European Journal of Applied Mathematics


Numerical study of liquid crystal elastomers by a mixed finite element method

C. LUOa1 and M. C. CALDERERa2

a1 Oxford Center for Collaborative Applied Mathematics, Mathematical Institute, University of Oxford, Oxford OX1 3LB, UK email:

a2 School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA


Liquid crystal elastomers present features not found in ordinary elastic materials, such as semi-soft elasticity and the related stripe domain phenomenon. In this paper, the two-dimensional Bladon–Terentjev–Warner 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 minimiser of the energy for the proposed model. Next we formulate the discrete model, and also prove that it possesses a minimiser of the energy. The inf-sup values of the discrete linearised 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 that might lead to the recovery of the stripe domain phenomenon.

(Received July 14 2011)

(Revised July 18 2011)

(Accepted July 19 2011)

(Online publication August 22 2011)

Key words:

  • Liquid crystal elastomer;
  • Semi-soft elasticity;
  • Variational methods;
  • Mixed finite element method;
  • Inf-sup condition