Uniqueness of Equilibrium with Sufficiently Small Strains in Finite Elasticity

Daniel Eli Spector*, Scott J. Spector

*Corresponding author for this work

Research output: Contribution to journalArticle

Abstract

The uniqueness of equilibrium for a compressible, hyperelastic body subject to dead-load boundary conditions is considered. It is shown, for both the displacement and mixed problems, that there cannot be two solutions of the equilibrium equations of Finite (Nonlinear) Elasticity whose nonlinear strains are uniformly close to each other. This result is analogous to the result of John (Commun Pure Appl Math 25:617–634, 1972), who proved that, for the displacement problem, there is a most one equilibrium solution with uniformly small strains. The proof in this manuscript utilizes Geometric Rigidity, a new straightforward extension of the Fefferman–Stein inequality to bounded domains, and an appropriate adaptation, for Elasticity, of a result from the Calculus of Variations. Specifically, it is herein shown that the uniform positivity of the second variation of the energy at an equilibrium solution implies that this mapping is a local minimizer of the energy among deformations whose gradient is sufficiently close, in BMO∩L1 , to the gradient of the equilibrium solution.

Original languageEnglish
Pages (from-to)409-449
Number of pages41
JournalArchive for Rational Mechanics and Analysis
Volume233
Issue number1
DOIs
StatePublished - 1 Jul 2019

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