Asymptotic Methods Applied to Materials science – AMAM

Mathematical models for material instabilities, phase transitions, plasticity, fracture, or micromagnetics are such examples of equations whose analysis is the source of challenging issues for mathematicians. In particular, recent works have been devoted to the justification of effective theories (asymptotic analysis), and to the study of existence, regularity and singular behavior of solutions to nonconvex, nonlocal problems (etc.). Among this very large field, we will focus on multiscale structures in nonlinear partial differential equations. These multiscale structures may have several different origins. They may appear through explicit small parameters in the model (as it is the case in homogenization or in models for micromagnetics). They may also be more subtle and appear as the result of a competition between two terms in an energy functional (as it is the case for the bulk and surface energies in models for phase transitions). For both types of problems, the suitable mathematical tools are typically compactness arguments, Young measures, Gamma-convergence, viscosity solutions, gradient flow structures, etc. Some of these mathematical tools have already given very deep insights in some problems such as phase transitions and micromagnetics, and still need to be further developed.

The aim of this proposal is to create a team of five young mathematicians with different and complementary expertise in order to share and exchange their skills and to efficiently take part in the development of tools for nonlinear analysis in order to study models from materials science.

Our scientific program can be split into three major parts:

• Nonstandard Homogenization: First of all, we wish to study nonstandard homogenization problems, namely problems with nonstandard spatial structure assumptions (beyond the periodic setting) as well as problems with nonstandard operator structure (beyond the continuous setting). Our main goals are to study and apply the concept of homogenization structure introduced by Nguetseng, and to address the homogenization of some discrete systems.

• Micromagnetics: Due to their multiscale nature, ferromagnetic materials exhibit complex microstructures such as magnetic domains, domain walls (Néel walls, Bloch walls, cross-tie walls) or vortices (Bloch-lines). We plan to study variational models used to predict the morphology of a given specimen at different scales.

• Defect Mechanics: We attend to analyze the evolution of damage, cracks and dislocations in various settings as quasi-static evolution or minimizing movements. Moreover we wish to stress relationships between : damage/homogenization, fracture/Mumford-Shah, dislocations/Ginzburg-Landau vortices.