Effective theories for metric gradient flows in solid mechanics

非线性耗散演化方程提出了各种具有挑战性的数学问题，从存在理论到解的近似，再到依赖于小参数的演化系统的有效行为。在这种情况下，希尔伯特或一般度量空间中梯度流的变分方法通过提供有效的建模、分析和模拟工具而具有重要意义。我们提出了一个关于固体力学中具有弹性能量和粘性耗散的梯度流动的研究项目。在第一部分中，我们得到了多井非线性弹性能和随时间无关的耗散的存在性结果和有效理论。特别地，我们研究了这类演化问题与几何线性对应物和相变的锐界面模型的关系。项目的第二部分致力于用降维方法推导粘弹性细杆和细带的有效理论。与先进的工具将解决的问题变分法包括公制梯度流的现代理论,进化Gamma-convergence几何刚度和定量估计。除了在材料科学上的应用外，该项目还将通过深化对应变率相关耗散势演化方程的理解，以及将关于多井能量和降维的静态结果扩展到演化框架，进一步发展数学理论。

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.