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
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
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
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
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
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)
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
b. Smoothly evolving surfaces . . . . . . . . . . . . . . . . . . . 97b1. Time derivative following S . Normal time derivative . . 97b2. Velocity fields for the boundary curve ∂G of a smoothly
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
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.
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