A circular inhomogeneity incorporating surface/interface strain gradient elasticity

We derive a rigorous solution to the anti-plane shear problem of a circular elastic inhomogeneity with an imperfect interface described by surface strain gradient elasticity. The loadings considered include a uniform remote stress field and a screw dislocation applied either in the matrix or in the inhomogeneity. By using analytical continuation, the problem is reduced to a third-order differential equation for a single analytic function. The analysis indicates that in the absence of the screw dislocation, the internal stress field is uniform and is reliant on two size-dependent parameters arising from surface strain gradient elasticity. The size-dependent image force acting on the screw dislocation is derived. The conditions for the existence of equilibrium positions for the dislocation in the matrix or in the inhomogeneity are obtained. Several numerical examples are presented to illustrate the solution. As an application of the solution, the stabilities of a misfit screw dislocation dipole in the matrix and a screw dislocation pair in the inhomogeneity are also discussed.