Lorina Varvaruca
University of Bath
mapel@maths.bath.ac.uk



On the Quasi-convexity of a functional in Nonlinear Elasticity

Abstract: Let $ \Omega\subset\mathbb{R}^3$ denote the reference configuration of a nonlinear elastic material, and consider deformations $ u$$ :\Omega\to \mathbb{R}^3$. We determine a sufficient condition for the stored energy density $ W({\bf F})=\vert{\bf F}\vert^p+h(\det{\bf F})$, where $ h$ is a smooth and convex function, and $ 2< p<3$, to be $ W^{1,p}$-quasiconvex at $ \lambda {\bf I}$ on the restricted class of deformations $ u$ that satisfy condition (INV), $ \det\nabla$   $ u$$ >0$ a.e., and which open a single hole anywhere in the material. The condition takes the form $ \lambda^{3-p}h'(\lambda^3)\leq \Upsilon_p$, where $ \Upsilon_p>0$ is explicitly determined, and is in a certain sense optimal for the class of energy densities under consideration. The proof makes use of isoperimetric inequalities and of estimates on radial $ p$-harmonic functions.