Quasistatic evolution of Orlicz-Sobolev nematic elastomers

TL;DR

This paper extends the analysis of nematic elastomers to quasistatic evolution within Orlicz-Sobolev spaces, proving existence of energetic solutions under general growth conditions and physical confinement, with new compactness and differentiability results.

Contribution

It introduces a quasistatic evolution framework for nematic elastomers in Orlicz-Sobolev spaces, extending previous models and proving existence results with novel compactness and differentiability theorems.

Findings

Proved existence of energetic solutions for quasistatic nematic elastomer evolution.

Established Poincaré and trace inequalities in modular form for coercivity.

Generalized classical differentiability results to Orlicz-Sobolev maps.

Abstract

We investigate the variational model for nematic elastomer proposed by Barchiesi and DeSimone with the director field defined on the deformed configuration under general growth conditions on the elastic density. This leads us to consider deformations in Orlicz-Sobolev spaces. Our work builds upon a previous paper by Henao and the Second Author, and extends their analysis to the quasistatic setting. The overall strategy parallels the one devised by the First author in the case of Sobolev deformations for a similar model in magnetoelasticity. We prove two existence results for energetic solutions in the rate-independent setting. The first result concerns quasistatic evolutions driven by time-dependent applied loads. For this problem, we establish suitable Poincar\'{e} and trace inequalities in modular form to recover the coercivity of the total energy. The second result ensures the existence of quasistatic evolution for both time-depend applied loads and boundary conditions under physical confinement. In its proof, we follow the approach advanced by Francfort and Mielke based on a multiplicative decomposition of the deformation gradient and we implement it for energies comprising terms defined on the deformed configuration. Both existence results rely on a compactness theorem for sequences of admissible states with uniformly bounded energy which yields the strong convergence of the composition of the nematic fields with the corresponding deformations. While proving it, we show the regular approximate differentiability of Orlicz-Sobolev maps with suitable integrability, thus generalizing a classical result for Sobolev maps due to Goffman and Ziemer.