A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics.

*(English)*Zbl 1161.74054Summary: This paper proposes a three-dimensional meshfree method for arbitrary crack initiation and propagation that ensures crack path continuity for nonlinear material models and cohesive laws. The method is based on a local partition of unity. An extrinsic enrichment of the meshfree shape functions is used with discontinuous and near-front branch functions to close the crack front and improve accuracy. The crack is hereby modeled as a jump in the displacement field. The initiation and propagation of a crack is determined by the loss of hyperbolicity or the loss of material stability criterion. The method is applied to several static, quasi-static and dynamic crack problems. The numerical results very precisely replicate available experimental and analytical results.

##### MSC:

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

74R20 | Anelastic fracture and damage |

74R10 | Brittle fracture |

##### Keywords:

extended element-free Galerkin method; partition of unity enrichment; material stability criterion
PDF
BibTeX
Cite

\textit{T. Rabczuk} et al., Comput. Mech. 40, No. 3, 473--495 (2007; Zbl 1161.74054)

Full Text:
DOI

##### References:

[1] | Alonso A, Valli A (1997) A domain decomposition approach for heterogenous time-harmonic maxwell equations. Comput Methods Appl Mech Eng 143:97–112 · Zbl 0883.65096 |

[2] | Alonso A, Valli A (1999) An optimal domain decomposition preconditioner for low-frequency maxwell equations. Math Comput 68:607–631 · Zbl 1043.78554 |

[3] | Areias PMA, Belytschko T (2005) Analysis of three-dimensional crack initiation and propagation using the extended finite element method. Int J Numer Methods Eng 63:760–788 · Zbl 1122.74498 |

[4] | Arrea M, Ingraffea AR (1982) Mixed-mode crack propagation in mortar and concrete. Technical Report 81-13, Department of Structural Engineering Cornell University Ithaka |

[5] | Beck R, Hiptmair R, Hoppe RHW, Wohlmuth B (2000) Residual based a posteriori error estimators for eddy current computation. Math Model Numer Anal 34:159–182 · Zbl 0949.65113 |

[6] | Bellec J, Dolbow JE (2003) A note on enrichment functions for modelling crack nucleation. Commun Numer Methods Eng 19:921–932 · Zbl 1047.74536 |

[7] | Belytschko T, Chen H, Xu J, Zi G (2003) Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment. Int J Numer Methods Eng 58(12):1873–1905 · Zbl 1032.74662 |

[8] | Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37:229–256 · Zbl 0796.73077 |

[9] | Belytschko T, Moes N, Usui S, Parimi C (2001) Arbitrary discontinuities in finite elements. Int J Numer Methods Eng 50(4):993–1013 · Zbl 0981.74062 |

[10] | Bordas S (2003) Extended finite element and level set methods with applications to growth of cracks and biofilms. PhD Thesis, Northwestern University |

[11] | Bordas S, Moran B (2006) Extended finite element and level set method for damage tolerance assessment of complex structures: an object-oriented approach. EFM (in press) |

[12] | Bordas S, Legay A (2005) Enriched finite element short course: class notes. In: The extended finite element method, a new approach to numerical analysis in mechanics: course notes. Organized by S. Bordas and A. Legay through the EPFL school of continuing education, Lausanne, Switzerland |

[13] | Cervenka J (1994) Discrete crack modeling in concrete structures. PhD Thesis, University of Colorado |

[14] | Chevrier P, Klepaczko JR (1999) Spall fracture: Mechanical and microstructural aspects. Eng Fract Mech 63:273–294 |

[15] | Chopp DL, Sukumar N (2003) Fatigue crack propagation of multiple coplanar cracks with the coupled extended finite element/fast marching method. Int J Eng Sci 41:845–869 · Zbl 1211.74199 |

[16] | Daux C, Moes N, Dolbow J, Sukumar N, Belytschko T (2000) Arbitrary branched and intersection cracks with the extended finite element method. Int J Numer Methods Eng 48:1731–1760 · Zbl 0989.74066 |

[17] | Demkovicz L, Vardapetyan L (1998) Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements. Comput Methods Appl Mech Eng 152:103–124 · Zbl 0994.78011 |

[18] | Devloo P, Oden TJ, Pattani P (1988) An h-p adaptive finite element method for the numerical simulation of compressible flow. Comput Methods Appl Mech Eng 70(2):203–235 · Zbl 0636.76064 |

[19] | Duflot M (2006) A meshless method with enriched weight functions for three-dimensional crack propagation. Int J Numer Methods Eng 65(12):1970–2006 · Zbl 1114.74064 |

[20] | Galdos R (1997) A finite element technique to simulate the stable shape evolution of planar cracks with an application to a semi-elliptical surface crack in a bimaterial finite solid. Int J Numer Methods Eng 40:905–917 · Zbl 0886.73063 |

[21] | Gasser TC, Holzapfel GA (2005) Modeling 3D crack propagation in unreinforced concrete using PUFEM. Comput Methods Appl Mech Eng 194:2859–2896 · Zbl 1176.74180 |

[22] | Gravouil A, Moes N, Belytschko T (2002) Non-planar 3D crack growth by the extended finite element and level sets–Part II: Level set update. Int J Numer Methods Eng 53:2569–2586 · Zbl 1169.74621 |

[23] | Hiptmair R (1998) Multigrid method for maxwell’s equations. SIAM J Numer Anal 36:204–225 · Zbl 0922.65081 |

[24] | Houston P, Perugia H, Schötzau D (2005) Energy norm a posteriori error estimation for mixed discontinuous galerkin approximations of the maxwell operator. Comput Methods Appl Mech Eng 194:499–510 · Zbl 1063.78021 |

[25] | Johnson GR, Cook WH (1983) A constitutive model and data for metals subjected to large strains, high strain rates, and high temperatures. In: Proceedings of 7th International Symposium on Ballistics |

[26] | Kalthoff JF, Winkler S (1987) Failure mode transition at high rates of shear loading. Int Conf Impact Load Dyn Behav Mater 1:185–195 |

[27] | Krysl P, Belytschko T (1999) The element free Galerkin method for dynamic propagation of arbitrary 3-D cracks. Int J Numer Methods Eng 44(6):767–800 · Zbl 0953.74078 |

[28] | Lemaitre J (1971) Evaluation of dissipation and damage in metal submitted to dynamic loading. In: Proceedings ICM 1 |

[29] | Li S, Simonson Bo C (2004) Meshfree simulation of ductile crack propagation. Int J Comput Methods Eng Sci Mech 6:1–19 |

[30] | Liu WK, Jun S, Li S, Adee J, Belytschko T (1995) Reproducing kernel particle method for structural dynamics. Int J Numer Methods Eng 38:1665–1679 · Zbl 0840.73078 |

[31] | Liu WK, Li S, Belytschko T (1997) Moving least square reproducing kernel method. (I) methodology and convergence. Comput Methods Appl Mech Eng 143:113–154 · Zbl 0883.65088 |

[32] | Liu Y, Murakami S, Kanagawa Y (1994) Mesh-dependence and stress singularity in finite element analysis of creep crack growth by continuum damage mechanics approach. Eur J Mech A/Solids 13:395–417 · Zbl 0825.73743 |

[33] | Lo SH, Dong CY, Cheung YK (2005) Integral equation approach for 3d multiple crack problems. Eng Fract Mech 72:1830–1840 |

[34] | Loehner R (2001) Applied CFD techniques: an introduction based on finite element methods. Wiley, New York |

[35] | Martha LF, Wawrzynek PA, Ingraffea AR (1993) Arbitrary crack representation using solid modeling. Eng Comput 9:63–82 |

[36] | Moes N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):133–150 · Zbl 0955.74066 |

[37] | Moes N, Gravouil A, Belytschko T (2002) Non-planar 3-D crack growth by the extended finite element method and level sets, Part I: Mechanical model. Int J Numer Methods Eng 53(11):2549–2568 · Zbl 1169.74621 |

[38] | Monk P (1994) On the p- and hp-extension of nedelec’s curl-conforming elements. J Comput Appl Math 53:117–137 · Zbl 0820.65066 |

[39] | Monk P (1998) A posteriori error indicators for maxwell’s equations. J Comput Appl Math 100:173–190 · Zbl 1023.78004 |

[40] | Ogden RW (1984) Non-linear elastic deformations. Halsted, New York |

[41] | Oliver J, Huespe AE, Snchez PJ (2006) A comparative study on finite elements for capturing strong discontinuities: E-fem vs. x-fem. Comput Methods Appl Mech Eng (in press) · Zbl 1144.74043 |

[42] | Rabczuk T, Areias PMA, Belytschko T (2006) A simplified meshfree method for shear bands with cohesive surfaces. Int J Numer Methods Eng (submitted) |

[43] | Rabczuk T, Belytschko T (2005) Adaptivity for structured meshfree particle methods in 2D and 3D. Int J Numer Methods Eng 63(11):1559–1582 · Zbl 1145.74041 |

[44] | Rabczuk T, Belytschko T, Xiao SP (2004) Stable particle methods based on lagrangian kernels. Comput Methods Appl Mech Eng 193:1035–1063 · Zbl 1060.74672 |

[45] | Rachowicz W, Demkovicz L (2000) An hp-adaptive finite element method for electromagnetics–Part I: Data structure and constrained approximation. Comput Methods Appl Mech Eng 187:307–335 · Zbl 0979.78031 |

[46] | Rachowicz W, Demkovicz L (2002) An hp-adaptive finite element method for electromagnetics-part ii: a 3d implementation. Int J Numer Methods Eng 53:147–180 · Zbl 0994.78012 |

[47] | Simkins DC Jr, Li S (2006) Meshfree simulations of ductile failure under thermal-mechanical loads. Comput Mech 3:235–249 |

[48] | Sukumar N, Moran B, Black T, Belytschko T (1997) An element-free Galerkin method for three-dimensional fracture mechanics. Comput Mech 20:170–175 · Zbl 0888.73066 |

[49] | Teng X, Wierzbicki T, Hiermaier S, Rohr I (2005) Numerical prediction of fracture in the Taylor test. Int J Solids Struct 42:1919–1948 · Zbl 1096.74517 |

[50] | Vardapetyan L, Demkovicz L (1999) hp-adaptive finite elements in electromagnetics. Comput Methods Appl Mech Eng 169:331–344 · Zbl 0956.78013 |

[51] | Ventura G, Xu J, Belytschko T (2002) A vector level set method and new discontinuity approximations for crack growth by EFG. Int J Numer Methods Eng 54(6):923–944 · Zbl 1034.74053 |

[52] | Xu G, Bower FP, Ortiz M (1994) An analysis of non-planar crack growth under mixed mode loading. Int J Solids Struct 31:2167–2193 · Zbl 0946.74570 |

[53] | Xu G, Ortiz M (1993) A variational boundary integral method for the analysis of 3D cracks of arbitrary geometry modelled as continuous distributions of dislocation loops. Int J Numer Methods Eng 36:3675–3701 · Zbl 0796.73067 |

[54] | Xu X-P, Needleman A (1994) Numerical simulations of fast crack growth in brittle solids. J Mech Phys Solids 42:1397–1434 · Zbl 0825.73579 |

[55] | Zi G, Song J-H, Budyn E, Lee S-H, Belytschko T (2004) A method for growing multiple cracks without remeshing and its application to fatigue crack growth. Model Simul Mater Sci Eng 12(1):901–915 |

[56] | Zienkiewicz OC, Zhu JZ (1992) The superconvergent patch recovery and a posteriori error estimates. Part I: The recovery technique. Int J Numer Methods Eng 33:1331–1364 · Zbl 0769.73084 |

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.