Abstract
The XFEM is a powerful method to handle strong discontinuities in a finite element environment, especially in the study of the final stages of material failure, modelling the propagation of cracks, suppressing the need of remeshing. Nevertheless, for some materials undergoing large strain processes without noticeable volume changes, the discretization technique employed must not only describe the material behaviour but also correctly address the incompressibility constraints. In order to develop a robust formulation for this type of problems, an approach based on the analyses of the underlying sub-space of incompressible deformations embedded in the XFEM approximation is used, in the context of both infinitesimal and finite strains. This study motivated the extension of the conventional formulations of B-bar and F-bar to include the XFEM enrichment functions, whose performance is evaluated through some numerical examples and compared with competing methods such as the enhanced strain formulation.
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References
Belytschko T, Black T (1999) Elastic crack growth in finite elements with minimal remeshing. Int J Numer Methods Eng 45: 601–620
Moës N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46: 131–150
Simone A, Wells G, Sluys L (2003) From continuous to discontinuous failure in a gradient-enhanced continuum damage model. Comput Methods Appl Mech Eng 192(41–42): 4581–4607
Benvenuti E (2008) A regularized XFEM framework for embedded cohesive interfaces. Comput Methods Appl Mech Eng 197: 4367–4378
Seabra M, de Sa JC, Andrade F, Pires F (2011) Continuous-discontinuous formulation for ductile fracture. Int J Mater Form 4(3): 271–281
Cazes F, Coret M, Combescure A, Gravouil A (2009) A thermodynamic method for the construction of a cohesive law from a non local damage model. Int J Solids Struct 46(6): 1476–1490
Belytschko T, Moës N, Usui S, Parimi C (2001) Arbitrary discontinuities in finite elements. Int J Numer Methods Eng 50(4): 993–1013
Sukumar N, Prevost JH (2003) Modeling quasi-static crack growth with the extended finite element method part I: computer implementation. Int J Solids Struct 40(26): 7513–7537
Strouboulis T, Copps K, Babuška I (2001) The generalized finite element method. Comput Methods Appl Mech Eng 190(32–33): 4081–4193
Elguedj T, Gravouil A, Combescure A (2006) Appropriate extended functions for XFEM simulation of plastic fracture mechanics. Comput Methods Appl Mech Eng 195(7–8): 501–515
Prabel B, Combescure A, Gravouil A, Marie S (2007) Level set XFEM non-matching meshes: application to dynamic crack propagation in elastic-plastic media. Int J Numer Methods Eng 69(8): 1553–1569
Iarve E (2003) Mesh independent modelling of cracks by using higher order shape functions. Int J Numer Methods Eng 56(6): 869–882
Ventura G (2006) On the elimination of quadrature subcells for discontinuous functions on the extended finite element method. Int J Numer Methods Eng 66(5): 767–795
Benvenuti E, Tralli A, Ventura G (2008) A regularized XFEM model for the transition from continuous to discontinuous displacements. Int J Numer Methods Eng 74(6): 911–944
Holdych D, Noble D, Secor R (2008) Quadrature rules for triangular and tetrahedral elements with generalized functions. Int J Numer Methods Eng 73(9): 1310–1327
Belytschko T, Gracie R, Ventura G (2009) A review of extended/generalized finite element methods for material modeling. Model Simul Mater Sci Eng 17(043001)
Ventura G, Gracie R, Belytschko T (2009) Fast integration and weight function blending in the extended finite element method. Int J Numer Methods Eng 77(1): 1–29
Natarajan S, Mahapatra D, Bordas S (2010) Integrating strong and weak discontinuities without integration subcells and example applications in a XFEM/GFEM framework. Int J Numer Methods Eng 83(3): 269–294
Mousavi S, Sukumar N (2010) Generalized quadrature rules for discontinuities and crack singularities in the extended finite element method. Comput Methods Appl Mech Eng 199(49–52): 3237–3249
Nagtegaal J, Parks D, Rice J (1974) On numerical accurate finite element solutions in the fully plastic range. Comput Methods Appl Mech Eng 4(2): 153–177
Malkus D, Hughes T (1978) Mixed finite element methods-reduced and selective integration techniques: a unification of concepts. Comput Methods Appl Mech Eng 15(1): 63–81
Hughes T (1980) Generalization of selective integration procedures to anisotropic and nonlinear media. Int J Numer Methods Eng 15(9): 1413–1418
Cesar de Sa J (1986) Numerical modeling of incompressible problems in glass forming and rubber technology. PhD Thesis, University College of Swansea
Cesar de Sa J, Jorge RN (1999) New enhanced strain elements for incompressible problems. Int J Numer Methods Eng 44: 229–248
Cesar de Sa J, Areias P, Jorge RN (2001) Quadrilteral elements for the solution of elasto-plastic finite strain problems. Int J Numer Methods Eng 51(8): 883–917
Simo JC, Taylor RL, Pister KS (1985) Variational and projection methods for the volume constraint in finite deformation elasto-plasticity. Comput Methods Appl Mech Eng 51(1-3): 177–208
de Souza Neto EA, Peric D, Dutko D, Owen DRJ (1996) Design of simple low order finite elements for large strain analysis of nearly incompressible solids. Int J Solids Struct 33(20-22): 3277–3296
de Souza Neto EA, Peric D, Owen D (2008) Computational methods for plasticity: theory and applications. Wiley, New York
Simo JC, Armero F (1992) Geometrically non-linear enhanced strain mixed methods and the method of incompatible modes. Int J Numer Methods Eng 33(7): 1413–1449
Dolbow JE, Devan A (2004) Enrichment of enhanced assumed strain approximations for representing strong discontinuities: addressing volumetric incompressibility and the discontinuous patch test. Int J Numer Methods Eng 59(1): 47–67
Legrain G, Moës N, Huerta A (2008) Stability of incompressible formulations enriched with X-FEM. Comput Methods Appl Mech Eng 197(21-24): 1835–1849
Stolarska M, Chopp DL, Moës N, Belytschko T (2001) Modelling crack growth by level sets in the extended finite element method. Int J Numer Methods Eng 51(8): 943–960
Dolbow J, Moës N, Belytschko T (2000) Discontinuous enrichment in finite elements with a partition of unity method. Finite Elem Anal Des 36(3-4): 235–260
Chessa J, Wang H, Belytschko T (2003) On the construction of blending elements for local partition of unity enriched finite elements. Int J Numer Methods Eng 57(7): 1015–1038
Fries TP (2008) A corrected XFEM approximation without problems in blending elements. Int J Numer Methods Eng 75(5): 503–532
Fries TP, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84(3): 253–304
Guibas L, Knuth D, Sharir M (1992) Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica 7: 381–413
Natarajan S, Bordas S, Mahapatra DR (2009) Numerical integration over arbitrary polygonal domains based on schwarz-christoffel conformal mapping. Int J Numer Methods Eng 80(1): 103–134
Driscoll A, Trefthen N (1996) Algorithm 756: a matlab toolbox for schwarz-christoffel mapping. ACM Trans Math Softw 22: 168–186
Moran B, Ortiz M, Shih F (1990) Formulation of implicit finite element methods for multiplicative finite deformation plasticity. Int J Numer Methods Eng 29(3): 438–514
Simo JC, Taylor RL (1991) Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms. Comput Methods Appl Mech Eng 85(3): 273–310
Khoei AR, Biabanaki S, Anahid M (2008) Extended finite element method for three-dimensional large plasticity deformations on arbitrary interfaces. Comput Methods Appl Mech Eng 197(9–12): 1100–1114
Loehnert S, Mueller-Hoeppe DS, Wriggers P (2011) 3d corrected XFEM approach and extension to finite deformation theory. Int J Numer Methods Eng 86(4–5): 431–452
Legrain G, Moës N, Verron E (2005) Stress analysis around crack tips in finite strain problems using the eXtended finite element method. Int J Numer Methods Eng 63(2): 290–314
Brink U, Stein E (1996) On some mixed finite element methods for incompressible and nearly incompressible finite elasticity. Comput Mech 19(1): 105–119
Piltner R, Taylor R (1999) A systematic construction of b-bar functions for linear and non-linear mixed-enhanced finite elements for plane elasticity problems. Int J Numer Methods Eng 44: 615–639
Korelc J (2009) Automation of primal and sensitivity analysis of transient coupled problems. Comput Mech 44(5): 631–639
Korelc J (2002) Multi-language and multi-environment generation of nonlinear finite element codes. Eng Comput 18(4): 312–327
Korelc J (2009) AceGen—Users manual. www.fgg.uni-lj.si/symech/
Korelc J (2009) AceFem—Users manual. www.fgg.uni-lj.si/symech/
Wang P (1986) FINGER: a symbolic system for automatic generation of numerical programs in finite element analysis. J Symb Comput 2(3): 305–316
Korelc J (1997) Automatic generation of finite element code by simultaneous optimization of expressions. Theor Comput Sci 187(1–2): 231–248
Flory P (1961) Thermodynamic relations for highly elastic materials. Trans Faraday Soc 57: 829–838
Simo J, Taylor R, Pister K (1985) Variational and projection methods for volume constraint in finite deformation elasto-plasticity. Comput Methods Appl Mech Eng 51: 177–208
Holzapfel G (1996) On large strain viscoelasticity: continuum formulation and finite element applications to elastomeric structures. Int J Numer Methods Eng 39(22): 3903–3926
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Seabra, M.R.R., Cesar de Sa, J.M.A., Šuštarič, P. et al. Some numerical issues on the use of XFEM for ductile fracture. Comput Mech 50, 611–629 (2012). https://doi.org/10.1007/s00466-012-0694-x
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DOI: https://doi.org/10.1007/s00466-012-0694-x