Mathematical Problems in Engineering

Volume 2015, Article ID 637254, 10 pages

http://dx.doi.org/10.1155/2015/637254

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

^{1}Mathematics Department, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia^{2}Institute for Mathematical Research, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia^{3}College of Foundation and General Studies, Universiti Tenaga Nasional, 43000 Kajang, Selangor, Malaysia^{4}Faculty of Science and Technology, Universiti Sains Islam Malaysia (USIM), 71800 Negeri Sembilan, Malaysia

Received 22 September 2014; Accepted 8 December 2014

Academic Editor: Trung Nguyen-Thoi

Copyright © 2015 N. M. A. Nik Long et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The interaction between the inclined and curved cracks is studied. Using the complex variable function method, the formulation in hypersingular integral equations is obtained. The curved length coordinate method and suitable quadrature rule are used to solve the integral equations numerically for the unknown function, which are later used to evaluate the stress intensity factor. There are four cases of the mode stresses; Mode I, Mode II, Mode III, and Mix Mode are presented as the numerical examples.

#### 1. Introduction

For two-dimensional crack, Panasyuk et al. [1], Cotterell and Rice [2], Shen [3], and Martin [4] used perturbation method to obtain the elastic stress intensity factor for a variety of crack positions. Formulation in terms of singular, hypersingular, or Fredholm integral equations for solving single [5] and multiple cracks problems [6] in various sets of cracks positions was proposed later. These integral equations are solved numerically. Numerical solution of the curved crack problem using polynomial approximation of the dislocation distribution was achieved by taking the crack opening displacement (COD) as the unknown and the resultant forces as the right term in the equations [7].

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.

In this paper, the interaction between inclined and curved cracks is formulated into the hypersingular integral equations using the complex potential method. This approach has been considered by Guo and Lu [11]. Then, by the curved length coordinate method, the cracks are mapped into a straight line, which require less collocation points, and hence give faster convergence. In order to solve the equations numerically, the quadrature rules are applied and we obtained a system of algebraic equations for solving the unknown coefficients. The obtained unknown coefficients will later be used in calculating the SIF.

#### 2. Complex Variable Function Method

The complex variable function method is used to formulate the hypersingular integral equation for the interaction between an inclined crack and a curved crack. Let and be two complex potentials. Then the stress , the resultant function , and the displacement are related to and as [12]where is shear modulus of elasticity, for plane strain, and for plane stress; is Poisson’s ratio and . The derivative in a specified direction (DISD) is defined aswhere denotes the normal and tangential tractions along the segment . Note that the value of depends not only on the position of point , but also on the direction of the segment [5].

The complex potential in plane elasticity is obtained by placing two point dislocations with intensities and at points and , yieldingMaking substitutions and by and in (6) and performing integration on the right side of (6) givewhere denotes the crack configuration. Substituting (7) into (4) and letting approach and , which are located on the upper and lower sides of the crack faces, then using the Plemelj equations, and rewriting as , the following result is obtained [5]:where denotes the crack opening displacement (COD) for both cracks. It is well known that the COD possesses the following properties:

#### 3. Hypersingular Integral Equation

The hypersingular integral equation for an inclined or a curved crack problem is obtained by placing two point dislocations at points and . It is given by [5]where and is the dislocation distribution along the curved crack. In (10), the first integral with h.p. denotes the hypersingular integral and it must be interpreted in Hadamart sense [8].

Now consider the interaction between inclined and curved cracks problem (see Figure 1). For the crack-1, if the point dislocation is placed at points and , is the dislocation doublet distribution along crack-1, and the traction is applied on the , then the hypersingular integral equation for crack-1 iswhere denotes the traction influence on crack-1 caused by dislocation doublet distribution, , on crack-1 andThe influence from the dislocation doublet distribution on crack-2 gives where denotes the traction influence on crack-1 caused by dislocation doublet distribution, , on crack-2 and