applied mathematical sciences978-0-387-22656... · 2017. 8. 28. · ordinary differential...
TRANSCRIPT
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
Springer New York Berlin Heidelberg Barcelona Hong Kong London Milan Paris Singapore Tokyo
Applied Mathematical Sciences
1. John: Partial Differential Equations, 4th ed. 2. Sirovich: Techniques of Asymptotic Analysis. 3. Hale: Theory of Functional Differential
Equations, 2nd ed. 4. Percus: Combinatorial Methods. 5. von Mises/Friedrichs: Fluid Dynamics. 6. Freiberger/Grenander: A Short Course in
Computational Probability and Statistics. 7. Pipkin: Lectures on Viscoelasticity Theory. 8. Giacagiia: Perturbation Methods in Non-linear
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
Theory, Vol. II. 25. Davies: Integral Transforms and Their
Applications, 2nd ed. 26. Kushner/Clark: Stochastic Approximation
Methods for Constrained and Unconstrained Systems.
27. de Boor: A Practical Guide to Splines. 28. Keilson: Markov Chain Models—Rarity and
Exponentiality. 29. de Veubeke: A Course in Elasticity. 30. Shiatycki: Geometric Quantization and Quantum
Mechanics. 31. Reid: Sturmian Theory for Ordinary Differential
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:
Data Assimilation Methods. 37. Saperstone: Semidynamical Systems in Infinite
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
Applications to Partial Differential Equations. 45. Glashoff/Gustafson: Linear Operations and
Approximation: An Introduction to the Theoretical Analysis and Numerical Treatment of Semi-Infinite Programs.
46. Wilcox: Scattering Theory for Diffraction Gratings.
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
Hyperfunctions. 56. Chang/Howes: Nonlinear Singular Perturbation
Phenomena: Theory and Applications. 57. Reinhardt: Analysis of Approximation Methods
for Differential and Integral Equations. 58. Dwoyer/Hussaini/Voigt (eds): Theoretical
Approaches to Turbulence. 59. Sanders/Verhulst: Averaging Methods in
Nonlinear Dynamical Systems. 60. Ghil/Childress: Topics in Geophysical
Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics.
(continued following index)
Applied Mathematical Sciences (continued from page ii)
61. Sattinger/Weaver: Lie Groups and Algebras with Applications to Physics, Geometry, and Mechanics.
62. LaSalle: The Stability and Control of Discrete Processes.
63. Grasman: Asymptotic Methods of Relaxation Oscillations and Applications.
64. Hsu: Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems.
65. Rand/Armbruster: Perturbation Methods, Bifurcation Theory and Computer Algebra.
66. Hlavdcek/Haslinger/Necasl/Lovlsek: Solution of Variational Inequalities in Mechanics.
67. Cercignani: The Boltzmann Equation and Its Applications.
68. Temam: Infinite Dimensional Dynamical Systems in Mechanics and Physics.
69. Golubitsky/Stewart/Schaeffer: Singularities and Groups in Bifurcation Theory, Vol. II.
70. Constantin/Foias/Nicolaenko/Temam: Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations.
71. Catlin: Estimation, Control, and the Discrete Kalman Filter.
72. Lochak/Meunier: Multiphase Averaging for Classical Systems.
73. Wiggins: Global Bifurcations and Chaos. 74. Mawhin/Willem: Critical Point Theory and
Hamiltonian Systems. 75. Abraham/Marsden/Ratiu: Manifolds, Tensor
Analysis, and Applications, 2nd ed. 76. Lagerstrom: Matched Asymptotic Expansions:
Ideas and Techniques. 77. Aldous: Probability Approximations via the
Poisson Clumping Heuristic. 78. Docorogna: Direct Methods in the Calculus of
Variations. 79. Herndndez-Lerma: Adaptive Markov Processes. 80. Lawden: Elliptic Functions and Applications. 81. Bluman/Kumei: Symmetries and Differential
Equations. 82. Kress: Linear Integral Equations. 83. Bebernes/Eberly: Mathematical Problems from
Combustion Theory. 84. Joseph: Fluid Dynamics of Viscoelastic Fluids. 85. Yang: Wave Packets and Their Bifurcations in
Geophysical Fluid Dynamics. 86. Dendrinos/Sonis: Chaos and Socio-Spatial
Dynamics. 87. Weder: Spectral and Scattering Theory for Wave
Propagation in Perturbed Stratified Media. 88. Bogaevski/Povzner: Algebraic Methods in
Nonlinear Perturbation Theory.
89. O'Malley: Singular Perturbation Methods for Ordinary Differential Equations.
90. Meyer/Hall: Introduction to Hamiltonian Dynamical Systems and the N-body Problem.
91. Straughan: The Energy Method, Stability, and Nonlinear Convection.
92. Naber: The Geometry of Minkowski Spacetime. 93. Colton/Kress: Inverse Acoustic and
Electromagnetic Scattering Theory. 94. Hoppensleadt: Analysis and Simulation of
Chaotic Systems. 95. Hackbusch: Iterative Solution of Large Sparse
Systems of Equations. 96. Marchioro/Pulvirenti: Mathematical Theory of
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
Differential Equations. 100. Sirovich (ed): Trends and Perspectives in
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