A nonlinear bending theory for nematic LCE plates

Abstract

In this paper, we study an elastic bilayer plate composed of a nematic liquid crystal elastomer in the top layer and a nonlinearly elastic material in the bottom layer. While the bottom layer is assumed to be stress-free in the flat reference configuration, the top layer features an eigenstrain that depends on the local liquid crystal orientation. As a consequence, the plate shows non-flat deformations in equilibrium with a geometry that non-trivially depends on the relative thickness and shape of the plate, material parameters, boundary conditions for the deformation, and anchorings of the liquid crystal orientation. We focus on thin plates in the bending regime and derive a two-dimensional bending model that combines a nonlinear bending energy for the deformation, with a surface Oseen-Frank energy for the director field that describes the local orientation of the liquid crystal elastomer. Both energies are nonlinearly coupled by means of a spontaneous curvature term that effectively describes the nematic-elastic coupling. We rigorously derive this model as a Γ-limit from three-dimensional, nonlinear elasticity. We also devise a new numerical algorithm to compute stationary points of the two-dimensional model. We conduct numerical experiments and present simulation results that illustrate the practical properties of the proposed scheme as well as the rich mechanical behavior of the system.

Benchmarking Numerical Algorithms for Harmonic Maps into the Sphere

“…This situation admits smooth minimizing harmonic maps in each homotopy class. We start the solvers from the nodal interpolation of the function (11).…”

“…On a macroscopic level popular liquid crystal models are based on the Oseen-Frank energy, which is the Dirichlet energy of the field of molecule orientations. The emergence of new technologies in the manufacturing of liquid crystal compound materials as well as new ideas for technical applications [18,30,47,49] have lead to an ongoing interest also in corresponding simulation schemes [11,36,45].…”

A geometrically nonlinear Cosserat (micropolar) curvy shell model via Gamma convergence

“…A notorious property of the Γ-limit model based on classical elasticity is its de-coupling of the limit into either a membrane-like (scaling with h) or bending-like problem (scaling with h 3 ), see e.g. [12,45].…”

Reduced Membrane Model for Liquid Crystal Polymer Networks: Asymptotics and Computation

“…Depending on the density of crosslinks, the director field may be either totally free [19,20] or subject to a Frank elasticity term [21,22,23,24]. This kind of blueprinted configuration describes the situation where the LCs are unconstrained or constrained only on a low-level by rubbery polymers.…”

“…The metric condition giving zero stretching energy becomes a constraint in the bending model. Some existing bending models include theory derived via formal asymptotics [25], a von Karman plate model derived in [38] using asymptotics, a rigorous Gamma convergence theory for a model of bilayer materials composed of LCEs and a classical isotropic elastic plate [24], or a plate model where the LC dramatically changes its orientation through the thickness [39]. Moreover, reduced 1D models for LCNs/LCEs have been explored as well; we refer to [40] for a rod model and to [41,42] for ribbon models.…”