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Derivation of a new analytical solution for a general two‐dimensional finite‐part integral applicable in fracture mechanics
Author(s) -
Mayrhofer K.,
Fischer F. D.
Publication year - 1992
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620330509
Subject(s) - mathematics , hypergeometric function , mathematical analysis , gauss , cartesian coordinate system , singular integral , singular point of a curve , function (biology) , integral equation , numerical analysis , boundary (topology) , generalized hypergeometric function , exact solutions in general relativity , geometry , physics , quantum mechanics , evolutionary biology , biology
An exact expression is derived for the general finite‐part integralover an inclined ellipticaldomain Ω. r denotes the distance of a point in Ω to the singular point \documentclass{article}\pagestyle{empty}\begin{document}$\left({x,y} \right).f = x_{^0 }^i y_0^j \sqrt {Z\left({x_{0,} y_0 }\right)}$\end{document} is a general function of the Cartesian co‐ordinates x 0 ,y 0 . The boundary of the region Ω represents the equation Z(x 0 , y 0 )=O. These integrals appear during the numerical solution of plane crack problems in three‐dimensional elasticity where they are the dominant part of a hypersingular integral equation. The availability of exact expressions for the integrals with arbitrary integers i and j will increase the accuracy of the numerical results and, simultaneously, lead to quicker numerical results. The considered finite‐part integral can be expressed in closed form as function of complete elliptical integrals or Gauss hypergeometric functions, respectively. Formuias for special cases and some i , j values and their numerical verification are given in Appendices II and III.