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。它可以是不同物理机制的结果，也可以由不同的刺激触发--这一特性使这种材料对于制造形状变化可控的功能材料很有兴趣。规模从尺寸在(亚)毫米范围的聚合物薄膜到几米的大型复合结构。后者的一个例子是自成型木材双层，它目前正变得与复杂的、自由形式的木材建筑相关。在之前的资助期间，我们研究了具有微非均匀预应变的弹性复合板。从三维非线性弹性理论出发，严格推导了具有有效预应变的均质板模型，有效预应变触发了自发弯曲。推导出的模型以一种复杂的方式预测依赖于三维复合材料规格的非平坦平衡形状。我们用数值方法和解析方法相结合的方法来探索这种“微结构-形状关系”。到目前为止，预应变一直是固定的、规定的数量。在第二个资金期，我们将考虑由服从附加本构定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与速率无关的演化和形变梯度的乘法分解的模型。我们将集中讨论两个有代表性的模型，即膨胀模型(考虑自形成的木材双层的干燥过程)和有限弹塑性。我们计划从三维模型中严格地推导出有效的、静态的和随时间变化的板模型，通过(渐进的)Gamma收敛来降低复杂性，并将开发可靠的数值方法来模拟简化的模型。此外，我们计划结合分析方法和数值模拟来探索模型的力学行为。

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.