Discrete-to-continuum limits for strain-alignment-coupled systems:Magnetostrictive solids, ferroelectric crystals and nematicelastomers

In the framework of linear elasticity, we study the limit of a classof discrete free energies modeling strain-alignment-coupled systemsby a rigorous coarse-graining procedure, as the number of moleculesdiverges. We focus on three paradigmatic examples: magnetostrictivesolids, ferroelectric crystals and nematic elastomers, obtaining inthe limit three continuum models consistent with those commonlyemployed in the current literature. We also derive the correspondentmacroscopic energies in the presence of displacement boundaryconditions and of various kinds of applied external fields.