A numerical solution technique of hypersingular integral equation for curved cracks

Abstract

SUMMARYIn this paper, a hypersingular integral equation for curved cracks in plane elasticity is formulated and presented. This paper describes a new numerical technique for solution of deep curved cracks in plane elasticity. In this method, the crack curve length is taken as the co-ordinate in the hypersingular integral equation of the curved crack problems. The curved crack conÿguration maps on the real axis with interval (−a; a), where '2a' is the arc length of the crack. The original hypersingular integral equation is converted into other hypersingular integral equation which is formulated on the curve length coordinate, or on (−a; a). The hypersingular integral equation is solved numerically. Numerical examples prove that higher e ciency has been achieved in the suggested method.

Interaction between inclined and curved cracks problem in plane elasticity

“…Note that the value of J not only depends on the position of point z, but also on the direction of the segment [7]. The complex potential in plane elasticity is obtained by placing two point dislocations with intensity and -at point and , yield…”

“…Substituting (7) into (4) and letting approaches and , which are located on the upper and lower sides of the crack faces, then using the Plemelj equations [7], and rewriting as , the following result is obtained where denotes the crack opening displacement (COD) for the both cracks. It is well known that the COD possesses the following properties:…”

“…(10), the first integral with h.p. denotes the hypersingular integral and it must be interpreted in Hadamart sense [7]. Now consider the interaction between inclined and curved cracks problem (see Fig.…”

“…The curved length coordinate method where the crack is mapped on a real axis provides an effective way to solve the integral equations for the curved crack [7]. Boundary element method, which avoids singularities of the resulting algebraic system of equation [8] and the dual boundary element method have also been used successfully [9].…”

Mode Stresses for the Interaction between an Inclined Crack and a Curved Crack in Plane Elasticity

“…The curved length coordinate method [8] where the crack is mapped on a real axis provides an effective way to solve the integral equations for the curved crack. Boundary element method, which avoids singularities of the resulting algebraic system of equation [9], and the dual boundary element method [10] have also been considered successfully.…”