Energy-minimizing nematic elastomers

Abstract

We prove weak lower semi-continuity and existence of energy-minimizers for a free energy describing stable deformations and the corresponding directorconfiguration of an incompressible nematic liquid-crystal elastomer subject to physically realistic boundary conditions. The energy is a sum ofthe trace formula developed by Warner, Terentjev and Bladon (coupling the deformation gradient and the director field) and the Landau-de Gennesenergy in terms of the gradient of the director field and the bulk term for the director with coefficients depending on temperature. A key step in our analysis is to prove that the energy density has a convex extension to non-unit length director fields.Our results apply to the setting of physical experimentsin which a thin incompressible elastomer in $\mathbb{R}^3$ is clamped on its sidesandstretched perpendicular to its initial director field, resulting inshape-changes and director re-orientation.