Comparison of two numerical procedures for solution of the integro-differential equation of flat crack problem

•Two approaches are considered: the direct and hypersingular ones.•Stress and displacement fields are presented as interrelated polynomials.•The effective numerical integration procedure for arbitrary shape cracks is developed.•3 functions of form for displacement are analyzed and Oore–Burns FoF is the best one.

General approach for numerical treatment of the integro-differential equation of flat crack problem is considered. It consists in presenting the crack surface loading as a set of polynomial functions of two Cartesian coordinates while the corresponding crack surface displacements are chosen as similar polynomials multiplied by the function of form (FoF) to reflect the required singularity of their behavior. Two methods of getting the relations matrixes between these two sets are examined: the first is classical one (when initially the Laplace operator is analytically applied to the integral part of equation and later the resulting hypersingular equation is considered); and the second one is so called direct method (values of the integral are calculated at chosen points of the crack surface, and then these are fitted by polynomial series with subsequent application of the Laplace operator to them).In both these methods the most efforts are devoted to the choice of the FoFs. Three different types of them are tested as to accuracy of results. The first is a usual one when the conditional center of crack is chosen and the FoF is taken as a square root of 1 minus squared relative polar radius of considered surface point. What is unusual here is investigation of shift of the center even for the circular crack. The second one is presentation of the FoF as a square root of products of equations of straight and circular lines of crack boundary. And the third one is based on new idea to use the Burns–Oore FoF, previously suggested in their famous 3D weight function method.Comprehensive investigation of the accuracy of above methods with different combination of FoFs on examples of circular, elliptic, semicircular and square cracks are performed.