Models of thin interphases with variable moduli in plane-strain elasticity

Two approximate models of thin interphases are formulated in plane-strain elasticity for the case in which the elastic moduli of the interphase are non-constant. The models make it possible to determine the elastic fields in the adjacent media without solving the fields within the interphase itself. They are of O(h) accuracy, where h is the constant thickness of the interphase. The first model, in which the geometry of the interphase is left intact, is characterized by conditions on the displacements and tractions pertaining to the adjacent media that are evaluated at both sides of the interphase. In the second model, the interphase is replaced by an interface that comes into direct contact with the media that are adjacent to the interphase, and on which appropriate interface jump conditions are derived for the displacements and tractions. The conditions characterizing the models depend on the moduli of both the interphase and the adjacent media. It is shown that in systems containing interphases for which the Reciprocal Theorem of Elasticity holds in the exact setting, this theorem will continue to be true when the interphases are replaced by either one of the developed approximate models. The consistency of those models in regard to the so-called 'self-adjointness' property of the system is therefore established.