##
**Non-planar 3D crack growth by the extended finite element and level sets. I: Mechanical model. II: Level set update.**
*(English)*
Zbl 1169.74621

Summary: A methodology for solving three-dimensional crack problems with geometries that are independent of the mesh is described. The method is based on the extended finite element method, in which the crack discontinuity is introduced as a Heaviside step function via a partition of unity. In addition, branch functions are introduced for all elements containing the crack front. The branch functions include asymptotic near-tip fields that improve the accuracy of the method. The crack geometry is described by two signed distance functions, which in turn can be defined by nodal values. Consequently, no explicit representation of the crack is needed. Examples for three-dimensional elastostatic problems are given and compared to analytic and benchmark solutions. The method is readily extendable to inelastic fracture problems.

In part II, we present a level set method for treating the growth of non-planar three-dimensional cracks. The crack is defined by two almost-orthogonal level sets (signed distance functions). One of them describes the crack as a two-dimensional surface in a three-dimensional space, and the second is used to describe the one-dimensional crack front, which is the intersection of the two level sets. A Hamilton-Jacobi equation is used to update the level sets. A velocity extension is developed that preserves the old crack surface and can accurately generate the growing surface. The technique is coupled with the extended finite element method which approximates the displacement field with a discontinuous partition of unity. This displacement field is constructed directly in terms of the level sets, so the discretization by finite elements requires no explicit representation of the crack surface. Numerical experiments show the robustness of the method, both in accuracy and in treating cracks with significant changes in topology.

In part II, we present a level set method for treating the growth of non-planar three-dimensional cracks. The crack is defined by two almost-orthogonal level sets (signed distance functions). One of them describes the crack as a two-dimensional surface in a three-dimensional space, and the second is used to describe the one-dimensional crack front, which is the intersection of the two level sets. A Hamilton-Jacobi equation is used to update the level sets. A velocity extension is developed that preserves the old crack surface and can accurately generate the growing surface. The technique is coupled with the extended finite element method which approximates the displacement field with a discontinuous partition of unity. This displacement field is constructed directly in terms of the level sets, so the discretization by finite elements requires no explicit representation of the crack surface. Numerical experiments show the robustness of the method, both in accuracy and in treating cracks with significant changes in topology.

### MSC:

74S05 | Finite element methods applied to problems in solid mechanics |

74S30 | Other numerical methods in solid mechanics (MSC2010) |

74R10 | Brittle fracture |

PDF
BibTeX
XML
Cite

\textit{N. Moës} et al., Int. J. Numer. Methods Eng. 53, No. 11, 2549--2586 (2002; Zbl 1169.74621)

Full Text:
DOI

### References:

[1] | Belytschko, Elastic crack growth in finite elements with minimal remeshing, International Journal for Numerical Methods in Engineering 45 (5) pp 601– (1999) · Zbl 0943.74061 |

[2] | Moës, A finite element method for crack growth without remeshing, International Journal for Numerical Methods in Engineering 46 pp 131– (1999) · Zbl 0955.74066 |

[3] | Dolbow, Discontinuous enrichment in finite elements with a partition of unity method, Finite Elements in Analysis and Design 36 pp 235– (2000) · Zbl 0981.74057 |

[4] | Melenk, The partition of unity finite element method: Basic theory and applications, Computer Methods in Applied Mechanics and Engineering 39 pp 289– (1996) · Zbl 0881.65099 |

[5] | Daux, Arbitrary branched and intersecting cracks with the extended finite element method, International Journal for Numerical Methods in Engineering 48 pp 1741– (2000) · Zbl 0989.74066 |

[6] | Belytschko, Arbitrary discontinuities in finite elements, International Journal for Numerical Methods in Engineering 50 pp 993– (2001) · Zbl 0981.74062 |

[7] | Sukumar, Modeling holes and inclusions by level sets in the extended finite element method, Computer Methods in Applied Mechanics and Engineering 190 pp 6183– (2001) · Zbl 1029.74049 |

[8] | Stolarska, Modelling crack growth by level sets and the extended finite element method, International Journal for Numerical Methods in Engineering 51 (8) pp 943– (2001) · Zbl 1022.74049 |

[9] | Sukumar, Extended finite element method for three-dimensional crack modelling, International Journal for Numerical Methods in Engineering 48 (11) pp 1549– (2000) · Zbl 0963.74067 |

[10] | Sukumar N Chopp DL Moran B Extended finite element method and fast marching method for three-dimensional fatigue crack propagation 2001 |

[11] | Carter, Automated 3d crack growth simulation, International Journal for Numerical Methods in Engineering 47 pp 229– (2000) · Zbl 0988.74079 |

[12] | Neto, An algorithm for three-dimensional mesh generation for arbitrary regions with cracks, Engineering with Computers 17 (2001) pp 75– (2001) · Zbl 1002.68527 |

[13] | Dhondt, Automatic 3-d mode in crack propagation calculations with finite elements, International Journal for Numerical Methods in Engineering 41 (4) pp 739– (1998) · Zbl 0902.73072 |

[14] | Gerstle, Finite and boundary element modeling of crack propagation in two- and three-dimensions, Engineering with Computers 2 pp 167– (1987) |

[15] | Nishioka, Analytical solution for embedded cracks, and finite element alternating method for elliptical surface cracks, subjected to arbitrary loading, Engineering Fracture Mechanics 17 pp 247– (1983) |

[16] | Duarte, A generalized finite method for the simulation of three-dimensional dynamic crack propagation, Computer Methods in Applied Mechanics and Engineering 190 pp 2227– (2001) · Zbl 1047.74056 |

[17] | Krysl, Element free Galerkin method for dynamic propagation of arbitrary 3-d cracks, International Journal for Numerical Methods in Engineering 44 (6) pp 767– (1999) · Zbl 0953.74078 |

[18] | Bolzon, Finite elements with embedded displacement discontinuity: a generalized formulation, International Journal for Numerical Methods in Engineering 49 (10) pp 1227– (2001) · Zbl 1004.74067 |

[19] | Jirasek, Embedded crack model: Part I basic formulation, International Journal for Numerical Methods in Engineering 50 (6) pp 1269– (2001) · Zbl 1013.74068 |

[20] | Remacle J-F Karamete BK Shephard M Algorithm oriented mesh database 349 359 2000 |

[21] | Remacle J-F Klaas O Flaherty J Shephard M Parallel algorithm oriented mesh database 2001 |

[22] | Moran, Crack tip and associated domain integrals from momentum and energy balance, Engineering Fracture Mechanics 127 pp 615– (1987) |

[23] | Gosz, Domain integral formulation for stress intensity factor computation along curved three-dimensional interface cracks, International Journal of Solids and Structures 35 pp 1763– (1998) · Zbl 0936.74057 |

[24] | Gosz M Moran B An interaction energy integral method for the computation of mixed-mode stress intensity factors along non-planar crack fronts in three dimensions 2001 |

[25] | Destuynder, Some remarks on elastic fracture mechanics (quelques remarques sur la mécanique de la rupture élastique), Journal de Mécanique théorique et appliquée 2 (1) pp 113– (1983) |

[26] | Belytschko, Nonlinear Finite Elements for Continua and Structures (2000) |

[27] | 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 |

[28] | Martynenko, Stress state near the vertex of a spherical notch in an unbounded elastic medium, Soviet Applied Mechanics 14 (2) pp 15– (1978) · Zbl 0435.73005 |

[29] | Singh, Universal crack closure integral for sif estimation, Engineering Fracture Mechanics 60 (2) pp 133– (1998) |

[30] | Kassir, Three-dimensional stress distribution around an elliptical crack under arbitrary loadings, IMA Journal of Applied Mathematics 33 pp 601– (1966) |

[31] | Remacle J-F Geuzaine C www.geuz.org/gmsh 1998 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.