partition of unity; enrichment function; level sets; signed distance function; fast marching method; stress intensity factor; crack propagation
Abstract :
[en] A numerical technique for non-planar three-dimensional linear elastic crack growth simulations is proposed. This technique couples the extended finite element method (X-FEM) and the fast marching method (FMM). In crack modeling using X-FEM, the framework of partition of unity is used to enrich the standard finite element approximation by a discontinuous function and the two-dimensional asymptotic crack-tip displacement fields. The initial crack geometry is represented by two level set functions, and Subsequently signed distance functions are used to maintain the location of the crack and to compute the enrichment functions that appear in the displacement approximation. Crack modeling is performed without the need to mesh the crack, and crack propagation is simulated without remeshing. Crack growth is conducted using FMM; unlike a level set formulation for interface capturing, no iterations nor any time step restrictions are imposed in the FMM. Planar and non-planar quasi-static crack growth simulations are presented to demonstrate the robustness and versatility of the proposed technique. Copyright (C) 2008 John Wiley & Sons, Ltd.
Disciplines :
Engineering, computing & technology: Multidisciplinary, general & others Mathematics
Author, co-author :
Sukumar, N.; Department of Civil and Environmental Engineering, University of California, Davis, CA 95616, U.S.
Chopp, D. L.; Department of Engineering Sciences & Applied Mathematics, Northwestern University, Evanston, IL 60208, U.S.A.
Béchet, Eric ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Conception géométrique assistée par ordinateur
Moes, N.; Laboratoire de Mécanique et Matériaux, Ecole Centrale de Nantes, 1 Rue de la No´e, 44321 Nantes, France
Language :
English
Title :
Three-dimensional non-planar crack growth by a coupled extended finite element and fast marching method
Publication date :
2008
Journal title :
International Journal for Numerical Methods in Engineering
ISSN :
0029-5981
eISSN :
1097-0207
Publisher :
John Wiley & Sons, Inc, Chichester, United Kingdom
Sethian JA. Fast marching methods. SIAM Review 1999; 41(2):199-235.
Chopp DL. Some improvements of the fast marching method. SIAM Journal on Scientific Computing 2001; 23(1):230-244.
Sukumar N, Moës N, Moran B, Belytschko T. Extended finite element method for three-dimensional crack modelling. International Journal for Numerical Methods in Engineering 2000; 48(11):1549-1570.
Sukumar N, Chopp DL, Moran B. Extended finite element method and fast marching method for three dimensional fatigue crack propagation. Engineering Fracture Mechanics 2000; 70(1):29-48.
Moës N, Gravouil A, Belytschko T. Non-planar 3D crack growth by the extended finite element and level sets. Part I: mechanical model. International Journal for Numerical Methods in Engineering 2002; 53(11):2549-2568.
Gravouil A, Moës N, Belytschko T. Non-planar 3D crack growth by the extended finite element and level sets. Part II: level set update. International Journal for Numerical Methods in Engineering 2002; 53(11):2569-2586.
Stolarska M, Chopp DL, Möes N, Belytschko T. Modelling crack growth by level sets in the extended finite element method. International Journal for Numerical Methods in Engineering 2001; 51(8):943-960.
Ventura G, Budyn E, Belytschko T. Vector level sets for description of propagating cracks in finite elements. International Journal for Numerical Methods in Engineering 2003; 58(10):1571-1592.
Chopp LD, Sukumar N. Fatigue crack propagation of multiple coplanar cracks with the coupled extended finite element/fast marching method. International Journal of Engineering Science 2003; 41(8):845-869.
Prabel B, Combescure A, Gravouil A, Marie S. Level set X-FEM non-matching meshes: application to dynamic crack propagation in elastic-plastic media. International Journal for Numerical Methods in Engineering 2007; 69(8):1553-1569.
Duflot M. A study on the representation of cracks with level sets. International Journal for Numerical Methods in Engineering 2007; 70(11):1261-1302.
Dhondt G. Automatic 3-D mode I crack propagation calculations with finite elements. International Journal for Numerical Methods in Engineering 1998; 41(4):739-757.
Carter BJ, Wawrzynek PA, Ingraffea AR. Automated 3-d crack growth simulation. International Journal for Numerical Methods in Engineering 2000; 47:229-253.
Mi Y, Aliabadi MH. Three-dimensional crack growth simulations using BEM. Computers and Structures 1994; 52(5):871-878.
Forth CS, Keat WD. Three dimensional nonplanar fracture model using the surface integral method. International Journal of Fracture 1996; 77:243-262.
Bonnet M, Maier G, Polizzotto G. Symmetric Galerkin boundary element method. Applied Mechanics Review 1998; 51:669-704.
Nikishkov GP, Atluri SN. SGBEM-FEM alternating method for analyzing 3D non-planar cracks and their growth in structural components. CMES - Computer Modeling in Engineering and Sciences 2001; 2(3):401-422.
Yoshida K, Nishimura N, Kobayashi S. Application of new fast multipole boundary integral equation method to crack problems in 3D. Engineering Analysis with Boundary Elements 2001; 25(4-5):239-247.
Kolk K, Weber W, Kuhn G. Investigation of 3D crack propagation problems via fast BEM formulations. Computational Mechanics 2005; 37(1):32-40.
Xu G, Ortiz M. A variational boundary integral equation method for the analysis of 3d cracks of arbitrary geometry modelled as continuous distribution of dislocation loops. International Journal for Numerical Methods in Engineering 1993; 31:3675-3701.
Xu G, Bower AF, Ortiz M. An analysis of non-planar crack growth under mixed mode loading. International Journal of Solids and Structures 1994; 31(16):2167-2193.
Gao H, Rice JR. Somewhat circular tensile cracks. International Journal of Fracture 1997; 33(3):155-174.
Lai YS, Movchan AB, Rodin GJ. A study of quasi-circular cracks. International Journal of Fracture 2002; 113:1-25.
Lazarus V. Brittle fracture and fatigue propagation paths of 3D plane cracks under uniform remote tensile loading. International Journal of Fracture 2003; 122(1-2):23-46.
Favier E, Lazarus V, Leblond J-B. Coplanar propagation paths of 3D cracks in infinite bodies loaded in shear. International Journal of Solids and Structures 2006; 43(7-8):2091-2109.
Lazarus V. Mixed mode stress intensity factors for deflected and inclined corner cracks in finite-thickness plates. International Journal of Fatigue 2007; 29(2):305-317.
Areias PMA, Belytschko T. Analysis of three-dimensional crack initiation and propagation using the extended finite element method. International Journal for Numerical Methods in Engineering 2005; 63(5):760-788.
Ferrié E, Buffière J-Y, Ludwig W, Gravouil A, Edwards L. In situ visualization using X-ray microtomography and 3D simulation using the extended finite element method. Acta Materialia 2006; 54(4):1111-1122.
Bordas S, Moran B. Enriched finite elements and level sets for damage tolerance assessment of complex structures. Engineering Fracture Mechanics 2006; 73(9):1176-1201.
Gasser TC, Holzapfel GA. 3D crack propagation in unreinforced concrete. A two-step algorithm for tracking 3D crack paths. Computer Methods in Applied Mechanics and Engineering 2006; 195(37-40):5198-5219.
Mergheim J, Kuhl E, Steinmann P. Towards the algorithmic treatment of 3d strong discontinuities. Communications in Numerical Methods in Engineering 2007; 23(2):97-108.
Rabczuk T, Belytschko T. A three-dimensional large deformation meshfree method for arbitrary evolving cracks. Computer Methods in Applied Mechanics and Engineering 2007; 196(29-30):2777-2799.
Chopp DL. Another look at velocity extensions in the level set method. SIAM Journal on Scientific Computing 2007; in review.
Osher S, Sethian JA. Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics 1988; 79(1):12-49.
Adalsteinsson D, Sethian JA. The fast construction of extension velocities in level set methods. Journal of Computational Physics 1999; 48(1):2-22.
Melenk JM, Babuška I. The partition of unity finite element method: basic theory and applications. Computer Methods in Applied Mechanics and Engineering 1996; 139:289-314.
Babuška I, Melenk JM. Partition of unity method. International Journal for Numerical Methods in Engineering 1997; 40:727-758.
Möes N, Dolbow J, Belytschko T. A finite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 1999; 46(1):131-150.
Strouboulis T, Copps K, Babuška I. The generalized finite element method. Computer Methods in Applied Mechanics and Engineering 2001; 190(32-33):4081-4193.
Belytschko T, Black T. Elastic crack growth in finite elements with minimal remeshing. International Journal for Numerical Methods in Engineering 1999; 45(5):601-620.
Adalsteinsson D, Sethian JA. A fast level set method for propagating interfaces. Journal of Computational Physics 1995; 118(2):269-277.
Chopp DL. Computing minimal surfaces via level set curvature flow. Journal of Computational Physics 1993; 106(1):77-91.
Eshelby JD. Energy relations and the energy momentum tensor in continuum mechanics. In Inelastic Behavior of Solids, Kanninen MF, Adler WF, Rosenfeld AR, Jafee RT (eds). McGraw-Hill: New York, 1970; 77-114.
Shih CF, Moran B, Nakamura T. Energy release rate along a three-dimensional crack front in a thermally stressed body. International Journal of Fracture 1986; 30:79-102.
Gosz M, Moran B. An interaction energy integral method for computation of mixed-mode stress intensity factors along non-planar crack fronts in three dimensions. Engineering Fracture Mechanics 2002; 69(3):299-319.
Banks-Sills L, Wawrzynek PA, Carter B, Ingraffea AR, Hershkovitz I. Methods for calculating stress intensity factors in anisotropic materials: part II - arbitrary geometry. Engineering Fracture Mechanics 2007; 74(8): 1293-1307.
Erdogan F, Sih GC. On the crack extension in plates under plane loading and transverse shear. Journal of Basic Engineering 1963; 85:519-527.
Geuzaine C, Remacle JF. Gmsh: A Three-dimensional Finite Element Mesh Generator with Built-in Pre- and Post-processing Facilities. Available at: http://www.geuz.org/gmsh, 2007.