Effective theories for metric gradient flows in solid mechanics

固体力学中度量梯度流的有效理论

• 批准号: 454756334
• 负责人: Professor Dr. Manuel Friedrich
• 金额: --
• 依托单位: Professur für Angewandte Analysis
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/454756334?language=en
• 关键词: Effective; theories; metric; gradient; flows

Nonlinear dissipative evolution equations present a variety of challenging mathematical problems ranging from existence theory, to approximation of solutions, to the effective behavior of evolutionary systems depending on a small parameter. In this context, the variational approach to gradient flows in Hilbert or general metric spaces is of overriding importance by providing efficient tools for modeling, analysis, and simulations.We propose a research project on gradient flows in solid mechanics featuring elastic energies and viscous dissipations. In the first part, we derive existence results and effective theories for nonlinear elastic energies with multiple wells and for dissipations complying with time-dependent frame indifference. In particular, we study the relation of such evolutionary problems to geometrically linear counterparts and to sharp-interface models for phase transformations. The second part of the project is devoted to the derivation of effective theories for thin viscoelastic rods and ribbons by means of dimension reduction. The problems will be tackled with advanced tools from the calculus of variations including the modern theory of metric gradient flows, evolutionary Gamma-convergence, and quantitative geometric rigidity estimates. Besides its applications to Materials Science, the proposed project will further develop the mathematical theory by deepening the understanding of evolution equations with strain-rate dependent dissipation potentials as well as by extending static results about multiwell energies and dimension reduction to an evolutionary framework.

非线性耗散演化方程提出了各种具有挑战性的数学问题，从存在性理论，到近似的解决方案，到依赖于一个小参数的演化系统的有效行为。在这种情况下，变分方法的梯度流动在希尔伯特或一般度量空间是压倒一切的重要性，提供有效的工具，建模，分析和simulation.We提出了一个研究项目，在固体力学梯度流动具有弹性能量和粘性耗散。在第一部分中，我们得到了多威尔斯井非线性弹性能量和耗散满足时变框架无差异的存在性结果和有效理论。特别是，我们研究的关系，这样的进化问题的几何线性对应物和尖锐的界面模型的相变。本项目的第二部分致力于通过降维方法推导粘弹性细杆和薄带的有效理论。这些问题将通过变分法的先进工具来解决，包括现代度量梯度流理论，进化伽玛收敛和定量几何刚性估计。 除了其在材料科学中的应用外，该项目还将通过加深对具有应变率依赖的耗散势的演化方程的理解以及通过将关于多阱能量和降维的静态结果扩展到演化框架来进一步发展数学理论。