Calculus of Variations and Geometric Measure Theory

D. Spector - S. Spector

Uniqueness of Equilibrium with Sufficiently Small Strains in Finite Elasticity

created by spector on 19 Apr 2018
modified on 25 Mar 2019



Inserted: 19 apr 2018
Last Updated: 25 mar 2019

Journal: Arch. Ration. Mech. Anal.
Pages: 38
Year: 2018


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 Fritz John (Comm.\ Pure Appl.\ Math.\ \textbf{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 $BMO\cap\, L^1$, to the gradient of the equilibrium solution.

Keywords: uniqueness, nonlinear elasticity, Finite Elasticity, Equilibrium Solutions, Local Fefferman-Stein Inequality, Geometric Rigidity, BMO Local Minimizers, Small Strains