Research Article | Open Access
Nik Mohd Asri Nik Long, Lee Feng Koo, Zainidin K. Eshkuvatov, "Computation of Energy Release Rates for a Nearly Circular Crack", Mathematical Problems in Engineering, vol. 2011, Article ID 741075, 17 pages, 2011. https://doi.org/10.1155/2011/741075
Computation of Energy Release Rates for a Nearly Circular Crack
This paper deals with a nearly circular crack, in the plane elasticity. The problem of finding the resulting shear stress can be formulated as a hypersingular integral equation over a considered domain, and it is then transformed into a similar equation over a circular region, , using conformal mapping. Appropriate collocation points are chosen on the region to reduce the hypersingular integral equation into a system of linear equations with unknown coefficients, which will later be used in the determination of energy release rate. Numerical results for energy release rate are compared with the existing asymptotic solution and are displayed graphically.
The determination of energy release rate, a measurement of energy necessary for crack initiation in fracture mechanics, has stirred a huge interest among researchers, and different approaches have been applied. Williams and Isherwood  proposed an approximate method in terms of a mean stress to approximate the strain-energy release rates of finite plates. Sih  proposed the energy density theory as an alternative approach for fracture prediction. Hayashi and Nemat-Nasser  modelled the kink as a continuous distribution of infinitesimal edge dislocations to obtain the energy release rate at the onset of kinking of a straight crack in an infinite elastic medium subjected to a predominantly Mode I loading. Further, a similar method to  has also been adopted by Hayashi and Nemat-Nasser  to obtain the energy release rate for a kinked from a straight crack under combined loading based on the maximum energy release rate criterion. Gao and Rice  extended Rice's work  in finding the energy release rate for a plane crack with a slightly curved front subject to shear loading. While, Gao and Rice  and Gao  considered a penny-shaped crack as a reference crack in solving the energy release rate for a nearly circular crack subject to normal and shear loads. Jih and Sun  employed the finite element method based on crack-closure integral in calculating the strain energy release rate elastostatic and elastodynamic crack problems in finite bodies whereas Dattaguru et al.  adopted the finite element analysis and modified crack closure integral technique in evaluating the strain energy release rate. Poon and Ruiz  applied the hybrid experimental-numerical method for determining the strain energy release rate. Wahab and de Roeck  evaluated the strain energy release rate from three-dimensional finite element analysis with square-root stress singularity using different displacement and stress fields based on the Irwin's crack closure integral method . Guo et al.  used the extrapolation approach in order to avoid the disadvantages of self-inconsistency in the point-by-point closed method to determine the energy release rate of complex cracks. Xie et al.  applied the virtual crack closure technique in conjunction with finite element analysis for the computation of energy release rate subject to kinked crack, while interface element based on similar approach also adopted by Xie and Biggers  in calculating the strain energy release rate for stationary cracks subjected to the dynamic loading.
In this paper, we focus our work on obtaining the numerical results for energy release rate for a nearly circular crack via the solution of hypersingular integral equation and compare our computational results with Gao's .
2. Formulation of the Problem
Consider the infinite isotropic elastic body containing a flat circular crack, , as in Figure 1, located on the Cartesian coordinate with origin , and lies in the plane . Let the radius of the crack, be and .
If the equal and opposite shear stresses in the and directions, and , respectively, are applied to the crack plane, and it is assumed that the direction is traction free, then in the view of shear load, the entire plane, must free from the normal stress, that is and the stress field can be found by considering the above problem subjected to the following mixed boundary condition on its surface, : where is stress tensor, is shear modulus, is denoted as Poisson's ratio, and is the entire . Also, the problem satisfies the regularity conditions at infinity where is the distance Martin  showed that the problem of finding the resultant force with condition (2.2) can be formulated as a hypersingular integral equation where is the unknown crack opening displacement, , , the , and the angle is defined by The cross on the integral means the hypersingular, and it must be interpreted as a Hadamard finite part integral [18, 19]. Equation (2.5) is to be solved subject to on where is boundary of . For the constant shear stress in direction, we have and , hence, (2.5) becomes
Polar coordinates and are chosen so that the loadings and can be written as a Fourier series where the Fourier components are -complex. The -complex crack opening displacement, and , have similar expressions Without loss of generality, we consider . Using Guidera and Lardner , the dimensionless function and can be expressed as where the -complex coefficients are known, are unknown, and is an orthogonal Gegenbauer polynomial of degree and index , which is defined recursively by  with the initial values and . For a constant shear loading, , the solution for a circular crack is obtainable.
3. Nearly Circular Crack
Let be an arbitrary shaped crack of smooth boundary with respect to origin , such that is defined as where the boundary of , is given by . Let with such that the unit disc is By the properties of Reimann mapping theorem , a circular disc is mapped conformally onto using . This approach works for a general smooth star-shaped domain, . For a particular application, let be an analytic function, simply connected in the domain , is nonzero and bounded for all , which maps a unit circle, in the -plane into a nearly circular domain in the -plane where is a real parameter and is the boundary of . This domain has a smooth, regular boundary for . As one or more cusps develop; see Figure 2 with various choices of .
Let and define and as where and so that . Next, we define and as Set Substituting (3.5), (3.6), (3.7), and (3.8) into (2.7) gives where the kernel and are  This hypersingular integral equation over a circular disc is to be solved subject to on , and the is a Cauchy-type singular kernel with order , and the kernel is weakly singular with , as (see the appendix).
We are going to solve (3.9) numerically. Write as a finite sum where is defined by Introduce where . The relationship between these two functions, , and can be expressed as where is Kronecker delta and Both functions and have square-root zeros at .
Krenk  showed that where
In (3.22), we have used the following notation: , , and .
In evaluating the multiple integrals in (3.22), we have used the Gaussian quadrature and trapezoidal formulas for the radial and angular directions, with the choice of collocation points and defined as follows: where and are abscissas for and , respectively, and is the number of collocation points in radial and angular directions, respectively. This effort leads to the system of linear equations where is a square matrix, and and are vectors, to be determined.
4. Energy Release Rate
The energy release rate (measured in ), by Irwin's relation subject to shear load is defined as [7, 8] where , Young's modulus, a measurement of the stiffness of an isotropic elastic material and the relationship of , and , is and and , the sliding and tearing mode stress intensity factor, respectively, are defined as [5, 7, 8] where are constants.
Table 1 shows that our numerical scheme converges rapidly at a different point of the crack with only a small value of are used.
Figures 3, 4, 5, and 6 show the variations of against for , , , and , respectively. It can be seen that the energy release rate has local extremal values when the crack front is at or . Similar behavior can be observed for the solution of for a different and at , displayed in Figures 7 and 8. Our results agree with those obtained asymptotically by Gao , with the maximum differences for are , , , and for , , , and , respectively.
In this paper, the hypersingular integral equation over a nearly circular crack is formulated. Then, using the conformal mapping, the equation is transformed into hypersingular integral equation over a circular crack, which enable us to use the formula obtained by Krenk . By choosing the appropriate collocation points, this equation is reduced into a system of linear equations and solved for the unknown coefficients. The energy release rate for the mentioned crack subject to shear load is presented graphically. Our computational results seem to agree with the asymptotic solution obtained by Gao .
The Singularity of the Kernel and
At , we have Differentiate with respect to , we have Let where , , , and are real. As and as , we see that and are as ().
Apply leads to Hence,
Martin  showed that where , , and are constants and as .
For , expand at , and truncating at second order, (3.4) gives where Next, substituting (A.5) and (A.7) into (3.4) gives where Using (A.22) and (A.20) yields Hence, as , then . It is not difficult to see that , , , and , respectively; then (3.11) becomes Applying similar procedures as in gives Thus,
The authors would like to thank the reviewers for their very constructive comments to improve the quality of the paper. This project is supported by Ministry of Higher Education Malaysia for the Fundamental Research Grant scheme, project no. 01-04-10-897FR and the second author received a NSF scholarship.
- J. G. Williams and D. P. Isherwood, “Calculation of the strain-energy release ratesof cracked plates by an approximate method,” The Journal of Strain Analysis for Engineering Design, vol. 3, no. 1, pp. 17–22, 1968.
- G. C. Sih, “Strain-energy-density factor applied to mixed mode crack problems,” International Journal of Fracture, vol. 10, no. 3, pp. 305–321, 1974.
- K. Hayashi and S. Nemat-Nasser, “Energy release rate and crack kinking,” International Journal of Solids and Structures, vol. 17, no. 1, pp. 107–114, 1981.
- K. Hayashi and S. Nemat-Nasser, “Energy release rate and crack kinking under combined loading,” Journal of Applied Mechanics, Transactions ASME, vol. 48, no. 3, pp. 520–524, 1981.
- H. Gao and J. R. Rice, “Shear stress intensity factors for a planar crack with slightly curved front,” Journal of Applied Mechanics, Transactions ASME, vol. 53, no. 4, pp. 774–778, 1986.
- J. R. Rice, “First order variation in elastic fields due to variation in location of a planar crack front,” Journal of Applied Mechanics, Transactions ASME, vol. 52, no. 3, pp. 571–579, 1985.
- H. Gao and J. R. Rice, “Somewhat circular tensile cracks,” International Journal of Fracture, vol. 33, no. 3, pp. 155–174, 1987.
- H. Gao, “Nearly circular shear mode cracks,” International Journal of Solids and Structures, vol. 24, no. 2, pp. 177–193, 1988.
- C. J. Jih and C. T. Sun, “Evaluation of a finite element based crack-closure method for calculating static and dynamic strain energy release rates,” Engineering Fracture Mechanics, vol. 37, no. 2, pp. 313–322, 1990.
- B. Dattaguru, K. S. Venkatesha, T. S. Ramamurthy, and F. G. Buchholz, “Finite element estimates of strain energy release rate components at the tip of an interface crack under mode I loading,” Engineering Fracture Mechanics, vol. 49, no. 3, pp. 451–463, 1994.
- C. Y. Poon and C. Ruiz, “Hybrid experimental-numerical approach for determining strain energy release rates,” Theoretical and Applied Fracture Mechanics, vol. 20, no. 2, pp. 123–131, 1994.
- M. M.A. Wahab and G. de Roeck, “3D finite element computation of strain energy release rate using different displacement and stress fields,” Engineering Fracture Mechanics, vol. 49, no. 5, pp. 659–665, 1994.
- G. R. Irwin, “Analysis of stresses and strains near the end of a crack traversing a plate,” Journal of Applied Mechanics, Transactions ASME, vol. 24, no. 3, pp. 361–364, 1957.
- M.-L. Guo, Q.-Y. Meng, and B. Wang, “Research for the strain energy release rate of complex cracks by using point-by-point closed extrapolation approach,” Applied Mathematics and Mechanics, vol. 24, no. 4, pp. 421–426, 2003.
- D. Xie, A. M. Waas, K. W. Shahwan, J. A. Schroeder, and R. G. Boeman, “Computation of energy release rates for kinking cracks based on virtual crack closure technique,” CMES: Computer Modeling in Engineering and Sciences, vol. 6, no. 6, pp. 515–524, 2004.
- D. Xie and S. B. Biggers Jr., “Calculation of transient strain energy release rates under impact loading based on the virtual crack closure technique,” International Journal of Impact Engineering, vol. 34, no. 6, pp. 1047–1060, 2007.
- P. A. Martin, “Mapping flat cracks onto penny-shaped cracks: shear loadings,” Journal of the Mechanics and Physics of Solids, vol. 43, no. 2, pp. 275–294, 1995.
- J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Dover, New York, NY, USA, 1999.
- P. A. Martin, “Exact solution of a simple hypersingular integral equation,” Journal of Integral Equations and Applications, vol. 4, no. 2, pp. 197–204, 1992.
- J. T. Guidera and R. W. J. Lardner, “Penny-shaped cracks,” Journal of Elasticity, vol. 5, no. 1, pp. 59–73, 1975.
- A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, McGraw-Hill, New York, NY, USA, 2nd edition, 1953.
- Z. Nehari, Conformal Mapping, McGraw-Hill, New York, NY, USA, 1st edition, 1952.
- S. Krenk, “A circular crack under asymmetric loads and some related integral equations,” Journal of Applied Mechanics, Transactions ASME, vol. 46, no. 4, pp. 821–826, 1979.
- P. A. Martin, “Mapping flat cracks onto penny-shaped cracks, with application to somewhat circular tensile cracks,” Quarterly of Applied Mathematics, vol. 54, no. 4, pp. 663–675, 1996.
Copyright © 2011 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.