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Accurate fracture modelling using meshless methods, the visibility criterion and level sets: formulation and 2D modelling. (English) Zbl 1235.74346

Summary: Fracture modelling using numerical methods is well-advanced in 2D using techniques such as the extended finite element method (XFEM). The use of meshless methods for these problems lags somewhat behind, but the potential benefits of no meshing (particularly in 3D) prompt continued research into their development. In methods where the crack face is not explicitly modelled (as the edge of an element for instance), two procedures are instead used to associate the displacement jump with the crack surface: the visibility criterion and the diffraction method. The visibility criterion is simple to implement and efficient to compute, especially with the help of level set coordinates. However, spurious discontinuities have been reported around crack tips using the visibility criterion, whereas implementing the diffraction method in 3D is much more complicated than the visibility criterion. In this paper, a tying procedure is proposed to remove the difficulty with the visibility criterion so that crack tip closure can be ensured while the advantages of the visibility criterion can be preserved. The formulation is based on the use of level set coordinates and the element-free Galerkin method, and is generally applicable for single or multiple crack problems in 2D or 3D. The paper explains the formulation and provides verification of the method against a number of 2D crack problems.

MSC:

74S20 Finite difference methods applied to problems in solid mechanics
74R10 Brittle fracture

Software:

Gmsh
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Full Text: DOI

References:

[1] Gingold, Smoothed particle hydrodynamics-theory and application to non-spherical stars, Monthly Notices of the Royal Astronomical Society 181 pp 375– (1977) · Zbl 0421.76032
[2] Atluri, A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics, Computational Mechanics 22 pp 117– (1998) · Zbl 0932.76067
[3] Sukumar, Natural neighbour Galerkin methods, International Journal for Numerical Methods in Engineering 50 pp 1– (2001) · Zbl 1082.74554
[4] Fries T Matthies H Classification and overview of meshfree methods 2004 · Zbl 1354.76105
[5] Nguyen, Meshless methods: a review and computer implementation aspects, Mathematics and Computers in Simulation 79 pp 763– (2008) · Zbl 1152.74055
[6] Belytschko, Element-free Galerkin methods, International Journal for Numerical Methods in Engineering 37 pp 229– (1994) · Zbl 0796.73077
[7] Belytschko, Fracture and crack-growth by element free Galerkin methods, Modelling and Simulation in Materials Science and Engineering 2 pp 519– (1994)
[8] Belytschko, Element-free Galerkin methods for static and dynamic fracture, International Journal for Numerical Methods in Engineering 32 pp 2547– (1995) · Zbl 0918.73268
[9] Belytschko, Crack-propagation by element-free Galerkin methods, Engineering Fracture Mechanics 51 pp 295– (1995)
[10] Organ, Continuous meshless approximations for nonconvex bodies by diffraction and transparency, Computational Mechanics 18 pp 225– (1996) · Zbl 0864.73076
[11] Belytschko, Meshless methods: an overview and recent developments, Computer Methods in Applied Mechanics and Engineering 139 pp 3– (1996) · Zbl 0891.73075
[12] Krysl, The element free Galerkin method for dynamic propagation of arbitrary 3-d cracks, International Journal for Numerical Methods in Engineering 44 pp 767– (1999) · Zbl 0953.74078
[13] Belytschko, Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering 45 pp 601– (1999) · Zbl 0943.74061
[14] Bordas, A simple error estimator for extended finite elements, Communications in Numerical Methods in Engineering 24 pp 961– (2008) · Zbl 1156.65093
[15] Zi, Extended meshfree methods without branch enrichment for cohesive cracks, Computational Mechanics 40 pp 367– (2007) · Zbl 1162.74053
[16] Zi, New crack-tip elements for XFEM and applications to cohesive cracks, International Journal for Numerical Methods in Engineering 57 pp 2221– (2003) · Zbl 1062.74633
[17] Bordas, IUTAM Symposium on Discretization Methods for Evolving Discontinuities pp 21– (2007) · Zbl 1209.74068
[18] Bordas, Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment, Engineering Fracture Mechanics 75 pp 943– (2008)
[19] Rabczuk, On three-dimensional modelling of crack growth using partition of unity methods, Computers and Structures 88 pp 1391– (2010)
[20] Gravouil, 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 53 pp 2569– (2002) · Zbl 1169.74621
[21] Ventura, A vector level set method and new discontinuity approximations for crack growth by EFG, International Journal for Numerical Methods in Engineering 54 pp 923– (2002) · Zbl 1034.74053
[22] Lu, A new implementation of the element free Galerkin method, Computer Methods in Applied Mechanics and Engineering 113 pp 397– (1994) · Zbl 0847.73064
[23] Zhuang, Aspects of the use of orthogonal basis functions in the element free Galerkin method, International Journal for Numerical Methods in Engineering 81 (3) pp 366– (2010) · Zbl 1183.74376
[24] Belytschko, Lecture Notes in Computational Science and Engineering, in: Meshfree Methods for Partial Differential Equations pp 37– (2003)
[25] Melenk, The partition of unity finite element method: basic theory and applications, Computer Methods in Applied Mechanics and Engineering 139 pp 289– (1996) · Zbl 0881.65099
[26] Babuška, Lecture Notes in Computational Science and Engineering, in: Meshfree Methods for Partial Differential Equations pp 1– (2003)
[27] Rabczuk, Simulations of instability in dynamic fracture by the cracking particles method, Engineering Fracture Mechanics 76 pp 730– (2009)
[28] Béchet, Improved implementation and robustness study of the X-FEM for stress analysis around cracks, International Journal for Numerical Methods in Engineering 64 pp 1033– (2005) · Zbl 1122.74499
[29] Cai, A new partition of unity finite element free from the linear dependence problem and possessing the delta property, Computer Methods in Applied Mechanics and Engineering 199 pp 1036– (2010) · Zbl 1227.74065
[30] Fleming M The element-free Galerkin method for fatigue and quasi-static fracture 1997
[31] Fries, The intrinsic partition of unity method, Computational Mechanics 40 pp 803– (2007) · Zbl 1162.74049
[32] Yoon, Enriched meshfree collocation method with diffuse derivatives for elastic fracture, Computers and Mathematics with Applications 51 pp 1349– (2006) · Zbl 1141.74046
[33] Zhang, Analyzing 2d fracture problems with the improved element-free Galerkin method, Engineering Analysis with Boundary Elements 32 pp 241– (2008) · Zbl 1244.74240
[34] Duflot, A meshless method with enriched weight functions for fatigue crack growth, International Journal for Numerical Methods in Engineering 59 pp 1945– (2004) · Zbl 1060.74664
[35] Duflot, A meshless method with enriched weight functions for three-dimensional crack propagation, International Journal for Numerical Methods in Engineering 65 pp 1970– (2006) · Zbl 1114.74064
[36] Muravin, Multiple crack weight for solution of multiple interacting cracks by meshless numerical methods, International Journal for Numerical Methods in Engineering 67 pp 1146– (2006) · Zbl 1113.74094
[37] Rooke, Compendium of Stress Intensity Factors (1976)
[38] Houlsby, A tying scheme for imposing displacement constraints in finite element analysis, Communications in Numerical Methods in Engineering 16 pp 721– (2000) · Zbl 0993.74065
[39] Sethian, Level Set Methods and Fast Marching Methods (1999) · Zbl 0929.65066
[40] Duflot, A study of the representation of cracks with level sets, International Journal for Numerical Methods in Engineering 70 pp 1261– (2007) · Zbl 1194.74516
[41] Osher, Level Set Methods and Dynamic Implicit Surfaces (2003)
[42] Dolbow, An introdution to programming the meshless element-free Galerkin method, Archives of Computational Methods in Engineering 5 pp 207– (1998)
[43] Liu, Meshfree Methods: Moving Beyond the Finite Element Method (2003)
[44] Lee, An improved crack analysis technique by element-free Galerkin method with auxiliary supports, International Journal for Numerical Methods in Engineering 56 pp 1291– (2003) · Zbl 1106.74426
[45] Rice, A path independent integral and the approximate analysis of strain concentration by notches and cracks, Journal of Applied Mechanics 35 pp 379– (1968)
[46] Bordas, Derivative recovery and a posteriori error estimate for extended finite elements, Computer Methods in Applied Mechanics and Engineering 196 pp 3381– (2007) · Zbl 1173.74401
[47] Portela, The dual boundary element method-effective implementation for crack problems, International Journal for Numerical Methods in Engineering 33 pp 1269– (1992) · Zbl 0825.73908
[48] Shih, Energy-release rate along a 3-dimensional crack front in a thermally stressed body, International Journal of Fracture 30 pp 79– (1986)
[49] Nikishkov, Calculation of fracture-mechanics parameters for an arbitrary 3-dimensional crack, by the equivalent domain integral method, International Journal for Numerical Methods in Engineering 24 pp 1801– (1987) · Zbl 0625.73118
[50] Spencer, Continuum Mechanics (2003)
[51] Geuzaine, Gmsh: a three-dimensional finite element mesh generator with built-in pre-and post-processing facilities, International Journal for Numerical Methods in Engineering 79 pp 1309– (2009) · Zbl 1176.74181
[52] Rabczuk, Adaptivity for structured meshfree particle methods in 2D and 3D, International Journal for Numerical Methods in Engineering 63 pp 1559– (2005) · Zbl 1145.74041
[53] Liu, An adaptive procedure based on background cells for meshless methods, Computer Methods in Applied Mechanics and Engineering 191 pp 1923– (2002) · Zbl 1098.74738
[54] The Stress Intensity Factors Handbook (1987)
[55] Askes, Non-singular stresses in gradient elasticity at bi-material interface with transverse crack, International Journal of Fracture 156 (2) pp 217– (2009) · Zbl 1273.74423
[56] Eringen, Crack-tip problem in non-local elasticity, Journal of the Mechanics and Physics of Solids 25 pp 339– (1977) · Zbl 0375.73083
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