In this review article general results are presented concerning problems in the fracture dynamics of elastic bodies with allowance for unilateral crack edge contact interaction with friction.. A formulation has been made of the elastodynamic contact problem with unilateral restrictions for bodies with cracks under arbitrary dynamic loading . A specific case of liarmonic loading important to these applications has also been considered. The mathematical aspects of the elastodynamics problem for bodies with cracks and with unilateral restrictions in the form of inequalities on the crack edges have been considered in brief. A variational formulation of the problem has been given. Boundary variational inequalities and boundary functionals have been derived. The boundary integral equations (BIE) method in a Laplace transform domain has been used as a solution for the elastodynamic problem for bodies with cracks. Singularities of the kernels in these integral equations have been studied. Two regularization methods of the potentials with "strongly" singular kernels have been considered. The first is based on its transformation into integro-differentional equations. The second consists of the utilization of the BIE with hypersingular integrals, which are considered in the sense of a finite part, according to Hadamard. An algorithm for the solution of the elastodynamic unilateral contact problem for bodies with cracks has been elaborated. The algorithm is based on finding a saddle point of a sub-differentional boundary functional. It has been shown that the algorithm may be considered as a compressive operator, which acts in corresponding functional spaces. This means that the algorithm is convergent. Numerical methods have been elaborated for the solution to elastodynamic contact problems with unilateral restrictions and friction for bodies with cracks. The problem has been solved, for plane liarmonic tension-compression wave propagation in a plane with one and two colinear finite length cracks and with allowance for unilateral contact interaction of the crack edges. Dependence of the solution accuracy on the approximation of coordinates and time, and also of numbers of terms in the expansion of the stress-strain state components into Fourier series, has been investigated. Numerical results have also been presented. Quantitative and qualitative effects caused by contact interaction of the crack edges have been investigated. This review contains 170 references.
This article considers weakly singular, singular and hypersingular integrals, which arise when the boundary integral equation (BIE) methods are used to solve problems in science and engineering. For their regularization, an approach based on the theory of distribution and application of the Green theorem has been used. The expressions, which allow an easy calculation of the weakly singular, singular and hypersingular integrals, have been constructed. Such approach may be easily generalized and applied to the calculation of multidimensional integrals with singularities of various types.
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