Thin inclusion in a homogeneous elastic medium

Thin hard inclusions are often used as a filler in composite materials. for the same volume concentration of fillers, rigid laminas increase the elastic moduli by 1.5–2 times more than spherical inclusions or fibers with the same elastic properties. On the other hand, a medium containing thin soft inclusions or cracks is an appropriate model of damaged materials. In both cases, the elastic moduli of thin inclusions differ essentially from the moduli of the matrix material. The properties of such inclusions are characterized by two dimensionless parameters. The “geometrical” parameter δ1 is the ratio of the minimal and maximal sizes of the inclusion, and the “physical” parameter δ2 is the ratio of the characteristic elastic modulus of the inclusion and the medium. The parameter δ1 is always small, and the parameter δ2 is either small (soft inclusion), or large (hard inclusion). The main terms of the asymptotic expansions of the elastic fields in the vicinity of thin inclusions with respect to these parameters are the objectives of this chapter. The problem of construction of these terms is reduced to integral equations on the middle surface of a thin inclusion. For thin ellipsoidal inclusions, the solutions of these equations are obtained in explicit forms.