Bending plates of nematic liquid crystal elastomers

可编程执行器是由弹性材料制成的装置，其形状可以通过外部刺激来控制。由于广泛的技术应用，特别是在小尺度上，它们在工程、物理和应用数学中受到了相当大的关注。精确理解它们的行为对于它们的设计和可靠使用至关重要。这需要有效的数学模型和精确的数值模拟方法。提出的项目侧重于向列薄膜类，即向列液晶弹性体的薄片。从使用三维超弹性框架的耦合模型和液晶模型（例如，Oseen-Frank）开始，将推导出不同的有效二维板模型，该模型确定弹性薄膜与描述液晶排列的相应取向场相关的可能的大变形。模型的特征是曲率量的出现，强制直纹曲面的约束，单位或有界长度的向量场，以及定义两个感兴趣的量之间耦合的非线性。这些特殊的特性需要使用和发展适当的数值方法，以便在有效的实现中模拟相关的影响，并为新的纳米工具的开发做出可靠的预测。通过伽玛收敛的概念建立三维和降维非线性板弯曲模型之间的数学严格联系，对所提出的有限元离散化进行彻底的收敛分析，避免不合理的规则假设，并从理论上和实验上研究有效的迭代求解策略。

Programmable actuators are devices made of elastic materials whose shape can be controlled via external stimulation. Because of a wide range of technological applications particularly at small scales, they have received considerable attention in engineering, physics, and applied mathematics. A precise understanding of their behavior is crucial for their design and reliable use. This requires effective mathematical models and accurate numerical simulation methods. The proposed project focuses on the class of nematic thin films which are thin sheets of nematic liquid crystal elastomers. Starting from a coupled model using the frameworks of 3D hyperelasticity, and liquid crystal models (e.g., Oseen-Frank), different effective two-dimensional plate models will be derived that determine a possibly large deformation of the elastic thin film in connection with a corresponding orientation field describing the arrangement of the liquid crystals. Characteristic for the models are the occurrence of curvature quantities, constraints that enforce ruled surfaces, vector fields of unit or bounded length, and nonlinearities that define the coupling between the two quantities of interest. These special features require the use and development of appropriate numerical methods in order to simulate relevant effects within efficient implementations and to make reliable predictions for the development of new nanotools. A mathematically rigorous connection between the three-dimensional and the dimensionally reduced nonlinear plate bending model will be established via the concept of Gamma-convergence, a thorough convergence analysis for the proposed finite element discretization will be carried out avoiding unjustified regularity assumptions, and efficient iterative solution strategies will be investigated theoretically and experimentally.