Geometric nonlinearities in PDEs for Cosserat elasticity:how they affect the regularity of solutions

Cosserat 弹性偏微分方程中的几何非线性：它们如何影响解的规律性

• 批准号: 441380936
• 负责人: Professor Dr. Andreas Gastel
• 金额: --
• 依托单位: Fachgebiet Geometrische Analysis
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/441380936?language=en
• 关键词: Geometric; nonlinearities; PDEs; Cosserat; elasticity

Models for shells or solids in Cosserat micropolar elasticity involve micro-rotations which contribute to the energy. If the corresponding term in the equations is not linearized, the Euler-Lagrange equations of the model are a critically nonlinear system of partial differential equations. It consists of equations describing the balance of forces, coupled with equations describing the balance of angular momentum. The latter coincide with the system for harmonic maps to SO(3) (known from geometric analysis and mathematical physics), plus an additional coupling term. For harmonic maps, there is a rich regularity theory, including interesting results about the structure of singularities. Since the harmonic maps system plays a dominating role in the Cosserat models mentioned above, there are interesting implications to be expected about possible singularities in Cosserat solids and shells. Based on first theorems about regularity by the applicant and Neff, and on examples for singular behavior of Cosserat solids by the applicant and Hüsken from the first phase of the priority program, we will extend regularity methods towards more realistic settings. This involves models allowing for more parameters instead of the "uniconstant approximations" used so far, which will require new arguments, because the structural similarity to harmonic map type systems will be much weaker than before. It is not clear if existing methods are strong enough to cover the more general equations. This also involves the study of boundary conditions useful for applications, which involve both fixed and free boundary parts. Even for harmonic maps, boundary regularity has not been proven in this setting. Finally, we hope that regularity estimates can help to establish the existence of minimizers in Cosserat theory for choices of parameters for which that has not been settled so far.

Cosserat 微极弹性中的壳或固体模型涉及贡献能量的微旋转。如果方程中的相应项不是线性化的，则模型的欧拉-拉格朗日方程是一个临界非线性偏微分方程组。它由描述力平衡的方程以及描述角动量平衡的方程组成。后者与 SO(3) 的调和映射系统（从几何分析和数学物理已知）以及附加的耦合项一致。对于调和映射，有丰富的规律性理论，包括有关奇点结构的有趣结果。由于调和映射系统在上述 Cosserat 模型中起着主导作用，因此可以预期 Cosserat 实体和壳中可能存在的奇点会产生有趣的影响。基于申请人和 Neff 提出的关于正则性的第一定理，以及申请人和 Hüsken 在优先计划第一阶段提出的 Cosserat 固体奇异行为的示例，我们将把正则性方法扩展到更现实的环境。这涉及允许更多参数的模型，而不是迄今为止使用的“不恒定近似”，这将需要新的参数，因为与调和映射类型系统的结构相似性将比以前弱得多。目前尚不清楚现有方法是否足够强大以涵盖更一般的方程。这还涉及对应用有用的边界条件的研究，其中涉及固定和自由边界部分。即使对于调和图，边界规律性也没有在这种情况下得到证明。最后，我们希望正则性估计能够帮助确定 Cosserat 理论中最小化器的存在，以用于迄今为止尚未解决的参数选择。