M. A. Grekov and N. F. Morozov

Solution of the Kirsch Problem in View of Surface Stresses

abstract:
The Kirsch problem on the tension of an elastic plane with a circular hole free from external traction is considered. It is assumed that complementary surface stresses are applied at the boundary. Based on Kolosov-Muskhelishvili's method, the solution of the problem is reduced to the solution of a singular integro-differential equation for an unknown surface stress. A solution to the obtained equation is derived in an explicit form and shows that stress concentration at the boundary depends on the elastic properties of a surface and bulk material, and the radius of a hole as well if surface stresses are taken into account.
The paper is an example of the modern applications of Muskhelisvili's outstanding achievements to the problems of the nanomechanics.

Mathematics Subject Classification: 30E20, 30E25, 74A50

Key words and phrases: Kirsch problem, surface stress, singular integro-differential equation, stress concentration