Arch. Rational Mech. Anal. 159 (2001) 335–361 Digital Object Identifier (DOI) 10.1007/s002050100161 Interfacial Energies for Incoherent Inclusions PaoloCermelli&GiovanniLeoni Communicated by M.E.Gurtin Abstract We study a variational problem describing an incoherent interface between a rigid inclusion and a linearly elastic matrix. The elastic material is allowed to slip relative to the inclusion along the interface, and the resulting mismatch is penalized by an interfacial energy term that depends on the surface gradient of the relative displacement. The competition between the elastic and interfacial energies induces a threshold effect when the interfacial energy density is non-smooth: small inclusions are coherent (no mismatch); sufficiently large inclusions are incoherent. We also show that the relaxation of the energy functional can be written as the sum of the bulk elastic energy functional and the tangential quasiconvex envelope of the interfacial energy functional. 1. Introduction In two-phase solids, for instance metal alloys, one of the phases is often segre- gated into inclusions distributed in the other phase’s matrix. Since the phases generally have different crystalline structure and composition, their stress-free states are often related by a transformation strain involving a dilation or a contraction. Therefore, the presence of new-phase particles generates a stress field in the matrix, and elastic energy accumulates in the system. The stored elastic energy increases with the size of the inclusions, and when these are large enough the elastic energy may be decreased by allowing the particles to break away from the matrix, by a process which essentially involves the nucleation or migration of dislocations at the interface. This process has an energetic cost, and it is the competition between this cost and the stored elastic energy which determines the threshold effect. This phenomenon is quite common in metal alloys, and it has been demonstrated experimentally in a variety of systems, such as Cu-Co, Fe-Cu, Cu-Al, Mg-Al (see