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

TL;DR

This paper establishes a straightforward sufficient condition for quasiconvexity of elastic energy functionals in materials that can cavitate, providing explicit bounds on deformation parameters ensuring energy minimization at linear deformations.

Contribution

It introduces a simple sufficient condition for quasiconvexity of stored-energy functions allowing cavitation, with explicit bounds on deformation parameters.

Findings

Provides an explicit upper bound on the stretch parameter  for quasiconvexity.

Formulates a sufficient condition for quasiconvexity at linear deformations.

Ensures energy minimization occurs at linear deformations under certain conditions.

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 M\"{u}ller 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 $I(u) \geq I(u_{\lambda})$ 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.