Applications of Two Scale Gamma Convergence to Composite Non-classical Materials

Effective macroscopic models for composite non-classical materials are studied using the tool of Γ−convergence. The novelty is that the topology of convergence is taken to be in the two-scale topology due to the increased information kept in the limit leading to novel insights in the problems. Firstly, the case of phase separation within a periodically heterogeneous fluidi is studied. The preferred phases are allowed to be spatially-dependent and even discontinuous. An effective energy scale and an effective energy is derived through Γ−convergence techniques. As a corollary, stronger results on the sharp interface limits of Modica Mortola functionals with spatially dependent phases are obtained. Finally, the case of a thin composite metamaterial is considered. The metamaterial is modelled through a high contrast model. An effective energy is derived in the membrane limit under both polynomial growth conditions and the noninterpenetrability conditions of finite nonlinear elasticity.