An isoperimetric estimate and W$^{1,p}$-quasiconvexity in nonlinear elasticity

Abstract.

A class of stored energy densities that includes functions of the form $W(\vec F) = a|\vec F|^p + g(\vec F,{\rm adj}\vec F) +h({\rm det} \vec F)$ with $a>0$, g and h convex and smooth, and $2<p<3$ is considered. The main result shows that for each such W in this class there is a $k>0$ such that, if a 3 by 3 matrix $\vec F_0$ satisfies $h'({\rm det} \vec F_0) |\vec F_0|^{3-p} \le k$, then W is $W^{1,p}$-quasiconvex at $\vec F_0$ on the restricted set of deformations $\vec u$ that satisfy condition (INV) and ${\rm det} \nabla\vec u >0$ a.e. (and hence that are one-to-one a.e.). Condition (INV) is (essentially) the requirement that $\vec u$ be monotone in the sense of Lebesgue and that holes created in one part of the material not be filled by material from other parts. The key ingredient in the proof is an isoperimetric estimate that bounds the integral of the difference of the Jacobians of $\vec F_0\vec x$ and $\vec u$ by the $L^p$-norm of the difference of their gradients. These results have application to the determination of lower bounds on critical cavitation loads in elastic solids.