MECHANICS OF MATERIAL FORCES
Advances in Mechanics and Mathematics VOLUME 11
Series Editors: David Y. Gao Virginia Po...

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MECHANICS OF MATERIAL FORCES

Advances in Mechanics and Mathematics VOLUME 11

Series Editors: David Y. Gao Virginia Polytechnic Institute and State University, U.S.A. Ray W. Ogden University of Glasgow, U.K.

Advisory Editors: I. Ekeland University of British Columbia, Canada S. Liao Shanghai Jiao Tung University, P.R. China K.R. Rajagopal Texas A&M University, U.S.A. T. Ratiu Ecole Polytechnique, Switzerland W. Yang Tsinghua University, P.R. China

MECHANICS OF MATERIAL FORCES

Edited by PAUL STEINMANN University of Kaiserslautem, Germany GERARD A. MAUGIN Universite Pierre et Marie Curie, Paris, France

Spri ringer

Library of Congress Cataloging-in-Publication Data Mechanics of material forces / edited by Paul Steinmann, Gerard A. Maugin. p. cm. — (Advances in mechanics and mathematics ; 11) Includes bibliographical references. ISBN 0-387-26260-1 (alk. paper) - ISBN 0-387-26261-X (ebook) 1. Strength of materials. 2. Strains and stresses. 3. Mechanics, Applied. 1. Steinmann, Paul, Dr.-tng. 11. Maugin, G. A. (Gerard A.), 1944- 111. Series. TA405.M512 2005 620.1'123-dc22 2005049040 AMS Subject Classifications: 74-06, 74Axx, 74R99, 74N20, 65Z05 iSBN-10: 0-387-26260-1 ISBN-13: 978-0387-26260-4 e-lSBN-10: 0-387-26261-X e-lSBN-13: 978-0387-26261-1 © 2005 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science-t-Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 springeronline.com

SPIN 11367529

Contents

Preface

ix

Contributing Authors

Part I

4d Formalism

1 On establishing balance and conservation laws in elastodynamics George Herrmann^ Reinhold Kienzler 2 Prom mathematical physics to engineering science Gerard A. Maugin

Part II

xiii

3

13

Evolving Interfaces

3 The unifying nature of the configurational force balance Eliot Fried^ Morton E. Gurtin

25

4 Generalized Stefan models Alexandre Danescu

33

5 Explicit kinetic relation from "first principles" Lev Truskinovsky, Anna Vainchtein

43

Part III

Growth & Biomechanics

6 Surface and bulk growth unified Antonio DiCarlo

53

vi

Contents

7 Mechanical and t her mo dynamical modelling of tissue growth using domain derivation techniques Jean Francois Ganghoffer

65

Material forces in the context of biotissue remodelling 77 Krishna Garikipati, Harish Narayanan, Ellen M. Arruda, Karl Grosh, Sarah Calve

Part IV

Numerical Aspects

9

Error-controlled adaptive finite element methods in nonlinear elastic fracture mechanics Marcus Riiter, Erwin Stein 10 Material force method. Continuum damage & thermo-hyperelasticity Ralf Denzer, Tina Liehe, Ellen Kuhl, Franz Josef Barth, Paul Steinmann 11 Discrete material forces in the finite element method Ralf Mueller, Dietmar Gross 12 Computational spatial and material settings of continuum mechanics. An arbitrary Lagrangian Eulerian formulation Ellen Kuhl^ Harm Askes^ Paul Steinmann

Part V

87

95

105

115

Dislocations & Peach-Koehler-Forces

13 Self-driven continuous dislocations and growth Marcelo Epstein 14 Role of the non-Riemannian plastic connection in finite elastoplasticity with continuous distribution of dislocations Sanda Cleja-Tigoiu 15 Peach-Koehler forces within the theory of nonlocal elasticity Markus Lazar

129

141

149

Contents Part VI

vii Multiphysics & Microstructure

16 On the material energy-momentum tensor in electrostatics and magnetostatics Carmine Trimarco 17 Continuum thermodynamic and variational models for continua with microstructure and material inhomogeneity Bob Svendsen 18 A crystal structure-based eigentransformation and its work-conjugate material stress Chien H. Wu

Part VII

161

173

181

Fracture & Structural Optimization

19 Teaching fracture mechanics within the theory of strength-of-materials Reinhold Kienzler, George Herrmann

193

20 Configurational thermomechanics and crack driving forces Cristian Dascalu, Vassilios K. Kalpakides

203

21 Structural optimization by material forces Manfred Braun

211

22 On structural optimisation and configurational mechanics Franz-Joseph Barthold

219

Part VIII

Path Integrals

23 Configurational forces and the propagation of a circular crack in an elastic body Vassilios K. Kalpakides^ Eleni K. Agiasofitou 24 Thermoplastic M integral and path domain dependence Pascal Sans en ^ Philippe Dufrenoy, Dieter Weichert

231

241

viii Part IX

Contents Delamination & Discontinuities

25 Peeling tapes Paolo Podio-Guidugli

253

26 Stability and bifurcation with moving discontinuities Claude Stolz^ Rachel-Marie Pradeilles-Duval

261

27 On Fracture Modelling Based on Inverse Strong Discontinuities Ragnar Larsson, Martin Fagerstrom

269

Part X

Interfaces & Phase Transition

28 Maxwell's relation for isotropic bodies Miroslav Silhavy

281

29 Driving force in simulation of phase transition front propagation Arkadi Berezovski^ Gerard A. Maugin

289

30 Modeling of the thermal treatment of steel with phase changes Serguei Dachkovski, Michael Bohm

299

Part XI

Plasticity & Damage

31 Configurational stress tensor in anisotropic ductile continuum damage mechanics Michael Briinig 32 Some class of SG Continuum models to connect various length scales in plastic deformation Lalaonirina Rakotomanana 33 Weakly nonlocal theories of damage and plasticity based on balance of dissipative material forces Helmut Stumpf, Jerzy Makowski^ Jaroslaw Gorski, Klaus Hackl

311

319

327

Preface

The notion dealt with in this volume of proceedings is often traced back to the late 19th-century writings of a rather obscure scientist, C.V. Burton. A probable reason for this is that the painstaking deciphering of this author's paper in the Philosophical Magazine (Vol.33, pp. 191-204, 1891) seems to reveal a notion that was introduced in mathematical form much later, that of local structural rearrangement. This notion obviously takes place on the material manifold of modern continuum mechanics. It is more or less clear that seemingly different phenomena - phase transition, local destruction of matter in the form of the loss of local ordering (such as in the appearance of structural defects or of the loss of cohesion by the appearance of damage or the extension of cracks), plasticity, material growth in the bulk or at the surface by accretion, wear, and the production of debris - should enter a common framework where, by pure logic, the material manifold has to play a prominent role. Finding the mathematical formulation for this was one of the great achievements of J.D. Eshelby. He was led to consider the apparent but true motion or displacement of embedded material inhomogeneities, and thus he began to investigate the "driving force" causing this motion or displacement, something any good mechanician would naturally introduce through the duahty inherent in mechanics since J.L. d'Alembert. He rapidly remarked that what he had obtained for this "force" - clearly not a force in the classical sense of Newton or Lorentz, but more a force in the sense of those "forces" that appear in chemical physics (thermodynamical forces), was related to the divergence of a quantity known since David Hilbert in mathematical physics, namely, the so-called energy-momentum tensor of field theory, or to be more accurate, the purely spatial part of this essentially fourdimensional object. Similar expressions were to appear in the study of the "force" that causes the motion of a dislocation (Peach and Koehler) and the driving force acting on the tip of a progressing macroscopic crack (Cherepanov, Rice). Simultaneously, this tensor, now rightfully called an Eshelby stress tensor, but identified by others as a tensor of

X

Preface

chemical potentials (a tensorial generalization of the Gibbs energy), was appearing in the theory of fluid mixtures (Ray Bowen, M. Grinfeld) and with its jump at discontinuity surfaces. It would be dangerous to pursue this historical disquisition further. If we did, we would enter a period where most active contributors to the field are still active at the time of this writing and are in fact contributors to the present book. Still, a few general trends can be emphasized. The late 1980s and the early 1990s witnessed a more comprehensive approach to the notion of local structural rearrangement (the local change of material configuration resulting from some physico-chemical process) and the recognition of the tensor introduced by Eshelby in the treatment of interface phenomena. With these works, a whole industry started to develop along different paths, sustained by different viewpoints, and deaUng with various applications, while identifying a general background. Although certainly not an entirely new field per 5e, it started to organize itself with the introduction of a nomenclature. This is often the result of personal idiosyncratsies. But two essential expressions have come to be accepted, practically on an equal footing: that of material forces^ and that of configurational forces, both of which expressions denote the thermodynamical forces that drive the development of the above-mentioned structural rearrangements and the defect motions. While we are not sure that these two terms are fully equivalent, it became obvious to the co-editors of this volume in the early 2000s that time was ripe for a symposium fully pledged to this subject matter. The format of the EUROMECH Colloquia was thought to be well adapted because of its specialized focus and the relatively small organizational effort required for its set up. Midway between Paris and central Germany was appropriate for the convenience of transport and some neutrality. All active actors, independently of their peculiar viewpoints, were invited, and most came, on the whole resulting in vivid discussions between gentlemen. The present book of proceedings well reflects the breadth of the field at the time of the colloquium in 2003, and it captures the contributions in the order in which they were presented, all orally, at the University of Kaiserslautern, Germany, May 21-24, 2003, on the occasion of the EUROMECH Colloquium 445 on the "Mechanics of Material Forces". The notion of material force acting on a variety of defects such as cracks, dislocations, inclusions, precipitates, phase boundaries, interfaces and the hke was extensively discussed. Accordingly, typical topics of interest in continuum physics are kinetics of defects, morphology changes, path integrals, energy-release rates, duality between direct and inverse motions, four-dimensional formalism, etc. Especially remarkable are the initially unexpected recent developments in the field of numerics, which

Preface

xi

are generously illustrated in these proceedings. Further developments have already taken place since the colloquium, and many will come, no doubt. Students and researchers in the field will therefore be happy to find in the present volume useful basic material accompanied by fruitful hints at further fines of research, since the book offers, with a marked open-mindedness, various interpretations and applications of a field in full momentum. Last but not least our sincere thanks go to everybody who helped to make this EUROMECH Colloquium a success, in particular to the coworkers and students at the Lehrstuhl fiir Technische Mechanik of the University of Kaiserslautern who supported the Colloquium during the time of the meeting. Very special thanks go to Ralf Denzer who took the responsibility, and the hard and at times painful work, to assemble and organize this book.

Paris, June 2004. Kaiserslautern, June 2004,

Gerard A. Maugin Paul Steinmann

Contributing Authors

Franz-Joseph Barthold University of Dortmund, Germany Arkadi Berezovski Tallinn Technical University, Estonia Manfred Braun Universitat Duisburg-Essen, Germany Michael Briinig Universitat Dortmund, Germany Sanda Cleja-Tigoiu University of Bucharest, Romania Serguei Dachkovski University of Bremen, Germany Alexandre Danescu Ecole Centrale de Lyon, Prance Cristian Dascalu Universite Joseph Fourier, Grenoble, France Ralf Denzer University of Kaiserslautern, Germany Antonio DiCarlo Universita degli Studi "Roma Tre", Italy Marcelo Epstein The University of Calgary, Canada Jean-Francois GanghofFer Ecole Nationale Superieure en Electricite et Mecanique, Nancy, France Krishna Garikipati University of Michigan, Ann Arbor, USA

xiv

Contributing Authors

Morton E. Gurtin Carnegie Mellon University, Pittsburgh, USA George Herrmann Stanford University, USA Vassilios K. Kalpakides University of loannina, Greece Reinhold Kienzler University of Bremen, Germany Ellen Kuhl University of Kaiserslautern, Germany Ragnar Larsson Chalmers University of Technology, Goteborg, Sweden Markus Lazar Universite Pierre et Marie Curie, Paris, Prance Gerard A. Maugin Universite Pierre et Marie Curie, Paris, Prance Ralf Miiller Technische Universitat Darmstadt, Germany Paolo Podio-Guidugli Universita di Roma "Tor Vergata", Italy Lalaonirina Rakotomanana Universite de Rennes, Prance Pascal Sansen Ecole Superieure d'Ingenieurs en Electrotechnique et Electronique, Amiens, Prance Miroslav Silhavy Mathematical Institute of the AV CR, Prague, Czech Republic Erwin Stein University of Hannover, Germany Claude Stolz Ecole Poly technique, Palaiseau, Prance Helmut Stumpf Ruhr-Universitat Bochum, Germany Bob Svendsen University of Dortmund, Germany

Contributing Authors Carmine Trimarco University of Pisa, Italy Anna Vainchtein University of Pittsburgh, USA Chien H. Wu University of Illinois at Chicago, USA

xv

4D FORMALISM

Chapter 1 ON ESTABLISHING BALANCE AND C O N SERVATION LAWS IN ELASTODYNAMICS George Herrmann Division of Mechanics and Computation, Stanford University [email protected]

Reinhold Kienzler Department of Production Engineering, University of Bremen [email protected]

Abstract

By placing time on the same level as the space coordinates, governing balance and conservation laws are derived for elastodynamics. Both Lagrangian and Eulerian descriptions are used and the laws mentioned are derived by subjecting the Lagrangian (or its product with the coordinate four-vector) to operations of the gradient, divergence and curl. The 4 x 4 formalism employed leads to balance and conservation laws which are partly well known and partly seemingly novel.

Keywords: elastodynamics, 4 x 4 formalism, balance and conservation laws

Introduction The most v^idely used procedure to establish balance and conservation laws in the theory of fields, provided a Lagrangian function exists, is based on the first theorem of Noether [1], including an extension of Bessel-Hagen [2]. If such a function does not exist, e. g., for dissipative systems, then the so-called Neutral-Action method (cf. [3], [4]) can be employed. For systems which do possess a Lagrangian function, the Neutral-Action method leads to the same results as the Noether procedure including the Bessel-Hagen extension. In addition to the above two methods, a third procedure consists in submitting the Lagrangian L to the differential operators of grad^ div

4

George Herrmann and Reinhold Kienzler

and curl^ respectively. The latter two operations are applied to a vector xL where x is the position vector. This procedure has been successfully used in elastostatics in material space (cf. [4]), leading to the pathindependent integral commonly known as J , M and L. Thus it appeared intriguing to investigate the results of the application of the three differential operators to the elastodynamic field, considering the time not as a parameter, as is usual, but as an independent variable on the same level as the three space coordinates. To make all four coordinates of the same dimension, the time is multiplied by some characteristic velocity, as is done in the theory of relativity. Results are presented both in terms of a Lagrangian and an Eulerian formulation.

1. 1.1.

ELASTODYNAMICS IN LAGRANGIAN DESCRIPTION LAGRANGIAN FUNCTION

We consider an elastic body in motion with mass density po = Po{X^) in a reference configuration. We identify the independent variables ^^ {fi ~ 0,1,2,3) as

^^ = i = Cot, (1) ^^ =X\

J = 1,2,3 .

The independent variable i and not t is used in order for all independent variables to have the same dimensions, where CQ is some velocity. In the special theory of relativity, CQ = c = velocity of light is used, to obtain a Lorentz-ivariant formulation in Minkowski space. In the non-relativistic theory, CQ may be chosen arbitrarily, e. g., as some characteristic wave speed or it may be normalized to one. The independent variables ^^ = X'^ (J = 1,2,3) are the space-coordinates in the reference configuration of the body. Although ^^ = t and ^'^ = X^ have the same dimensions, the time t is an independent variable and is not related to the space-coordinates by a proper time r as in the theory of relativity. Therefore, we will deal with Galilean-invariant objects, for "vectorsand matrices that will be called (cf. [5]) "tensors", although the formulation is not covariant. Special care has to be used when dealing with div and curl^ where timeand space-coordinates are coupled. The current coordinates are designated by x'^ and play the role of dependent variables or fields. They are defined by the mapping or motion

On establishing balance and conservation laws in elastodynamics x' = x'iiX'),

i = 1,2,3.

5 (2)

The derivatives are dx'^

dx^

= -v" XO fixed

dx^

.

(3)

r

= F'j

(4)

with the physical velocities v^ and the deformation gradient F'^j as the Jacobian of the mapping (2). For later use, the determinant of the Jacobian is introduced JF

= det

[F'J\

>0.

(5)

The Lagrangian function that will be treated further is thus identified as L = L{i,X';x\v\rj)

.

(6)

In more specific terms, we postulate the Lagrangian to be the kinetic potential L = T-po{W

+ V)

(7)

with the densities of the • kinetic energy

T = po{X'^)v^Vi ,

• strain energy

poW = poW{i,X'^;F'j)

• force potential

poV = poV{i^X'^\x'^) ,

,

(8)

The various derivatives of L allow their identifications in our specific case as indicated

Volume force^

dL Q^j -

Advances in Mechanics and Mathematics VOLUME 11

Series Editors: David Y. Gao Virginia Polytechnic Institute and State University, U.S.A. Ray W. Ogden University of Glasgow, U.K.

Advisory Editors: I. Ekeland University of British Columbia, Canada S. Liao Shanghai Jiao Tung University, P.R. China K.R. Rajagopal Texas A&M University, U.S.A. T. Ratiu Ecole Polytechnique, Switzerland W. Yang Tsinghua University, P.R. China

MECHANICS OF MATERIAL FORCES

Edited by PAUL STEINMANN University of Kaiserslautem, Germany GERARD A. MAUGIN Universite Pierre et Marie Curie, Paris, France

Spri ringer

Library of Congress Cataloging-in-Publication Data Mechanics of material forces / edited by Paul Steinmann, Gerard A. Maugin. p. cm. — (Advances in mechanics and mathematics ; 11) Includes bibliographical references. ISBN 0-387-26260-1 (alk. paper) - ISBN 0-387-26261-X (ebook) 1. Strength of materials. 2. Strains and stresses. 3. Mechanics, Applied. 1. Steinmann, Paul, Dr.-tng. 11. Maugin, G. A. (Gerard A.), 1944- 111. Series. TA405.M512 2005 620.1'123-dc22 2005049040 AMS Subject Classifications: 74-06, 74Axx, 74R99, 74N20, 65Z05 iSBN-10: 0-387-26260-1 ISBN-13: 978-0387-26260-4 e-lSBN-10: 0-387-26261-X e-lSBN-13: 978-0387-26261-1 © 2005 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science-t-Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 9 8 7 6 5 4 3 2 1 springeronline.com

SPIN 11367529

Contents

Preface

ix

Contributing Authors

Part I

4d Formalism

1 On establishing balance and conservation laws in elastodynamics George Herrmann^ Reinhold Kienzler 2 Prom mathematical physics to engineering science Gerard A. Maugin

Part II

xiii

3

13

Evolving Interfaces

3 The unifying nature of the configurational force balance Eliot Fried^ Morton E. Gurtin

25

4 Generalized Stefan models Alexandre Danescu

33

5 Explicit kinetic relation from "first principles" Lev Truskinovsky, Anna Vainchtein

43

Part III

Growth & Biomechanics

6 Surface and bulk growth unified Antonio DiCarlo

53

vi

Contents

7 Mechanical and t her mo dynamical modelling of tissue growth using domain derivation techniques Jean Francois Ganghoffer

65

Material forces in the context of biotissue remodelling 77 Krishna Garikipati, Harish Narayanan, Ellen M. Arruda, Karl Grosh, Sarah Calve

Part IV

Numerical Aspects

9

Error-controlled adaptive finite element methods in nonlinear elastic fracture mechanics Marcus Riiter, Erwin Stein 10 Material force method. Continuum damage & thermo-hyperelasticity Ralf Denzer, Tina Liehe, Ellen Kuhl, Franz Josef Barth, Paul Steinmann 11 Discrete material forces in the finite element method Ralf Mueller, Dietmar Gross 12 Computational spatial and material settings of continuum mechanics. An arbitrary Lagrangian Eulerian formulation Ellen Kuhl^ Harm Askes^ Paul Steinmann

Part V

87

95

105

115

Dislocations & Peach-Koehler-Forces

13 Self-driven continuous dislocations and growth Marcelo Epstein 14 Role of the non-Riemannian plastic connection in finite elastoplasticity with continuous distribution of dislocations Sanda Cleja-Tigoiu 15 Peach-Koehler forces within the theory of nonlocal elasticity Markus Lazar

129

141

149

Contents Part VI

vii Multiphysics & Microstructure

16 On the material energy-momentum tensor in electrostatics and magnetostatics Carmine Trimarco 17 Continuum thermodynamic and variational models for continua with microstructure and material inhomogeneity Bob Svendsen 18 A crystal structure-based eigentransformation and its work-conjugate material stress Chien H. Wu

Part VII

161

173

181

Fracture & Structural Optimization

19 Teaching fracture mechanics within the theory of strength-of-materials Reinhold Kienzler, George Herrmann

193

20 Configurational thermomechanics and crack driving forces Cristian Dascalu, Vassilios K. Kalpakides

203

21 Structural optimization by material forces Manfred Braun

211

22 On structural optimisation and configurational mechanics Franz-Joseph Barthold

219

Part VIII

Path Integrals

23 Configurational forces and the propagation of a circular crack in an elastic body Vassilios K. Kalpakides^ Eleni K. Agiasofitou 24 Thermoplastic M integral and path domain dependence Pascal Sans en ^ Philippe Dufrenoy, Dieter Weichert

231

241

viii Part IX

Contents Delamination & Discontinuities

25 Peeling tapes Paolo Podio-Guidugli

253

26 Stability and bifurcation with moving discontinuities Claude Stolz^ Rachel-Marie Pradeilles-Duval

261

27 On Fracture Modelling Based on Inverse Strong Discontinuities Ragnar Larsson, Martin Fagerstrom

269

Part X

Interfaces & Phase Transition

28 Maxwell's relation for isotropic bodies Miroslav Silhavy

281

29 Driving force in simulation of phase transition front propagation Arkadi Berezovski^ Gerard A. Maugin

289

30 Modeling of the thermal treatment of steel with phase changes Serguei Dachkovski, Michael Bohm

299

Part XI

Plasticity & Damage

31 Configurational stress tensor in anisotropic ductile continuum damage mechanics Michael Briinig 32 Some class of SG Continuum models to connect various length scales in plastic deformation Lalaonirina Rakotomanana 33 Weakly nonlocal theories of damage and plasticity based on balance of dissipative material forces Helmut Stumpf, Jerzy Makowski^ Jaroslaw Gorski, Klaus Hackl

311

319

327

Preface

The notion dealt with in this volume of proceedings is often traced back to the late 19th-century writings of a rather obscure scientist, C.V. Burton. A probable reason for this is that the painstaking deciphering of this author's paper in the Philosophical Magazine (Vol.33, pp. 191-204, 1891) seems to reveal a notion that was introduced in mathematical form much later, that of local structural rearrangement. This notion obviously takes place on the material manifold of modern continuum mechanics. It is more or less clear that seemingly different phenomena - phase transition, local destruction of matter in the form of the loss of local ordering (such as in the appearance of structural defects or of the loss of cohesion by the appearance of damage or the extension of cracks), plasticity, material growth in the bulk or at the surface by accretion, wear, and the production of debris - should enter a common framework where, by pure logic, the material manifold has to play a prominent role. Finding the mathematical formulation for this was one of the great achievements of J.D. Eshelby. He was led to consider the apparent but true motion or displacement of embedded material inhomogeneities, and thus he began to investigate the "driving force" causing this motion or displacement, something any good mechanician would naturally introduce through the duahty inherent in mechanics since J.L. d'Alembert. He rapidly remarked that what he had obtained for this "force" - clearly not a force in the classical sense of Newton or Lorentz, but more a force in the sense of those "forces" that appear in chemical physics (thermodynamical forces), was related to the divergence of a quantity known since David Hilbert in mathematical physics, namely, the so-called energy-momentum tensor of field theory, or to be more accurate, the purely spatial part of this essentially fourdimensional object. Similar expressions were to appear in the study of the "force" that causes the motion of a dislocation (Peach and Koehler) and the driving force acting on the tip of a progressing macroscopic crack (Cherepanov, Rice). Simultaneously, this tensor, now rightfully called an Eshelby stress tensor, but identified by others as a tensor of

X

Preface

chemical potentials (a tensorial generalization of the Gibbs energy), was appearing in the theory of fluid mixtures (Ray Bowen, M. Grinfeld) and with its jump at discontinuity surfaces. It would be dangerous to pursue this historical disquisition further. If we did, we would enter a period where most active contributors to the field are still active at the time of this writing and are in fact contributors to the present book. Still, a few general trends can be emphasized. The late 1980s and the early 1990s witnessed a more comprehensive approach to the notion of local structural rearrangement (the local change of material configuration resulting from some physico-chemical process) and the recognition of the tensor introduced by Eshelby in the treatment of interface phenomena. With these works, a whole industry started to develop along different paths, sustained by different viewpoints, and deaUng with various applications, while identifying a general background. Although certainly not an entirely new field per 5e, it started to organize itself with the introduction of a nomenclature. This is often the result of personal idiosyncratsies. But two essential expressions have come to be accepted, practically on an equal footing: that of material forces^ and that of configurational forces, both of which expressions denote the thermodynamical forces that drive the development of the above-mentioned structural rearrangements and the defect motions. While we are not sure that these two terms are fully equivalent, it became obvious to the co-editors of this volume in the early 2000s that time was ripe for a symposium fully pledged to this subject matter. The format of the EUROMECH Colloquia was thought to be well adapted because of its specialized focus and the relatively small organizational effort required for its set up. Midway between Paris and central Germany was appropriate for the convenience of transport and some neutrality. All active actors, independently of their peculiar viewpoints, were invited, and most came, on the whole resulting in vivid discussions between gentlemen. The present book of proceedings well reflects the breadth of the field at the time of the colloquium in 2003, and it captures the contributions in the order in which they were presented, all orally, at the University of Kaiserslautern, Germany, May 21-24, 2003, on the occasion of the EUROMECH Colloquium 445 on the "Mechanics of Material Forces". The notion of material force acting on a variety of defects such as cracks, dislocations, inclusions, precipitates, phase boundaries, interfaces and the hke was extensively discussed. Accordingly, typical topics of interest in continuum physics are kinetics of defects, morphology changes, path integrals, energy-release rates, duality between direct and inverse motions, four-dimensional formalism, etc. Especially remarkable are the initially unexpected recent developments in the field of numerics, which

Preface

xi

are generously illustrated in these proceedings. Further developments have already taken place since the colloquium, and many will come, no doubt. Students and researchers in the field will therefore be happy to find in the present volume useful basic material accompanied by fruitful hints at further fines of research, since the book offers, with a marked open-mindedness, various interpretations and applications of a field in full momentum. Last but not least our sincere thanks go to everybody who helped to make this EUROMECH Colloquium a success, in particular to the coworkers and students at the Lehrstuhl fiir Technische Mechanik of the University of Kaiserslautern who supported the Colloquium during the time of the meeting. Very special thanks go to Ralf Denzer who took the responsibility, and the hard and at times painful work, to assemble and organize this book.

Paris, June 2004. Kaiserslautern, June 2004,

Gerard A. Maugin Paul Steinmann

Contributing Authors

Franz-Joseph Barthold University of Dortmund, Germany Arkadi Berezovski Tallinn Technical University, Estonia Manfred Braun Universitat Duisburg-Essen, Germany Michael Briinig Universitat Dortmund, Germany Sanda Cleja-Tigoiu University of Bucharest, Romania Serguei Dachkovski University of Bremen, Germany Alexandre Danescu Ecole Centrale de Lyon, Prance Cristian Dascalu Universite Joseph Fourier, Grenoble, France Ralf Denzer University of Kaiserslautern, Germany Antonio DiCarlo Universita degli Studi "Roma Tre", Italy Marcelo Epstein The University of Calgary, Canada Jean-Francois GanghofFer Ecole Nationale Superieure en Electricite et Mecanique, Nancy, France Krishna Garikipati University of Michigan, Ann Arbor, USA

xiv

Contributing Authors

Morton E. Gurtin Carnegie Mellon University, Pittsburgh, USA George Herrmann Stanford University, USA Vassilios K. Kalpakides University of loannina, Greece Reinhold Kienzler University of Bremen, Germany Ellen Kuhl University of Kaiserslautern, Germany Ragnar Larsson Chalmers University of Technology, Goteborg, Sweden Markus Lazar Universite Pierre et Marie Curie, Paris, Prance Gerard A. Maugin Universite Pierre et Marie Curie, Paris, Prance Ralf Miiller Technische Universitat Darmstadt, Germany Paolo Podio-Guidugli Universita di Roma "Tor Vergata", Italy Lalaonirina Rakotomanana Universite de Rennes, Prance Pascal Sansen Ecole Superieure d'Ingenieurs en Electrotechnique et Electronique, Amiens, Prance Miroslav Silhavy Mathematical Institute of the AV CR, Prague, Czech Republic Erwin Stein University of Hannover, Germany Claude Stolz Ecole Poly technique, Palaiseau, Prance Helmut Stumpf Ruhr-Universitat Bochum, Germany Bob Svendsen University of Dortmund, Germany

Contributing Authors Carmine Trimarco University of Pisa, Italy Anna Vainchtein University of Pittsburgh, USA Chien H. Wu University of Illinois at Chicago, USA

xv

4D FORMALISM

Chapter 1 ON ESTABLISHING BALANCE AND C O N SERVATION LAWS IN ELASTODYNAMICS George Herrmann Division of Mechanics and Computation, Stanford University [email protected]

Reinhold Kienzler Department of Production Engineering, University of Bremen [email protected]

Abstract

By placing time on the same level as the space coordinates, governing balance and conservation laws are derived for elastodynamics. Both Lagrangian and Eulerian descriptions are used and the laws mentioned are derived by subjecting the Lagrangian (or its product with the coordinate four-vector) to operations of the gradient, divergence and curl. The 4 x 4 formalism employed leads to balance and conservation laws which are partly well known and partly seemingly novel.

Keywords: elastodynamics, 4 x 4 formalism, balance and conservation laws

Introduction The most v^idely used procedure to establish balance and conservation laws in the theory of fields, provided a Lagrangian function exists, is based on the first theorem of Noether [1], including an extension of Bessel-Hagen [2]. If such a function does not exist, e. g., for dissipative systems, then the so-called Neutral-Action method (cf. [3], [4]) can be employed. For systems which do possess a Lagrangian function, the Neutral-Action method leads to the same results as the Noether procedure including the Bessel-Hagen extension. In addition to the above two methods, a third procedure consists in submitting the Lagrangian L to the differential operators of grad^ div

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George Herrmann and Reinhold Kienzler

and curl^ respectively. The latter two operations are applied to a vector xL where x is the position vector. This procedure has been successfully used in elastostatics in material space (cf. [4]), leading to the pathindependent integral commonly known as J , M and L. Thus it appeared intriguing to investigate the results of the application of the three differential operators to the elastodynamic field, considering the time not as a parameter, as is usual, but as an independent variable on the same level as the three space coordinates. To make all four coordinates of the same dimension, the time is multiplied by some characteristic velocity, as is done in the theory of relativity. Results are presented both in terms of a Lagrangian and an Eulerian formulation.

1. 1.1.

ELASTODYNAMICS IN LAGRANGIAN DESCRIPTION LAGRANGIAN FUNCTION

We consider an elastic body in motion with mass density po = Po{X^) in a reference configuration. We identify the independent variables ^^ {fi ~ 0,1,2,3) as

^^ = i = Cot, (1) ^^ =X\

J = 1,2,3 .

The independent variable i and not t is used in order for all independent variables to have the same dimensions, where CQ is some velocity. In the special theory of relativity, CQ = c = velocity of light is used, to obtain a Lorentz-ivariant formulation in Minkowski space. In the non-relativistic theory, CQ may be chosen arbitrarily, e. g., as some characteristic wave speed or it may be normalized to one. The independent variables ^^ = X'^ (J = 1,2,3) are the space-coordinates in the reference configuration of the body. Although ^^ = t and ^'^ = X^ have the same dimensions, the time t is an independent variable and is not related to the space-coordinates by a proper time r as in the theory of relativity. Therefore, we will deal with Galilean-invariant objects, for "vectorsand matrices that will be called (cf. [5]) "tensors", although the formulation is not covariant. Special care has to be used when dealing with div and curl^ where timeand space-coordinates are coupled. The current coordinates are designated by x'^ and play the role of dependent variables or fields. They are defined by the mapping or motion

On establishing balance and conservation laws in elastodynamics x' = x'iiX'),

i = 1,2,3.

5 (2)

The derivatives are dx'^

dx^

= -v" XO fixed

dx^

.

(3)

r

= F'j

(4)

with the physical velocities v^ and the deformation gradient F'^j as the Jacobian of the mapping (2). For later use, the determinant of the Jacobian is introduced JF

= det

[F'J\

>0.

(5)

The Lagrangian function that will be treated further is thus identified as L = L{i,X';x\v\rj)

.

(6)

In more specific terms, we postulate the Lagrangian to be the kinetic potential L = T-po{W

+ V)

(7)

with the densities of the • kinetic energy

T = po{X'^)v^Vi ,

• strain energy

poW = poW{i,X'^;F'j)

• force potential

poV = poV{i^X'^\x'^) ,

,

(8)

The various derivatives of L allow their identifications in our specific case as indicated

Volume force^

dL Q^j -

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