International Journal of Fracture 63: 229-245, 1993. © 1993 Kluwer Academic Publishers. Printed in the Netherlands. 229 Numerical solutions of singular integral equations having Cauchy-type singular kernel by means of expansion method NAO-AKI NODA and TADATOSHI MATSUO Department of Mechanical Engineering, K yushu Institute of Technology, Kitakyushu 804, Japan Received 5 January 1993; accepted in revised form 4 August 1993 Abstract. This paper is concerned with numerical solutions of singular integral equations with Cauchy-type singular kernel. It is well-known that this type of singular integral equations appears in the analysis of crack problems using the continuously distributed dislocation method. In addition, it also appears in the analysis of notch problems using the body force method. In the present analysis, the unknown function of densities of dislocations and body forces are approximated by the product of the fundamental density functions and polynomials. The accuracy of stress intensity factors and stress concentration factors obtained by the present method is verified through the comparison with the exact solution and the reliable numerical solution obtained by other researchers. The present method is found to give good convergency of the numerical results for notch problem as well as internal and edge crack problems. 1. Introduction In the analysis of stress intensity factors for crack problems, various numerical methods, such as conformal mapping technique, Laurent series expansion method, body force method, continuously distributed dislocation method and finite element method have been applied to different types of problems depending on their peculiarity. Among those methods, the continuously distributed dislocation method has been used by a lot of researchers: in the analysis, a crack is represented by a distribution of infinitesimal dislocations and the problem is reduced to the singular integral equations having Cauchy-type singular kernel. To solve this type of singular integral equations, Erdogan [1, 2], Theocaris-Ioakimidis [3], Boiko-Kerpenko [4] and other researchers have discussed several numerical methods. Recently, Fujimoto [-5] has applied Boiko's method to internal crack problems and has shown that internal crack problems are solved with higher accuracy than previous research has shown. On the other hand, in previous papers [-6, 7, 8], numerical solutions of the singular integral equation in the crack analysis using the body force method, which has the singularity of the form r -z, have been discussed. Then, an approximation of the unknown function by the product of the fundamental density function and Chebyshev polynomials is found to give more accurate results compared with previous research. In this paper, the Cauchy-type singular integral equations for crack and notch problems are solved using a similar approxi- mation, namely, by the product of the fundamental density function and polynomials. Then, the accuracy of stress intensity factors obtained by the present method are compared with the results given by the hypersingular integral equations in the previous paper. Moreover, problems of a cruciform crack, an internal crack and an edge crack are solved and compared with exact solutions and reliable numerical solutions obtained by quadrature methods. The