Abstract

Hypersingular integrals of the type $I_{\alpha} = \int_{-1}^{1} \hpsngAbs \frac{T_n^{m-{1/2}}}{^\alpha}ds |r| 4$ and $m>3$ is provided. The integrals are also evaluated as $|r|>1$ in order to calculate stress intensity factors. Examples involving crack problems are given and discussed with emphasis on the linkage between mathematics and mechanics of fracture. The examples include classical linear elastic fracture mechanics, functionally graded materials, and gradient elasticity theory. An appendix, with closed form solutions for a broad class of integrals, supplements the paper.