Rate-independent evolution of prestrained plates

预应变板的速率无关演化

• 批准号: 431469590
• 负责人: Professor Dr. Stefan Neukamm
• 金额: --
• 依托单位: Institut für Numerische Mathematik
• 依托单位国家: 德国
• 项目类别: Research Units
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/431469590?language=en
• 关键词: Rate; independent; evolution; prestrained; plates

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.

预应变对弹性薄板和薄膜的机械行为有巨大影响。它可能是不同物理机制的结果，并且可以由不同刺激触发——这种特性使得此类材料对于制造具有受控形状变化的功能材料很有趣。尺度范围从尺寸在（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型木质双层，它目前正与复杂的、自由形式的木结构建筑相关。在之前的资助期间，我们研究了具有微异质预应变的弹性复合材料板。从三维非线性弹性出发，我们严格推导了均质弯曲板模型，该模型具有触发自发弯曲的有效预应变。导出的模型以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。我们通过结合数值和分析方法探索了这种“微观结构-形状关系”。到目前为止，预应变是固定的、规定的量。在第二个资助期间，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速率无关演化和变形梯度乘法分解的模型。我们将重点关注两个代表性模型，即膨胀模型（考虑到自成型木材双层的干燥过程）和有限弹塑性。我们计划通过三维模型的（进化）伽玛收敛来严格推导有效的、静态的、时间相关的板模型，并降低复杂性，并且我们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。