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A three-dimensional meshfree method for continuous multiple-crack initiation, propagation and junction in statics and dynamics. (English) Zbl 1161.74054

Summary: 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
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[1] Alonso, A.; Valli, A., A domain decomposition approach for heterogenous time-harmonic maxwell equations, Comput Methods Appl Mech Eng, 143, 97-112 (1997) · Zbl 0883.65096
[2] Alonso, A.; Valli, A., An optimal domain decomposition preconditioner for low-frequency maxwell equations, Math Comput, 68, 607-631 (1999) · Zbl 1043.78554
[3] Areias, PMA; Belytschko, T., Analysis of three-dimensional crack initiation and propagation using the extended finite element method, Int J Numer Methods Eng, 63, 760-788 (2005) · 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., Residual based a posteriori error estimators for eddy current computation, Math Model Numer Anal, 34, 159-182 (2000) · Zbl 0949.65113
[6] Bellec, J.; Dolbow, JE, A note on enrichment functions for modelling crack nucleation, Commun Numer Methods Eng, 19, 921-932 (2003) · Zbl 1047.74536
[7] Belytschko, T.; Chen, H.; Xu, J.; Zi, G., Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment, Int J Numer Methods Eng, 58, 12, 1873-1905 (2003) · Zbl 1032.74662
[8] Belytschko, T.; Lu, YY; Gu, L., Element-free Galerkin methods, Int J Numer Methods Eng, 37, 229-256 (1994) · Zbl 0796.73077
[9] Belytschko, T.; Moes, N.; Usui, S.; Parimi, C., Arbitrary discontinuities in finite elements, Int J Numer Methods Eng, 50, 4, 993-1013 (2001) · 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, Spall fracture: Mechanical and microstructural aspects, Eng Fract Mech, 63, 273-294 (1999)
[15] Chopp, DL; Sukumar, N., Fatigue crack propagation of multiple coplanar cracks with the coupled extended finite element/fast marching method, Int J Eng Sci, 41, 845-869 (2003) · Zbl 1211.74199
[16] Daux, C.; Moes, N.; Dolbow, J.; Sukumar, N.; Belytschko, T., Arbitrary branched and intersection cracks with the extended finite element method, Int J Numer Methods Eng, 48, 1731-1760 (2000) · Zbl 0989.74066
[17] Demkovicz, L.; Vardapetyan, L., Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements, Comput Methods Appl Mech Eng, 152, 103-124 (1998) · Zbl 0994.78011
[18] Devloo, P.; Oden, TJ; Pattani, P., An h-p adaptive finite element method for the numerical simulation of compressible flow, Comput Methods Appl Mech Eng, 70, 2, 203-235 (1988) · Zbl 0636.76064
[19] Duflot, M., A meshless method with enriched weight functions for three-dimensional crack propagation, Int J Numer Methods Eng, 65, 12, 1970-2006 (2006) · Zbl 1114.74064
[20] Galdos, R., 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 (1997) · Zbl 0886.73063
[21] Gasser, TC; Holzapfel, GA, Modeling 3D crack propagation in unreinforced concrete using PUFEM, Comput Methods Appl Mech Eng, 194, 2859-2896 (2005) · Zbl 1176.74180
[22] Gravouil, A.; Moes, N.; Belytschko, T., 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 (2002) · Zbl 1169.74621
[23] Hiptmair, R., Multigrid method for maxwell’s equations, SIAM J Numer Anal, 36, 204-225 (1998) · Zbl 0922.65081
[24] Houston, P.; Perugia, H.; Schötzau, D., Energy norm a posteriori error estimation for mixed discontinuous galerkin approximations of the maxwell operator, Comput Methods Appl Mech Eng, 194, 499-510 (2005) · 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., Failure mode transition at high rates of shear loading, Int Conf Impact Load Dyn Behav Mater, 1, 185-195 (1987)
[27] Krysl, P.; Belytschko, T., The element free Galerkin method for dynamic propagation of arbitrary 3-D cracks, Int J Numer Methods Eng, 44, 6, 767-800 (1999) · 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., Meshfree simulation of ductile crack propagation, Int J Comput Methods Eng Sci Mech, 6, 1-19 (2004) · Zbl 1094.92029
[30] Liu, WK; Jun, S.; Li, S.; Adee, J.; Belytschko, T., Reproducing kernel particle method for structural dynamics, Int J Numer Methods Eng, 38, 1665-1679 (1995)
[31] Liu, WK; Li, S.; Belytschko, T., Moving least square reproducing kernel method. (I) methodology and convergence, Comput Methods Appl Mech Eng, 143, 113-154 (1997) · Zbl 0883.65088
[32] Liu, Y.; Murakami, S.; Kanagawa, Y., 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 (1994) · Zbl 0825.73743
[33] Lo, SH; Dong, CY; Cheung, YK, Integral equation approach for 3d multiple crack problems, Eng Fract Mech, 72, 1830-1840 (2005)
[34] Loehner, R., Applied CFD techniques: an introduction based on finite element methods (2001), New York: Wiley, New York
[35] Martha, LF; Wawrzynek, PA; Ingraffea, AR, Arbitrary crack representation using solid modeling, Eng Comput, 9, 63-82 (1993)
[36] Moes, N.; Dolbow, J.; Belytschko, T., A finite element method for crack growth without remeshing, Int J Numer Methods Eng, 46, 1, 133-150 (1999) · Zbl 0955.74066
[37] Moes, N.; Gravouil, A.; Belytschko, T., 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 (2002) · Zbl 1169.74621
[38] Monk, P., On the p- and hp-extension of nedelec’s curl-conforming elements, J Comput Appl Math, 53, 117-137 (1994) · Zbl 0820.65066
[39] Monk, P., A posteriori error indicators for maxwell’s equations, J Comput Appl Math, 100, 173-190 (1998) · Zbl 1023.78004
[40] Ogden, RW, Non-linear elastic deformations (1984), New York: 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., Adaptivity for structured meshfree particle methods in 2D and 3D, Int J Numer Methods Eng, 63, 11, 1559-1582 (2005) · Zbl 1145.74041
[44] Rabczuk, T.; Belytschko, T.; Xiao, SP, Stable particle methods based on lagrangian kernels, Comput Methods Appl Mech Eng, 193, 1035-1063 (2004) · Zbl 1060.74672
[45] Rachowicz, W.; Demkovicz, L., An hp-adaptive finite element method for electromagnetics—Part I: Data structure and constrained approximation, Comput Methods Appl Mech Eng, 187, 307-335 (2000) · Zbl 0979.78031
[46] Rachowicz, W.; Demkovicz, L., An hp-adaptive finite element method for electromagnetics-part ii: a 3d implementation, Int J Numer Methods Eng, 53, 147-180 (2002) · Zbl 0994.78012
[47] Simkins, DC Jr; Li, S., Meshfree simulations of ductile failure under thermal-mechanical loads, Comput Mech, 3, 235-249 (2006) · Zbl 1162.74052
[48] Sukumar, N.; Moran, B.; Black, T.; Belytschko, T., An element-free Galerkin method for three-dimensional fracture mechanics, Comput Mech, 20, 170-175 (1997) · Zbl 0888.73066
[49] Teng, X.; Wierzbicki, T.; Hiermaier, S.; Rohr, I., Numerical prediction of fracture in the Taylor test, Int J Solids Struct, 42, 1919-1948 (2005) · Zbl 1096.74517
[50] Vardapetyan, L.; Demkovicz, L., hp-adaptive finite elements in electromagnetics, Comput Methods Appl Mech Eng, 169, 331-344 (1999) · Zbl 0956.78013
[51] Ventura, G.; Xu, J.; Belytschko, T., A vector level set method and new discontinuity approximations for crack growth by EFG, Int J Numer Methods Eng, 54, 6, 923-944 (2002) · Zbl 1034.74053
[52] Xu, G.; Bower, FP; Ortiz, M., An analysis of non-planar crack growth under mixed mode loading, Int J Solids Struct, 31, 2167-2193 (1994) · Zbl 0946.74570
[53] Xu, G.; Ortiz, M., 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 (1993) · Zbl 0796.73067
[54] Xu, X-P; Needleman, A., Numerical simulations of fast crack growth in brittle solids, J Mech Phys Solids, 42, 1397-1434 (1994) · Zbl 0825.73579
[55] Zi, G.; Song, J-H; Budyn, E.; Lee, S-H; Belytschko, T., A method for growing multiple cracks without remeshing and its application to fatigue crack growth, Model Simul Mater Sci Eng, 12, 1, 901-915 (2004)
[56] Zienkiewicz, OC; Zhu, JZ, The superconvergent patch recovery and a posteriori error estimates. Part I: The recovery technique, Int J Numer Methods Eng, 33, 1331-1364 (1992) · Zbl 0769.73084
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