In the past decades, a better understanding of engineering materials and processes has emerged, leading to numerous technological applications as the design of tailor-made materials having specific properties or optimal solutions of engineering problems. This evolution would not have been possible without fundamental contributions from the theoretical sciences, in particular solid mechanics and mathematics, which offer both analytical and numerical tools for the solution of complex problems. 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 too, the theories of homogenization and scale transition, relaxation, Gamma convergence and variational time evolution. Classical application areas involve models in the framework of nonlinear elasticity, finite plasticity, diffusion 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 proposed course will approach the aforementioned topics from different perspectives and not only from one point of view. The different perspectives refer to continuum modeling techniques and the associated algorithmic treatments as well as to the different types of applications. Mathematics and especially the calculus of variations are essential in the understanding of multiscale problems, micro structured materials and localization phenomena. New solution concepts have to be introduced in order to treat the associated models. Solutions to macroscopic boundary value problems become infimizing sequences whose limits are probability measures. This is a rapidly developing area of research with essential progress made only over the last two decades, which is why this is still a relatively young field of research with many unsolved problems. Professors Manuel Friedrich and Dorothee Knees will give lectures to lay the foundation. The mechanics 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 the behavior of complex materials and processes. Professors Laura De Lorenzis, Sanjay Govindjee, Laurent Stainier, and Klaus Hackl will contribute lectures to survey the theoretical and numerical fundamentals as well as problem classes of interest.