Analytic solution for a circular nano-inhomogeneity in a finite matrix

The Gurtin-Murdoch model has found wide applications in analyzing the mechanical behaviors of nanocomposites with surface/interface effect. In the existing literature, the matrix is usually assumed to be infinite and the surface/interface effect is considered only at the inhomogeneity-matrix interface. This assumption is indeed valid as the matrix is usually at macroscale rather than nanoscale. However, if the size of the matrix decreases to the nanoscale too, the surface/interface effect will have to be considered at the outer boundary of the matrix. In this paper, the plane deformation of a circular nano-inhomogeneity embedded inside a finite circular matrix (which implies the matrix is also at nanoscale) is investigated. The stress boundary conditions are given at the inhomogeneity-matrix interface and the outer boundary of the matrix by the G-M model. The analytic solution for the stress field is finally obtained through the complex variable method. The results show that the stress field inside the inhomogeneity is still uniform (size-dependent) when the surface/interface effect is considered. In addition, the stress field inside the bulk (including the inhomogeneity and the matrix) can be influenced not only by the size and elastic constant of the inhomogeneity, but also by those of the matrix.