Bending plates of nematic liquid crystal elastomers
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 focusses 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.