Applied Mathematical Sciences Volume 137

Editors J.E. Marsden L. Sirovich

Advisors S. Antman J.K. Hale P. Holmes T. Kambe J. Keller K. Kirchgassner B.J. Matkowsky C.S. Peskin

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Systems. 9. Friedrichs: Spectral Theory of Operators in

Hilbert Space. 10. Stroud: Numerical Quadrature and Solution of

Ordinary Differential Equations. 11. Wolovich: Linear Multivariate Systems. 12. Berkovitz: Optimal Control Theory. 13. Bluman/Cole: Similarity Methods for Differential

Equations. 14. Yoshizawa: Stability Theory and the Existence of

Periodic Solution and Almost Periodic Solutions. 15. Braun: Differential Equations and Their

Applications, 3rd ed. 16. Lefschetz: Applications of Algebraic Topology. 17. Collatz/Wetterling: Optimization Problems. 18. Grenander: Pattern Synthesis: Lectures in Pattern

Theory, Vol. I. 19. Marsden/McCracken: Hopf Bifurcation and Its

Applications. 20. Driver: Ordinary and Delay Differential

Equations. 21. Courant/Friedrichs: Supersonic Flow and Shock

Waves. 22. Rouche/Habets/Laloy: Stability Theory by

Liapunov's Direct Method. 23. Lamperti: Stochastic Processes: A Survey of the

Mathematical Theory. 24. Grenander: Pattern Analysis: Lectures in Pattern

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Methods for Constrained and Unconstrained Systems.

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Equations. 32. Meis/Markowilz: Numerical Solution of Partial

Differential Equations. 33. Grenander: Regular Structures: Lectures in

Pattern Theory, Vol. III.

34. Kevorkian/Cole: Perturbation Methods in Applied Mathematics.

35. Carr: Applications of Centre Manifold Theory. 36. Bengtsson/Ghil/Kallen: Dynamic Meteorology:

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Dimensional Spaces. 38. Lichtenberg/Lieberman: Regular and Chaotic

Dynamics, 2nd ed. 39. Piccini/Stampacchia/Vidossich: Ordinary

Differential Equations in R". 40. Naylor/Sell: Linear Operator Theory in

Engineering and Science. 41. Sparrow: The Lorenz Equations: Bifurcations,

Chaos, and Strange Attractors. 42. Guckenheimer/Holmes: Nonlinear Oscillations,

Dynamical Systems, and Bifurcations of Vector Fields.

43. Ockendon/Taylor: Inviscid Fluid Flows. 44. Pazy: Semigroups of Linear Operators and

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47. Hale et al: An Introduction to Infinite Dimensional Dynamical Systems—Geometric Theory.

48. Murray: Asymptotic Analysis. 49. Ladyzhenskaya: The Boundary-Value Problems

of Mathematical Physics. 50. Wilcox: Sound Propagation in Stratified Fluids. 51. Golubitsky/Schaeffer: Bifurcation and Groups in

Bifurcation Theory, Vol. I. 52. Chipot: Variational Inequalities and Flow in

Porous Media. 53. Majda: Compressible Fluid Flow and System of

Conservation Laws in Several Space Variables. 54. Wasow: Linear Turning Point Theory. 55. Yosida: Operational Calculus: A Theory of

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Incompressible Nonviscous Fluids. 97. Lasota/Mackey: Chaos, Fractals, and Noise:

Stochastic Aspects of Dynamics, 2nd ed. 98. de Boor/HSllig/Riemenschneider: Box Splines. 99. Hale/Lunel: Introduction to Functional

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Applied Mathematics. 101. Nusse/Yorke: Dynamics: Numerical

Explorations. 102. Chossat/Jooss: The Couette-Taylor Problem. 103. Chorin: Vorticity and Turbulence. 104. Farkas: Periodic Motions. 105. Wiggins: Normally Hyperbolic Invariant

Manifolds in Dynamical Systems. 106. Cercignani/Illner/Pulvirenti: The Mathematical

Theory of Dilute Gases. 107. Antman: Nonlinear Problems of Elasticity. 108. Zeidler: Applied Functional Analysis:

Applications to Mathematical Physics. 109. Zeidler: Applied Functional Analysis: Main

Principles and Their Applications. 110. Diekmannfvan GilsNerduyn Lunel/Walther:

Delay Equations: Functional-, Complex-, and Nonlinear Analysis.

111. Visintin: Differential Models of Hysteresis. 112. Kuznetsov: Elements of Applied Bifurcation

Theory. 113. Hislop/Sigal: Introduction to Spectral Theory:

With Applications to SchrSdinger Operators. 114. Kevorkian/Cole: Multiple Scale and Singular

Perturbation Methods. 115. Taylor: Partial Differential Equations I, Basic

Theory. 116. Taylor: Partial Differential Equations n,

Qualitative Studies of Linear Equations. 117. Taylor: Partial Differential Equations HI,

Nonlinear Equations.

Applied Mathematical Sciences

(continued from previous page)

118. Godlewski/Raviart: Numerical Approximation of Hyperbolic Systems of Conservation Laws.

119. Wu: Theory and Applications of Partial Functional Differential Equations.

120. Kirsch: An Introduction to the Mathematical Theory of Inverse Problems.

121. Brokate/Sprekels: Hysteresis and Phase Transitions.

122. Gliklikh: Global Analysis in Mathematical Physics: Geometric and Stochastic Methods.

123. Le/Schmilt: Global Bifurcation in Variational Inequalities: Applications to Obstacle and Unilateral Problems.

124. Polak: Optimization: Algorithms and Consistent Approximations.

125. Amold/Khesin: Topological Methods in Hydrodynamics.

126. Hoppensteadt/Izhikevich: Weakly Connected Neural Networks.

127. Isakov: Inverse Problems for Partial Differential Equations.

128. Li/Wiggins: Invariant Manifolds and Fibrations for Perturbed Nonlinear Schrodinger Equations.

129. Mtiller: Analysis of Spherical Symmetries in Euclidean Spaces.

130. Feintuch: Robust Control Theory in Hilbert Space.

131. Ericksen: Introduction to the Thermodynamics of Solids, Revised ed.

132. Ihlenburg: Finite Element Analysis of Acoustic Scattering.

133. Vorovich: Nonlinear Theory of Shallow Shells. 134. Vein/Dale: Determinants and Their Applications

in Mathematical Physics. 135. Drew/Passman: Theory of Multicomponent

Fluids. 136. Cioranescu/Saint Jean Paulin: Homogenization

of Reticulated Structures. 137. Gurtin: Configurational Forces as Basic Concepts

of Continuum Physics. 138. Haller: Chaos Near Resonance. 139. Sulem/Sulem: The Nonlinear Schrodinger

Equation: Self-Focusing and Wave Collapse.

Morton E. Gurtin

Configurational Forces as Basic Concepts of Continuum Physics

Springer

Morton E. Gurtin Department of Mathematics Carnegie Mellon University Pittsburgh, PA 15213 USA

Editors

J.E. Marsden L. Sirovich Control and Dynamical Systems, 107-81 Division of Applied Mathematics California Institute of Technology Brown University Pasadena, CA 91125 Providence, RI 02912 USA USA

Mathematics Subject Classification (1991): 73bxx, 73m25, 73a05

With seven illustrations.

Library of Congress Cataloging-in-Publication Data Gurtin, Morton E.

Configurational forces as basic concepts of continuum physics / Morton E. Gurtin.

p. cm. — (Applied mathematical sciences ; 137) Includes bibliographical references. ISBN 0-387-98667-7 (cloth : alk. paper) 1. Field theory (Physics) 2. Configuration space. I. Title.

II. Series: Applied mathematical sciences (Springer-Verlag New York Inc.) ; v. 137. QA1.A647 vol. 137 [QC173.7] 510 s—dc21 [530.14] 98-55407

© 2000 Springer-Verlag New York, 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-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, 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 known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

ISBN 0-387-98667-7 Springer-Verlag New York Berlin Heidelberg SPIN 10698130

For my grandchildren Katie, Grant, and Liza

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Contents

1. Introduction 1a. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . 1b. Variational definition of configurational forces . . . . . . . . . 2c. Interfacial energy. A further argument for a configurational

force balance . . . . . . . . . . . . . . . . . . . . . . . . . . . 5d. Configurational forces as basic objects . . . . . . . . . . . . . 7e. The nature of configurational forces . . . . . . . . . . . . . . . 9f. Configurational stress and residual stress.

Internal configurational forces . . . . . . . . . . . . . . . . . . 10g. Configurational forces and indeterminacy . . . . . . . . . . . . 11h. Scope of the book . . . . . . . . . . . . . . . . . . . . . . . . 12i. On operational definitions and mathematics . . . . . . . . . . . 12j. General notation. Tensor analysis . . . . . . . . . . . . . . . . 13

j1. On direct notation . . . . . . . . . . . . . . . . . . . . 13j2. Vectors and tensors. Fields . . . . . . . . . . . . . . . 13j3. Third-order tensors (3-tensors). The operation T : � . . 15j4. Functions of tensors . . . . . . . . . . . . . . . . . . . 16

A. Configurational forces within a classical context 19

2. Kinematics 21a. Reference body. Material points. Motions . . . . . . . . . . . . 21b. Material and spatial vectors. The sets Espace and Ematter . . . . . 22c. Material and spatial observers . . . . . . . . . . . . . . . . . . 23d. Consistency requirement. Objective fields . . . . . . . . . . . 23

viii Contents

3. Standard forces. Working 25a. Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25b. Working. Standard force and moment balances as consequences

of invariance under changes in spatial observer . . . . . . . . . 26

4. Migrating control volumes. Stationary and time-dependentchanges in reference configuration 29a. Migrating control volumes P � P (t). Velocity fields for ∂P (t)

and ∂P̄ (t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29b. Change in reference configuration . . . . . . . . . . . . . . . . 31

b1. Stationary change in reference configuration . . . . . . 31b2. Time-dependent change in reference configuration . . . 32

5. Configurational forces 34a. Configurational forces . . . . . . . . . . . . . . . . . . . . . . 34b. Working revisited . . . . . . . . . . . . . . . . . . . . . . . . 35c. Configurational force balance as a consequence of invariance

under changes in material observer . . . . . . . . . . . . . . . 36d. Invariance under changes in velocity field for ∂P (t).

Configurational stress relation . . . . . . . . . . . . . . . . . . 37e. Invariance under time-dependent changes in reference.

External and internal force relations . . . . . . . . . . . . . . . 38f. Standard and configurational forms of the working.

Power balance . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6. Thermodynamics. Relation between bulk tension and energy.Eshelby identity 41a. Mechanical version of the second law . . . . . . . . . . . . . . 41b. Eshelby relation as a consequence of the second law . . . . . . 42c. Thermomechanical theory . . . . . . . . . . . . . . . . . . . . 44d. Fluids. Current configuration as reference . . . . . . . . . . . . 45

7. Inertia and kinetic energy. Alternative versions of the second law 46a. Inertia and kinetic energy . . . . . . . . . . . . . . . . . . . . 46b. Alternative forms of the second law . . . . . . . . . . . . . . . 47c. Pseudomomentum . . . . . . . . . . . . . . . . . . . . . . . . 47d. Lyapunov relations . . . . . . . . . . . . . . . . . . . . . . . . 48

8. Change in reference configuration 50a. Transformation laws for free energy and standard force . . . . 50b. Transformation laws for configurational force . . . . . . . . . 51

9. Elastic and thermoelastic materials 53a. Mechanical theory . . . . . . . . . . . . . . . . . . . . . . . . 54

a1. Basic equations . . . . . . . . . . . . . . . . . . . . . 54

Contents ix

a2. Constitutive theory . . . . . . . . . . . . . . . . . . . 54b. Thermomechanical theory . . . . . . . . . . . . . . . . . . . . 56

b1. Basic equations . . . . . . . . . . . . . . . . . . . . . 56b2. Constitutive theory . . . . . . . . . . . . . . . . . . . 57

B. The use of configurational forces to characterizecoherent phase interfaces 61

10. Interface kinematics 63

11. Interface forces. Second law 66a. Interface forces . . . . . . . . . . . . . . . . . . . . . . . . . 66b. Working . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67c. Standard and configurational force balances at the interface . . 68d. Invariance under changes in velocity field for S (t). Normal

configurational balance . . . . . . . . . . . . . . . . . . . . . 69e. Power balance. Internal working . . . . . . . . . . . . . . . . 70f. Second law. Internal dissipation inequality for the interface . . 71g. Localizations using a pillbox argument . . . . . . . . . . . . . 72

12. Inertia. Basic equations for the interface 74a. Relative kinetic energy . . . . . . . . . . . . . . . . . . . . . 74b. Determination of bS and eS . . . . . . . . . . . . . . . . . . 75c. Standard and configurational balances with inertia . . . . . . . 77d. Constitutive equation for the interface . . . . . . . . . . . . . . 78e. Summary of basic equations . . . . . . . . . . . . . . . . . . . 79f. Global energy inequality. Lyapunov relations . . . . . . . . . . 80

C. An equivalent formulation of the theory.Infinitesimal deformations 81

13. Formulation within a classical context 83a. Background. Reason for an alternative formulation

in terms of displacements . . . . . . . . . . . . . . . . . . . . 83b. Finite deformations. Modified Eshelby relation . . . . . . . . . 84c. Infinitesimal deformations . . . . . . . . . . . . . . . . . . . . 86

14. Coherent phase interfaces 88a. General theory . . . . . . . . . . . . . . . . . . . . . . . . . . 88b. Infinitesimal theory with linear stress-strain relations in bulk . . 89

x Contents

D. Evolving interfaces neglecting bulk behavior 91

15. Evolving surfaces 93a. Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

a1. Background. Superficial stress . . . . . . . . . . . . . 93a2. Superficial tensor fields . . . . . . . . . . . . . . . . . 94

b. Smoothly evolving surfaces . . . . . . . . . . . . . . . . . . . 97b1. Time derivative following S . Normal time derivative . . 97b2. Velocity fields for the boundary curve ∂G of a smoothly

evolving subsurface of S . Transport theorem . . . . 99b3. Transformation laws . . . . . . . . . . . . . . . . . . 100

16. Configurational force system. Working 101a. Configurational forces. Working . . . . . . . . . . . . . . . . . 101b. Configurational force balance as a consequence of invariance

under changes in material observer . . . . . . . . . . . . . . . 102c. Invariance under changes in velocity fields. Surface tension.

Surface shear . . . . . . . . . . . . . . . . . . . . . . . . . . . 103d. Normal force balance. Intrinsic form for the working . . . . . . 104e. Power balance. Internal working . . . . . . . . . . . . . . . . 105

17. Second law 108

18. Constitutive equations 110a. Functions of orientation . . . . . . . . . . . . . . . . . . . . . 110b. Constitutive equations . . . . . . . . . . . . . . . . . . . . . . 111c. Evolution equation for the interface . . . . . . . . . . . . . . . 113d. Lyapunov relations . . . . . . . . . . . . . . . . . . . . . . . . 114

19. Two-dimensional theory 115a. Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115b. Configurational forces. Working. Second law . . . . . . . . . . 116c. Constitutive theory . . . . . . . . . . . . . . . . . . . . . . . . 118d. Evolution equation for the interface . . . . . . . . . . . . . . . 119e. Corners . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120f. Angle-convexity. The Frank diagram . . . . . . . . . . . . . . 120g. Convexity of the interfacial energy and evolution

of the interface . . . . . . . . . . . . . . . . . . . . . . . . . . 124

E. Coherent phase interfaces with interfacial energyand deformation 127

20. Theory neglecting standard interfacial stress 129a. Standard and configurational forces. Working . . . . . . . . . 129

Contents xi

b. Power balance. Internal working . . . . . . . . . . . . . . . . 131c. Second law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

c1. Second law. Interfacial dissipation inequality . . . . . . 132c2. Derivation of the interfacial dissipation inequality

using a pillbox argument . . . . . . . . . . . . . . . . 132d. Constitutive equations . . . . . . . . . . . . . . . . . . . . . . 133e. Construction of the process used in restricting

the constitutive equations . . . . . . . . . . . . . . . . . . . . 135f. Basic equations with inertial external forces . . . . . . . . . . 135

f1. Standard and configurational balances . . . . . . . . . 135f2. Summary of basic equations . . . . . . . . . . . . . . 136

g. Global energy inequality. Lyapunov relations . . . . . . . . . . 137

21. General theory with standard and configurational stresswithin the interface 138a. Kinematics. Tangential deformation gradient . . . . . . . . . . 138b. Standard and configurational forces. Working . . . . . . . . . 139c. Power balance. Internal working . . . . . . . . . . . . . . . . 142d. Second law. Interfacial dissipation inequality . . . . . . . . . . 144e. Constitutive equations . . . . . . . . . . . . . . . . . . . . . . 145f. Basic equations with inertial external forces . . . . . . . . . . 147g. Lyapunov relations . . . . . . . . . . . . . . . . . . . . . . . . 147

22. Two-dimensional theory with standard and configurational stresswithin the interface 149a. Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149b. Forces. Working . . . . . . . . . . . . . . . . . . . . . . . . . 150c. Power balance. Internal working. Second law . . . . . . . . . . 152d. Constitutive equations . . . . . . . . . . . . . . . . . . . . . . 155e. Evolution equations for the interface . . . . . . . . . . . . . . 156

F. Solidification 157

23. Solidification. The Stefan condition as a consequence of theconfigurational force balance 159a. Single-phase theory . . . . . . . . . . . . . . . . . . . . . . . 159b. The classical two-phase theory revisited. The Stefan condition

as a consequence of the configurational balance . . . . . . . . 160

24. Solidification with interfacial energy and entropy 163a. General theory . . . . . . . . . . . . . . . . . . . . . . . . . . 163b. Approximate theory. The Gibbs-Thomson condition as a

consequence of the configurational balance . . . . . . . . . . . 166c. Free-boundary problems for the approximate theory.

Growth theorems . . . . . . . . . . . . . . . . . . . . . . . . . 167

xii Contents

c1. The quasilinear and quasistatic problems . . . . . . . . 167c2. Growth theorems . . . . . . . . . . . . . . . . . . . . 168

G. Fracture 173

25. Cracked bodies 175a. Smooth cracks. Control volumes . . . . . . . . . . . . . . . . 175b. Derivatives following the tip. Tip integrals. Transport theorems . 177

26. Motions 182

27. Forces. Working 184a. Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184b. Working . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186c. Standard and configurational force balances . . . . . . . . . . 186d. Inertial forces. Kinetic energy . . . . . . . . . . . . . . . . . . 188

28. The second law 190a. Statement of the second law . . . . . . . . . . . . . . . . . . . 190b. The second law applied to crack control volumes . . . . . . . . 191c. The second law applied to tip control volumes. Standard form

of the second law . . . . . . . . . . . . . . . . . . . . . . . . 191d. Tip traction. Energy release rate. Driving force . . . . . . . . . 193e. The standard momentum condition . . . . . . . . . . . . . . . 194

29. Basic results for the crack tip 196

30. Constitutive theory for growing cracks 198a. Constitutive relations at the tip . . . . . . . . . . . . . . . . . 198b. The Griffith-Irwin function . . . . . . . . . . . . . . . . . . . 199c. Constitutively isotropic crack tips. Tips with constant mobility . 200

31. Kinking and curving of cracks. Maximum dissipation criterion 201a. Criterion for crack initiation. Kink angle . . . . . . . . . . . . 202b. Maximum dissipation criterion for crack propagation . . . . . 204

32. Fracture in three space dimensions (results) 208

H. Two-dimensional theory of corners and junctionsneglecting inertia 211

33. Preliminaries. Transport theorems 213a. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . 213b. Transport theorems . . . . . . . . . . . . . . . . . . . . . . . 214

Contents xiii

b1. Bulk fields . . . . . . . . . . . . . . . . . . . . . . . . 214b2. Interfacial fields . . . . . . . . . . . . . . . . . . . . . 215

34. Thermomechanical theory of junctions and corners 218a. Motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218b. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219c. Forces. Working . . . . . . . . . . . . . . . . . . . . . . . . . 220d. Second law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221e. Basic results for the junction . . . . . . . . . . . . . . . . . . 222f. Weak singularity conditions. Nonexistence of corners . . . . . 222g. Constitutive equations . . . . . . . . . . . . . . . . . . . . . . 223h. Final junction conditions . . . . . . . . . . . . . . . . . . . . 224

I. Appendices on the principle of virtual work forcoherent phase interfaces 225

A1. Weak principle of virtual work 227a. Virtual kinematics . . . . . . . . . . . . . . . . . . . . . . . . 227b. Forces. Weak principle of virtual work . . . . . . . . . . . . . 228c. Proof of the weak theorem of virtual work . . . . . . . . . . . 229

A2. Strong principle of virtual work 232a. Virtually migrating control volumes . . . . . . . . . . . . . . . 232b. Forces. Strong principle of virtual work . . . . . . . . . . . . . 233c. Proof of the strong theorem of virtual work . . . . . . . . . . . 234d. Comparison of the strong and weak principles . . . . . . . . . 236

References 239

Index 247