Rate-independent evolution of prestrained plates

预应变对弹性薄板和薄膜的力学行为有很大的影响。它可以是不同的物理机制的结果，并可以由不同的刺激触发，这使得这种材料感兴趣的功能材料的制造与控制形状变化的属性。规模范围从（亚）毫米范围内的聚合物薄膜到数米的大型复合结构。后者的一个例子是自成形木材双层，目前正成为复杂的，自由形式的木材建筑相关。在上一个资助期，我们研究了弹性复合材料板与微观非均匀预应变。从三维非线性弹性力学出发，严格推导了一个均匀化的弯曲板模型，该模型具有一个有效的预应变，可以触发自发弯曲。导出的模型预测非平坦的平衡形状，依赖于在一个复杂的方式的三维复合材料的规格。我们探讨了这种“微观结构形状关系”相结合的数值和分析方法。到目前为止，预应变一直是一个固定的，规定的数量。在第二个资助期内，我们将考虑由服从附加本构律的场变量控制的预应变。我们将考虑变分，静态平衡问题，以及涉及速率无关的演变和乘法分解的变形梯度的模型。我们将集中在两个代表性的模型，即膨胀模型（与干燥过程中的自我形成的木材双层铭记）和有限的弹塑性。我们计划严格推导出有效的，静态的和随时间变化的板块模型，通过（进化）伽马收敛从三维模型降低复杂性，我们将开发可靠的数值方法来模拟简化模型。此外，我们计划探索模型的力学行为结合分析方法和数值模拟。

Prestrain has a huge impact on the mechanical behavior of thin elastic plates and films. It can be the result of different physical mechanisms and can be triggered by different stimuli—a property that makes such materials interesting for the fabrication of functional materials with a controlled shape change. Scales range from polymer films of sizes in the (sub-)millimeter range to large composite structures of several meters. An example for the latter are self-shaping wood bilayers, which are currently becoming relevant for complex, free-form timber architecture. In the previous funding period we studied elastic composite plates with a microheterogeneous prestrain. From three-dimensional nonlinear elasticity, we rigorously derived a homogenized bending plate model with an effective prestrain that triggers spontaneous bending. The derived model predicts non-flat equilibrium shapes that depend on the specification of the three-dimensional composite in a complex way. We explored this “microstructure–shape relation” by combining numerical and analytical methods. So far, the prestrain has been a fixed, prescribed quantity. In the second funding period we will consider prestrain that is governed by a field variable that obeys an additional constitutive law. We shall consider variational, static equilibrium problems as well as models that involve a rate-independent evolution and a multiplicative decomposition of the deformation gradient. We will focus on two representative models, namely a swelling model (with the drying process of self-forming wood bilayers in mind) and finite elasto-plasticity. We plan to rigorously derive effective, static and time-dependent plate models with reduced complexity via (evolutionary) Gamma-convergence from three-dimensional models and we shall develop reliable numerical methods to simulate the reduced models. Furthermore, we plan to explore the mechanical behavior of the models combining analytical approaches and numerical simulations.