Research Article  Open Access
N. M. A. Nik Long, M. R. Aridi, Z. K. Eshkuvatov, "Mode Stresses for the Interaction between an Inclined Crack and a Curved Crack in Plane Elasticity", Mathematical Problems in Engineering, vol. 2015, Article ID 637254, 10 pages, 2015. https://doi.org/10.1155/2015/637254
Mode Stresses for the Interaction between an Inclined Crack and a Curved Crack in Plane Elasticity
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 twodimensional 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 crack1, if the point dislocation is placed at points and , is the dislocation doublet distribution along crack1, and the traction is applied on the , then the hypersingular integral equation for crack1 iswhere denotes the traction influence on crack1 caused by dislocation doublet distribution, , on crack1 andThe influence from the dislocation doublet distribution on crack2 gives where denotes the traction influence on crack1 caused by dislocation doublet distribution, , on crack2 and
Note that since , all three integrals in (14) are regular and note that and satisfy (9). By superposition of the dislocation doublet distribution, , along crack1 (12) and the dislocation doublet distribution, , along crack2 (14), we obtained the following hypersingular integral equation for crack1 which is as follows:where is the traction applied at point of crack1, which is derived from the boundary condition. The first three integrals in (16) represent the effect on crack1 caused by the dislocation on crack1 itself, whereas the second three integrals represent the effect of the dislocations on crack2.
Similarly, the hypersingular integral equation for crack2 iswhere is the traction applied at point of crack2 and In (17), the first three integrals represent the effect on crack2 caused by the dislocation on crack2 itself, and the second three integrals represent the effect of the dislocation on crack1. Equations (16) and (17) are to be solved for and .
Mapping the two cracks configurations on a real axis with intervals and , respectively, the mapping functions and are expressed as where and . In solving the integral equations, we used the following integration rules [13] for the hypersingular and regular integrals, respectively;where is a given regular function, , whereHere is a Chebyshev polynomial of the second kind, defined by and can be evaluated usingwhereand and are defined from (19) and (20), respectively.
4. Stress Intensity Factor
The stress intensity factor (SIF) for the two cracks can be calculated, respectively, as follows:where and are obtained by solving (19) and (20), simultaneously.
In order to show that the suggested method can be used for solving more complicated curved cracks problems, several numerical examples are presented. For verification purposes, we observe that if the two cracks are far apart, we have and approach infinity. These lead to the second three integrals vanish in (16) and (17). Then (16) and (17) become an equation for an inclined and a curved crack, respectively. For the curved crack with the length , we compare the result with the exact solution with the remote traction , given by Cotterell and Rice [2]:where is the tangent angle at the direction of crack tip.
The numerical results are tabulated in Table 1. It can be seen that maximum error is less than .

4.1. Example 1: Mode I
Consider an inclined crack in upper position of a curved crack (Figure 2(a)); the traction applied is and the calculated results for SIF at the crack tips , , , and are, respectively, expressed as Figure 3(a) shows the nondimensional SIF for an inclined crack when is changing for . It can be seen that as varies within the range , the values of , , , and are varied significantly due to shielding effect. Whereas Figure 3(b) shows the nondimensional SIF for a curved crack when is changing for , the values of , , , and are varied significantly for the considered domain.
(a)
(b)
(c)
(d)
(e)
(a)
(b)
(c)
(d)
The effect of the distance between both cracks, , is also studied by taking and the results are shown in Figures 3(c) and 3(d) for and , respectively. As the two cracks are close together, the nondimensional SIF at the crack tip becomes higher.
4.2. Example 2: Mode II
Consider the problem in Figure 2(b); the traction applied is and the calculated results for SIF at the crack tips , , , and are, respectively, expressed as Figures 4(a) and 4(b) show the nondimensional SIF for an inclined crack when is changing for and the nondimensional SIF for a curved crack when is changing for , respectively. It can be seen that as and vary within the range , the values of , , , , , , , and are varied significantly for the considered domain.
(a)
(b)
(c)
(d)
The effect of the distance between both cracks, , is also studied by taking and the results are shown in Figures 4(c) and 4(d) for and , respectively. As the two cracks are close together, the nondimensional SIF at the crack tip becomes higher.
4.3. Example 3: Mode III
Consider the problem in Figure 2(c); the traction applied is and the calculated results for SIF at the crack tips , , , and are, respectively, expressed as Figures 5(a) and 5(b) show the nondimensional SIF for an inclined crack when is changing for and the nondimensional SIF for a curved crack when is changing for , respectively. It can be seen that as and vary within the range , the values of , , , , , , , and are varied significantly for the considered domain.
(a)
(b)
(c)
(d)
The effect of the distance between both cracks, , is also studied by taking and the results are shown in Figures 5(c) and 5(d) for and , respectively. As the two cracks are close together, the nondimensional SIF at the crack tip becomes higher.
4.4. Example 4: Mix Mode
Consider that the inclined crack is located at the lower position of the curved crack (Figure 2(d)). The traction applied is and the calculated results for SIF at the crack tips , , , and are, respectively, expressed asFigures 6(a) and 6(b) show the interaction of both cracks by evaluating the nondimensional SIF at the crack tips and when for and , respectively, whereas Figures 6(c) and 6(d) show the interaction of both cracks by evaluating the nondimensional SIF at the crack tips and when for and , respectively. From these results, we see that and at the crack tip are higher than .
(a)
(b)
(c)
(d)
4.5. Example 5
Consider that the inclined crack is located at the right position of the curved crack (Figure 2(e)). The traction applied is and the calculated results for SIF at the crack tips , , , and are, respectively, expressed as Figures 7(a) and 7(b) show the interaction of both cracks by evaluating the nondimensional SIF at the crack tips and when for and , respectively, whereas Figures 7(c) and 7(d) show the interaction of both cracks by evaluating the nondimensional SIF at the crack tips and when for and , respectively. As the decreases, the nondimensional SIF becomes higher.
(a)
(b)
(c)
(d)
5. Conclusion
In this paper, the different types of loading modes have been applied to the inclined and curved cracks in plane elasticity. We obtained different results of nondimensional SIF due to the different loading modes. We also observed that the SIF increases as both cracks become closer.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors would like to thank the Ministry of Science, Technology and Innovation (MOSTI), Malaysia, for the Science Fund Vot no. 5450657.
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Copyright
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.