Quasiconvexity and uniqueness of equilibrium solutions in nonlinear elasticity

Abstract

In this paper, we prove the uniqueness of the solution to certain simple displacement boundary value problems in the nonlinear theory of homogeneous hyperelasticity for a body occupying a star-shaped reference configuration Ω ⊂ Rn whose boundary ∂ Ω is subjected to an affine deformation i.e., there exists a constant n×n matrix F and a constant n vector b such that x ↦ Fx + b for all x∈ ∂ Ω. We consider all smooth equilibrium configurations satisfying this boundary condition. Clearly, the homogeneous deformation x ↦ Fx + b, for all \( x \in \bar{\Omega } \), is one such solution. Our aim is to prove under suitable hypotheses that it is the only such solution.