Non-circular nano-inclusions with interface effects that achieve uniform internal strain fields in an elastic plane under anti-plane shear

Abstract

This paper verifies the existence of a single non-circular nano-inclusion with interface effect that achieves a uniform internal strain field in an elastic plane under uniform remote anti-plane shear loadings. The uniform strain field inside such a non-circular inclusion is prescribed via perturbations of the uniform strain field inside the analogous circular inclusion, and the unknown (non-circular) inclusion shape is characterized by a conformal mapping whose unknown coefficients are determined by a system of nonlinear equations. Numerical examples show various shapes of non-circular nano-inclusions with interface effects that achieve uniform internal strain fields. It is found that such a non-circular inclusion shape depends on the inclusion size and the specific uniform internal strain field. In particular, for given interface shear modulus and residual tension, it is shown that the minimum inclusion size required to guarantee the existence of such a non-circular inclusion usually increases as the shear modulus of the inclusion approaches that of the matrix. Moreover, the relationship between the shear traction jump across the interface and the curvature of the interface is discussed in detail for such non-circular inclusions.