Stability and convergence analysis of singular integral equations for unequal arms branch crack problems in plane elasticity

• Published in: Applied Mathematical Modelling 2022-03, Vol.103, p.731-749
• Main Authors: Ghorbanpoor, R., Saberi-Nadjafi, J., Long, N.M.A. Nik, Erfanian, M.
• Format: Article
• Language: English
• Subjects: Branch crack problem; Complex potential method; Complex variables; Convergence; Elasticity; Error analysis; Integral equations; Mathematical analysis; Singular integral equation; Singular integral equations; Stability analysis; Stress intensity factors
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2021.11.009

•The unequal arms branch crack problem subjected to traction in plane elasticity was solved using singular integral equation.•To problem is formulated by making used of a point dislocation at origin and distribution dislocation at branches.•We proved the stability, convergence, order of convergence of method and got the estimate error for approximate solution.•To justify the correctness of results, we compared our results with the existing results, and obtained good agreement.•The COD was obtained without evaluating any integration and it depends on length, angle and the number of branch arms. In this paper, an unequal arms branch crack problem in a plane elasticity is treated. Using distribution dislocation function and complex variable potential method, the problem is formulated into a singular integral equation. The appropriate integration scheme, in which a point dislocation is set at the origin and the distribution dislocation, is applied through all arms of the branch crack to solve the obtained singular integral equations numerically. Stability, convergence, the order of convergence, and the error term of the solution are analyzed. Some numerical examples are examined to describe the behavior of stress intensity factors at the endpoints of each branch crack.