NOTICE: this is the author’s version of a work that was accepted for publication in International Journal of Plasticity. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in International Journal of Plasticity 127 (2020) 102631, DOI: 10.1016/j.ijplas.2019.11.010
All documents in ORBi are protected by a user license.
Ductile fracture; Cohesive band model; Damage to crack transition; Discontinuous Galerkin; Porous plasticity
Abstract :
[en] The failure process of ductile porous materials is simulated by representing the damage nucleation, growth and coalescence stages up to crack initiation and propagation using a physically-based constitutive model. In particular, a non-local damage to crack transition
framework is developed to predict the fracture under various loading conditions while minimising case-dependent calibration process. The formulation is based on a discontinuous Galerkin method, making it computationally efficient and scalable. The initial stable damage
process is simulated using an implicit non-local damage model ensuring solution uniqueness beyond the onset of softening relying on the Gurson-Tvergaard-Needleman (GTN) model.
Once the coalescence criterion is satisfied, which can physically arise before or during the
softening stage, a cohesive band is introduced. Within the cohesive band, a void coalescence-based governing law is solved, accounting for the stress triaxiality state and material history,
in order to capture the near crack tip failure process in a micro-mechanically sound way. Two
coalescence models are then successively considered and compared. First, with a view to
model verification towards literature results, a numerical coalescence model detects crack
initiation at loss of ellipticity of a local model, and the crack opening is governed by ad-hoc parameters of the GTN model. Alternatively, the Thomason criterion is used to detect crack nucleation during the softening stage while the Thomason coalescence model governs the crack opening process. This latter model is able to reproduce slant and cup-cone failure modes in plane-strain and axisymmetric specimens, respectively.
S.C. Aduloju, and T.J. Truster A variational multiscale discontinuous galerkin formulation for both implicit and explicit dynamic modeling of interfacial fracture Comput. Methods Appl. Mech. Eng. 343 2019 602 630 10.1016/j.cma.2018.08.025 http://www.sciencedirect.com/science/article/pii/S0045782518304201
F. Aldakheel Micromorphic approach for gradient-extended thermo-elastic-plastic solids in the logarithmic strain space Continuum Mech. Therm. 29 2017 1207 1217 10.1007/s00161-017-0571-0 http://link.springer.com/10.1007/s00161-017-0571-0
F. Aldakheel, P. Wriggers, and C. Miehe A modified gurson-type plasticity model at finite strains: formulation, numerical analysis and phase-field coupling Comput. Mech. 62 2018 815 833 10.1007/s00466-017-1530-0
G. Barenblatt The Mathematical Theory of Equilibrium Cracks in Brittle Fracture 1962 Elsevier 55 129 10.1016/S0065-2156(08)70121-2 volume 7 of Advances in Applied Mechanics http://www.sciencedirect.com/science/article/pii/S0065215608701212
Z.P. Bažant, T.B. Belytschko, and T.P. Chang Continuum theory for strain-softening J. Eng. Mech. 110 1984 1666 1692
G. Becker, and L. Noels A full-discontinuous galerkin formulation of nonlinear Kirchhoff-love shells: elasto-plastic finite deformations, parallel computation, and fracture applications Int. J. Numer. Methods Eng. 93 2012 80 117 10.1002/nme.4381 https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.4381 https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.4381
A. Benzerga Micromechanics of coalescence in ductile fracture J. Mech. Phys. Solids 50 2002 1331 1362 10.1016/S0022-5096(01)00125-9 http://www.sciencedirect.com/science/article/pii/S0022509601001259
A. Benzerga, J. Besson, and A. Pineau Anisotropic ductile fracture: Part ii: theory Acta Mater. 52 2004 4639 4650 10.1016/j.actamat.2004.06.019 http://www.sciencedirect.com/science/article/pii/S135964540400357X
A.A. Benzerga, and J.B. Leblond Ductile fracture by void growth to coalescence H. Aref, E. van der Giessen, Advances in Applied Mechanics 2010 Elsevier 169 305 10.1016/S0065-2156(10)44003-X volume 44 of Advances in Applied Mechanics http://www.sciencedirect.com/science/article/pii/S006521561044003X
A.A. Benzerga, and J.B. Leblond Effective yield criterion accounting for microvoid coalescence J. Appl. Mech. 81 2014 031009 10.1115/1.4024908
A.A. Benzerga, J.B. Leblond, A. Needleman, and V. Tvergaard Ductile failure modeling Int. J. Fract. 201 2016 29 80 10.1007/s10704-016-0142-6
J. Besson Damage of ductile materials deforming under multiple plastic or viscoplastic mechanisms Int. J. Plast. 25 2009 2204 2221 10.1016/j.ijplas.2009.03.001 http://www.sciencedirect.com/science/article/pii/S0749641909000357
J. Besson Continuum models of ductile fracture: a review Int. J. Damage Mech. 19 2010 3 52 10.1177/1056789509103482
J. Besson, D. Steglich, and W. Brocks Modeling of crack growth in round bars and plane strain specimens Int. J. Solids Struct. 38 2001 8259 8284 10.1016/S0020-7683(01)00167-6 http://www.sciencedirect.com/science/article/pii/S0020768301001676
J. Besson, D. Steglich, and W. Brocks Modeling of plane strain ductile rupture Int. J. Plast. 19 2003 1517 1541 10.1016/S0749-6419(02)00022-0 http://www.sciencedirect.com/science/article/pii/S0749641902000220
R. de Borst, and C.V. Verhoosel Gradient damage vs phase-field approaches for fracture: similarities and differences Comput. Methods Appl. Mech. Eng. 312 2016 78 94 10.1016/j.cma.2016.05.015. phase Field Approaches to Fracture http://www.sciencedirect.com/science/article/pii/S0045782516303796
E. Bosco, V.G. Kouznetsova, and M.G.D. Geers Multi-scale computational homogenization-localization for propagating discontinuities using x-fem Int. J. Numer. Methods Eng. 102 2015 496 527 10.1002/nme.4838 https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.4838 https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.4838
W. Brocks, D.Z. Sun, and A. Hönig Verification of the transferability of micromechanical parameters by cell model calculations with visco-plastic materials Int. J. Plast. 11 1995 971 989 10.1016/S0749-6419(95)00039-9 http://www.sciencedirect.com/science/article/pii/S0749641995000399
M. Brünig, S. Gerke, and M. Schmidt Damage and failure at negative stress triaxialities: experiments, modeling and numerical simulations Int. J. Plast. 102 2018 70 82 10.1016/j.ijplas.2017.12.003 http://www.sciencedirect.com/science/article/pii/S0749641917305764
G. Camacho, and M. Ortiz Computational modelling of impact damage in brittle materials Int. J. Solids Struct. 33 1996 2899 2938 10.1016/0020-7683(95)00255-3 http://www.sciencedirect.com/science/article/pii/0020768395002553
F. Cazes, M. Coret, A. Combescure, and A. Gravouil A thermodynamic method for the construction of a cohesive law from a nonlocal damage model Int. J. Solids Struct. 46 2009 1476 1490 10.1016/j.ijsolstr.2008.11.019 http://www.sciencedirect.com/science/article/pii/S0020768308004903
F. Cazes, A. Simatos, M. Coret, and A. Combescure A cohesive zone model which is energetically equivalent to a gradient-enhanced coupled damage-plasticity model Eur. J. Mech. A Solid. 29 2010 976 989 10.1016/j.euromechsol.2009.11.003 http://www.sciencedirect.com/science/article/pii/S0997753809001338
C. Chu, and A. Needleman Void nucleation effects in biaxially stretched sheets J. Eng. Mater. Technol. 102 1980 249 256
J. Chung, and G. Hulbert A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-α method J. Appl. Mech. 60 1993 371 375 10.1115/1.2900803
C. Comi, S. Mariani, and U. Perego An extended fe strategy for transition from continuum damage to mode i cohesive crack propagation Int. J. Numer. Anal. Methods Geomech. 31 2006 213 238 10.1002/nag.537 https://onlinelibrary.wiley.com/doi/abs/10.1002/nag.537 https://onlinelibrary.wiley.com/doi/pdf/10.1002/nag.537
L. Cortese, F. Nalli, and M. Rossi A nonlinear model for ductile damage accumulation under multiaxial non-proportional loading conditions Int. J. Plast. 85 2016 77 92 10.1016/j.ijplas.2016.07.003 http://www.sciencedirect.com/science/article/pii/S0749641916301036
A. Cuitino, and M. Ortiz A material-independent method for extending stress update algorithms from small-strain plasticity to finite plasticity with multiplicative kinematics Eng. Comput. 9 1992 437 451
S. Cuvilliez, F. Feyel, E. Lorentz, and S. Michel-Ponnelle A finite element approach coupling a continuous gradient damage model and a cohesive zone model within the framework of quasi-brittle failure Comput. Methods Appl. Mech. Eng. 237-240 2012 244 259 10.1016/j.cma.2012.04.019 http://www.sciencedirect.com/science/article/pii/S0045782512001466
D. Dugdale Yielding of steel sheets containing slits J. Mech. Phys. Solids 8 1960 100 104 10.1016/0022-5096(60)90013-2 http://www.sciencedirect.com/science/article/pii/0022509660900132
M. Dunand, and D. Mohr Optimized butterfly specimen for the fracture testing of sheet materials under combined normal and shear loading Eng. Fract. Mech. 78 2011 2919 2934 10.1016/j.engfracmech.2011.08.008 http://www.sciencedirect.com/science/article/pii/S0013794411003067
A. Esmaeili, P. Steinmann, and A. Javili Non-coherent energetic interfaces accounting for degradation Comput. Mech. 59 2017 361 383 10.1007/s00466-016-1342-7
J. Faleskog, X. Gao, and C.F. Shih Cell model for nonlinear fracture analysis-i. micromechanics calibration Int. J. Fract. 89 1998 355 373 10.1023/A:1007421420901 URL
S. Forest Micromorphic approach for gradient elasticity, viscoplasticity, and damage J. Eng. Mech. 135 2009 117 131 10.1061/(ASCE)0733-9399(2009)135:3(117) http://ascelibrary.org/doi/10.1061/%7b%25%7d28ASCE%7b%25%7d290733-9399%7b%25%7d282009%7b%25%7d29135%7b%25%7d3A3%7b%25%7d28117%7b%25%7d29
M. Geers, R. de Borst, W. Brekelmans, and R. Peerlings Strain-based transient-gradient damage model for failure analyses Comput. Methods Appl. Mech. Eng. 160 1998 133 153 10.1016/S0045-7825(98)80011-X http://www.sciencedirect.com/science/article/pii/S004578259880011X
C. Geuzaine, and J.F. Remacle Gmsh: a 3-d finite element mesh generator with built-in pre-and post-processing facilities Int. J. Numer. Methods Eng. 79 2009 1309 1331 10.1002/nme.2579 https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.2579 https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.2579
A.L. Gurson Continuum theory of ductile rupture by void nucleation and growth: Part i-yield criteria and flow rules for porous ductile media J. Eng. Mater. Technol. 99 1977 2 15 10.1115/1.3443401
C.F. Guzmán, S. Yuan, L. Duchêne, E.I.S. Flores, and A.M. Habraken Damage prediction in single point incremental forming using an extended gurson model Int. J. Solids Struct. 151 2018 45 56 10.1016/j.ijsolstr.2017.04.013. special Issue with a Selection of Papers from the NUMISHEET 2016 Conference http://www.sciencedirect.com/science/article/pii/S0020768317301646
A. Hillerborg, M. Modéer, and P.E. Petersson Analysis of crack formation and crack growth in concrete by means of fracture mechanics and finite elements Cement Concr. Res. 6 1976 773 781 10.1016/0008-8846(76)90007-7 http://www.sciencedirect.com/science/article/pii/0008884676900077
A. Huespe, A. Needleman, J. Oliver, and P. Sánchez A finite thickness band method for ductile fracture analysis Int. J. Plast. 25 2009 2349 2365 10.1016/j.ijplas.2009.03.005 http://www.sciencedirect.com/science/article/pii/S0749641909000461
A. Huespe, A. Needleman, J. Oliver, and P. Sánchez A finite strain, finite band method for modeling ductile fracture Int. J. Plast. 28 2012 53 69 10.1016/j.ijplas.2011.05.010 http://www.sciencedirect.com/science/article/pii/S0749641911000866
G.R. Johnson, and W.H. Cook Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures Eng. Fract. Mech. 21 1985 31 48 10.1016/0013-7944(85)90052-9 https://linkinghub.elsevier.com/retrieve/pii/0013794485900529
S. Keralavarma, and S. Chockalingam A criterion for void coalescence in anisotropic ductile materials Int. J. Plast. 82 2016 159 176 10.1016/j.ijplas.2016.03.003 http://www.sciencedirect.com/science/article/pii/S0749641916300274
S.M. Keralavarma A multi-surface plasticity model for ductile fracture simulations J. Mech. Phys. Solids 103 2017 100 120 10.1016/j.jmps.2017.03.005 http://www.sciencedirect.com/science/article/pii/S0022509616307359
J. Koplik, and A. Needleman Void growth and coalescence in porous plastic solids Int. J. Solids Struct. 24 1988 835 853 10.1016/0020-7683(88)90051-0 http://www.sciencedirect.com/science/article/pii/0020768388900510
J.B. Leblond, and G. Mottet A theoretical approach of strain localization within thin planar bands in porous ductile materials Compt. Rendus Mec. 336 2008 176 189 10.1016/j.crme.2007.11.008. duality, inverse problems and nonlinear problems in solid mechanics http://www.sciencedirect.com/science/article/pii/S1631072107002215
J. Leclerc, L. Wu, V.D. Nguyen, and L. Noels A damage to crack transition model accounting for stress triaxiality formulated in a hybrid nonlocal implicit discontinuous galerkin-cohesive band model framework Int. J. Numer. Methods Eng. 113 2018 374 410 10.1002/nme.5618.nme.5618
J. Lemaitre Local approach of fracture Eng. Fract. Mech. 25 1986 523 537 10.1016/0013-7944(86)90021-4 http://www.sciencedirect.com/science/article/pii/0013794486900214
C. Linder, and F. Armero Finite elements with embedded strong discontinuities for the modeling of failure in solids Int. J. Numer. Methods Eng. 72 2007 1391 1433 10.1002/nme.2042 https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.2042 https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.2042
Z. Liu, W. Wong, and T. Guo Void behaviors from low to high triaxialities: transition from void collapse to void coalescence Int. J. Plast. 84 2016 183 202 10.1016/j.ijplas.2016.05.008 http://www.sciencedirect.com/science/article/pii/S0749641916300821
J. Mediavilla, R.H.J. Peerlings, and M.G.D. Geers Discrete crack modelling of ductile fracture driven by non-local softening plasticity Int. J. Numer. Methods Eng. 66 2005 661 688 10.1002/nme.1572 https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.1572 https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.1572
J. Mergheim, E. Kuhl, and P. Steinmann A hybrid discontinuous galerkin/interface method for the computational modelling of failure Commun. Numer. Methods Eng. 20 2004 511 519 10.1002/cnm.689 https://onlinelibrary.wiley.com/doi/abs/10.1002/cnm.689 https://onlinelibrary.wiley.com/doi/pdf/10.1002/cnm.689
C. Miehe, F. Aldakheel, and A. Raina Phase field modeling of ductile fracture at finite strains: a variational gradient-extended plasticity-damage theory Int. J. Plast. 84 2016 1 32 10.1016/j.ijplas.2016.04.011 http://www.sciencedirect.com/science/article/pii/S0749641916300602
C. Miehe, D. Kienle, F. Aldakheel, and S. Teichtmeister Phase field modeling of fracture in porous plasticity: a variational gradient-extended Eulerian framework for the macroscopic analysis of ductile failure Comput. Methods Appl. Mech. Eng. 312 2016 3 50 10.1016/j.cma.2016.09.028 https://linkinghub.elsevier.com/retrieve/pii/S0045782516305412
N. Moës, J. Dolbow, and T. Belytschko A finite element method for crack growth without remeshing Int. J. Numer. Methods Eng. 46 1999 131 150
A. Mota, J. Knap, and M. Ortiz Fracture and fragmentation of simplicial finite element meshes using graphs Int. J. Numer. Methods Eng. 73 2008 1547 1570 10.1002/nme.2135
K. Nahshon, and J. Hutchinson Modification of the gurson model for shear failure Eur. J. Mech. A Solid. 27 2008 1 17 10.1016/j.euromechsol.2007.08.002 http://www.sciencedirect.com/science/article/pii/S0997753807000721
G.D. Nguyen, A.M. Korsunsky, and J.P.H. Belnoue A nonlocal coupled damage-plasticity model for the analysis of ductile failure Int. J. Plast. 64 2015 56 75 10.1016/j.ijplas.2014.08.001 http://www.sciencedirect.com/science/article/pii/S0749641914001491
Nguyen, V.D., Pardoen, T., Noels, L., Submitted. An implicit nonlocal approach of ductile failure incorporating void growth, internal necking coalescence, and shear coalescence mechanisms. J. Mech. Phys. Solids.
K.L. Nielsen, and V. Tvergaard Ductile shear failure or plug failure of spot welds modelled by modified gurson model Eng. Fract. Mech. 77 2010 1031 1047 10.1016/j.engfracmech.2010.02.031 http://www.sciencedirect.com/science/article/pii/S0013794410001128
L. Noels, and R. Radovitzky A general discontinuous galerkin method for finite hyperelasticity. formulation and numerical applications Int. J. Numer. Methods Eng. 68 2006 64 97 10.1002/nme.1699 https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.1699 https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.1699
T. Pardoen, and J. Hutchinson An extended model for void growth and coalescence J. Mech. Phys. Solids 48 2000 2467 2512 10.1016/S0022-5096(00)00019-3 http://www.sciencedirect.com/science/article/pii/S0022509600000193
T. Pardoen, F. Scheyvaerts, A. Simar, C. Tekoǧlu, and P.R. Onck Multiscale modeling of ductile failure in metallic alloys Compt. Rendus Phys. 11 2010 326 345 10.1016/j.crhy.2010.07.012. computational metallurgy and scale transitions http://www.sciencedirect.com/science/article/pii/S1631070510000770
G. Paulino, W. Celes, R. Espinha, and Z. Zhang A general topology-based framework for adaptive insertion of cohesive elements in finite element meshes Eng. Comput. 24 2008 59 78 10.1007/s00366-007-0069-7. 10.1007/s00366-007-0069-7
R. Peerlings, M. Geers, R. de Borst, and W. Brekelmans A critical comparison of nonlocal and gradient-enhanced softening continua Int. J. Solids Struct. 38 2001 7723 7746 10.1016/S0020-7683(01)00087-7 http://www.sciencedirect.com/science/article/pii/S0020768301000877
R.H.J. Peerlings, R. de Borst, W.A.M. Brekelmans, and J.H.P. De Vree Gradient enhanced damage for quasi-brittle materials Int. J. Numer. Methods Eng. 39 1996 3391 3403 10.1002/(SICI)1097-0207(19961015)39:19<3391::AID-NME7>3.0.CO;2-D
A. Pineau, A. Benzerga, and T. Pardoen Failure of metals i: brittle and ductile fracture Acta Mater. 107 2016 424 483 10.1016/j.actamat.2015.12.034 http://www.sciencedirect.com/science/article/pii/S1359645415301403
L.H. Poh, and G. Sun Localizing gradient damage model with decreasing interactions Int. J. Numer. Methods Eng. 110 2016 503 522 10.1002/nme.5364 https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.5364 https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.5364
M. Prechtel, G. Leugering, P. Steinmann, and M. Stingl Towards optimization of crack resistance of composite materials by adjustment of fiber shapes Eng. Fract. Mech. 78 2011 944 960 10.1016/j.engfracmech.2011.01.007 http://www.sciencedirect.com/science/article/pii/S0013794411000117
R. Radovitzky, A. Seagraves, M. Tupek, and L. Noels A scalable 3d fracture and fragmentation algorithm based on a hybrid, discontinuous galerkin, cohesive element method Comput. Methods Appl. Mech. Eng. 200 2011 326 344 10.1016/j.cma.2010.08.014 http://www.sciencedirect.com/science/article/pii/S0045782510002471
D. Reddi, V. Areej, and S. Keralavarma Ductile failure simulations using a multi-surface coupled damage-plasticity model Int. J. Plast. 118 2019 190 214 10.1016/j.ijplas.2019.02.007 http://www.sciencedirect.com/science/article/pii/S0749641918307149
J.J.C. Remmers, R. de Borst, C.V. Verhoosel, and A. Needleman The cohesive band model: a cohesive surface formulation with stress triaxiality Int. J. Fract. 181 2013 177 188 10.1007/s10704-013-9834-3
F. Reusch, B. Svendsen, and D. Klingbeil Local and non-local gurson-based ductile damage and failure modelling at large deformation Eur. J. Mech. A Solid. 22 2003 779 792 10.1016/S0997-7538(03)00070-6 http://www.sciencedirect.com/science/article/pii/S0997753803000706
F. Reusch, B. Svendsen, and D. Klingbeil A non-local extension of gurson-based ductile damage modeling Comput. Mater. Sci. 26 2003 219 229 10.1016/S0927-0256(02)00402-0. 11th International Workshop on Computational Mechanics of Materials http://www.sciencedirect.com/science/article/pii/S0927025602004020
C.C. Roth, T.F. Morgeneyer, Y. Cheng, L. Helfen, and D. Mohr Ductile damage mechanism under shear-dominated loading: in-situ tomography experiments on dual phase steel and localization analysis Int. J. Plast. 109 2018 169 192 10.1016/j.ijplas.2018.06.003 http://www.sciencedirect.com/science/article/pii/S0749641917306940
F. Scheyvaerts, T. Pardoen, and P. Onck A new model for void coalescence by internal necking Int. J. Damage Mech. 19 2010 95 126 10.1177/1056789508101918
R. Shen, H. Waisman, and L. Guo Fracture of viscoelastic solids modeled with a modified phase field method Comput. Methods Appl. Mech. Eng. 346 2019 862 890 10.1016/j.cma.2018.09.018 http://www.sciencedirect.com/science/article/pii/S0045782518304699
T. Siegmund, and W. Brocks A numerical study on the correlation between the work of separation and the dissipation rate in ductile fracture Eng. Fract. Mech. 67 2000 139 154 10.1016/S0013-7944(00)00054-0 http://www.sciencedirect.com/science/article/pii/S0013794400000540
C. Tekoǧlu, J.W. Hutchinson, and T. Pardoen On localization and void coalescence as a precursor to ductile fracture Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 373 2015 20140121 10.1098/rsta.2014.0121 https://royalsocietypublishing.org/doi/abs/10.1098/rsta.2014.0121 https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2014.0121
C. Tekoǧlu, J.B. Leblond, and T. Pardoen A criterion for the onset of void coalescence under combined tension and shear J. Mech. Phys. Solids 60 2012 1363 1381 10.1016/j.jmps.2012.02.006 http://www.sciencedirect.com/science/article/pii/S0022509612000373
P. Thomason A three-dimensional model for ductile fracture by the growth and coalescence of microvoids Acta Metall. 33 1985 1087 1095 10.1016/0001-6160(85)90202-0 http://www.sciencedirect.com/science/article/pii/0001616085902020
P. Thomason Three-dimensional models for the plastic limit-loads at incipient failure of the intervoid matrix in ductile porous solids Acta Metall. 33 1985 1079 1085 10.1016/0001-6160(85)90201-9 http://www.sciencedirect.com/science/article/pii/0001616085902019
M. Torki, C. Tekoglu, J.B. Leblond, and A. Benzerga Theoretical and numerical analysis of void coalescence in porous ductile solids under arbitrary loadings Int. J. Plast. 91 2017 160 181 10.1016/j.ijplas.2017.02.011 http://www.sciencedirect.com/science/article/pii/S0749641916302595
M.E. Torki A unified criterion for void growth and coalescence under combined tension and shear Int. J. Plast. 119 2019 57 84 10.1016/j.ijplas.2019.02.002 http://www.sciencedirect.com/science/article/pii/S0749641918303152
V. Tvergaard Influence of voids on shear band instabilities under plane strain conditions Int. J. Fract. 17 1981 389 407 10.1007/BF00036191
V. Tvergaard, and A. Needleman Analysis of the cup-cone fracture in a round tensile bar Acta Metall. 32 1984 157 169 10.1016/0001-6160(84)90213-X http://www.sciencedirect.com/science/article/pii/000161608490213X
B. Vandoren, and A. Simone Modeling and simulation of quasi-brittle failure with continuous anisotropic stress-based gradient-enhanced damage models Comput. Methods Appl. Mech. Eng. 332 2018 644 685 10.1016/j.cma.2017.12.027 http://www.sciencedirect.com/science/article/pii/S0045782517307776
Y. Wang, and H. Waisman An arc-length method for controlled cohesive crack propagation using high-order xfem and irwin's crack closure integral Eng. Fract. Mech. 199 2018 235 256 10.1016/j.engfracmech.2018.05.018 http://www.sciencedirect.com/science/article/pii/S0013794418301358
L. Wu, G. Becker, and L. Noels Elastic damage to crack transition in a coupled non-local implicit discontinuous galerkin/extrinsic cohesive law framework Comput. Methods Appl. Mech. Eng. 279 2014 379 409 10.1016/j.cma.2014.06.031 http://www.sciencedirect.com/science/article/pii/S0045782514002175
L. Wu, D. Tjahjanto, G. Becker, A. Makradi, A. Jérusalem, and L. Noels A micro-meso-model of intra-laminar fracture in fiber-reinforced composites based on a discontinuous galerkin/cohesive zone method Eng. Fract. Mech. 104 2013 162 183 10.1016/j.engfracmech.2013.03.018 http://www.sciencedirect.com/science/article/pii/S0013794413001252
Q. Xu, and Z. Lu An elastic-plastic cohesive zone model for metal-ceramic interfaces at finite deformations Int. J. Plast. 41 2013 147 164 10.1016/j.ijplas.2012.09.008 http://www.sciencedirect.com/science/article/pii/S0749641912001386
Z. Xue, M. Pontin, F. Zok, and J. Hutchinson Calibration procedures for a computational model of ductile fracture Eng. Fract. Mech. 77 2010 492 509 10.1016/j.engfracmech.2009.10.007 https://linkinghub.elsevier.com/retrieve/pii/S0013794409003233
Y. Zhang, E. Lorentz, and J. Besson Ductile damage modelling with locking-free regularised gtn model Int. J. Numer. Methods Eng. 113 2018 1871 1903 10.1002/nme.5722 https://onlinelibrary.wiley.com/doi/abs/10.1002/nme.5722 https://onlinelibrary.wiley.com/doi/pdf/10.1002/nme.5722
Z. Zhang, C. Thaulow, and J. Odegard A complete gurson model approach for ductile fracture Eng. Fract. Mech. 67 2000 155 168 10.1016/S0013-7944(00)00055-2 http://www.sciencedirect.com/science/article/pii/S0013794400000552
