Published by AMSS Press, Wuhan, China Acta Mechanica Solida Sinica, Vol. 24, No. 5, October, 2011 ISSN 0894-9166 NUMERICAL SOLUTIONS FOR A NEARLY CIRCULAR CRACK WITH DEVELOPING CUSPS UNDER SHEAR LOADING ⋆⋆ N. M. A. Nik Long 1,2⋆ L. F. Koo 1 Z. K. Eshkuvatov 1,2 ( 1 Department of Mathematics, Faculty of Science, University Putra Malaysia, Serdang 43400, Selangor, Malaysia) ( 2 Institute for Mathematical Research, University Putra Malaysia, Serdang 43400, Selangor, Malaysia) Received 17 October 2010, revision received 12 July 2011 ABSTRACT In this paper, we study the behavior of the solution at the crack edges for a nearly circular crack with developing cusps subject to shear loading. The problem of finding the resulting force can be written in the form of a hypersingular integral equation. The equation is then trans- formed into a similar equation over a circular region using conformal mapping. The equation is solved numerically for the unknown coefficients, which will later be used in finding the stress intensity fac- tors. The sliding and tearing mode stress intensity factors are evaluated for cracks and displayed graphically. Our results seem to agree with the existing asymptotic solution. KEY WORDS nearly circular crack, numerical method, Galerkin method, stress intensity factors, hypersingular integral equation, shear loading, conformal mapping I. INTRODUCTION The existence of cracks may compromise the strength and toughness of structures. The fracture me- chanics theory is employed to analyze structures and machine components with cracks in order to obtain an efficient design while the mathematical modeling is used to predict crack propagation behavior. Hence, crack problems have not only mathematical interests but also are of practical importance in fracture mechanics. Thus, the use of mathematical modeling in crack problems has been exploded during the past decade from its original acceptance in epidemiologic research; the method is now commonly employed in many fields of study such as in manufacturing, engineering, physics and architecture. A problem of current interest in fracture mechanics is the determination of the stress intensity factor for the prediction of the stress state along the crack edges and crack tips caused by a load or stress and it has attracted the interest of many researches [1–9] . Bui [10] obtained a singular integral equation for the crack displacement discontinuity by using the single layer and double layer potential. The penny, elliptical and square shaped cracks under normal pressure are solved numerically. Integral transform method was adopted by Kassir [11,12] to solve the plane crack problem. Murakami [13] applied the body force method to analyse the stress intensity factors of the arbitrarily shaped plate with an inner or edge crack. Similar methods were also used by Makoto et al. [14] in solving an infinite solid containing an embedded plane crack of an arbitrary shape. Further, Qin et al. [15] applied the hypersingular integral equation method based on the body force method to calculate the mixed mode stress intensity factor of 3D cracks under shear loading. ⋆ Corresponding author. E-mail: nmasri@math.upm.edu.my ⋆⋆ This project is supported by the Ministry Of Higher Education Malaysia for the Fundamental Research Grant scheme, project No. 01-04-10-897FR and the NSF scholarship. The authors would like to thanks the reviewers for the constructive comments to improve the quality of the paper.