To one Belokon’s problem

Introduction. Prof A.V. Belokon’s (1941-2013; former Rector of Rostov State University and President of Southern Federal University) PhD thesis was devoted to asymptotical methods in contact problems of the elasticity theory for bodies of cylindrical shape (1969). In the present paper, a contact problem of the elasticity theory on torsion of an infinite hollow cylinder by a rigid insert is investigated. The outer cylinder surface is rigidly fixed. The insert of a finite length is inside the cylinder. In 1971, this problem was formulated and analyzed by A.V. Belokon. He reduced it to an integral equation with respect to the unknown contact stress by using the Fourier integral transformation. A.V. Belokon derived a complete solution to this problem for the case of thick-walled cylinders when the kernel symbol of the integral equation can be approximated by the function corresponding to the torsion of the space with a cylindrical cavity. In the present paper, the case of thin-walled cylinders being complementary to Belokon’s case is mainly considered. Materials and Methods. The cylinder material is supposed to be linearly elastic. The method of integral transformations is used to solve the problem. The singular asymptotic method is applied to solve the integral equation. Research Results. On the basis of studying the properties of the integral equation kernel symbol function, a new special easily factorable approximation applicable for any cylinder thickness is suggested. The Monte-Carlo method is used to determine optimal approximation parameters. Calculations are mainly made for thin-walled cylinders. As a result, an analytic asymptotical solution to the integral equation is obtained. Discussion and Conclusions. The new solution can be effective for relatively long rigid inserts whose length is bigger than the internal diameter of the cylinder. The method based on new approximation remains applicable also for the cases when a cylinder can be regarded as a cylindrical shell. The asymptotical solution can be recommended to engineers for the strength analysis of elastic machine parts of the cylindrical form twisted by a rigid insert.