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 $\mathrm{\Gamma }$-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.

在本文中，我们研究了由顶层向列液晶弹性体和底层非线性弹性材料组成的弹性双层板。虽然假定底层在平面参考配置中没有应力，但顶层具有取决于局部液晶取向的本征应变。因此，该板显示出与几何形状平衡的非平面变形，该几何形状在很大程度上取决于板的相对厚度和形状、材料参数、变形的边界条件以及液晶取向的锚定。我们专注于弯曲状态下的薄板，并推导出结合变形的非线性弯曲能量的二维弯曲模型，具有用于描述液晶弹性体局部取向的导向场的表面 Oseen-Frank 能量。两种能量都通过有效描述向列-弹性耦合的自发曲率项进行非线性耦合。我们严格推导出这个模型作为$\mathrm{\Gamma }$- 三维非线性弹性极限。我们还设计了一种新的数值算法来计算二维模型的固定点。我们进行了数值实验并给出了仿真结果，这些结果说明了所提出方案的实际特性以及系统丰富的机械行为。