On singular models of a thin inclusion in a homogeneous elastic medium

Abstract

The equilibrium of an unbounded homogeneous and anisotropic elastic medium containing an inclusion, one of whose characteristic dimensions is much less than the other two is considered. It is assumed that the elastic moduli of the medium and the inclusion differ substantially. The principal terms of the expansion of the elastic fields in the neighbourhood of the thin inclusion are constructed in asymptotic series in small parameters of the problem: the ratio of the characteristic linear dimensions of the inclusion and the ratio of the characteristic values of the elastic moduli of the medium and the inclusion. The problem of constructing the principal terms of these expansions reduces to solving two-dimensional pseudo-differential equations derived by using the procedure of matching the external and internal asymptotic expansions /1, 2/. The results obtained enable two singular models of thin inclusions to be formulated for the cases when their elastic moduli are substantially greater and substantially less than the elastic modulus of the medium. It is shown that one of these models is equivalent to the model of a thin inclusion proposed in /3, 4/.