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A variational model for describing the morphology of two-phase continua byallowing for the interplay between coherent and incoherent interfaces isintroduced. Coherent interfaces are characterized by the microscopicalarrangement of atoms of the two materials in a homogeneous lattice, withdeformation being the solely stress relief mechanism, while at incoherentinterfaces delamination between the two materials occurs. The model is designedin the framework of the theory of Stress Driven Rearrangement Instabilities,which are characterized by the competition between elastic and surface effects.The existence of energy minimizers is established in the plane by means of thedirect method of the calculus of variations under a constraint on the number ofboundary connected components of the underlying phase, whose exterior boundaryis prescribed to satify a graph assumption, and of the two-phase compositeregion. Both the wetting and the dewetting regimes are included in theanalysis.

Existence of minimizers for a two-phase free boundary problem with coherent and incoherent interfaces