Research Article  Open Access
Numerical Solution for an Epicycloid Crack
Abstract
A flat crack, , is lying in a threedimensional homogenous isotropic elastic solid subjected to shear loading. A mathematical formulation is developed based on the mixed boundary values for such that the problem of finding the resulting force can be written in the form of hypersingular integral equation. Employing conformal mapping, the integral equation is transformed to a similar equation over a circular region, . By making a suitable representation of hypersingular integral equation, the problem is reduced to solve a system of linear equations. Numerical solution for the shear stress intensity factors, maximum stress intensity, and strain energy release rate is obtained. Our results give an excellent agreement to the existing asymptotic solutions.
1. Introduction
Crack problems play an important role in engineering application due to the fact that the presence of cracks may compromise the strength and toughness of structures. Hence, great efforts [1–10] have been made in solving the crack problems and it has been widely investigated since the pioneer work by Sneddon [11] for a pennyshaped crack. Integral transform method to the solution of a Fredholm integral equation of second kind and numerical approach was implemented by Kassir [12, 13] in solving the rectangular crack problem, while the classic collocation and Galerkin methods were applied by Ioakimidis [14] for solving the plane crack problem subjected to normal load, whereas a perturbation analysis and the complex potential method [15] were performed by Cotterell and Rice [16] to obtain the stress intensity factors for the curved and kinked crack subject to arbitrary tractions in an explicit and simple form. Recently, Wu [17] proposed the dual boundary element method to solve the antiplane crack problem, whilst Georgiadis and Gourgiotis [18] advocated distributed dislocation technique in solving crack problems within Cosserat elasticity with constrained rotations. Motivated by the work of Lazzarin and Zappalorto [19], Lazzarin et al. [20] investigated the stress fields close to a rectangular hole in a plate of finite thickness.
Ioakimidis [21] introduced the concept of finitepart integrals and derived the hypersingular integral equation for a flat crack subjected to tensile pressure, where the unknown function is the crack displacement discontinuity while the righthand terms are the applied tractions on the crack faces. This equation can be numerically solved effectively by using the Gaussian quadrature rules for finitepart integral. Hence, this concept had been advocated widely for the solution of crack problem and some of them can be found in [22–26].
In this paper, the epicycloid crack problem is formulated into solving the hypersingular integral equation numerically for finding the stress intensity factors, maximum stress intensity and energy release rate for the crack subject to shear loading. Our computational results agree with the existing asymptotic solution.
2. Statement of Problem and Basic Equations
Consider an arbitrary shaped crack, , embedded in threedimensional unbounded isotropic elastic body, . Let the Cartesian coordinate with origin and lie in the plane . Assume that is a point in and the body force is absent. Let the crack edges be deformed by the application of equal and opposite constant shear stresses in the and directions, and , and it is assumed that the direction is tractionfree; see Figure 1. Hence, in view of the shear load, the entire plane must be free from the normal stress; that is, Thus, the stress field can be found by considering the halfspace, , subject to the following mixed boundary condition on its surface : where , , and denote the stress tensor, is shear modulus, and is Poisson’s ratio and the usual regularity requirements at the location away from the crack region, where , , , are displacement vectors and is given by where are the material moduli defined by where . The is the Kronecker delta, defined as 1 if and 0 if . And (5) must satisfy Hooke’s law linear elasticity symmetry conditions: The displacement vector, , is represented by Somigliana formula [27, 28]: where the component of Green function is and is the displacement discontinuity in across the crack, where is the unit normal vector, which is assumed to point into . Equation (9) is known as Kelvin’s pointload solution. Substitute (5) and (9) into (7) and perform the integration by parts with respect to and , yielding a system of Cauchy principlevalue integral equations [29]: for , where The resulting boundary terms which involved and are evaluated at the crack edge, assuming that there are no tractions applied onto the boundary. Consequently, and are zero; that is, these equations are to be solved subject to where is the boundary of . Integrating (11) by parts and using condition (13) and making use of the relationship between Cauchy principlevalue integral and hypersingular integral equations [30, 31], yield [29, 32] and the angle is defined by and . The cross on the integral of (15) and (16) means the hypersingular, and it must be interpreted as a Hadamard finite part integral [31, 33, 34]. Multiplying (16) with complex and adding to (15) lead to where , is the unknown crack opening displacement, and the bar denotes the conjugation of and . Equation (17) is to be solved subject to on and can be used for general crack problems under shear loading which is equivalent to those equations obtained in [27, 35]. Suppose the constant shear stress is applied on opposite crack surfaces at direction, and then the general solution of (17) can be reduced into a single hypersingular integral equation:
3. Conformal Mapping and Epicycloid Cracks
Suppose that is a pennyshaped crack, with radius so that the crack occupies the region where and are polar coordinates, , and .
Now, let be a simply connected domain in the plane defined as whose boundary has the polar equation , where is bounded and piecewise continuous and is a small positive parameter. Define with such that the circular unit disc, , is defined as Using the properties of Riemann Mapping theorem [36], a circular disc is mapped conformally onto by where , , , and . Let and the analytic function in (22) is known to exist for any simply connected domain . Further, we assume that is nonzero and bounded for all . Define , , , and as Let such that, for small , and .
A similar integral equation with (18) can be obtained by substituting (23) and (24) into (18); that is, where and are Cauchy type singular and weak singular kernel, respectively [29]: This transformed hypersingular integral equation (25) over a circular disc is solved subject to on .
4. Numerical Treatment
Define such that the orthogonal polynomials and are satisfying the following relationship [37, Page 1054, 8.939.8]: where the respective weight function is and Write as a finite sum Substituting (30) into (25) yields where The following formula [38] is useful in deriving (31): where To determine the unknown coefficients, , multiply (31) by and integrate over and using (28), leads to with the following notations: In evaluating the multiple integral in (35), we have used the Gaussian quadrature and trapezoidal formulas for the radial and angular directions with appropriate choice of collocation points and . This effort leads to the system of linear equations, for the unknown coefficients of , where is a square matrix and and are vectors, and, solved numerically using LAPACK routine F07ASF (ZGETRS) in Numerical Algorithms Group (NAG).
5. Stress Intensity Factors, Maximum Stress Intensity, and Energy Release Rate
The sliding mode, , and the tearing mode, , stress intensity factors are defined as [39, 40] where are constants. The maximum stress intensity, , is defined as while the energy release rate, , by Irwin’s relation subjected to shear loading is calculated from stress intensity factors and defined as where , Young’s modulus, is a measurement of the stiffness of an isotropic elastic material and the relationship between , , and is Let and , followed by substituting (30) into (37), which leads to where as . Introduce where is defined recursively by with and . Substituting (42) into (41) yields where the unknown coefficients, , are obtained from (35).
6. Results and Discussion
Consider the conformal mapping [15] where is an integer and must satisfy . The domain is circular if and has a smooth, regular boundary for . As , one or more cusps develop; see Figure 2 for various and , respectively.
(a)
(b)
Tables 1, 2, and 3 show that our numerical scheme converges rapidly at a different point of the crack with only a small value of used. Tables 1 and 2 show numerical scheme for for with and , respectively. Based on these two tables, it is evident that the convergence of stress intensity factors becomes slow as increases, whilst Table 3 presents the numerical scheme for , for with .



Figures 3, 4, 5, and 6 display the comparison of asymptotic and numerical solutions for and stress intensity factors, maximum stress intensity, , and strain energy release rate, , respectively, for at and . As demonstrated in these figures, our results seem to agree with those obtained by Gao [40] except at the cusps. As the cusps become sharper, the analytical result by Gao [40] does not work, and this gives rise to the difference between our and Gao’s [40] result. It can be seen that the stress intensity factors have local extremal values when the crack front is at or . Figure 7 presents the comparison of asymptotic and numerical solutions for and stress intensity factors, maximum stress intensity, , and strain energy release rate, , respectively, for at . Figure 8 shows the variations of , , , and against for various of . Similar behavior can be observed for the solution of , , , and , for a different parameter of for , displayed in Figure 9.
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7. Conclusion
The present work dealt with the epicycloid crack with the application of shear loading in fracture mechanics. To this end, the numerical solution for the stresses in such specimens is derived, based on the solution of the hypersingular integral equation, and the conformal mapping technique is adopted to transform the hypersingular integral equation over a circular region such that the equation is reduced into a system of linear equations and solved for the unknown coefficients. The stress intensity factors, maximum stress intensity, and strain energy release rate for the epicycloid crack subject to shear load are presented graphically. The proposed model and the obtained numerical results are in good agreement when compared to Gao [40].
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This project is supported by the Universiti Putra Malaysia for the Research University Grant scheme Project no. 0502121834RU.
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Copyright © 2014 Nik Mohd Asri 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.