Hypersingular Cauchy-type integrals in crack problems with hypersingular stress fields
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Date
1990-02
Authors
Ioakimidis, Nikolaos
Journal Title
Journal ISSN
Volume Title
Publisher
Kluwer Academic Publishers
Abstract
Cauchy-type integrals with –1/2 power singularities at the endpoints a and b of the integration interval [a, b] proved to be extremely useful in crack problems in fracture mechanics for straight cracks [a, b] in a plane elastic medium and, particularly, for the computation of the stress intensity factors at the two crack tips t = a and t = b. Here hypersingular Cauchy-type integrals (interpreted in the sense of finite-part integrals) with –3/2 strong power singularities at the same endpoints a and b (the crack tips for a crack problem) are introduced in the complex plane z = x+iy with z a point outside the crack [a, b]. It is proved that near these endpoints t = a and t = b there exist (i) a strong, –3/2, singularity, (ii) a less strong, –2/2, singularity and (iii) an ordinary inverse-square-root weak, –1/2, singularity. The hypersingular Cauchy-type integrals introduced here are appropriate for crack problems with hypersingular stress fields near their tips with a –3/2 order of singularity for the stress components and a –1/2 order of singularity for the displacement components. Hypersingular stress fields near crack tips were introduced and extensively used by Bueckner and Stern for the computation of stress intensity factors at crack tips based on the classical Betti’s reciprocal work theorem.