Singular Integral Equations for Some Contact Problems of Elasticity Theory for Bodies with Cracks

Abstract

This chapter is devoted to the mathematical method of modeling of the deformation and fracture process for elements of moving and fixed joints (tribojoints). In this chapter, we deduce singular integral equations of some contact problems of the elasticity theory for bodies with cracks. The Kolosov-Muskhelishvili complex potentials for the analyzed problems are represented in the form of integral representations with Cauchy-type kernels with respect to the derivatives of the discontinuities of displacements on the crack contours. In the general case, the problems are reduced to systems of singular integral equations of the first kind. We propose singular integral equations (SIE) for the elastic half-plane weakened by a system of curvilinear cracks under the action of various model contact loads applied to the boundary of the half-plane. We briefly describe the Gauss-Chebyshev method of mechanical quadratures that enables one to efficiently construct the numerical solutions of these SIE. We also deduce the relations for the stress intensity factors at the crack tips can be expressed via the solutions of SIE for inside and edge curvilinear cracks in the half-plane. In this chapter, we present both known results available from the literature and new results.