Applied Math Seminar

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Finite Element Approximation of a Membrane Model for Liquid Crystal Polymeric Networks

Speaker: Lucas Bouck, University of Maryland

Location: Warren Weaver Hall 1302

Date: Friday, May 5, 2023, 10 a.m.

Synopsis:

Liquid crystal polymeric networks (LCNs) 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 LCNs are natural candidates for soft robotics applications. Starting from the classical 3D trace formula energy of Bladon, Warner and Terentjev (1994), we derive a 2D membrane energy as the formal asymptotic limit of the 3D energy and characterize the zero energy deformations. The membrane energy lacks certain convexity properties, which presents challenges for the design of a numerical method. We discretize the problem with a finite element method and add a higher order bending energy regularization to address the lack of convexity. We prove that minimizers of the discrete energy converge to zero energy states of the membrane energy in the spirit of Gamma convergence. For minimizing the discrete problem, we employ an energy stable gradient flow scheme. We present computations showing the geometric effects that arise from liquid crystal defects and as well as computations of nonisometric origami.