Контактная задача об осесимметричном кручении упругого слоя посредством цилиндрического штампа

Abstract

The paper studies the axis-symmetric contact problem between an elastic layer and a rigid cylindrical circular stamp under torque. The stamp adheres to the upper boundary of the layer whereas the lower boundary of the layer is rigidly fastened. With the application of Hankel integral transform solving the problem reduces to solving the first kind Fredholm integral equation (IE) with symmetrical kernel, represented as a sum of its principal part, Weber-Sonin integral, and the regular kernel. It is estimated that once its height attains a certain level, the layer actually deforms as a semi-space. In the process, through Abel IE method, the solution of the well-known Reissner-Sagoci problem is obtained once again and the original first kind Fredholm IE is reduced to the second kind Fredholm IE. Concurrently, using the collocation method combined with Gauss type quadrature formulas for integral estimation, the original IE reduces to a finite system of linear algebraic equations. To obtain this quadrature formula, properties of Gegenbauer and Chebyshev orthogonal polynomials are used. In the enough wide range of change of characteristic elastic and geometrical parameters of the problem numerical analysis is performed and patterns of changes of tangential contact stresses under the stamp as well as the angle of twist of the stamp are identified.