Effect of a Type of Loading on Stresses at a Planar Boundary of a Nanomaterial

Abstract

A two-dimensional model of an elastic body at nanoscale is considered as a half-plane under the action of a periodic load at the boundary. An additional surface stress, and constitutive equations of the Gurtin–Murdoch surface linear elasticity are assumed. Using Goursat–Kolosov complex potentials and Muskhelisvili technique, the solution of the boundary value problem in the case of an arbitrary load is reduced to a hypersingular integral equation in an unknown surface stress. For the case of a periodic load, the solution of this equation is found in the form of Fourier series. The influence of the surface stress on the stresses at the boundary of the half-plane under the tangential and normal periodic loading is analyzed. In particular, it is found out the size effect which becomes apparent in the dependence of the stresses on a length of the load period of the order 10  nm. Moreover, the tangential stresses appear under the action of the normal loads.