[en] The state of maturity of the micromechanics of ductile fracture is such that it is possible, today, to simulate extensive crack growth in 3D using a sophisticated physics-based description of the mechanisms of nucleation, growth and coalescence of voids within a non-local formulation. Here, different phenomena related to ductile crack growth are addressed in the context of fracture mechanics specimens. Structural integrity assessments of many critical components rely indeed on predictive models of crack growth from pre-existing sharp defects. Two similar extended Gurson models are used, after comparison to cross-verify their numerical implementation, to generate results about the effect of plate thickness, plastic anisotropy and strain hardening. The variation of the fracture toughness increasing and then decreasing with thickness down to the plane strain regime is captured owing to the 3D nature of the simulations, capturing the crack tip necking phenomenon. The non-local formulation introduces a length scale that sets the range over which the fracture toughness depends on thickness. The thickness effect disappears when using homothetic geometries. The effect of plasticity anisotropy is shown to be particularly important when a crack grows in sheets exhibiting significant crack tip necking through increasing or not the plastic dissipation in the neck. A large strain hardening capacity enhances very much the fracture toughness, an effect that is amplified in 3D when crack tip necking takes place. These findings constitute only a limited set of answers to many remaining questions in the very important field of ductile tearing, setting an ambitious roadmap for the years to come to the solid mechanics community. These questions are particularly important in the context of modern technologies such as H2 storage, additive manufacturing, new high strength metallic alloys development, and generation IV nuclear fission and fusion reactors to name a few.
Research Center/Unit :
A&M - Aérospatiale et Mécanique - ULiège
Disciplines :
Mechanical engineering
Author, co-author :
Pardoen, Thomas
Kaniadakis, Antonio
Nguyen, Van Dung ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Computational & Multiscale Mechanics of Materials (CM3)
Noels, Ludovic ; Université de Liège - ULiège > Département d'aérospatiale et mécanique > Computational & Multiscale Mechanics of Materials (CM3)
Javangorough, Sara
Besson, Jacques
Language :
English
Title :
Modelling and 3D simulation of ductile crack growth with non-local Gurson-based formulation
This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement No. 101097433).
Commentary :
NOTICE: this is the author’s version of a work that was accepted for publication in European Journal of Mechanics - A/Solids. 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 European Journal of Mechanics - A/Solids 114 (2025), DOI: 10.1016/j.euromechsol.2025.105772
Anderson, T.L., Fracture mechanics: Fundamentals and applications, third edition. 2005, CRC Press, 10.1201/9781420058215.
ASTM E1820, Standard Test Method for Measurement of Fracture Toughness. 2019, ASTM International, 10.1520/E1820-24 ASTM E1820-24.
Bai, Y., Wierzbicki, T., A new model of metal plasticity and fracture with pressure and lode dependence. Int. J. Plast. 24:6 (2008), 1071–1096, 10.1016/j.ijplas.2007.09.004.
Bažant, Z.P., Jirásek, M., Nonlocal integral formulations of plasticity and damage: Survey of progress. J. Eng. Mech. 128:11 (2002), 1119–1149, 10.1061/(asce)0733-9399(2002)128:11(1119).
Benzerga, A., Micromechanics of coalescence in ductile fracture. J. Mech. Phys. Solids 50:6 (2002), 1331–1362, 10.1016/s0022-5096(01)00125-9.
Benzerga, A.A., Besson, J., Plastic potentials for anisotropic porous solids. Eur. J. Mech. A Solids 20:3 (2001), 397–434, 10.1016/s0997-7538(01)01147-0.
Benzerga, A.A., Leblond, J.-B., Ductile fracture by void growth to coalescence. Aref, H., van der Giessen, E., (eds.) Advances in Applied Mechanics Advances in Applied Mechanics, vol. 44, 2010, Elsevier, 169–305, 10.1016/S0065-2156(10)44003-X URL https://www.sciencedirect.com/science/article/pii/S006521561044003X.
Besson, J., Damage of ductile materials deforming under multiple plastic or viscoplastic mechanisms. Int. J. Plast. 25:11 (2009), 2204–2221, 10.1016/j.ijplas.2009.03.001.
Besson, J., McCowan, C., Drexler, E., Modeling flat to slant fracture transition using the computational cell methodology. Eng. Fract. Mech. 104 (2013), 80–95, 10.1016/j.engfracmech.2013.02.032.
Broek, D., Elementary engineering fracture mechanics. 1982, Springer Dordrecht, 10.1007/978-94-009-4333-9.
Chen, Y., Lorentz, E., Besson, J., Crack initiation and propagation in small-scale yielding using a nonlocal GTN model. Int. J. Plast., 130, 2020, 102701, 10.1016/j.ijplas.2020.102701 URL https://www.sciencedirect.com/science/article/pii/S0749641919304085.
Chen, J., Yuan, H., A micro-mechanical damage model based on gradient plasticity: algorithms and applications. Internat. J. Numer. Methods Engrg. 54:3 (2002), 399–420, 10.1002/nme.431.
Cheng, S., Garnier, J., Marini, B., Madi, Y., Besson, J., Size and thickness effects on the ductile fracture toughness of a 316 l(n) austenitic stainless steel in as-received and aged conditions. Fatigue Fract. Eng. Mater. Struct. 48:7 (2025), 3168–3184, 10.1111/ffe.14665 arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1111/ffe.14665. URL https://onlinelibrary.wiley.com/doi/abs/10.1111/ffe.14665.
Choisez, L., Ding, L., Marteleur, M., Idrissi, H., Pardoen, T., Jacques, P.J., High temperature rise dominated cracking mechanisms in ultra-ductile and tough titanium alloy. Nat. Commun., 11(1), 2020, 10.1038/s41467-020-15772-1.
Chu, C.C., Needleman, A., Void Nucleation Effects in Biaxially Stretched Sheets. J. Eng. Mater. Technol. 102:3 (1980), 249–256, 10.1115/1.3224807.
Cotterell, B., Pardoen, T., Atkins, A., Measuring toughness and the cohesive stress–displacement relationship by the essential work of fracture concept. Eng. Fract. Mech. 72:6 (2005), 827–848, 10.1016/j.engfracmech.2004.10.002.
Das, R., Special issue: structural integrity of additively manufactured materials—developments in damage, fracture, fatigue and failure assessments. Int. J. Fract. 235:1 (2022), 1–2, 10.1007/s10704-022-00651-1.
De Borst, R., Sluys, L., Muhlhaus, H., Pamin, J., Fundamental issues in finite element analyses of localization of deformation. Eng. Comput. 10:2 (1993), 99–121, 10.1108/eb023897.
Depraetere, R., De Waele, W., Cauwels, M., Depover, T., Verbeken, K., Hertelé, S., Modeling of hydrogen-charged notched tensile tests of an X70 pipeline steel with a hydrogen-informed Gurson model. Materials, 16(13), 2023, 4839, 10.3390/ma16134839.
El-Naaman, S., Nielsen, K., Observations on mode I ductile tearing in sheet metals. Eur. J. Mech. A Solids 42 (2013), 54–62, 10.1016/j.euromechsol.2013.04.007.
Elguedj, T., Bazilevs, Y., Calo, V., Hughes, T., And projection methods for nearly incompressible linear and non-linear elasticity and plasticity using higher-order NURBS elements. Comput. Methods Appl. Mech. Engrg. 197:33–40 (2008), 2732–2762, 10.1016/j.cma.2008.01.012.
Ernst, H., Paris, P., Landes, J., Estimations on J-integral and tearing modulus T from a single specimen test record. Fracture Mechanics, 1981, ASTM International100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, 476–502, 10.1520/stp28814s.
Fabrègue, D., Pardoen, T., A constitutive model for elastoplastic solids containing primary and secondary voids. J. Mech. Phys. Solids 56:3 (2008), 719–741, 10.1016/j.jmps.2007.07.008.
Faleskog, J., Gao, X., Shih, C., Cell model for nonlinear fracture analysis - I. Micromechanics calibration. Int. J. Fract. 89 (1998), 355–373, 10.1023/A:1007421420901.
Forest, S., Micromorphic approach for gradient elasticity, viscoplasticity, and damage. J. Eng. Mech. 135:3 (2009), 117–131, 10.1061/(ASCE)0733-9399(2009)135:3(117).
Frómeta, D., Cuadrado, N., Rehrl, J., Suppan, C., Dieudonné, T., Dietsch, P., Calvo, J., Casellas, D., Microstructural effects on fracture toughness of ultra-high strength dual phase sheet steels. Mater. Sci. Eng.: A, 802, 2021, 140631, 10.1016/j.msea.2020.140631.
Gaffard, V., Gourgues-Lorenzon, A., Besson, J., High temperature creep flow and damage properties of 9Cr1MoNbV steels: Base metal and weldment. Nucl. Eng. Des. 235:24 (2005), 2547–2562, 10.1016/j.nucengdes.2005.07.001.
Gludovatz, B., Hohenwarter, A., Thurston, K.V.S., Bei, H., Wu, Z., George, E.P., Ritchie, R.O., Exceptional damage-tolerance of a medium-entropy alloy CrCoNi at cryogenic temperatures. Nat. Commun., 7(1), 2016, 10.1038/ncomms10602.
Gologanu, M., Leblond, J.-B., Devaux, J., Approximate models for ductile metals containing non-spherical voids—Case of axisymmetric prolate ellipsoidal cavities. J. Mech. Phys. Solids 41:11 (1993), 1723–1754, 10.1016/0022-5096(93)90029-F URL https://www.sciencedirect.com/science/article/pii/002250969390029F.
Gurson, A.L., 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. 99:1 (1977), 2–15, 10.1115/1.3443401.
Håkansson, P., Wallin, M., Ristinmaa, M., Thermomechanical response of non-local porous material. Int. J. Plast. 22:11 (2006), 2066–2090, 10.1016/j.ijplas.2005.08.003.
Hilhorst, A., Jacques, P.J., Pardoen, T., Towards the best strength, ductility, and toughness combination: High entropy alloys are excellent, stainless steels are exceptional. Acta Mater., 260, 2023, 119280, 10.1016/j.actamat.2023.119280 URL https://www.sciencedirect.com/science/article/pii/S1359645423006109.
Hilhorst, A., Leclerc, J., Pardoen, T., Jacques, P.J., Noels, L., Nguyen, V.-D., Ductile fracture of high entropy alloys: From the design of an experimental campaign to the development of a micromechanics-based modeling framework. Eng. Fract. Mech., 275, 2022, 108844, 10.1016/j.engfracmech.2022.108844 URL https://www.sciencedirect.com/science/article/pii/S0013794422005628.
Hill, R., A theory of the yielding and plastic flow of anisotropic metals. Proc. R. Soc. Lond. Ser. A. Math. Phys. Sci. 193:1033 (1948), 281–297.
Hütter, G., Linse, T., Mühlich, U., Kuna, M., Simulation of ductile crack initiation and propagation by means of a non-local Gurson-model. Int. J. Solids Struct. 50:5 (2013), 662–671, 10.1016/j.ijsolstr.2012.10.031.
Hütter, G., Linse, T., Roth, S., Mühlich, U., Kuna, M., A modeling approach for the complete ductile–brittle transition region: cohesive zone in combination with a non-local Gurson-model. Int. J. Fract. 185:1–2 (2013), 129–153, 10.1007/s10704-013-9914-4.
Ismail, K., Perlade, A., Jacques, P.J., Pardoen, T., Outstanding cracking resistance of fibrous dual phase steels. Acta Mater., 207, 2021, 116700, 10.1016/j.actamat.2021.116700.
Jirásek, M., Nonlocal models for damage and fracture: Comparison of approaches. Int. J. Solids Struct. 35:31–32 (1998), 4133–4145, 10.1016/s0020-7683(97)00306-5.
Kaniadakis, A., Nguyen, V.-D., Besson, J., Pardoen, T., Strain hardening effect on ductile tearing under small scale yielding plane strain conditions. J. Mech. Phys. Solids, 2025, 106171, 10.1016/j.jmps.2025.106171.
Kim, J., Gao, X., Srivatsan, T.S., Modeling of void growth in ductile solids: effects of stress triaxiality and initial porosity. Eng. Fract. Mech. 71:3 (2004), 379–400, 10.1016/s0013-7944(03)00114-0.
Kong, X., Helfen, L., Hurst, M., Hänschke, D., Missoum-Benziane, D., Besson, J., Baumbach, T., Morgeneyer, T.F., 3D in situ study of damage during a ’shear to tension’ load path change in an aluminium alloy. Acta Mater., 231, 2022, 117842, 10.1016/j.actamat.2022.117842.
Koplik, J., Needleman, A., Void growth and coalescence in porous plastic solids. Int. J. Solids Struct. 24:8 (1988), 835–853, 10.1016/0020-7683(88)90051-0.
Leblond, J.B., Perrin, G., Devaux, J., Bifurcation effects in ductile metals with nonlocal damage. J. Appl. Mech. 61:2 (1994), 236–242, 10.1115/1.2901435.
Leblond, J.B., Perrin, G., Devaux, J., An improved gurson-type model for hardenable ductile metals. Eur. J. Mech. A Solids 14 (1995), 499–527 URL https://api.semanticscholar.org/CorpusID:118050919.
Lecarme, L., Maire, E., Kumar K.C., A., De Vleeschouwer, C., Jacques, L., Simar, A., Pardoen, T., Heterogenous void growth revealed by in situ 3-D X-ray microtomography using automatic cavity tracking. Acta Mater. 63 (2014), 130–139, 10.1016/j.actamat.2013.10.014 URL https://www.sciencedirect.com/science/article/pii/S1359645413007659.
Lin, M., Yu, H., Ding, Y., Wang, G., Olden, V., Alvaro, A., He, J., Zhang, Z., A predictive model unifying hydrogen enhanced plasticity and decohesion. Scr. Mater., 215, 2022, 114707, 10.1016/j.scriptamat.2022.114707.
Linse, T., Hütter, G., Kuna, M., Simulation of crack propagation using a gradient-enriched ductile damage model based on dilatational strain. Eng. Fract. Mech. 95 (2012), 13–28, 10.1016/j.engfracmech.2012.07.004.
Lopes Pinto, D., El Ouazani Tuhami, A., Osipov, N., Madi, Y., Besson, J., Simulation of hydrogen embrittlement of steel using mixed nonlocal finite elements. Eur. J. Mech. A Solids, 104, 2024, 105116, 10.1016/j.euromechsol.2023.105116.
Madou, K., Leblond, J.-B., A gurson-type criterion for porous ductile solids containing arbitrary ellipsoidal voids—I: Limit-analysis of some representative cell. J. Mech. Phys. Solids 60:5 (2012), 1020–1036, 10.1016/j.jmps.2011.11.008.
Madou, K., Leblond, J.B., A gurson-type criterion for porous ductile solids containing arbitrary ellipsoidal voids—II: Determination of yield criterion parameters. J. Mech. Phys. Solids 60:5 (2012), 1037–1058, 10.1016/j.jmps.2012.01.010.
Marteleur, M., Leclerc, J., Colla, M.S., Nguyen, V.D., Noels, L., Pardoen, T., Ductile fracture of high strength steels with morphological anisotropy, part I: Characterization, testing, and void nucleation law. Eng. Fract. Mech., 244, 2021, 107569, 10.1016/j.engfracmech.2021.107569.
Morgeneyer, T., Besson, J., Flat to slant ductile fracture transition: Tomography examination and simulations using shear-controlled void nucleation. Scr. Mater. 65:11 (2011), 1002–1005, 10.1016/j.scriptamat.2011.09.004 URL http://dx.doi.org/10.1016/j.scriptamat.2011.09.004.
Morin, L., Leblond, J.B., Benzerga, A.A., Kondo, D., A unified criterion for the growth and coalescence of microvoids. J. Mech. Phys. Solids 97 (2016), 19–36, 10.1016/j.jmps.2016.01.013 URL http://dx.doi.org/10.1016/j.jmps.2016.01.013.
Nahshon, K., Hutchinson, J., Modification of the Gurson model for shear failure. Eur. J. Mech. A Solids 27:1 (2008), 1–17, 10.1016/j.euromechsol.2007.08.002 URL https://www.sciencedirect.com/science/article/pii/S0997753807000721.
Nguyen, V.D., Pardoen, T., A ductile fracture model accounting for plastic anisotropy and its application in ductile tearing of thin metallic plates. 2025 (in preparation).
Nguyen, V.D., Pardoen, T., Noels, L., A nonlocal approach of ductile failure incorporating void growth, internal necking, and shear dominated coalescence mechanisms. J. Mech. Phys. Solids, 137, 2020, 103891, 10.1016/j.jmps.2020.103891 URL https://www.sciencedirect.com/science/article/pii/S0022509619309275.
Onck, P., van der Giessen, E., Micromechanics of creep fracture: simulation of intergranular crack growth. Comput. Mater. Sci. 13:1–3 (1998), 90–102, 10.1016/s0927-0256(98)00049-4 URL http://dx.doi.org/10.1016/S0927-0256(98)00049-4.
Papasidero, J., Doquet, V., Mohr, D., Ductile fracture of aluminum 2024-T351 under proportional and non-proportional multi-axial loading: Bao–Wierzbicki results revisited. Int. J. Solids Struct. 69–70 (2015), 459–474, 10.1016/j.ijsolstr.2015.05.006 URL http://dx.doi.org/10.1016/j.ijsolstr.2015.05.006.
Pardoen, T., Hachez, F., Marchioni, B., Blyth, P., Atkins, A., Mode I fracture of sheet metal. J. Mech. Phys. Solids 52:2 (2004), 423–452, 10.1016/S0022-5096(03)00087-5 URL https://www.sciencedirect.com/science/article/pii/S0022509603000875.
Pardoen, T., Hutchinson, J., An extended model for void growth and coalescence. J. Mech. Phys. Solids 48:12 (2000), 2467–2512, 10.1016/S0022-5096(00)00019-3 URL https://www.sciencedirect.com/science/article/pii/S0022509600000193.
Pardoen, T., Hutchinson, J., Micromechanics-based model for trends in toughness of ductile metals. Acta Mater. 51:1 (2003), 133–148, 10.1016/S1359-6454(02)00386-5 URL https://www.sciencedirect.com/science/article/pii/S1359645402003865.
Pardoen, T., Marchal, Y., Delannay, F., Thickness dependence of cracking resistance in thin aluminium plates. J. Mech. Phys. Solids 47:10 (1999), 2093–2123, 10.1016/s0022-5096(99)00011-3 URL http://dx.doi.org/10.1016/S0022-5096(99)00011-3.
Pardoen, T., Scheyvaerts, F., Simar, A., Tekoğlu, C., Onck, P.R., Multiscale modeling of ductile failure in metallic alloys. Comptes Rendus. Phys. 11:3–4 (2010), 326–345, 10.1016/j.crhy.2010.07.012 URL http://dx.doi.org/10.1016/j.crhy.2010.07.012.
Peerlings, R., Borst, de, R., Brekelmans, W., Geers, M., Gradient-enhanced damage modelling of concrete fracture. Mech. Cohesive- Frict. Mater. 3:4 (1998), 323–342, 10.1002/(SICI)1099-1484(1998100)3:4<323::AID-CFM51>3.0.CO;2-Z.
Peerlings, R.H.J., de Borst, R., Brekelmans, W.A.M., de Vree, J.H.P., Gradient enhanced damage for quasi-brittle materials. Internat. J. Numer. Methods Engrg. 39:19 (1996), 3391–3403, 10.1002/(sici)1097-0207(19961015)39:19<3391::aid-nme7>3.0.co;2-d.
Peerlings, R., Geers, M., de Borst, R., Brekelmans, W., A critical comparison of nonlocal and gradient-enhanced softening continua. Int. J. Solids Struct. 38:44–45 (2001), 7723–7746, 10.1016/s0020-7683(01)00087-7.
Petti, J.P., Dodds, R.H., Ductile tearing and discrete void effects on cleavage fracture under small-scale yielding conditions. Int. J. Solids Struct. 42:13 (2005), 3655–3676, 10.1016/j.ijsolstr.2004.11.015.
Pineau, A., Benzerga, A., Pardoen, T., Failure of metals I: Brittle and ductile fracture. Acta Mater. 107 (2016), 424–483, 10.1016/j.actamat.2015.12.034 URL https://www.sciencedirect.com/science/article/pii/S1359645415301403.
Ramaswamy, S., Aravas, N., Finite element implementation of gradient plasticity models part II: Gradient-dependent evolution equations. Comput. Methods Appl. Mech. Engrg. 163:1–4 (1998), 33–53, 10.1016/s0045-7825(98)00027-9.
Reusch, F., Hortig, C., Svendsen, B., Nonlocal modeling and simulation of ductile damage and failure in metal matrix composites. J. Eng. Mater. Technol., 130(2), 2008, 10.1115/1.2840967.
Reusch, F., Svendsen, B., Klingbeil, D., Local and non-local gurson-based ductile damage and failure modelling at large deformation. Eur. J. Mech. A Solids 22:6 (2003), 779–792, 10.1016/s0997-7538(03)00070-6.
Reusch, F., Svendsen, B., Klingbeil, D., A non-local extension of gurson-based ductile damage modeling. Comput. Mater. Sci. 26 (2003), 219–229, 10.1016/s0927-0256(02)00402-0.
Rice, J.R., Paris, P.C., Merkle, J.G., Some further results of J-integral analysis and estimates. Progress in Flaw Growth and Fracture Toughness Testing, 1973, ASTM International100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, 231–245, 10.1520/stp49643s.
Rivalin, F., Besson, J., Pineau, A., Di Fant, M., Ductile tearing of pipeline-steel wide plates. Eng. Fract. Mech. 68:3 (2001), 347–364, 10.1016/s0013-7944(00)00108-9.
Roubaud, F., Morin, L., Remmal, A., Marie, S., Leblond, J.-B., A gurson-type layer model for ductile porous solids containing ellipsoidal voids withx isotropic and kinematic hardening. Eur. J. Mech. A Solids, 104, 2024, 105114, 10.1016/j.euromechsol.2023.105114.
Ruggieri, C., Dodds, R.H., A local approach to cleavage fracture modeling: An overview of progress and challenges for engineering applications. Eng. Fract. Mech. 187 (2018), 381–403, 10.1016/j.engfracmech.2017.12.021.
Samal, M., Seidenfuss, M., Roos, E., Dutta, B., Kushwaha, H., Experimental and numerical investigation of ductile-to-brittle transition in a pressure vessel steel. Mater. Sci. Eng.: A 496:1–2 (2008), 25–35, 10.1016/j.msea.2008.06.046.
Scheyvaerts, F., Pardoen, T., Onck, P., A new model for void coalescence by internal necking. Int. J. Damage Mech. 19 (2010), 126–195 URL https://api.semanticscholar.org/CorpusID:137294886.
Shahani, A., Rastegar, M., Botshekanan Dehkordi, M., Moayeri Kashani, H., Experimental and numerical investigation of thickness effect on ductile fracture toughness of steel alloy sheets. Eng. Fract. Mech. 77:4 (2010), 646–659, 10.1016/j.engfracmech.2009.12.017.
Shinohara, Y., Madi, Y., Besson, J., Anisotropic ductile failure of a high-strength line pipe steel. Int. J. Fract. 197:2 (2016), 127–145, 10.1007/s10704-015-0054-x.
Siegmund, T., Brocks, W., A numerical study on the correlation between the work of separation and the dissipation rate in ductile fracture. Eng. Fract. Mech. 67:2 (2000), 139–154, 10.1016/s0013-7944(00)00054-0.
Tanguy, B., Bouchet, C., Bugat, S., Besson, J., Local approach to fracture based prediction of the δt56j and shifts due to irradiation for an A508 pressure vessel steel. Eng. Fract. Mech. 73:2 (2006), 191–206, 10.1016/j.engfracmech.2005.05.006.
Thaulow, C., Østby, E., Nyhus, B.a., Zhang, Z.L., Skallerud, B., Constraint correction of high strength steel. Eng. Fract. Mech. 71:16–17 (2004), 2417–2433, 10.1016/j.engfracmech.2004.01.003.
Tuhami, A.E.O., Feld-Payet, S., Quilici, S., Osipov, N., Besson, J., A two characteristic length nonlocal GTN model: Application to cup–cone and slant fracture. Mech. Mater., 171, 2022, 104350, 10.1016/j.mechmat.2022.104350.
Tvergaard, V., Interaction of very small voids with larger voids. Int. J. Solids Struct. 35:30 (1998), 3989–4000, 10.1016/s0020-7683(97)00254-0.
Tvergaard, V., Crack growth predictions by cohesive zone model for ductile fracture. J. Mech. Phys. Solids 49:9 (2001), 2191–2207, 10.1016/s0022-5096(01)00030-8.
Tvergaard, V., Legarth, B.N., Effect of plastic anisotropy on crack growth resistance under mode 1 loading. Int. J. Fract. 130:1 (2004), 411–425, 10.1023/b:frac.0000049498.15818.33.
Tvergaard, V., Needleman, A., Analysis of the cup-cone fracture in a round tensile bar. Acta Metall. 32:1 (1984), 157–169, 10.1016/0001-6160(84)90213-X URL https://www.sciencedirect.com/science/article/pii/000161608490213X.
Tvergaard, V., Needleman, A., Effects of nonlocal damage in porous plastic solids. Int. J. Solids Struct. 32:8–9 (1995), 1063–1077, 10.1016/0020-7683(94)00185-y.
Van Der Giessen, E., Onck, P., Van Der Burg, M., Some effects of random microstructural variations on creep rupture. Eng. Fract. Mech. 57:2–3 (1997), 205–226, 10.1016/s0013-7944(97)00009-x.
Xia, L., A computational approach to ductile crack growth under large scale yielding conditions. J. Mech. Phys. Solids 43:3 (1995), 389–413, 10.1016/0022-5096(94)00069-h.
Zybell, L., Hütter, G., Linse, T., Mühlich, U., Kuna, M., Size effects in ductile failure of porous materials containing two populations of voids. Eur. J. Mech. A Solids 45 (2014), 8–19, 10.1016/j.euromechsol.2013.11.006.
