Variational approaches and methods for dissipative material models with multiple scales

In a first part we consider evolutionary systems given as generalized gradient systems and discuss various variational principles that can be used to construct solutions for a given system or to derive the limit dynamics for multiscale problems via the theory of evolutionary Gamma-convergence. On the one hand we consider a family of viscous gradient system with quadratic dissipation potentials and a wiggly energy landscape that converge to a rate-independent system. On the other hand we show how the concept of Balanced-Viscosity solution arise in the vanishing-viscosity limit. As applications we discuss, first, the evolution of laminate microstructures in finite-strain elastoplasticity and, second, a two-phase model for shape-memory materials, where H-measures are used to construct the mutual recovery sequences needed in the existence theory. © Springer International Publishing Switzerland 2015.