Numerical Study of Liquid Crystal Elastomers by Mixed Finite Element Methods

The goal of this article is the application of the finite element method to study and simulate the unique behaviors of liquid crystal elastomers, such as semi-soft elasticity and stripe-domain instability. We start with an energy functional that consists of the Bladon-Warner-Terentjev stored energy of the liquid crystal elastomer, the Oseen-Frank energy of liquid crystals, plus Lagrange multiplier terms associated with the incompressibility constraint on the displacement, and the unit length constraint of the director. We investigate existence, uniqueness, well-posedness and convergence of the numerical methods. We find the semi-soft elasticity phenomenon, and we relate it to the rotation of the directors. We carry out numerical simulations of a stripe domain subject to extensional loading on the boundary.