Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Mapping flat cracks onto penny-shaped cracks, with application to somewhat circular tensile cracks

Author: P. A. Martin
Journal: Quart. Appl. Math. 54 (1996), 663-675
MSC: Primary 73M25; Secondary 73B50
DOI: https://doi.org/10.1090/qam/1417230
MathSciNet review: MR1417230
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Abstract: Consider a three-dimensional homogeneous isotropic elastic solid containing a flat pressurized crack, $ \Omega $. The problem of finding the resulting stress distribution can be reduced to a hypersingular integral equation over $ \Omega $ for the crack-opening displacement. Here, this equation is transformed into a similar equation over a circular region $ D$, using a conformal mapping between $ \Omega $ and $ D$. This new equation is then regularized analytically by using an appropriate expansion method (Fourier series in the azimuthal direction and series of orthogonal polynomials in the radial direction). Analytical results for regions that are approximately circular are also obtained. The method will generalize to other scalar problems and to vector problems (such as shear loading of the crack).

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DOI: https://doi.org/10.1090/qam/1417230
Article copyright: © Copyright 1996 American Mathematical Society

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