Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T09:07:26.029Z Has data issue: false hasContentIssue false
  • English
  • Français

Hyperelasticity for crystals

Published online by Cambridge University Press:  16 July 2009

Michel Chipot
Michel Chipot
Affiliation:
Université de Metz, Département de Mathématique, Ile du Saulcy, 57045 Metz – Cedex, France
Get access Rights & Permissions [Opens in a new window]

Abstract

The goal of this note is to explain the process of computing the lowest energy level achieved by an elastic crystal subject to homogeneous boundary deformation. This analysis follows mainly the lines of Chipot and Kinderlehrer or Fonesca but we expect that it will lead to some new results for some other energy functional having different invariance properties. The mathematical interest of the result also lies in the fact that the lowest energy level computed this way coincides with the relaxed energy functional (see Fonesca). This relaxed energy is determined by a known thermodynamic quantity, Ericksen' subenergy. Within this review we will also analyse some of the mathematical and physical aspects of these highly oscillatory problems.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 1 , Issue 2 , June 1990 , pp. 113 - 129
Copyright
Copyright © Cambridge University Press 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ball, J. 1977 Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal. 63, 337–403.CrossRefGoogle Scholar
Ball, J. & James, R. 1987 Fine phase mixtures as minimizers of energy. Arch. Rat. Mech. Anal. 100, 13–52.CrossRefGoogle Scholar
Ball, J. & Murat, F. 1984 W1.p quasiconvexity and variational problems for multiple integrals. J. Fund. Anal. 58, 225–253.CrossRefGoogle Scholar
Ciarlet, P. G. 1983 Three Dimensional Elasticity, Tata Institute Lecture Notes. Springer.CrossRefGoogle Scholar
Chipot, M. & Kinderlehrer, D. 1988 Equilibrium configuration of crystals. Arch. Rat. Mech. Anal. 103, 3, 237–277.CrossRefGoogle Scholar
Chipot, M., Kinderlehrer, D. & Vergara-Caffarelli, G. 1986 Smoothness of linear laminates. Arch. Rat. Mech. Anal. 96, 81–96.CrossRefGoogle Scholar
Dacorogna, B. 1982 Weak Continuity and Weak Lower Semicontinuity of Nonlinear Functionals. Springer Lecture Notes in Math. 922.CrossRefGoogle Scholar
Dacorogna, B. 1985 Remarques sur les notions de polyconvexité, quasiconvexité et convexité de rang I. Journal de Maths. Pures et Appl. 64, 403–438.Google Scholar
Ericksen, J. L. 1987 Twinning of crystals I. In Metastability and Incompletely Posed problems. I.M.A. Volume Math. Appl. 3 (ed. Antman, S., Ericksen, J. L., Kinderlehrer, D. & Muller, I.), pp. 77–96. Springer.Google Scholar
Ericksen, J. L. 1981 Some simpler cases of the Gibbs phenomenon for thermoelastic solids. J. Thermal Stresses 4, 13–30.CrossRefGoogle Scholar
Fonseca, I. 1987 Variational methods for elastic crystals. Arch. Rat. Mech. Anal. 97, 189–220.CrossRefGoogle Scholar
Fonseca, I. 1988 The lower quasiconvex envelope of the stored energy function of an elastic crystal. J. Math. P. Appl. 67, 175–195.Google Scholar
Fonseca, I. & Tartar, L. 1989 The displacement problem for elastic crystals. Proc. Roy. Soc. Edinburgh 113A, 159–180.CrossRefGoogle Scholar
Kinderlehrer, D. 1987 Twinning of crystal II. Metastability and Incompletely Posed Problems. I.M.A. Vol. Math. Appl. 3 (ed. Antman, S., Ericksen, J. L., Kinderlehrer, D. & Muller, I.), pp.185–211. Springer.Google Scholar
Kinderlehrer, D. 1987 Remarks about the equilibrium configurations of crystals. Proc. Symp. Material Instabilities in continuum mechanics (ed. Ball, J. M.). Oxford.Google Scholar
Saks, S. 1937 Theory of the Integral. Warszawa-Lwow, Stechert.Google Scholar