LCE板的非线性弯曲理论
A nonlinear bending theory for nematic LCE plates
摘要：
本文研究了一种双层弹性平板，它的顶层是液晶弹性体，底层是非线性弹性材料。虽然底层被假定为在平面参考配置中是无应力的，但顶层的特征在于取决于局部液晶取向的本征应变。因此，板显示出与几何形状平衡的非平坦变形，该几何形状非平凡地取决于板的相对厚度和形状、材料参数、变形的边界条件和液晶取向的锚定。我们专注于薄板的弯曲制度，并推导出一个二维的弯曲模型，结合了非线性弯曲变形的能量，与表面Oseen-Frank能量的导演场，描述了液晶弹性体的局部取向。这两种能量是非线性耦合的自发曲率项，有效地描述了向列-弹性耦合。我们严格推导出这个模型作为一个从三维，非线性弹性的Γ-极限。我们还设计了一种新的数值算法来计算二维模型的稳定点。我们进行了数值实验，并提出了模拟结果，说明了所提出的计划，以及丰富的系统的力学行为的实用性能。
英文摘要：
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.
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