Effective theories for metric gradient flows in solid mechanics

点击翻译按钮获取中文摘要

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.