Multi-value microstructural descriptors for complex materials: analysis of ground states

A multi-scale and multi-field description of material microstructures which distinguish at every point multiple microstructural individuals is analyzed paying attention to the determination of equilibrium configurations. Rigid bodies with microstructure represented by means of Q−valued maps from the reference place to the manifold of microstructural shapes are considered first. In that case, just microenergetics is considered and macroscopic motion neglected. Then, the stage is enlarged up to include deformable bodies suffering large strains, endowed with energies of Ginzburg-Landau type with respect to the microstructural descriptor fields. In both cases, conditions for semicontinuity of the relevant energies and the existence of ground states are provided.