Interfacial energies for incoherent inclusions

We study a variational problem describing an incoherent interface between a rigid inclusion and a linearly elastic matrix. The elastic material is all owed to slip relative to the inclusion along the interface, and the resulti ng 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 interf acial energy density is non-smooth: small inclusions are coherent (no misma tch); sufficiently large inclusions are incoherent. We also show that the r elaxation of the energy functional can be written as the sum of the bulk el astic energy functional and the tangential quasiconvex envelope of the inte rfacial energy functional.