A newly developed approach without crack surface discretization for modeling 2D solids with large number of cracks in linear elastic fracture mechanics is proposed with the eigen crack opening displacement (COD) boundary integral equations in this paper. The eigen COD is defined as a crack in an infinite domain under fictitious traction acting on the crack surface. Respect to the computational accuracies and efficiencies, the multiple crack problems in finite and infinite plates are solved and compared numerically using three different kinds of boundary integral equations (BIEs): 1) the dual BIEs require crack surface discretization; 2) the BIEs with numerical Green’s functions (NGF) without crack surface discretization, but have to solve a complementary matrix; 3) the eigen crack opening displacement (COD) BIEs in the present paper. With the concept of eigen COD, the multiple crack problems can be solved by using a conventional displacement discontinuity boundary integral equation in an iterative fashion with a small size of system matrix as that in the NGF approach, but without troubles to determine the complementary matrix. Solution of the stress intensity factors of multiple crack problems is solved and compared in some numerical examples using the above three computational algorithms. Numerical results clearly demonstrate the numerical models of eigen COD BIEs have much higher efficiency, providing a newly numerical technique for multiple crack problems. Not only the accuracy and efficiency of computation can be guaranteed, but also the overall properties and local details can be obtained. In conclusion, the numerical models of eigen COD BIEs realize the simulations for multiple crack problems with large quantity of cracks.
Dolado, J.S. and Breugel, K.V. (2011) Recent Advances in Modeling for Cementitious Materials. Cement and Concrete Research, 41, 711-726. https://doi.org/10.1016/j.cemconres.2011.03.014
Wu, J.Y. and Xu, S.L. (2011) An Augmented Multicrack Elastoplastic Damage Model for Tensile Cracking. International Journal of Solids and Structures, 48, 2511-2528. https://doi.org/10.1016/j.ijsolstr.2011.05.001
Xiao, H.T. and Yue, Z.Q. (2011) A Three-Dimensional Displacement Discontinuity Method for Crack Problems in Layered Rocks. International Journal of Rock Mechanics and Mining Sciences, 48, 412-420. https://doi.org/10.1016/j.ijrmms.2011.02.005
Chmelik, F., Anton Trnik, A., Stubna, I. and Pesick, J. (2011) Creation of Microcracks in Porcelain during Firing. Journal of the European Ceramic Society, 31, 2205-2209. https://doi.org/10.1016/j.jeurceramsoc.2011.05.045
Hu, G.L., Ramesh, K.T., Cao, B.Y. and McCauley, J.W. (2011) The Compressive Failure of Aluminum Nitride Considered as a Model Advanced Ceramic. Journal of the Mechanics and Physics of Solids, 59, 1076-1093. https://doi.org/10.1016/j.jmps.2011.02.003
Sobelman, O.S., Gibeling, J.C., Stover, S.M., Hazelwood, S.J., Yeh, O.C., Shelton, D.R. and Martina, R.B. (2004) Do Microcracks Decrease or Increase Fatigue Resistance in Cortical Bone. Journal of Biomechanics, 37, 1295-1303. https://doi.org/10.1016/j.jbiomech.2003.12.034
Dhanasekar, M., Han, J.J. and Qin, Q.H. (2007) Interaction of Surface and Cracks in Railhead. Computer Assisted Mechanics and Engineering Sciences, 14, 79-89.
Chen, Y.Z. (2007) Integral Equation Methods for Multiple Crack Problems and Related Topics. Applied Mechanics Review, 60, 172-194. https://doi.org/10.1115/1.2750671
Brebbia, C.A., Telles, J.C.F. and Wrobel, L.C. (1984) Boundary Element Techniques—Theory and Applications in Engineering. Springer, Berlin. https://doi.org/10.1007/978-3-642-48860-3
Saez, A., Gallego, R. and Dominguez, J. (1995) Hypersingular Quarter-Point Boundary Elements for Crack Problems. International Journal for Numerical Methods in Engineering, 38, 1681-1701. https://doi.org/10.1002/nme.1620381006
Blandford, G.E., Ingraffea, A.R. and Liggett, J.A. (1981) Two-Dimensional Stress Intensity Factor Computation Using the Boundary Element Method. International Journal for Numerical Methods in Engineering, 17, 387-404. https://doi.org/10.1002/nme.1620170308
Aliabadi, M.H. (1997) Boundary Element Formulations in Fracture Mechanics. Applied Mechanics Reviews, 50, 83-96. https://doi.org/10.1115/1.3101690
Hong, H.K. and Chen, J.T. (1988) Derivations of Integral Equations of Elasticity. ASCE Journal of Engineering Mechanics, 114, 1028-1044. https://doi.org/10.1061/(ASCE)0733-9399(1988)114:6(1028)
Chen, J.T. and Hong, H.K. (1999) Review of Dual Boundary Element Methods with Emphasis on Hypersingular Integrals and Divergent Series. Applied Mechanics Reviews, 52, 17-33. https://doi.org/10.1115/1.3098922
Portela, A., Aliabadi, M.H. and Rooke, D.P. (1992) Dual Boundary Elements Analysis of Cracked Plates: Singularity Subtraction Technique. International Journal of Fracture, 55, 17-28. https://doi.org/10.1007/BF00018030
Greengard, L.F. and Rokhlin, V. (1987) A Fast Algorithm for Particle Simulations. Journal of Computational Physics, 73, 325-348. https://doi.org/10.1016/0021-9991(87)90140-9
Liu, Y.J. (2009) Fast Multipole Boundary Element Method—Theory and Applications in Engineering. Cambridge University Press, London. https://doi.org/10.1017/CBO9780511605345
Erdogan, F. (1983) Stress Intensity Factors. ASME Journal of Applied Mechanics, 50, 992-1002. https://doi.org/10.1115/1.3167212
Ang, W.T. and Clements, D.L. (1987) A Boundary Integral Equation Method for the Solution of a Class of Crack Problems. Journal of Elasticity, 17, 9-21. https://doi.org/10.1007/BF00042444
Telles, J.C.F., Castor, G.S. and Guimaraes, S. (1995) A Numerical Green’s Function Approach for Boundary Elements Applied to Fracture Mechanics. International Journal for Numerical Methods in Engineering, 38, 3259-3274. https://doi.org/10.1002/nme.1620381906
Telles, J.C.F. and Guimaraes, S. (2000) Green’s Function: A Numerical Generation for Fracture Mechanics Problems via Boundary Elements. Computer Methods in Applied Mechanics and Engineering, 188, 847-858. https://doi.org/10.1016/S0045-7825(99)00366-7
Kachanov, M. (1987) Elastic Solids with Many Cracks: A Simple Method of Analysis. International Journal of Solids and Structures, 23, 23-43. https://doi.org/10.1016/0020-7683(87)90030-8
Kachanov, M. (1993) Elastic Solids with Many Cracks and Related Problems. Advances in Applied Mechanics, 30, 259-445. https://doi.org/10.1016/S0065-2156(08)70176-5
Feng, X.Q. and Yu, S.W. (2010) Damage Micromechanics for Constitutive Relations and Failure of Microcracked Quasi-Brittle Materials. International Journal of Damage Mechanics, 19, 911-948. https://doi.org/10.1177/1056789509359662
Guo, Z. and Ma, H. (2011) Solution of Stress Intensity Factors of Multiple Cracks in Plane Elasticity with Eigen COD Formulation of Boundary Integral Equation. Journal of Shanghai University (English Edition), 15, 173-179. https://doi.org/10.1007/s11741-011-0716-1
Ma, H., Guo, Z., Dhanasekar, M., Yan, C. and Liu, Y.J. (2013) Efficient Solution of Multiple Cracks in Great Number Using Eigen COD Boundary Integral Equations with Iteration Procedure. Engineering Analysis with Boundary Elements, 37, 487-500. https://doi.org/10.1016/j.enganabound.2012.12.007
Ma, H., Guo, Z., Yan, C. and Dhanasekar, M. (2013) Numerical Solution of Stress Intensity Factors of Multiple Cracks in Great Number with Eigen COD Boundary Integral Equations. Australian Journal of Mechanical Engineering, 11, 1-10. https://doi.org/10.7158/M12-054.2013.11.1
Ma, H., Yan, C. and Qin, Q.H. (2009) Eigenstrain Formulation of Boundary Integral Equations for Modeling Particle-Reinforced Composites. Engineering Analysis with Boundary Elements, 33, 410-419. https://doi.org/10.1016/j.enganabound.2008.06.002
Ma, H., Fang, J.B. and Qin, Q.H. (2011) Simulation of Ellipsoidal Particle-Reinforced Materials with Eigenstrain Formulation of 3D BIE. Advances in Engineering Software, 42, 750-759. https://doi.org/10.1016/j.advengsoft.2011.05.013
Ma, H. and Qin, Q.H. (2007) Solving Potential Problems by a Boundary-Type Meshless Method—The Boundary Point Method Based on BIE. Engineering Analysis with Boundary Elements, 31, 749-761. https://doi.org/10.1016/j.enganabound.2007.03.001
Ma, H., Zhou, J. and Qin, Q.H. (2010) Boundary Point Method for Linear Elasticity Using Constant and Quadratic Moving Elements. Advances in Engineering Software, 41, 480-488. https://doi.org/10.1016/j.advengsoft.2009.10.006
Tada, H., Paris, P. and Irwin, G. (1985) The Stress Analysis of Cracks Handbook. Professional Engineering Publishers, Suffolk.
Chen, W.H. and Chang, C.S. (1989) Analysis of Two-Dimensional Mixed-Crack Problems for Finite Element Alternating Method. Computers & Structures, 6, 1451-1458. https://doi.org/10.1016/0045-7949(89)90485-9
Wang, G.S. and Feng, X.T. (2001) The Interaction of Multiple Rows of Periodical Cracks. International Journal of Fracture, 110, 73-100. https://doi.org/10.1023/A:1010824317724