Numerical study of liquid crystal elastomers by a mixed finite element method - Luo, C; Calderer, M (2011)
Liquid crystal elastomers (LCE) present features not found in ordinary elastic materials, such as semisoft elasticity and the related stripe domain phenomenon. In this paper, the two-dimensional (2D) Bladon-Terentjev-Warner (BTW) model and the one-constant Oseen-Frank energy expression are combined to study the liquid crystal elastomer. We also impose two material constraints, the incompressibility of the elastomer and the unit director norm of the liquid crystal. We prove existence of minimizer of the energy for the proposed model. Next, we formulate the discrete model, and also prove that it possesses a minimizer of the energy. The inf-sup values of the discrete linearized system are then related to the smallest singular values of certain matrices. Next, the existence and uniqueness of the Lagrange multipliers associated with the two material constraints are proved under the assumption that the inf-sup conditions hold. Finally, numerical simulations of the clamped-pulling experiment are presented for elastomer samples with aspect ratio 1 or 3. The semi-soft elasticity is successfully recovered in both cases. The stripe domain phenomenon, however, is not observed, which might be due to the relative coarse mesh employed in the numerical experiment. Possible improvements are discussed which might lead to the recovery of the stripe domain phenomenon.