Multiscale problems in materials science

The focus of our project is the mathematically rigorous investigation of models for so-called active materials, for example elastic media with additional relevant electrical or magnetic responses. Active materials have been the focus of an intense research effort in applied sciences. A major reason for the interest in this topic is the well-known fact that the macroscopic behavior of a composite is intimately related to the characteristics of its microconstituents, but that it may at the same time be very dissimilar from the individual response of its components. We are particularly interested in a specific class of active materials, obtained by embedding small, almost rigid electro- or magneto-responsive inclusions into a rubber-like soft matrix. The resulting structure is called a reinforced active material. The proposed research is structured into three work packages (WPs): WP1: Effective models for reinforced active materials via homogenization. WP2: Optimal design problems for magnetoelastic materials. WP3: Elasticity models with impenetrability constraints. All three aim at a thorough mathematical analysis of the underlying application-oriented models, which will strongly benefit from joining the expertise of our two groups in close collaboration. The theoretical work will be accompanied and supplemented by related numerical experiments.