On the Application of a Novel Approach to Solving Integral Equations of Plane Contact Problems in the Theory of Elasticity and Fracture Mechanics

From a new point of view, the issue of constructing effective solutions to the governing integral equations of plane contact problems of the theory of elasticity and crack mechanics is reconsidered when interfacial cracks exist in a piecewise-homogeneous elastic plane. A new approach is used, based on the relationship between the integral with a logarithmic kernel and the integral with the Cauchy kernel. As a result, the original governing Fredholm integral equations (IEs) of the first kind with a Hermitian-logarithmic kernel, which describe plane contact problems of elasticity theory taking into account cohesion or friction forces and problems of interfacial cracks, are reduced to singular integral equations (SIEs) with a Cauchy kernel. The solutions of these SIEs are constructed using the method of Jacobi orthogonal polynomials and the well-known numerical-analytical method of reducing them to finite systems of linear algebraic equations (SLAE) by applying Gauss-type quadrature formulas for computing singular integrals. To compare the effectiveness of the various methods used, the well-known solutions of conventional SIEs of the above problems of elasticity theory, obtained by the method of the Riemann boundary value problem, are also provided. A comparative numerical analysis of the solutions of the governing IEs is carried out and the characteristic patterns of changes in the main characteristics of the problems under consideration are revealed.