In the past decades a wealth of new solid materials has emerged, designed to have very specific properties in order to offer optimal solutions to engineering problems. This evolution would not have been possible without fundamental contributions from the theoretical sciences, in particular solid mechanics and mathematics. Within this general framework, mathematical concepts from the broad context of variational analysis have proven to be successful. This spectrum of methods includes, but is not limited to, the theories of homogenization, relaxation, Gamma convergence and variational time evolution. Classical application areas involve models in the framework of nonlinear elasticity, finite plasticity and phase transformations in general and the analysis of fracture, damage, motion of dislocations, formation of microstructure and the impact of these effects on material behavior in particular.

The role of pattern formation and microstructures becomes more and more noticeable with a decreasing size of the material specimen considered. These scale effects play a major role in modern micromechanical applications. Microstructure is indeed crucial, since material behavior typically is the result of the interaction of complex substructures on several length scales. The macroscopic behavior is then determined by appropriate averages over the (evolving) microstructure. Effects controlling the lifetime and deterioration of specimens, too, depend strongly on the microstructure. What is needed are models which are more closely related to physics and material science and which are able to take into account the microstructural behavior of the material.

Mathematics and especially the calculus of variations are essential in the understanding of microstructural pattern formation in the presence of nonconvex potentials (i.e., potential energy functionals lacking weakly lower semi-continuity). Solutions to macroscopic boundary value problems become infimizing sequences whose description requires the identification of the quasiconvex envelope of the non(quasi)convex energy density, whose infimizers are interpreted as the associated microstructural patterns.

The mechanics of materials side of this course aims to exploit the above mathematical concepts towards formulating and validating constitutive theories and associated numerical tools for the prediction of complex material behavior. Since exact solutions of quasiconvex hulls are a rare find, approximate solutions from the theory of relaxation via, e.g., rank-one-convexification and sequential lamination, via convexification, via polyconvex envelopes, or via time-incremental variational formulations have served to describe the intricate micromechanical processes leading to the well-documented macroscopic inelastic material behavior.

The course is addressed particularly at doctoral students and young researchers (postdoctoral scholars, research associates and assistant faculty). The lectures will target an audience that has a proficient background in graduate-level mathematics and solid mechanics (typical of Master’s graduates in, e.g., mechanical and civil engineering, applied mechanics, applied physics, applied mathematics, and related disciplines).