Title: Relaxation of nonlinear elastic energies involving deformed configuration and applications to nematic elastomers,
Abstract: We start from a variational model for nematic elastomers that involves two
energies: mechanical and nematic. The first one consists of a nonlinear elastic
energy which is influenced by the orientation of the molecules of the nematic
elastomer. The nematic energy is an Oseen--Frank energy in the deformed
configuration. The constraint of the positivity of the determinant of the
deformation gradient is imposed. The functionals are not assumed to have the
usual 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 the
tangential quasiconvexification of the nematic term. The main assumptions are
that the quasiconvexification of the mechanical term is polyconvex and that the
deformation is in the Sobolev space $W^{1,p}$ (with $p>n-1$ and $n$ the
dimension of the space) and does not present cavitation.