A periodic set of edge dislocations in an elastic semi-infinite solid with a planar boundary incorporating surface effects

The 2-D problem of interacting periodic set of edge dislocations and point forces with planar traction-free surface of semi-infinite elastic solid at the nanoscale is considered. Complex variable based technique and Gurtin-Murdoch model of surface elasticity, which leads to the hypersingular integral equation in surface stress, are used. The solution of this equation and explicit formulas for stress field (Green functions) are obtained in terms of Fourier series. The detailed numerical investigation of stress field induced by the dislocations at the nanometer distance from the surface and the force acting on each dislocation in classical and non-classical (with surface stress) solutions is presented. It is shown that formulas derived for the periodic set of dislocations can be applied to the analysis of the interaction of a single dislocation with the surface as well. The fundamental solutions obtained in the work can be used for applying the boundary integral equation method to an analysis of defects such as cracks and inhomogeneities, periodically distributed at the nanometer distance from the boundary.