Modified Kirsch Problem Incorporating Surface Stresses under Plane Stress

Abstract

We consider the Kirsch problem, taking into account the surface stresses at the boundary of the circular hole and on the front surfaces of the plate, in the framework of the original Gurtin–Murdoch model. The boundary conditions on the cylindrical surface of a circular hole in a nanoplate are derived in terms of a complex variable in the case of the plane stress state. The solution of the two-dimensional problem for an infinite plane with a circular hole under remote loading is explicitly obtained. Based on the analytical solution, we investigated the dependence of the elastic stress field on the nanosised plate thickness and dimension of the hole. Numerical examples are given in the paper to illustrate quantitatively the effect of the plate thickness at the nanoscale on the stress field at and near the cylindrical surface. The results are presented graphically as the dependence of the components of the stress tensor on the polar angle.

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