STRESS-STRAIN STATE OF AN ELASTIC BODY WITH A NEARLY CIRCULAR INCLUSION INCORPORATING INTERFACIAL STRESS

In modern industry, to produce various structure elements, composite materials containing cutouts and foreign inclusions are widely used. To provide the integrity of a construction, it is necessary to study in details the distribution of stresses occurring in it in the result of force actions. Concerning the circular holes and inclusions, in practice, the ideal circles do not exist, and this fact should be taken into account when calculating. In the case when the boundary form slightly differs from the circular, to solve the problem, it is possible to apply the approximate analytical method that is called the perturbation method. 

The plane problem on a nanoinclusion in an infinite elastic body under arbitrary remote loading is considered. It’s assumed that the shape of the inclusion is weakly deviated from the circular one and the complementary interfacial stresses are acting at the boundary. In contrast with previously constructed methods for solving such problems, the solution is built without the use of conformal mapping. Contact of the inclusion with the matrix satisfies to the ideal conditions of cohesion. To solve this problem, Gurtin – Murdoch surface elasticity model is used. Based on Goursat – Kolosov complex potentials and the boundary perturbation technique, the solution of the problem is reduced to the singular integro-differential equation for any-order approximation. The algorithm of solving this integral equation is constructed in the form of a power series. The solution in the first-order approximation for the periodic shape of the inclusion determined by the cosine function is obtained. With the help of software package, for the inclusion and the matrix the graphic dependence of maximum hoop stresses upon the radius of basic circular inclusion under uniaxial tension are built. The size effect in the form of the dependence of the stress distribution at the interface on the size of the inclusion is demonstrated.