# Relaxation of nonlinear elastic energies related to Orlicz-Sobolev nematic elastomers

## TL;DR

This paper analyzes the relaxation of a complex energy model for nematic elastomers, combining nonlinear elasticity and nematic orientation energies within an Orlicz-Sobolev framework, extending previous results to broader function spaces.

## Contribution

It extends the relaxation analysis of nematic elastomer energies to Orlicz-Sobolev spaces with specific growth conditions, incorporating polyconvexity and orientation-preserving map properties.

## Key findings

- Derived relaxation formulas for the energy model.
- Extended previous results to new Orlicz space settings.
- Ensured the regularity and non-cavitation conditions for deformations.

## Abstract

We compute the relaxation of the total energy related to a variational model for nematic elastomers, involving a nonlinear elastic mechanical energy depending on the orientation of the molecules of the nematic elastomer, and a nematic Oseen--Frank energy in the deformed configuration. The main assumptions are that the quasiconvexification of the mechanical term is polyconvex and that the deformation belongs to an Orlicz-Sobolev space with an integrability just above the space dimension minus one, and does not present cavitation. We benefit from the fine properties of orientation-preserving maps satisfying that regularity requirement proven in \cite{HS} and extend the result of \cite{MCOl} to Orlicz spaces with a suitable growth condition at infinity.