Nonnodal Extended Finite-Element Method for Crack Modeling with Four-Node Quadrilateral Elements
Publication: Journal of Engineering Mechanics
Volume 145, Issue 10
Abstract
A nonnodal extended finite-element method (NXFEM) was developed to predict two-dimensional dynamic failure with four-node quadrilateral elements. A new nonnodal enrichment scheme is presented such that the enriched quadrilateral finite elements satisfy the linear completeness and represent a strong discontinuity, i.e., crack. The enrichment bases for both solutions and the gradient fields are included in the finite-element approximation through the nonnodal enrichment scheme. The partition of unity is also naturally satisfied without adopting additional neighbor blending elements. To facilitate an implementation of the NXFEM into pre-existing finite-element analysis software, a generalized notation is proposed which is universally applicable to both quadrilateral and triangular finite elements. The developed method was initially verified with a convergence study on benchmark near-crack-tip field problems. The effectiveness of the method was further demonstrated with mixed-mode dynamic failure problems.
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References
Adalsteinsson, D., and J. A. Sethian. 1995. “A fast level set method for propagating interfaces.” J. Comput. Phys. 118 (2): 269–277. https://doi.org/10.1006/jcph.1995.1098.
Aragón, A. M., and A. Simone. 2017. “The discontinuity-enriched finite element method.” Int. J. Numer. Methods Eng. 112 (11): 1589–1613. https://doi.org/10.1002/nme.5570.
Areias, P., M. Msekh, and T. Rabczuk. 2016. “Damage and fracture algorithm using the screened Poisson equation and local remeshing.” Eng. Fract. Mech. 158 (Jun): 116–143. https://doi.org/10.1016/j.engfracmech.2015.10.042.
Areias, P., and T. Rabczuk. 2017. “Steiner-point free edge cutting of tetrahedral meshes with applications in fracture.” Finite Elem. Anal. Des. 132 (Sep): 27–41. https://doi.org/10.1016/j.finel.2017.05.001.
Areias, P., T. Rabczuk, and D. Dias-da Costa. 2013. “Element-wise fracture algorithm based on rotation of edges.” Eng. Fract. Mech. 110 (Sep): 113–137. https://doi.org/10.1016/j.engfracmech.2013.06.006.
Areias, P., J. Reinoso, P. Camanho, J. C. de Sá, and T. Rabczuk. 2018. “Effective 2D and 3D crack propagation with local mesh refinement and the screened Poisson equation.” Eng. Fract. Mech. 189 (Feb): 339–360. https://doi.org/10.1016/j.engfracmech.2017.11.017.
Asareh, I., T.-Y. Kim, and J.-H. Song. 2018a. “A linear complete extended finite element method for dynamic fracture simulation with non-nodal enrichments.” Finite Elem. Anal. Des. 152 (Nov): 27–45. https://doi.org/10.1016/j.finel.2018.09.002.
Asareh, I., Y.-C. Yoon, and J.-H. Song. 2018b. “A numerical method for dynamic fracture using the extended finite element method with non-nodal enrichment parameters.” Int. J. Impact Eng. 121 (Nov): 63–76. https://doi.org/10.1016/j.ijimpeng.2018.06.012.
Babuška, I., and J. M. Melenk. 1997. “The partition of unity method.” Int. J. Numer. Methods Eng. 40 (4): 727–758. https://doi.org/10.1002/(SICI)1097-0207(19970228)40:4%3C727::AID-NME86%3E3.0.CO;2-N.
Belytschko, T., and T. Black. 1999. “Elastic crack growth in finite elements with minimal remeshing.” Int. J. Numer. Methods Eng. 45 (5): 601–620. https://doi.org/10.1002/(SICI)1097-0207(19990620)45:5%3C601::AID-NME598%3E3.0.CO;2-S.
Belytschko, T., H. Chen, J. Xu, and G. Zi. 2003. “Dynamic crack propagation based on loss of hyperbolicity and a new discontinuous enrichment.” Int. J. Numer. Methods Eng. 58 (12): 1873–1905. https://doi.org/10.1002/nme.941.
Belytschko, T., N. Moës, S. Usui, and C. Parimi. 2001. “Arbitrary discontinuities in finite elements.” Int. J. Numer. Methods Eng. 50 (4): 993–1013. https://doi.org/10.1002/1097-0207(20010210)50:4%3C993::AID-NME164%3E3.0.CO;2-M.
Belytschko, T., and R. Mullen. 1976. “Mesh partitions of explicit-implicit time integration.” In Formulations and computational algorithms in finite element analysis, 673–690. Cambridge, MA: MIT Press.
Belytschko, T., and R. Mullen. 1978. “Stability of explicit-implicit mesh partitions in time integration.” Int. J. Numer. Methods Eng. 12 (10): 1575–1586. https://doi.org/10.1002/nme.1620121008.
Bordas, S., T. Rabczuk, and G. Zi. 2008. “Three-dimensional crack initiation, propagation, branching and junction in non-linear materials by an extended meshfree method without asymptotic enrichment.” Eng. Fract. Mech. 75 (5): 943–960. https://doi.org/10.1016/j.engfracmech.2007.05.010.
Chau-Dinh, T., G. Zi, P.-S. Lee, T. Rabczuk, and J.-H. Song. 2012. “Phantom-node method for shell models with arbitrary cracks.” Comput. Struct. 92 (Feb): 242–256. https://doi.org/10.1016/j.compstruc.2011.10.021.
Chessa, J., H. Wang, and T. Belytschko. 2003. “On the construction of blending elements for local partition of unity enriched finite elements.” Int. J. Numer. Methods Eng. 57 (7): 1015–1038. https://doi.org/10.1002/nme.777.
Dolbow, J., N. Moës, and T. Belytschko. 2000. “Discontinuous enrichment in finite elements with a partition of unity method.” Finite Elem. Anal. Des. 36 (3–4): 235–260. https://doi.org/10.1016/S0168-874X(00)00035-4.
Dolbow, J., N. Moës, and T. Belytschko. 2001. “An extended finite element method for modeling crack growth with frictional contact.” Comput. Methods Appl. Mech. Eng. 190 (51–52): 6825–6846. https://doi.org/10.1016/S0045-7825(01)00260-2.
Fries, T.-P. 2008. “A corrected XFEM approximation without problems in blending elements.” Int. J. Numer. Methods Eng. 75 (5): 503–532. https://doi.org/10.1002/nme.2259.
Fries, T.-P., and T. Belytschko. 2010. “The extended/generalized finite element method: An overview of the method and its applications.” Int. J. Numer. Methods Eng. 84 (3): 253–304. https://doi.org/10.1002/nme.2914.
Kalthoff, J., and S. Winkler. 1988. “Failure mode transition at high rates of shear loading.” In Vol. 1 of Impact loading and dynamic behavior of materials. 185–195. Oberursel, Germany: DGM Informationsgesellschaft mbH.
Kim, T. Y., J. Dolbow, and T. Laursen. 2007. “A mortared finite element method for frictional contact on arbitrary interfaces.” Comput. Mech. 39 (3): 223–235. https://doi.org/10.1007/s00466-005-0019-4.
Leon, S., D. Spring, and G. Paulino. 2014. “Reduction in mesh bias for dynamic fracture using adaptive splitting of polygonal finite elements.” Int. J. Numer. Methods Eng. 100 (8): 555–576. https://doi.org/10.1002/nme.4744.
Lua, J., T. Zhang, E. Fang, and J.-H. Song. 2016. “Explicit phantom paired shell element approach for crack branching and impact damage prediction of aluminum structures.” Int. J. Impact Eng. 87 (Jan): 28–43. https://doi.org/10.1016/j.ijimpeng.2015.07.007.
Melenk, J. M., and I. Babuška. 1996. “The partition of unity finite element method: Basic theory and applications.” Comput. Methods Appl. Mech. Eng. 139 (1–4): 289–314. https://doi.org/10.1016/S0045-7825(96)01087-0.
Menouillard, T., J. Rethore, A. Combescure, and H. Bung. 2006. “Efficient explicit time stepping for the extended finite element method (X-FEM).” Int. J. Numer. Methods Eng. 68 (9): 911–939. https://doi.org/10.1002/nme.1718.
Moës, N., J. Dolbow, and T. Belytschko. 1999. “A finite element method for crack growth without remeshing.” Int. J. Numer. Methods Eng. 46 (1): 131–150. https://doi.org/10.1002/(SICI)1097-0207(19990910)46:13.0.CO;2-J.
Paulino, G. H., K. Park, W. Celes, and R. Espinha. 2010. “Adaptive dynamic cohesive fracture simulation using nodal perturbation and edge-swap operators.” Int. J. Numer. Methods Eng. 84 (11): 1303–1343. https://doi.org/10.1002/nme.2943.
Rabczuk, T., and T. Belytschko. 2004. “Cracking particles: A simplified meshfree method for arbitrary evolving cracks.” Int. J. Numer. Methods Eng. 61 (13): 2316–2343. https://doi.org/10.1002/nme.1151.
Rabczuk, T., and T. Belytschko. 2007. “A three-dimensional large deformation meshfree method for arbitrary evolving cracks.” Comput. Methods Appl. Mech. Eng. 196 (29–30): 2777–2799. https://doi.org/10.1016/j.cma.2006.06.020.
Rabczuk, T., G. Zi, S. Bordas, and H. Nguyen-Xuan. 2010. “A simple and robust three-dimensional cracking-particle method without enrichment.” Comput. Methods Appl. Mech. Eng. 199 (37–40): 2437–2455. https://doi.org/10.1016/j.cma.2010.03.031.
Rittel, D., R. Levin, and H. Maigre. 1996. “A study of mixed-mode dynamic crack initiation in PMMA.” Mech. Res. Commun. 23 (5): 475–481. https://doi.org/10.1016/0093-6413(96)00047-X.
Rittel, D., and H. Maigre. 1996. “An investigation of dynamic crack initiation in PMMA.” Mech. Mater. 23 (3): 229–239. https://doi.org/10.1016/0167-6636(96)00014-2.
Sethian, J. A. 1996. “A fast marching level set method for monotonically advancing fronts.” Proc. Nat. Acad. Sci. 93 (4): 1591–1595. https://doi.org/10.1073/pnas.93.4.1591.
Soghrati, S., A. M. Aragón, C. Armando Duarte, and P. H. Geubelle. 2012. “An interface-enriched generalized FEM for problems with discontinuous gradient fields.” Int. J. Numer. Methods Eng. 89 (8): 991–1008. https://doi.org/10.1002/nme.3273.
Song, J.-H., P. M. Areias, and T. Belytschko. 2006. “A method for dynamic crack and shear band propagation with phantom nodes.” Int. J. Numer. Methods Eng. 67 (6): 868–893. https://doi.org/10.1002/nme.1652.
Song, J.-H., and T. Belytschko. 2009a. “Cracking node method for dynamic fracture with finite elements.” Int. J. Numer. Methods Eng. 77 (3): 360–385. https://doi.org/10.1002/nme.2415.
Song, J.-H., and T. Belytschko. 2009b. “Dynamic fracture of shells subjected to impulsive loads.” J. Appl. Mech. 76 (5): 051301. https://doi.org/10.1115/1.3129711.
Song, J.-H., and Y.-C. Yoon. 2014. “Multiscale failure analysis with coarse-grained micro cracks and damage.” Theor. Appl. Fract. Mech. 72 (Aug): 100–109. https://doi.org/10.1016/j.tafmec.2014.04.005.
Sukumar, N., D. L. Chopp, N. Moës, and T. Belytschko. 2001. “Modeling holes and inclusions by level sets in the extended finite-element method.” Comput. Methods Appl. Mech. Eng. 190 (46–47): 6183–6200. https://doi.org/10.1016/S0045-7825(01)00215-8.
Sun, H., H. Waisman, and R. Betti. 2014. “A multiscale flaw detection algorithm based on XFEM.” Int. J. Numer. Methods Eng. 100 (7): 477–503. https://doi.org/10.1002/nme.4741.
Tabarraei, A., J.-H. Song, and H. Waisman. 2013. “A two-scale strong discontinuity approach for evolution of shear bands under dynamic impact loads.” Int. J. Multiscale Comput. Eng. 11 (6): 543–563. https://doi.org/10.1615/IntJMultCompEng.2013005506.
Ventura, G., E. Budyn, and T. Belytschko. 2003. “Vector level sets for description of propagating cracks in finite elements.” Int. J. Numer. Methods Eng. 58 (10): 1571–1592. https://doi.org/10.1002/nme.829.
Wang, Y., and H. Waisman. 2017. “Material-dependent crack-tip enrichment functions in XFEM for modeling interfacial cracks in bimaterials.” Int. J. Numer. Methods Eng. 112 (11): 1495–1518. https://doi.org/10.1002/nme.5566.
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Published In
Journal of Engineering Mechanics
Volume 145 • Issue 10 • October 2019
Copyright
©2019 American Society of Civil Engineers.
History
Received: Oct 5, 2018
Accepted: Mar 13, 2019
Published online: Aug 6, 2019
Published in print: Oct 1, 2019
Discussion open until: Jan 6, 2020
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