Thin Sheets of Liquid Crystal Elastomers: Modeling, Approximation, and Computation

Liquid crystal elastomers (LCEs) are materials where a nematic liquid crystal is coupled with a rubbery material. When actuated with heat or light, the interaction of the liquid crystal with the rubber creates complex shapes. Thin bodies of LCEs are natural candidates for soft robotics applications. Starting from the classical 3D trace energy formula of Bladon, Warner and Terentjev (1994), we derive a 2D plate theory as the asymptotic limit of the 3D energy using the tools of Gamma-convergence. The resulting bending problem consists of minimizing an energy that is a function of the curvatures of the parameterized surface, subject to a metric constraint. This metric constraint is the leading order behavior that drives large deformations of LCEs. Numerically, we solve this metric constraint by numerical minimization of a formally derived stretching energy with the bending energy serving as a regularization for the discrete problem. We prove that minimizers of the discrete energy converge to zero energy states of the stretching energy in the spirit of Gamma convergence. We solve the discrete minimization problem via an energy stable gradient flow scheme. We present computations showing the geometric effects that arise from liquid crystal defects as well as computations of nonisometric origami, both within and beyond theory. The plate theory derivation is joint work with P. Plucinsky and D. Padilla-Garza and the computational work is joint with R.H. Nochetto and S. Yang.

Contatto:
marco.verani@polimi.it