Gamma-convergence results for nematic elastomer bilayers: relaxation and actuation

TL;DR

This paper analyzes the asymptotic behavior of thin nematic elastomer bilayers, deriving Gamma-limits in different regimes, and explores their actuation capabilities under electric fields, revealing complex energy landscapes and mechanical responses.

Contribution

It extends plane strain models by computing Gamma-limits for nematic bilayers, capturing spontaneous stress relaxation, shape-morphing, and actuation effects with electric fields.

Findings

Gamma-limits derived for different regimes showing shape-morphing

Identification of min-max equilibrium solutions under electric fields

Demonstration of energy relaxation and actuation potential

Abstract

We compute effective energies of thin bilayer structures composed by soft nematic elastic-liquid crystals in various geometrical regimes and functional configurations. Our focus is on order-strain interaction in elastic foundations composed of an isotropic layer attached to a nematic substrate. We compute Gamma-limits as the layers thickness vanishes in two main scaling regimes exhibiting spontaneous stress relaxation and shape-morphing, allowing in both cases out-of-plane displacements. This extends the plane strain modelling of [*], showing the asymptotic emergence of fully coupled macroscopic active-nematic foundations. Subsequently, we focus on actuation and compute asymptotic configurations of an active plate on nematic foundation interacting with an applied electric field. From the analytical standpoint, the presence of an electric field and its associated electrostatic work turns the total energy into a non-convex and non-coercive functional. We show that equilibrium solutions are min-max points of the system, that min-maximising sequences pass to the limit and, that the limit system can exert mechanical work under applied electric fields. [*]: P. Cesana and A. A. Le\'on Baldelli. "Variational modelling of nematic elastomer foundations". In: Mathematical Models and Methods in Applied Sciences 14 (2018)