A simple sufficient condition for the quasiconvexity of elastic stored-energy functions in spaces which allow for cavitation

Abstract

In this note we formulate a sufficient condition for the quasiconvexity at $x\mapsto \lambda x$ of certain functionals $I(u)$ which model the stored-energy of elastic materials subject to a deformation $u$. The materials we consider may cavitate, and so we impose the well-known technical condition (INV), due to Müller and Spector, on admissible deformations. Deformations obey the condition $u(x)=\lambda x$ whenever $x$ belongs to the boundary of the domain initially occupied by the material. In terms of the parameters of the models, our analysis provides an explicit upper bound on those $\lambda >0$ such that $I(u)\ge I(u_{\lambda })$ for all admissible $u$, where $u_{\lambda }$ is the linear map $x\mapsto \lambda x$ applied across the entire domain. This is the quasiconvexity condition referred to above.