Kalker's method applied to some improper integrals in fracture mechanics

by J. Jger, Blattwiesenstr. 7, D- 76227 Karlsruhe, Germany

Eng. Fracture Mech., 1996, Vol. 54, 229-237

email: j_jaeger@t-online.de

Abstract: This paper discusses some methods of contact mechanics, which can be applied to fracture problems. First, improper integrals with a singularity of the order r-3 are treated, which some authors call hypersingular integrals. Compared with other publications, a simplified formulation is achieved by application of Kalker's analytical method for singularities of the order r-1. The  result is written in terms of hypergeometric functions, which are recursively reduced to standard elliptic integrals. Similar to the superposition of single forces in contact mechanics, a superposition of single displacements is used in fracture mechanics and the methods of contact mechanics are applied to fracture mechanics. In contrast to the modelling of the crack surface as a polynomial in x and y, a discrete displacement function with constant values on an equidistant rectangular mesh is more promising. The integration is performed as a matrix product and the cyclic structure of this matrix is used to reduce the required computer memory.