Browse > Article
http://dx.doi.org/10.12989/sem.2015.56.4.589

Computation of 2-D mixed-mode stress intensity factors by Petrov-Galerkin natural element method  

Cho, Jin-Rae (Department of Naval Architecture and Ocean Engineering, Hongik University)
Publication Information
Structural Engineering and Mechanics / v.56, no.4, 2015 , pp. 589-603 More about this Journal
Abstract
The mixed-mode stress intensity factors of 2-D angled cracks are evaluated by Petrov-Galerkin natural element (PG-NE) method in which Voronoi polygon-based Laplace interpolation functions and CS-FE basis functions are used for the trial and test functions respectively. The interaction integral is implemented in a frame of PG-NE method in which the weighting function defined over a crack-tip integral domain is interpolated by Laplace interpolation functions. Two Cartesian coordinate systems are employed and the displacement, strains and stresses which are solved in the grid-oriented coordinate system are transformed to the other coordinate system aligned to the angled crack. The present method is validated through the numerical experiments with the angled edge and center cracks, and the numerical accuracy is examined with respect to the grid density, crack length and angle. Also, the stress intensity factors obtained by the present method are compared with other numerical methods and the exact solution. It is observed from the numerical results that the present method successfully and accurately evaluates the mixed-mode stress intensity factors of 2-D angled cracks for various crack lengths and crack angles.
Keywords
2-D angled crack; mixed-mode stress intensity factor (SIF); interaction integral; Petrov-Galerkin natural element (PG-NE) method; crack length and angle;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 Yvonnet, J., Ryckelynck, D., Lorong, P. and Chinesta, F. (2004), "A new extension of the natural element method for non-convex and discontinuous problems: the constrained natural element method (C-NEM)", Int. J. Numer. Meth. Eng., 60(8), 1451-1474.   DOI
2 Zhang, Z., Liew, K.M., Cheng, Y. and Lee, Y.Y. (2008), "Analyzing 2D fracture problems with the improved element-free Galerkin method", Eng. Anal. Bound. Elem., 32, 241-250.   DOI
3 Anderson, T.L. (1991), Fracture Mechanics: Fundamentals and Applications, 1st Edition, CRC Press.
4 Babuska, I. and Melenk, J.M. (1997), "The partition of unity method", Int. J. Numer. Meth. Eng., 40, 727-758.   DOI
5 Barsoum, R.S. (1976), "On the use of isoparametric finite elements in linear fracture mechanics", Int. J. Numer. Meth. Eng., 10, 25-38.   DOI
6 Belytschko, T., Lu, Y.Y., Gu, L. and Tabbara, M. (1995), "Element-free Galerkin methods for static and dynamic fracture", Int. J. Solid. Struct., 32(17-18), 2547-2570.   DOI
7 Bhardwaj, G., Singh, I.V. and Mishra, B.K. (2015), "Stochastic fatigue crack growth simulation of interfacical crack in bi-layered FGMs using XIGA", Comput. Meth. Appl. Mech. Eng., 284, 186-229.   DOI
8 Braun, J. and Sambridge, M. (1995), "A numerical method for solving partial differential equations on highly irregular evolving grids", Nature, 376, 655-660.   DOI
9 Cherepanov, G.P. (1967), "The propagation of cracks in a continuous medium", J. Appl. Math. Mech., 31(3), 503-512.   DOI
10 Chinesta, F., Cescotto, S., Cueto, E. and Lorong, P. (2011), Natural Element Method for the Simulation of Structures and Processes, John Wiley & Sons, New Jersey.
11 Cho, J.R. and Lee, H.W. (2007), "2-D frictionless dynamic contact analysis of large deformable bodies by Petrov-Galerkin natural element method", Comput. Struct., 85, 1230-1242.   DOI
12 Ching, H.K. and Batra, R.C. (2001), "Determination of crack tip fields in linear elastostatics by the meshless local Petrov-Galerkin (MLPG) method", Comput. Model. Eng. Sci., 2(2), 273-289.
13 Cho, J.R. and Lee, H.W. (2006a), "A Petrov-Galerkin natural element method securing the numerical integration accuracy", J. Mech. Sci. Tech., 20(1), 94-109.   DOI
14 Cho, J.R. and Lee, H.W. (2006b), "2-D large deformation analysis of nearly incompressible body by natural element method", Comput. Struct., 84, 293-304.   DOI
15 Cho, J.R., Lee, H.W. and Yoo, W.S. (2013), "Natural element approximation of Reissner-Mindlin plate for locking-free numerical analysis of plate-like thin elastic structures", Comput. Meth. Appl. Mech. Eng., 256, 17-28.   DOI
16 Cho, J.R. and Lee, H.W. (2014), "Calculation of stress intensity factors in 2-D linear fracture mechanics by Petrov-Galerkin natural element method", Int. J. Numer. Meth. Eng., 98, 819-839.   DOI
17 Dolbow, J. and Gosz, M. (2002), "On the computation of mixed-mode stress intensity factors in functionally graded materials", Int. J. Solid. Struct., 39(9), 2557-2574.   DOI
18 Fan, S.C., Liu, X. and Lee, C.K. (2004), "Enriched partition-of-unity finite element method for stress intensity factors at crack tips", Comput. Struct., 82, 445-461.   DOI
19 Fleming, M., Chu, Y.A., Moran, B. and Belytschko, T. (1997), "Enriched element-free Galerkin methods for crack tip fields", Int. J. Numer. Meth. Eng., 40, 1483-1504.   DOI
20 Henshell, R.D. and Shaw, K.G. (1975), "Crack tip elements are unnecessary", Int. J. Numer. Meth. Eng., 9, 495-507.   DOI
21 Hibbitt, H.D. (1977), "Some properties of singular isoparametric elements", Int. J. Numer. Meth. Eng., 11, 180-184.   DOI
22 Irwin, G.R. (1957), "Analysis of stresses and strains near the end of a crack traveling a plate", J. Appl. Mech., 24, 361-364.
23 Liu, X.Y., Xiao, Q.Z. and Karihaloo, B.L. (2004), "XFEM for direct evaluation of mixed mode SIFs in homogeneous and bi-materials", Int. J. Numer. Meth. Eng., 59, 1103-1118.   DOI
24 Moes, N., Dolbow, J. and Belytschko, T. (1999), "A finite element method for crack growth without remeshing", Int. J. Numer. Meth. Eng., 46, 131-150.   DOI
25 Nie, Z.F., Zhou, S.J., Han, R.J., Xiao, L.J. and Wang, K. (2011), " $C^1$ natural element method for strain gradient linear elasticity and its application to microstructures", Acta Mechanica Sinica, 28(1), 91-103.   DOI
26 Pant, M., Singh, I.V. and Mishra, B.K. (2011), "A novel enrichment criterion for modeling kinked cracks using element free Galerkin method", Int. J. Mech. Sci., 68, 140-149.
27 Pena, E., Martinez, M.A., Calvo, B. and Doblare, M. (2008), "Application of the natural element method to finite deformation inelastic problems in isotropic and fiber-reinforced biological soft tissues", Comput. Meth. Appl. Mech. Eng., 197(21-24), 1983-1996.   DOI
28 Rabczuk, T. and Belytschko, T. (2004), "Cracking particles: a simplified meshfree method for arbitrary evolving cracks", Int. J. Numer. Meth. Eng., 61, 2316-2343.   DOI
29 Rao, B.N. and Rahman, S. (2000), "An efficient meshless method for fracture analysis of cracks", Comput. Mech., 26, 398-408.   DOI
30 Rice, J.R. (1968), "A path independent integral and the approximate analysis of strain concentration by notches and cracks", J. Appl. Mech., 35, 379-386.   DOI
31 Rice, J.R. and Tracey, D.M. (1973), Computational fracture mechanics. Numerical and Computer Methods in Structural Mechanics, Eds. Fenves, S.J. et al., Academic Press.
32 Rooke, D.O. and Cartwright, D.J. (1976), Compendium of Stress Intensity Factors, The Hillingdon Press.
33 Shi, J., Ma, W. and Li, N. (2013), "Extended meshless method based on partition of unity for solving multiple crack problems", Meccanica, 43(9), 2263-2270.
34 Singh, I.V., Mishra, B.K., Bhattacharya, S. and Patil, R.U. (2012), "The numerical simulation of fatigue crack growth using extended finite element method", Int. J. Fatig., 36, 109-119.   DOI
35 Strang, G. and Fix, G.J. (1973), An Analysis of the Finite Element Method, Prentice-Hall, New Jersey.
36 Sukumar, N., Moran, B. and Belytschko, T. (1998), "The natural element method in solid mechanics", Int. J. Numer. Meth. Eng., 43, 839-887.   DOI
37 Tong, P., Pian, T.H.H. and Lasry, S.J. (1973), "A hybrid element approach to crack problems in plane elasticity", Int. J. Numer. Meth. Eng., 7, 297-308.   DOI
38 Xiao, Q.Z., B.L. Karihaloo, B.L. and Liu, X.Y. (2004), "Direct determination of SIF and higher order terms of mixed mode cracks by a hybrid crack element", Int. J. Fract., 125, 207-225.   DOI
39 Yau, J.F., Wang, S.S. and Corten, H.T. (1980), "A mixed-mode crack analysis of isotropic solids using conservation laws of elasticity", J. Appl. Mech., 47, 335-341.   DOI