Relaxation of nonlinear elastic energies involving deformed configuration and applications to nematic elastomers

摘要

We start from a variational model for nematic elastomers that involves twoenergies: mechanical and nematic. The first one consists of a nonlinear elasticenergy which is influenced by the orientation of the molecules of the nematicelastomer. The nematic energy is an Oseen--Frank energy in the deformedconfiguration. The constraint of the positivity of the determinant of thedeformation gradient is imposed. The functionals are not assumed to have theusual polyconvexity or quasiconvexity assumptions to be lower semicontinuous.We instead compute its relaxation, that is, the lower semicontinuous envelope,which turns out to be the quasiconvexification of the mechanical term plus thetangential quasiconvexification of the nematic term. The main assumptions arethat the quasiconvexification of the mechanical term is polyconvex and that thedeformation is in the Sobolev space $W^{1,p}$ (with $p>n-1$ and $n$ thedimension of the space) and does not present cavitation.