Forschungsprojekt ::

Variational methods for multiscale problems: The interplay of rigidity and anisotropic geometries

Projektbeschreibung

The calculus of variations provides powerful mathematical methods for tackling problems in materials sciences, where modern multiscale modeling is often based on variational principles. Applications include the theory of composite materials or microstructure formation in solids. In the 1970s the Italian school around De Giorgi introduced the concept of Γ-convergence, an abstract framework for limit passages in parameter-dependent variational problems, which naturally appear in multiscale analysis. In this research project, I develop novel explicit representation theorems for homogenization and relaxation via Γ-convergence. These allow to characterize the effective behavior of non-convex variational problems with heterogeneities. State-of-the-art results for integral functionals are usually formulated in a general setting and given by implicit formulas. The drawback is that the latter correspond to infinite-dimensional minimization problems, and are hence hard to compute. My goal is to identify a new class of non-standard homogenization problems whose specific mathematical structure facilitates to derive (more) explicit analytical solutions. Precisely, I choose variational models with heterogeneities of layered geometry and admissible functions that are close to rigid body motions in parts of their domain. On a technical level, challenges arise from the occurrence of functionals that are highly anisotropic and subject to constraints in the form of (approximate) differential inclusions. To overcome these issues I prove new asymptotic rigidity theorems and exploit the essentially one-dimensional character of the problem to enable localization. In the second period, I go beyond stationary problems, addressing relevant aspects of quasi-static evolution and investigating novel strategies for explicit relaxation of time-incremental problems.