Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-10-31T23:32:22.708Z Has data issue: false hasContentIssue false

On homotopy conditions and the existence of multiple equilibria in finite elasticity

Published online by Cambridge University Press:  14 November 2011

K. D. E. Post
Affiliation:
Institut für Mathematik–Angewandte Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10 099 Berlin, Germany
J. Sivaloganathan
Affiliation:
School of Mathematical Sciences, University of Bath, Bath BA2 7AY, U.K.

Abstract

In this paper we study homotopy classes of deformations and their properties under weak convergence. As an application, we give an analytic proof (in two and three dimensions) of the existence of infinitely many local minimisers for a displacement boundary-value problem from finite elasticity, posed on a nonconvex domain, under the constitutive assumption of polyconvexity.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1997

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

1Ball, J. M.. Constitutive inequalities and existence theorems in nonlinear elastostatics. In Nonlinear Analysis and Mechanics: Heriot-Watt Symposium Vol. 1, ed. Knops, R. J., 187241 (London: Pitman, 1977).Google Scholar
2Ball, J. M.. Minimisers and the Euler–Lagrange equations. In Proceedings of ISIMM Conference Paris (Berlin: Springer, 1983).Google Scholar
3Ball, J. M., Currie, J. C. and Olver, P. J.. Null Lagrangians, weak continuity and variational problems of arbitrary order. J. Fund. Anal. 41 (1981), 135–74.CrossRefGoogle Scholar
4Ball, J. M. and Murat, F.. W 1.P-Quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984), 225–53.CrossRefGoogle Scholar
5Baumann, P., Owen, N. C. and Phillips, D.. Maximum principles and apriori estimates for a class of problems from nonlinear elasticity. Anal. Nonlineaire 8 (1991), 119–57.Google Scholar
6Ciarlet, P. G.. Mathematical Elasticity Vol. 1: Three-Dimensional Elasticity (Amsterdam: North Holland, 1988).Google Scholar
7Ciarlet, P. G. and Necas, J.. Injectivity and self-contact in nonlinear elasticity. Arch. Rational Mech. Anal. 97 (1987), 171–88.CrossRefGoogle Scholar
8Evans, L. C. and Gariepy, R. F.. Measure Theory and the Fine Properties of Functions (Boca Raton, Florida: CRC Press, 1992).Google Scholar
9Greenberg, M. J.. Lectures on Algebraic Topology (Reading, MA: W. A. Benjamin, 1967).Google Scholar
10James, R. D. and Spector, S. J.. Remarks on W 1,p-quasiconvexity, interpenetration of matter, and function spaces for elasticity. Ann. Inst. H. Poincaré, Anal. Non Linéaire 9 (1992), 263–80.CrossRefGoogle Scholar
11John, F.. Uniqueness of nonlinear equilibrium for prescribed boundary displacements and sufficiently small strains. Comm. Pure Appl. Math. 25 (1972), 617–34.CrossRefGoogle Scholar
12Morrey, C. B.. Multiple integrals in the calculus of variations (Berlin: Springer, 1966).CrossRefGoogle Scholar
13Müller, S. and Spector, S. J.. An existence theory for nonlinear elasticity that allows for cavitation. Arch. Rational Mech. Anal. 131 (1995), 166.CrossRefGoogle Scholar
14Ogden, R. W.. Nonlinear Elastic Deformations (Chichester: Ellis Horwood Ltd., Halstead Press, Wiley, 1984).Google Scholar
15Schwartz, J. T.. Nonlinear Functional Analysis (New York: Gordan and Breach, 1969).Google Scholar
16Sivaloganathan, J.. The generalised Hamilton–Jacobi inequality and the stability of equilibria in nonlinear elasticity. Arch. Rational Mech. Anal. 107 (1989), 347–69.CrossRefGoogle Scholar
17Spivak, M.. A Comprehensive Introduction to Differential Geometry, Vol. 1 (Berkeley, CA: Publish or Perish Inc, 1979).Google Scholar
18Struwe, M.. Variational Methods (Berlin: Springer, 1990).CrossRefGoogle Scholar
19Sverak, V.. Regularity properties of deformations with finite energy. Arch. Rational Mech. Anal. 100 (1988), 105–27.CrossRefGoogle Scholar
20Truesdell, C. and Noll, W.. The Non-Linear Field Theories of Mechanics, Handbuch der Physik III/3 (Berlin: Springer, 1965).Google Scholar