 kHz wave source injected into the left inlet of the waveguide. In the plot, darker color means larger amplitude. The acoustic wave propagation inside the waveguide has a simple pattern. From the dispersion, the defect mode is marked as point A in Fig. 11(a). The characteristic displacement field is plotted in Fig. 11(c). The 3D vector plot shows a full view of the waveguide, and the 2D figure shows the cross section at the center of the waveguide. The figures identify the wave as a flexural mode. Though fewer bands exist in the waveguides of 6 mm width in the first complete band gap, there are still numerous defect modes in the second complete band gap of 125---162 kHz. The band structure of an even narrower waveguide of width w=4 mm is shown in Fig. 12. The waveguide results from an additional space of one quarter of a lattice constant between adjacent unit cells. Because of the extremely narrowed width, there are only six bands inside the waveguide. Both the two complete band gaps have simple defect bands, and in addition, the range of 93---101 kHz in the first complete band gap allows no defect modes. For instance, 95 kHz waves cannot propagate

31


into the waveguide. In short, the number of the defect bands can be depressed by constructing a very narrow waveguide; however, the width of the waveguide should not be too small in practice for acoustic wave propagation. Although waveguides with a narrow width have fewer defect bands, these waveguides may no longer consists of a multiplicity of unit cells. This is inconvenient for the construction of an acoustic circuit with PCs of single periodicity, especially when creating a bent waveguide. This can be circumvented by inserting scatters in the center of waveguides, as shown in Fig. 13(a). The waveguide has a width of 10 mm (the same with the lattice constant) and the scatters in the center of the waveguide have a smaller diameter of 4 mm. The band structure of the waveguide is shown in Fig. 13(b). In the dispersion diagram, there are 11 defect bands in the two complete band gaps. However, a single-mode band ranges from 130 to 142 kHz in the second complete band gap. The propagation of the 136 kHz acoustic wave is shown in Fig. 13(c), and the corresponding mode is marked as point A in the dispersion curve. The characteristic displacement field of this mode is shown in Fig. 13(d). From the displacement patterns, the wave is also identified as a flexural mode and the polarization varies along the x2 direction. Since the polarization plane shows symmetry with respect to the central plane (the

Phononic-Crystal Plate Waveguide with scatters of d=4 mm, Cylinder:Steel / Base:Epoxy, w=10 mm

190 180 170

w

d

Frequency (kHz)

160 150 140

A

130

(c)

120 110

x1 x1

100

x2

90

Supercell

a

x3

b

80

a'

70

b'

60 _

(a)

x3

(b)

Γ

Reduced Wave Vector

_ X

a-a'

(d) b-b'

Fig. 13. (a) Top view of the phononic waveguide with scatters and the supercell used in the FDTD calculation. (b) Band structure of Lamb waves in the steel/epoxy phononic waveguide of width w=10 mm with the scatters of diameter d=4 mm. (c) Amplitude distribution of the 136 kHz acoustic wave [point A in (b)]. (d) Displacement fields of the eigenmode labeled by A in (b).57

32


x1-x3 plane) of the waveguide, the mode can be excited by the line source of the x3 polarization. 3.2.2. Stubbed PC Plate Waveguide The plate with a periodic stubbed surface can result in complete band gaps of Lamb waves, as described in Sec. 3.1.3. Therefore, the structure can be further applied to the guided propagation of Lamb waves. Consider a line-defect waveguide produced by removing one row of the stubs from the perfect periodic structure. The geometrical parameters of the defect-contained structure are the same as the scales described in Sec. 3.1.3 (i.e., a=10 mm, h1=1 mm, d=7 mm, and h2=10 mm) but one row of the stubs has been removed. The waveguide structure [see Fig. 14(b)] was analyzed using the FE method,74 and the band structure is shown in Fig. 14(a). In the figure, it can be clearly observed that some additional frequency bands, which correspond to the existence of defect modes localized in the waveguide, appear inside the complete band gap. Specifically, the intensity of the displacement field the defect mode with frequency in the complete band gap marked by the red circle in Fig. 14(a) with the reduced wave number ka/π=0.25 and frequency f=126 kHz is

Max

(a)

180

(b)

Frequency (kHz)

160 Min Max

140

(c)

120 Min Max

100

(d)

80 Min

Fig. 14. (a) Band structures along the ΓX direction of Lamb waves in the stubbed phononic plate containing a waveguide produced by removing one row of the stubs. (b)–(d) are the eigenmode shapes corresponding to the red circle marked in (a) with different viewpoints, respectively.74


33

shown in Figs. 14(b)---14(d) from three different views, respectively. In the figures, it is obvious that most of the elastic-wave energy is well confined within the area of the line-defect waveguide. 3.3. Cavity

Another approach utilizing PCs to confine acoustic energy by band gaps is to create a cavity in the structure. The confined energy can result in resonance with frequencies associated with the dimensions of the cavity. As a result, this leads to possible applications of phononic cavities to acoustic resonator devices. To be compatible with micro fabricating processes for the device, silicon based structures could be adopted.24 The phononic plate can be constructed by etching periodic circular holes in a silicon plate. Thus, the resonance conditions of the cavity should be considered. To design a resonant cavity utilizing the phononic structure, the band gap property of the PC needs to be considered. Figure 15(a) shows the dispersion curves calculated by the FE method for a 2D air/silicon PC plate of square lattice. The lattice constant a is 20 µm, the radius of the air cylindrical hole 8.9 µm, and the plate thickness h=12 µm. With the filling fraction F=0.622, there exist (i) a partial band gap from 148.8 to 174.1 MHz forbidden the Lamb wave propagation along the ΓX direction and (ii) a complete band gap from 148.8 to 160 MHz. Designing a Lamb wave resonator device could begin by calculating the proper position of the wave sources. The interference of Lamb waves encountering the PC structure was calculated by the FE simulations. The frequency of the Lamb wave was chosen to be 160 MHz and the wavelength λ of the corresponding lowest anti-symmetric mode A0 in a 12µm plate is equal to 25 µm. In the simulation, 10 layers of air holes were assumed. Two pairs of normal line forces with interlaced positive and negative polarization were used as external excitation. The interval between the adjacent line forces was one half of the wavelength. The distance between the PC structure and the wave source is defined as D. Two examples of different distances D are shown in Fig. 15(b). They present constructive and destructive interferences, respectively. The color shows

34


(a)

Dispersion of Air/Si PC Plate

D=1.178λ

y

A0 S0

D=1.43λ

x y

150 SH1 SH0

100 M Γ

X

S0

50

A0 SH0

0

x

(c)

A0

Normalized Amplitude

Frequency (MHz)

SH0

SH1

200

(b)

6

2 0 -2 -8

Γ

X

Constructive Destructive

4

-7

-6

-5

-4

-3

-2

-1

0

1

2

3

4

Arc length (a)

Fig. 15. (a) Dispersion of Lamb waves along the ΓX direction in the air/silicon PC plate with h=12 µm, a=20 µm, and F=0.622. (b) Constructive and destructive interference situations. (c) u3 component along the dashed lines in (b) of the 160 MHz Lamb wave.24

the amplitude of the displacement component u3. In the first case of Fig. 15(b), the amplitude is enhanced significantly as the reflective distance D is equal to 1.178λ. The amplitude of the u3 component is almost twice the case without the PC structure. Alternatively, when the reflective distance D is equal to 1.43λ, the normalized amplitude of the region of interest is almost vanishes, and results in a destructive interference [the second case of Fig. 15(b)]. Comparison of the amplitudes of the constructive and destructive interferences is shown in Fig. 15(c). Consequently, the distance D between the line force and the PC structure accordingly should be designed. 4. Demonstrations of Band Gaps and Potential Applications

This section describes measurement of band gaps and resonance characteristics found in the surface stubbed phononic plate. In this presentation, the demonstration of micro cavities based on silicon


35

phononic structures for potential applications to high frequency acoustic wave devices is also included. Band gaps of Lamb waves and resonance peaks in the surface-stubbed phononic plate were detected and characterized using the laser ultrasonic technique and piezoelectric transducers. Details are described in Sec. 4.1. In Sec. 4.2, the design and fabrication of a micro cavity for Lamb waves are explained. In the designs, the cavities can serve as acoustic wave resonator devices where the phononic structures act as high efficient reflective gratings. The performance of the devices in frequency responses was experimentally determined, and the results showed significant improvements of the performance. 4.1. Measurements of Band Gaps in a Stubbed PC Plate77

The experimental scheme is shown in Fig. 16(a). The stubbed phononic plate was fabricated with aluminum using mechanical machining. The geometrical parameters of the structure are the same as the scales considered for Fig. 9, and there are 16×10 stubs on one side of the plate surfaces. In the experiments, an Nd: YAG (yttrium aluminum garnet) pulsed laser was utilized to generate broadband elastic waves in the specimen, and an optical interferometer (with He-Ne laser) was employed to measure the out-of-plane component of the displacement of the waves. The signals were received on the base plate at positions with four stubs away from the wave source [see Fig. 16(a)]. Other than the optical interferometer which might be insensitive to the in-plane vibrations, point piezoelectric transducers (the contact spots are about 1.5 mm in diameter) with different polarizations were, respectively, used to detect the out-of-plane and in-plane vibrations of the waves. The measured point was located at the center of the top surface of the stub six rows apart from where the pulsed laser source was applied. Then the measured time-domain signals were Fourier transformed. The digitized sampling rate for the signal from the detectors was 50 MHz; so the transformed spectra were obtained in a good resolution.

36


The left-hand side of Figs. 16(b) and 16(c) show the measured reference spectra by the optical interferometer for waves propagating along the ΓX and ΓM directions, respectively, and the corresponding band structures are attached on their right-hand side. The reference spectrum is defined as the ratio of the values in the spectrum measured on the stubbed phononic plate to that measured on the uniform thin plate of thickness 1 mm. In the figures, the blue regions indicate the range of the complete band gap, which is from 114 to 143 kHz. Both the measured reference spectra in the ΓX and ΓM directions exhibit very low intensities in that range which agree well with the prediction by the band structure. In Fig. 16(b), three apparent dips in the spectral ranges which correspond to the three partial band gaps (the green regions) also can be clearly observed, indicating much conformity to the locations and width of the partial band gaps appearing in the band structure. In addition to the observations on the band gaps, several peaks can be obviously found in the spectra. These peaks are resonances corresponding to the band-edge frequencies, such as at 19 kHz (the bottom edge of the first partial band gap), 100 kHz, and 109 kHz (the bottom and upper edges of the second partial band gap, respectively), where the slopes of the frequency bands approach zero and associate with high density of states of Lamb modes. Moreover, the peak frequency at 109 kHz has a value of 1.46, which is

250

Frequency (kHz)

200

250

ΓX

(b)

150

100

(b)

150

100

50

50

(a)

ΓM

200

Frequency (kHz)

(a)

0 0 0.3 0.6 0.9 1.2 1.5

Γ

X

(c)

0

0 0.2 0.4 0.6 0.8 1

Γ

M

Fig. 16. (a) Schematic of the experimental setup for measuring the Lamb waves in the stubbed phononic plate along the ΓX and ΓM directions. (b) and (c) are the measured reference spectra (normal displacement) for waves propagating along the ΓX and ΓM directions, respectively.77

37


larger than unity, which means an amplified out-of-plane vibration relative to that in the uniform thin plate, induced by the resonance. In Figs. 16(b) and 16(c), in the frequency range between 143 and 193 kHz, the reference spectra have values about 0.15, which means that elastic waves with frequency in this range may be propagating but with perceptible attenuation. Two nearly flat bands exist in the ΓX direction around 190 kHz. However, these bands are mainly governed by the T0 plate mode (with polarization perpendicular to the sagittal plane), and thus, result in no peak in the spectrum. Figure 17(a) shows the reference spectra measured on the top surface of the stub by using the longitudinal and transverse-polarization piezoelectric transducers. For the measurement of the in-plane vibration, the polarization of the transverse transducer was set to be parallel to the wave vector corresponding to the ΓX direction. In the figure, both the reference spectra measured by longitudinal and transverse transducers show, again, very low intensities within the complete band gap, i.e., the range from 114 to 143 kHz. As a result, the experiment confirmed the existence of the complete band gap in such a periodic stubbed plate. Moreover, the in-plane measurement (shown by the red dashed line)

250 On the plate

Frequency (kHz)

200

150

(g)

Transverse signal on the top of stub Longitudinal signal on the top of stub

(f)

(b)

(c)

(d)

(e)

(f)

(g)

(e) (d)

100

(c)

50

(b)

0

0 1 2 3 4 5 6

Γ

X

Fig. 17. (a) Reference spectra measured on the top of the stub (normal and transverse displacements by using point piezoelectric transducers). (b)–(g) are the eigenmodes labeled on the right panel of (a) by the FE method.77

38


exhibits more resonant peaks with values larger than one. To give an insight into the resonances in the structure, several calculated eigenmode shapes associated with the peak frequencies are, respectively, shown in Figs. 17(b)–17(g). Figure 17(b) displays the eigenmode associated with the lowest outstanding peak around 19 kHz. The measured spectra in Fig. 17(a) simultaneously demonstrate notably enhanced in-plane and out-of-plane vibrations at this frequency. From the mode shape it can be observed that the resonance forms by concentrating energy in the stubs, and thus, results in a very slow group velocity. The second peak appearing at 25 kHz corresponding to the joint cutoff frequency of the mode in the bands in red and green colors is a band-edge state. Next, the eigenmode shape with frequency 60 kHz where the third apparent peak appears is shown in Fig. 17(c). This mode is mainly governed by the antisymmetric mode but with significant in-plane vibration in the stub so the spectrum measured by the transverse transducer exhibits a peak there. The fourth peak appearing at 100 kHz is a mode in the flat part of the band (in green color) evolved from the S0 and A1 plate modes. This mode is also a band-edge state, and its mode shape is shown in Fig. 17(d) where notable transverse vibrations can also be found in the stub. Adjacent to the peak at 100 kHz, the next peak appears at 109 kHz. The associated mode shape is shown in Fig. 17(e). The vibration of this mode is very similar to the mode in Fig. 17(d). For the bottom edge state of the complete band gap, the mode shape is calculated and plotted in Fig. 17(f). It is found that this mode generates no transverse vibration on the top surface of the stub. For frequencies close to 205 kHz, there exists a small peak and the corresponding mode shape is shown in Fig. 17(g). The vibration shows that strong dilatational vibration exists in the stub and out-of-plane vibration exists in the base plate. However, this mode in the band in purple color is mainly governed by the T0 plate mode and the excitation by the pulsed laser source is not that effective as are the antisymmetric modes. Therefore, relatively low peak intensity appears around 205 kHz when detecting by the optical interferometer and longitudinal transducer [Figs. 16(b) and 17(a)], and no significant intensity is observed when detecting by the transverse transducer applied on the stub.


39

4.2. Micro Phononic Cavity for Lamb Wave Resonator

This section focuses on a study of Lamb wave resonators using phononic structures as reflective gratings. Note that in traditional acoustic wave resonators, metal strips are usually adopted as reflective gratings to improve insertion losses of IDTs. To achieve a high reflection, gratings may consist of hundreds of metal strips. Moreover, the distance between the IDT and the grating has to be optimized to result in coherent 15 reflective waves in the resonator. If PCs are applied to replace the metal gratings and formed a phononic cavity, the reflection efficiency can be improved and the size of the resonator can be considerably reduced. A schematic of the Lamb wave resonator using the PC gratings is shown in Fig. 18(a). The Lamb wave device was constructed on a silicon plate. The PC reflector on the plate was introduced that air cylinders form a square lattice with band gap of 148.8–174.1 MHz [refer to the band structure shown in Fig. 15(a)]. As can be seen, the Lamb wave device also included a pair of IDTs and a piezoelectric film on top of the silicon plate to generate elastic waves.

-20

IDT

Insertion Loss (dB)

PC reflective grating

ZnO silicon plate

(a)

(b)

-30 -40 -50 -60 150

155 160 Frequency (MHz)

165

Fig. 18. (a) Schematic of the Lamb wave resonator combined with PC gratings. (b) The measured transmission of the two-port Lamb wave resonator with the PC gratings.24

40

T.-T. Wu, J.-C. Hsu and J.-H. Sun Table 2. Parameters of the Two-Port Lamb Wave Resonator IDT line width (µm) IDT pair

6.2 40

IDT aperture

80λ

IDT thickness (nm)

150

PC grating layers

15

IDT-IDT interval (µm)

756.4

IDT-PC distance (D)

2.178λ

PC-PC distance (µm)

2886.5

Based on the simulation, two-port Lamb wave resonators which utilize the PC reflective gratings were designed and fabricated. The design parameters are listed in Table 2. The fabrication processes include the deposition of metals for developing IDTs and bottom electrodes, deposition of piezoelectric ZnO film for generating elastic waves, lithography for patterning the IDTs and PCs, dry etching for air PC holes, and wet back etching for obtaining the thin plate structure. A silicon-oninsulator (SOI) wafer with a 12 µm thick Si device layer was chosen to control the plate thickness of the Lamb wave device precisely. In addition to the device layer, the thicknesses of the handle layer and the oxide layer of the SOI wafer are 300 µm and 0.5 µm, respectively. The oxide layer was used as blocking material during the dry etching process. At the beginning, a gold layer of thickness 100 nm was first evaporated on the device side of the SOI wafer. Then a 0.9 µm thick piezoelectric ZnO layer was deposited and patterned using RF sputtering and wet etching. The IDTs structures were pattered by lithography before the metallization. Aluminum film of 150 nm thickness was evaporated on the photoresist structure and then IDTs were formed using a lift-off process. The width of the IDTs is 6.2 µm. Then, 15 layers of air holes were obtained by patterning and dry etching using plasma etching. As shown in the figure, the PC reflectors are placed on the outer sides of the two IDT ports. Finally, the lower handle layer and oxide layer were removed by KOH and BOE wet etching, respectively. The frequency response of the Lamb wave resonator with phononic reflective gratings was

Lamb Waves in Phononic Band Gap Structures cavity length

(a)

Frequency (MHz)

(b)

41

170

165

160

155

150

3

3.5

4 20

20.5

21

cavity length (a)

Fig. 19. (a) Schematic of cavities in PCs. (b) Resonant frequencies of anti-symmetric modes inside the phononic cavities.24

measured by a vector network analyzer and the result is shown in Fig. 18(b). The frequency response shows a main lobe of around 2.5 MHz width and the central frequency is around 157 MHz. The shift of central frequency may be caused by the difference of the actual plate thickness to the simulated case. Furthermore, the signal shows three sharp peaks at the main peak frequency region. The frequencies of the three peaks are 157.2, 158.2 and 159.1 MHz and the largest quality factor is 2259 at the third peak. The difference between these frequencies is about 1 MHz. The appearance of three peaks is due to a large size of the resonant cavity consisting of two PC gratings. The resonance of Lamb waves inside the cavity could be understood by the Fabry-Perot resonant condition. It is worth noting that the multiwavelength of different frequencies may satisfy the resonant condition in one resonant cavity. Thus, theoretically there can be infinite resonant modes in a cavity. Figure 19 shows the resonant frequency of antisymmetric Lamb waves inside the PC cavities of different cavity length. As show in Fig. 19(a), the cavity length is defined as the additional size inserted into the PC plate. Fig. 19(b) shows the variation of the resonant frequency as cavity length changes from 3a---4a and 20a---21a. The

42


concerned frequency range is 150---170 MHz inside the band gap. In Fig. 19(b), the cavities with length 3.58 a and 20.49 a have the 160 MHz resonant frequency. Further, the cavity with 20.49a allows another two resonant modes at 153.98 and 165.78 MHz inside the 150-170 MHz range. This shows the difference of two neighboring resonant frequencies depends on the cavity size. On the experimental case, the calculated frequency difference narrows down to about 0.95 MHz. Obviously, a cavity of larger size allows a smaller frequency difference. Therefore, to design a single resonant mode resonator, it is necessary to estimate the interval frequency range of two resonant modes and the width of the main peak controlled by the IDT pair number. 5. Conclusions

Theoretical and experimental studies of Lamb waves in 2D phononic plates are reviewed. The theoretical formulation and methods to analyze the phononic structures are summarized, including the PWE methods, FDTD method, and FE method. Analyzing the phononic plate structures using these methods shows that band gaps of Lamb waves exist in the structures. Further analysis of the phononic waveguides and resonant cavity based on phononic plates with complete band gaps shows that efficient confinement and localization of the acoustic energy can be achieved, and the characteristics of the waveguides and cavity are described. The transport efficiency of the wave energy in straight and bent waveguides was evaluated where an improved design was proposed. The conditions of constructive interference of the resonant modes in long phononic cavities were determined as well. Experimental result show that the detected signals of Lamb waves generated by a pulsed laser in a centimeter-scale stubbed phononic plate exhibited complete band gap between 143 and 193 kHz. Moreover, phononic cavities had been designed and applied to serve as the reflective gratings of Lamb wave resonators. Experimental result showed that, with proper designed band gap frequencies and cavity lengths, the insertion losses and quality factors of the Lamb wave resonators can be significantly improved.


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CHAPTER 2 ON GENERALIZATION OF THE PHASE RELATIONS IN THE METHOD OF REVERBERATION-RAY MATRIX†

W. Q. Chen1, Y.-H. Pao2,‡, Y. Q. Guo3 and J. Q. Jiang4 1

Department of Engineering Mechanics, Zhejiang University Yuquan Campus, Hangzhou, Zhejiang 310027, China E-mail: [email protected] (corresponding author)

2

College of Civil Engineering and Architecture, Zhejiang University Yuquan Campus, Hangzhou, Zhejiang 310027, China E-mail: [email protected]

3

Key Laboratory of Mechanics on Disaster and Environment in Western China Ministry of Education, and School of Civil Engineering and Mechanics Lanzhou University Lanzhou, Gansu 730000, China E-mail: [email protected] 4

Department of Civil Engineering, Zhejiang University Zijingang Campus, Hangzhou, Zhejiang 310058, China E-mail: [email protected]

The method of reverberation-ray matrix (MRRM) has been developed and successfully applied to analyses of dynamic responses of framed structures and transient wave propagation in layered media. However, attention has been paid mainly to solving various practical problems, and there is a lack of study on the theoretical aspects of the method

† This work is supported by the National Project of Scientific and Technical Supporting Programs Funded by Ministry of Science & Technology of China (No. 2009BAG12A01A03-2) and the National Basic Research Program of China (No. 2009CB623204). It is also partly supported by the National Natural Science Foundation of China (Nos. 10972196 and 10725210). ‡ Professor Emeritus, National Taiwan University and Cornell University, USA.

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itself. In this paper, we intend to explore the local phase relations in MRRM based on the theory of first-order linear differential equations with variable coefficients. By using the coordinate transform, a generalized form of the local phase relations is presented, which includes the effect of inhomogeneous terms appearing in the differential equations. These terms may correspond to distributive external loads in a mechanics problem. When the coefficients of the differential equations are constant, the local phase relations will take a simple form. Furthermore, to ensure numerical stability, the phase relations should be rearranged properly, and a modification of the scattering relations is also suggested. These aspects are illustrated by considering the wave propagation in an arbitrarily anisotropic elastic laminate.

1. Introduction In the recent decade, a new method, called the method of reverberationray matrix (abbreviated as MRRM hereafter), has been proposed by Professor Pao and his associates for the analysis of dynamic responses of framed structures2,12,21 and wave propagation in stratified media.22,24 Numerous applications1,23,26-29,34-37 show that the MRRM exhibits certain advantages in vibration and wave propagation analyses when compared to other methods such as the method of transfer matrix (MTM),18 the method of exact dynamic stiffness matrix (MEDSM),31 the finite element method (FEM)39, the spectral element method (SEM)4, the traveling wave approach (TWA),30 etc. Just like the MTM and MEDSM, the MRRM is also based on the exact solutions of the governing equations, and hence there is no need to divide a single uniform member in a structure (or a single layer in a stratified medium) further into elements of small size, as in the FEM. However, the MRRM is basically quite different from the MTM and MEDSM: (1) the unknowns in the MRRM are the wave amplitudes in the traveling wave solutions of the governing equations, instead of the state variables in the MTM or the displacements in the MEDSM; and (2) there is a unique characteristic in the MRRM that two local coordinate systems are employed simultaneously for a single element (a structural member in a framed structure or a layer in a stratified medium). For a comprehensive comparison between the

On Generalization of the Phase Relations

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MRRM and MTM, the reader may be referred to the paper of Pao et al.20 Recently, Pao and Chen19 summarized the most important features of the MRRM for the dynamic analysis of three-dimensional framed structures. In the formulations of MRRM, two important kinds of relations are needed in the construction of the reverberation-ray matrix: The (local) phase relations represent the phase differences between different wave components in the two local coordinate systems persistent to each single element, and the (local) scattering relations reflect the compatibility and equilibrium conditions at each node of the system (an end of a member in a structure or a surface/interface in a stratified medium). The treatment of the scattering relations is trivial, but the use of two local coordinates leads to a much simpler form of the compatibility and equilibrium conditions at the node, which does not involve the length of the element since each node always coincides with the origin of a local coordinate system. Thus, the scattering relations in the MRRM should be free of the numerical instability phenomenon associated with the large product of element length and frequency as encountered in the conventional MTM.18 To the MRRM, the local phase relations seem to be theoretically more important and also interesting. Most available studies1-3,9,12,21-24,26-29,32-37 deal with concentrated external loads only, which act on the nodes of the system. Then the governing equations for any element are homogeneous, and only the complementary traveling wave solutions are used. In such cases, the phase relations are also homogeneous. In practice, however, we frequently encounter cases of distributive loadings. It is therefore necessary to extend the current formulations of MRRM so that the distributive loads can be taken into consideration. For a simple bar model, Jiang and Chen16 showed that it is possible to develop the formulations of MRRM when there are distributive or moving loads acting on the bar. In such a case, the phase relations become inhomogeneous. Recently, Jiang et al.15 further presented the procedure of deriving the inhomogeneous phase relations for a Timoshenko beam subject to distributive loads. A detailed discussion can also be found in Ref. [14].

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In the available studies,2,12,21,22,24 the governing equations are usually assumed to have constant coefficients. If a non-uniform element is encountered, then we must divide it into many sub-elements so that in each sub-element the geometric and material properties can be regarded as constant.15,16,32 The constant coefficients in the governing differential equations enable us to use complementary traveling wave solutions in a simple form. Nevertheless, whether it is possible or not to establish the formulations of MRRM for non-uniform elements directly is always a concern of the related studies. On the other hand, a solid theoretical basis of the MTM has already been put forward using the concept of propagator by Gilbert and Backus,7 who demonstrated that, for any linear differential system the formulations of MTM can be established, at least theoretically. Inspired by their work, this paper intends to consider a general linear differential system as well, but is focused on the MRRM. The layout of the paper is as follows. Section 2 presents some basic formulations for first-order linear ordinary differential equations, whose coefficients may be functions of the independent variable. In particular, the solution of the inhomogeneous differential system is presented in two different forms, both will be employed for relevant analysis purposes. The property of the propagator7 is also utilized in our analysis. As mentioned above, the MRRM employs two local coordinate systems (the dual coordinate system) for each element. The relations of various vectors and matrices, including the propagator, in this dual coordinate system are discussed in Sec. 3 through coordinate transform. We then, in Sec. 4, derive the generalized phase relations for any linear differential equations with variable coefficients as well as inhomogeneous terms. Section 5 considers the particular case of homogeneous linear differential equations for wave propagation in arbitrarily anisotropic elastic laminated plates. Some particular features of the linear system with constant coefficients are emphasized. Phase relations are shown to take a very simple form. A proper rearrangement of the phase relations and a modification of the scattering relations are suggested in the consideration of numerical stability. The paper finally ends with a summary drawn in Sec. 6.


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2. System of First-Order Differential Equations and the Solution Consider the following set of first-order ordinary differential equations: d f ( z) = C( z )f ( z ) + g ( z ) , ( z0 ≤ z ≤ z1 ) , dz

(1)

where f(z) is the vector to be determined, whose n elements are unknown functions of z, C( z ) is the coefficient matrix of order n × n , and g(z) is the inhomogeneous vector, which is known. For a mechanics problem, Eq. (1) is usually called the state equation of the physical system, and f(z), C( z ) and g(z) are the state vector, the system matrix, and source (input) vector, respectively. Since for most mechanics problems we usually encounter partial differential equations rather than ordinary ones, proper methods should be employed to transform the original set of governing equations to the form as given in Eq. (1). For example, for dynamic problems of framed structures, Fourier (or Laplace) transforms may be used to eliminate the time-dependence and Eq. (1) corresponds to that in the frequency domain19,20; and for wave propagation in infinite layered media, Fourier (or Hankel) transforms may be used so that Eq. (1) represents the corresponding equation in the joint wave numberfrequency domain.22,24 In the first example, the state vector, system matrix and source vector will also depend on the frequency, in addition to the independent spatial variable z, while in the second example, they all are further related to the wave numbers. The latter will be considered in great detail in Sec. 5. It is also noted that, for a general mechanics problem, n must be an even number, and f(z) shall consist of n/2 (generalized) displacement components and n/2 (generalized) force (stress) components. The complete solution to Eq. (1) can be expressed as6 z

f ( z ) = B( z , z0 )f ( z0 ) + ∫ B( z , ζ )g (ζ )d ζ , z0

(2)

where B( z, z0 ) = M ( z )M −1 ( z0 ) is the propagator,7 M ( z ) is any fundamental matrix, whose columns are n linearly independent complementary solutions of Eq. (1), and f ( z0 ) is the state vector at z0 , i.e. the initial state vector. We have the following property of the propagator (see Ref. [6])

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B( z2 , z1 )B( z1 , z0 ) = B( z2 , z0 ) , B( z1 , z0 )B( z0 , z1 ) = I .

(3)

The first term on the right-hand side of Eq. (2) corresponds to the complementary solution, while the second is the particular solution of Eq. (1). The complementary solution, denoted by fc , can also be written as  a1    a  fc ( z ) = M ( z )  2  ≡ M ( z ) A , ⋮  an 

(4)

where ai are arbitrary constants to be determined, and A is the corresponding unknown amplitude vector. By comparing Eq. (4) with Eq. (2), we obtain A = M −1 ( z0 )f ( z0 ) or f ( z0 ) = M ( z0 ) A ,

(5)

which gives the relation between the amplitude vector A and the initial state vector f ( z0 ) . Making use of Eq. (5), the complete solution can be rewritten as z

f ( z ) = M ( z ) A + ∫ M ( z )M −1 (ζ )g (ζ ) d ζ , z0

(6)

which will be employed in Sec. 4 to derive the generalized phase relations for the general physical system that can be described by the linear differential equations as shown in Eq. (1). If the coefficient matrix C is a constant matrix, we have M ( z ) = exp(Cz ) = V exp( Λz )V −1 ,

(7)

where V = [V1 , V2 , ⋯ , Vn ] , with Vi being the eigenvector corresponding to the eigenvalue λi of C ,a and Λ = diag(λ1 , λ2 , ⋯, λn ) is an n-th order diagonal matrix. The complete solution to Eq. (1) then becomes z

f ( z ) = V exp[ Λ( z − z0 )]V −1f ( z0 ) + ∫ V exp[ Λ( z − ζ )]V −1g(ζ )d ζ z0

(8)

or z

f ( z ) = V exp( Λz )V −1 A + ∫ V exp[ Λ ( z − ζ )]V −1g (ζ )d ζ , z0

(9)

a For the sake of simplicity, it is assumed in this paper that all eigenvalues of the constant matrix C are distinct, as in cases of most dynamic problems.


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where A = V exp(− Λz0 )V −1f ( z0 ) .

(10)

On the other hand, for a constant coefficient matrix C , the complementary solution to Eq. (1) may also be written as n

f c ( z ) = ∑ ai′Vi exp(λi z ) = V exp( Λz ) A′ ,

(11)

i =1

which has been employed to study the elastic surface waves in a general semi-finite anisotropic medium by Ingebrigtsen and Tonning.13 Here A′ is a new amplitude vector and ai′ are the corresponding elements. Clearly, we can derive the following relation between the two amplitude vectors A′ = V −1A = exp(− Λz0 )V −1f ( z0 ) .

(12)

The complementary solution in the form of Eq. (11) will be used in Sec. 5. There are still many other choices of the form of the complementary solution5 because of the flexibility in defining the amplitude vector through the initial state vector. 3. Dual Coordinate System and Coordinate Transform As a unique characteristic of the MRRM,12,19-22 we shall use two local (or dual) coordinate systems for a single element in the physical system. As mentioned earlier, a practical problem is usually governed by a set of partial differential equations, and hence transform methods may be employed to reduce the governing equations to ordinary differential equations. Let z be the only retained independent variable so that the transformed equations can be recast into the state equation in Eq. (1). Furthermore, assume that the z-direction coincides with the depth direction of a stratified medium or the axial direction of a beam, as depicted in Fig. 1. No matter which case is considered, we will use a one-dimensional element as shown in Fig. 2, to represent the problem described by Eq. (1). The two ends (nodes) of the element, denoted as I and J, correspond to the upper and lower surfaces of a layer in Fig. 1(a) or the left and right end sections of a beam in Fig. 1(b). This element will be referred to as element IJ, which has the length h IJ . Two local

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coordinate systems are used with z IJ pointing from I to J, and z JI pointing from J to I. Thus z IJ and z JI are opposite in direction. In the following, the superscript IJ or JI will be affixed to all physical quantities to denote the corresponding coordinate system ( z IJ or z JI ) and the associated element, except those that are independent of the element and the coordinate systems. According to Eq. (2), the complete solution to Eq. (1) with respect to coordinate z IJ is, f IJ ( z IJ ) = B IJ ( z IJ ,0)f IJ (0) + ∫

z IJ 0

B IJ ( z IJ , ζ )g IJ (ζ )d ζ .

(13)

Note that the origin of z IJ has been taken to coincide with end I so that we have z0IJ = 0 . Similarly, in the coordinate z JI we have f JI ( z JI ) = B JI ( z JI ,0)f JI (0) + ∫

z JI 0

B JI ( z JI , ζ )g JI (ζ )d ζ .

(14)

Solutions (13) and (14) may be called dual solutions of the inhomogeneous differential system represented by Eq. (1).

z z (a)

(b)

Figure 1. The z coordinate of (a) a layer in a stratified medium, and (b) a beam in a framed structure.

I

zIJ

zJI

J

hIJ Figure 2. The dual coordinate system for a typical element in the system.

The unknown state vector f and the inhomogeneous vector g usually take different values in different coordinates. Note that, although the x and y-axes are not shown in Figs. 1 and 2, their directions also have an


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influence on the values of these vectors and the associated transformation matrices. For a practical physical problem, since the state vector f contains both (generalized) displacements and (generalized) forces or stress components, as will be seen in Sec. 5, its transformation does not follow the rule for a vector in Euclidean space. We denote the transformation matrix between f JI and f IJ at a same point ( z JI = h IJ − z IJ ) as Tf , i.e. f JI (h IJ − z IJ ) = Tf f IJ ( z IJ ) .

(15)

Note that the transformation matrix only depends on the direction cosines between the corresponding coordinate axes of the two local coordinate systems. Similarly, we assume the transformation matrix between g JI and g IJ to be Tg , and that between C JI and CIJ to be TC , i.e. g JI (h IJ − z IJ ) = Tg g IJ ( z IJ ) , C JI (h IJ − z IJ ) = TC C IJ ( z IJ ) .

(16)

Then we can deduce the following relations directly from Eq. (1) CIJ ( z IJ ) = −Tf−1TC CIJ ( z IJ )Tf = −Tf−1C JI (h IJ − z IJ )Tf

(17a)

C JI (h IJ − z IJ ) = −Tf C IJ ( z IJ )Tf−1 ,

(17b)

Tf−1Tg = −I , or Tg = −Tf ,

(18)

or and where I is an identity matrix. The minus sign in the above equations results from the fact z IJ = h IJ − z JI for any point between I and J. Equations (17) and (18) show that the transforms for the state vector, the inhomogeneous vector as well as the system matrix are not independent of each other. The transform of f is the negative of that of g ; both are similar to, but not the same as that of, a first-rank tensor. The transform of C resembles that of a second-rank tensor, and the transformation matrix TC is a combined result of the left-multiplied transformation matrix −Tf and the right-multiplied matrix Tf−1 . Once a differential system is mathematically formulated, its propagator is determinate as well. Hence, we will pay attention to the behavior of the propagator matrix under the coordinate transform discussed above. To do so, we consider the homogeneous system with its

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solution given by f c ( z ) = B( z, z0 )f ( z0 ) , which is rewritten here for the local dual coordinate system f IJ ( z IJ ) = B IJ ( z IJ ,0)f IJ (0) , f JI ( z JI ) = B JI ( z JI ,0)f JI (0) .

(19)

With the aid of Eq. (15), we can obtain the following relation B IJ ( z IJ , h IJ ) = Tf−1B JI (h IJ − z IJ ,0)Tf .

(20)

This relation of the propagator holds for the inhomogeneous system as well. In the derivation of Eq. (20), we have utilized the property of the propagator as shown in Eq. (3). Incidentally, the transform in Eq. (20) can also be obtained directly using the Magnus expansion.17 Moreover, with Eq. (20) we can easily deduce Eq. (13) from Eq. (14) and vice versa. 4. Generalized Phase Relations The transformation rules presented in the previous section can be used to derive the relation between two amplitude vectors A IJ and A JI , if the consistency in physical reality (or compatibility) of the dual solutions is noticed. To this end, according to Eq. (6), we rewrite the dual solutions in Eqs. (13) and (14) as:

{ {

f IJ ( z IJ ) = M IJ ( z IJ ) A IJ + ∫ f JI ( z JI ) = M JI ( z JI ) A JI + ∫

z IJ 0

z JI 0

} ζ }.

[M IJ (ζ )]−1 g IJ (ζ )d ζ ,

(21)

[M JI (ζ )]−1 g JI (ζ )d

(22)

In general, the fundamental matrix M ( z ) also varies with the coordinate system. It should be noted that, although all columns of the matrix M ( z ) satisfy Eq. (1), the transform of these columns between different coordinate systems does not follow the same rule as that of f ( z ) because M ( z ) are associated with arbitrary constants. Since B( z, z0 ) = M ( z )M −1 ( z0 ) , we deduce from Eq. (20) M IJ ( z IJ ) = Tf−1M JI (h IJ − z IJ )[M JI (0)]−1 Tf M IJ (h IJ ) ,

and in view of Eqs. (15)-(18), we obtain from Eq. (22)

(23)

59


{

f IJ ( z IJ ) = Tf−1M JI (h IJ − z IJ ) A JI + ∫

h IJ − z IJ 0

[M JI (ζ )]−1 g JI (ζ ) d ζ

}

= M IJ ( z IJ )[M IJ ( h IJ )]−1 Tf−1M JI (0) ×

{

× A JI − ∫

IJ

h −z 0

(24)

}

IJ

[M JI (ζ )]−1 Tf g IJ ( h IJ − ζ ) d ζ .

The above equation can be further transformed, by variable substitution, to f IJ ( z IJ ) = M IJ ( z IJ )[M IJ ( h IJ )]−1 Tf−1M JI (0) ×

{

× A JI + ∫ IJ

z IJ h IJ

[M JI ( h IJ − ζ )]−1 Tf g IJ (ζ ) d ζ

}

(25)

IJ

= M (z ) ×

{

× [M IJ (h IJ )]−1 Tf−1M JI (0) A JI + ∫

z IJ h IJ

}

[M IJ (ζ )]−1 g IJ (ζ ) d ζ .

Combining this result with Eq. (21) gives A IJ = [M IJ (h IJ )]−1 Tf−1M JI (0) A JI − ∫

h IJ 0

[M IJ (ζ )]−1 g IJ (ζ ) d ζ .

(26)

This equation establishes the connection between the two amplitude vectors of the dual solutions in the dual coordinate system. As we will show in Sec. 5, this relation just expresses the phase differences between various waves existing in element IJ for dynamic response (wave propagation) problems. We here refer to them as generalized (local) phase relations because Eq. (26) is suitable for all problems that can be represented mathematically by Eq. (1). As seen from Eq. (26), the generalized phase relations are quite different from those reported earlier.1-3,9,12,21-24,26-29,32-37 First, the current phase relations are derived in a mathematically more strict sense by adopting the concept of coordinate transforms, while those in the previous works are obtained based on a direct physical interpretation. Second, the current phase relations are also suitable for systems with variable coefficients, which have not been tackled before. Third, the phase relations given by Eq. (26) include inhomogeneous terms, which result directly from the inhomogeneous vector g in Eq. (1). The vector g evidently corresponds to the external stimuli of a given mechanics problem. Note that, in most studies on the MRRM,1-3,9,12,21-24,26-29,32-37 only concentrated external loads applied on

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the nodes (joints) are considered. When a concentrated load is applied in the interior of an element, one may use subdivision to make the load act just on the additional node, and hence the analysis procedure outlined in Pao et al.2,12,21,22,24 remains valid. Only recently, Jiang and Chen16 and Jiang et al.15 demonstrated through simple physical models (bar or beam) that problems involving distributive loads can be readily dealt with within the framework of MRRM. Their analyses show that consideration of distributive loads will lead to inhomogeneous phase relations, as just shown here for a more general differential system. If the physical problem with n dimensions involves one element only as illustrated in Fig. 1, there will be n / 2 conditions at each end of the element, providing a total of n linear or nonlinear algebraic equations. Combining these conditions with Eq. (26), we are able to solve for the total 2n unknowns in A IJ and A JI . For problems involving many elements, compatibility (of displacements) and equilibrium conditions (of forces) at every node should be taken into consideration. These conditions give another set of relations between the amplitude vectors; these are called (local) scattering relations in dynamic response/wave propagation problems. The treatment of scattering relations is more trivial than the phase relations, as to be seen in the next section for the problem of wave propagation in a layered medium. If the boundary conditions are also linear, then the whole problem is linear. For linear problems, we can group all local phase relations for each element and all local scattering relations at each node to form the global phase relations and the global scattering relations for the whole system. Then, according to Pao et al.,2,12,21,22,24 we can construct the reverberation-ray matrix, which is particularly useful in the calculation of transient responses along with the Neumann series expansion. To avoid numerical instability associated with the classical MTM, the local phase relations given in Eq. (26) should be rearranged in a proper way in the construction of the reverberation-ray matrix.19,20 The scattering relations are also suggested to take a slightly different form from the available works2,12,21,22,24 to ensure unconditional numerical stability. These aspects will be discussed in the next section for the problem of wave propagation in anisotropic elastic laminates.8,10

61


5. Wave Propagation in Anisotropic Laminates To illustrate the analysis for any linear differential system with constant coefficients, we consider in this section the problem of wave propagation in an arbitrarily anisotropic laminated plate of N homogeneous elastic layers (Fig. 3) subject to no external loads. The bonds between two adjacent layers are assumed to be perfect. The z coordinate of the local dual coordinate system for the i-th layer, with the upper and lower surfaces denoted as I and J respectively, are also depicted in Fig. 3. All local coordinates (x, y, z) are right-handed, with their x-axes (not shown in the figure) identical to the X-axis of the global coordinate system (X, Y, Z) as shown in the panel on the left.

1 2

X Y zIJ

i

zJI zJK

I (i) i J (i+1)

N Z Figure 3. An N-layered plate and the local dual coordinate system for a typical layer. i(ωt − k x x − k y y )

Consider a plane harmonic wave with a common factor e in ω k k all physical quantities. Here is the circular frequency, and x and y are the wave numbers in the x and y-directions, respectively. There are several ways to establish the state Eq. (1) for such a problem of wave propagation in anisotropic elastic media. Here we follow the derivation put forward by Ingebrigtsen and Tonning13 but omit details. We choose u x , u y , u z , σ xz , σ yz , and σ zz as the primary variables, and their amplitudes which vary with the z coordinate only, constitute the state vector f ( z ) = [u x , u y , u z , σ xz , σ yz , σ zz ]T

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in Eq. (1), where the overbar indicates the amplitude, the transformed counterpart in the joint wave number-frequency domain, −1 −1   i C33 L ⋮ C33   C =  ⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯ ⋯ ⋯⋯⋯ , (27) −1 −1   − ρω 2 I 3 + k x2C11 + k y2 C22 + k x k y (C12 + C21 ) − LT C33 L ⋮ i LT C33  

in which L = k x C31 + k y C32 , and  c1i1 j  Cij = c2i1 j  c3i1 j 

c1i 2 j c2 i 2 j c3i 2 j

c1i 3 j   c2 i 3 j  . c3i 3 j 

(28)

In Eqs. (27) and (28), ρ and cijkl are the mass density and elastic constants of the anisotropic material, respectively, both assumed to be constant within a single layer, and I n is an identity matrix of order n × n . Obviously, we have Cij = CTji because of the symmetry property of the elastic tensor. For the current problem, the transformation matrix in Eq. (15) for the state vector is 1 0 0 0 0 −1 0 0  0 0 −1 0 Tf = J 2 =   0 0 0 −1 0 0 0 0  0 0 0 0

0 0 0 0  0 0 . 0 0 1 0  0 1 

(29)

where J 2 represents a diagonal matrix with components being ±1 only. We have J 2 = J 2−1 = J 2T . It is straightforward to verify that C JI = − J 2 C IJ J 2 .

(30)

as a result of the coordinate transform of elastic constants, which agrees with Eq. (17) which has been derived based on the set of linear differential equations, Eq. (1). Assume that λ IJ is an eigenvalue of CIJ , i.e. (C IJ − λ IJ I 6 ) VλIJ = 0 . IJ

(31)

with Vλ being the corresponding eigenvector. Multiplying Eq. (31) on the left by J 2 , gives


J 2 CIJ VλIJ − λ IJ J 2 VλIJ = J 2 CIJ J 2 J 2 VλIJ − λ IJ J 2 VλIJ = (J 2CIJ J 2 − λ IJ I 6 )J 2 VλIJ .

63

(32)

In view of Eq. (30), we immediately conclude that −λ IJ is the eigenvalue of C JI with the corresponding eigenvector J 2 VλIJ .b Thus, for monoclinic materials (materials have one plane of symmetry parallel to the x − y plane such that C JI = C IJ = C ), as a special case of this result, we know that if λ is an eigenvalue of C , −λ is also an eigenvalue of it. The generalized local phase relations in Eq. (26), which are suitable for a general differential system as in Eq. (1), are derived based on the dual solutions given in Eqs. (21) and (22). When the system has constant coefficients we prefer to use, in view of Eqs. (9) and (11), the following formulae f IJ ( z IJ ) = V IJ exp( Λ IJ z IJ ) A IJ + exp(CIJ z IJ ) ∫ f JI ( z JI ) = V JI exp( Λ JI z JI ) A JI + exp(CJI z JI )∫

z IJ 0 z JI 0

exp(−C IJ ζ )g IJ (ζ )d ζ , (33) exp(−CJI ζ )g JI (ζ )d ζ . (34)

These differ from Eqs. (21) and (22) only by the arbitrary constants involved, as noticed in Sec. 1. For the current problem, the integral terms in Eqs. (33) and (34) should vanish since there are no sources at all (i.e. g = 0 ). Using Eqs. (33) and (34), and noticing Eq. (20), we obtain the local phase relations as A IJ = exp(− Λ IJ h IJ )(V IJ ) −1 Tf−1V JI A JI ,

(35)

which can, of course, be derived from Eq. (26) by proper substitution. According to the above discussion on eigenvalues and eigenvectors, we can make the following arrangement of eigenvalues Λ IJ = diag(λ1IJ , λ2IJ , ⋯, λ6IJ ) , Λ JI = diag(λ1JI , λ2JI , ⋯, λ6JI ) = diag(−λ1IJ , − λ2IJ , ⋯, − λ6IJ ) ,

b

(36)

This conclusion is extendable to the general differential system, Eq. (1). The statement is then modified as, if λ IJ is an eigenvalue of CIJ ( z IJ ), with eigenvector VλIJ , then −λ IJ is an eigenvalue of C JI (h IJ − z IJ ), with eigenvector Tf VλIJ . In this case, the coordinate z IJ is seen as a parameter.

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W. Q. Chen et al.

so that V JI = J 2 V IJ .

(37)

Equation (35) then can be significantly simplified to A IJ = exp(− Λ IJ h IJ ) A JI .

(38)

Equation (38) is actually valid for any linear differential system in Eq. (1) with constant coefficients. The local scattering relations are straightforward to derive; they are obtained directly from the compatibility and equilibrium conditions at the nodes. For the current problem, using the solutions given in Eq. (33) or (34), and noticing the continuity conditions at the interface, we can derive V JI A JI = J 2 V JK A JK , V IJ A JI = V JK A JK ,

(39)

where Eq. (37) has been utilized. The local scattering relations at the surfaces can also readily be obtained according to different boundary conditions imposed there,8,22,24 as will be shown later. As can be seen from Eq. (39), the element length does not enter the local scattering relations, which therefore are free from numerical instability as encountered by the conventional MTM.18 With the local phase relations for all layers and scattering relations at all interfaces/surfaces as shown in Eq. (38) and (39), respectively, we are able to construct the reverberation-ray matrix in the MRRM. However, this should be done in a proper way to avoid numerical instability and other problems. First, let us check the eigenvalues ( η ) of the matrix E = i C , which satisfy the following characteristic equation13 C33η 2 − (LT + L)η − ρω 2 I 3 + k x2C11 + k y2C22 + k x k y (C12 + C21 ) = 0 . (40)

Once η is determined from this equation, the corresponding eigenvalue of C , λ , is readily calculated as λ = − i η . The eigenvectors of C are thus identical to those of E . If ω , k x and k y are all real, Eq. (40) is a sextic algebraic equation about η with real coefficients. Thus, we have the following four possibilities for the six λ s: (a) all are complex ( λ j = a j + i b j , λ j +3 = −a j + i b j , j = 1, 2,3 , where a j > 0 and b j are real numbers); (b) all are pure imaginary ( λ j = i b j , j = 1, 2,⋯,6 , where b j are


65

real numbers); (c) four are complex and two are purely imaginary, in the forms as in (a) and (b), respectively; (d) two are complex and four are purely imaginary. The purely imaginary λ corresponds to a homogeneous wave, and the complex one to an inhomogeneous wave. According to Synge,25 the solutions may be divided into two different catalogues for each situation. One half of the complex solutions (corresponding to λ j = a j + i b j ) represent inhomogeneous waves which grow exponentially with z, while the other half decay exponentially. One half of the pure imaginary solutions (corresponding to λ j = i b j , with b j < 0 ) represent homogeneous waves with energy transport in the positive z-direction, while the others ( b j > 0 ) in the negative z-direction. Hence, let A −IJ and λ jIJ− ( j = 1,2,3) be, respectively, the amplitudes and eigenvalues of the exponentially growing waves or the arriving plane waves with energy transport in the negative z IJ -direction, and A +IJ and λ jIJ+ ( j = 1,2,3) those of the exponentially decaying waves or the departing plane waves with energy transport in the positive z IJ -direction. For a semi-infinite space, A +IJ should, therefore, correspond to the ‘physically acceptable solutions’, and A −IJ to the ‘physically unacceptable solutions’, as discussed by Ingebrigtsen and Tonning.13 For the current problem, both types of solutions are acceptable and should be retained in the analysis. The property λ jJI = −λ jIJ ( j = 1,2,⋯,6) as shown in Eq. (36) indicates that the solution in the coordinate z JI corresponding to the eigenvalue λ jJI and the one in the coordinate z IJ corresponding to the eigenvalue λ jIJ represent waves of the same mode but traveling in opposite directions. Thus, we can recast Eq. (38) into  A −IJ   exp( − Λ −IJ h IJ )   A +JI  0 =  IJ   ,  0 exp( − Λ +IJ h IJ )   A −JI   A +  

(41)

Λ −IJ = diag(λ1IJ− , λ2IJ− , λ3IJ− ) , Λ +IJ = diag(λ1IJ+ , λ2IJ+ , λ3IJ+ ) .

(42)

where By rearrangement, we obtain IJ  IJ  0 exp(− Λ −IJ h IJ )   A +IJ   A −   IJ  A +   JI  =    JI  = P  JI  . IJ IJ 0  A −   exp( Λ + h )   A +   A + 

(43)

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W. Q. Chen et al.

where P IJ is the local phase matrix. Equation (43) expresses the right phase relations between arriving homogeneous waves and departing homogeneous waves in the layer IJ. If inhomogeneous waves exist, the corresponding elements in the phase matrix P IJ involve exponential functions whose indices have negative real parts only. Thus, the large product of element length and frequency encountered in the MTM, which causes numerical instability,18 is eliminated from the proper phase relations shown in Eq. (43). For monoclinic materials, since Λ +IJ = − Λ −IJ , we simply have IJ 0 exp(− Λ −IJ h IJ )   A +IJ   A −    JI  =    JI  . IJ IJ 0  A −   exp(− Λ − h )   A + 

(44)

By grouping all local phase relations for each layer together, we get  A12   A12  − +    12 21 21    P ⋯ 0 A+   A −      ⋮  ⋮   ⋮ = ⋮ ⋱  A ( N −1) N   0 ⋯ P ( N −1) N   A ( N −1) N   +  −    N ( N −1) N ( N −1)  A −   A + 

(45a)

a = Pd ,

(45b)

or which expresses the global phase relations, where a and d are the global arriving wave vector and the global departing wave vector, respectively, and P is the global phase matrix. Accordingly, the local scattering relations at the interface, Eq. (39), can be rewritten as  V1IJ−  IJ  V2−

V1IJ+   A +JI   V1JK − =  V2IJ+   A −JI   V2JK−

  A −JK  V1JK + , JK   JK  V2+   A + 

(46)

which, in turn, gives  V1IJ−  IJ  V2 −

  A +JI   −V1IJ+ −V1JK +  =  −V2JK+  A +JK   −V2IJ+ S +J

or

  A −JI  V1JK −    V2JK−   A −JK  S −J

(47a)

On Generalization of the Phase Relations JI  JI   A +  J  A−  S +J  JK  = S −  JK  ,  A +   A − 

67

(47b)

where V1IJ− and V1IJ+ , etc. are the partitioned matrices given by  V IJ V IJ =  1IJ−  V2−

V1IJ+  . V2IJ+ 

(48)

For invertible matrix S +J Eq. (47b) may also be written as JI  A +JI   A −  J −1 J  J  JK  = (S + ) S −  JK  = S  A +   A − 

 A −JI   JK  ,  A − 

(49)

where S J is the local scattering matrix.22,24 However, in some particular cases, the matrix inversion is impossible because S +J may be singular.8 Thus, for the free wave propagation problem, it is suggested to use Eq. (47) directly. Note that Eq. (47) is only valid for J = 2, 3,⋯ , N − 1 , i.e. at the interfaces between two adjacent layers. At the uppermost and lowermost surfaces, where J = 1 and J = N , if free surface conditions are assumed, we have 12 12 N ( N −1) V212+ A12 A +N ( N −1) = −V2N−( N −1) A −N ( N −1) + = − V2− A − , V2+

(50a)

1 12 N N ( N −1) S1+ A12 = S −N A −N ( N −1) . + = S− A− , S+ A+

(50b)

or Combing Eq. (47b) for J = 2, 3,⋯ , N − 1 with Eq. (50b) gives the global scattering relations  S1+   06× 3  ⋮   06×3 0  3×3

  S1− 03×6 ⋯ 03×6 03×3   A12 +   2 21  S + ⋮ 06× 6 06× 3   A +   06×3   ⋮ ⋱ ⋮ ⋮  ⋮  =  ⋮   06×6 ⋯ S +N −1 06×3   A (+N −1) N   06×3   03×6 ⋯ 03×6 S +N   A +N ( N −1)   03×3

 03×6 ⋯ 03×6 03×3   A12 −   S −2 ⋮ 06× 6 06× 3   A 21 −    ⋮ ⋱ ⋮ ⋮  ⋮   06×6 ⋯ S −N −1 06×3   A (−N −1) N    03×6 ⋯ 03×6 S −N   A −N ( N −1) 

(51a)

or Sd d = Saa .

(51b)

The following equation then can be derived from Eqs. (45b) and (51b) as

68

W. Q. Chen et al.

(S d − S a P)d = 0 ,

(52)

from which we obtain the secular equation Sd − S a P = 0 .

(53)

From this relation the dispersion characteristics of the waves in an arbitrarily anisotropic elastic laminated plate can be determined. Obviously, the reverberation-ray matrix in the MRRM2,12,21,22,24 can be obtained as R = S −d1S a P , which contains no large product of element length and frequency as mentioned earlier. The secular equation then takes the form I − R = 0 as in Refs. [2,12,21,22,24]. However, the matrix inversion will constitute a severe numerical problem when S d is singular or nearly singular. On the other hand, the modified form as given by Eq. (53) is numerically unconditionally stable.8 For illustration, we display in Fig. 4 the dispersion curves of a twolayered cross-ply laminate with surfaces traction-free and interface perfectly bonded. The constituent 0° and 90° laminas are assumed to have equal thickness and the material properties are given in Table 1. Waves propagating in the X-direction are considered for illustration. Denote the frequency, wave number and phase velocity by ω , k and va respectively. The corresponding dimensionless quantities are defined as Ω = ω H / (2π vs ) , γ = kH / (2π ) and V = va / vs , where H is the total thickness of the laminate and vs = (c55 ρ )0° is the shear wave velocity with (c 55 )0° and ( ρ )0° the stiffness coefficient and material density of the 0° lamina. It is interesting to note from Fig. 4 that all Lamb-type propagated and attenuated wave modes as well as the asymptotic lines can be predicted based on Eq. (53). Numerically stable results at high frequencies can be found in Ref. [10]. Table 1. Material properties of the 0° or 90° lamina. Lamina 0° 90° Lamina 0° 90°

ρ 1200 1200 c23 0.0692 0.0644

c11 1.6073 0.1392 c33 0.1392 0.1392

c12 0.0644 0.0644 c44 0.035 0.0707

c13 0.0644 0.0692 c55 0.0707 0.035

Note: The unit of density is kg/m3 and that of stiffness is 1011 N/m2.

c22 0.1392 1.6073 c66 0.0707 0.0707

69


Dimensionless circular frequency Ω

2.0

1.6

1.2

0.8

0.4

0.0 -2.0

-1.5 -1.0 -0.5 0.0 Dimensionless imaginary wave number γI

0.5 1.0 1.5 2.0 Dimensionless real wave number γR

(a) Frequency vs. wave number

6.0

Dimensionless phase velocity va/vs

5.0

4.0

3.0

2.0

1.0

0.0 0.0

0.5

1.0 Dimensionless wavenumber γR

1.5

(b) Phase velocity vs. wave number

Figure 4. Dispersion spectra of a [0°/90°] laminated plate.

2.0

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W. Q. Chen et al.

The derivation and numerical results presented in this section shows that, with the general method described in Sec. 4 to derive the local phase relations, we are able to deal with any linear differential system with constant coefficients within the framework of MRRM. In fact, the wave propagation analysis for arbitrarily anisotropic laminates can be as simple as that for isotropic24 or transversely isotropic29 materials. Investigation of wave propagation in piezoelectric laminates has also been conducted recently using the modified MRRM.11 6. Summary This paper presents a solid theoretical basis for the establishment of the phase relations for a general linear differential system based on the concept of coordinate transforms. The generalized phase relations are inhomogeneous when the differential equations are inhomogeneous. Hence, theoretically, we are able to deal with any distributive dynamic loading case within the framework of the method of reverberation-ray matrix. Recent efforts using simple bar and beam models have indicated such a feasibility, although for practical problems care should be taken of the particular solutions in order to gain an easy numerical scheme for the MRRM. For concentrated loads, previous studies usually assume that they are applied just on the nodes of a mechanics system. When they are applied in the interior of an element, then the load point can be taken as an additional node so that the conventional MRRM2,12,21,22,24 can be adopted as well. In this case, we will obtain homogeneous phase relations for the two elements connected to this additional node, but the local scattering relations at the node are inhomogeneous2,12,21,22,24. In fact, by eliminating the unknown amplitudes related to the two coordinate systems of which the origins coincide with the node, one may derive the relations between the amplitudes related to the other two coordinate systems, which are also pertinent to the two elements. These are just the generalized phase relations of the undivided element and are inhomogeneous! The detailed verification of the equivalence between the two formulations for a bar element subject to a concentrated load can be found in Jiang14. Interestingly, following the same idea of eliminating the


71

unknown amplitudes at the internal nodes, we can arrive at a numerically very efficient recursive formula for the MRRM.38 The phase relations can be simplified significantly for a linear differential system with constant coefficients. It is demonstrated in this paper by considering the problem of wave propagation in a general anisotropic elastic laminated plate. To ensure unconditional numerical stability, the phase relations as well as the scattering relations must be organized in a proper way. Specifically, any exponentially growing factor should be eliminated from the phase matrix, while the matrix inversion operation should be avoided in obtaining the scattering relations. For the linear system with constant coefficients, we have for simplicity confined ourselves to the case of distinct eigenvalues. This usually happens for most dynamic problems. But such a formulation excludes cases such as static deformations of structures. Therefore, it seems important to consider the cases when there are equal eigenvalues. Acknowledgments The junior authors (W.Q. Chen, Y.Q. Guo and J.Q. Jiang) are very grateful to Professor Pao for his advice and guidance. Numerous discussions with him have greatly helped them in understanding the method of reverberation-ray matrix as well as associated topics. Professor Pao once proposed to use “reverberator” instead of the reverberation-ray matrix that is currently employed. The property of the “reverberator” itself has not been discussed in this paper, and needs further investigation. A part of this paper is based upon unpublished notes of a lecture (co-authored by Y.H. Pao, W.Q. Chen, and J.Q. Jiang), delivered by Y.H. Pao at the 4th International Symposium on Environmental Vibration: Prediction, Monitoring and Evaluation (ISEV2009), held at Beijing on October 28-30, 2009. The lecture was invited by Professor H. Xia of Beijing Jiao Tong University, General Chairman of the Symposium, to whom the authors are indebted.

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References 1. G. Q. Cai and G. H. Nie, Reverberation-ray analysis and program implementation on beam structure in space, Computer Aided Engineering 18, 12 (2009). (in Chinese) 2. J. F. Chen and Y.-H. Pao, Effects of causality and joint conditions on method of reverberation-ray matrix, AIAA Journal 41, 1138 (2003). 3. W. Q. Chen, H. M. Wang and R. H. Bao, On calculating dispersion curves of waves in functionally graded elastic plate, Composite Structures 81, 233 (2007). 4. J. F. Doyle, Wave Propagation in Structures (2nd ed.) (Springer, New York, 1997). 5. J. W. Dunkin, Computations of modal solutions in layered elastic media at high frequencies, Bulletin of the Seismological Society of America 55, 335 (1965). 6. F. R. Gantmacher, The Theory of Matrices (II) (Chelsea Publishing Company, New York, 1960). 7. F. Gilbert and G. E. Backus, Propagator matrices in elastic wave and vibration problems, Geophysics 31, 326 (1966). 8. Y. Q. Guo, The Method of Reverberation-Ray Matrix and its Applications, Ph.D. Thesis (Zhejiang University, Hangzhou, 2008). (in Chinese) 9. Y. Q. Guo and W. Q. Chen, Dynamic analysis of space structures with multiple tuned mass dampers, Engineering Structures 29, 3390 (2007). 10. Y. Q. Guo and W. Q. Chen, On free wave propagation in anisotropic layered media, Acta Mechanica Solida Sinica 21, 500 (2008). 11. Y. Q. Guo, W. Q. Chen and Y. L. Zhang, Guided wave propagation in multilayered piezoelectric structures, Science in China (Series G) 52, 1094 (2009). 12. S. M. Howard and Y.-H. Pao, Analysis and experiments on stress waves in planar trusses, ASCE Journal of Engineering Mechanics 124, 884 (1998). 13. K. A. Ingebrigtsen and A. Tonning, Elastic surface waves in crystal, Physical Reviews 184, 942 (1969). 14. J. Q. Jiang, Reverberation-Ray Analysis of Inhomogeneous Elastodynamic Equations and Nondestructive Inspection of Structures, Ph.D. Thesis (Zhejiang University, Hangzhou, 2008). (in Chinese) 15. J. Q. Jiang, Y. X. Bao and W. Q. Chen, Extension and application of reverberationray matrix method to dynamic responses of structures, Journal of Zhejiang University (Engineering Science) 43, 1063 (2009). (in Chinese) 16. J. Q. Jiang and W. Q. Chen, Reverberation-ray analysis of moving or distributive loads on a non-uniform elastic bar, Journal of Sound and Vibration 319, 320 (2009). 17. W. Magnus, On the exponential solution of differential equations for a linear operator, Communications on Pure and Applied Mathematics 7, 649 (1954). 18. E. C. Pestel and F. A. Leckie, Matrix Methods in Elasto Mechanics (McGraw-Hill, New York, 1963). 19. Y.-H. Pao and W. Q. Chen, Elastodynamic theory of framed structures and reverberation-ray matrix analysis, Acta Mechanica 204, 61 (2009).


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20. Y.-H. Pao, W. Q. Chen and X. Y. Su, The reverberation-ray matrix and transfer matrix analyses of unidirectional wave motion, Wave Motion 44, 419 (2007). 21. Y.-H. Pao, D. C. Keh and S. M. Howard, Dynamic response and wave propagation in plane trusses and frames, AIAA Journal 37, 594 (1999). 22. Y.-H. Pao, X. Y. Su and J. Y. Tian, Reverberation matrix method for propagation of sound in a multilayered liquid, Journal of Sound and Vibration 230, 743 (2000). 23. Y.-H. Pao and G. J. Sun, Dynamic bending strains in planar trusses with pinned or rigid joints, ASCE Journal of Engineering Mechanics 129, 324 (2003). 24. X. Y. Su, J. Y. Tian and Y.-H. Pao, Application of the reverberation-ray matrix to the propagation of elastic waves in a layered solid, International Journal of Solids and Structures 39, 5447 (2002). 25. J. L. Synge, Flux of energy for elastic waves in anisotropic media, Proceedings of the Royal Irish Academy 58, 13 (1956). 26. J. Tian and Z. Xie, A hybrid method for transient wave propagation in a multilayered solid, Journal of Sound and Vibration 325, 161 (2009). 27. J. Y. Tian and X. Y. Su, The transient waves in frame structures, Explosion Shock Waves 21, 98 (2001). (in Chinese) 28. J. Y. Tian and X. Y. Su, The vibration suppression of joint mass-damping in frame structures, Acta Mechanica Solida Sinica 23, 47 (2002). (in Chinese) 29. J. Y. Tian, W. X. Yang and X. Y. Su, Transient elastic waves in a transversely isotropic laminate impacted by axisymmetric load, Journal of Sound and Vibration 289, 94 (2006). 30. A. H. von Flotow, Disturbance propagation in structural networks, Journal of Sound and Vibration 106, 433 (1986). 31. F. W. Williams and W. H. Wittrick, An automatic computational procedure for calculating natural frequencies of skeletal structures, International Journal of Mechanical Sciences 12, 781 (1970). 32. W. Yan, W. Q. Chen, J. B. Cai and C. W. Lim, Quantitative structural damage detection using high frequency piezoelectric signatures via the reverberation matrix method, International Journal for Numerical Methods in Engineering 71, 505 (2007). 33. W. Yan, C. W. Lim, W. Q. Chen and J. B. Cai, A coupled approach for damage detection of framed structures using piezoelectric signature, Journal of Sound and Vibration 307, 802 (2007). 34. Y. Y. Yu, Y. X. Bao and Y. M. Chen, Damage detection of frame structures based on the reverberation matrix method, China Civil Engineering Journal 38, 53 (2005). (in Chinese) 35. Y. Y. Yu, Y. X. Bao and Y. M. Chen, Discussion on quality testing of frame structure embedded partially in soil, Acta Mechanica Sinica 38, 339 (2006). (in Chinese) 36. Y. Y. Yu and Y. M. Chen, Flexural wave of finite pile in a non-uniform soil, Acta Mechanica Solida Sinica 26, 429 (2005). (in Chinese)

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37. Y. Y. Yu, Y. H. Pao and Y. M. Chen, Wave analysis of damaged frame structures, Journal of Vibration Engineering 17, 20 (2004). (in Chinese) 38. J. Zhu, W. Q. Chen, G. R. Ye and C. F. Lü, Recursive formulations for the method of reverberation-ray matrix and the application, Science in. China (Series G) 52, 293 (2009). 39. O. C. Zienkiewicz, The finite element method: from intuition to generality, Applied Mechanics Reviews 23, 249 (1970).

CHAPTER 3 SURFACE-WAVE NONLINEARITY MEASURED WITH EMAT FOR FATIGUED STEELS

Hirotsugu Ogi* and Masahiko Hirao* *Graduate

School of Engineering Science, Osaka University Toyonaka, Osaka 560-8531, Japan E-mail: [email protected]

A nonlinear acoustic measurement technique is studied for fatigue damage monitoring. An electromagnetic acoustic transducer (EMAT) magnetostrictively couples to a surface-shear wave resonance along the circumference of a rod specimen during rotating bending fatigue of carbon steels. Excitation of the EMAT at the half of the resonance frequency caused the standing wave containing only the secondharmonic component, which was received by the same EMAT to determine the second-harmonic amplitude. Thus measured surfacewave nonlinearity always showed two distinct peaks approximately at 60% and 85% of the total fatigue life. We attribute the earlier peak to the crack nucleation and growth, and the later peak to an increase of free dislocations associated with crack extension in the final stage of life. This non-contact resonance-EMAT measurement can monitor the evolution of the surface-shear-wave nonlinearity throughout the metal’s fatigue life and is able to detect the pertinent precursors of the eventual failure.

1. Introduction Acoustic nonlinearity holds the potential of becoming the primary means of characterizing fatigue in materials, because it is capable of probing the process of crack nucleation and growth,1,10,11,21 and dislocation movement.2,4-6 The sensitivity to microstructural attributes during the incubation period is often higher than that of the linear properties 75

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(velocity and attenuation). Richardson21 analyzed the second-harmonic amplitude generated by the opening and closing effect of an interface subjected to a bias pressure with a passing acoustic wave. An optimum pressure exists for the second-harmonic generation. Buck et al.1 and Morris et al.10 demonstrated that Richardson’s model is basically applicable to fatigue cracks in metals; the second-harmonic amplitudes showed maxima with a low external compressive stress. Thus, small fatigue cracks act as generators of harmonics, and the efficiency should exhibit a maximum during crack growth because they are partly closed by compressive residual stress and large cracks are fully open, producing no higher harmonics. Concerning dislocations, Hikata et al.4 studied the harmonic generation caused by their anelastic vibration and showed that the second-harmonic amplitude increases with the increase of dislocation density and loop length. We recently revealed that a dislocation-structure change occurs in the later stages of rotating bending fatigue, during which much mobile dislocations temporally arise.13,17 These theoretical and experimental studies predict a nonlinearity peak responding to this event. We then expect two nonlinear peaks during a fatigue life; one associated with cracks and the other with dislocations. However, such a study has not been reported in the literature. We note two key points. First, surface waves should be used to focus on the specimen surfaces, where damage progresses. Second, a contactless acoustic transduction should be used to avoid background nonlinearity caused by coupling agents and the transducer itself. Among the previous nonlinear acoustic studies for the characterization of materials, few satisfied these requirements and none for detecting the fatigue damage. A notable exception is reported by Hurley8, who used a comb transducer for excitation and an interferometric laser detector to accurately measure Rayleigh wave harmonics in aluminum; it is a quasi-contactless technique. In the present study, we apply an electromagnetic acoustic transducer (EMAT) to monitoring the surface-shear-wave nonlinearity throughout the metal’s fatigue life. The use of an EMAT makes the

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77

contactless transduction possible. However, EMATs lack a large enough transduction efficiency, which the nonlinear measurement needs. To overcome this limitation, we choose using acoustic resonance so as to excite and enhance the standing wave of the second-harmonic component around a rod specimen. Coherent superposition produces a highly magnified amplitude of the second harmonics. Thus measured surface-wave nonlinearity detected the two peaks in the fatigue life for the first time, one caused by microcracking and the other by dislocations. 2. Materials We used commercial steel rods containing 0.25 and 0.35 mass percent carbon. They were heated at 880 °C for one hour and cooled in air. Specimens were 600 mm long. The diameter was 14 mm at the center; it smoothly increased from 14 mm to 20 mm at the end portions with a large curvature to bring about damage and failure at the minimum diameter. We prepared the measuring surface by electropolishing. 3. Resonance-EMAT for Nonlinear Acoustics 3.1. Spectroscopy Measurement The EMAT consists of a solenoid coil to introduce the bias magnetic field H0 in the specimen’s axial direction and a meander-line coil, having equal meandering period of 0.9 mm, to induce the dynamic field Hω in the circumference direction (see Fig. 1). When a sinusoidal current is applied to the meander-line coil, the total field Ht oscillates about the axial direction at the same frequency as the driving current and produces the shearing vibration through the magnetostriction effect to excite the surface shear wave propagating along the circumference with the axial polarization.16 This is called the axial shear wave.9 After the excitation, the same meander-line coil receives the axial shear wave through the reversed magnetostrictive effect.

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Fig. 1. Axial-shear-wave EMAT consisting of a solenoid coil and a meander-line coil surrounding the cylindrical surface. The magnetostriction mechanism causes the axially polarized surface SH wave in the cylindrical specimen.

Driving the meander-line coil with long tone bursts causes interference among the axial shear waves traveling around the cylindrical surface, and a frequency scan detects resonance peaks at unequal frequency intervals, at which all the waves overlap coherently to produce large amplitudes. We used the first resonance mode around fr=3.9 MHz, whose amplitude distribution has the maximum at the surface and a steep radial gradient; the penetration depth is estimated to be 0.5 mm.13 We defined the maximum amplitude of the first resonance peak as the fundamental amplitude, A1 (Fig. 2). We then excited the axial shear wave by driving the EMAT at the half of the resonance frequency ( fr /2), keeping the input power unchanged. In this case, the driving frequency does not satisfy the resonance condition and the fundamental component does not make a detectable signal. However, the second-harmonic component having the double frequency ( fr) satisfies the resonance condition, and the resonance spectrum of the received signal contains a peak at the original resonance frequency as shown in Fig. 2. We defined this peak height as the second-harmonic amplitude, A2, to calculate the nonlinearity A2/A1. The magnitude of A2/A1 varied in the order of 10-3. These measurements were made possible with the system for nonlinear acoustic phenomena (SNAP) manufactured by RITEC Inc., which provided the burst signals to the meander-line coil and detected the resonance peaks.


79

3.2. Second Power Law In metals without flaws, the higher harmonics arise from the nonlinear elasticity caused by lattice anharmonicity and the anelasticity due to dislocation movement. These two effects are inseparable in actual nonlinear measurements. Both generate the higher harmonics, among which the second harmonic usually dominates. The amplitude is then given by the square of the fundamental wave amplitude,4 that is, the second power law. The present definition of nonlinearity A2/A1 is based on the second power law and the proportionality between A1( fr) and the true fundamental amplitude, say A1’( fr/2). The second harmonic amplitude A2 is of course proportional to the square of the true fundamental amplitude A1’, that is, A2∝A1’2. This component lasts in a very short time after excitation and vanishes through mutual interference in the off-resonance condition. Within this short time, it generates the second harmonics following the second power law. But, we cannot measure A1’. Instead, we measured A1 and A2, both at fr, by changing the driving voltage. Figure 3 shows the result for 0.25 mass%C steel before and after applying 1.5% tensile plastic deformation. It demonstrates the linear relationship between A2 and A12, proving that A1∝A1’. (The larger slope after the deformation is caused by the dislocation multiplication. The slope is proportional to the dislocation density Λ times the fourth power of the loop length L; A2/A12 ∝ΛL4.5) Normalization of A2 in terms of A1 removes the influences of liftoff, frequency dependence of the transduction efficiency, and other anomalies. 3.3. Background Nonlinearity by Magnetostrictive Effect Magnetostrictively coupling EMAT entails background nonlinearity caused by the metal’s magnetostrictive response to the applied magnetic field. We have to estimate this effect. The preceding study15 derived a relationship between the bulk-shear-wave amplitude and the metal’s magnetostrictive characteristics by assuming isovolume magnetostriction and reversible magnetization rotation. Similar approach is possible in the present situation by assuming a flat surface. The resultant surface-shear-

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0.01

A1 0.008 0.006

A2

0.5

0.004 0.002

0

3.88

3.9

3.92

Amplitude (a.u.)

Amplitude (a.u.)

1

0 3.94

Frequency (MHz) Fig. 2. Resonance spectra for the fundamental and the second-harmonic components.

4 as annealed 1.5% deformed

A2 (a.u.)

3

2

1

0 0

0.5

1

A12 (a.u.)

1.5

Fig. 3. Verification of the second power law before and after plastic elongation. The second-harmonic amplitude is proportional to the square of the fundamental amplitude.

10-2

2

10-3

0

10-4

-2

10-5 0

2 4 Static Field (104 A/m)

81

A2/A1

measurement approximation A2/A1

4

-6

Magnetostriction (10 )


6

Fig. 4. Measured magnetostriction (solid circle), the fitted function (solid line), and the calculated nonlinearity in transmission (broken line). A 0.25 mass%C steel was used after demagnetization.

wave amplitude, USH, is related to the magnetostriction εM and its field derivative (dε M dH ) as U SH ∝

 jωt 3µ   dε M  2 2  sin θ cos θe ε M cos 2θ cos θ + 3H 0  H0   dH  

(1)

Here, µ denotes the shear modulus, θ=tan-1(Hω/H0) and Hω=hωexp( jωt) with the dynamic-field amplitude hω. When the static field H0 is much larger than the dynamic field Hω, i.e., │θ(t)│ 0.

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In order that p, so defined, solves the homogeneous wave equation for z = 0 and is causal, set ρ − ξ2 − η2 . ζ± = ±ζ, ζ = κ Of course, for the plane wave defined by (ξ, η, ζ) to propagate, the argument of the square root must be positive, meaning that the plane wave propagation direction makes a nonzero angle with z = 0. The pressure field must be a generalized solution of the wave equation, since the right hand side in (5.3) is singular. Applying the definition of weak solution [Lax (2006)] and making the standard integration-by-parts argument using smooth test functions, the connection between P± and F follows: 1 F (t). (5.8) P+ (t) = P− (t) = 2ζ Plane waves have infinite energy, of course, but the energy per unit volume is finite if the waveform is square-integrable. It follows from (5.8) that the energy of the plane wave acoustic field in a spatial volume at time T is proportional to the mean-square of the derivative of F over a related volume and time < T . The presence of ζ in the denominator of (5.8) implies that in general the energy density in the plane wave field is not bounded by the mean-square of the source waveform F (t), as ζ may be arbitrarily small (positive). As it turns out, neither does the energy of the field bound the mean-square of the source. Examples illustrating the failure of any such bounds may be found in Symes (1983). However, the anomalously large or small fields involve high-frequency pulses propagating near (“grazing”) z = 0, that is, to phenomena confined to a small zone around the source support plane. In contrast, the energy of the solution over any finite subvolume of {x : z ≥ zmin > 0} for any positive minimum depth zmin is bounded by a fixed multiple of the source mean-square. Even with this restriction to wave energy propagating into the region |z| > 0, however, the energy contributed to the acoustic field per unit volume for a unit mean square source field f (t, x, y) should be larger for f that are rich in plane wave components with high angle of incidence (small ζ). This effect will be evident in the examples to come. Further insight into the nature of Λ may be gained through study of the map from data to field at the (final) time T: set Γ[κ, ρ]f (x) = (p(x, T ), v(x, T )).

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Given a scalar field φ(x) and a vector field ψ(x), define (q(x, t), w(x, t)) to be the dual acoustic field satisfying 1 ∂q = −∇ · w, κ ∂t ∂w = −∇q, ρ ∂t q(T, x) = φ(x),

(5.9) (5.10) (5.11)

w(T, x) = ψ(x).

(5.12)

Compare Γf with (φ, ψ) using the bilinear form associated with the energy: 1 Γf, (φ, ψ)E ≡ dx pq + ρv · w (T, x) (5.13) κ

T

=

d dx dt

dt −∞

T

=

dt

dx

−∞

+ =

∂v 1 ∂p q+ρ ·w κ ∂t ∂t

dt

dx

−∞

+ =

dt

dx

−∞

+

−∞

1 ∂p q − ∇p · w κ ∂t

1 ∂p q + p∇ · w κ ∂t

dt

+

(t, x)

(t, x)

(t, x)

1 ∂q p + q∇ · v (t, x) κ ∂t

T

=

1 ∂q p − ∇q · v (t, x) κ ∂t

T

1 pq + ρv · w (t, x) κ

1 ∂q ∂w p+ρ · v (t, x) κ ∂t ∂t

T

dx

1 ∂p +∇·v q κ ∂t

1 ∂q − ∇ · w p (t, x) κ ∂t

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T

=

dt

dx (f (t, x, y)δ(z)) q(x, y, z)

−∞

T

dt

=

dx

dyf (t, x, y)q(t, x, y, 0).

−∞

It follows that the formal adjoint of Γ (with respect to the energy as metric for the acoustic Cauchy data (output), and the RMS metric for the input (RHS data), is given by ΓT [κ, ρ](φ, ψ) = q(t, x, y, 0), 0 ≤ t ≤ T. Suppose (φ, ψ) = (p, v)t=T = Γ[κ, ρ]f . Then the uniqueness theorem for the wave equation Cauchy problem [Lax (2006)] implies that actually q = p in the time range 0 ≤ t ≤ T , from which we deduce that Λ[κ, ρ] = ΓT [κ, ρ]Γ[κ, ρ].

(5.14)

Thus Λ is self-adjoint and positive (semi-)definite. The analysis presented in [Symes (1983)] implies that Λ has arbitrarily large and small positive eigenvalues. However, the very large and very small eigenvalues are related to solutions which vanish away from z = 0. Therefore, if two acoustic models κ, ρ and κ0 , ρ0 are identical near z = 0, then the eigenvectors corresponding to very large and very small eigenvalues of Λ[κ, ρ] and Λ[κ0 , ρ0 ] should be the same, and should combine to produce near-zero eigenvalues of ∆[κ, ρ, κ0 , ρ0 ]. That is, these extreme source effects should not affect the size of the operator ∆, which can be regarded as a bounded self-adjoint operator. A natural measure of size of ∆ is its operator norm, as mentioned in the Introduction, that is, the least upper bound of the absolute value of its spectrum. 5.3. Numerical Example The power method is the simplest method for estimating an approximate eigenpair f, λ approximating the spectral upper bound. The power method applies to bounded self-adjoint positive (semi)definite operators, whereas the difference operator ∆ is only bounded self-adjoint. Therefore I apply the power method to ∆2 , which is positive semidefinite. To update a current

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estimate f, λ to an improved estimate f + , λ+ , g = ∆f +

λ f

+

f+

T

(5.15)

= g g

(5.16)

= ∆g

(5.17)

← f + / (f + )T f +

(5.18)

The last step normalizes the eigenvector estimate f + to have unit norm. To make this prescription into an iterative algorithm for approximating the spectral upper bound, replace f, λ with f + , λ+ and repeat until the increase λ+ − λ falls below a user-chosen tolerance, or a fixed number of iterations are accomplished. Implementation of this algorithm requires an approximate solver for the acoustic wave system (5.1,5.2). I have used IWAVE, a staggered-grid finite difference solver developed in my group, which supports high (up to 14th) order difference approximation and parallel execution, amongst other features [Terentyev (2009)]. For the small 2D example described here, 4th order spatial difference approximation and serial execution sufficed. Implementation of algorithms such as the power method also requires an algebraic realization of simulators such as IWAVE in the form of operators, with all of the features that operators (linear, in this case) commonly possess. I have employed the operator interfaces defined by the Rice Vector Library [Padula et al. (2009)], via wrapper code which my group also uses to construct various inversion algorithms [Sun and Symes (2010a,b)]. Figures 5.1 and 5.2 show bulk modulus (κ) and density (ρ) fields used in this example. The values of density are typical of shallow sediments overlain by a water layer. The reference bulk modulus and density were constants approximating those of sea water (κ0 = 2250 MPa, ρ0 = 1000 kg/m3 ). The bulk modulus is actually computed as κ = ρv2 , where v ≡ 1.5km/s is a homogenous “water” velocity - this accounts for the very similar appearance of κ and ρ. That is, the wave velocities in the target and reference materials are the same. This choice produces an easier material parameter estimation problem, as the wave velocity is by far the most difficult parameter to estimate. Even though this paper will not address this material parameter inverse problem directly, the intent of the work reported here is to produce good selections of sources for this inverse problem, so the example is set up to produce a relatively easy instance. The domain depicted in the figures is the computational domain used in this example. Absorbing boundary conditions of PML type [Cohen (2002)]

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0

x (km) 4

2

6

z (km)

0

1

3000

Fig. 5.1.

4000 MPa

5000

Bulk modulus model used in numerical example.

0

2

x (km) 4

6

z (km)

0

1

1000

Fig. 5.2.

1500 kg/m^3

2000

Density model used in numerical examples.

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simulate a full (2D) Euclidean space, consistent with the theory. The source plane is z = 40 m, in the coordinate system of the figures. The distributed source is represented by a grid function in the horizontal direction with the same sampling (20 m) as the fields. Data were simulated and recorded for 2 s, at a sample rate of 2 ms. The initial source distribution is a tapered plane wave at normal incidence, depicted in Figure 5.3. The Rayleigh quotient for this source is 173.9 (dimensionless, as it is a ratio of norms of similar vectors). After 20 iterations of the power method, the Rayleigh quotient had grown to 3785.5, with increase of well under 1% for the last five iterations. Thus this final Rayleigh quotient can be taken as a reasonably accurate estimate of the operator norm of ∆[κ, ρ, κ0 , ρ0 ]2 . Clearly the initial source estimate is far from an eigenvector with maximum eigenvalue. The final optimized source f is displayed in Figure 5.4. Note the very high angle waves which dominate this source. The output, that is ∆f , is displayed in Figure 5.5. High angle waves also predominate in this display. As noted before, it is possible to confine an oscillating pulse near the surface with arbitrarily large Rayleigh quotient, but such solutions would be identical for both the target (κ, ρ) and reference (κ0 , ρ0 ) models, hence would cancel to yield a near-null vector for the difference operator ∆. Instead,

x (km)

t (s)

0

0

2

4

6

1

2 Fig. 5.3. Initial truncated plane wave source used in numerical example. Waveform is trapezoidal bandpass filter with corner frequencies 2.5, 5, 12.5, and 20 Hz, tapered to zero at the edges of the domain with a cosine shoulder.

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x (km)

t (s)

0

0

2

4

6

1

2 Fig. 5.4. Source estimate, approximating eigenvector with largest eigenvalue of ∆2 , produced by 20 iterations of the power method starting with the truncated plane wave source of Figure 5.4. Note dominance of high-angle waves - these are the traces on z = 40 of nearly horizontally propagating waves.

x (km)

t (s)

0

0

2

4

6

1

2 Fig. 5.5. Output of ∆ applied to the optimized source of Figure 5.4, that is, the difference of responses between (κ, ρ) (Figures 5.1, 5.2) and the homogeneous background model. Displays high-angle energy, propagating just far enough into z > 0 to interrogate the water bottom with maximum energy.

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the data of Figure 5.5 is that of a solution propagating nearly horizontally but penetrating the half-space z > 0 sufficiently to interact with the largest feature differentiating the two models, that is, the water bottom at depth 400 m. It is somewhat easier to appreciate the wave geometry of this example through examination of time slices of the field. Figures 5.6 and 5.7 show time slices of the pressure field produced by the optimized source (Figure 5.4) in the homogeneous reference model, at 1 s and 2 s. It is clear that most of the energy in the acoustic field is involved in near-horizontal propagation, but that the angle is steep enough to cause a substantial reflection from the water bottom interface. To assess the impact of this emphasis on high-angle waves on inversion for the material parameters, it is natural to examine the gradient of the least-squares objective (in 2D) (κtrial , ρtrial ) →

T

dt 0

dx|Λ[κtrial , ρtrial ]f − d|2 .

This gradient is known as the reverse-time migration image in reflection seismology, and may be regarded as a scaled, bandlimited image of the model difference [Tarantola (1987); Symes (2008)]. I take d = Λ[κ, ρ]f as the data created by the target model (Figures 5.1, 5.2). Figure 5.8 shows the bulk modulus component of the resulting gradient at κtrial = κ0 , ρtrial = ρ0 for the plane wave source (Figure 5.3). The first interface (water bottom) is correctly positioned, as the sound velocity in the water layer is correct in the reference model. Deeper features are also correct in overall shape, though mispositioned due to the incorrect velocity in the reference model below the water bottom. The gradient for the optimized source, while almost three orders of magnitude larger, is visually much less appealing (Figure 5.9). It does provide a bandlimited image of the central part of the water bottom, hence has captured some of the difference between the two models. However it does not possess the aperture of the initial source image. The reduced bandwidth evident in the figure is easy to understand: the horizontal interface has been interrogated with high-angle waves, which it “sees” as being lower in frequency content as a consequence of their larger vertical velocity. It is not possible to dismiss this direction in model space as unconstructive, but it is far from obvious that the optimized source will lead to more rapidly convergent inversion iterations.

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0

2

x (km) 4

6

z (km)

0

1

Fig. 5.6. Pressure field response to optimized source (Figure 5.4), propagating in homogeneous model (κ0 , ρ0 ), at t = 1 s.

0

2

x (km) 4

6

z (km)

0

1

Fig. 5.7. Pressure field response to optimized source (Figure 5.4), propagating in homogeneous model (κ0 , ρ0 ), at t = 2 s.

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z (km)

0

1

Fig. 5.8. Bulk modulus component of least-squares gradient (also known as reverse time migration image) at homogeneous reference model (κ0 , ρ0 ), for initial source estimate (Figure 5.3). This field represents the bulk modulus perturbation that most rapidly increases the least squares difference between the pressure trace responses to current trial estimate (κ0 , ρ0 ) and the target model (κ, ρ). As is evident, it is also an image of the zones of rapid change in the model.

0

2

x (km) 4

6

z (km)

0

1

Fig. 5.9. Bulk modulus component of least squares gradient at homogeneous reference model, for optimized source (Figure 5.4) produced by 20 iterations of the power method.

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5.4. Conclusion In the preceding pages, I have explained a translation of Isaacson’s distinguishability algorithm to the reflected wave problem, and explored its properties. The emphasis on high-angle energy, both predicted by the theory and evident in the numerical experiments, appears unlikely to be appropriate for application to waveform inversion. The sources produced by approximating the largest eigenvalue of the difference simulation operator (∆ above) certainly produce the largest differences between responses per unit input energy, but do not seem to result in particularly good material parameter updates. At least one obvious avenue for improvement of this algorithm is evident. The phrase “per unit energy” is the key: the mean square of the distributed source field f (t, x, y) is not an energy, nor does it correspond directly to the actual energy in the field. A natural avenue for improving the correspondence between “best” source and useful interrogation of the acoustic medium may be the use of the field energy as a source metric. Within the power method, this simply amounts to a redefinition of the source normalization. A straightforward way to implement this renormalization of the difference operator may be via the identity (5.14) linking the simulation operator Λ to the “final state” operator Γ. As discussed in the theory section, the (in principle unboundedly) large eigenvalues of Λ, or singular values of Γ, correspond to solutions which do not significantly leave the plane of the source. Thus insertion of a window function wR in Γ, restricting attention to a zone R contained in z > 0, converts Γ into a bounded operator. If the “energy” of the source is measured as actual energy in R: 2 1 T p + ρv · v (T, x) dt dx wR (x) ER (f )2 = 2 0 κ then the tendency of high angle energy to generate bigger fields is automatically compensated, and moreover the source selection is focused on producing large fields in the window region R. A further investigation along these lines is in progress. Acknowledgments I am grateful to the sponsors of The Rice Inversion Project for their longterm support of my research, and to the organizers of the International

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Symposium on Engineering Mechanics for the opportunity to present my evolving work on source synthesis. My understanding of the theoretical basis of this subject relies in the work I performed in the early 1980’s as a member of Prof. Y.-H. Pao’s research group at Cornell University. My appreciation of the central importance of source mechanisms in everything having to do with wave motion dates from my time with Prof. Pao and reflects his strong influence on my work. References Beasley, C. J., Chambers, R. E. and Jiang, Z. (1998). An new look at simultaneous sources, in Expanded Abstracts (Society of Exploration Geophysicists), pp. 133–135. Berkhout, G. (2008). Changing the mindset in seismic data acquisition, The Leading Edge 27, pp. 924–938. Blacquiere, G., Berkhout, G. and Verschuur, E. (2009). Survey design for blended acquisition, in Expanded Abstracts (Society of Exploration Geophysicists), pp. 56–61. Borcea, L., Papanicolaou, G. and Tsogka, C. (2007). Optimal waveform design for array imaging, Inverse Problems 23, pp. 1973–2020. Cohen, G. C. (2002). Higher Order Numerical Methods for Transient Wave Equations (Springer, New York). Diebold, J. and Stoffa, P. (1981). The traveltime equation, tau-p mapping, and inversion of common midpoint data, Geophysics 46, pp. 238–254. Fink, M. and Prada, C. (2001). Acoustic time-reversal mirrors, Inverse Problems 17, pp. R1–R38. Isaacson, D. (1987). Distinguishability of conductivities by electric current computed tomography, IEEE Transactions on Medical Imaging MI-5, pp. 91– 95. Krebs, J. R., Anderson, J. E., Hinkley, D., Neelamani, R., Lee, S., Baumstein, A. and Lacasse, M.-D. (2009). Fast full-waveform seismic inversion using encoded sources, Geophysics 74, pp. WCC177–WCC188. Lax, P. D. (2006). Hyperbolic Partial Differential Equations (Courant Lecture Notes) (American Mathematical Society, Providence, RI). Padula, A. D., Symes, W. W. and Scott, S. D. (2009). A software framework for the abstract expression of coordinate-free linear algebra and optimization algorithms, ACM Transactions on Mathematical Software 36, pp. 8:1–8:36. Romero, A., Ghiglia, D. C., Ober, C. C. and Morton, S. A. (2000). Phase encoding of shot records in prestack migration, Geophysics 65, pp. 426–436. Sun, D. and Symes, W. W. (2010a). IWAVE implementation of Born simulation, Tech. Rep. 10-05, Department of Computational and Applied Mathematics, Rice University, Houston, Texas, USA, URL http://www.caam.rice.edu. Sun, D. and Symes, W. W. (2010b). IWAVE implementation of the adjoint state method, Tech. Rep. 10-06, Department of Computational and Ap-

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plied Mathematics, Rice University, Houston, Texas, USA, URL http: //www.caam.rice.edu. Symes, W. W. (1983). Trace theorem for solutions of the wave equation and the remote determination of acoustic sources, Math. Methods in Appl. Sciences 5, pp. 131–152. Symes, W. W. (2008). Approximate linearized inversion by optimal scaling of prestack depth migration, Geophysics 73, pp. R23–35. Tarantola, A. (1987). Inverse Problem Theory (Elsevier). Terentyev, I. (2009). A software framework for finite difference simulation, Tech. Rep. 09-07, Department of Computational and Applied Mathematics, Rice University, Houston, Texas, USA, URL http://www.caam.rice.edu. Treitel, S., Gutowski, P. and Wagner, D. (1982). Plane-wave decomposition of seismograms, Geophysics 47, pp. 1375–1401. Verschuur, D. J. and Berkhout, G. (2009). Target-oriente least-squares imaging of blended data, in Expanded Abstracts (Society of Exploration Geophysicists), pp. 2889–2894.

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CHAPTER 6 AN INTRODUCTION TO AN ADAPTIVE DATA ANALYSIS METHOD

Norden E. Huang Research Center for Adaptive Data Analysis National Central University Chungli, Taiwan, ROC E-mail: [email protected] The existing methods of data analysis, either the probability theory or the spectral analysis, are all developed by mathematicians or based on their rigorous mathematical rules. For analyzing data from the real physical world, we have to face the reality of nonstationarity and nonlinearity in the processes. The traditional analysis methods are based on rigorous approach, but cannot fully accommodate these conditions. A new adaptive data analysis was introduced by Huang et al (1998), which was designated by NASA as the Hilbert-Huang Transform (HHT). It consists of Empirical Mode Decomposition (EMD) and the Hilbert Spectral Analysis (HSA) methods; both were introduced recently by Huang et al. (1996, 1998, 1999 and 2003). The method is adaptive, and specifically designed for analyzing data from nonlinear and nonstationary processes. Since its introduction over ten years ago, the HHT has been applied to a wide range of applications, covering (among many others) biology, geophysics, ocean research, engineering, radar and medicine (see, for example, Huang and Attok-Okine, 2005; Huang and Shen, 2005; Huang and Wu, 2008). Yet, up to this time, a rigorous mathematical foundation is still lacking. Under this condition, progresses are still empirically based. An introduction and some of the recent advances on Ensemble Empirical Mode Decomposition (Wu and Huang, 2009), Instantaneous frequency and trend computations, Time-dependent Intrinsic Correlation (Chen,

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2010) and its extension to multi-dimensional data (Wu et al. 2009) are summarized and briefly discussed. These advances have made the HHT method much more robust and mature.

1. Introduction Data analysis is indispensable to every science and engineering endeavor. In fact collecting and analyzing data; synthesizing and theorizing on the analyzed results are the core activities of scientific studies. As a result, data analyzing is a key link in this continuous loop of these activities. As Poincaré had famously put it: Science is built up of facts, as a house is built of stones; but an accumulation of facts is no more a science than a heap of stones is a house. Here ‘facts’ are in fact data. Yet, just data are still useless. Undigested or un-analyzed data are just like a heap of stones. To convert the facts as part of the edifice of science, we have to analyze, synthesize and even theorize on the data. The logic of this argument seems so plain, yet our present practice is anything but logical. Instead of analyzing data, we process them. There are fundamental differences between data processing and data analysis. All the data processing methods are based on well established algorithms, which are developed through rigorous mathematics approaches. To be rigorous, we are forced to idealize the real physical conditions consummating with the mathematical assumptions used in establishing the algorithms. Unfortunately, the physical conditions are never ideal. Indeed, Einstein had once stated: ‘As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.’ In pursuing mathematical rigor, we are forced to live in a pseudo-real world, in which all processes are idealized to conform to mathematician’s

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conditions and requirements. There is always a conflict between reality and the ideal world. The most prevailing assumptions in the data processing algorithms are linearity and stationarity. All the statistical and probabilistic studies are based on these assumptions. Yet, most physical processes, either in sciences or engineering studies, are both nonstationary and nonlinear. As research becomes increasingly sophisticated, the inadequacies of the traditional data analysis methods based on the linear and stationary assumptions have become glaringly clear; consequently, the algorithms have become woefully inadequate. The only viable way is to break away from the traditional limitations and establish a new paradigm in the data analysis method. Take the powerful spectral analysis as an example. The foundation of the traditional spectral analysis methods were built on a priori bases, be it Fourier or wavelet analyses. Indeed an a priori basis is needed for a rigorous analysis. Unfortunately, the a priori basis would exact a heavy toll on the methodology: Once the a priori basis is chosen, the results produced from the processing would be limited to a simple convolution computation or an integral transform. The consequences are the uncertainty on time-frequency resolutions and spurious harmonics. To overcome these inherited limitations, we have to explore new approaches, for example to use adaptive basis; i.e., with the basis founded on and derived from the data. This approach is still in its infancy. The general mathematical frame work has not been established. For lack of rigorous mathematical foundation, this approach is operated empirically. Yet some preliminary results had amply demonstrated the usefulness of this approach to analyze data from nonlinear and nonstationary processes. In this paper, an empirically based data analysis will be introduced. Some simple applications will be presented also, which will be followed by a general discussion and a sketch for the future developments. 2. The Hilbert-Huang Transform The Hilbert-Huang Transform (HHT) is the NASA designated name for the combination of Empirical Mode Decomposition (EMD) and Hilbert Spectral Analysis (HSA) methods developed recently by Huang et al.

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(1996, 1998, 1999, 2003 and 2010) and Wu and Huang (2009). The method is adaptive, and specifically for analyzing data from nonlinear and nonstationary processes. The motivations for HHT are obvious: First, the natural physical processes are mostly nonlinear and nonstationary. The available methods are either for linear but nonstationary (such as the wavelet analysis, Wagner-Ville, and various short-time Fourier spectrograms as summarized by Priestley (1988), Cohen (1995), Daubechies (1992) and Flandrin (1999)), or nonlinear but stationary and statistically deterministic processes (such as the various phase plane representations and time-delayed imbedded methods as summarized by Tong (1990), Diks (1997) and Kantz and Schreiber (1997)). Even on the more recent treatment on nonlinear time series analysis by Fan and Yao (2005), the treatment is limited to deal with stochastic properties of the time series when they were consisted of various ARIMA models. Second, the nonlinear processes need special treatment. Other than periodicity, we want to learn the detailed dynamics in the processes from the data. One of the typical characteristics of nonlinear processes, proposed by Huang et al. (1998), is the intra-wave frequency modulation, which indicates that the instantaneous frequency would change within one oscillation cycle. Let us examine a very simple nonlinear system given by the non-dissipative Duffing equation as d2x + x + ε x 3 = γ cos ω t . dt 2

(1)

This equation could be re-written as d2x + x 1 + ε x 2 = γ cos ω t . dt 2

(

)

(2)

Then the quantity within the parenthesis in equation (2) can be regarded as a variable spring constant, or a variable pendulum length. With this view, we can see that the frequency should be changing from location to location, and time to time, even within one oscillation cycle. Indeed with ε set at -1, the virtue pendulum would have its length shortening as it swings away from the central equilibrium position. Therefore, the period would be longer near x = 0, and shorter near x at


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Fig. 1. The solution of Duffing equation with ε = -1.

extrema positions. The numerical solution for the equation is shown in Figure 1. As pointed out by Huang et al. (1998), this intra-frequency frequency variation is the hallmark of nonlinear systems. In the past, the best we could do with these distorted wave data was to represent it with harmonics; hence the term harmonic distortion. Harmonic distortion is the result of perturbation analysis, which is obtained by imposing a linear structure on a nonlinear system. Those harmonics only have mathematical meaning: to fit the profiles of a distorted wave; they have no physical meaning. Take deep water surface waves for example. The solution based on small perturbation was derived by Stokes in 1847 with many harmonic terms. As the solution was based on the assumption of a wave with permanent form, all the harmonics must propagate at the same phase velocity as the fundamental, an impossible physical condition for the harmonics to meet, for water waves are dispersive: the wave speed is a function of frequency or wave number. Therefore, physically, the higher harmonics should lag behind the fundamental. The physically meaningful way to describe the system should be in terms of the instantaneous frequency as discussed by Huang et al. (1998, 2009). To illustrate the effect of instantaneous frequency, let us consider the mathematical model as in Huang (1998):

x (t ) = cos (ω t + ε sin 2ω t ) .

(3)

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Fig. 2. The wave profile for the mathematical model with ε = 0.3.

For ε set at 0.3, we would have the oscillation given in Figure 2. It has been shown by Huang et al (1998) that the wave form given in Equation (3) is equivalent to

ε  ε x (t ) ≈ 1 −  cos ω t + cos 3ω t + ... ,  2 2

(4)

while Equation (3) gives an instantaneous frequency of

Ω (t ) = ω (1 + 2ε cos 2ω t ) .

(5)

Equation (4) gives a series of spurious harmonics. Indeed the pointed crests and troughs of the waves in Figure 2 clearly representing the instantaneous frequency fluctuation. As the instantaneous frequency is defined through a derivative, it is strictly local and can be used to describe the detailed variation of frequency, including the intra-wave frequency variation. The easiest way to compute the instantaneous frequency is by the Hilbert transform. However, as discussed by Huang et al (2009),


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computing instantaneous frequency through the Hilbert Transform is limited by the Bedrosian (1963) and Nuttall (1966) theorems. As a result, it can only give an approximation. There are many other methods that could give the exact instantaneous frequency. The first is the quadrature method (Huang et al, 2009), which consists of an empirical AM-FM signal separation, such that x (t ) ; x (t ) = a (t ) cos θ (t ) ⇒ cos θ (t ) = (6) a (t ) therefore,

a 2 (t ) − x 2 ( t ) ; a( t )

sin θ (t ) = 1 − cos 2 θ (t ) =

(7) a ( t ) − x (t ) . x( t ) 2

⇒ θ (t ) = arc tan

2

With the phase function given, the instantaneous frequency could be computed easily. An alternative to this approach is proposed by Wu and Huang (2009) by examining the following equation: cos θ i +1 − cos θ i −1 cos(θ i + ∆θ i ) − cos (θ i − ∆θ i ) = sin θ i sin θ i =

[ cos θ i cos ∆θ i − sin θ i sin ∆θ i ] − [ cos θ i cos ∆θ i + sin θ i sin ∆θ i ] sin θ i

(8)

= 2 sin ∆θ i ≈ 2 ∆θ i . Once the ∆θi as a function of time is know, the instantaneous frequency can be computed by dividing it by ∆t. Still another approach is proposed by Hou et al (2009). In this approach, no normalization is required. Let the data be x(t). Using Taylor’s expansion, we can write

x ( t j +1 ) = ( a (t j ) + ∆ t a' (t j ) ) cos (θ (t j ) + ∆ t θ ' (t j ) ) + ... ; x ( t j −1 ) = ( a (t j ) − ∆ t a' (t j ) ) cos (θ (t j ) − ∆t θ ' (t j ) ) + ... . Therefore, we have x (t j +1 ) + x (t j −1 ) = cos ( ∆ t θ ' ( t j ) ) + o( ∆ t ) . 2 x(t j )

(9)

(10)

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Thus, we should have the instantaneous frequency as the derivative of the phase function given as

θ ' (t j ) =

x (t j +1 ) + x (t j −1 ) 1 arccos . 2 x (t j ) ∆t

(11)

No matter which method, we have to reduce the function to the Intrinsic Mode Function (IMF), defined as any function having the same numbers of extrema and zero-crossings or at most differed by 1; moreover, the function should have symmetric envelopes defined by all the maxima and minima separately. To reduce an arbitrary function as a sum of IMFs calls for the Empirical Mode Decomposition (EMD) method, an adaptive decomposition method proposed by Huang et al (1998, 1999). The EMD is implemented through the following steps: For any data, we first identify all the local extrema, and then connect all the local maxima by a cubic spline line as the upper envelope. Then, we repeat the procedure for the local minima to produce the lower envelope. The upper and lower envelopes should cover all the data between them. Their mean is designated as m1,1, and the difference between the data and m1,1 is the first proto-IMF (PIMF) component, h1,1:

x (t ) − m1,1 = h1,1 .

(12)

By construction, this PIMF, h1,1, should satisfy the definition of an IMF, but the change of its reference frame from a rectangular to a curvilinear coordinate can cause anomalies, where multi-extrema between successive zero-crossings could still exist, mostly due to some inflection points having become local extrema. To eliminate such anomalies, the sifting process has to be repeated as many times as necessary to get rid of the riding waves. In the subsequent sifting process steps, h1,1 is treated as the data. With the iteration of the steps, we have h1,1 ( t ) − m1,2 ( t ) = h1,2 (t ) ;

⋅⋅⋅ ⋅⋅⋅ h1,k −1 (t ) − m1,k (t ) = h1,k (t ) ; ⇒ h1,k (t ) = c1 (t ) ,

(13)


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After each repetition, we will check the results against a ‘stoppage criterion.’ Various criteria had been proposed by Huang et al (1998, 2003), Flandrin (2004) and Wu and Huang (2004), for example. Whenever the stoppage criterion is satisfied, the PIMF is designated as an IMF, c1(t). The ‘stoppage criterion’ actually determines the number of sifting steps to produce an IMF; it is thus of critical importance in a successful implementation of the EMD method. Furthermore, the spline function used is also of critical importance, for x (t ) − m1,1 ( t ) = h1,1 ( t ) ; h1,2 ( t ) = h1,1 ( t ) − m1,2 ( t ) = x ( t ) − ( m1,1 + m1,2 ) ; ⋅⋅⋅ ⋅⋅⋅ h1,k (t ) = h1,k −1 (t ) − m1,k (t ) = x (t ) − ( m1,1 + m1,2 + ... + m1,k ) ;

(14)

⇒ c1 (t ) = x (t ) − ( m1,1 + m1,2 + ... + m1,k ) . Thus, c1(t) is the result of the data minus a sum of spline functions. From Equation (14), we can see that if we subtract the first IMF, c1(t), from the data, the residual is the sum of spline functions. Mathematically, we have

x (t ) − c1 (t ) = r1 (t ) .

(15)

Therefore, the residual, r1(t), contains all the remaining information in the data other than the first IMF, c1(t). We can, therefore, repeat the above operation using r1(t) as the next set of data, and re-iterate the process, we should have r1 ( t ) − c 2 ( t ) = r2 ( t ) ; r2 (t ) − c 3 ( t ) = r3 ( t ) ; ⋅⋅⋅ ⋅⋅⋅

(16)

rn −1 (t ) − c n ( t ) = rn (t ) ; n

⇒ x (t ) −

∑ c (t ) = r ( t ) . j

n

j =1

By summing up Equations (15) and (16), we finally obtain the last equation in (16), which means that the data could be decomposed into a

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sum of IMFs and a residue, rn(t), which can be either a constant, a monotonic mean trend or a curve having only one extremum. Thus, we have succeeded in expanding the given function in terms of IMFs. With this operation, it is easily shown that k1

ci (t ) = ri −1 (t ) − ri (t ) =

∑m

i −1, j

j =1

k2

−

∑m

i, j

(17)

j =1

Thus all IMFs are combinations of spline functions. Therefore, the EMD is critically dependent on the spline function selected. Recent studies by Flandrin et al. (2004) and Wu and Huang (2004) have established that the EMD is a dyadic filter, and it is equivalent to an adaptive wavelet. Being adaptive, we have avoided the pitfalls of using an a priori-defined basis, and also avoided the spurious harmonics that would have resulted, had the a priori basis been employed. As the representation is dyadic, therefore, the maximum number of the IMFs is limited by log2N, with N as the total number of data points; thus, this expansion is a sparse representation. The components of the EMD are usually physically meaningful, if there is no scale mixing, defined as mixed characteristic scales in a single IMF component. To avoid the scale mixing, Wu and Huang (2009) had proposed an Ensemble EMD (EEMD), which is essentially the same EMD procedure, except that the procedure will be repeated n times each with a different white noise added to the data. The procedures are these: a. add a white noise series to the targeted data; b. decompose the data with added white noise into IMFs; c. repeat step 1 and step 2 again and again, but with different white noise series each time; and d. obtain the (ensemble) means of corresponding IMFs of the decompositions as the final result. The critical concept advanced in the EEMD is based on the following observations: a. A collection of white noise will cancel each other out in a time domain ensemble mean; therefore, only the signal can survive and persist in the final noise added signal ensemble mean.


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b. Finite, not infinitesimal, amplitude white noise is necessary to force the ensemble to exhaust all possible solutions; the finite magnitude noise will make the different scale signals to reside in the corresponding IMF dictated by the dyadic filter banks, and render the resulting ensemble mean more meaningful. c. The true and physically meaningful answer of the EMD is not the set without noise; it should be the ensemble mean of a large number of trials consisted of the noise added signal. This new method proposed in EEMD has utilized all these important statistical characteristics of noise. The critical conclusion here is that the final IMFs obtained is not the one with zero noise, but should be the ensemble mean of infinite many trials each with different perturbation of finite amplitude of noise. The EEMD indeed represents a major improvement over the original EMD. EEMD has fully utilized the statistical characteristics of noise assisted data analysis. As the level of the added noise is not of critical importance, as long as it is of finite amplitude to enable a fair ensemble of all the possibilities, the EEMD can be used without any subjective intervention; thus, it provides a truly adaptive data analysis method. By eliminating the problem of mode mixing, it also produces a set of IMFs that bears the full physical meaning for the signal, and a time-frequency distribution without transition gaps. EMD, with the Ensemble approach, has become less adaptive due to the fixed dyadic window dedicated by the added noise. Yet, the EEMD has become a more robust and mature tool for nonlinear and nonstationary time series analysis.

3. A Significant Product of HHT: The Determination of Trend Other than the decomposition, HHT could help in the determination of the trend and detrending the data set, which are important steps in data analysis. Yet definitions of “trend” tend to be imprecise, and no logical algorithm has been established for extracting the trend; it is similar in determining the regression of a data set (Fan and Yao, 2005). As a result, various ad hoc extrinsic methods have been used to determine the trend,

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or to effect a detrend operation. Such a confused state prompt Stock and Watson (1988) to lament: “one economist’s ‘trend’ can be another’s ‘cycle’.” A new procedure to determine trend based on HHT was proposed by Wu et al. (2007). They first gave the trend a precise definition as: Within the given data span, the trend is an intrinsically fitted monotonic function, or a function in which there can be at most one extremum. This definition is identical to the definition of the residue used in the Empirical Mode Decomposition method (Huang et al., 1996, 1998, 2003), where the residue had been treated as the overall trend. But the concept of the trend given here is far more general. Before proceeding to the details, there are several subtle, but important, points to be clarified. First, the trend should be an intrinsic property of the data; it should be part of the data, and driven by the same mechanisms that generate the observed or measured data. Most of the available methods, however, define trend by using an extrinsic approach, such as using pre-selected simple functional forms. Being intrinsic, therefore, requires that the method used in defining the trend must be adaptive, so that the trend extracted is derived from and based on the data. Second, the trend exists only within a given data span; therefore, it should be local and should be associated with a local length scale in terms of data length. Consequently, the trend can only be valid within that part of the data, which should be shorter than a full local wavelength. If there were two or more extrema of the same sense, the data would share that characteristic with a cycle. Thus the difficulty expressed by Stock and Watson (1988) could be resolved. Unfortunately, in most applications, the trend is taken as a simple straight line. Let us take the global warming phenomenon as an example. This case has been studied by Wu et al (2007). According to the 4th Assessment Report of the Intergovernmental Panel on Climate Change (IPCC, 2007), the global temperature anomaly is given in Figure 3, in which various trends are determined from the annual averaged data.


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Fig. 3. Instrumental climate data covering the period 1856 to 2007. Various trends and temperature increasing rates are determined for different data lengths: 25 (yellow), 50 (brown), 100 (black) and 150 (red) years.

Based on the slopes of the various trends, IPCC reached this conclusion: “Note that for shorter recent periods, the slope is greater, indicating accelerated warming.” If we use EMD to analyze the original monthly data, we would get 12 different sets of IMFs each representing a specific month of the years such as January, February and so on as shown in Figure 4. Clearly we have a clear trend given by the residue after all the oscillations were removed. From the twelve different monthly results, we could get an annual mean and the scattering range indicating the confidence limit of this decomposition. Having obtained this result, we will consider an alternative of the straight line trends given by IPCC. Instead of drawing the 25-year trend at the end of the data, we can draw 25-year trend centered at any point. The slope values would be a variable now. The result is plotted in Figure 5. Clearly, the running 25-years trends showed a fluctuating

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pattern with the period roughly 65 years based on this study; we concluded that the true warming rate of the climate is only half the rate as determined by the IPCC. Therefore, the recent global warming is partially contributed by natural fluctuation with a 65 years cycle corresponding to the multi-decadal oscillation as suggested by Schlesinger and Ramankutty (1994). IMFs of each downsample 0.5 IMF 1

0 -0.5 0.2

IMF 2

0 -0.2 0.2

IMF 3

0 -0.2 0.2

IMF 4

0

Nonlinear Trend

-0.2

0.5 0 -0.5 1840

1860

1880

1900

1920 1940 Time: Year

1960

1980

2000

2020

Fig. 4. The various IMFs representing each specific month of the year from the original monthly data. We could get the mean and the scattering bound from these results.

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An Introduction to an Adaptive Data Analysis Method Various Rates of Global Warming 0.025 0.02

Warming Rate: oC/Yr

0.015 0.01 0.005 0 -0.005 -0.01 -0.015 -0.02 1840

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1920 1940 Time : Year

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Fig. 5. The 25-year trends determined with any 25 period. The thin jaged dashed line gives the running value determined from the annual mean data. The mean of the corresponding slopes determined from the specific monthly data is given as a solid curve line with roughly 70 years cycle and the scattering given by the shaded band. The slopes from the various straight line trends are given by the corresponding colored dashed lines (25-year, 50-year, 100-year and 150-year all in dashed lines), each is roughly the mean values over the time span. The slope from the EMD determined trend is given by the solid line with the scattering band in gray that is only half the value as given by IPCC.

4. Some Applications In this section, we will use two examples to illustrate the usefulness of the adaptive data analysis approach in time series data. The first example demonstrates the prowess of the method to represent chirp signals and the second example demonstrates the application of the method in speech analysis. Finally, we also use a 2-dimensional image to illustrate the Multi-dimensional EEMD.

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Fig. 6. Comparisons among Fourier, wavelet and Hilbert Spectral analysis for a chirp wave. The labels for the prospective figures are these: time axes are from 0 to 1000 and frequency axes are from 0 to 0.01 Hz.

a. The Chirp Data Let us consider the following model equation   2π t  t  x (t ) = sin     , for t=0:1024 .   256  1024 

(18)

The wave shape and the analyzed results based on Fourier, Morlet wavelet, and Hilbert spectral analysis are given in Figure 6. The model wave clearly has frequency variation along the time axis. Yet the Fourier spectrum totally missed the temporal variation, for in the analysis both amplitude and frequency are constant over the whole time domain. Wavelet certainly could capture the temporal variation to some


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degree. Yet, the leakage and fixed basis limit the power of frequency resolution. Only Hilbert spectral analysis could give a clean and crisp frequency variation as a function of time.

b. Speech Signal Analysis Any utterance of speech could be transcribed as time variation of the pressure field. Figure 7 gives the recorded signal when the word, ‘Hello,’ is uttered. The recording was taken at a sampling rate of 22050 Hz. Again, various methods of analyses were used on this set of data. The results are given in Figure 8. Here the top row shows the results from the Fourier Spectrogram. For narrow and wide band representations. The narrow band result, on the left, is computed with a window size of 1024 data points; while the wide band result, on the right, is computed with a window size of 64 data points. The narrow band representation can certainly resolve the

Fig. 7. The signal of the word, ‘Hello,’ recorded at 22050 Hz.

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N. E. Huang Narrowband Spectrogram Hello 1024

4000

3500

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3000

F re q u e n c y : H z

F re q u e n c y: H z

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2500 2000 1500

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4000

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Wavelet Spectrum: Hello 5:2048

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Wideband Spectrogram Hello 64

Hilbert Spectrum: Hello

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 Time: second

Fig. 8. Fourier spectrograms, Wavelet and Hilbert spectral analyses for the ‘Hello’ signal. The figures in the top row are the result from Fourier Analysis (narrow band on the left, wide band on the right); the bottom left is Morlet wavelet and the right one is from HHT.

frequency better, but the time location is smeared. The wide band representation can resolve the time location better, but the frequency is smeared. The inability to represent both time and frequency to an arbitrary degree of precision is known as the uncertainty principle. This uncertainty principle is very different from the counter part in physics, known as the Heisenberg Uncertainty Principle, which has full physical meaning. The uncertainty principle here is purely the result of using integral transformation for the frequency determination. It is artificially inflected on the result because an improper analysis method is used.


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The Wavelet transform seems to be an improvement over the Fourier transform in this representation. Yet, the resolution is still poor. Furthermore, the continuous Wavelet analysis used here is over redundant and not energy conserving. The discrete Wavelet analysis, unfortunately, could not be used to recognize or extract physical features of the data. The solution for the true time-frequency analysis is undoubtedly the Hilbert Spectral analysis. Here the frequency is determined by differentiation; therefore, its value and location is precise. The adaptive basis also enables the frequency to be a function of time and totally eliminated the need of the spurious harmonics. The intra-wave frequency modulation fully represents the nonlinear characteristics of the speech sound production processes.

5. Conclusion HHT is a new method in data analysis. Its power is in the adaptive approach, which results in the adaptive basis. This offers a new and valuable view of nonstationary and nonlinear data analysis methods. With the recent developments on the normalized Hilbert transform, the confidence limit, and the statistical significance test, the Ensemble EEMD and the more faithful instantaneous frequency computation methods have made HHT a more robust tool for data analysis, and it is now ready for a wide variety of applications. The development of HHT, however, is not complete yet. There is still the need for a more rigorous mathematical foundation for the general adaptive methods for data analysis. As any new method in mathematics, such as calculus, it always started from a more or less empirical approach. We hope the final establishment of this method on rigorous and solid mathematical foundation would not keep us wait for too long.

Acknowledgements The author would like to acknowledge the generous help he received from his colleagues, specifically Professors Zhaohua Wu of FSU and Xianyao Chen of FIO, Qingdao, for allowing the author to quote their

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work freely. He would also like to thank the supports by a grant from Federal Highway Administration, DTFH61-08-00028, and grants from NSC, NSC95-2119-M-008-031-MY3, NSC97-2627-B-008-007, and finally a grant from NCU 965941 that have made the conclusion of this study possible.

References 1. Bedrosian, E., On the Quadrature Approximation to the Hilbert Transform of Modulated Signals. Proc. IEEE, 51, 868-869 (1963). 2. Cohen, L., Time-frequency Analysis, 299pp. Prentice Hall, Englewood Cliffs, NJ (1995). 3. Diks, C., Nonlinear Time Series Analysis, 209pp. World Scientific Press, Singapore (1997). 4. Daubechies, I., Ten Lectures on Wavelets, 357pp. Philadelphia SIAM (1992). 5. Fan, J. and Q. Yao, Nonlinear Time Series: Nonparametric and Parametric methods. 551pp. Springer, New York (2005). 6. Flandrin, P, Time-Frequency / Time-Scale Analysis. 386pp. Academic Press, San Diego, CA (1999). 7. Flandrin, P., Rilling, G. and Gonçalves, P., Empirical mode decomposition as a filter bank. IEEE Signal Proc. Lett. 11 (2): 112-114 (2004). 8. Hou, T. Y., M. P. Yan and Z. Wu, A variant of EMD Method for Multi-Scale data. Adv. Adap. Data Analy., 1, 483-516 (2009). 9. Huang N. E., S. R. Long, and Z. Shen, Frequency Downshift in Nonlinear Water Wave Evolution. Advances in Appl. Mech. 32, 59-117 (1996). 10. Huang, N. E., Shen, Z., Long, S. R., Wu, M. C., Shih, S. H., Zheng, Q., Tung, C. C. and Liu, H. H., The empirical mode decomposition method and the Hilbert spectrum for non-stationary time series analysis, Proc. Roy. Soc.London, A454, 903-995 (1998). 11. Huang, N. E., Z. Shen, R. S. Long, A New View of Nonlinear Water Waves – The Hilbert Spectrum, Ann. Rev. Fluid Mech. 31, 417-457 (1999). 12. Huang, N. E., Wu, M. L., Long, S. R., Shen, S. S. P., Qu, W. D., Gloersen, P. and Fan, K. L., A confidence limit for the empirical mode decomposition and the Hilbert spectral analysis, Proc. of Roy. Soc. London, A459, 2317-2345 (2003). 13. Huang, N. E. and N. O. Attoh-Okine, Hilbert-Huang Transforms in Engineering, 313 pp, CRC Taylor and Francis Group (2005). 14. Huang, N. E. and S. S. P. Shen, Hilbert-Huang Transform and Its Applications. Ed. N. E. huang and S. S. P. Shen, 311pp, World Scientific, New Jersey (2005). 15. Huang, N. E. and Z. Wu, A review on Hilbert-Huang transform: Method and its applications to geophysical studies, Rev. Geophys., 46, doi:10.1029/2007RG000228 (2008).


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16. Huang, N. E., Wu, Z., Long, S. R., Arnold, K. C., Blank, K., Liu, T. W., On Instantaneous Frequency, Adv. Adap. Data Analy. 1,. 177-229 (2009). 17. IPCC, Intergovernmental Panel on Climate Change, The Fourth Assessment Report “Climate Change 2007” (2007). Available http://www.ipcc.ch/ 18. Kantz, H. and T. Schreiber, Nonlinear Time Series Analysis, 304pp. Cambridge University Press, Cambridge (1997). 19. Nuttall, A. H., On the quadrature approximation to the Hilbert Transform of modulated signals, Proceedings of IEEE, 54, 1458-1459 (1996). 20. Priestley, M. B., Nonlinear and nonstationary time series analysis, 237pp. Academic Press, London (1988). 21. Schlesinger, M. E. and N. Ramankutty, An oscillation in the global climate system of period 65-70 years. Nature, 367, 723-726 (1994). 22. Stock, J. H. and M.W. Watson, Testing for Common Trends. J. Am. Statist. Assoc., 83, 1097-1109 (1988). 23. Tong, H., Nonlinear Time Series Analysis, 552pp. Oxford University Press, Oxford (1990). 24. Wu, Z. and Huang, N. E., A study of the characteristics of white noise using the empirical mode decomposition method, Proc. Roy. Soc. London, A460, 1597-1611 (2004). 25. Wu, Z., N. E. Huang, S. R. Long, and C.-K. Peng, On the trend, detrending, and variability of nonlinear and nonstationary time series. Proc. Natl. Acad. Sci., 104, 14,889-14,894 (2007). 26. Wu Z. and N. E. Huang, Ensemble Empirical Mode decomposition: A Noise-Assisted Data Analysis Method. Adv. Adap. Data Analy. 1, 1-42 (2009). 27. Wu Z., N. E. Huang and X. Chen, The Multi-Dimensional Ensemble Empirical Mode Decomposition Method. Adv. Adap. Data Analy. 1, 339-372 (2009).


CHAPTER 7 COMPUTATIONAL FLUID DYNAMICS BASED ON THE UNIFIED COORDINATES — AN EXPOSE∗

W. H. Hui Emeritus Professor of Applied Mathematics, University of Waterloo, Canada and Hong Kong University of Science and Technology, Hong Kong E-mail: [email protected] Computational Fluid Dynamics (CFD) uses large scale numerical computation to solve problems of fluid flow. Traditionally, it uses either the Eulerian or the Lagrangian coordinate system. These two systems are numerically non-equivalent, but each has its advantages as well as drawbacks. A unified coordinate system (UC) has recently been developed which combines the advantages of both Eulerian and Lagrangian systems, while avoiding their drawbacks. This paper gives an expose of the CFD using the unified coordinates. Specifically, it will be shown that: (1) Governing equations of fluid flow in any moving coordinates can be written as a system of closed conservation PDEs; consequently, the effects of moving mesh on the flow are fully accounted for. (2) The system of Lagrangian gas dynamics equations is written in conservation PDE form for the first time. (3) The Lagrangian gas dynamics equations in 2-D and 3-D are shown to be theoretically non-equivalent to the Eulerian ones. (4) The UC is superior to both Eulerian and Lagrangian systems in that contact discontinuities are resolved sharply without mesh tangling. (5) Additionally, the UC is superior to the Eulerian as it avoids the tedious ∗

In Honor of Prof. Y. H. Pao on his 80th Birthday. I first met Professor Y. H. Pao in May 1985 and, at his invitation, spent one semester in 1990 as a Visiting Professor in his Institute of Applied Mechanics, National Taiwan University to begin my new research area: computational fluid dynamics, especially searching for a new coordinate system to do better computation in CFD. It is fitting to celebrate Professor Pao’s 80th birthday by giving an account on what my collaborators and I have achieved in this area since then. 159

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and time-consuming task of mesh generation: the mesh in UC is automatically generated by the flow. Many examples are given to demonstrate these properties of the UC.

1. CFD as Numerical Solution to Nonlinear Hyperbolic PDEs – An Overview of Major Developments The objective of this paper is to give an exposé of the recently developed unified coordinate system (UC) in CFD mostly by the author and his collaborators;13-28,30,32,34-40,42,44,46,47,51 details can be found in the upcoming monograph bearing the same title by W. H. Hui and K. Xu.29 To put it in perspective we shall first give an overview of the major developments of CFD as numerical solution to the initial value problem of nonlinear hyperbolic partial differential equations before discussing the role of coordinates in CFD in Sect. 2. The theoretical foundation was laid by Riemann in his pioneering work (Riemann43, 1862) where he introduced the concept of Riemann invariants and posed the special initial value problem – since been known as the Riemann problem. It turns out that the Riemann problem plays a central role in all numerical methods of CFD. Nothing very significant happened during the following six decades until Richardson proposed weather prediction by numerical process (L. F. Richardson, Cambridge University Press, 1922). Even without an electronic computer, wanting to find numerical solutions to nonlinear hyperbolic PDEs immediately raises many interesting theoretical and practical questions, and progresses are made in answering them. The first of these is the discovery of the CFL condition (Courant, Friedrichs and Lewy5, 1928). It simply says that in a time-marching process to find a numerical solution, marching too fast causes numerical instability and destroys the solution. Practical methods for computing solutions with shock discontinuities are developed: the artificial viscosity method which smears shock discontinuities (von Neumann & Richtmyer49, 1950); the Godunov method which reduces the general initial value problem to a sequence of Riemann problems with cell-averaging data (Godunov9, 1959); the Glimm random choice method which also reduces the general initial

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value problem to a sequence of Riemann problems but with data of randomly chosen representative states (Glimm8, 1965; Chorin4, 1968); and the shock-fitting method (G. Moretti41, Polytechnic Institute of Brooklyn Report, 1972). The last two methods are not easy to extend the three-dimensional flow. A very important discovery was made by Lax & Wendroff 33 (1960) that in order to numerically capture shock discontinuities correctly, the governing PDE must be written in conservation form to begin with. This is easily done in Eulerian coordinates (in any dimensions) and also for one-dimensional flow in Lagrangian coordinates. But for a long time, it was not known how to use Lagrangian coordinates to write the governing PDEs for multidimensional flows in conservation form. This problem was solved by Hui et al (Ref. 18, 1999). To extend Godunov’s method to higher order accuracy, the important concepts of limiters and TVD were introduced which avoid non-physical oscillations in high resolution schemes (Boris and Book2, 1973; Van Leer48, 1973). From the onset of CFD, it was known that the numerical solution to a given flow depends on the coordinates (mesh) used to compute it; hence great efforts have been devoted to search for the optimum coordinate system: the Particle-in-Cell method (Harlow10, 1955); the ArbitraryLagrangian-Eulerian method (Hirt et al12, 1974); various moving mesh methods (Brackbill and Saltzman3, 1982); and the unified coordinate method (Hui et al18, 1999). To compute a flow past a body, which is the central problem in fluid dynamics, it is necessary to construct a body-fitted mesh prior to computing the flow. Even after decades of research, mesh-generation remains tedious and time-consuming. The unified coordinate approach to CFD has opened up a way of automatic mesh-generation (Hui et al25, 2005). 2. The Role of Coordinates in CFD 2.1. Theoretical Issues For more than 200 years, two coordinate systems have existed for describing fluid flow: The Eulerian system is fixed in space, whereas the

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Lagrangian system follows the fluid. An immediate question is ‘are they equivalent to each other theoretically?’ This question must have been asked by numerous researchers in fluid dynamics, and the answer presumably was positive. Surprisingly, the first mathematical proof of equivalency, meaning the existence of a one-to-one map between the two sets of weak solutions obtained by using the two systems, was given as 50 late as 1987 by Wagner and holds only for one-dimensional flow (Note: in the presence of a vacuum, the definition of weak solution for the Lagrangian equations must be strengthened to admit test functions which are discontinuous at the vacuum). Even more surprisingly, for two- and three-dimensional flows, Hui et al18,20 showed that they are not equivalent to each other theoretically (see Sect. 6). 2.2. Computational Issues Computationally, Eulerian and Lagrangian systems are not equivalent even for one-dimensional (1-D) flow. However, we shall show19,21,22,32 in Sect. 4 that for 1-D flow, the Lagrangian system plus a shock–adaptive Godunov scheme14,34 is superior to the Eulerian system. The situation in two- and three-dimensional flows is more complicated: Eulerian and Lagrangian systems have advantages and drawbacks. In general, the Eulerian method is relatively simple, because the gas dynamics equations can be easily written in conservation PDE form, which provides the theoretical foundation for shock-capturing computation. However, it has two drawbacks: (a) it smears contact discontinuities badly and (b) it needs generating a body-fitted mesh prior to computing flow past a body, but mesh generation is tedious, time-consuming and requires specialized training. The Lagrangian method, by contrast, resolves contact discontinuities (including material interfaces and free surfaces) sharply, because they coincide with Lagrangian coordinates. It, too, has two drawbacks: (a) it may break down due to cell deformation, because a Lagrangian computational cell is literally a fluid particle with finite — though small — size and hence deforms with the fluid and (b) the gas dynamics equations could not be written in conservation partial differential


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equations (PDE) form, rendering numerical computation complicated. In this regard, we note that as late as 1999 Serre45 stated ‘Writing the equations of gas dynamics in Lagrangian coordinates is very complicated if (dimension) D ≥ 2’. Interestingly enough, it was shown18 in the same year that using the unified coordinates it is quite easy to derive the Lagrangian gas dynamics equations in conservation PDE form (see Sect. 6). This motivates us to search for an ‘optimal’ coordinate system, leading to the unified coordinate system. 2.3. The “Optimal Coordinate System” Can we have a coordinate system that combines the advantages of Eulerian and Lagrangian systems, while avoiding their drawbacks? Such a system would be ‘optimal’ in some sense (whether or not a system is optimal depends on the criteria, which are necessarily subjective). Specifically, we want the system to possess the following properties for compressible flow computation: (a) Conservation PDE form exists, as in Eulerian; (b) Contact discontinuities are sharply resolved, as in Lagrangian; (c) Mesh can be automatically generated to fit given body shapes; (d) Mesh is orthogonal; (e) …… The unified coordinate system satisfies these requirements as shown in Sects. 4 to 7. 3. The Unified Coordinate System We introduce arbitrary coordinates, (λ , ξ ,η , ζ ) , via a transformation from Cartesian (t , x, y , z ) as follows: dt = dλ , dx = Udλ + Adξ + Ldη + Pdζ , dy = Vdλ + Bdξ + Mdη + Qdζ , dz = Wdλ + Cdξ + Ndη + Rdζ .

(1)

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For (1), we request ξ  DQ   η  = 0 , Dt   ς 

where

DQ

Dt

≡

∂ + Q ⋅ ∇x . ∂t

(2)

So the coordinates (ξ ,η , ζ ) , and hence the computational cells, move with velocity Q = (U , V , W ). It should be noted that the time transformation in (1) is not trivial: while both t and λ are physical time, they are not equal to each other. For instance, in the Lagrangian case (see Eqs. (10) and (8) below), t is Eulerian time while λ is Lagrangian time. On the other hand, for 2-D steady flow, (1) reduces to31 dx = hu dλ + Adξ ,  dy = hv dλ + Bdξ

(1’)

where q = (u, v ) is the fluid velocity of two-dimensional flow and h is arbitrary. In this case, the Eulerian time t disappears by definition of steady flow, but λ still exists, representing the time travelled by the fluid particle along the streamline ξ = const. There are two special cases: Eulerian when Q = 0 and Lagrangian when Q = q . In the general case, we have a coordinate system with three degrees of freedom: U , V and W are arbitrary. On the other hand, the nine coefficients A, B,…, R in the transformation are not arbitrary, but must satisfy a set of compatibility conditions for dx, dy and dz to be total differentials. These conditions are:

∂A ∂U ∂L ∂U ∂P ∂U = , = , = , ∂λ ∂ξ ∂λ ∂η ∂λ ∂ζ ∂B ∂V ∂M ∂V ∂Q ∂V = , = , = , ∂λ ∂ξ ∂λ ∂η ∂λ ∂ζ ∂C ∂W ∂N ∂W ∂R ∂W = , = , = . ∂λ ∂ξ ∂λ ∂η ∂λ ∂ζ

(3a)

∂A ∂L = , ∂η ∂ξ ∂B ∂M = , ∂η ∂ξ ∂C ∂N = , ∂η ∂ξ

(3b)

∂A ∂P = , ∂ζ ∂ξ ∂B ∂Q = , ∂ζ ∂ξ ∂C ∂R = , ∂ζ ∂ξ

∂L ∂P = , ∂ζ ∂η ∂M ∂Q = , ∂ζ ∂η ∂N ∂R = . ∂ζ ∂η


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We note that of the eighteen conditions in (3), only nine are independent. We shall take the first nine conditions, (3a), which all involve λ-derivatives and are called time-evolution (also called geometric conservation laws), to be the independent conditions; the remaining nine, called differential constraints, then hold for all time provided they do initially. The coordinates (λ , ξ ,η , ζ ) are called unified in the sense that (a) they combine the geometric conservation laws with the physical conservation laws; and (b) they unify the Lagrangian and Eulerian coordinates (including ALE, but simpler). One way to choose the mesh-moving velocity Q = (U ,V ,W ) is as follows:

1-D

2-D 3-D

Dq ξ

Dt Dq ξ

Dt Dq ξ

Dt Dqη

Dt

=0

thus U = u , but A is arbitrary (in the classical Lagrangian case: U = u , A = 1 / ρ ), plus adaptive Godunov scheme (Lepage & Hui34, 1995).

=0

plus mesh-angle (hence orthogonality) preserving.

=0

plus mesh-skewness preserving.

=0

Here q is the fluid velocity, u is its x -component, and ρ is the fluid density. With the above choice for the mesh-moving velocity (U ,V , W ) , we see that (a) UC is a generalization of Lagrangian coordinates; (b) Since ξ is a material coordinate, UC resolves contacts sharply without mesh tangling; (c) ξ is a level-set function, as it is a material function.

4. One-Dimensional Flow It is shown32 that UCs (Lagrangian coordinates plus shock-adaptive Godunov scheme34) are superior to the Lagrangian system which, in turn, is superior to the Eulerian system.

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We begin with the Euler equations of gas dynamics in Cartesian coordinates (t , x ) for a γ-law gas  ρu  ρ  ∂  ∂  (4a) ρu + ρ u 2 + p  = 0, ∂t    ρe

e=

 ∂x   u ( ρ e + p ) 

1 2 1 p u + . 2 γ −1 ρ

(4b)

Here u is velocity, p pressure, ρ density and e specific total energy. System (4) is hyperbolic and in conservation form. We transform (4) to the Lagrangian coordinates (λ , ξ ) by dt = dλ ,  dx = udλ + 1 dξ ,  ρ

(5)

to get   ∂  ∂λ  

1  ρ ∂ u +  ∂ξ e  

 −u   p  = 0.    up 

(6)

Example 1. A Riemann problem System (6) is also hyperbolic and in conservation form, and Wagner50 showed that weak solutions to (4) and to (6) are equivalent theoretically. However, they give different numerical solutions, as demonstrated in Figs. 1a and 1b19,21 for a Riemann problem, whose solution consists of two shocks and a contact discontinuity in between. It is seen that, while shock resolutions are similar, Lagrangian computation resolves the contact much better than Eulerian. This is because a contact line is a material line and hence coincides with the Lagrangian coordinate, whereas it does not coincide with the Eulerian coordinate. The error near the contact discontinuity (Fig. 1b) in Lagrangian computation can be remedied using the shock-adaptive Godunov scheme34 instead of the classical Godunov scheme. In the shock-adaptive


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A Riemann Problem (Godunov-MUSCL scheme) (a)

(Eulerian)

ρ

(b)

(Lagrangian)

ρ

(c)

(UC)

ρ

Fig. 1. Numerical solutions to a Riemann problem. Solid line: exact, dots: computed. (a) Eulerian, (b) Lagrangian, (c) UC (Lagrangian with shock-adaptive Godunov scheme).

Godunov scheme shocks are fitted, using the Riemann solution with no extra cost, so we can replace the energy conservation equation by the entropy conservation equation which holds in the smooth flow regions. Accordingly, (6) becomes  1  − u   ∂  ρ  ∂   u +  p  = 0. ∂λ  p  ∂ξ    γ   0   ρ 

(7)

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Computations using (7) resolve both shock and contact discontinuities sharply, as shown in Fig. 1c. Indeed, UC can cure all known defects of Eulerian and Lagrangian shock-capturing methods32, namely: (a) contact smearing; (b) slow moving shocks; (c) sonic-point glitch; (d) wall-overheating†; † (e) start-up errors ; † (f) low-pressure flow ; (g) strong rarefaction waves†. We therefore conclude that UC computation is superior to both Lagrangian and Eulerian computation, and is numerically optimal for 1-D flow. Furthermore, UC reduces the mathematical problem of 2-D steady supersonic flow to that of 1-D unsteady flow by marching in the flow direction (instead of time), so it is also numerically optimal for that flow.

5. Multi-Dimensional Flow For simplicity, we consider 2-D flow, for which the Euler equations of gas dynamics in Eulerian coordinates are ∂E ∂F ∂G + + = 0, ∂t ∂x ∂y

where ρv ρu     ρ     2   ρuv   ρu + p   ρ u   E =  , F =  ρuv  , G =  ρv 2 + p  .     ρv    ρu  e + p    ρv e + p    ρe        ρ   ρ      

†

also shared in Lagrangian computation.

(8)


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Under transformation (1), we obtain ∂ E ∂ F ∂G + + = 0, ∂λ ∂ξ ∂η

(9)

where ρX ρY  ρJ            Ju Xu pM Yu pB + − ρ ρ ρ        ρJv      ρXv − pL ρYv + pA        ρJe   ρXe + p(uM − vL )  ρYe + p(vA − uB ) E= , F =  , G =   0 −U  A       B      0 −V       0 −U  L      M      0 V −      

with J = AM – BL, X = (u - U )M – (v - V )L and Y = (v – V )A – (u – U )B. The first four equations of (9) are the physical conservation laws and the last four the geometric conservation laws. We note that this system of equations is: (a) a closed system of PDE in conservation form, and (b) hyperbolic in λ, except for the special case of Lagrangian coordinates when U = u and V = v, which will be discussed in Sect. 6.

Remarks: The consequences of having a closed system of governing equations in conservation form are: • Effects of moving mesh on the flow are fully accounted for, making the UC superior to other existing moving mesh methods. • System (9) can be solved as easily as the Eulerian system with one extra work: at each time step the mesh velocity (U, V ) is either given or computed. 6. Lagrangian Case For U = u and V = v, we get ∂ E ∂ F ∂G + + = 0, ∂λ ∂ξ ∂η

(10)

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where 0 0  ρJ            pM  ρJu     − pB   ρJv   − pL    pA        ρJe   p(uM − vL )  p (vA − uB ) E= , F =  , G =  . −u 0  A       B      −v 0       −u 0  L       M      v − 0      

Remarks: • The gas dynamics equations in Lagrangian coordinates are here written in conservation form for the first time18. • It thus provides a foundation for developing Lagrangian schemes as moving mesh schemes6,7. • The system of Lagrangian gas dynamics equations is only weakly hyperbolic18: all eigenvalues are real, but there is no complete set of linearly independent eigenvectors. This is also true for the 3-D case20. Hence, the Lagrangian system of coordinates is not equivalent to the Eulerian system. 7. Automatic Mesh-Generation As one of its coordinates is a material coordinate, the UC provides the basis for automatic mesh-generation for flow past a body. We shall illustrate the computational procedure via an example. In all examples below γ = 1.4 is used, unless otherwise stated.

Example 2. Mach 0.8 steady flow past a NACA 0012 airfoil28 We compute the solution to equations (9) by time marching in λ. The computation procedure is as follows: Initialization stage — automatic generation of body-fitted mesh in a computational window.


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Given the grid sizes, ∆x and ∆y , and the number of cells, M × N , in the window, a flow-chart of the computations looks as follows: (1) Begin with a column of N orthogonal cells, representing the given uniform flow in the x-direction (Fig. 2a). This gives the initial values of ( A, B, M , L ) = (1, 0, 0,1) . We also take (U ,V ) = (u, v ) initially. (2) Compute the solution to (9) by marching in time λ using operator splitting: splitting into two 1-D systems in λξ and λη . Each of them is solved using the standard Godunov/ MUSCL scheme with minmod limiter. (Details are given in27). After one time step ∆λ , this column of cells moves to the right by a distance equal to U∆λ . (3) After several time steps when the initial column of cells has moved to the right by a distance equal to ∆x , add one new column of cells on the left that is identical to the initial column. (4) Repeat this process of adding cell columns on the left of the computational region until the leading column of cells meets the body surface and afterwards (Fig. 2b), we then impose the boundary condition of zero normal relative fluid velocity on the body surface by solving a boundary Riemann problem. (5) Continue this process until after the columns of cells cover the whole body surface and further downstream, when we have M columns of cells in the window (Fig. 2c). This completes the initialization stage, and we now have an airfoil-fitted mesh and a flow field around it in the window of M × N cells. The flow-generated mesh (Fig. 2d) is seen body-fitted and orthogonal, as predicted. It is also fairly uniform in the x -direction. The flow field computed (Fig. 2e) so far is, however, only a rough approximation to the correct one, partly because it has not reached the steady state and partly because the downstream boundary condition used in the transient times, e.g., in Fig. 2b, are obviously inaccurate as the computational regions at those times have not yet reached the full window. To compute further, one could use the body-fitted orthogonal mesh generated so far to perform an Eulerian computation with the associated flow field as an initial solution. This can be easily done by

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Fig. 2. Mach 0.8 flow past a NACA 0012 airfoil at zero angle of attack. (a) – (c) flow-generated meshes, (d) close view of preliminary mesh, (e) preliminary surface pressure, (f) close view of final mesh, (g) final surface pressure.


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putting U = V = 0 in the subsequent iterations towards a steady state. In this way, the UC approach plays the role of mesh generation for Eulerian computation. An alternative and better way is to continue the UC computation in the Main stage — iteration with flow-adjusted mesh (6) To iterate the computation towards a steady state, whenever we add a new column of cells on the left we also simultaneously delete the right-most column of cells from the computation window, thus keeping the window approximately in the same size. To improve the solution, we may also use26 the information of the flow field at every time step, e.g., the surface pressure gradient, so that the mesh is refined in regions of high pressure gradient (Fig. 2f).

Example 3 Figure 3 shows a sample computation28 for a Mach 2.2 steady flow of air-SF6 past a NACA 0012 airfoil at an angle of attack of 8° using the procedure explained above. It is seen that without any special treatment, the interface between the two fluids is sharply resolved.

Example 4 This is an unsteady supersonic flow, M ∞ = 3.0 , past a diamond-shaped airfoil with 10° vertex angle which is oscillating about its vertex according to θ = 2° sin 30t , where θ is the instantaneous pitching angle. In Eulerian computation it is necessary to generate a body-fitted mesh at every time step, which is time-consuming. Mesh-generation at every time step is completely avoided using the unified coordinate 25 approach. The flow-generated meshes at different times, computed using UC, are plotted in Fig. 4. It is seen that at all times of the oscillation, the flow-generated meshes are body-fitted and are almost orthogonal, as predicted, although the mesh on the expansion side of the airfoil is coarsened.

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

Fig. 3. Mach 2.2 flow of air-SF6 over a NACA 0012 airfoil at 8° angle of attack. Flow-generated meshes and density (and pressure) contours at different times are shown here.

175


t=

t =0

t=

6 T 10

t = 10T

3 t = (10 + )T 4

1 T 10

t=

t=

8 T 10

1 t = (10 + )T 4

3 T 10

t =T

2 t = (10 + )T 4

t = 11T

Fig. 4. Flow-generated meshes for oscillating diamond-shape airfoil. Apex angle = 10°, pitching motion about the apex: θ(t) = 2°sin(2πt/T ), the period of oscillation T = 2π/30, free stream Mach number M = 3.0.

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Fig. 5. Fixed computational mesh around a rectangular plate.

(b)

Fig. 6. Trajectories of the falling rectangular plate. (a) Computed trajectory, (b) The tumbling phase: from experiment (black) and as computed (red).


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8. Aerodynamics of Falling Leaves This is the problem of unsteady incompressible aerodynamics of a freely falling leave. There is rich dynamic behavior, such as fluttering and tumbling (Refs. 1, 31). It is a problem of interaction between the unsteady motion of the leave and the air flow around it, and is rather cumbersome to compute using the Eulerian coordinate system. Numerical and experimental studies have been presented in Ref. 1 and, more recently, 31 Jin and Xu have elegantly used the unified coordinate formulation, 52 together with a gas-kinetic BGK solver , to obtain very accurate results. In Ref. 31 the computational mesh is fixed rigidly with the leave (a plate) (Fig. 5), hence the mesh velocity is determined according to the translational and rotational velocity of the plate which, in turn, is

Fig. 7. Computed vortex fields of a falling rectangular plate at four instants during a full rotation.

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determined by the unsteady aerodynamic forces and the gravitational force according to Newton’s laws of motion. In the UC formulation, Eqs. in (9), the effects of the mesh movement are fully accounted for in Eqs. (3) as pointed out earlier, while the effects of viscosity are fully included in the gas-kinetic BGK solver. Computed trajectories of the falling rectangular plate are plotted in Fig. 6b and are seen to agree well 1 with the experimental measurements in . The computed vortex fields of the same motion are also given in Fig. 7, showing the complex vortex shedding of a falling rectangular plate at four different instants during a full rotation. 9. Conclusions (1) For 1-D flow, UC is superior to Eulerian and Lagrangian systems and completely satisfactory. Also, with UC, 2-D steady supersonic flow computation is reduced to that of 1-D unsteady flow, hence is also completely satisfactory. (2) The governing equations in any moving coordinates can be written as a system of closed conservation PDE; consequently, effects of moving mesh on the flow are fully accounted for (hence it is better than any moving-mesh method). (3) The Lagrangian system of equations of gas dynamics is written in conservation PDE form for the first time, thus providing a foundation for developing Lagrangian schemes as moving mesh schemes. (4) The Lagrangian system of gas dynamics equations in 2-D and 3-D are shown to be only weakly hyperbolic, in contrast to the Eulerian system which is fully hyperbolic; hence the two systems are not equivalent to each other. (5) The UCs retain the advantages of the Lagrangian system: contact discontinuities are resolved sharply. (6) Finally, in using the UC there is no need to generate a body-fitted mesh prior to computing flow past a body: the mesh is automatically generated by the flow.


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References 1. Andersen, A., Pesavento, U. and Wang, Z.J., “Unsteady aerodynamics of fluttering and tumbling plates.” J. Fluid Mech., 541, 65 (2005). 2. Boris, J.P. and Book, D.L., “Flux corrected transport I, SHASTA, a fluid transport algorithm that works.” J. Comp. Phys., 11, 38 (1973). 3. Brackbill, J.U. and Saltzman, J.S., “Adaptive zoning for singular problems in two dimensions.” J. Comput. Phys., 46, 342 (1982). 4. Chorin, A.J., “Random choice solution of hyperbolic systems.” J. Comput. Phys., 22, 517 (1968). 5. Courant, R., Friedrichs, K.O. and Lewy, H., Physik. Math. Ann., 100, 32 (1928). 6. Despres, B. and Mazeran, C., “Symmetrization of Lagrangian gas dynamics in dimensions two and multidimensional solvers.” Comptes Rendus (Mecanique), 331, 475 (2003). 7. Despres, B. and Mazeran, C., “Lagrangian gas dynamics in two dimensions and Lagrangian systems.” Arch. Rational Mech. Anal., 178, 327 (2005). 8. Glimm, J., “Solutions in the large for nonlinear hyperbolic systems of equations.” Comm. Pure and Appl. Math, 18, 697 (1965). 9. Godunov, S.K., “Difference method of numerical computations of discontinuous solutions in hydrodynamic equations.” Math. Sbornik, 47, 271 (1959). 10. Harlow, F.H., Los Alamos Scientific Laboratory Report, LAMS-1956, (1955). 11. Hafez, M.M., Osher, S. and Whitlow, W., “Improved finite different schemes for transonic potential calculations.” AIAA Paper 84-0092, AIAA 22nd Aerospace Sciences Meeting, (1984). 12. Hirt, C.W., Amsden, A.A., and Cook, J.L., “An arbitrary Lagrangian-Eulerian computing method for all flow speeds.” J. Comput. Phys., 14, 227 (1974). 13. Hui, W.H., and Loh, C.Y., “A new Lagrangian method for steady supersonic flow computation, Part II: slip-line resolution.” J. Comput. Phys., 103, 450 (1992). 14. Hui, W.H., and Loh, C.Y., “A new Lagrangian method for steady supersonic flow computation, Part III: strong shocks.” J. of Comput. Phys., 103, 465 (1992). 15. Hui, W.H., and Zhao, Y.C., “A generalized Lagrangian method for solving the Euler equations.” in “Nonlinear Hyperbolic Problems: Theoretical, Applied, and Computational Aspects”, (Eds.) Donato, A. and Oliveri, F., Vieweg, (1993), p336. 16. Hui, W.H., and Chu, D.L., “Optimal grid for the steady Euler equations.” Comput. Fluid Dyn. J., 4, 403 (1996). 17. Hui, W.H., and He, Y., “Hyperbolicity and optimal coordinates of the three-dimensional steady Euler equations.” SIAM J. Appl. Maths., 57, 893 (1997). 18. Hui, W.H., Li, P.Y., and Li, Z.W., “A unified coordinate system for solving the two-dimensional Euler equations.” J. Comput. Phys., 153, 596 (1999). 19. Hui, W.H., and Koudriakov, S., “Role of coordinates in the computation of discontinuities in one-dimensional flow.” Comput. Fluid Dyn. J., 8, 495 (2000).

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20. Hui, W.H., and Kudriakov, S., “A unified coordinate system for solving the three-dimensional Euler equations.” J. Comput. Phys., 172, 235 (2001). 21. Hui, W.H., and Kudriakov, S., “On contact overheating and other computational difficulties of shock-capturing methods.” Comput. Fluid Dyn. J., 10, 192 (2001). 22. Hui, W.H., and Kudriakov, S., “Computation of shallow water waves using the unified coordinates.” SIAM J. Sci. Comput., 23, 1615 (2002). 23. Hui W.H. and Zhao, G.P., “Capturing contact discontinuities using the unified coordinates.” in “Proceedings of the 2nd MIT Conference on Computational Fluid and Solid Mechanics”, (Ed: K J Bathe) vol. 2, (2003), p 2000. 24. Hui, W.H., Wu, Z.N. and Gao, B., “Preliminary extension of the unified coordinate approach to computation of viscous flows.” accepted by J. Sci. Comput., in print, (2006). See also Gao, B. and Wu, Z.N., Physically related coordinate system for compressible flow, Modern Phys. Letters, B19, 1455 (2005). 25. Hui, W.H., Zhao, G.P., Hu, J.J. and Zheng, Y., “Flow-generated-grids — gridless computation using the unified coordinates.” “Proceedings of the 9th International Conference on Numerical Grid Generation”, San Jose, CA, 11-18 June, (2005), p 111. 26. Hui, W.H. and Hu, J.J., “Space-marching gridless computation of steady supersonic/hypersonic Flow.” Int’l J. of Comput. Fluid Dyn., 20, 55 (2006). 27. Hui, W.H., “The unified coordinate system in computational fluid dynamics, Review Article.” Communications in Computational Physics, 2, 577 (2007). 28. Hui, W.H., Hu, J.J. and Shyue, K.M., “Role of coordinates in Computational Fluid Dynamics.” Proceedings of the 14th Canadian CFD Conference, Session # 6, Queen’s University, Kingston ON Canada, 16-18 July, 2006. To be published in the Int’l J. of Comput. Fluid Dyn., (2007). 29. Hui, W.H. and Xu, K., “Computational Fluid Dynamics based on the Unified Coordinates.” Springer, (2010). 30. Jia, P., Jiang, S. and Zhao, G.P., “Two-dimensional compressible multi-material flow calculations in a unified coordinate system.” Computers & Fluids, 35, 168 (2006). 31. Jin, C. and Xu, K., “A unified moving grid gas-kinetic method in Eulerian space for viscous flow computation.” J. Comput. Phys., 218, 68 (2006). 32. Kudriakov, S. and Hui, W.H., “On a new defect of shock-capturing methods.” Journal of Computational Physics, 227, 2105 (2008). 33. Lax, P.D. and Wendroff, B., “Systems of conservation laws.” Comm.Pure Appl. Math., 13, 217 (1960). 34. Lepage, C.Y., and Hui, W.H., “A shock-adaptive Godunov scheme based on the generalized Lagrangian formulation.” J. Comput. Phys., 122, 291 (1995). 35. Liou, M.S., “An extended Lagrangian method.” J. Comput. Phys., 118, 294 (1995). 36. Loh, C.Y., and Hui, W.H., “A new Lagrangian method for steady supersonic flow computation Part I: Godunov scheme.” J. Comput Phys., 89, 207 (1990).


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37. Loh, C.Y. and Liou, M.S., “A new Lagrangian method for solving the 2-D steady flow equations for real gas.” J. Comput. Phys., 104, 150 (1993). 38. Loh, C.Y. and Liou, M.S., “A new Lagrangian method for three-dimensional steady supersonic flows.” J. Comput. Phys., 113, 224 (1994). 39. Loh, C.Y., Liou, M.S. and Hui, W.H., “An investigation of random choice method for 3-D steady supersonic flow.” Int’l J. for Numer. Methods in Fluids., 29, 97 (1999). 40. Loh, C.Y. and Hui, W.H., “A new Lagrangian method for time-dependent inviscid flow computation.” SIAM J. on Sci. Comput., 22, 330 (2001). 41. Moretti, G., “Thoughts and Afterthoughts About Shock Computations.” Polytechnic Institute of Brooklyn PIBAL Report, 72 (Dec. 1972). 42. Niu, Y.Y., Lin, Y.H., Hui, W.H. and Chang, C.C., “Development of a moving artificial compressibility solver on unified coordinates.” Int’l J. for Numerical Methods in Fluids, 65, 1029 (2009). 43. Riemann, G.F.B. “Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite”, Abh. König. Gesell. Wiss. Göttingen, 8, 43 (1860). 44. Rogers, C., Schief, W.K. and Hui, W.H., On complex-lamellar motion of a Prim gas, J. Math Anal. Appl., 266, 55 (2002). 45. Serre, D., Systems of Conservation Laws, Cambridge University Press, (1999). 46. So, R.M.C., Liu, Y. and Lai, Y.G., “Mesh shape preservation for flow-induced vibration problems.” Journal of Fluids and Structures, 18, 287 (2003). 47. Tai, Y.C. and Kuo, C.Y., “A new model of granular flows over general topography with erosion and deposition.” Acta Mech., 199, 71 (2008). 48. Van Leer, B., “Towards the ultimate conservative difference scheme I. The quest of monotonicity.” Springer Lecture Notes in Physics, 18, 163 (1773). 49. von Neumann, J. and Richtmyer, R.D., “A method for the numerical calculation of hydrodynamic shocks.” J. Appl. Phys., 21, 232 (1950). 50. Wagner, D.H., “Equivalence of Euler and Lagrangian equations of gas dynamics for weak solutions.” J. Differential Equations, 68, 118 (1987). 51. Wu, Z.N., “A note on the unified coordinate system for computing shock waves.” J. Comput Phys., 180, 110 (2002). 52. Xu, K., “A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method.” J. Comput. Phys., 171, 289 (2001).


CHAPTER 8 TOWARDS GREEN’S FUNCTION RETRIEVAL FROM IMPERFECTLY PARTITIONED AMBIENT WAVE FIELDS: TRAVEL TIMES, ATTENUATIONS, SPECIFIC INTENSITIES, AND SCATTERING STRENGTHS

Richard L. Weaver Department of Physics, University of Illinois at Urbana-Champaign 1110 W. Green St., Urbana Illinois 61801, USA E-mail: [email protected] Recent theorems have established that fully diffuse wave fields permit, on cross-correlating the noisy signal detected at different positions, retrieval of the Green’s function, and in particular, retrieval of wave travel times. Application to seismology has led to the creation of maps of seismic surface wave velocity with exceptional resolution. What is perhaps perplexing is that the ambient seismic noise fields that are used for this are rarely fully diffuse; the conditions for the theorems are not satisfied. It is shown here that this may be understood theoretically by a simple asymptotic argument. It is further shown that incompletely diffuse noise fields may permit us to robustly recover more than just travel times, but also ray arrival amplitudes, the ambient field’s specific intensity, the strength and density of its scatterers if any, and most importantly attenuation. A theoretical basis for this is presented.

1. Introduction Interest in diffuse elastic waves has seen an extraordinary growth over the last decade, driven largely by applications in seismology and exploration geophysics. One key to this growth was the observation22,23,25 that fully diffuse wave fields permit, on cross-correlating the ostensibly random material displacements at different positions, retrieval of the Green’s function, where the Green’s function is the signal one would 183

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have at one receiver if the other were replaced with an impulsive force. Further theoretical arguments24,17,13,20 and laboratory demonstrations7,1,8 in support of this conclusion quickly appeared. The essential theorem states that the (time derivative of the) cross correlation between fully diffuse white noise fields ψ at two points x , and y C xy′ (τ ) = limT →∞

(1) d 1 d < ψ ( x , t )ψ ( y , t + τ ) >t ψ ( x , t )ψ ( y, t + τ ) dt = ∫ T dτ T dτ

is equal to the (time symmetrized) Green’s function G between points x and y . C ′xy (τ ) = E (G ( x , y ,τ ) − G ( x , y, −τ )) , (2)

where E is a measure of the energy density in the diffuse field. The white noise condition is not onerous. If the diffuse field is band limited, then the cross correlation is merely the Green’s function convolved with the spectrum of the ambient field, i.e. the factor E becomes a convolution. The time-symmetrization is not onerous either, as G is causal, so the two terms in Eq. (2) have distinct supports in τ . The amount of time T over which one must integrate to obtain sufficiently converged C has received considerable attention. In practice seismologists find that a period of months usually suffices. Theory suggests that the amount of time needed scales with the distance between stations, but is greater in the presence of attenuation. Considerable thought has gone into creatively weighting the time-average to prevent periods of high energy density from dominating the average. One method is to replace the ambient field with its sign before cross correlating. If, as in elastodynamics, the fields are vector-valued, the above expressions generalize easily. The relevance of this to seismology and exploration geophysics,21,6 ocean acoustics12 and structural acoustics,2,15 was immediately apparent. Applications in long-period seismology have been particularly striking. Tomographic maps of Rayleigh wave velocity, in which different frequencies reveal information from different depths, have been

Towards Green’s Function Retrieval

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constructed with extraordinary spatial resolution.16,14 The literature also reports retrieval of waveforms due to Love waves and bulk waves. High spatial resolution arises because the method frees the seismologist from needing a controlled source (impractical in seismology where one needs an earthquake or a nuclear explosion and in exploration geophysics where one uses chemical explosives or thumper trucks.) Dense large arrays consisting of up to hundreds of seismic stations are a recent development; the world-wide availability of their daily records for periods of years lends itself naturally to the necessary kind of data processing. The early work made it clear that the theoretical basis for the relation is restrictive; the notion of a fully diffuse wave field is a demanding one and few natural fields are fully diffuse to the required degree. In practice, ambient surface seismic waves at frequencies below 1 Hz are not fully diffuse. They appear to have their sources in localized ocean storms that are often distant from the seismic detection network.18,4,10 Therefore the fields at the detectors retain strong directionality. (Scattering from seismic heterogeneities mitigates this somewhat but not entirely). The fields are diffuse in the sense that they lack coherent wave fronts and are superpositions of uncorrelated waves traveling in different directions. They are not fully diffuse in that their distributions of intensity are not isotropic, and further in that the energy amongst different wave types, bulk and surface, is probably not equipartitioned. Nonetheless numerous researchers report highly robust retrieval of travel times between distant stations, both in the lab and in the earth’s surface. It transpires that an isotropic distribution of intensity is not a prerequisite for retrieving travel times. Retrieval of Green’s functions per se does require both isotropy and equipartition amongst wave types. But both experience16,14 and theory26,3,5 show that retrieval of travel times is robust against these failures of full diffusivity. This is the basis for the extraordinary quality of the maps of seismic Rayleigh wave velocity in spite of the failures of the correlations C′ to correspond exactly to G . This raises the further question of whether it is possible using the imperfectly diffuse fields with which seismology has to deal and for which C ′ ≠ G , to retrieve not only travel times, but also meaningful measures of amplitude. If so, we could extract maps of attenuation and

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scattering strength as well as seismic velocity. It is towards this goal that the present communication is directed.

2. Imperfectly Diffuse Fields Ambient seismic fields fail to be fully equipartitioned in at least two ways. They often fail to be equipartitioned amongst all wave types. The ambient field is dominated by surface waves. For this reason, retrieval of bulk waves from correlations has been more difficult than retrieval of surface waves. (This is in part also due to the lower amplitude of typical P and S contributions to Green’s functions between surface stations.) Retrieval of surface wave ray arrivals has proven far more feasible, however the ambient field's surface waves also fail to be fully equipartitioned in practice: the ambient surface waves are generally not distributed isotropically, some directions have greater intensity than others. Recently Weaver et al26 investigated the effect of such a distribution, or “ponderosity” B (θ ) , on retrieval of ray arrival times. Here we summarize that argument, and extend it to the effect on ray arrival amplitudes. Consider two receivers, one at the origin, the other a distance x from the origin along the x-axis. We distribute incoherent sources s (θ , t ) over an annular region of (large) radius R around the receivers, with < s > = 0 , and < s (θ ) s * (θ ′ ) > = B (θ )δ (θ − θ ′ ) . These give rise to a fieldψ (r , t ) whose Fourier transform

ψɶ (r ,ω ) = ∫ψ ( r , t ) exp ( −iω t ) dt is a superposition of cylindrical waves from the many sources s . At a point on the x-axis it is

ψɶ ( x, ω ) = ∫ s (θ ) dθ exp (iω R − iω x cosθ ) / Ri (ω − iε ) ,

(3)

the angle θ being defined relative to the strike line connecting the receivers. The cylindrical waves have been written in a form valid for asymptotically large ω R . Thus the derivation is restricted to noise whose sources are in the far field. ε is an infinitesimal positive quantity. Units are used such that wave speed c is unity.


187

The time derivative of the correlation between receivers at r = 0 and ˆ r ′ = ix has Fourier transform Cɶ 0,' x (ω ) ≡ iω ∫ C0, x (τ ) exp( −iωτ )dτ = iω < ψɶ (0, ω ) *ψɶ ( x,ω ) > .

(4)

The relevant expectation is

< ψɶ (0) *ψɶ ( x) >=< ∫ s(θ ')dθ '∫ s(θ )dθ exp( −iω x cos θ ) / R ω 2 + ε 2 > = ∫ B(θ )dθ exp(−iω x cos θ ) / R ω 2 + ε 2 .

(5)

On returning to the time domain and removing the uninteresting factor of

R, C0,' x (τ ) = ∫ B (θ ) dθ i sgn(ω )exp(−iω x cos θ ) exp(iωτ )d ω .

(6)

In practice, source fields and correlation waveforms have finite bandwidth. In this case Eq. (6) must be modified to include a smooth factor of spectral weight, assumed independent of angle,

C0,' x (τ ) = ∫ B (θ )dθ i sgn(ω ) exp(−iω x cos θ ) exp(iωτ ) S (ω )d ω

(7)

In accord with typical seismic applications we take the spectrum to be band-limited with central frequency ω o , in the form of a sum of (assumed non-overlapping) parts at positive and negative frequency, S (ω ) = Aɶ (ω − ω 0 ) + Aɶ (ω + ω 0 ) ,

(8)

where Aɶ (ω ) is the Fourier transform of a real envelope function A (t ) that has its maximum at t = 0 . Without loss of generality we take the maximum to be unity. The autocorrelation, C′ at x = 0 , is then readily and exactly evaluated to be ' C0,0 (τ ) = ∫ B(θ )dθ

∫ i sgn(ω ) exp(iωτ )S (ω )dω

= −2{ ∫ B(θ )dθ } A(τ )sin ω oτ ,

(9)

which shows that, unlike the cross correlation below, the auto-correlation receives significant contributions from waves in all directions θ .

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At x ≠ 0 , and if ponderosity B is independent of angle, i.e. if ambient energy is incident uniformly from all directions, the integral (7) over θ is recognized as giving a Bessel function and one exactly recovers the usual 2-d Green’s function. If B is θ -dependent, then the integral can be evaluated analytically for asymptotically large ω x . One recognizes that the dominant contribution comes from the vicinity of θ = 0 , i.e. from the ambient waves that are incident in the on-strike direction from the origin towards x . Then, to leading order we have C0,' x (τ ) ~ ∫ {B (0) + 12 B "(0)θ 2 + 241 B ''''(0)θ 4 + ..} ×

exp( −iω x{1 − 12 θ 2 }) i sgn(ω ) exp(iωτ ) S (ω ) dω dθ .

(10)

Higher order corrections to the exponent (e.g. exp(−iω xθ 4 / 24) ) will affect the asymptotic corrections, but such effects will be of the same (small) order for the leading term B (0) and the first correction proportional to its derivatives. Thus we neglect them. Asymptotically, then C ′ becomes (c.c. representing complex conjugate) ∞

C0,' x (τ ) ~ i ∫ exp(iωτ ) Aɶ (ω − ω o )d ω 2π / | ω | x × 0

(11)

1 1 exp( −iω x + iπ / 4)[ B(0) − B ''(0) − B ''''(0) 2 2 + ..] + c.c. 2iω x 8ω x At leading order, where we neglect the B '' and B '''' terms, the correlation becomes C0,' x (τ ) / B(0) 2π / | ω o | x ∞

~ exp(−iω o x + iπ / 4) i ∫ exp(iωτ ) Aɶ (ω − ωo )d ω + c.c. 0

∞

~ exp(iω o (τ − x ) + iπ / 4)i ∫ exp(i β (τ − x )) Aɶ ( β )d β + c.c. 0

= exp(iω o (τ − x ) + iπ / 4)iA(τ − x ) + c.c. = −2 A (τ − x ) sin(ω o (τ − x ) + 3π / 4).

(12)


189

C' is therefore the usual Green’s function at asymptotically large ωx. It has an arrival at time x / c = x , and an amplitude proportional to the noise intensity B (0) on strike. The higher order term B '' can be seen in Eq. (11) to affect the phase of the ray arrival. This has been discussed at length elsewhere26,3,5, where it is established that the practical consequences for retrieval of ray travel times are usually negligible. Both the term in B '' and B '''' affect the amplitude of the apparent ray arrival. A small bit of algebra establishes that the amplitude B (0) in Eq. (12) should be replaced by

  B '' (0)2 B '''' ( 0)  1  B (0) 1 +  −  2 2 + .. . 2 B ( 0)  8ωo x   B (0) 

(13)

At practical station separations of more than two wavelengths, the difference is usually negligible.

3. Retrieval of Attenuation While retrieval of travel times is largely independent of the uniformity of ponderosity, retrieval of attenuation requires that it be considered. A full examination of the amplitude of a correlation waveform will supplement Eq. (12) with a factor related to attenuation, and another for the so-called ‘site effect’. The amplitude of the ray arrival in the correlation between the fields at site i and site j is then, on neglecting the asymptotically negligible corrections to B(0) indicated in Eq. (13),

X ij = 2 si s j Bi (nî → j ) 2π / ω o | xi − x j | exp(−α | xi − x j |) .

(14)

If the attenuation were to vary spatially, one would replace the exponent with an integral over the line between sites i and j . Here Bi is the ponderosity in the direction from i towards j evaluated at the position of station i . Here we have neglected additional geometric factors related to focusing and defocusing due to spatially varying wave speeds. In principle these effects can be important, and they can be accounted for once one has the velocity map, although Prieto et al9 suggested that they average out.

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The expression above may be supplemented by the corresponding expression for the amplitude of the autocorrelation

X ii = 2si2 ∫ Bi (θ ) dθ .

(15)

Site factors si , in which the sensitivity of a seismic station to an incident seismic wave depends on the geology near the station, are commonly assumed to be scalar and real. The assumption of reality means that they do not affect apparent ray arrival times. They do, however, affect the amplitude of a seismic station record and in principle must be accounted for if one is attempting to compare amplitudes between different stations in order to recover attenuation. If the seismic signal at a station i is contaminated by so-called ‘local noise’ that does not propagate into the interior and from there to other stations, then the autocorrelation includes an additional term related to the energy of the local noise. The goal is to recover the attenuations, and by implication the B and the site factors, from measurements of the amplitudes of the ray arrivals X in the cross correlations amongst all the stations of a region. This goal is not new. It has been suggested that the cross correlations between two seismic stations could be normalized by the noise auto-correlation at the nominal receiver, by considering quantities like X ij / X ii X jj ,

(16)

which has the virtue of eliminating the site factors s . Unfortunately this normalizes with respect to the angle-integrated B , not to the B on strike. Thus, this construction does not correct for non-isotropy in the ambient noise fields. Another suggestion is to identify a linear array, along a direction nˆ , of several seismic stations for which the amplitudes of cross correlations X for j > i should be xj

X i < j = 2 si s j Bi ( nî → j ) 2π / ω o | xi − x j | exp(− ∫ α dx). xi

(17)

Taking N to be the number of stations along this array, one notes that there are at most N ( N − 1) / 2 such ray-amplitudes and up to 3N


191

unknowns to be extracted (the site factors, the B and the local attenuations). Rays in direction − nˆ can provide up to N ( N − 1) / 2 additional amplitudes and N additional unknowns B . It is not difficult then to show that one needs from 5 to 7 stations along this line before one has an over-determined set of conditions. A model for the spatial dependence of the α, and a model that assumes B( nˆ ) diminishes exponentially with distance like exp(-2α x), and also geometrically due to spreading from its distant source, but is not replenished by scattering into direction nˆ from other directions, might help constrain the parameters and reduce the demand for a large number of stations. The approach is intuitive and may well suffice in many circumstances. It does, however, rely on identifying a linear array of sufficient length, and/or upon assumptions on how attenuation and noise vary in space. More problematic in this researcher’s eyes however is that it does not make use of all the information that is in principle available. How might one use sets of seismic stations that are not aligned along a linear array? Can one make use of autocorrelations Xii ? How quickly ought one permit the diffuse intensity implicit in the above formulation to vary in direction and from place to place? Is other information extractable? It is the contention here that certain conditions on diffuse intensity, in particular how it is permitted to vary in space and direction, can be introduced that will further constrain the model and thus lead to more reliable recovery of seismic attenuation and diffuse intensity. The appropriate condition is that the noise field ought to obey a Radiative Transfer Equation19,11 or RTE, nˆ ⋅ ∇B (r , nˆ ) + 2α (r ) B (r , nˆ ) = p2(πr ) ∫ B (r , nˆ ')dnˆ '+ P(r , nˆ ) (18) The ponderosity B of the earlier sections is thus interpretable as the socalled ‘specific intensity’ of Radiative Transfer Theory. It is the diffuse Rayleigh wave intensity in direction nˆ at position r . In Eq. (18) α is the usual attenuation. It includes intrinsic absorption, and also losses due to scattering to bulk waves and to other modes, and also scattering of Rayleigh waves into Rayleigh waves going in other directions. The left side indicates that intensity decays like exp(-2α x); this is the coherent loss. The right side shows that the specific intensity is

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augmented by sources P (which are generally negligible for long period seismology on the continents and may therefore be confined to the boundaries of the region of interest) and by scattering, with strength p , into direction nˆ from direction nˆ ′. Here p is the (assumed for now isotropic) scattering strength. Scattering p contributes to attenuation; indeed on recognizing that 2α = p + intrinsic losses + scattering into other modes, one can demand the inequality 2α ≥ p ≥ 0 . The RTE governs how quickly specific intensity B can vary, in directions or position. Arbitrary differences in ambient intensity at different stations are not permitted. It is proposed that Eqs. (14), (15) and (18) together with measurements of ray amplitudes X (as many as 104 in an array of 100 stations) and reasonable models for attenuation and scattering p, constitute a large global and overdetermined set of conditions for recovering the noise field B, the site factors s, the attenuation, the diffuse scattering strength p and the source distribution P. Inasmuch as the X are always determined with some readily estimated uncertainty, this may be formulated as a nonlinear least square minimization in a space of hundreds of parameters. Unless good first guesses are given, such minimizations are notoriously poor at converging to the global best fit. Towards that end the above procedures entailing linear arrays may be of use to generate starting points. Preliminary numerical trials on a set of six stations and 36 amplitudes with spatially constant B (θ ) have shown that, even with random initial guesses, a simple conjugate gradient minimization sufficed to find the correct solution.

Acknowledgments The author’s work in diffuse field seismology has been supported by the National Science Foundation, in particular by grant no. EAR-05-43328.

References 1. A. Derode, E. Larose, M. Tanter, J. de Rosny, A. Tourin, M. Campillo and M. Fink, “Recovering the Green’s function from field-field correlations in an open scattering medium,” J. Acoust. Soc. Am. 113, 2973 (2003).


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2. A. Duroux, K. G. Sabra, J. Ayers, M. Ruzzene, “Extracting guided waves from cross-correlations of elastic diffuse fields: Applications to remote structural health monitoring,” J. Acoust. Soc. Am., 122, 204 (2010). 3. B. Froment, M. Campillo, P. Roux, P. Gouédard, A. Verdel and R. L. Weaver, “Estimation of the effect of non-isotropically distributed energy on the apparent arrival time in correlations” Geophysics 75, SA85 (2010). 4. P. Gerstoft and T. Tanimoto, “A year of microseisms in southern California,” Geophs. Res. Lett. 34 L20304 (2007). 5. O. Godin, “Accuracy of the deterministic travel time retrieval from crosscorrelations of non-diffuse ambient noise,” J. Acoust. Soc. Am., 126 EL183 (2009). 6. P. Gouedard, L. Stehly, F. Brenguier, M. Campillo, Y. C. de Verdiere, E. Larose, L. Margerin, P. Roux, F. J. Sanchez-Sesma, N. M. Shapiro and R. L. Weaver, “Cross-correlation of random fields: mathematical approach and applications,” Geophys. Prospecting, 56, 375 (2008). 7. E. Larose, A. Derode, M. Campillo and M. Fink, “Imaging from one-bit correlations of wideband diffuse wave fields,” J. Appl. Phys. 95, 8393 (2004). 8. A. E. Malcolm, J. A. Scales and B. A. van Tiggelen, “Extracting the Green function from diffuse, equipartitioned waves,” Phys. Rev. E 70, 015601(R) (2004). 9. G. A. Prieto, J. F. Lawrence and G. C. Beroza, “Anelastic Earth structure from the coherency of the ambient seismic field,” J. Geophys. Res. 114, B07303 (2009). 10. J. Rhie and B. Romanowicz, “Excitation of Earth’s continuous free oscillations by atmosphere–ocean–seafloor coupling,” Nature, 431, 552 (2004). 11. L. Ryzhik, G. Papanicolaou and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion, 24, 327 (1996). 12. P. Roux and W. A. Kuperman, “Extracting coherent wave fronts from acoustic ambient noise in the ocean,” J. Acoust. Soc. Am. 116, 1995 (2004). 13. P. Roux, K. G. Sabra, W. A. Kuperman and A. Roux, “Ambient noise cross correlation in free space: Theoretical approach,” J. Accoust. Soc. Am., 117, 79 (2005). 14. K. G. Sabra, P. Gertoft, P. Roux, W. A. Kuperman and M. C. Fehler, “Surface wave tomography from microseisms in Southern California,” Geophys. Res. Lett. 32, L14311 (2005). 15. K. G. Sabra, A. Srivastava, F. L. di Scalea and I. Bartoli, “Structural health monitoring by extraction of coherent guided waves from diffuse fields,” J. Acoust. Soc. Am. 123, EL8-EL13 (2008). 16. N. M. Shapiro, M. Campillo, L. Stehly and M. H. Ritzwoller, “High-resolution surface-wave tomography from ambient seismic noise,” Science, 307, 1615 (2005). 17. R. Snieder, “Extracting the Green’s function from the correlation of coda waves: A derivation based on stationary phase,” Phys. Rev. E, 69, 046610 (2004). 18. L. Stehly, M. Campillo and N. M. Shapiro, “A study of the seismic noise from its long-range correlation properties,” J. Geophys. Res., 111, B10306 (2006).

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19. J. A. Turner and R. L. Weaver, “Radiative Transfer of Ultrasound,” J. Acoust. Soc. Am., 96, 3654 (1994). 20. K. Wapenaar, “Retrieving the elastodynamic Green’s function of an arbitrary inhomogeneous medium by cross correlation,” Phys. Rev. Lett., 93, 254301 (2004). 21. K. Wapenaar, D. Draganov and J. Robertsson (eds) Seismic Interferometry supplement to Geophysics 71 (2006). 22. R. L. Weaver and O. I. Lobkis, “Ultrasonics without a source, Thermal fluctuation correlations at MHz frequencies,” Phys. Rev. Lett., 87 art. no. 134301 (2001). 23. R. L. Weaver and O. I. Lobkis, “On the emergence of the Greens function in the correlations of a diffuse field,” J. Acoust. Soc. Am., 110, 3011 (2001). 24. R. L. Weaver and O. I. Lobkis, “Diffuse waves in open systems and the emergence of the Greens’ function,” J. Accoust. Soc. Am., 116, 2731-4 (2004). 25. R. L. Weaver, “Perspectives Geophysics: Information from Seismic Noise,” Science, 307, 1568 (2005). 26. R. L. Weaver, B. Froment and M. Campillo, “On the correlation of non-isotropically distributed ballistic scalar diffuse waves,” J. Acoust. Soc. Am, 126 1817 (2009).

CHAPTER 9 STUDY ON TWO SCALE DESIGN OPTIMIZATION OF STRUCTURES AND MATERIALS WITH PERIODIC MICROSTRUCTURE*

Gengdong Cheng1,* and Jun Yan1,* *State

Key Laboratory of Structural Analysis of Industrial Equipment1 Dalian University of Technology, Dalian, China, 116024 E-mail: [email protected]

Two scale design optimization for structures and materials with periodic microstructure is studied in the present paper. The aim is to obtain optimal configurations of macro scale structures and microstructure of materials for optimizing structural performance. The link between macro effective material properties and material micro structure is established by a homogenization approach. Structural topology optimization and inverse homogenization is applied to obtain the optimum structural topology design and optimum microstructure. Global structural performance such as minimum compliance design and maximizing fundamental frequency and local performance such as stress concentration are investigated. The proposed method and computational model are validated by a set of numerical examples. A number of issues for further development in two scale design are addressed.

1. Introduction Light weight structures can reduce the long-run energy consumption of aeronautic crafts. With shortage of energy and resources and intense competitions light weight structures have attracted more and more This paper is dedicated to Professor Yih-Hsing Pao on the occasion of his 80th birthday. This work is supported by the program (90816025, 10902018) of NSFC. 195

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attention in various industries. For high speed vehicles, structural weight is the decisive factor for their essential performances such as speed, efficiency and maneuverability. Optimum design of light weight structures has been the focus in the area of structural optimization. Structural topology optimization3,4 is a powerful approach to create innovative light weight structural design. A well known alternative approach to reduce structural weight is to use light material such as Al-alloy, Al-Mg alloy, Ti-Alloy and composite materials in the aeronautic, aerospace and car industry. With the rapid development in manufacturing techniques, more ultra-light porous metal materials such as truss-like materials, linear cellular metals (LCMs) and metal foams are emerging in the engineering practice. Because of their porous periodic microstructure in the size of tiny structure, they are called structural materials. They have received increasing attention for their high stiffness-weight and strength-weight ratios together with the design potential for multifunctional applications9. More importantly, their properties can be engineered to some extent to suit the application by changing their microstructure. As is pointed out by Ashby1 that innovation, often, takes the form of replacing a component made of one material by another one, then redesigning the product to exploit, to the maximum, the potential offered by the change. The emergence of various structured materials provides a vast opportunity for optimum design of structures and components by concurrent design of material and structures. It can be expected that the global optimum design, or the ideal light weight structure, should have the best material for every component, and the ideal light weight structural component should have the best material for every material point. That is, the material of ideal structures varies point-wise in an optimum way. In reality, material of natural biological structures such as animal’s bone and plant’s stalk has a point-wise varying microstructure8. Manufactured by new technology such as laser melting deposition, functional gradient material varies its composition gradually, which alleviates thermal stresses and enhances fracture toughness. Rodrigues et al24, Coelho et al.6 propose a hierarchical approach for the optimization design of two-dimensional and three-dimensional macro structures composed of porous materials

Study on Two Scale Design Optimization

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which are inhomogeneous at the macro scale (material microstructure varies point-wise). Nevertheless, it is not only expensive but extremely difficult to control the material microstructure variation point-wise. On the other hand, material with macro homogeneous microstructure such as metal foam material is routinely produced nowadays although random variation is unavoidable. The present paper introduces our attempt to achieve ultra-light structures composed of ultra-light structured materials by utilizing topology optimization in both structural and material scales. To improve the possibility of practical applications, manufacturing factors are strongly underlined by assuming homogeneity of the material micro-structures at the macro scale. Multi scale analysis is an important research area and has received great interest in recent years13,19,21. Mechanical performance of structures made of structured material can be obtained by multi scale analysis. Developments in multi scale analysis and structural optimization provide the basis for two scale design optimizations of structures and materials with periodic microstructure. Two scale design optimization of ultra-light structures composed of structured materials is studied in the present paper. The aim is to obtain optimal configurations of macro scale structures and microstructure of materials for optimizing structural performance. Global structural performance such as minimum compliance and maximum fundamental frequency design and local performance such as stress concentration are investigated. The present paper is organized as follows. In Section 2, a concurrent multi scale topology optimization formulation is proposed. Macro densities in the structural design domain and micro densities in the material unit cell are introduced as design variables to independently represent the macro structure and material microstructure topology. For global performance, the homogenization method is applied to establish effective material properties from the material microstructure. Penalization approaches together with new filter techniques are adopted at both scales to ensure clear topologies, i.e. SIMP2 (Solid Isotropic Material Penalization) at micro-scale and PAMP (Porous Anisotropic Material Penalization) at macro scale. Optimization and sensitivity analysis with respect to both macro and micro densities in two scales are

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integrated into one system. The problem formulation and computational model are illustrated in more detail by minimum compliance design under mechanical load. Minimum compliance design under thermomechanical loads and maximum frequency design are presented with less detail. In Section 3, concurrent optimization of material and structure for the local performance is studied. Optimum stress distribution around a hole is investigated for hollowed plates composed of a truss-like material with a micropolar continuum representation. Relative density and cell size distribution of truss-like materials are determined by optimization. The relation between stresses in two scales is discussed. In Section 4, a number of issues for further development in two scale design optimization are discussed. 2. Two Scale Design Optimization for Global Structural Performance Let us consider design optimization of ultra-light structures composed of structured materials. The Design domain and boundary conditions of the macro structure to be designed are given. The objective and constraints include the structural weight and its global performances such as the structural compliance and fundamental vibration frequency. Both the macro structural topology, including size and shape, and the micro structural topology of the structured material is to be designed. This is a two scale design optimization. Considering today’s reality of manufacture, we will assume that the structure are made of a material with macro homogeneous porous micro structure. The mathematical formulation and basic approach of solving the two scale design optimization is described as follows. Figure 1 illustrates a macro structure composed of a porous anisotropic material with homogeneous microstructure and its material micro-structure in unit cell. There are two materials involved in our study, the base material and the porous anisotropic material. The base material, black in the figure, could be any type of solid materials such as aluminum and alloy. The porous anisotropic material, grey in the figure, sometimes merely called ‘material’ for short in the following discussion, is assumed to be made of the base material and to have periodic

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microstructure free of any restriction (e.g. cavities in linear cellular materials and ranked laminates with regular shape). For the structural analysis and optimization, the macro design domain Ω and micro design domain Y , i.e. the unit cell, is meshed into Porous anisotropic material

Characterization

Magnify

Base material

y2

x2 F

A unit cell

y1

x1

Fig. 1. Structure composed of a porous anisotropic material and its unit cell.†

Base material

Porous anisotropic material B

D B Thermal expan. coeff. α Elasticity Matrix

H

D B Effective thermal expan. coeff. α Elasticity matrix

∆T

MI

α

B

D = ρ ⋅D αB Solid Isotropic Material Penalization †

α

H

D MA =Ρ⋅ D MA α H B β =Ρ⋅ D ⋅α Porous Anisotropic Material Penalization

F

The topology optimizations of structures are normally based on relatively coarse resolution of the

design Fig. 2. Penalization-based concurrent optimization two classes of density computational grid, which then naturally lead to step-likewith boundaries of computed domains. The variables. figures in this paper represent the resolution at which the computations have been done actually.

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N and n elements, respectively, see Figure 2. To describe the macro structural topology and material micro structural topology, the element in macro-scale and micro-scale is assigned a density Ρi and ρ j , respectively. Their values are either zero or one. Elements with zero material density, white in the figure, are void. The macro element of unit density is occupied by the real porous anisotropic material. The micro element of unit density is composed of the base solid material. In this way, topology optimization of macro structure and micro structure in a unit cell is replaced by finding the optimum element density distribution in the two scales. In the terminology of mathematical programming, this is a 0-1 programming because the design variables Ρ i and ρ j can only have the values zero and one. To reduce the computational burden, the 0-1 programming is transformed into mathematical programming with continuous design variables by relaxing the constraint on density. That is, intermediate densities, i.e., 0 < Ρ i < 1 , 0 < ρ j < 1 are allowed. Physically, the element of intermediate density is artificial. In the well known SIMP approach15 of structural topology optimization, the elasticity matrix of material with artificial density is assumed to be a mathematical function of the elasticity matrix of its parent material of unit density. To ensure clear topology, elements of intermediate density should not exist in the final optimum design; a penalty on the elasticity matrix of the artificial material of intermediate density is introduced. In micro scale this is done by defining

D MI = ρ α ⋅ D B .

(1)

D MI , D B are elasticity matrices for the base material of artificial density

ρ and base solid material of unit density, respectively. α is a penalty exponent which is set equal to 3 in this research. For the given base material, D B is prescribed. Since the real material is solid base material, this basic idea represented by equation (1) is the well known approach SIMP. In macro scale, a similar penalization is adopted

D MA = Pα ⋅ D H .

(2)


201

D MA and D H are the effective elasticity matrices for the porous anisotropic material of artificial density P, and the porous material of unit density, respectively. Since it penalizes the intermediate density of the porous anisotropic material, this approach is called as PAMP (Porous Anisotropic Material with Penalty)16. The key step in two scale design optimization is to establish a relation between objective/constraint function and the design variables, in particular the relation between the structural performance at the macro scale and the micro element density at the micro scales. To this end, D H serves as a crucial link between the two scales: on the one hand it is a representation of effective material properties depending on the microstructural configuration, and on the other hand it is also involved in the macrostructural analysis. The computation of D H could follow the classical homogenization procedures by implementing the following two steps2: Firstly, analyze the unit cell subjected to periodic boundary conditions and body forces corresponding to uniform strain fields k⋅u =

∫

bT D MI dY ,

(3)

b T ⋅ D MI ⋅ bdY .

(4)

Y

k=

∫

Y

Here k is the stiffness matrix of the microstructure in a unit cell with periodic boundary conditions, u is the microstructural nodal displacement vector, b is the strain/displacement matrix and D MI is defined in Eq. (1) and depends on the micro element density ρ , which in turn describes the microstructure in a unit cell. Secondly, compute the effective elasticity matrix by performing the integration over the domain of a unit cell DH =

1 Y

∫D

MI

⋅ ( I − b ⋅ u )dY .

(5)

Y

By virtue of Eqs. (1)-(5), we have established the link between the effective elasticity matrices of porous anisotropic material with artificial density and microstructural topology in a unit cell. For the structural performance other than mechanical performance such as heat conductivity and seepage capacity, a similar homogenization

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method is available for computing the corresponding effective properties from the microstructure in a unit cell. The present approach can directly be extended to two scale design optimizations for structural thermal performance and seepage performance, respectively. 2.1. Minimum Compliance Design of Structure Subject to Mechanical Loads The first problem is the optimum design of elastic structures composed of macro homogeneous porous anisotropic materials. The macro structure with Ω denoting its design domain is subjected to external forces F . The structural compliance is minimized with specific base material volume. With the SIMP and PAMP approaches, the formulation of minimum compliance design can be expressed as16,

∫

Minimize : C = F ⋅ Ud Ω ,

(6)

Ω

ρ PAM ⋅ Constraint I : ς =

∫

Pd Ω

Ω MA

≤ ς , ρ PAM =

∫ ρdy = ς Y

V V MI Constraint II : 0 < δ ≤ P ≤ 1, 0 < δ ≤ ρ ≤ 1 .

MI

,

(7) (8)

Here, C denotes the structural compliance and U is the structural nodal displacement under static load F . ρ PAM *P and ρ PAM are the relative densities of the artificial and real porous anisotropic material, respectively. Constraint I sets a limit on the total available base material by defining relative density ς and its upper bound ς . V MI , V MA are the areas of the macro and micro design domains, respectively. Constraint II sets bounds for the density variables at two scales to avoid numerical singularities, where δ is a small predetermined positive number, chosen here as δ = 0.001 here. The structural nodal displacement U can be obtained from the following structural governing equation K⋅U = F

(9)

Here the structural stiffness matrix K is the assemblage of the element stiffness matrices k e , which depend on the effective elasticity matrix


203

DeMA = Peα DH of macro element composed of porous anisotropic material with artificial density K = AeN=1k e ,

ke =

∫

Ωe

Pe :

BT ⋅ D MA ⋅ Bd Ω =Peα

∫

Ωe

BT ⋅ D H ⋅ Bd Ω . (10)

Based on Eqs. (1-6), (9) and (10), the sensitivity of the objective function with respect to the artificial densities of macro element and micro element could be derived as ∂C α ⋅ Ci =− , ∂Ρ i Ρi

(11)

∂C ∂D H = − Peα ⋅ UTe ⋅( BT ⋅ ⋅ Bd Ω) ⋅ U e , ∂ρ j ∂ρ j e =1

(12)

N

∑

∫

Ωe

where the derivative of D

H

with respect to ρ j is given by

H

∂D = αρ αj −1 ∫ (I − b ⋅ u j )T ⋅ D B ⋅ (I − b ⋅ u j )dY . ∂ρ j Yj

(13)

Now, we have completed the structural analysis and sensitivity analysis. With the sensitivity information available, gradient-based optimization algorithms such as MMA (Moving Asymptotic Algorithm), SLP (Sequential Linear Programming) and SQP (Sequential Quadratic Programming) can be applied to find the optimum design of Eqs. (6-8). To illustrate the formulation, an example problem is shown here. The MBB beam in Figure 3 is loaded with a concentrated vertical force of P = 1000 at the centre of the top edge and is supported on rollers at the bottom-right corner and on simple supports at the bottom-left corner. The base material is assumed to have Young’s modulus E = 2.1 × 105 and Poisson’s ratio v = 0.3 . Geometric parameters are L = 4 and h = 1 . As we are only interested in qualitative results, the dimensions and loads for this problem are chosen in non-dimensional units. Due to the axial symmetry of the problem, only the right half part is considered as macro design domain. The mesh is 50×25 for the macro design domain and 25×25 for the microstructure (8-node planar isoparametric element). OPTIMUM SEARCH is implemented by the optimization package DOT using the SQP algorithm.

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Table 1 shows the optimal designs with varying ς (available base material) and specified ς MI = 0.4. The resulting microstructures are all porous anisotropic, the topology varies considerably among the different P = 1000

Micro design domain A unit cell

?

h =1

?

Macro design domain

L=4

Fig. 3. MBB Beam.

Mixed cell

Triangular cell

Fig. 4. Two microstructures for Linear Cellular Material9.

cases. When ς = 0.25, the microstructure after a rotation of 45 degrees closely resembles the so-called ‘Triangular cell’ (see Figure 4) which is said to have superior in-plane mechanical properties9. Alternatively, when ς = 0.075, the microstructure after a rotation of 45 degrees is somewhat similar to the ‘Mixed cell’ (see Figure 4), which is considered as another competitive microstructure for linear cellular material. The resemblance between optimized results and some existing superior microstructures implies that the proposed method indeed generates optimum micro-topology for porous materials. In macro-scale, the design also changes with the increase of ς . For the compliance value, it is reasonable that better system performance is achieved with more available base material, see the third column of Table 1.


205

Table 1. Results for varying ς (available base material) and specified ς MI = 0.4.

ς

Compliance

0.075

9855

0.12

5676

0.18

3707

0.25

2234

Structural topology

Microstructural topology

Table 2. Results for varying ς MI (specified material volume fraction of microstructure) and specified ς = 0.12.

ς MI

Compliance

0.2

8880

0.3

6210

0.4

5676

Structural topology

Microstructural topology

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One interesting problem is the trade-off between macro-scale design and micro-scale design, namely the allocation of material between the two scales which can be characterized by ς /ς MI and ς MI . To illustrate the discussion, the two-scale design problem will be solved again with predetermined available base material ς = 0.12 and varying ς MI . As shown in Table 2, a larger value of ς MI leads to a lower system compliance with more base material consumption in micro-scale design and less porous material consumption in macro-scale design. This means in this case, it is advantageous to strengthen the porous material rather than enhancing the macro structure. 2.2. Minimum Compliance Design of Structure Subject to Mechanical and Thermal Loads

The second problem is optimum design of thermo-elastic structures composed of macro homogeneous porous anisotropic materials. The optimization aims at minimum compliance design under mechanical and thermal loads by optimizing configurations of the macro structures and the micro structures of the material. The base material volume is given. The structure is subjected to external forces F and an uniform temperature increase ∆T . With SIMP and PAMP approaches, the formulation of the minimum compliance design can be expressed as29,

∫

∫

Ω

Ω

Minimize : C = F ⋅ Ud Ω + β MA ⋅ ε ⋅ ∆Td Ω

ρ PAM ⋅ Constraint I : ς =

V

∫

Ω MA

Pd Ω

≤ ς , ρ PAM =

∫

(14)

ρ dy

Y

V MI

Constraint II : 0 < δ ≤ P ≤ 1, 0 < δ ≤ ρ ≤ 1 .

,

(15) (16)

Here, C denotes the structural compliance and U is the structural nodal displacement vector under thermal and static mechanical load. ε , β MA represent structural strain and the effective thermal stress matrix of the porous material in macro scale, respectively, and β MA = Pα ⋅ D H ⋅ α H .

207


A Unit Cell

？

? Micro design domain

∆T

H=47.7cm

Macro Design Domain

L=72cm F=500000

Fig. 5. Numerical example of beam with two fixed ends.

According to reference17 no matter what configuration of the micro H B unit cell has, we always have α = α for porous anisotropic materials composed of one pure base material, α B being the matrix of the coefficient of thermal expansion of the base material. The other constraints have been explained in 2.1. To illustrate the formulation, an example is given here. The initial design domain and boundary conditions of the structure to be designed are shown in Fig. 5. A vertical load F = 500000 N and a uniform temperature increment ∆T are applied. The volume constraint is equal to 5% of the total volume. Due to the symmetry, only the left half part is considered as macro design domain. The mesh is 60×30 for the macro design domain and 25×25 for the micro structure (8-Node Planar Element). Optimum design is obtained by the SQP algorithm. Influences of the temperature differential on the two scale design results are listed in Table 3. Two observations from Table 3 are very interesting, more or less, out of expectation. First, it can be seen from the second row of Table 3 that the optimum material was solid if no thermal load exists. In other words, no porous material was needed for minimum compliance design if the structure was only subjected to a mechanical load, and both macro structural topology and micro material topology were allowed to be optimized. However, the structure made of porous material gives

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G. Cheng and J. Yan Table 3. Influences of temperature differential on the concurrent design results.

∆T

Structural topology

Microstructural topology Unit Cell

4 × 4 arrays

0°C

1°C

20°C

40°C

lower compliance if both thermal load and mechanical load are applied. Second, as temperature differential increased from 1° to 40° the macro structural topologies were greatly changed. More test examples confirmed that in most cases isotropic solid material was the best configuration for microstructure to improve the stiffness of the structure with only mechanical loads, except the uniaxially stressed structures. Moreover, this assertion is valid under the following two conditions: (1) only macro uniform material, even with microstructure, is allowed and both macro structural topology and micro


209

structural topology are optimized. (2) The optimization aims at minimum compliance under mechanical loads. The reason for superiority of the solid configuration could be understood with the sensitivity analysis. When ∆T = 0 , the sensitivity of compliance to two designing densities can be written as  ∂C α T ui  BT DiMA B d Ω e  ui , =− ∂Pi Pi  e  Ω

∫

N

∂C = − uTr ∂ρ j r =1

∑

H   T α ∂D r r B B P d Ω  u r . r  r  ∂ρ j Ω

∫

(17)

(18)

It could be observed from the above two equations that the sensitivity of compliance to macro/micro density was always negative, which means that the compliance decreased with an increase of available base material volume. At the same time, the sensitivity of compliance to micro density was the sum of the corresponding values over all N macro elements. Moreover, its counterpart at the macro scale is related only with one local macro element. Then generally, the sensitivity of the micro scale was much larger than that of the macro scale in absolute value. This fact means that it was more efficient to distribute material of the micro scale than that of the macro scale to decrease the structural compliance, or the isotropic solid material was the best choice for the microstructure of the unit cells. It should be pointed out that the Michell truss is not contradicting the above statement because its microstructure varies point-wise. Furthermore, the above statement is consistent with the prediction of Lakes (1993)15, who suggested that the stiffness to weight ratio of framework-type structures decreases with increasing structural hierarchy. For structures with both static and thermal loads, it was found that PAM porous materials ( ρOpt < 1 ) could help to reduce the compliance of structures. Some explanations of the cellular configuration could also be given by sensitivity analysis. Observing the sensitivity analysis for compliance optimization with both mechanical and thermal loads, Eq. (19) and Eq. (20), it was found out that the sensitivity of the compliance w.r.t macro/micro density was not always negative because

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they were composed of two terms corresponding to static and thermal loads, respectively. That is, the sensitivity of compliance to densities might be positive, which means hat the compliance might increase as the available base material volume increases. In numerical implementations, it was found that there was indeed an optimum material volume fraction. The optimum macro/micro volume fractions and configurations remained relatively stable and the volume constraint, constraint Ι in Eq. (15), was no longer active after the optimum volume fraction was reached.   ∂C α α = − ui T  BT DiMA Bd Ωe  ui + 2 ⋅ ui T BT βi MA ∆Td Ωe , ∂ Pi Pi Pi  e  Ω Ωe

∫

N

∂C = − uiT ∂ρ j i =1

∑ ∫

∫

α

BT P i

Ωi

∂DiH Bd Ωi u i + 2 ⋅ ∂ρ j

N

∑ ∫ uTi

i =1

α

BT P i

Ωi

(19)

∂DiH B α ∆Td Ωi . (20) ∂ρ j

Since these two observations were little out of expectation, the optimum design of truss structures with artificial porous material under similar condition was studied. The results confirmed our observations. Due to the space limitation, they are not shown here. 2.3. Maximum Fundamental Frequency Design

The third problem shown here is the two-scale topology design for maximization of fundamental frequency of vibrating elastic structures composed of porous anisotropic material. With above approaches, the problem can be formulated as a max-min problem as follows20: Find : X = {P, ρ} ,

{

Obj : max min

j =1,…, J

{λ

j

(21)

= ω 2j

}} ,

(22)

S .t. : Kφ j = λ j Mφ j , φ Tj Mφk = δ jk , k , j = 1,… , J ,

ς=

ρ PAM ⋅

∫

Ω MA

Pd Ω

≤ ς , ρ PAM =

∫

ρdY

= ς MI , V V MI 0 < P ≤ Pi ≤ P, i = 1,… , NE ; 0 < ρ ≤ ρ l ≤ ρ , l = 1,… , n. Y

(23)

211


In equations (21)-(23), ω 2j denotes the jth eigenfrequency and φ j the corresponding eigenvector, and K and M are the symmetric and positive definite stiffness and mass matrices. Figure 6 shows an example problem, a beam-like structure with L = 80m, H = 50m and M0 = 216000kg for maximization of the fundamental frequency. Table 4 lists the topological design of the macro and micro structure with the variation of the available base material at a certain specific micro volume ς MI = 40%.

Fig. 6. Numerical example of a cantilever beam. Table 4. Topological design of macro and micro structure for maximizing fundamental frequency. ς

λ1

10%

156.81

20%

487.99

25%

593.81

Macro structure

The first mode

Micro structure Cell

4 × 4 array

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3. Two-Scale Optimization for Local Performance

Structural local performance such as local stress concentration and fracture is very important for structural safety. Structural optimizations with local performance have been well studied. One of many difficulties is the large number of constraints. The difficulty of a singular optimum is typical for the structural topology optimization with stress constraint; it needs special constraint relaxation5,10. For two scale structure/material design optimization, an additional difficulty is how to define stress. Homogenized stress could be computed from a macro structural analysis based on the effective material properties. However, from a physical viewpoint, micro stress could be equally interesting. In the following, the optimum stress distribution around a hole is investigated for a hollow plate composed of a linear cellular metal, see Fig. 7. Instead of the classical continuum, the linear cellular metal is homogenized with a micropolar continuum representation14,30 which introduces independent rotational degrees of freedom and couple stresses in addition to the usual Cauchy-type stresses at each material point for a higher calculation precision of stresses. Two classes of design variables, a relative density and cell size distribution of LCMs are defined in one element. This is different from the traditional topology optimization with only an element material density as design variable. The finite element method is applied to discretize the micropolar continuum. The material distribution or the arrangements of the cells is uniform in each element and varies from element to element. Concurrent designs of materials and structures are carried out through three different optimization formulations. They are minimizations of the stress around the hole, the highest stress within the specified point-set and the ratio of stress over the corresponding yield strength. The third formulation actually maximizes the strength reservation and seems more rational. The three formulations are as follows30, Model I: To minimize the stress around a hole in order to reduce the stress concentration. Its mathematical formulation is Find

S.t

( )

ρi , Li , Min

∫

Ω

Ψ = σC

ρ i d Ω = S0 ,

0 < ρ ≤ ρi ≤ 0.3;

2

Li (1 − 1 − ρi ) ≥ h ;

L ≤ Li ≤ L

(24)

213


Model II: To minimize the maximum stress at a specified point-set about 2

which we care most, i.e., Min

Ψ =  Max σ i  , V0  i ∈V0 

is the set of

specified points. Model III: To maximize the strength reservation of the part around the hole, or equivalently, to minimize the ratio of stress around the hole over the effective yield strength. The effective yield strength for LCMs with square unit cells can be written as σ pl σ ys = 0.5 ρ , and the

objective can be written as Min

( )

Ψ = σC

2

(0.5ρC ) .

Here the

subscripts p and s refer to LCMs and cell wall material, respectively. σ C and ρ C denote the stresses and relative density of LCMs at stress concentration points, such as point P shown in Fig. 5. Model II and Model III have the same constraints as model I.

pressure = 1.0

φ =0 X2

P X1 R = 20

1 element ○

φ =0

Fig. 7. Model of geometry and FEM for a quarter of a circular plate with circular hole.

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G. Cheng and J. Yan

A hollowed circular plate of radius 400mm with FEM mesh is shown in Fig. 7. The plate is made of orthotropic LCMs with square unit cells, which is schematically shown in the inner square of Fig. 7. The outer edge is subjected to a prescribed uniform pressure p = 1.0 . We assume that the orientations of square unit cells coincide with the global coordinates. The relative density and cell size of initial design is uniform over the domain, ρ = 0.1 and cell size is L = 5mm . The prescribed material volume is 10% of the design space. h = 0.1mm , ρ = 0.01 and L = 0.1mm; L = 15mm . Due to symmetry we only study a quarter plate. 45 elements along the axial direction are chosen. The maximum stress σ C = σ yP appears at point P in the X 2 direction when the plate is composed of homogeneous cells. This stress is to be reduced. And the nodes on the horizontal edge along the X 1 axis are chosen as specified point-set V0 in model II. We assume that all elements on the same circumference have identical material properties. Max The results are summarized in Table 5. σ X2 is the maximum stress in the X2 direction of point-set V0 . Results in Table 5 show that for the uniform initial design the stress around the hole is 7.57 and the objective values are 57.37, 57.37 and 1147.41 for Models I, II and III, respectively. In Model I, the stress around the central hole is minimized to 0.14. However, the maximum stress emerges at another point on the horizontal axis and reaches 5.14, which is certainly not desired. In Model II, the maximum stress in the specified point set V0 can be reduced to 2.52. It must be mentioned that the above stress is the stress at the equivalent continuum. A mapping method has been studied to obtain stress at the micro scale. Table 5. Comparison of stress between uniform design and optimum design. Model

Initial objective function value

Objective function value after optimization

σ XMax 2

Model I

57.37 (stress around the hole 7.57)


5.14

Model II



2.52

Model III



4.77


215

4. Concluding Remarks Two scale design optimizations of macro structure and micro-structure of structured materials for structural global and local performances were studied in this paper. To improve the possibility of practical applications, manufacturing factors were strongly underlined by assuming homogeneity of the material microstructures at the macro scale. Structural topology optimization approaches such as SIMP, PAMP, homogenization-based two scale analysis method, inverse homogenization method for material microstructure design and new filter techniques were integrated to develop their mathematical formulation and obtain optimum structures and materials. Numerical examples illustrated the effectiveness of the strategy. In addition, a number of interesting observations were obtained from the numerical examples. For example, for the cases in which only mechanical loads apply, microstructure of isotropic solid material is best for achieving the minimum structural compliance; while for the cases in which both mechanical and thermal loads apply, the configuration of porous material is more effective to reduce the system compliance. The fundamental frequency and the stress distribution of structures composed of porous materials also can be optimized with the presented two scale optimization method. The method was also extended to more realistic applications considering multifunctional performances, e.g. active or passive heat transfer26, seepage flow rate28, or incorporated non-local effects of material model18. The two scale topology optimization suffers more from the checkerboard and gray element than the classic single scale optimization. In order to prevent these numerical instabilities and obtain a clear design, various regularization methods such as perimeter control, linear and nonlinear filters have been applied in our study. Due to the space limitation, they were not detailed here. Interesting readers are referred to relevant work12,25,27. Further development in this direction is also needed. Many challenges remain for multi scale design optimization. For example, the current homogenized continuum models suffer from their inability to capture the correct underlying deformation and local strength mechanisms, a more advanced multi scale continuum theory is needed.

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Many excellent works7,13,19,21,31 on multi scale analysis has deepened our understanding of material strength and failure mechanisms down to crystal, molecular or even smaller scale. Based on such a theory, it is expected that one can integrate the more realistic constraints, such as local fracture, plasticity failure into the multi scale design optimization. Material microstructure design at the smallest level could be considered too. This would be extremely exciting. Another important aspect in multi scale design optimization is to consider the size effect. The present work assumes that the unit cell is infinitely small, but if the two scales of structure and microstructure are comparable, size effects need to be considered. Especially for the design optimization problem with wave absorption or wave transmissivity constraint/objective, wave propagation in the structure composed of structured materials must be studied. When the frequency of the external excitations is high, the wave length approaches the intrinsic size of the microstructures of structured materials; so, the actual size of the microstructure will affect the results of the analysis and optimization greatly. The excellent work11,22,23 of Prof. Pao and his colleagues in dynamics and wave propagation in truss and frame structures may provide the benchmark test for approximate numerical methods. References 1. M.F. Ashby and D.R.H. Jones, Engineering Materials 2, Butterworth Heinemann (1998).

2. M.P. Bendsoe, M.P., O.Sigmund, Topology Optimization: Theory Methods and Applications. Springer-Verlag. Berlin (2003).

3. M.P. Bendsoe, N. Kikuchi, Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 71(2): 197 (1988). 4. G.D. Cheng, On non-smoothness in optimal design of solid elastic plates. International Journal of Solids and Structures, 17: 795 (1981). 5. G.D. Cheng, X. Guo, Epsilon-relaxed approach in structural topology optimization. Structural Optimization, 13: 258 (1997). 6. P.G. Coelho, P.R. Fernandes, J.M.Guedes, A hierarchical model for concurrent material and topology optimisation of three-dimensional structures. Structural and Multidisciplinary Optimization, 35(2): 107 (2008).


217

7. J. Fish and K. Shek, Multiscale analysis of composite materials and structures. Composites Science and Technology, 60: 2547 (2000).

8. P. Fratzl and R. Weinkamer, Nature's hierarchical materials. Progress in Materials Science, 52: 1263 (2007).

9. L.J. Gibson and M.F. Ashby, Cellular Solids: Structure and Properties. Cambridge, Cambridge University Press (1997).

10. X. Guo, G. D. Cheng, A new approach for the solution of singular optimum in structural topology optimization. Acta Mechanica Sinica, 13: 171 (1997).

11. Y.Q. Guo, W.Q. Chen, Y.H. Pao, Dynamic analysis of space frames: The method of

12.

13.

14.

15. 16. 17.

18. 19. 20.

21.

22. 23. 24.

reverberation-ray matrix and the orthogonality of normal modes. Journal of Sound and Vibration, 317: 716 (2008). J.K. Guest, J.H. Prevost and T. Belytschko, Achieving minimum length scale in topology optimization using nodal design variables and projection functions. International Journal for Numerical Methods in Engineering, 61 (2): 238 (2004). S. Hao, W.K. Liu, B. Moran, V. Franck, G.B. Olson, Multiple-scale constitutive model and computational framework for the design of ultra-high strength, Computer Methods in Applied Mechanics and Engineering, 193(17-20): 1865 (2004). R.S. Kumar and M.C. Dowell, D.L., Generalized continuum modeling of 2-D periodic cellular solids. International Journal of Solids and Structures, 41:7399 (2004). R. Lakes, Materials with structural hierarchy. Nature, 361: 511 (1993). L. Liu, J. Yan, G.D. Cheng, Optimum structure with homogeneous optimum trusslike material. Computers and Structures, 86: 1417 (2008). S.T. Liu, G.D. Cheng, Homogenization-based method for predicting thermal expansion coefficients of composite materials. Journal of Dalian University of Technology, 35(5): 451 (1995). S.T. Liu, W.Z. Su, Topology optimization of couple-stress material structures. Structural and Multidisciplinary Optimization, 40: 319 (2010). W.K. Liu and C. McVeigh, Predictive multiscale theory for design of heterogeneous materials. Computational Mechanics, 42(2): 147 (2008). B. Niu, J. Yan, G.D. Cheng, Optimum structure with homogeneous optimum cellular material for maximum fundamental frequency. Structural and Multidisciplinary Optimization, 39: 115 (2009). H.S. Park and W.K. Liu, An introduction and tutorial on multiple-scale analysis in solids. Computer Methods in Applied Mechanics and Engineering, 193: 1733 (2004). Y.H. Pao, D.C. Keh, S.M. Howard, Dynamic response and wave propagation in plane trusses and frames. AIAA Journal 37: 594 (1999). Y.H. Pao, W.Q. Chen, X.Y. Su, The reverberation-ray matrix and transfer matrix analyses of unidirectional wave motion. Wave Motion, 44: 419 (2007). H. Rodrigues, J.M. Guedes and M.P. Bendsoe, Hierarchical optimization of material and structure. Structural and Multidisciplinary Optimization, 24:1 (2002).

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25. O. Sigmund, Morphology-based black and white filters for topology optimization. Structural and Multidisciplinary Optimization, 33 (4-5): 401 (2007).

26. B. Wang, G.D. Cheng, L.Jiang, Design of multi-tubular heat exchanger for optimum efficiency of heat dissipation. Engineering Optimization, 40:767 (2008).

27. S. Xu, Y. Cai and G.D. Cheng, Volume preserving nonlinear density filter based on

28.

29.

30.

31.

heaviside functions. Structural and Multidisciplinary Optimization, 41 (4): 495 (2010). S.L. Xu, G.D. Cheng, Optimum material design of minimum structural compliance under seepage constraint. Structural and Multidisciplinary Optimization, 41 (4): 575 (2010). J. Yan, G.D. Cheng, L. Liu, A uniform optimum material based model for concurrent optimization of thermoelastic structures and materials. International Journal for Simulation and Multidisciplinary Design Optimization, 2: 259 (2008). J. Yan, Liu, L., G.D. Cheng, and S.T. Liu, Concurrent material and structural optimization of hollow plate with truss-like material. Structural and Multidisciplinary Optimization, 35: 153 (2008). S. Zhang, R. Khare, Q. Lu, T. Belytschko, A bridging domain and strain computation method for coupled atomistic-continuum modeling of solids. International Journal for Numerical Methods in Engineering, 70: 913 (2007).

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CHAPTER 10 A CONTINUUM FORMULATION OF LAVA FLOWS FROM FLUID EJECTION TO SOLID DEPOSITION∗

Kolumban Hutter† and Olivier Baillifard‡ c/o Laboratory of Hydraulics, Hydrology and Glaciology Swiss Federal Institute of Technology, Zurich, Switzerland ∗ This

paper is dedicated to Prof. Y.-H. Pao on the occasion of his 80th birthday.

A continuum formulation of lava flow down the mountain side of a volcano is presented. Three regimes are differentiated: Hot, temperate and cold lava. Hot lava is treated as a non-linear viscous heat conducting fluid; temperate lava as a binary solid-fluid frictional mixture and cold lava as a frictional heat conducting granular solid. These regions are separated from one another by mass-exchanging phase change surfaces. At the base, erosion, deposition processes and at the free surface heat transfer by radiation and sensible and latent heat are accounted for. The paper has fundamental character as a model for this kind of gravity flow is presented.

10.1. Introduction 10.1.1. Extended Summary A continuum formulation of lava flow down the mountain side of a volcano is presented. The lava, ejected from a volcanic vent, is treated as a non-Newtonian, strongly thermo-mechanically coupled density preserving fluid, which loses heat at the free surface and the ground by radiation and conduction, respectively. The moving lava will partly solidify at the † Retired

Professor of Mechanics from Darmstadt University of Technology, Germany. Corresponding author: [email protected] ‡ Now at: Ch. du Byai 19, CH-1934 Bruson, [email protected]

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free surface, and this solidification will move into the flowing mass further down. Thus, a new region will form, a binary mixture of solid and fluid constituents at the freezing temperature. This mixture is called temperate lava. Both constituents can be modeled as non-Newtonian viscous fluids. In the temperate region the solid volume fraction will grow until at some distance downstream, close to the deposition area all lava will be solidified. The depositing lava will be treated as a one-constituent body with solid properties. This cold lava will also be modeled as a very viscous nonNewtonian material, but having some plastic (rate independent) properties. The three regimes of hot, temperate and cold lava are bounded by the free and ground surfaces and two interfaces, bounding (i) the hot fluid and the fluid-solid mixture and (ii) the solid-fluid mixture and the cold solid. Of these last two interfaces the first, called hot-temperate transition surface (HTS), has a very low solid fraction on its mixture side, the second, called cold-temperate transition surface (CTS), has a high solid fraction on its mixture side. All boundary and transition surfaces are treated as non-material surfaces. The free surface may be subjected to precipitation of non-buoyant particles of ejected magma, and wind stress and atmospheric pressure. Heat exchange, will involve radiation, sensible and latent heat. At the basal surface, erosion of solids and deposition of fluid and solid constituents give rise to induced topographic changes by the flow. Across the HTS and CTS, mass is exchanged by solidification. They are defined as singular surfaces at which the temperature and the tangential components of the velocities are continuous. 10.1.2. Description of the Physical Problem Avalanches, debris and mudflows are natural phenomena which occur in many regions on our globe. Volcanic eruptions also generate avalanching motions of debris. When first expelled from a volcanic vent, lava is a liquid at temperatures from 1000 K to 1500 Ka . Although lava is quite viscous, with about 100’000 times the viscosity of water, it can flow great distances before cooling and solidifying, because of both its thixotropic and shear thinning properties. The correct description of such lava flow is faced with very difficult thermo-mechanical complexities: a Sources:

www.wikipedia.com and Pinkerton & al. 2002 [74].

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A Continuum Formulation of Lava Flows

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• Mass, momentum and heat may be exchanged (i) at the ground via mass and momentum fluxes and conduction of heat and (ii) at the free surface by precipitation of granular material from the air-borne dust cloud both associated with a momentum and energy inflow. • Phase changes may occur between the fluidized lava and its solid phase, and when conditions are right, also the rocky particles may melt and solidify again. These interior phase changes take place because the moving lava exchanges heat with the atmosphere and the ground. • Lava is made up of a variety of minerals dependent on the tectonic setting of the volcano from which it erupts. This complicates its thermo-dynamic description considerably. Indeed, the fluid-solid phase change processes depend in their details upon how the material is composed. If the material is a pure substance like H2 O, then (H2 O)solid and (H2 O)fluid transform into each other at a well defined temperature. This remains so also for polycrystalline ice that is e.g. formed from fallen snow which transforms into a solid ice polycrystal and may be isotropic or have any other distinguished symmetry according to its history of deformation. The crystallographic compound of ice has a well defined melting temperature of a pure substance. Not so for lava! It consists of several different minerals, a compound substance. The phase change processes in lava occur, in general, selectively because the various chemicals or crystallites respond differently in a transition from the fluid to the solid thermodynamic state. So, in reality there exists a temperature interval TL ≤ T ≤ TU within which lava exhibits properties between a solid and a fluid. At temperature T = TL (L for ‘lower’) or below, the composition of the cell elements comprising lava exhibits properties of a solid. Similarly, at the temperature T = TU (U for ‘upper’) or above, all elements comprising lava exhibit fluid properties. For T ∈ T := [TL , TU ] the material is in an aggregation state which neither exhibits fluid nor solid behavior. This thermodynamic behavior for a polycrystal will be simplified. We assume T to be very small on the temperature scale to be effectively reduced to a single point. This is tantamount to interpreting lava as a pure substance such as water/ice and not a polycrystal consisting of different crystallographic elements. More formally: it will be assumed that our ‘model lava’ is a pure substance having a well defined melting/freezing point.

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Thus, lava in a gravity flow may in disjoint regions be a very hot fluid, a mixture of a fluid intermingled with solidified components or a solid body. The mixture will be assumed to be saturated. The solid components within the mixture will, in general consist of particles of different sizes; however, this size difference will be ignored. So, our model lava as a fluid-solid mixture will be assumed to be a saturated binary continuous mixture of which all particles have the same nominal size. Important quantities in the subsequent developments will be the constituent volume fractions, νf and νs , of the fluid and the solid, respectively: να is defined as the volume fraction of constituent α(= s, f ) per unit volume element, να = ∆volα /∆volmix . With this definition, the saturation condition reads α να = 1 and for the binary mixture νs + νf = 1. 10.1.3. Gravity Flow Specifics When lava is flowing out of a volcanic vent, then its state is likely a fluid, Fig. 10.1. The fluid is bounded by rigid rock at the walls of the vent and by the equally rigid bottom of the volcano’s sloping mountain sides, and from above by the free surface, of which the form and position follows from the mass flow of magma M˙ and the pressure conditions at the lava-atmosphere interface. Since heat loss at the free surface is intensive, solidification will first occur at a point (P in Fig. 10.1) in the sloping regime of the free surface. From this point, a surface entering the flow region will mark the transition of the single-constituent region of fluid behavior into the two-constituent region of simultaneous existence of fluid and solid lava. This surface inside the flowing mass separates a pure fluid layer overlain by a fluid-solid mixture, but eventually it will reach the bottom surface at point Q (in Fig. 10.1). Downstream of this interior ‘separatrix’ the gravity flow is at phase change: The fluid and solid ‘particles’ have the same freezing or melting temperature dictated by the ‘Clausius-Clapeyron relation’. At this point it is advantageous to define the following denotations: • Single constituent fluid lava will be called hot lava. • The binary mixture consisting of fluid and solid lava constituents at the phase change temperature will be called temperate lava. • Single constituent solid lava will be called cold lava. The energy balance in the region of temperate lava does not adjust the temperature but solidify some fluid or melt solid grains. These phase transition processes will release or consume latent heat and govern the mass

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223

P Atmosphere

Vent

Free surface

Hot lava

Temperate lava S

Basal surface

HTS Rock & soil bed

CTS Cold lava

Fluidsolid mixture Q R

Solid

M

Fig. 10.1. The picture shows an idealized situation where hot lava leaves the vent ˙ . At PQ and beyond some of the fluid lava solidifies and forms a with a mass rate M fluid-solid mixture at the freezing/melting temperature. At SR the solid volume fraction reaches νs = 1. So, beyond, the lava is a solid. The different material constituents are called hot, temperate and solid lava and the interfaces are the hot-temperate transition surface (HTS) and the cold-temperate transition surface (CTS). (Principal sketch)

production of solid mass and the annihilation of fluid mass or vice-versa. The region of temperate lava exists where the solid volume fraction νs lies in the interval 0 < νs < 1; νs = 1 is likely first reached at point S (see Fig. 10.1) of the free surface. Downstream of this point a surface layer of cold lava develops and a new internal surface is formed which eventually will reach the rigid bed in point R (of Fig. 10.1). It is now advantageous to introduce the following nomenclature: • The interior interface separating hot lava from temperate lava (surface PQ in Fig. 10.1) is called hot-temperate transition surface (HTS). • The interior interface separating temperate lava from cold lava (surface SR in Fig. 10.1) is called cold-temperate transition surface (CTS). Remarks • The existence of a region of temperate lava is a conjecture. A simpler assumption would be, if the HTS and CTS would coincide so that fluid lava would directly change over to solid lava on the single interface HTS = CTS.

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• Because we assume the ‘model lava’ to be a pure substance, the fluid and solid constituents of temperate lava have the same temperature at the melting/freezing point. • HTS and CTS are non-material surfaces at which phase change processes take place. Therefore we request 1. Across HTS and CTS the temperature is continuous. 2. The phase change processes at the HTS and CTS are reversible. 3. On the basis of the viscous nature of lava flow, the tangential velocities of the solid and fluid constituents on the temperate side of the interfaces HTS and CTS are requested to be the same. 4. The tangential velocities of the fluid on the HTS and of the solid at the CTS are continuous. There are two other singular surfaces, which bound the flow, (i) the free surface and (ii) the basal surface. If precipitation of solid mass from ejected magma is ignored, then the free surface is material. Alternatively, the basal surface may be evolving in time and space because soil material may be eroded from the ground and entrained into the lava mass or lava may be deposited to the ground. 10.1.4. Earlier Theoretical Approaches Effusive eruptions produce lava flows with features that mainly depend on magma rheological properties, effusion rates, and topography (Hulme 1974 [36], Hess 1980 [33], Dobran & al. 1990 [17]). Such flows typically have very complex behavior, such as (i) non-Newtonian rheology, (ii) transitions between different flow regimes (iii) change in the topography during the flow, (iv) lava cooling governed by the coupling of energy transport in the flowing lava with heat transfer from the lava surface into the atmosphere by convection and radiation (Griffiths 2000 [24]), (v) dramatic changes in the kinematic and dynamic behavior with temperature variations because of the strong temperature dependence of the viscosity (Wylie & Lister 1998 [96], Costa & Macedonio 2003 [11]). A complete simulation of all these phenomena does not exist, and current models describe only a part of the observed phenomena. In avalanche and debris-flow research thermal effects are ignored, Pudasaini & Hutter 2004 [76]. The approaches summarized by them aim at describing the velocity distribution, stress and strain rates in the course of an avalanche sliding down a mountain topography. However, none of these describe the temperature behavior inside the avalanche. This simplification of the physical model description works well in snow avalanches, landslides and debris flows, but temperature variations

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will become important for hot materials such as lava or pyroclastic flows, particularly when phase changes occur. 10.1.4.1. Deterministic Models based on Transport Theory Lava flow is currently modeled by a set of highly nonlinear differential equations due to thermomechanical coupling, non-Newtonian behavior, presence of a free surface, and nonlinear boundary conditions in the energy equation (heat loss by radiation). During cooling, levees form laterally, and the flow may exhibit regime transitions. Current computational models commonly assume lava flows as a Bingham liquid with a non-null yield strength, (e.g. Miyamoto & Sasaki 1997 [64]; Robson 1967 [80]; Walker 1967 [91]). Its advantage is that the stress-strain rate relation is linear when the shear stress is sufficiently large. So the equations of motion for a Newtonian fluid can still be used. Isothermal Models Isothermal models of lava flows are a reasonable approximation in describing a limited segment of the flow, where the temperature can be considered approximately uniform (Dragoni 1989 [18]). If one assumes a flow rate and an initial temperature of the liquid at the eruption vent, the temperature decrease due to heat radiation and the consequent change in the rheological parameters can be computed along the flow. The equations of motion are solved by imposing a slow downslope change in the flow parameters, yielding flow thicknesses and velocities as functions of the distance from the eruption vent (Park & Iversen 1984 [72], Dragoni 1989 [18], Ishihara & al. 1989 [46]). Three-Dimensional Models The LavaSIM code (Fujita & al. 2004, [21]) is a three-dimensional CFD model for lava flow simulations based on the algorithm described by Hidaka & Ujita 2001 [34] and Hidaka & al. 2002 [35]. LavaSIM is able to account for three-dimensional convection, spreading and solidification, temperature dependent viscosity of magma and minimum spreading thickness. The model was applied to simulate lava flows at Mt. Fudji. Currently, the high complexity of the dynamics of actual lava flows has not permitted satisfactory development and efficient three-dimensional CFD codes. Thus, simplified models are usually adopted, e.g., 2D models and cellular automata (CA), see Crisci & al. 1986 [13]; Barca & al. 1987 [4]; DiGregorio & Serra, 1999 [23]; Cannataro & al. 1995 [7].

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Two-dimensional Models based on Depth-Averaged Equations Costa & Macedonio 2005 [12] presented depth-averaged equations obtained by integrating mass, momentum and energy equations over depth from the bottom to the free surface. Such shallow water equations have been applied to hazard assessments from flood simulation (Burguete & al. 2002 [6]) and tsunami propagation (Heinrich & al. 2001 [31]). For lava flows the viscosity is strongly temperature dependent. Costa & Macedonio 2005 [12] adopted a heuristic equation for the depth-averaged temperature using a simple exponential viscosity temperature relation and applied it to the 1991-1993 Etna eruption. The model showed fair agreement with field observations. Simplified Models Flowgo (Harris & Rowland 2001 [28]) is a self adaptative numerical model that follows lava elements down an open channel. Its basis is the estimation of lava velocity as a Bingham fluid flowing in a channel. At each step in the calculation, heat loss and gain are calculated in order to determine their effects on lava rheology. The model was applied and calibrated by simulating flow fields at Mauna Loa, Kilauea and Etna. Its principal limits are the drastic assumptions of a one-dimensionality and steadiness. 10.1.4.2. Probabilistic Models based on the Maximum Slope Macedonio & al. 1990 [58] assume that topography plays the major role in controlling the lava flow path. The identification of the different zones which are potentially invaded by lava is performed by computing the probability of invasion by using a Monte-Carlo algorithm. The flow is allowed to propagate along random paths, but cannot propagate upward. It must be noted that this scheme does not solve the physical equations but allows fast computations. The program was used to model the eruptions of Mt. Etna of 1983, 1985, 1987 and 1989. 10.2. Kinematics, Balance Laws, Saturation and Constituent Density Constraints Henceforth, scalars will be written as italics, vectors and tensors will be set in bold-faced letters, but if advantageous, indexed Cartesian tensor notation paired with Einstein’s summation convention will be used. Basic continuum mechanical concepts will be explained on the basis of mixture

Chapter10

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theory, from which the single phase expressions are deduced by reduction to just one component or restricting considerations to the mixture as a whole. Particular components of the mixture are identified by lower case Greek subscript labels α, β, etc. 10.2.1. Kinematics of Multi-Phase Mixtures We start from configurations and motions of a three-dimensional non-empty continuum B, the so-called material body. As we are dealing with mixtures, we must consider configurations one for each constituent and motions of a collection of n non-empty continua Bα (for each constituent one), that constitute the body B. An element, Xα , of continuum Bα is understood as a name tag for a specific material particle of constituent Kα . An open set of these elements is called Qα , and its surface is ∂Qα . Every Bα has a set of configurations {καθ }θ∈I with I ⊂ R that are bijective mappings from Bα into connected and compact regions, {Rαθ }θ∈I , in the Euclidian space, E 3 . In other words, one specific καθ assigns a vector X αθ to the material particle Xα at a fixed time θ, i.e., καθ : Bα → E 3 , X αθ = καθ (Xα ) ∈ Rαθ ,

for θ fixed .

(10.1)

X αθ is called the position vector of the corresponding material particle and Rαθ is the region occupied by constituent Kα . Now, let us choose any element from the set of configurations and call it a reference configuration, κα0 . Rαθ and X αθ corresponding to κα0 are then Rα0 and X α0 , respectively. κα0 can be different for every constituent, Truesdell 1984 [89]. The present configuration can be obtained by sequentially applying the mappings κ−1 α0 and καt (see Fig. 10.2), where καt is the mapping from Bα to the present configuration. However, all constituents are located at the same present configuration Rt , i.e. all Rαt fall together to one Rt and thus, in the present configuration we have only one position vector x. This can be expressed as x = καt ◦ κ−1 or x = χα (X α0 , t) , (10.2) α0 (X α0 ) , where καt ◦ κ−1 α0 denotes a mapping that results from sequentially applying and then καt . The symbol χα denotes a vector-valued function of first κ−1 α0 X α0 and t with value x. χα is called the motion of the constituent material body Bα . With these definitions we can now specify the constituent velocity v α , the material derivative dα (·)/dt following the motion of constituent Kα , the constituent acceleration aα , the constituent velocity gradient Lα ,

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Fig. 10.2. Material body Bα of constituent Kα . An open set Qα ⊂ Bα becomes in the reference configuration the material region Ωα0 ⊂ Rα0 ⊂ E 3 with boundary ∂Ωα0 , and the material particle Xα is mapped onto X α0 ∈ Rα0 . Similarly, in the present configuration, Qα is mapped into the open set Ωαt := ωα ∈ Rαt ⊂ E 3 with boundary ∂Ωαt := ∂ωα , and the material particle Xα is mapped onto x ∈ Rαt . {Eαi } (i = 1, 2, 3) is a basis for Kα in the reference configuration, and {ei } (i = 1, 2, 3) is a basis for Kα in the present configuration. Note all different Xα0 (α = 1, 2 , . . . , n) are mapped in the present configuration onto the same point x. So ωα and ∂ωα are the same region ω and boundary ∂ω for all α.

its symmetric, Dα , and skewsymmetric, W α , parts, the constituent deformation gradient F α and the constituent left Cauchy-Green deformation tensor , B α . They are defined as ∂ χα (X α0 , t) d = χ (X α0 , t) v ˆα (X α0 , t) := ∂t dt α X α0 dα χα χ−1 α (x, t) , t = ´α (x, t) , (10.3) =: v ˜α (x, t) =: x dt ∂ 2 χα (X α0 , t) d2 a ˆ α (X α0 , t) := = 2 χα (X α0 , t) 2 ∂t dt X α0 dα2 χα χ−1 α (x, t) , t =: a ˜α (x, t) , = (10.4) d2 t Lα := ∇v α = Dα + W α , F α :=

∂ χα (X α0 , t) , ∂X α0

D α := sym (∇v α ) , B α := F α F T α,

W α := skw (∇v α ) .

Bα = BT α .

(10.5)

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´ is an abbreviation for dα (·)/dt. Moreover, |X In these formulae (·) α0 α indicates that the indexed quantity is held fixed among the variables that are indexed; the functions v ˆα and v ˜α take the same values for the same X α0 , t and x evaluated from (10.2)2 , but they are different functions, as they depend on different position vectors. The same is true for the accelerations. 10.2.2. Balance Equations Let ψ α be the true density of a physical quantity. It stands, in particular, for the mass density, momentum density, energy density and entropy density per unit volume of constituent Kα at every point within a region ¯α , which is defined as the ωα . This density must be distinguished from ψ physical quantity per unit mixture volume. In the sequel we shall denote densities carrying an overbar as partial densities and those without a bar as true densities. The two are related by ¯ = να ψ , ψ α α

(10.6)

in which να is the volume fraction density of constituent α. The physical quantity contained in ωα is given by additivity over the volume as ¯α (x, t)dv . Gα (t) = (10.7) ψ ωα

We now postulate the balance law dGα (10.8) = Pα + Sα + Fα , dt where Pα , Sα , Fα are the production, supply and flux densities of Gα , which are given by ψ ¯ψ ¯ψ ¯ α (x, t) + γ Pα (t) = (x, t) dv , S (t) = σ π α α α (x, t)dv , ωα

ωα

(10.9) Fα (t) =

∂ωα

¯ ψ (x, t) · nda , φ α

(10.10)

¯ψ ¯ψ ¯ψ ¯ψ in which π α, γ α, σ α and φ α are partial densities per unit mixture volume and unit mixture area, respectively. The partial production density has ¯ψ been divided into two contributions. We interpret henceforth π α as the ψ ¯ ¯ α is the production (self) production rate of ψ α by constituent Kα , whilst γ ¯α by all constituents other than Kα . For mass, linear and angular rate of ψ

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230 Table 10.1. Balance Mass Momentum Moment of momentum Total energy Entropy

Chapter10

Densities for the constituent balance relations. ψ

ψα

πα

ψ

φα

ψ

ψ

σα

γα

ρα

0

0

0

ρα cα

ρα vα

0

tα

bα

mα

x × ρα vα ρα εα + 12 vα·vα

0

x × tα

x × bα

Mα

0

tα vα − qα

rα + bα · vα

eα

ρη σα

ρη γα

ρα ηα

ρη πα

φρη α

momentum, energy and entropy these quantities are defined in columns 3 ¯ψ and 6 of Table 10.1. Analogously, σ α is the partial density of the supply ¯α , a source of ψ ¯α outside the body volume ωα . rate of ψ ¯α through the boundary of the body The partial density of the flux of ψ ψ is defined per unit area on the surface ∂ωα and is denoted by φ¯ α . Thus, the total flux through the boundary ∂ωα into the body volume ωα is the surface integral (10.10). In writing down the flux quantity (10.10), Cauchy’s Lemma has been used to express the flux of ψ through the surface as a linear function of the unit vector perpendicular to the surface. The flux and supply terms are listed in columns 4 and 5 of Table 10.1. ¯ ψ (x, t, n) indicates that the flux density The bar in the representation φ α ¯ψ = in (10.10) is referred to a unit mixture area. To assume the relation φ α ψ να φ α , we are postulating in fact that areal and volume fractions are the same. With this we have ψ ψ ψ ψ ψ ¯ψ ¯ ψ , γ ¯α , π ¯ψ . (10.11) ψ α , φα , σ α ¯ α = να ψ α , π α , φ α , σ α , γ α The local balance law for a physical quantity of constituent Kα takes the form ¯α ∂ψ ¯ψ ¯ ¯ψ ¯ψ ¯ψ (10.12) −∇· φ α − ψα ⊗ vα − π α −σ α −γ α = 0 . ∂t The requirement that the mixture as a whole obeys local balance laws as if it were a single material implies the local balance law ∂ψ (10.13) = π ψ + ∇ · φψ − ψ ⊗ v + σ ψ , ∂t for the mixture variables which are related to the constituent quantities via

ψ ¯α , ¯α − ψ ¯α ⊗ v α , ψ := ψ φ φψ − ψ ⊗ v := (10.14)

ψ ψ ψ ψ π := π ¯α , σ := σ ¯α .

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Here and henceforth ρ :=

ρ¯α ,

ρv :=

231

ρ¯α v α .

(10.15)

define the mixture density and the barycentric velocity. The physical balance laws can be written as follows: ρα v α ) − ρ¯α cα = 0 , Rα := ∂ ρ¯α + ∇ · (¯

• mass :

(10.16)

¯α = 0, ¯α−b ρα v α ) − ∇ · (¯tα − ρ¯α v α ⊗ v α ) − m • momentum : Mα := ∂ (¯ (10.17) • energy :

E := ∂ (ρε) + ∇ · (q + ρεv) − t · ∇v − ρr = 0 ,

(10.18)

• entropy :

H := ∂ (ρη) + ∇ · (φρη + ρηv) − σ ρη = π ρη ,

(10.19)

in which we require tα = tT α , which holds true if exchange of angular momentum of constituent α is only due to exchange of linear momentum of the same constituent. In the above

¯ α = 0, (10.20) uα := v α − v , ρ¯α uα = 0 , ρ¯α cα = 0, m α

t :=

tI := q :=

¯tα −

ρ¯α uα ⊗ uα = tI + tD ,

¯tα ,

q¯ α −

α

tD := −

¯tα uα +

(10.21)

ρ¯α uα ⊗ uα ,

ρ¯α εα +

1 2

(10.22)

(uα · uα ) uα = q I + q D , (10.23)

q I :=

q¯ α , q D := −

¯tα uα +

ρ¯α εα +

1 2

(uα · uα ) uα . (10.24)

So, the local balance equations for mass, linear- and angular momentum, energy and entropy for the mixture as a whole are ∂ρ + ∇ · (ρv) = 0, ∂t

(10.25)

∂ (ρv) + ∇ · (ρv ⊗ v) = ∇ · t + ρg, ∂t

(10.26)

∂ (ρ(x × v)) + ∇ · (x × ρv ⊗ v) = ∇ · (x × t) + ρ(x × g), (10.27) ∂t ∂ 1 ( 2 ρv·v+ρ)+∇· ( 12 ρv · v + )v = −(∇·q)+∇·(vt)+ρ(v·g+r) (10.28) ∂t

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∂ρη + ∇ · (ρηv) = −∇ · φρη + πρη + ρςρη . ∂t

(10.29)

Notice that (10.27) can be replaced by the statement t = tT . 10.2.3. Saturation and Constituent Density Constraints Of the three variables ρ¯α , ρα and να , related by ρ¯α = να ρα , να is treated as an internal variable, for which the following balance law for the volume fractions να is postulated Nα =

∂να ¯ α = 0. + ∇ · (να v α ) − n ∂t

(10.30)

Here, n ¯ α is the volume fraction production rate density, for which a constitutive equation must be formulated. Because of the saturation condition, α να = 1, the volume fraction balance law for α = n is given by Nn := −

n−1

n−1

β=1

β=1

∂ ¯n = 0 . νβ + ∇ · v n − ∇ · νβ v n − n ∂t

(10.31)

Obviously, only να , α = 1, . . . , n − 1 are independent fields. In a binary mixture, N2 = −

∂ ν1 + ∇ · ((1 − ν1 )v 2 ) − n ¯ 2 = 0, ∂t

N1 =

∂ ν1 + ∇ · (ν1 v 1 ) − n ¯1 = 0 ∂t

(10.32)

(10.33)

are obtained. In a single constituent material ‘density preserving’ is defined as ρ˙ = 0, i. e., the density does not change along particle trajectories. Via mass balance this is equivalent to volume preserving. In a mixture, we commonly define density preserving of constituent α as the statement ρα = const., α = m + 1, . . . , n (0 ≤ m ≤ n) If the constituent densities ρα are independent constitutive variables, these constraint conditions reduce the number of independent constitutive variables by (n − m). In particular, comparison of the constituent mass balance laws with the volume fraction balance laws implies ¯α , c¯α = n

or

cα = nα ,

for all density preserving constituents.

(α = m + 1, . . . , n)

(10.34)

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10.3. Constitutive Laws for Different Aggregation States of Lava The basic concept about the gravity current in this work is that hot lava is a single constituent fluid, viscous and thermodynamically coupled. The material properties of this fluid ensue from a thermodynamic analysis of a heat conducting fluid, which is nearly density preserving. Similarly, cold lava will also be treated as a single-constituent material; however it possesses the properties of a solid, likely with some granular structure. In such materials the formation of an equilibrium heap at zero motion can be achieved if elastic and/or frictional, i.e. plastic effects are accounted for. Most likely, this material is porous with constant true density. By contrast, temperate lava will be treated as a binary mixture of nearly incompressible fluid and solid constituents at the same temperature. That the fluid and solid constituents possess the same temperature is a consequence that hot and cold lava are pure substances and temperate lava is at the melting point. Therefore, only the energy equation for the mixture as a whole must be formulated, and this energy equation is an evolution equation for the (volumetric) mass production rate density cs . Below we discuss in due order the postulations for the constitutive behavior of hot (§10.3.1), cold (§10.3.2) and temperate lava (§10.4). 10.3.1. Material Equations for Hot Lava Hot lava is treated here as a one component body ejected from the volcano and flowing down the mountain side. We assume that this material is a viscous, heat conducting and nearly density preserving fluid. A viscoelastic heat conducting compressible fluid is given by Ψ = Ψ(ρ, D, θ, ∇θ), where Ψ ∈ {, η, q, t}, D = 12 (L + LT ). If elastic effects are ignored, a dependence on ρ is dropped: Ψ = Ψ(D, θ, ∇θ), where Ψ ∈ {, η, q, t}. This is the constitutive class of viscous heat conducting fluids. Hot lava is (nearly) density preserving, however it is not recommended to model it at this stage of the development as strictly density preserving; the reason is that there would be no Clausius-Clapeyron equation in this case. Because at the hot-temperate transition surface the pressure dependence of the melting/freezing temperature may be important, we shall treat hot lava as a compressible fluid for which the compressibility in the bulk may be negligibly small, but density jumps across phase change surfaces will be accounted for.

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Constitutive relations must also satisfy the second law of thermodynamics. It can be shown that the second law implies the following properties: Define the Helmholtz free energy as ψ := − θη and decompose the Cauchy stress tensor t = −p1 + tE into a pressure tensor −p1 and the so-called extra stress tensor tE . Then, for a viscoelastic heat conducting fluid we have 1 1 2 ∂ψ d + pd , , p = ρ , dη = {, ψ, η, p} = fcts(ρ, θ), η = − ∂ψ ∂θ ∂ρ θ ρ {tE , q} = fcts(ρ, θ, D, ∇θ),

tE |E = 0,

q |E = 0.

(10.35) Here, and q |E are the values of t and q in thermodynamic equilibrium. We now wish to specify the material equations for tE , q and ψ if these are of the class Ψ(ρ, D, θ, ∇θ). • Cauchy stress tensor: An isotropic representation of tE of this class is given by tE |E

E

tE = α1+βD +γD 2 +δ ∇θ ⊗∇θ +ξ sym(∇θD ⊗∇θ)+ζ sym(∇θD 2 ⊗∇θ) (10.36) where α, β, γ, δ, ξ, ζ are scalar coefficients which are functions of S, where S is given by S = {ID , IID , IIID , ∇θ, Isym(∇θD⊗∇θ) , IIsym(∇θD⊗∇θ) , IIIsym(∇θD⊗∇θ) , Isym(∇θ D2 ⊗∇θ) , IIsym(∇θD2 ⊗∇θ) , IIIsym(∇θD2 ⊗∇θ) , θ, ρ }.

(10.37)

The symbols I, II, III are the principal invariants of the subscripted second order tensors. For simplicity we assume that the Cauchy stress tensor does not depend on the temperature gradient. Thus, (10.36), (10.37) reduce to tE = α1 + βD + γD 2 ,

(10.38)

where α, β, γ are functions of SRed = {ID , IID , IIID , θ, ρ}.

(10.39)

It is advantageous to separate the spherical and deviatoric parts of the extra stress and stretching tensors

tE = −pN 1 + tE , D = e1 ˙ + D ,

pN := − 31 (tr tE ), e˙ := 13 (tr D),

tr tE = 0, tr D = 0.

(10.40)

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Substituting these in (10.38) and separating the spherical and deviatoric components yields pN = −α − β e˙ − γ e˙ 2 − 13 γ(tr D2 ), (10.41)

˙ + γ(D )2 . tE = − 13 γ(tr D 2 )I + (β + 2γ e)D

We now simplify these relations by conjecturing that tE and D are collinear, thus γ ≡ 0. Therefore, the non-equilibrium pressure pN and the extra-stress deviator are given by

tE = βD ,

pN = − (α + β e) ˙ ,

(10.42)

in which α and β are functions of e, ˙ IID and IIID such that α|E = 0. Further simplification is achieved by assuming that ˙ α=α ˜ (e, ˙ IID , ρ, θ)e,

α ˜ + β = ζ(e, ˙ ρ, θ),

1 2β

= µ(IID , ρ, θ).

(10.43)

With this, the parameterizations (10.42) take the forms pN = −ζ(e, ˙ ρ, θ)e, ˙

tE = 2µ(IID , ρ, θ)D .

(10.44)

ζ and µ are, respectively, a non-linear bulk and shear viscosity. Note, a dependence of µ on IIID is omitted. We wish to make the following remarks: (i) A density dependence of

the bulk and shear viscosity is generally omitted, pN = −ζ(e, ˙ θ)e, ˙ tE = 2µ(IID , θ)D . (ii) The assumption that the coefficient γ of the constitutive relation is negligibly small has strong support in the literature. Polycrystal melts of metals and alloys are modeled with constitutive relations of the class (10.44). Similarly, polymers and flows of glycerine with particle suspensions and glacier ice, which is a polycrystalline material, very viscous heat conducting fluid, is described by a law of class (10.44). (iii) To ignore a dependence of µ on the third invariant has ample support in the plastic deformation of (polycrystalline) metals and alloys. It has also formal support in deformation experiments of ice. (iv) Because hot lava is a melt of a polycrystal we believe that (10.44) may have a good chance to be a realistic stress model, simply on the basis of analogy. (v) The law (10.44)

is non-linear, in general, despite the fact that tE is affine to D. Linearity only prevails when µ = const. The phenomenological parameter µ is a function of two scalars, IID and θ. It can be product-decomposed as follows: µ(IID , θ) = g(IID )B(θ),

(10.45)

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where good choices for g and B are g = (IID )

n−1 2

,

B=

β20

Q . exp − kB θ

(10.46)

Here Q is the activation energy, kB the Boltzmann constant if Q is given in eV and θ the Kelvin temperature. Moreover, n is the power law exponent. • Fourier heat law: The most general isotropic law for the heat flux vector of the class Ψ(ρ, D, θ, ∇θ) has the form (10.47) q = − κ + κ1 D + κ2 D2 ∇θ, where (κ, κ1 , κ2 ) are scalar functions of the parameter set S in (10.37). We now postulate that the heat flux vector does not depend on the strain rate tensor D: q = −κ (θ, ∇θ) ∇θ,

(10.48)

in which κ = κ(θ, ∇θ) is the heat conductivity. However, a dependence on ∇θ is unlikely. • Thermodynamic quantities: We postulate the internal energy of the compressible fluid in the form θ ¯ θ¯ cv (θ)d (10.49) (ρ, θ) = 0 (ρ) + θ0

with a specific heat which depends linearly on temperature cv (θ) = c1v + c2v θ,

(10.50)

implying (ρ, θ) = c1v (θ − θ0 ) +

c2v 2 (θ − θ02 ) + 0 (ρ). 2

(10.51)

θ − θ0 θ0 θ − θ0 ψ 0 c2v + (θ − θ0 )2 − 0 (ρ) + (ρ)θ, (10.52) − ψ = c1v θ ln θ θ0 2 θ0 θ0 η

0 − c2v (θ − θ0 ) + η0 , = −c1v ln θθ0 + 2 θ−θ θ0

η0 := η|θ=θ0 = − ψθ00 (ρ) + ψ0 = 0 − η0 θ0 ,

0 (ρ) , θ0

(10.53)

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ψ˜0 (v) θ ∂˜0 (v) θ − θ0 − , ∂v θ0 v θ0 (10.54) in which v := 1/ρ is the specific volume. We introduce the isothermal compressibility as p = ρ2

∂ψ = ρ2 ∂ρ

κθ = −

θ ∂ψ0 ∂0 θ − θ0 (ρ) − (ρ) θ0 ∂ρ ∂ρ θ0

=

1 ∂v 1 1 1 . =− =− 2 v ∂p v ∂p/∂v ∂ ˜0 θ − θ0 ∂ 2 ψ˜0 θ v − ∂v2 θ0 ∂v 2 θ0

(10.55)

With the choices (1)

(2)

˜0 = c0 + ˜0 (v − v0 ) + 12 ˜0 (v − v0 )2 , (1) (2) ψ˜0 = d0 + ψ˜0 (v − v0 ) + 12 ψ˜0 (v − v0 )2 ,

(10.56)

(1) (1) (2) (2) in which c0 , d0 , ˜0 , ψ˜0 , ˜0 , ψ˜0 are constants, we obtain

1 κθ = − θ − θ 0 (2) (2) θ v ˜0 − ψ˜0 θ0 θ0

(10.57)

and

(1) (2) 0 ˜(1) + ψ˜(2) (v − v0 ) θ , − ψ p = ˜0 + ˜0 (v − v0 ) θ−θ 0 0 θ0 θ0 (1) (1) (2) 1 1 ˜(2) ˜ η0 = − θ0 (d0 − c0 ) + (ψ0 − ˜0 )(v − v0 ) + 2 (ψ0 − ˜0 )(v − v0 )2 . (10.58) The number of parameters in these expressions can be reduced by as(2) (2) (1) (1) (2) (2) 0 , ψ˜0 )|| with (˜ 0 , ψ˜0 ) = suming that (i) ||(˜ 0 , ψ˜0 )(v − v0 )|| ||(˜ (1) ˜(1) O(1/ε), 0 < ε 1, with ε so small that ˜0 , ψ0 can be ignored in com(2) (2) (2) (2) 0 for some positive parison to (˜ 0 , ψ˜0 )(v − v0 ). (ii)c0 = d0 , (iii)ψ˜0 = λ˜ 3 λ. With these restrictions we get for (v − v0 ) = O(ε ) κθ = O(ε), (2)

p = ˜0 (v − v0 ) (2)

η0 =

˜ − 21 θ00

θ−θ0 θ0

− λ θθ0

= O(ε),

(10.59)

2

(λ − 1)(v − v0 ) .

10.3.2. Material Equations for Cold Lava In what follows solid lava will be modelled as an isotropic elasto-visco-plastic one-constituent continuum with vanishing porosity. Viscoplastic behaviour will be modeled as a hypoplastic frictional material, first formulated in a thermodynamic setting by Svendsen & al. 1999 [86]. According to this procedure “the phenomenological generalization of the Mohr-Coulomb

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model for internal friction in a granular material at low energy and high grain volume fraction, [here νs = 1] [...] is based upon a Euclidian frame indifferent, stress-like, symmetric-tensor-valued spatial internal variable Z. [...] In the current phenomenological setting, Z is modeled constitutively by an incremental relation of the formb ◦

Z:= Z˙ − [Ω, Z] = Φ

or

◦

Z :=Z −Φ = 0.

(10.60)

Here, Φ represents the constitutive part of (10.60), and the left hand side is a so-called ‘co-rotational’ objective time derivative of the spatial tensor field Z, Ω being the corresponding spin. For example, Ω is given by W in the Jaumann case (relevant, e.g. to the case of hypoplasticity)”, Svendsen & al. 1999 [86]. To this we add that the evolution equation (10.60) has no flux and no external source terms. Apart from (10.60) the governing equations are the physical balance laws • momentum:

ρv˙ − ∇ · t − ρg

• energy:

ρ˙ − ∇ · q − tr (tD) − r = 0,

= 0, (10.61)

in which ρ.v, t, g, , q, D, r are the density, velocity vector, Cauchy stress tensor t = tT , gravity vector g, internal energy, heat flux vector, stretching tensor D = sym∇v, and radiation density, later be set to zero. If x = χ(X, t) then F = ∂χ/∂X is the deformation gradient and L := F˙ F −1 = ∇v = D+W is the spatial gradient of the velocity field and W = skwL the vorticity tensor. Balance of mass has not been written in equations (10.61), because ρ = ρR (det F )−1 , where ρR is the reference density. Thus, since F is necessarily one of the independent constitutive variables and ρR will be taken as a constant, ρ can be treated as known, once F is determined. Constitutive equations are postulated to be of the class Ψ = ˙ ∇θ), where B = F F T is the left Cauchy-Green deΨ(B, D, Z, θ, θ, formation tensor. The second law of thermodynamics is based here on the entropy inequality ρη˙ + ∇ · φ − ς ≥ 0,

(10.62)

in which η is the entropy and φ the entropy flux, whilst ς is the entropy supply and η and φ are constitutive variables. Imagine that concrete, b We use the Lie, [·, ·], and Jacobi, ·, · brackets defined by [A, B] := AB − BA, A, B := AB + BA.

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explicit constitutive relations for {t, q, , η, φ, Φ} are prescribed and assume that they are substituted in (10.61) and (10.60). Then, (10.61) and (10.60) define 10 functional evolution equations for the 10 unknown fields {x, θ and Z}. The entropy inequality must be satisfied for all fields {x, θ and Z}, which satisfy the balance laws (10.61), (10.60) and constitutive relations for {t, q, , η, φ, Φ}. A theorem of Liu then says that the second law of thermodynamics is fulfilled, if the extended entropy inequality ρη˙ + ∇ · φ − ς − λv · {ρv˙ − ∇ · t − ρg} ◦

−λ {ρ˙ + ∇ · q − tr (tD) − r} − tr {ΛZ (Z −Φ)} = π ≥ 0

(10.63)

is satisfied for arbitrary differentiable fields. Here, λv , λ , ΛZ are Lagrangean multipliers, which scalarly premultiply the balance statements (10.61) and (10.60). The exploitation of this extended inequality has been performed by Svendsen et al. 1999 [86]. We quote here their results in a number of propositions. Proposition 10.1. For any admissible thermodynamic process of the outlined constitutive model of cold lava the following relations hold true: (1) The Lagrange multipliers are given by λv = 0,

ΛZ = PZ ,

ˆ (θ, θ), ˙ λ = λ

(10.64)

˙ is a materially independent (universal) function of the ˆ (θ, θ) where λ empirical temperature and its time-derivative. It is called coldness function. (2) Entropy, internal energy, entropy flux and heat flux vectors are related by the one-forms ρdη = λ ρd + P,

where

dφ = λ dq + F , ∂λ q · d∇θ, P = Pθ dθ + PB · dB + PZ · dZ − ∂ θ˙ ∂λ ∂λ ˙ dθ}. F = q{ dθ + ∂θ ∂ θ˙

(10.65) (10.66) (10.67)

(3) If the rotation tensor Ω = W = skwL, then [PB , B] = skw(∇θ ⊗ Pg ),

[PZ , Z] = 0,

(10.68)

but if Ω = W , then [PB , B] + [PZ , Z] = skw(∇θ ⊗ Pg ).

(10.69)

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(4) Heat flux vector and entropy flux vector are collinear, φ = λ q,

(10.70)

ˆ (θ) = 1/θ corresponds to the Duhem relation, which in the case λ = λ ¨ller 1985 [66]. see Mu (5) With the above properties the residual entropy inequality takes the form ∂λ ˙ sym(q ⊗ ∇θ) · D π := Pθ θ + PB , B + λ t + ∂ θ˙ ∂λ q · ∇θ + PZ · Φ ≥ 0. + (10.71) ∂θ Relations (10.65)1 and (10.66) can be combined to yield the Gibbs relation that is appropriate to this theory, namely ∂λ ρ −λ ρd − dη = Pθ dθ + PB · dB + PZ · dZ − q · d∇θ. (10.72) λ ∂ θ˙ This suggests to introduce the generalized Helmholtz free energy ψ = − η/λ and using this expression to eliminate from (10.72). This implies Proposition 10.2. (1) The generalized Helmholtz free energy is not a function of D, ∂ψ = 0. (10.73) ∂D (2) ψ determines all elements of the one-form P, the non-equilibrium entropy η and the heat flux vector as follows: ⎫ ⎧ ∂ψ ⎪ ∂ψ ⎪ ⎪ ⎪ ⎬ ⎨ ∂ 1 ˙ ∂ θ ∂θ , (10.74) − Pθ = ρλ 1 ⎪ ∂ ∂ 1 ∂θ λ ⎪ ⎪ ⎪ ⎭ ⎩ ∂θ λ ∂ θ˙ λ ∂ψ PB = −ρλ , (10.75) ∂B ∂ψ PZ = −ρλ , (10.76) ∂Z Pθ 1 ∂ψ ∂ψ 1 + , (10.77) η=− =− ∂ ∂ 1 ∂ θ˙ 1 ∂θ ρλ ˙ λ ∂θ λ ∂θ ∂ψ λ . (10.78) q = ρ ∂λ ∂∇θ ∂ θ˙

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We remark that all quantities (10.74) - (10.78) are known if the coldness function is known and the generalized Helmholtz free energy is satisfied. Moreover, if θ˙ is not among the independent constitutive variables, then it can be shown that Pθ = 0. Moreover, λ = 1/θ, where θ is the Kelvin temperature and ∂ 1 ∂ψ ( ) = 1 and η = − . ∂θ λ ∂θ (10.78) implies ∂ψ/∂∇θ = 0 for q to be regular, but q is now independent of ψ. Finally, the above proposition gives no information about Φ, the production rate density of Z. Thermodynamic equilibrium Further restrictions are obtained by exploring the imbalance (10.71) for thermodynamic equilibrium processes for which no entropy can be produced. If the independent constitutive variables are partitioned into ‘equilibrium variables’, e = (θ, B, Z), and ‘process vari˙ D) with n = 0 in equilibrium, then the entropy producables’, n = (∇θ, θ, Ê (e) = π ˆ (e, n = 0) = 0, which tion in equilibrium must vanish, viz., πE = π implies, owing to (10.71), ΦE = Φ(e, 0) = 0 ⇒ Φ = ΦN (e, n), ΦN |E = 0. Here, fE and fN denote equilibrium and non-equilibrium parts of f = fE + fN and (·)|E will signify evaluation of (·) at equilibrium. Since π in the imbalance (10.71) is a differentiable, non-negative function of the process variables n with minimum value at n = 0, we have ∂π =0 ∂n E

n·

and

∂ 2 π n ≥ 0. ∂n∂n E

(10.79)

These assignments imply Proposition 10.3. (1) It follows from the isotropy of the vectorial constitutive function ˙ ∇θ, B, Z, D) that the heat flux vector in thermodynamic equiˆ (θ, θ, q librium vanishes qE = 0

⇒

q = qN .

(10.80)

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(2) Performing (10.79)1 yields, in view of (10.80)1 , ∂π ∂Φ = 0 ⇒ ∂∇θ E ∂∇θ = 0, E ∂π ∂Φ = 0 ⇒ P | = −P · , θ E Z|E ∂ θ˙ E ∂ θ˙ E T (10.81) ∂π ∂Φ = 0 ⇒ λ t = −P , B − P , E B|E Z|E E ∂D E ∂ D |E P|E = −PZ|E · ∂∂Φ dθ + P · dB + P Z · dZ. B|E θ˙ E We shall now write 1 , (10.82) θ where from now on θ is identified with the Kelvin temperature. The equilibrium version of Propositions 10.2 & 10.3 then read: ˙ λ = λ E (θ) + λ N (θ, θ),

λ N (θ, 0) = 0,

λ E =

Proposition 10.4 (equilibrium). (1) The equilibrium Helmholtz free energy function is a function of θ, B, Z, viz., (10.83) ψE = ψÊ (θ, B, Z). (2) In thermodynamic equilibrium, with the help of (10.75) and (10.76), (10.68) reduce to ∂ψE ∂ψE ∂ψE ∂ψE B=B , Z=Z , if Ω = W (10.84) ∂B ∂B ∂Z ∂Z and ∂ψE ∂ψE ,B + , Z = 0, if Ω = W . (10.85) ∂B ∂Z (3) ρ ∂ψE ρ ∂ψE , PZ|E = − , PB|E = − θ ∂Z θ ∂B ∂ψE ∂Φ ∂ψE + · , ηE = − ∂θ ∂Z ∂ θ˙ E T ∂Φ ∂ψE ∂ψE ,B + ρ tE = ρ ∂B ∂D |E ∂Z

(10.86) (10.87) (10.88)

(4) The production rate density Φ of the stress like variable Z satisfies the conditions ∂Φ ˙ ∇θ, B, Z, D) = 0. (10.89) Φ = ΦN (θ, θ, ∂∇θ E

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The equilibrium Cauchy stress consists of two terms, ∂ψE ρ ,B is the elastic contribution, (10.90) ∂B T ∂Φ ∂ψE is the frictional equilibrium stress. (10.91) ρ ∂D |E ∂Z It is our thesis that the static stress in a volcanic heap is much more of frictional than elastic nature. So, in practical applications (10.91) must be kept, whilst (10.90) may be set to zero. So, (∂ψE /∂B) = 0 may perhaps be a tolerable assumption, whilst ∂Φ/∂D|E = 0, ∂ψE /∂Z = 0 are mandatory ˙ |E = 0. This will requirements. On the other hand, we may set (∂Φ/∂ θ) bring back relation (10.87) to the classical formula ηE = −∂ψE /∂θ. In anticipation of results to be derived in non-equilibrium we choose the Helmholtz free energy in equilibrium in the form E E (θ, IB , IIB ) + ψ12 (θ, IIIB , Z). ψE = ψ1E (θ, B, Z) = ψ11

(10.92)

If we choose E (θ, IB , IIB ) = C1 (IB − 3) + C2 (IIB − 3), ψ11 ρR E E ψ12 (θ, IIIB , Z) = ψ˜12 θ, ,Z , ρ

(10.93)

with constant C1 and C2 the elastic equilibrium stress tE E takes the form E E tE E = (tE )1 + (tE )2

= 2ρ (C1 + IB C2 )B − C2 B 2 + ρR

E ∂ ψ˜12 ∂(ρR /ρ)

! 1.

(10.94)

The first term is due to Mooney & Rivlin (cf. Rivlin & Saunders 1951 [77]) the last term is of frictional origin. If C1 = 12 µ ¯( 12 +β), C2 = 12 µ ¯( 12 −β), then the choice β = 1/2 leads to (tE E )1 = µB,

µ := ρ¯ µ.

(10.95)

which is known as Neo-Hookean behaviour. The last term in (10.94) ˜E gives a further contribution (tE E )2 which is due to ψ12 and defines an elastic E ˜ pressure due to friction. It only arises because ψ12 has been made to depend on (ρR /ρ).

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To determine the contribution of the frictional stress to the equiE E librium stress (10.87)2 ∂ψE /∂Z is needed. Writing ψ˜12 as ψ˜12 = E ψ˜12 (θ, ρR /ρ, IZ , IIZ , IIIZ ) one may deduce E E E E ∂ ψ˜12 ∂ ψ˜12 ∂ ψ˜12 ∂ ψ˜12 = + IZ + IIZ 1 ∂Z ∂IZ ∂IIZ ∂IIIZ E E E ∂ ψ˜12 ∂ ψ˜12 ∂ ψ˜12 Z+ − Z + IZ Z 2. (10.96) ∂IIZ ∂IIIZ ∂IIIZ With this representation and a given function Φ the third equilibrium stress contribution is computable as E ∂Φij ∂ ψ˜12 E , (10.97) (tE )3ij = ρ ∂Dkl ∂Zkl E

E which is determined once the constitutive parameterization for Φ and ψ˜12 are chosen.

There remains to choose a parametrization for the production rate density ˙ ∇θ, B, Z, D). The selection Φ(θ, θ, ˆ θ, ˙ ∇θ, B, Z)D + N ˙ ∇θ, B, Z)|D|, ˆ (θ, θ, Φ = L(θ, (10.98) (in which L is a fourth order tensor and N is second order with symmetries ˆ = 0) Lijkl = Ljikl = Lijlk , Nij = Nji ) is able to generate hypoelastic (N ˆ and hypoplastic (N = 0) stress relations. The first term on the righthand side of (10.98) is linear in D, the second is non-linear, since |D| := (D · D)1/2 . In the literature, restricted dependencies of (10.98) with L = ˆ ˆ (B, Z) have been proposed. It follows from (10.98) that L(B, Z), N = N ∂Φ ˆ +N ˆ (·) ⊗ D . = L(·) (10.99) ∂D |D| The equilibrium stress tE can then be formally computed by applying formula (10.87)2 . This leads to ∂ψE ∂ψE ∂ψE D , B + ρLT|E + N |E · . (10.100) tE = ρ ∂B ∂Z ∂Z |D| In this relation, the first term on the right-hand side is the elastic stress, the second represents the hypoelastic contribution and the third is the nonlinear hypoplastic part. With (10.92), (10.94) and (10.97) tE in (10.100) may also be written as ! ˜E ∂ ψ˜12 D E E E T ∂ ψ12 + N |E · . (10.101) tE = (tE )1 + (tE )2 + (tE )3 + ρL|E ∂Z ∂Z |D|

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Note that neither (10.99) nor (10.100) are differentiable with respect to D at D = 0 through the non-linear term involving the derivative D/|D| of the Euclidian norm, which is not defined in the limit as D → 0. For a procedure how this can be handled, see Schneider & Hutter 2009 [83]. The literature on hypoplasticity is vastc and concentrates on the parameterization of the operators L and N . These have the general structure ˆ B, Z) = c1 (θ, B)(tr Z)14 + c2 (θ, B) Z ⊗Z , L(θ, (tr Z ) 2 2 (Z ) Z ˆ N (θ, B, Z) = c3 (θ, B)(tr Z) tr Z + c4 (θ, B) dev , (tr Z )

(10.102)

used by Wu & al. 1996 [90] in modeling various types of soils. In the " ij δkl , and the maabove, 14 represents the fourth order unity tensor, 14 =δ terial coefficients c1−4 depend on B through the void ratio; a temperature dependence has been added here for the application of lava flow. Non-Equilibrium Constitutive Parameterization Proposition 10.2 determines non-equilibrium values for Pθ , PB , PZ , η ˙ and ψ(θ, B, Z, θ, ˙ q) are prescribed, whilst and q, if the functions λ (θ, θ) Propositions 10.3 and 10.4 give the corresponding expressions in equilibrium. In line with the results obtained up to now, we may choose (1) λ =

1 θ

˙ − λ N θ,

λ N |E ≥ 0,

(2) t = tE + tN ,

tE given in (10.87)2 & (10.100) tN |E = 0,

(3) q = q E + q N ,

q E = 0,

q N |E = 0,

(4) Φ = ΦE + ΦN ,

ΦE = 0,

ΦN |E = 0,

∂ ΦN ∂∇θ

= 0. E

(10.103)

The choice ˙ − ψ = ψ1 (θ, B, Z, θ)

1 λ

∂λ ∂ θ˙

ψ2 (θ, g 2 ),

g 2 := ∇θ · ∇θ

(10.104)

produces, via (10.78) 2 ∇θ = −κ(θ, g 2 )∇θ, q = −2ρ ∂ψ ∂g 2

κ|E ≥ 0,

(10.105)

in which κ := 2ρ∂ψ2 /∂g 2 is the heat conductivity. Obviously (10.105) is a non-linear version of Fourier’s law of heat conduction. c See

for example Kolymbas 1991 [49], Kolymbas 1993 [50], Wu & al. 1996 [90].

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To construct a parametrization of the non-equilibrium viscous stress we ˙ ∇θ, D). Its isotropic representamake the restrictive choice tN = ˆtN (θ, θ, tion takes the form tN = K1 ID 1 + K2 D + K3 D2 + K4 G + K5 sym(GD) + K6 sym(GD 2 ), (10.106) (j = 1, . . . , 6) are functions of S = where G = ∇θ ⊗ ∇θ and K j ˙ θ, θ, ID , IID , IIID , IG , IGD , IGD2 , in which IA , IIA , IIIA are the principal invariants of A. Notice that with (10.106) the requirement tN |E = 0 is automatically satisfied. To this we add the following remarks: First, most parametrizations in the literature ignore a dependence of tN on G and also omit the D 2 term of the ensuing Reiner-Rivlin constitutive law. This then yields tN = ζID 1 + 2µ(D − 13 ID 1).

(10.107) ˙ ID , IID , IIID . in which ζ = (K1 + K32 )(S ), µ = 12 K2 (S ), S = θ, θ, The coefficients ζ and µ may be interpreted as non-linear bulk and shear m viscosities. Granular shear experiments suggest µ ∝ IID , −1 < m < 0. Then, the dominant dependence on µ is likely on IID with infinite viscosity µ as IID → 0. Such power laws are sometimes referred to as infinite viscosity laws. Because of the singularities of ζ and µ it is computationally advantageous to postulate the constitutive relation for non equilibrium stress inversely as follows

(10.108) D = Fb (ItN )1 + Fs tN − 13 (ItN )1 ,

where Fb and Fs are now a bulk fluidity and a shear fluidity. Second, the above qualitative analysis suggests to postulate that ˆ ID ), ζ = ζ(e, µ=µ ˆ(e, IID , IIID ) (10.109) ˆ Fs = Fˆs (e, IIt , IIIt ) Fb = Fb (e, It ), where D = D − 13 ID 1, tN = tN − 13 It 1, and e is a measure of the porosity, in soil science the so-called void ratio. Explicit formulae for ζ and µ are proposed in Schneider & Hutter 2009 [83], pp 174-182. 10.4. Theory for Temperate Lava 10.4.1. Preliminaries We shall consider temperate lava as a mixture of fluid and solid constituents at the freezing/melting temperature. This binary mixture should be regarded as frictional because such contacts will certainly be present in the

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vicinity of the CTS where the solid volume fraction is close to, or at, unity. A mixture theory of this complexity has been presented very recently by Schneider & Hutter 2009 [83] with a brief summarizing review by Hutter & Schneider 2010 [43] and an application to debris avalanches by Hutter & Schneider 2010 [44]. We can only give a brief account of this theory. A large part of the ensuing presentation of the model requires detailed notation and painstaking reproduction of rather complicated results from [43], [44]. We shall at first address general facts and results mentioned in [43], [44] which are valid for arbitrary unidentified constituents Kα (α = 1, ..., n is the identifier of the constituent). The balance laws of mass and linear momentum for each constituent and the energy and entropy balances of the mixture can be written in the forms ∂ ρ¯α + ∇ · (¯ ρα v α ) − ρ¯α cα = 0; Rα := (10.110) ∂t Mα :=

∂ ρ¯α v α ¯ α = 0. + ∇ · (¯ ρα v α ⊗ vα − ¯tα ) − ρ¯α g α − m ∂t

E :=

∂ρ + ∇ · (q + ρv) − t · ∇v − ρr = 0; ∂t

(10.111) (10.112)

∂ρη + ∇ · (φρη + ρηv) − σ ρη = πρη . (10.113) ∂t Volume fractions να are treated as generic variables, for which we postulate the flux free balance law (10.30). Moreover, just as for cold lava, we will also introduce for each constituent a frictional stress like variable according to ◦ ◦ ¯ ¯ α = 0, ¯ α Ωα ). ¯α − Φ ¯ α = ∂ Z α + (∇Z ¯α − Z ¯ α )v α − (Ωα Z Z Zα := Z ∂t (10.114) Formula (10.114)2 is an objective time derivative and Ωα = skwLα is the skew-symmetric spin tensor of constituent α. It is natural to apply the frictional tensors Z α only to the solid constituents while they are set to zero for the fluid ones. The mixture model which we envisage to be appropriate for temperate lava is also assumed to satisfy the saturation condition n

να = 1. (10.115) H :=

α=1

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It states that there is no material free space within the mixture. Equation (10.115) means that only n − 1 volume fractions are independent: νn = n−1 1 − α=1 να . Obviously, then ⎞ ⎞ ⎛ ⎛ n−1 n−1

∂ ⎝ ⎠ ¯ n = 0. (10.116) νβ + ∇ · v n − ∇ · ⎝ νβ v n ⎠ − n Nn := − ∂t β=1

β=1

We also assume that (n − m) constituents are density preserving, ρα = const,

α = m + 1, . . . , n

(0 ≤ m ≤ n),

(10.117)

i.e. these true densities remain constant. Condition (10.117) reduces the number of independent constitutive variables by (n − m). In particular, comparison of (10.110) with (10.117) yields c¯α = n ¯ α , or cα = nα (α = m + 1, . . . , n) for all density preserving constituents. This is an important result for temperate lava since for the solid cs (α = s) is the solid production rate and for the fluid cf = −cs is the annihilation (negative production) rate of the fluid due to freezing. Furthermore, in a mixture of which all constituents are density preserving the barycentric velocity is not necessarily solenoidal. This follows from ρ¯α = να ρα , because with constant true density the partial density can still vary owing to variations of the volume fractions. Since in this case (10.111) and (10.117) collapse to the same equations, one of them is simply omitted as a field equation. Schneider & Hutter [83] chose the constitutive equations to be of the form ˆ Ψ = Ψ(S), ˙ ∇θ, ρ, ∇ρ, ν , ∇ν, v , B, D, W ,Z S = θ, θ,

ν := ν1 , . . . , νn−1 ρ := ρ1 , . . . , ρm , := B 1 , . . . , B n , v := v 1 , . . . , v n , B := D 1 , . . . , Dn , := W 1 , . . . , W n , D W := Z 1 , . . . , Z n , Z in which every dependent constitutive quantity of the set

¯α ¯ α , q, φρη , tα , Φ ¯α, m Ψ = , η, cα , n is an isotropic function of the variables S.

(10.118)

(10.119)

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10.4.2. Entropy Principle The union of the balance laws of mass, linear momentum, volume fraction and stress like second order tensor variables and constitutive relations for the variables (10.119) form a set of non-linear partial differential equations for the variables ¯ 1, . . . , Z ¯ n. θ, ρ1 , . . . , ρm , ν1 , . . . νn−1 , v 1 , . . . , v n , Z ' () * ' () * compr.

(10.120)

saturation

These equations must be in conformity with the second law of thermodynamics, here formulated as an entropy inequality, ∂(ρη) − ∇ · (φρη − ρηv) − σ ρη ≥ 0. ∂t

(10.121)

Satisfaction of (10.121) subject to the constraints (10.110)-(10.114) and (10.116) is equivalent to the fulfillment of the extended inequality ∂(ρη) ∂t

−

m − ∇ · (φρη − ρηv) − σ ρη − α=1 λρα Rα n−1 λνα · Mα − λ E − α=1 λνα Nα − λνn Nn − λZ α · Zα ≥ 0

(10.122)

for arbitrary independent fields, Liu [56]. In this imbalance the various λ’s are Lagrange multipliers to be determined together with restrictions of constitutive quantities as results of the exploitation of (10.122). The sum n mation in (10.122) must (and will henceforth) be interpreted as α=1 . The explicit exploitation of inequality (10.122) is long and tedious; in principle, it works as follows: The source free balance laws (10.110)-(10.114) and (10.116) are substituted in (10.122) (with σρη = 0) together with the constitutive relations (10.119) of the form (10.118) and in the emerging inequality all differentiations of constitutive quantities are performed. In this final expression it is convenient to introduce the one-forms P=

K

PxI dxI := d(ρη) − λ d(ρ),

(10.123)

I=1

F=

K

I=1

F xI dxI := dφ(ρη) + λ (dq) −

(d¯tα )λνα ,

(10.124)

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in which xI (I = 1, . . . , K) are the constitutive variables, and the auxiliary variables ρα := λρα + λvα · v α , αρI := ρα + 12 λ (uα · uα ), ν ρ ν ανI := να + 12 ρα (uα · uα ) − k, α := ρα α + λα , T −1 k := 12 λ ρn (un · un ), Γ := λ ρ t + (η − λ )1 = Γ , s := −νn , s := sv n + ρn Γun , (10.125) as well as the so-called extra entropy flux vector

¯tα λvα . k := φρη + λ q − (10.126) After this formidable transformation the entropy inequality can be expressed as ∗

X(Y) · Y +Γ(Y) ≥ 0,

∗

∀ Y∈ RKD .

(10.127)

Here, Y collects the independent constitutive variables arranged as a vector ∗

of dimension K, Y are time and space derivatives of Y, again arranged as a vector of dimension KD > K; moreover X is a matrix of dimension K × KD and ‘·’ is the scalar product in RKD . Finally, Γ(Y) is a scalar function of ∗

Y. Since (10.127) is linear in Y, its solution is given by ∗

∗

X(Y) · Y ≡ 0 ∀ Y∈ RKD ,

and

Γ(Y) ≥ 0.

(10.128)

Relations (10.128)1 are called Liu-identities, whilst (10.128)2 is referred to as residual entropy inequality. 10.4.3. Inferences Implied by the Liu-Identities (10.128)1 represents a great number of identities. To be able to derive explicit results from them, the following assumptions are needed: (i) λ is a function of the empirical temperature and its total time derivative ˆ (θ, θ). ˙ The function λ ˆ (θ, θ) ˙ is called the coldness function. (ii) The λ = λ Lagrange multiplier of the linear momentum of constituent α is selected as λvα = −λ uα , in which uα := v α − v is the diffusion velocity. ¨ller Both assumptions have proven precursors in simpler theories, Mu 1985 [66]. It is also convenient to introduce the following quantities: (1) The Helmholtz-like free energy 1 G , ψ G := − η = ψIG + ψD λ

ψIG := I −

1 η, λ

G ψD := D ;

(10.129)

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(2) the partial thermodynamic pressure of constituent α p¯G α := ρρα

∂ψIG , ∂ρα

α = 1, . . . , m;

(10.130)

α = 1, . . . , n − 1;

(10.131)

(3) the configuration pressure βαG = ρ

∂ψIG , ∂να

(4) the saturation pressure νn 1 s = − = ρn un · v n − (ρn λρn + λνn ) ; λ λ λ (5) the inner parts of the constituent Gibbs free energies ς :=

µαGI := −

1 ρ λ αI

(α=1,...,m)

=

ψIG +

pG α . ρα

(10.132)

(10.133)

On the basis of the above assumptions and definitions Schneider & Hutter [83] prove Proposition 10.5. (i) The internal energy , entropy η and Helmholtz-like free energy ψIG ∇ν, D cannotdepend on ∇ρ, and W and are thus only function of ˙ ∇θ, ρ, ν , v , B, Z . Moreover, λvα = −λ uα also rules out SR = θ, θ, a dependence of ψIG on any of the v α : ψIG = ψIG (SR \ {v }). ˆ (θ) = 1/θ, where θ is now the Kelvin If θ˙ ∈ / S, then λ = λ temperature. Furthermore, = (SRR ) and η = η(SRR ), whilst G G ˙ ∇θ . ψI = ψI (SRR \ {v }), with SRR = SR \ θ, G G (ii) p¯G α , βα and µα I as defined in (10.130), (10.131) and (10.133) are all derivable from ψIG and are functions of SR \ {v }. Alternatively, using the auxiliary variables (10.125) and (10.132), (10.133) it follows that ⎧ ∂(ρψIG ) ∂(ρψG ) ⎪ ⎪ ρα ∂ραI = να ∂να + ια + ς I , α = 1, . . . , m, ⎨ G ρ¯α µαGI = να ∂(ρψI ) + ια + ς I , α = m + 1, . . . , n − 1, ∂να ⎪ ⎪ ⎩ α=n να (ια + ς I ), (10.134) where λν ς I = ς − 12 ρn (un · un ). (10.135) ια := α , λ

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(iii) All Lagrange multipliers except λνn can be expressed in terms of thermodynamic quantities. Specifically, with the Heaviside step function H, pG α + 12 uα · uα ), α = 1, . . . , m, (10.136) ρα λνα = (−ρα ρα )H(m − α) − λ (ς + (ρα − ρn )ψIG + βα (10.137)

λρα = λ (uα · v α − ψIG −

+ 12 (ρα uα · uα − ρn un · un )) = λ ια ,

α = 1, . . . , n − 1

(iv ) The coefficient functions of the one-form P and the entropy η are given as follows: Pθ˙ P∇ρ P∇να PD α PW α

= 0, = 0, = 0, = 0, = 0,

α = 1, . . . , m, α = 1, . . . , n − 1, α = 1, . . . , n, α = 1, . . . , n,

Pρα = ¯αρI ,

α = 1, . . . , m,

Pνα = ¯ανI + s,

α = 1, . . . , n − 1,

Pθ = −ρλ

∂ ∂θ

1 λ

η+

∂ψIG

⇒

∂θ

η=−

∂ψ G

G ∂ψI ∂θ

∂ ∂θ

P

+ λθρ

1 λ

.

I , P∇θ = −ρλ ∂∇θ ⎧ ∂ψ G ∂ρ ⎪ α = 1, . . . , m, ⎨ −ρλ ∂ BIα − λ να ψIG ∂ Bαα , PB α = ⎪ ⎩ −ρλ ∂ψIG , α = m + 1, . . . , n, ∂B

PZ¯ α (= λZ α ) = −ρλ

∂ψIG ¯ , ∂Z α

and

α

α = 1, . . . , n, (10.138)

η=−

∂ ∂ θ˙

1

1 λ

∂ψIG ∂ θ˙

,

(10.139)

provided θ˙ is an independent constitutive variable. If θ˙ ∈ / S, then Pθ = 0. Thus, from (10.138)8 , η=−

∂ ∂θ

1

1 λ

∂ψIG ∂θ

( λ1 =θ)

=

−

∂ψIG . ∂θ

(10.140)

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Moreover, the only undetermined field variable is a quantity proportional to −νn , defined in (10.132) s = −νn or ς = s/λ and ς is called saturation pressure. It enters the theory as a new independent field quantity and replaces the nth volume fraction, that is lost as an independent variable by the saturation condition. The Liu identities also involve expressions of the vector one-form F . Exploring these identities, Schneider & Hutter [83] prove Proposition 10.6. (i) The coefficient functions of the one form F are given as F ∇θ F ∇ρα F ∇νβ F Dα

= = = =

0, 0, α = 1, . . . , m, 0, β = 1, . . . , n − 1, 0, α = 1, . . . , n,

FWα F θ˙ F Z¯ α F Bα

= 0, α = 1, . . . , n, = P∇θ , = −uα ⊗ PZ¯ α , α = 1, . . . , n, = −uα ⊗ PB α , α = 1, . . . , n, (10.141) (ii) The extra entropy flux vector possesses the representation (10.142) k = − 12 λ K α (S)uα , 1 K α = ¯tα − tr (¯tα )1 + {F vα − tr (F vα )1} ; λ

(10.143)

(iii) the entropy flux vector is given by

⎧ 1 1 ⎨ −λ q + ( ¯tα − 3 tr (¯tα )1)+ 3 tr (¯tα )1 uα if k = 0, 3 ¯ 1 ¯ φρη = −λ q + 2 tα − 3 tr (tα )1 uα ⎩ 1 −1

0, + 2 (λ ) {F vα − tr (F vα )1} uα , if k = (10.144) (iv ) In view of (10.141)6 an alternative expression for P∇θ is given by ∂λ ∂λ ¯ P∇θ = F θ˙ = − q − 32 tα − 13 tr (¯tα )1 uα (10.145) ˙ ˙ ∂θ ∂θ

∂ − 21 λ (¯tα − tr (¯tα )1)uα ∂ θ˙

∂ − 21 (F v¯ α − tr (F v¯ α )1)uα , ∂ θ˙ which reduces to P∇θ = 0 when θ˙ ∈ / S.

So, k, φρη and P∇θ are given by combinations of the diffusion velocities with tensorial coefficients which are given by constitutive quantities. Moreover,

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the entropy flux vector is not collinear to the heat flux vector, not even when k = 0. Interestingly also, k, φρη depend on F vα and P∇θ on derivatives ˙ of it with respect to θ. Most interestingly also in view of (10.138)9 and (10.145), two different expressions for P∇θ are at our disposal, which obviously must be equal. This yields the following expression for the “heat flux vector”: Proposition 10.7. (i) If θ˙ ∈ S the heat flux vector in temperate lava is determinable from λ ∂ψIG

− q=ρ · Qα (S)uα , (10.146) ∂λ ∂∇θ α ∂ θ˙ 1 λ ∂ ¯ 3(¯tα − 13 tr (¯tα )1) + ∂λ Qα = (t − tr (¯tα )1) (10.147) ˙ α 2 ∂ θ ˙ ∂θ 1 ∂ (10.148) (F − tr (F vα )1) . + ∂λ ˙ vα ˙ ∂θ ∂θ

(ii) If the Helmholtz-like free energy is assumed to have the form ψIG

=

(ψIG )1 (SRR

\ {v α }) −

∂λ ∂ θ˙ ψ G (|∇θ|2 , θ), ρλ 2

(10.149)

then formula (10.146) produces the law ∂ψ2G q := −2ρ (|∇θ|2 , θ) ∇θ − ∂|∇θ|2 ' () * = κ, Fourier contribution

'

α

Qα (S)uα ()

(10.150)

*

Diffusive contribution

(iii) If θ˙ ∈ / S, then the Helmholtz-like free energy does not depend on ∇θ ˆ (θ) = 1/θ, P∇θ = 0, F ˙ = 0 [see Proposition 10.5, item (i)], λ = λ θ and the heat flux vector cannot be determined from the Liu identities. In this case information of the diffusive contribution to q is lost and a simple Fourier law ensues. The results (10.146)-(10.150) are significant, because they split q into a conductive Fourier contribution and a diffusive contribution, with the structure of flux of ‘power of working’. So, it is more appropriate to call q ‘energy flux’ with a (heat) conductive and a convective-diffusive contribution.

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10.4.4. Implications of the Residual Entropy Inequality The residual entropy inequality (10.128)2 , when written in long hand, shows that the entropy production Γ is a function of the variable set S, see (10.118). This set can be written as S = e ∪ n, where e := ˙ ∇θ, v , D, ∇ν, B, Z , n := θ, W are called the equilibrium θ, ρ, ν , ∇ρ, and non-equilibrium variables, respectively. A process is now called a thermodynamic equilibrium, if no entropy is produced, Γ = 0. This is the case, if ˙ (i) the non-equilibrium variables, n vanish i.e. n = θ, ∇θ, v , D, W = 0, (ii) the constitutive quantities of the frictional production rate densities, Φα , (α = 1, . . . , n), are zero, lim n→0 Φα = Φα |E = 0 (α = 1, . . . , n), (iii) the interaction rate densities for mass and volume fraction also vanish lim n→0 cα := cα |E = 0, lim n→0 nα := nα |E = 0, (α = 1, . . . , n). It follows that Γ|E = 0, while else Γ ≥ 0. So, Γ takes a minimum in thermodynamic equilibrium. Of necessity then, ∂Γ =0 ∂nI E

∂ 2 Γ ∀nI ∈ n, is non-neg. definite, {nI , nJ } ∈ n. ∂nI ∂nJ E (10.151) Evaluation of (10.151)1 is straightforward; it implies Proposition 10.8. The following equilibrium properties can be derived from (10.151)1 : • If nI = v β : m

1 − (λε )−1 k,ρ v ∇ρα ¯ β E = δαβ − ξ¯β να (ρα )−1 pG m α E α β E E α=1

+

n−1

α=1

−

δαβ − ξ¯β ζα E 1 − c,vβ E − (λε )−1 E k,να vβ E ∇να

n

¯ α ,v ρ ψIG ,Z¯ α E Φ β E

α=1

−

n

ρ¯α µGI E (cα ),v β E − ια E (¯ . n α ),v β E α

α=1

(10.152)

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• If nI = Dβ : ¯tβ = −ζ¯β 1 + 2ρ sym (ψIG ),B β B β + (λε )−1 k,vβ E E E E E + +

¯ α ),D ρ ψIG ,Z¯ α E (Φ β E

ρ¯α µαGI E (cα ),D β E − ια E (¯ nα ),D β E .

(10.153)

˙ • If nI = θ: Pθ E =

n

¯ α ), ˙ ρ ψIG ,Z¯ α E · (Φ θ E

λε E

α=1

n

ρ¯α µαGI E (cα ),θ˙ E − ια E (¯ +λ E nα ),θ˙ E . ε

α=1

(10.154) • If nI = ∇θ: n

¯ α ),∇θ ρ ψIG ,Z¯ α E · (Φ (λε ),θ E q E = λε E E α=1

n

ρ¯α µαGI E (cα ),∇θ E − ια E (¯ + λε E nα ),∇θ E . (10.155) α=1

in which the following abbreviations have been used:

−1 G ∆∗α 1 + 12 (uα · uα )1, (10.156) tD − (λε )−1 sym (P∇θ ⊗ ∇θ) − ψD D := ρ ζα :=

βαG − ρn ψIG + ς,

α = 1, . . . , n − 1,

−ρn ψIG + ς,

α = n,

c := ρn ∆∗n D + ζn 1 un ,

ρ¯α uα ⊗ uα . tD := −

(10.157) (10.158) (10.159)

α

¯ β |E exhibit dependences on ∇ρα The interaction forces in equilibrium, m (α = 1, . . . , m) and ∇νβ (β = 1, . . . , n − 1), including dependences on the thermodynamic configuration and saturation pressures via pG α (α = 1, . . . , m), ζγ , (γ = 1, . . . , n) and the extra entropy flux, k. If all constituents are density preserving, then no equilibrium interaction force contributions involving ∇ρα arise.

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The thermodynamic pressures do not enter the constituent equilibrium stresses; only the configuration pressure arises through the pressure like variable ζβ |E . Compressibility can be accounted for by the elastic stress 2ρsym((ψIG )Bβ )|E B β . This stress has both dilatation and shearing contributions. However, equilibrium stresses are less due to elastic and more due to frictional effects, which will be expressed through the production rate ¯ α. density Φ Let us next concentrate on the contributions to (10.152)-(10.155) due ¯ α , cα and n to the production rate densities, Φ ¯ α . These contributions all ¯ α , cα and n ¯ α with prefactors which appear as products of derivatives of Φ ¯ β|E , ¯tα|E , Pθ|E and q |E . These factors are the same in the expressions of m are ¯α ρ ψIG ,Z¯ α |E for the friction term Φ ρ¯α µGI |E

for the mass production cα

ια |E

for the volume fraction production n ¯α .

α

(10.160)

and they are, in turn, multiplied ¯ iβ |E for m for ¯tα |E for Pθ |E for q|E

¯ α ,v |E , with Φ β ¯ with Φα ,D β |E , ¯ α , ˙ |E , with Φ θ ¯ α ,∇θ |E , with Φ

(cα ),vβ |E , (¯ nα ),v β |E (cα ),Dβ |E , (¯ nα ),Dβ |E (cα ),θ˙ |E ,

(¯ nα ),θ˙ |E

(10.161)

(cα ),∇θ |E , (¯ nα ),∇θ |E

and subsequently summed to reveal the corresponding representations arising in (10.152)-(10.155). The two lists (10.160) and (10.161) disclose the thermodynamic structure of the various contributions to the equilibrium quantities very clearly and make the rationals of simplifications rather sim ¯ α , cα , n ple. For instance, if we have reason to assume that Φ ¯ α ,θ˙ = ˙ ¯ ¯ α ,∇θ = 0, while θ, ∇θ -dependences are kept in the ener0 , Φ α , cα , n

gies I , ψIG , η then, provided θ is identified with the Kelvin temperature (λ |E = 1/θ), then η|E = −∂ψIG /∂θE and q|E = 0, which are most welcome properties. A frictional contribution to the equilibrium stress is important for temperate lava, but corresponding contributions due to melting/freezing and

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Chapter10


¯ α /∂Dβ |E = volume fraction are unlikely. So, we must assume that ∂ Φ 0 for some α and β but may choose ∂cα /∂Dβ = 0, ∂ n ¯ α /∂Dβ = 0. Anal¯ α , cα and ogously, one may have reason to ignore any dependences of Φ i ¯ β|E are independent of n ¯ α on the velocities v α , which then implies that m ¯ α , cα and n Φ ¯ β . Caution is, however, recommended for temperate lava with such ad hoc assumptions since the melting, freezing rate is dictated by the energy equation. 10.4.5. Simplifying Assumptions for Lava Flows The preceding theoretical results hold for a rather general mixture theory with an arbitrary number of constituents. We now apply them to the simplified situation appropriate for lava flows. The simplifications we wish to impose are: • restriction to a binary mixture of solid (α = s) and fluid (α = f ) constituents, • both constituents are density preserving, ρs = const, ρf = const. ⇒ ns = cs , • mass production is exclusively due to melting/freezing such that cs > 0 ⇔ cf < 0 for solidification. Because both constituents are density preserving, mass and volume fraction balance laws collapse to the same equations. The two mass balance equations take the forms ∂νs + ∇ · (νs v s ) = νs cs , ∂t ∂νs − + ∇ · (v f ) − ∇ · (νs v f ) = −νf cs , ∂t and adding the two equations yields ∇ · νs v s + (1 − νs )v f = cs (νs − νf ) = (2νs − 1)cs . ' () * v vol

(10.162) (10.163)

(10.164)

So, the balance of mass equations can be used as (10.162) & (10.163) or as (10.162) & (10.164), respectively. Equation (10.164) states that the volume averaged mixture velocity v vol is only solenoidal when cs ≡ 0, i.e., when no melting and no freezing occur. • We ignore θ˙ as an independent constitutive variable. Then, according to Proposition 10.5 (i) ψIG = ψÎG (SRR \ {v}) = ψIG (θ, ν , v , B, Z), (ii) λ = 1/θ, where θ is the Kelvin temperature, (iii) η = −∂ψIG /∂θ (Pθ = 0), (iv) P∇θ =

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0, F θ˙ = 0. (v) The heat flux vector cannot be determined from the Liu identities, and generally it contains only a Fourier component, cf. Proposition 10.7 item (iii). It is in this case, however still meaningful to compose the ‘energy flux vector’ q = q c + q l of two components q c the conductive heat, and q l , the latent heat due to melting as explained below. • It will be assumed that only the solid constituent exhibits contact frictional properties described by the symmetric tensor variable Z s ; Z f = 0. • The extra entropy flux vector will be set to zero, k ≡ 0, cf (10.144). 10.4.6. Parameterization of the Constituent Equilibrium Stresses and Interaction Forces All constitutive quantities can additively be split into equilibrium and nonequilibrium parts, ¯ts,f = ¯ts,f |E + ¯ts,f |N ,

¯s = m ¯ s|E + m ¯ s|N . m

(10.165)

The equilibrium contributions of (10.165) are stated as (10.152) and (10.153) and reduce with the above simplifications to n−1

¯ β E = δαβ − ξ¯β ζα E 1 − c,vβ E ∇να m

(10.166)

α=1

n n

G ¯ ρ¯α µGI E (cα ),vβ E , − ρ ψI ,Z¯ α E Φα ,vβ E − α=1

α=1

¯tβ = −ζ¯β 1 + 2ρ sym (ψIG ),B β B β E E E +

α

¯ α ),D + ρ¯α µαGI E ρ ψIG ,Z¯ α E (Φ β E

(10.167) (cα ),D β E ,

∗α −1 G − (λε )−1 sym (P∇θ ⊗ ∇θ) − ψD 1, c := ρn ∆∗n D + ζn 1 un , ∆D := ρ ζα :=

βαG − ρn ψIG + ς, α = 1, . . . , n − 1, −ρn ψIG + ς,

α = n,

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with α, β = 1, 2 and n = 2; we shall identify α = 1 with the solid and α = 2 with the fluid. The solid stress in equilibrium consists of the contributions G ¯ts|E = −ζ¯s|E 1 + 2ρsym(ψI,B )B s + ρψ G ¯ (Φs ),D s |E ' () * ' () s * ' I,Zs|E() * ¯ tcs

+ '

ρ¯s µsGI

¯ tes

(cs ),Ds |E + ρ¯f µGI f |E ()

¯ tf ric

|E

(cf ),D s |E

¯ tsmelt

(10.168) *

Using (10.134)-(10.137), a somewhat lengthly calculation shows that G ¯tsmelt = νs ∂(ρψI ) + (ρs − ρf )ψIG + βs (cf ),D s |E . (10.169) ∂νs (10.168) and (10.169) require further modelling. To this end we choose a free energy function ψIG which allows ‘separation’ of the different influences and assume G G (B s ) + ψêf (B f ). ψIG = ψˆfGric (νs , Z s ) + ψês

(10.170)

Here the indices ‘fric’, ‘es’, and ‘ef’ stand for ‘friction’, ‘elastic-solid’ and ‘elastic-fluid’. The last two terms in (10.170) are thought to account for elastic contributions of the solid and fluid, respectively, and ψˆf ric subsumes the frictional dependences which are known to strongly depend on the packing of the solid grains. G G Solid elastic stress: If ψes = ψês (IB s , IIB s ) is expressed as a function of the first two invariants of B s (but not of IIIB s for simplicity), then it is relatively easy to show, see Hutter & Schneider 2010 [43], that ! G ˆG ˆG ∂ ψ ∂ ψês ∂ ψ es es ¯tes = 2ρ B s − 2ρ + IB s B2. (10.171) ∂IBs ∂IIBs ∂IIB s s G = C1 (IB s −3)+C2 (IIB s − For a Mooney-Rivlin elastic body we have ρψês 3), (cf. [77]) which yields the solid elastic stress as

¯tes = 2(C1 + C2 IB s )B s − 2C2 B 2s

C2 =0, 2C1 =µ

=⇒

¯tes = µB s . (10.172)

Fluid elastic stress: Since the fluid stress in equilibrium must be spherical G is on the third invariant of B f . Thus the only possible dependence of ψef G G ψef = ψêf (IIIB f ) and G ¯tef = 2ρsym(ψI,B )B f = 2ρ f

G G ∂ ψêf ∂ ψêf ∂IIIB f B f = 2ρIIIB f 1. ∂IIIB f ∂B f ∂IIIB f (10.173)

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This parameterization accounts for the compressibility of lava. Even though it must be small, it is responsible for the pressure dependence of the melting temperature. A fairly reasonable parameterization would be IIIB f , (C − IIIB f ) (10.174) which generates a stress singularity when IIIB f approaches C, a measure for the densest possible packing. Because 1 < IIIB f < C for a compacting fluid volume, (10.174) generates a negative stress ¯tef (= pressure). G ψêf = A ln(B(C − IIIB f )2 ),

C > 0,

⇒

¯tef = −4ρA

Frictional part of the solid stress: This stress is given by G ¯tcs = ρψI,Z (Ψ),D s |E , s |E

(10.175)

and the difficulty is to find ψÎG (ν, Z s ) such that the stress ¯tcs is given by an expression that agrees with the stress relations of hypoplasticity. To this end we choose, see Teufel [87], ¯tcs = ρδZ s ,

δ = const.

(10.176)

Substituting (10.176) into the evolution equation for Z s , (10.114), leads to 1 ˚ ¯ s ( 1 ¯tcs , ·). ¯tcs + ν ρs − ρf (∇ · v s )¯tcs = Φ (10.177) ρδ ρ ρδ This is an evolution equation for ¯tcs ; by postulating an adequate expression for Φs we may generate an evolution equation of hypoplastic behavior. We choose

1/2 ¯ s )|D s | , ¯ s = f1 (·) L(Z¯s )D s + f2 (·)N (Z |D s | := tr (D 2s ) , Φ (10.178) in which f1 and f2 are the so-called coefficients of barotropy and pyknotropy which may depend on S. More realistic is to restrict this dependence to νs . L is a fourth order tensor. If we additionally require homogeneity of ¯ s with respect to Z ¯ s , then (10.177) with (10.178) and (10.176) can be Φ reduced to the form ˚ ¯tcs +νs ρs − ρf (∇·v s )¯tcs = tr (·) {L(t¯cs )D s + f2 (·)N (¯tcs )|D s |} . (10.179) ρ This evolution equation for the frictional stress deviates from that proposed for hypoplastic behavior by Wu & Kolymbas [90] by the second term on the left-hand side. For a dry granular material (ρf = 0), (10.179) agrees with the proposal by Svendsen & al. [86]. Schneider & Hutter 2009 [83]

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show that an evolution equation for ¯tcs can be obtained which is free of the second term on the left hand side, if the evolution equation required to hold ¯ is not (10.114), but for Z

¯ s, ¯ s= Φ Z

˚ ¯ s − νs ρs − ρf (∇ · v s )Z¯s . ¯ s := Z Z ρ

(10.180)

The introduction of this new evolution equation for Z s does change the thermodynamic analysis, but only the solid equilibrium pressure changes and now takes the form G ¯ ζs → ζsnew = νs (βs − ρf ψIG + ς) − νs (ρs − ρf )(ψI, ¯ s · Z s ). Z

(10.181)

The second, new, term on the right hand side is a frictional pressure. We still need to prescribe the operators L and N; Wu & al. 1996 [90] write ¯ ¯ ¯ s ) = c1 (νs )tr (Z ¯ s )14 + c2 (νs ) Z s ⊗ Z s , L(νs , Z ¯ s) tr (Z 2 D ¯ 2 ¯ ¯ s ) = c3 (νs ) Z s + c4 (νs ) (Z s ) . N (νs , Z ¯ ¯ s) tr (Z s ) tr (Z

(10.182) (10.183)

with material coefficients c1 −c4 which are parameterized for soil. Therefore the laws (10.182)-(10.183) can only serve as a first hint when being applied to lava. In summary we have ¯s ¯ts|E = −ζ¯new 1 + µB s + ρδ Z ∂(ρψIG ) +νs + (ρs − ρf )ψIG + βsG (cs ),Ds |E , (10.184) ∂νs IIIB f ¯tf |E = −(1 − νs )(−ρf ψIG + ς) − 4ρA , (10.185) (C − IIIB f )

¯ ,v s )|E ¯ s|E = βsG (1 − ξ¯s ) − ρf ψfG + ς ∇νs + ρ(ψIG ),Z¯ s (Φ m ∂(ρψIG ) +νs + (ρs − ρf )ψIG + βsG (cs ),vs |E . (10.186) ∂νs ¯ s satisfies the evolution equation (10.180). However, an In these relations Z expression for cs has still not been given from which (cs ),Ds |E and (cs ),vs |E can be computed. This will be done last. Prior to this end, we need nonequilibrium expressions for the stresses and the interaction forces.

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10.4.7. Parameterization for the Non-Equilibrium Stresses and Interaction Forces A detailed treatment has been given by Schneider & Hutter 2009 [83] and 2010 [43]. Here, we restrict attention to the simplest possible case and assume ¯ts|N = ¯ts|N (D s , νs , θ), ¯tf |N = ¯tf |N (D f , νf , θ), m ¯ s|N = m ¯ s|N (v f −v s , νs , θ). More specifically, we postulate the quasi-linearizations ¯ts,f |N = κs,f IDs,f 1 + 2µs,f (D s,f ),

¯ s|N = mD (v f − v s ). (10.187) m

D s,f is the deviator of Ds,f ; moreover, κf,s are bulk viscosities and µs,f are shear viscosities, whilst mD is a permeability of a Darcy-type constitutive relation for the interaction force. The phenomenological coefficients in (10.187) are assumed to have the dependencies ˆs,f (θ, νs , ID s,f ), µs,f = µ ˆ s,f (θ, νs , IID s,f , IIID s,f ), κs,f = κ mD = m ˆ D (θ, |v f − vs |, νs ).

(10.188)

Apart from the dependences on temperature and volume fraction, the bulk viscosities only depend on IDs,f , which are measures of compaction or dilatation. Similarly, the shear viscosities primarily depend on IID s,f which are the ‘shearing’ invariants of Ds,f . Dependences on the third invariants IIID s,f are thought to be less significant and may, in a first attempt be omitted. Moreover, if we suppose the fluid to have Newtonian type behavior with vanishing bulk viscosity, then (10.188) reduce to κf ≡ 0,

µf = µ ˆ f (θ, νs ) = µ0f = const.

(10.189)

κs = κ ˆ s (θ, νs , ID s ),

µs = µ ˆs (θ, νs , II

(10.190)

D s

).

To parametrize the viscosities for the solid constituents, we introduce the scaled variable νscale :=

νs − νscrit , νsmax − νs

νscrit < νsmax .

(10.191)

in which νscrit is the solid volume fraction at the critical packing at which the mean distance of the particles does not give rise to solid-solid contact; νsmax , on the other hand, is the solid volume fraction at densest packing. We conjecture that for νs νscrit , the solid bulk viscosity is smallest and probably constant, whilst at densest packing the viscous dynamic stress associated with it, becomes infinite, so that densest packing can never be

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achievedd . According to (10.191) νscale = (νscrit ) = 0 and νscale (νsmax ) = ∞. We choose κs {1 + (νscale )m } νscale > 0, κs = m > 0, (10.192) κs νscale < 0, in which κs may be taken as a constant or as κs = κs (θ, IDs ). A first choice is certainly κs = constant. For the solid shear viscosity we write ηs (IID s ) µs = µs (θ, νs )ˆ

(10.193)

with a monotonic dependence of ηs on IID s . For a power law representation we have n 0 1. (10.195) µs = νscale < 0, µs 10.4.8. Parameterization of the Melting/Freezing Rate We are still missing a functional relation for the melting-freezing rate cf = −cs in the constituent mass balance relations. We now show that an explicit relation for cf , say, can be obtained from the energy equation for the mixture as a whole. Let us commence with the balance law of mass for the fluid constituent ∂ ρ¯f + ∇ · (¯ ρf v f ) = ρ¯f cf . (10.196) ∂t Introducing the mass fraction ξ¯f = ρ¯f /ρ, it is straightforward to show that this equation can also be written as ρ

dξ¯f = −∇ · j f + ρ¯f cf , dt

d The

j f := ρξ¯f (v f − v) = ρξ¯f uf = ρξ¯f νs (v f − v s ), (10.197)

corresponding parameterizations in Schneider & Hutter 2009 and Hutter & Schneider 2010 [43] assume κs (νsmax ) = 0, which is not realistic; κs (νsmax ) should rather become very large.

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265

where v = νs v s + (1 − νs )v f has been used. j f may be interpreted as the fluid mass flux due to melting and, consequently, q l = Lj f = Lξ¯f (v f − v)

(10.198)

is the latent heat flux due to it. It follows, the total energy flux of the mixture, q = q c + q l , is given by the conductive Fourier heat flux plus the latent heat flux due to melting, q = −κ∇θM + Lξ¯f (v f − v) = −κ∇θM + Lξ¯f uf .

(10.199)

The balance law of the internal energy of the mixture as a whole is given by d (10.200) ρ = −∇ · q + tr (ttot D), dt in which q is prescribed in (10.199), ttot is the mixture stress and D the stretching tensor of the barocentric velocity field, ρ¯f 1 ρ¯s ρf } , Ds + D f + {(1 − νs )(v s − vf ) ⊗ ∇ρ¯s − νs (v s − v f ) ⊗ ∇¯ ρ ρ ρ (10.201) so that the second term on the right-hand side of (10.200) is known. In temperate lava the growth of the internal energy can be written as D=

d dξ¯f dθM =L + cv (θM ) . dt dt dt

(10.202)

Here, cv is the specific heat of the mixture at constant volume, cv = νs csv + (1−νs)cfv , evaluated at the melting temperature and the second term on the right-hand side only contributes, if the melting temperature changes in time and space due to a corresponding pressure variation. [A term −p(1/ρ)· has been ignored]. Substituting (10.202) into (10.200) and then using (10.197) and (10.199) yields dξ¯f d dθM = ρL + ρcv (θM ) ρ dt dt dt dθM (10.197) = −L∇ · j f + Lρ¯f cf + ρcv (θM ) dt = −∇ · q + tr (ttot D) (10.199)

=

∇ · (κ∇θM ) − L∇ · j f + tr (ttot D).

Comparing the second and fourth line yields the desired result, 1 dθM ρ¯f cf = tr (ttot D) + ∇ · (κ∇θM ) − ρcv (θM ) . L dt

(10.203)

(10.204)

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This equation can be interpreted as follows: The energy available for melting, ρ¯f cf L, is composed of three terms, (i) the heat generated by stress power, (ii) the heat conducted to the point under consideration and (iii) the heat becoming consumed by the change of the melting temperature, the latter being negative for positive dθM /dt. The result is both interesting and disturbing, which we shall now explain. (i) We assume L > 0 and know vaguely that θM does not vary much. So we may assume that in (10.204) the two terms involving ∇θM and θ˙M are small. This means ρ¯f cf ≈ tr (ttot D)/L. (ii) It is clear that in thermodynamic equilibrium D = 0 and hence cf = 0. However, in a nonequilibrium process it is likely that tr (ttot D) > 0, so, since L > 0, cf > 0. (iii) Therefore, it is at least very likely that tr (ttot D)/L is positive. So, under this assumption only melting occurs. This is a bad result, because it says that, if a temperate lava region exists, it will eventually become hot lava and turn into a single constituent fluid. This means that lava in a cooling process does most likely not develop a temperate region. Only hot and cold regions probably exist which are separated by a singular surface, the phase change surface from hot to cold. However, if we have a heap of cold lava that is subject to hot fluid lava, then cf < 0 may well exist and then temperate lava may survive. For the present situation we cannot prove that the right-hand side of (10.204) is really positive. The only statement we are able to make is that this conclusion is likely. A proof would have to come from an explicit numerical computation. 10.5. Kinematic and Dynamic Boundary Conditions 10.5.1. General Jump Conditions Fig. 10.3 shows that the entire mass and the three regions of distinct thermodynamic states are separated from one another by many boundaries and interface surfaces. For each of these surfaces, kinematic and dynamic boundary conditions must be formulated. ¯αγ , ζ¯αγ , φ¯γα be the partial density of the physical quantity γα , its Let γ¯α , π production, supply and flux rates, see Section 2. Let, moreover, w be a velocity of the surface point on, and nσ the unit normal vector to, the surface σ. These variables satisfy the jump condition ¯ γα nσ = ℘¯γα , ¯ γα (v α − w) · nσ − φ α α

(10.205)

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267

z

- +- ++ -

Cold temperate transition surface, Fc(x,t)=0

fluid ++ - +

-+-+

- ++ --+

ns v HT

S

n

w

Free surface, Fs(x,t)=0

+ solid + fluid +- + C - -+ TS

Bottom surface, Fb(x,t)=0

solid

Hot temperate transition surface, Fh(x,t)=0

x-y

Fig. 10.3. Lava mass, sketched as a gravity flow down the mountain side of a volcano. This mass is divided into three regions of hot fluid, temperate fluid-solid mixture and cold solid lava. The moving masses are separated from one another by boundaries and/or internal interfaces as indicated.

in which ℘γα is the surface production rate density of γ. Summation of (10.205) over all constituents yields the jump condition for the mixture as a whole n

¯ γ nσ = ℘, ¯ γα (v α − w) · nσ − φ α

α=1

where

℘ :=

N

℘¯γα .

(10.206)

α=1

The corresponding jump conditions of mass, linear- and angular momentum, energy and entropy for each component are ¯ ρα (v α − w) · nσ = µ ¯α ; ρα v α ((v α − w) · nσ ) + ¯tα nσ = τ¯ α ; (¯ ρα v α ⊗ (v α − w))nσ + ¯tα nσ = ¯ ¯ α ) · nσ = ζ¯α ; ¯ ρα 12 v 2α + α (vα − w) + (v α¯tα − q ¯ · nσ = −¯ pσα ; ¯ ρα η(v α − w) − φ α (10.207)

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268

and for the mixture as a whole ρ(v − w)nσ = µ; (ρv ⊗ (v − w))nσ + tnσ = ρv((v − w) · nσ ) + tnσ = τ ; ρ 12 v 2 + (v − w) + (vt − q) · nσ = ζ; ρη(v − w) − φ · nσ = −pσ ≤ 0. (10.208) in which µ=

N

α=1

µ ¯ α = 0,

τ =

N

α=1

τ¯ α ,

ζ=

N

¯

α=1 ζα .

It is presently not physically transparent whether all these mixture surface quantities can a priori be set to zero. The different surfaces which arise in our description of lava flow are (see Fig. 10.3): (i) The free surface which is here treated as material. This surface can be decomposed into 3 sub-surfaces, one for each transition between the hot-, temperate- and cold lava, respectively, and the atmosphere (A). These surfaces are mathematically defined by F sHA (x, t) = 0, F sT A (x, t) = 0, F sCA (x, t) = 0. (ii) The two transition surfaces (HTS and CTS) F HT S (x, t) = 0, F CT S (x, t) = 0. (iii) The bottom surface which can also be decomposed into 3 sub-surfaces. F bHB (x, t) = 0, F bT B (x, t) = 0, F bCB (x, t) = 0. 10.5.2. Kinematic and Dynamic Boundary Conditions at the Free Surface Kinematic surface equation: The subsequent calculations require expressions for the exterior surface normal vector ns . If x, y, z are Cartesian coordinates and F s = z − hs (x, y, t) = 0, where hs is the height of the free surface above the (x, y)-plane, then, since the free surface is material, ∂F s + ∇F s v s · ns = 0. ∂t With the material velocity as the convected velocity, this leads directly to ∂hs ∂hs ∂hs dhs (x, y, t) dz − = u+ v−w+ =0 at F s = 0. dt dt ∂x ∂y ∂t (10.209) The dynamic boundary conditions for this surface must also be satisfied. These conditions follow from the jump conditions of the physical balance laws.

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269

Since the free surface, F s (x, t) = 0, is material for which v · n = w · n, there is no mass flow across it and the jump condition for the density is trivially satisfied: ρ(w − v) · ns = 0. For the momentum jump we now have ρv(w − v) · ns +tns = 0 () * '

⇒

tns = −pns + tE ns = 0.

=0

(10.210) The statement on the right says that the traction (stress vector) is continuous across the material singular surface. It is not difficult to show that patm ≈ 105 P a and τwind < 10 P a. Both are considerably smaller than a typical stress paval ≈ 109 P a. The traction at the atmosphere side is therefore negligibly small; equation (10.210) can be written as 0 = −paval · ns + tE ns

tns = 0.

or

(10.211)

For the energy jump condition at the free surface we have vt · ns − q · ns = 0 +

+

s

−

−

s

−

s

+

(10.212)

s

t n* − v t n* +q · n − q · n = 0. 'v ·() ' ·() ' () *

(tE ns )atm ≈0

=0 see (10.211)

:=Q⊥ atm

With this definition a positive value of Q⊥ atm corresponds to a flow of energy into the atmosphere. So, s rad q s · ns = Q⊥ atm = Qatm + Qatm + Qatm ,

(10.213)

where Qatm , Qsatm and Qrad atm are the latent, sensible and radiative heat flows, respectively. The latent heat flow is the heat transferred to the atmosphere when phase transitions occur at the free surface; that is, when hot lava solidifies, it releases an amount of energy to the atmosphere surface. The latent heat flux can be parametrized as Qatm = f (Θsurf aval , v∗ ),

freeze surf Θsurf aval := (θaval − θatm )

(10.214)

The wind velocity function f is usually taken to be linear or quadratic in v∗ . This contribution is likely small and can be ignored in a first approximation. The sensible heat flow at the free surface describes the energy transferred from the avalanche to the air by convection or conduction. Analogously to the latent heat flux, it is parametrized by surf surf , v∗ )(θaval − θaval ). Qsatm = f˜(θaval

(10.215)

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270

f˜(·) is a non-linear heat transfer coefficient. The last term arising in Q⊥ atm is the radiative heat flow, Qrad atm . This term describes the thermal (electromagnetic) radiation which is emitted from the surface of an object which is due to its temperature: Q

4 4 − Tatm ) = σα(T0 + θaval )4 − σα(T0 + θatm )4 σα(Tsurf

= Taylor

=

surf surf σα(T04 + 4T03 θaval + ...) − σα(T04 + 4T03θatm + ...).

Therefore, surf surf Qrad atm ≈ catm (θaval − θatm ),

catm = 4σαT03 .

(10.216)

T is the temperature given in degrees Kelvin, T0 ≈ 300K, α is the emissivity of the grey body and σ the Stephan-Boltzmann constant. In temperate lava, the emissivity is a combination of the two, likely with weighting factors depending on the solid volume fractions αtemp = (1 − νs )αf + νs αs . 10.5.3. Kinematic and Dynamic Boundary Conditions at the HTS Let F = 0 be the equation for the HTSe . As a non-material surface, it has the evolution equation ± ∇F M dF = ∇F (v − w)± · = ∇F a± ⊥ = ∇F ± , (10.217) dt ∇F ρ in which ± ± M := ρ± a± ⊥ = ρ (v − w) · n,

(10.218)

We define the surface of phase change as a special singular surface at which the temperature and tangential velocity are continuous. Thus θ = 0

and

v || := v − (v · n)n = 0,

(10.219)

Straightforward computations, in which (10.219) and (10.208)3 are com¨ hnk [41]) that bined, show (see e.g. Hutter & Jo n · T Esh n = 0 with T e We

Esh

:=

θη − −

1 2 (v

or

µ = 0.

± 1 − w) · (v − w) 1 + t ρ

(10.220)

(10.221)

assume here for simplicity that φ = q/θ. The more general case is treated elsewhere.

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µ := − θη − 12 (v − w) · (v − w) +

p⊥ ρ

±

,

Chapter10

271

p⊥ = −n · tn

(10.222)

must hold across any phase change surface. T Esh is called Eshelby tensor and µ surface chemical potential. In words, the normal component of the Eshelbi stress traction or the non-equilibrium chemical potential is continuous across the phase change surface. For general dynamical conditions further inferences are difficult to derive but when quadratic contributions in the velocity can be ignored, (10.208)2,3 and (10.220) reduce to θ = 0, v || = 0, ⊥

(and M = 0)

(10.223)

p = 0, µ = 0, in which p⊥ = −nσ · tnσ = 0, which is the continuity requirement of the normal component of the total stress. Notice that (10.223) holds for both thermodynamic equilibrium and non-equilibrium conditions, for a ¨ hnk [41]. derivation, see Hutter & Jo Finally, recall that the latent heat at a phase boundary is given by ⊥ 1 L|E := θη = |E + p|E 1/ρ. (10.224) L := θη = + p ρ Subsequently we shall identify the dynamic L with the equilibrium latent heat of fusion: L = L|E . As an application, consider a body consisting of a mixture of two compressible viscous heat-conducting bodies. We assume that on one side of the surface of phase change, fluid lava and solid lava will coexist while on the other side, only the fluid lava component will be present (see Fig. 10.4a). For the mixture as a whole, we have in particular q · nσ = −θηρ(v − w) · nσ where η = ηs + ηf . Only ηs will suffer a jump across the HTS. Denoting the mass fraction of the solid lava in the fluid-solid-lava mixture by ξ¯s we have, ξ¯s =

ρ¯s , ρ¯s + ρ¯f

η + = (1 − ξ¯s+ )ˆ ηf + ξ¯s+ ηˆs ,

η − = ηf .

(10.225)

Here, η + and η− are the specific entropies per unit mixture volume on the (+)- and (–)-sides of the singular surface, and ηs , ηf are the specific entropies of the pure substances ‘solid lava’ and ‘fluid lava’, respectively.

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272 a)

+ HTS

Solid + fluid lava

Fluid lava only

b)

Solid lava only

+ CTS

Solid + fluid lava

Fig. 10.4. Phase change surface, separating a solid-fluid mixture lava from, (a) fluid lava, and (b) solid lava.

Thus, (10.225) implies, since ξ¯s− = 0, θη = ξ¯s θ(ˆ ηs − ηˆf ) = −ξ¯s L. Consequently, because of reversibility from (10.208)4, we obtain q · n = −Lξ¯s ρ(v − w)nσ = −Lξ¯s M.

(10.226)

The same equation can be shown to hold also for the CTS (see Fig. 10.4b). In terms of the total mass density and the barycentric ve locity defined as ρ = α ρα , ρv = α ρα v α the solid-constituent mass balance can be written as ρξ¯˙s = −∇ · j s + ρξ¯s cs ,

j s · nσ + ρξ¯s (v − w) · nσ = 0, j s · nσ + ξ¯s M = 0.

(10.227) (10.228)

For j s = 0, (10.226) and (10.227) can be related by q · nσ = Lj s · nσ

(10.228)

=

−Lξ¯s M = −Lξ¯s ρ± a± ⊥.

(10.229)

This relation states that the jump in heat flow across the phase boundary is given by the latent heat times the solid mass flow across the surface. In a computational solution scheme equation (10.229) serves as an equation for the determination of a⊥ or M. This melting or freezing rate is needed in the evolution equation for the HTS. Indeed, with F HT S = z − hHT S (x, y, t), the kinematic equation takes the form + 2 HT S 2 ∂hHT S ∂hHT S ∂h ∂hHT S ∂hHT S − + + 1. + u+ v−w = a⊥ ∂t ∂x ∂y ∂x ∂y (10.230) This sign choice makes a⊥ positive for freezing, i.e. for a⊥ > 0 the hot lava flow direction is from the hot to the temperate region. An equation analogous to (10.229) also hold for hCT S .

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273

Lava (hot, temp or cold) Mb > 0

Mb < 0

zb-z=0 nb

+

soil or rock Fig. 10.5. Moving basal surface separating the soil or rock bed from hot, temperate or cold lava.

10.5.4. Kinematic and Dynamic Boundary Conditions at the Bottom For the basal boundary conditions, we want to take into account erosion and deposition processes. With F b (x, t) ≡ z b (x, y, t) − z = 0, the kinematic equation can then be written as, see Fig. 10.5, + b 2 b 2 b ∂z ∂z ∂z b ∂z b − ∂z b − M − + u + v −w = 1+ + (10.231) ∂t ∂x ∂y ∂x ∂y ρ¯b with (¯ ρf,s (v f,s − w) · nb )− = Mbf,s , (ρ(v − w) · nb )− = Mbf + Mbs = Mb .

(10.232)

We note that Mf > 0 (Ms > 0) means deposition of fluid (solid) lava to the ground and Mf < 0 (Ms < 0) means erosion of fluid (solid) lava from the ground. Let us now look in detail how the jump conditions for the basal surface look like: The linear momentum jump conditions at the base for the fluidand solid lava and for the mixture as a whole are v f,s Mbf,s + ¯tf,s nb = τ¯ f,s ,

vMb + tnb = 0,

(10.233)

where we used the fact that the sum of the constituent interface momentumexchange terms τ f,s is zero, i.e. τ¯ s + τ¯ f = 0, ensuring momentum balance in the mixture as a whole at the interface. The formulae (10.233) require basal stress and mass flow parameterizations to close the system. A first approach is to postulate the no-slip condition, requiring that the jumps of the tangential velocities vanish, (1 − nb ⊗ nb )v f,s base = 0,

(10.234)

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274

z

z

a) Fig. 10.6.

v

b)

v

Possible flow velocity distribution.

see Fig. 10.6a. For τ¯ s = 0, (10.233) then implies that the basal shear traction is continuous across the (moving) basal surface. This tangential − shear traction (1 − nb ⊗ nb )(¯tf,s nb ) then follows as part of the solution of the boundary value problem for the lava flow. We conjecture that the velocity distribution is more like that in Fig. 10.6b, namely a more or less uniform flow over most depth of the avalanching lava with a very strong shearing layer and postulate a frictional sliding law. Defining at the base ||

||

b b ¯ b ¯ ,v (T f,s ¯ f,s ) := (1−n ⊗n )(tf,s n , v f,s ),

b ¯ b p¯⊥ f,s := −n ·tf,s n , (10.235)

a frictional sliding law is a functional relation ¯ || := Tf,s (v || f,s , p⊥ , θ, . . .). T f,s Specific parameterizations, which are popular in geodynamics are ¯ f,s = −ρ¯f,s cf,s (| [v [ f,s]] |, p⊥ , θ)v || f,s , T

cf,s (·) = cf,s |v f,s | (10.236)

with dimensionless drag coefficients cf,s of order 10−3. At high solid concentration a Mohr-Coulomb contribution can be added on the right hand side of (10.236). The above formulae apply for hot, temperate and cold || || lava (with the assumption v f = vs at the base.) There remains the parameterization of the quantities τ s , Mbf and Mbs . As a first guess we set τ s = 0. Mbs and Mbf represent the solid and fluid mass flows through the surface. Deposition and entrainment of fluid and solid constituents occur according to whether > 0 → deposition, b (10.237) (v − − w) · n f,s < 0 → erosion.

Chapter10

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275

So, if H is the Heaviside function, we may write dep − b b eros Mf,s = H((v − f,s − w) · n ) Mf,s − H(−(v f,s − w) · n ) Mf,s . (10.238) ' ' () * () *

=

1 for deposition 0 for erosion

=

0 for deposition 1 for erosion

dep/eros

For the fluid we conjecture Mf = 0. It means physically that the basal surface of temperate lava is material with respect to the fluid. To dep/eros we assume that erosion is dormant as long as the parameterize Ms solid second stress deviator invariant evaluated at the base is below a prescribed threshold value. Basal shear is certainly a key factor in determining whether material ,can be torn off from - the ground surface. This is the reason why J2 := 12 tr (ts − 13 (tr ts )1)2 |base is the decisive variable. Let J2thres be a positive constant (with the dimension of (stress)2 ). Then we assume − b that deposition occurs when J2thres − J2 > 0 and (v − s − v f ) · n > 0. The deposition rate of the solid in temperate lava is postulated to be − − b dep b = H(J2thres −J2 )H((v − ρ¯s ((v − Mdep s s −v f )·n ) α s − v f ) · n ) (10.239) where H is the Heaviside function and αdep a parameter characterizing the free fall of the particles in the fluid. Similarly, if (J2thres − J2 ) < 0, we conjecture = −H(J2 − J2thres )cs |J2 − J2thres |, |v||s |, . . . Meros s = −H(J2 − J2thres )αeros |J2 − J2thres |n |v ||s |m ,

(10.240)

where αeros , n, m are constant parameters. Exchange of heat between the lava avalanche and the ground is formulated by use of the energy jump condition + 21 v·vM+q·nb −v·tnb = 0, in which M > 0 is a net deposition of mass. If we let v + base = 0, q + · nb = −Qgeoth then, ⊥ q − · nb = −v− · (tnb ) − Qgeoth + ( − 12 (v − · v − ))M. ⊥

(10.241)

For hot lava M = Mf = 0 was assumed. Thus, (10.241) implies −q − · nb + v − · (tnb ) − Qgeoth = 0. ' () * ' () * ' ⊥() * (i)

(ii)

(iii)

(10.242)

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276

For cold lava (10.241) takes the form −q− · nb + v− · (tnb ) − Qgeoth + − 12 (v − · v − ) M = 0. ⊥ ' () * ' () * ' () * ' () * (i)

(ii)

(iii)

(10.243)

(iv)

In the above the various contributions account for (i) conductive transfer of heat, (ii) frictional heat due to basal sliding, (iii) geothermal heat provided by the rock formation of the volcano, and (iv) added mass-heat transfer due to the deposited/eroded mass; for this case ≈ 0. For temperate lava, the term corresponding to (iv) in (10.243) is split up in two contributions dep melt − 12 (v − · v − ) eros Mbs + − 12 (v − · v − ) f reeze Mmelt (10.244) f reeze ' () * ' () * (iv)1

(iv)2

(iv)1 is the heat transfer contribution due to deposition/erosion of solid material, and the energy jump, dep eros ≈ 0, may be ignored. The second part, (iv)2 , is the corresponding contribution due to phase changes. Since s + (1 − νs )ˆ f we have + = ˆs , − = νs ˆ ⊥ 1 melt f reeze = −(1 − νs )(f − s ) = −(1 − νs ) L − p ≈ −νs L. (10.245) ρ ' () * negl.

It follows that the energy jump condition at the basal boundary with deposition/erosion plus melting/freezing taking place is given by −q − · nb + v− · (tnb ) −Qgeoth ⊥

−

1 − 2 (v

(10.246) melt 1 − − · v )Ms − νs L + 2 (v · v ) Mf reeze = 0, (10.247) −

in which νs follows from the solid mass balance in the form j s · nb + νs Ms + Mmelt f reeze = 0, with

− − j s = −ρs (v − s − v ) = −ρs us

(10.248) (10.249)

see (10.228). The only additional unknown in equations (10.246) and melt (10.248) is Mmelt f reeze . So, these equations serve to determine Mf reeze . This concludes the derivation of the boundary conditions at the basal surface.

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10.6. Conclusion, Outlook 10.6.1. Summary Lava is a multi-phase and chemically heterogeneous system. This entails a characteristic, non-Newtonian behavior, which is emphasized by the fact that the rheological parameters are strongly temperature dependent and therefore affected by progressive cooling of lava after effusion. From the time when effusion starts, a complex thermal interaction begins with the environment, producing the gradual cooling of lava. Thermal exchange processes include conduction toward the ground and the atmosphere, radiation into the atmosphere and convection in the atmosphere above the flow. Heat is produced in the flow as a result of viscous dissipation and latent heat of crystallization. Thermal and rheological boundary layers are formed progressively and thermically insulate an inner core of fluid lava, slowing down the process. In order to correctly model the process, the flow was divided into three regions: (i) a hot fluid region, (ii) a temperate region composed of a mixture of fluid and solid lava and (iii) cold solid lava region. Hot lava was treated as a non-Newtonian, thermomechanically coupled density preserving fluid; a viscoelastic-plastic heat conducting model was used to describe cold lava. To include the plastic response of cold lava, the hypoplastic class of material concepts, appropriate for the description of frictional behavior, was used and extended to a mixture of different constituents. The derivation of the sub-models of hot, temperate and cold lava showed that these models are ‘equipped’ each with a number of phenomenological coefficients. Some of these have been found in the literature; however, numerical values for others must still be determined. This by itself will be an important and time consuming procedure for which numerical computations, perhaps paired with inverse procedures will most likely be involved. This work is built on earlier known theoretical models of deformable continuous frictional bodies and is in large parts based upon specialized application of these concepts. It uses the thermodynamically consistent formulation of Schneider & Hutter 2009 [83] and Hutter & Schneider [43] [44]. Novel is the incorporation of phase change processes occurring as melting and solidification, which in earlier debris flow models have played no role. New, at least in emphasis, is also the incorporation and careful

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consideration of the thermal sub-processes which describe the cooling as an exchange phenomenon of heat between the lava mass and the atmosphere and the ground. All these required a setting of the model within a complete thermodynamic exploitation of the second law of thermodynamics. In this regard, the derivation of the solid mass production rate density due to solidification via the mixture energy equation demonstrated a particularly nice but also disturbing aspect of the model. The beauty is its determination from a fundamental law of physics, the concern with the deduced result is that the temperate region of the moving lava may likely not form in a pure solidification scenario. A strict proof of this statement could, however, not be given. It should also be mentioned that the early assumption that lava is treated as a pure substance is a very drastic limitation of the physical capability of the model. It eliminates the multi-crystal composition of lava and makes descriptions of segregation of the crystal composition and the development of induced anisotropies of the polycrystalline material impossible. 10.6.2. Outlook It transpires from the above that the content of this thesis presents a possible model of which the validation must still be performed. This step would entail the completion and preparation of the model for numerical implementation by presenting numerical values for all phenomenological coefficients. This involves not only the parametrization of the coefficients of the continuum models of hot, temperate and cold lava, but also concerns the entrainment-deposition parametrization and the heat exchange processes with the atmosphere and the ground. A major step in achieving many of these important informations is the development of computational software for lava flow in form of sub-models, and eventually the entire complex model. The requirements are dictated by the fact that HTS and CTS represent singular surfaces which may not exist at the beginning of the process but may be formed in time. So, shock capturing schemes must be used in the numerical implementation. Moreover, because the flow goes through large displacements, large shearing and tremendous distortion of a selected initial grid, the flexibility of an Eulerian and Lagrangean formulation will be of advantage.

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Acknowledgements The authors thank Profs. Ioana Luca, Yih-Chin Tai and an anonymous referee for constructive criticism of an earlier version of this manuscript. References [1] O. Baillifard. A Continuum Formulation of Lava Flows From Fluid Ejection to Solid Deposition. Master Thesis ETH Zurich, Switzerland, 2010. [2] O. Baillifard and K. Hutter. A continuum formulation of dense granular pyroclastic flows term paper for ethz d-phys. 2009. [3] D. Barca, G. Crisci, S. DiGregorio, S. Marabini, and F. Nicoletta. Nuovo modello cellulare per flussi lavici: Colate di Pantelleria. Bollettino del Gruppo Nazionale per la Vulkanologia: Roma Italy, 1988. [4] D. Barca, G. Crisci, S. DiGregorio, and F. Nicoletta. Lava flow simulation by cellular automata: Pantelleria’s example. International Applied Modeling and Simulation: Cairo, Egypt, 1987. [5] G. Blatter. Thermodynamik: Skript zur Vorlesung. ETHZ, 2007. [6] J. Burguete, P. Garcia-Navarro, and R. Aliod. Numerical simulation of runoff from extreme rainfall events in a mountain water catchment. Natural Hazards and Earth System Sciences, 2002. [7] M. Cannataro, S. D. Gregorio, S. Rongo, R. Spataro, W. Spezzano, and D. Talia. A parallel cellular automata environment on multicomputers for modeling and simulation. Parallel Computing, 1995. [8] C. Carathéodory. Untersuchungen u ¨ber die Grundlagen der Thermodynamik. Math. Ann. 69, 1909. [9] H. Carslaw and J. Jaeger. Conduction of heat solids. Cladenron Press, Oxford, 1947. [10] L. Collier, J. Neuberg, N. Lensky, V. Lyakhovsky, and O. Navon. Attenuation in gas-charged magma. J. Volcanology and Geoth. Res., 2006. [11] A. Costa and G. Macedonio. Viscous heating in fluids with temperaturedependent viscosity: Implications for magma flows. Nonlinear Processes in Geophysics, 2003. [12] A. Costa and G. Macedonio. Numerical simulation of lava flows based on depth-averaged equations. Geophysical Research Letter, 2005. [13] G. Crisci, S. DiGregorio, O. Pindaro, and G. Ranieri. Lava flow simulation by a discrete cellular model: First implementation. International Journal of Modeling and Simulation, 1986. [14] R. Davis and G. Mullenger. A rate-type constitutive model for soil with critical state. Anal. Geomech. 2, 255-282, 1978. [15] D. Dingwell. Recent experimental progress in the physical description of silicic magma relevant to explosive volcanism. Geol. Soc. London, 1994. [16] G. Dissertori. Physik 1 & 2: Skript zur Vorlesung. ETHZ, 2005. [17] F. Dobran and G. Macedonia. Modeling of volcanological processes and simulation of volcanic eruptions. Gruppo Nazionale per la Vulkanologia (GNV), Italy, 1990.

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[18] M. Dragoni. A dynamical model of lava flows cooling by radiation. Bull. Volcanol., 1989. [19] T. Erismann and G. Abele. Dynamics of Rockslides and Rockfalls. Springer, 2001. ´ esit¨ [20] G. Farkas. A Fourier-féle mechanikai elv alkamaz´ ai. Természet. Ert´ o 12, 457-472, 1894. [21] E. Fujita, M. Hidaka, S. Umino, and A. Goto. Lava flow simulation code: Lavasim and by VTFS-project. Transactions, American Geophys. Union, 2004. [22] G. M. Graf. Kontinuumsmechanik: Skript zur Vorlesung. ETHZ, 2009. [23] S. D. Gregorio and R. Serra. An empirical method for modelling and simulating some complex macroscopic phenomena by cellular automata. Future Generation Computer Systems, 1999. [24] R. Griffiths. The dynamics of lava flows. Annual review of fluid mechanics, 2000. [25] G. Gudehus. A comparison of some constitutive laws for soil under symmetric loading and unloading. Proceedings of the 3rd International Conference on Numerical Methods in Geomechanics, pp 1309-1323, 1979. [26] M. Gurtin. Thermodynamics of evolving phase boundaries in the plane. Oxford University Press, 1993. [27] W. Haeberli, C. Huggel, A. K¨ aa ¨b, S. Oswald, A. Polkvoj, I. Zotikov, and N. Osokin. The Kolka-Karmadon rock/ice slide of 20 September 2002 – an extraordinary event of historical dimensions in North Ossetia (Russian Caucasus). Journal of Glaciology 50, 2004. [28] A. Harris and S. Rowland. FLOWGO: A kinematic thermo-rheological model for lava flowing in a channel. Bulletin of Volcanology, 2001. [29] R. Hauser and N. Kirchner. A historical note on the entropy principle of M¨ uller and Liu. Continuum Mech. Thermodyn. 14, 223-226, 2002. [30] H. Hebert, F. Hemberger, and J. Fricke. Thermophysical properties of a volcanic rock material. High temperatures – Hight pressures, 2002. [31] P. Heinrich, A. Piatanesi, and H. Hébert. Numerical modelling of tsunami generation and propagation from submarine slumps: The Papua New Guinea event. Geophysical Journal International, 2001. [32] H. Herrmann. Introduction to Computational Physics: lecture notes. ETHZ, 2010. [33] P. Hess. Polymerization model for silicate melts. Princeton University Press, 1980. [34] M. Hidaka and H. Ujita. Verification for flow analysis capability in the model of three-dimensional natural convection with simultaneous spreading, melting and solidification for debris coolability analysis module in the severe accident analysis code ’SAMPSON’ (I). Journal of Nuclear Science and Technology, 2001. [35] M. Hidaka and H. Ujita. Verification for flow analysis capability in the model of three-dimensional natural convection with simultaneous spreading, melting and solidification for debris coolability analysis module in the severe accident analysis code ’SAMPSON’ (II). Journal of Nuclear Science and Technology, 2002.

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CHAPTER 11 A TRIBUTE TO A COLLEAGUE AND FRIEND: PROFESSOR Y. H. PAO RIGOROUS MECHANICS AND ELEGANT MATHEMATICS ON THE FORMULATION OF CONSTITUTIVE LAWS FOR COMPLEX MATERIALS: AN EXAMPLE FROM BIOMECHANICS

Van C. Mow*, Xin L. Lu* and Leo Q. Wan* *Department of Biomedical Engineering, Columbia University 351 Eng. Terra. 1210 Amsterdam Ave, New York, NY 10027 USA E-mail: [email protected]

Congratulations to Professor Y. H. Pao on this 80th birthday celebration. Lao Pao, as I have called him since I first met him in December of 1954! He was a graduate student at RPI and later Columbia University, and a school mate with my older brother, Dr. Harry C. C. Mow. He has been a constant friend to my family and me, a “brother”, and a professional colleague. Professor Pao’s works in mechanics have always represented rigorous developments in mechanics and elegant mathematical solutions, particularly in the analyses of elastodynamics problems. In 1971, he co-authored a monograph on Diffraction of Elastic and Dynamic Stress Concentrations with Harry. In this contribution for this occasion, I hope to demonstrate these same characteristics in my own biomechanics studies as a tribute to him. Again, congratulations, best wishes for a happy and prosperous future. Van C. Mow Abstract: A unified mechanical-physicochemical theory, i.e., a tertiary mixture theory, for modeling a class of biological materials characterized as hydrated-charged-soft tissues has been developed, along with applications to articular cartilage biomechanics studies are presented. Particular emphasis is placed on a linearization procedure 285

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V. C. Mow, X. L. Lu and L. Q. Wan

for this highly nonlinear constitutive law, and the application of the resultant equations toward determining tissue mechanical behavior and biochemical composition by mechanical indentation. For the first application, the tested specimen is modeled as a homogeneous and isotropic charged-hydrated-soft tissue; in our second application, the specimen will be modeled as a layered laminate and in-plane orthotropic material. In this presentation a descriptive explanation is presented to motivate the need for a tertiary mixture theory to represent the behavior of biological tissues that are comprised of interstitial water (~75%), a porous-permeable charged organic solid matrix of collagen fibers and proteoglycan supra-macromolecules (~25%), and dissolved electrolytes (e.g., Na+, Cl− of negligible%). The viscoelastic behavior of such hydrated soft tissues has two major components: 1) a flow independent component due to the intrinsic macromolecular viscoelasticity; and 2) a flow dependent component due to frictional dissipation from flow through the porous-permeable matrix. A mathematical method of solution is suggested with the introduction of transformation of variables resulting in a new “generalized correspondence principle” between charged-hydrated-soft tissues and linear-isotropic-elastic materials (i.e., elasticity theory). This principle makes the employment of triphasic theory as straightforward as using an elasticity theory.

1. Introduction Articular cartilage is a load-bearing soft tissue that is vital for the maintenance of normal joint functions. It can absorb mechanical shocks during joint motion, spread the applied load onto the underlying subchondral bone, and provide a relatively smooth surface for the decades-long, highly loaded and repetitive gliding motions between articulating bones. However, cartilage has a limited ability to repair itself when injuries and degenerative joint diseases such as osteoarthritis (OA) occur [17]. OA affects a large population (~27% of the U.S. population under the age of 70) and causes joint deformities, pain, reduced joint motion, and even total loss of joint function (Figure 1). Therefore, tremendous efforts not only in the U.S. but also around the world have been taken to regenerate or repair cartilage tissue with an ultimate aim of the restoration of cartilage and joint mechanical function [13].

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Fig. 1. An opened knee joint with glistening, smooth, and intact articular cartilage (left). In advanced OA, articular cartilage is entirely lost at weight-bearing sites over the joint surface and spurs grow out from the edge of the bone (right).

Articular cartilage is composed of an electrically charged and hydrated organic matrix of collagen and proteoglycan, where a sparse population of chondrocytes reside. Biomechanically the cartilage is widely considered as a mixture consisting of solid matrix and interstitial fluid, the interaction of which was formulized, and experimentally validated by the well-known biphasic theory [20, 21]. Normal cartilage contains on average approximately 75% water within the pores of the collagen-proteoglycan solid matrix that are estimated to be 50-65 Å in diameter; even though its water content is high, its hydraulic permeability is extremely low at approximately 10-15 m 4 / ( N ⋅ s ) . During joint movement, the cartilage undergoes volumetric deformation with interstitial fluid moving through the open-connected-pores of the solid matrix and carrying large mechanical loads, which, otherwise, would damage the solid matrix and be detrimental to embedded chondrocytes. The solid matrix of articular cartilage consists mainly of collagen fibrils and proteoglycans (Fig. 2) [20]. While the former is a slender, long, relatively stiff and neutral molecule, the latter is more like a globular, highly negatively charged, soft gel like substance with well defined and complex molecular architecture in solution. The charges on proteoglycans mainly derive from the presence of carboxyl (COO−) and sulfate groups (SO3−) along with the chondroitin and keratan sulfate chains. These negative fixed charges in such porous-permeable interstitial space require a high concentration of water, counter-ions

288


Fig. 2. A schematic diagram indicating the fiber-reinforced, collagen-proteoglycan matrix in cartilage.

(Na+) for electro-neutrality to be maintained and also introduce an imbalance of total mobile ion concentration between the fluid compartment inside the tissue and the bathing fluid outside the tissue. The physico-chemical colligative property of this difference in total ion concentration is the Donnan osmotic pressure, which causes the tissue to swell both dimensionally and by weight [18, 20]. To analytically model the above mentioned interstitial fluid pressurization, and osmotic swelling behaviors, together with dynamic electrochemical events such as streaming and diffusion potentials, the triphasic mixture theory was developed based on a continuum point of view with an additional phase (electrolyte ions) introduced along with the charges fixed to the solid porous-permeable matrix [12, 14]. This theory was first introduced in 1991, and expanded in 1997. The triphasic theory, is not only entirely consistent with the well recognized earlier biphasic theory, but it can also explain many complex mechano-electrochemical (MEC) phenomena inside the tissue, such as solid matrix deformation, fluid flow, electrolyte transport, electrical potential, and swelling pressure. Due to the complexity of the triphasic theory, to date, however, there are only a handful of mathematical solutions to triphasic theory available in the literature using complex


289

numerical computational methods, not to mention being used by bioengineers who are often lack a rigorous mathematical training. In this chapter, we present the triphasic theory, a novel application of the theory to determine the biochemical composition of cartilage, recent major advances derived from a linearization of the triphasic formulations, and applications in two distinct areas: cartilage indentation – a simple poke that arouses spatio-temporally dependent mechanical, chemical and electrical events, and tissue curling – an important phenomena of cartilage that is dictated by the inherent inhomogeneities and anisotropies. 2. Triphasic Mixture Theory: A Unified Model for Soft Tissue The triphasic theory was developed in 1991 under continuum mixture theory framework [14]. It is mainly based on the general theory for a mixture of N phases with one incompressible solid phase and N-1 incompressible fluid phases developed by Bowen [4, 5]. Considering the structure and composition of articular cartilage, it is natural to treat the collagen and charged proteoglycans as the solid phase, the water as fluid phase, and the free electrolytes as a single fluid phase. This selfcontained and self-consistent field theory can describe both the known mechanical behaviors and the key electrochemical and physicochemical behaviors of the charged soft tissues. The triphasic theory may be conceptually summarized with the following constitutive assumptions: • All three phase are assumed to be incompressible. The fluid phases include a water phase and an ion phase, capable of individual free movement. The ions only occupy negligible volume. • The negative charges fixed on proteoglycan are attached to the whole solid phase. The fixed charge density (FCD) depends on the volume of solid matrix. At any point inside the tissue, the net charge should be zero. • The driving forces for fluid phases are the gradients of the chemical potential of the fluid phase (including the fluid pressure) and the electrochemical potentials of the ion phase. • Frictional drag between different phases exists when there is relative velocity. The drag forces between ions and solid, ion and ion, are negligibly small.

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The triphasic constitutive equations are thermodynamically permissible. To include the fixed charges into the model, Helmholtz energy is assumed to be dependent on the FCD value. The incorporation of traditional chemical potential for electrolyte and polyelectrolyte solutions provides a theoretical bridge between the continuum mixture theory and physicochemical theory. 2.1. Continuity Equation and Electroneutrality Condition If ϕ α denotes the volumetric fraction of α phase in the mixture, then we have:

ϕ α = ρ α / ρTα ,

(1)

α

where ρT is the true density of α phase and ρα is the apparent density. According to mixture theory, the water volume fraction ϕ w determines the apparent densities of w, +, and – phases through:

ρ w = ρTwϕ w , ρ + = ϕ wc + M + , ρ − = ϕ wc − M − ,

(2) α

where Mα is the α-ion molecular weight (g/mol), and c is the concentration of ion in fluid phase. Generally, it is assumed that the solution is dilute enough so that the volumetric fractions of ion phases are negligible, i.e. ϕ + , ϕ − ≈ 0 , then we have:

ϕw +ϕs =1.

(3)

When no chemical reaction takes place, the balance of mass equation for each individual fluid phase is:

∂ρ α + div( ρ α vα ) = 0, ∂t

α = w, +, − ,

(4)

α

where v is the velocity of α phase. When using the condition of intrinsic incompressibility for both the solid phase and the fluid phase of cartilage and again neglecting the volume of cations and anions, the continuity equation for entire mixture can be written as:

div(ϕ s v s + ϕ w v w ) = 0 .

(5)

291


According to assumption 2, the net electric charge at every spatial point within the tissue must be zero to satisfy the electroneutrality condition:

c+ = c− + c F ,

(6)

F

where c is the fixed charge density (FCD) raised by the negative charges on proteoglycans. This density changes with the volume deformation of the solid matrix according to the following equation:

c F = c0F (1 −

trE

ϕ 0W

),

(7)

where E is the infinitesimal strain tensor which defines the deformation of the solid matrix. Here trE represents the dilatation of solid matrix, c0F , ϕ0w are the fixed charge density and water volume fraction at the initial reference state, respectively. The conservation of ions and fixed charges, and the electro-neutrality condition yield:

div ( I e ) = 0

(8)

where I e is the electrical current density given by:

I e = Fcϕ w [c + ( v + − v s ) − c − ( v − − v s )] ,

(9)

where Fc is the Faraday constant. 2.2. Momentum Equations The balance of momentum equations for the tissue and each fluid phase, under a quasi-static condition, with inertia neglected, are given by: (10) ∇ ⋅σ = 0,

− ρ α ∇µ α + ∑ fαβ ( v β − vα ) = 0,

(α ≠ solid phase)

(11)

β

where σ is the total stress acting in the tissue; µα is the chemical potential for water or the electrochemical potentials for cations and anions. The summation in Eq. (11) is carried out for β over the solid, water, cation, and anion phases ( α , β = s, w, +, − ), and fαβ is the diffusive drag coefficient (per unit of mixture volume) between α and β phases and satisfies the Onsager’s reciprocity relations: fαβ = fβα and fαβ ≡ 0 if α = β . Usually the diffusive drag between anions and cations, and

292


the diffusive drag between ions and the solid matrix are neglected (i.e., f +− ≈ f s + ≈ f s − ≈ 0 ). The remaining diffusive drag coefficients relate to the ion diffusivities ( D + , D − ) and interstitial fluid permeability k through:

f ws =

ϕw k

2

ϕ w RTc −

, f w− =

D−

, f w+ =

ϕ w RTc + D+

,

(12)

where R is the universal gas constant, T is the absolute temperature. 2.3. Constitutive Equations The constitutive equations for the total stress σ , chemical potential for water and electrochemical potentials for cations and anions are given by:

σ = − pI + σ e ,

µ w = µ 0w +

1

ρTw

[ p − RTΦ (c + + c − )] ,

(13) (14)

µ + = µ0+ +

Fc Ψ RT ln(γ + c + ) , + M+ M+

(15)

µ − = µ0− −

Fc Ψ RT ln(γ − c − ) , + M− M−

(16)

where σe is the elastic stress tensor for the solid matrix of cartilage tissue, and p is the interstitial fluid pressure, Fc is Faraday’s constant, Ψ is the electric potential across the tissue, Φ is the osmotic coefficient of the solution which is generally assumed as unity, γ+, γ- are the activity α coefficients of cation and anion inside the tissue, respectively. Also, µ0 is the chemical or electrochemical potential at a chosen reference state. If we assume the solid matrix as linear, isotropic elastic material undergoing infinitesimal deformation, then σ e turns out to be:

σ e = λs tr ( E ) I + 2µ s E .

(17)

Here, λs and µs are the Lames’ coefficients in elasticity of the porouspermeable solid matrix.


293

2.4. Boundary Conditions The complexity and variety of boundary conditions are responsible for much of the complications associated with a triphasic analysis. Before pursuing the triphasic solution, it is necessary to clarify the general boundary rules for triphasic mixture. It is natural to define the boundary of triphasic mixture to be fixed to the solid phase. When a triphasic soft tissue interfaces with an external bathing solution, this boundary allows free exchange of the interstitial fluid and ions with the external solution. The jump conditions of a triphasic mixture, corresponding to the balances of mass, momentum, and energy, with inertia neglected, are:

[[ v s ]] = 0 ,

(18)

[[ϕ w ( v w − v s )]] ⋅ n = 0 ,

(19)

[[ϕ wc + ( v + − v s )]] ⋅ n = 0 ,

(20)

[[ϕ wc − ( v − − v s )]] ⋅ n = 0 ,

(21)

[[σ ]] ⋅ n = 0 .

(22)

Assuming the interface is ideally singular with continuous temperature, the entropy jump condition can generate:

[[ µ α ]] = 0 ,

α = w, + , −

(23)

3. Determining the Proteoglycan Content: A Novel Mechanical Method 3.1. A Generalized Correspondence Principle of Triphasic Theory The objective of the following analysis is to establish a closed-form relationship between the displacements, stresses, strains, charge densities, pressures, and ionic concentrations within the loaded triphasic tissue at its final steady state. If an arbitrary piece of ideal triphasic medium subjected to a Heaviside loading function, the tissue would creep, due to fluid and ion exudation from the tissue, until it reaches equilibrium after the initial transient phase, both mechanically and thermodynamically. This configuration represents most mechanical tests and theoretical

294


analyses employed in the literature, such as confined compression, unconfined compression, and indentation tests, etc. In some models, it is also common to assume that the material properties of cartilage are homogeneous and that the solid matrix strain remains infinitesimally small. Since the strain is small, a regular perturbation method can be employed to perform the analysis [27]. The perturbation sequence for the unknown parameters such as cF, p and σ are:

Q = Q0 + ε 0Q1 + ε 02 Q2 + ... ,

(24)

where Q0 is the value of unknowns at reference state as defined previously, and the linearized perturbation amount for a specific unknown will be:

δ Q = Q − Q0 ≈ ε 0Q1 .

(25)

At thermodynamic equilibrium, three of the boundary conditions at the surface of triphasic material are:

µ w = µ w* , µ + = µ +* , µ − = µ −*.

(26)

By combining the constitutive equations for the electrochemical potentials and the electroneutrality condition, the boundary conditions in Eq. (26) require that: 2

k

c =

F 2

(c )

γ  + 4  ∗ c∗  . γ± 

(27)

Thus the linearized equation between the FCD and the sum of ion concentrations is:

c0F

δ ck =

(c ) F o

2

+4

γ∗ ∗ 2 (c ) γ±

δ cF .

(28)

Similarly, the chemical potential of water within the tissue must also be equal to the chemical potential in the bathing solution. This condition yields the following relationship on p:

δ p = Φ RT δ c k ,

(29)

where δ p is the linearized perturbation to the fluid pressure resulting k from the deformation, and δ c is the total difference of ions

295


concentration inside and outside the tissue. According to Eq. (7), the linearized change in fixed charge density is related to the elastic deformation of solid matrix by:

δ cF = −

c0F

ϕ0w

δe,

(30)

where δ e is the solid matrix dilatation relative to the initial swollen state. It is equal to the trace of the perturbation to the strain tensor. Here c0F , ϕ0w are the fixed charge density and water volumetric fraction at the reference state, respectively. Combining Eqs. (27-30) yields an important relationship between δ p and δ e :

 − δp= w  ϕ0 

Φ RT ( c0F )

(c ) F 0

2

+4

2

γ∗ ∗ 2 (c ) γ±

  δ e .  

(31)

This new and important equation relates the change of the osmotic pressure with the solid matrix dilatation e. The term inside the square brackets can be treated as the rate of change of the osmotic pressure with dilatation. In the loaded triphasic material, the total stress consists of both the solid matrix stress and the fluid pressure. The perturbation to the total tissue stress tensor resulting from a small solid matrix deformation is given by:

δ σ = −δ pI + δ σ e .

(32)

Substituting Eq. (31) into Eq. (32) and assuming a linear isotropic solid matrix for cartilage tissue, the total stress inside the tissue may be written as:

δ σ = 2µs*δ E + λs*δ eI,

(33)

where:

  * λs = λs + Π, and Π =  w  ϕ0 

Φ RT ( c0F )

(c )

µs* = µs .

F 0

2

+4

2

γ∗ ∗ 2 (c ) γ±

  ,  

(34)

(35)

296


Fig. 3. Comparison of the calculated FCD values (ordinate) from correspondence principle and those obtained from biochemical GAG assay (abscissa) using a commercial kit.

Again, µs and λs are the intrinsic shear modulus and Lame’ constant of * the solid matrix. Note that the parameter λs is independent of the spatial position within the deformed tissue. The situation for shear modulus is even simpler. It remains equal to its intrinsic value. More importantly, Eq. (33) is identical in mathematical form to the generalized Hooke’s law for an elastic material. Therefore, the mechanical response of a linear triphasic medium at final steady state will be indistinguishable from that * * of a linear elastic material exhibiting Lame’ coefficients λs and µs . Eqs. (34-35) define the generalized correspondence principle for triphasic theory. They make the employment of triphasic analysis as straightforward as using an elastic model to solve equilibrium problems, and yet being able to properly account for effects due to the FCD and the osmotic effect on the charged porous-permeable solid matrix. 3.2. Determination of Fixed Charge Density For osteoarthritic articular cartilage at its early stage, one of the changes is in the morphology and micro-structure of proteoglycans. This makes


297

proteoglycan/FCD an excellent biochemical marker for early OA diagnosis [6, 7, 20]. Numerous methods, such as chemical or MR imaging techniques were developed to determine proteoglycan content [3, 8, 19]. None of the techniques, however, can determine this biochemical composition change and simultaneously with mechanical properties of tissue at the same spatial location. According to correspondence principle, with the knowledge of both intrinsic and apparent mechanical properties, the FCD can be easily calculated [15]. A commonly employed method to obtain intrinsic properties of cartilage is to perform a mechanical test by submerging the tissue in a hypertonic environment (with high external solution concentration, such as 2.0 M NaCl) [9, 10]. In a hypertonic solution (2.0 M), the Donnan osmotic pressure induced by the FCD asymptotically approaches zero (as can be theoretically shown [15]). Thus, the intrinsic mechanical properties (without the effects of the osmotic swelling pressure) can be extracted by analyzing mechanical testing data obtained with the specimen submerged in 2.0 M solution. In contrast, when cartilage is equilibrated in physiological condition (0.15 M NaCl solution), the mechanical loading is supported by both the solid matrix mechanical stress and the osmotic pressure in the fluid phase. Therefore, the mechanical properties obtained directly has the osmotic effect lumped into it, which are known as the apparent properties. With the knowledge of both intrinsic and apparent mechanical properties, the fixed charge density can be calculated directly from Eq. (34). Figure 3 shows the comparison of the calculated FCD based on this strategy and those from independent biochemical GAG assay. The intrinsic and apparent mechanical properties were obtained from indentation creep tests [16]. A linear regression analysis showed that the results from correspondence principle and biochemical test are remarkably consistent with each other. This not only proves the validity and applicability of triphasic theory, but also provided an important and easy biomechanical alternative method for non-invasively determining the FCD in articular cartilage simultaneously with mechanical stiffness using a pure mechanical technique.

298


4. Curling Behavior of Articular Cartilage: Anisotropic Lamination Model and Triphasic Theory With the introduction of biomechanical complexities in tissue microstructure such anisotropy and inhomogeneity, the triphasic theory can be further used to describe a variety of experimentally known mechanical behaviors of classes of cartilage, and other biological tissues as blood vessels. Here we demonstrate that with the aforementioned simplification (i.e., the linearization of the triphasic theory) the application of triphasic theory in explaining tissue curling behavior would rather be easy and straightforward. It has been known for decades that adult articular cartilage sample will curl and warp toward its articular surface after removal from the underlying subchondral bone. This is particularly important for grafting cartilage a procedure used in common plastic surgery (e.g., rhinoplasty). The curvature is found to vary with the saline concentration in the external bathing solution [25]. This curling behavior is an evidence of inhomogeneous residual stresses and strains inside the solid matrix of articular cartilage, and is believed to be contributed to by both swelling properties and layered inhomogeneous ultrastructure of articular cartilage [24]. Residual stresses and strains play important roles in the biomechanical function of the load-bearing tissues such as vein, cardiac ventricle, and cartilage [11].

Fig. 4. The layer structure of articular cartilage strip with its dimensions and the predominant orientation of collagen fibrils.


299

The layered inhomogeneties of chemical contents and macromolecular organization in situ cause variations of osmotic pressure and solid matrix stresses throughout the depth of the tissue, and thus giving rise to the often-observed curling behaviors of articular cartilage. Many polarized light and electron microscopy studies have been reported on the variation of collagen fibrillar (or fibrous) architectural organization through the depth of articular cartilage [20]. Collagen fibrils are aligned tangentially to the surface in the superficial tangential zone (SZ), randomly in the middle zone (MZ), and vertically in the deep zone (DZ) (Figure 4). These micro-structural features of collagen seem to correlate well with the relative amounts of the substances found within the matrix, such as water and collagen contents and fixed charge density (FCD); in other words, composition also varies layer-wise throughout the tissue depth from the surface to the deep layer attached to the bone. As a result, mechanical properties and swelling effects of the tissue also show the depth dependency, leading to the curling behavior of articular cartilage. While the intuitive picture for the cartilage curling behavior has been known for many years [18], the successful predictions from a theoretical model with experimental data for cartilage curling behavior remained a big challenge. However, with the development of the triphasic theory, this problem can now be modeled. The focus of this research is therefore to develop a quantitative model based on the triphasic constitutive law to describe the swelling and curling behaviors of articular cartilage; and to compare the predicted deformation on the curling behavior of thin strips of cartilage specimens with previous experimental results. 4.1. Mathematical Modeling for Cartilage Curling Again, at the hypertonic state (asymptotically c* →∞), we assume the swelling effects associated with the FCD are negligible and the entire tissue (length a × width b × thickness h) is assumed to be flat initially as shown in Figure 5(A). All physical parameters after swelling are determined relative to those at this hypertonic reference state (HRS). The material properties and chemical parameters are assumed to vary with depth (i.e., between the three (~150µm originally thin-flat layers), but to be homogenous within each layer. If these three layers were separated

300


Fig. 5. Schematic explanations for the origin of curling behavior of articular cartilage. Three layers (A) have different swelling potentials (B), which lead to curling behavior of cartilage strips that space multiple layers (C).

Table 1. The intrinsic parametric values for articular cartilage, based on previous literatures. Position STZ MZ DZ

Thickness FCD Porosity λ (MPa) µ (MPa) (mm) (mEq/ml) 0.3 0.7 0.12 0.2 0.4 1.2 0.5

0.8 0.6

0.25 0.20

0.2 0.1

0.4 0.15

λ+1 (MPa)

λ-1 (MPa)

10

3.0

MZ=Isotropic 3.0

― 0.9

from each other, and allowed to swell independently, then each layer would experience a different deformation as shown in Figure 5(B) and remain straight [22]. However, if the layers were joined together before swelling, then the in-plane strains (in x-y plane) at the interface would have to match, which would produce curling (Figure 5(C)). The triphasic theory [14] is used to model the swelling behavior (Figure 5: A→B) of each layer. The total stress (σ) consists of elastic stresses inside the solid matrix (σs) and the osmotic pressure (p) as shown below: σ = –p I+σs

with p = π – П e

(36)

where π = ΦRT(ck –2c*) and Π = ΦRT(crF)2/(φrwck) [2, 27]. Here φrw and crF are the porosity and FCD at HRS, respectively. An orthotropic constitutive equation with tension-compression nonlinearity is used to describe the mechanical property of the solid matrix for each layer based on the collagen fibril microstructure [1, 22]. Since the cartilage strip is assumed to be very thin, i.e., h

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