| Equilibrium in optimal multiphase structures, breakable structures, or materials under phase transition is a solution to a nonconvex variational problem. In these problems, the material's properties sharply are changed when the stress passes a threshold; this leads to nonmonotonic constitutive relations (bistability). Typically, the problem possesses many local minima, and minimizing sequences are characterized by fine-scale spatial oscillations or microstructures. We discuss the choice of a proper variational functional for such problems and the variational techniques: Special necessary conditions and bounds. Dynamics of such materials is marked by intensive waves that are excited when the system passes from one locally stable equilibrium to another. To describe them, we consider chains or lattices of bistable elements. The model effectively describes nonlinear waves, including conditions of propagation of a wave ("house of cards" problem) and a homogenized state of the excited system. |