Calculus of Variations and Geometric Measure Theory

J. J. Bevan - C. I. Zeppieri

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

created by zeppieri1 on 06 Jul 2015
modified on 20 Oct 2016


Published Paper

Inserted: 6 jul 2015
Last Updated: 20 oct 2016

Journal: Calc. Var. Partial Differential Equations
Year: 2016


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 $\lambda_0>0$ such that for every $\lambda\in (0,\lambda_0]$ it holds 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.

Keywords: nonlinear elasticity, quasiconvexity, rigidity estimate, cavitation