An integral equation formulation of two- and three-dimensional nanoscale inhomogeneities

An integral equation formulation of two- and three-dimensional infinite isotropic medium with nanoscale inhomogeneities is presented in this paper. The Gurtin–Murdoch interface constitutive relation is used to model the continuity conditions along the internal interfaces between the matrix and inhomogeneities. The Poisson’s ratios of both the matrix and inhomogeneities are assumed to be the same. The proposed integral formulation only contains the unknown interface displacements and their derivatives. In order to solve the nanoscale inhomogeneities, the displacement integral equation is used when the source points are acting on the interfaces between the matrix and inhomogeneities. Thus, the resulting system of equations can be formulated so that the interface displacements can be obtained. Furthermore, the stresses at points being in the matrix and nanoscale inhomogeneities can be calculated using the stress integral equation formulation. Numerical results from the present method are in good agreement with those from the conventional sub-domain boundary element method and the analytical method.