Abstract
The aim of this paper is to develop a size-dependent Gurson type model. The approach is based on a micromechanical implementation of a local isotropic hardening able to account for different mechanisms responsible for size effects arising at the nanoscale (surface stress effects) and at the micronscale (strain gradient effects). The heterogeneity of hardening is accounted for by considering a finite number of spherical layers (Leblond et al. in Eur J Mech A 14:499–527, 1995; Morin et al. in Int J Solids Struct 118:167–178, 2017) in which hardening is described by a Taylor dislocation model. This introduces some strain gradient effect inducing a void size dependence. In the limit of a thin interphase, the model is shown to be very close to the imperfect coherent interface based model of Dormieux and Kondo (Int J Eng Sci 48:575–581, 2010) for nanoporous materials. In the case of micronscale voids, the model is assessed through comparison of its predictions with finite element cell calculations for different stress triaxiality. A good agreement is observed between the model predictions and numerical data from cell calculations performed by Niordson (Eur J Mech A 27:222–233, 2008).
Similar content being viewed by others
References
Aifantis EC (1984) On the microstructural origin of certain inelastic models. J Eng Mater Technol 106:326–330
Benzerga AA, Besson J (2001) Plastic potentials for anisotropic porous solids. Eur J Mech A 20:397–434
Benzerga AA, Leblond JB (2010) Ductile fracture by void growth to coalescence. Adv Appl Mech 44:169–305
Benzerga AA, Leblond JB (2014) Effective yield criterion accounting for microvoid coalescence. J Appl Mech 81:031009
Benzerga AA, Leblond JB, Needleman A, Tvergaard V (2016) Ductile failure modeling. Int J Fract 201:29–80
Brach S, Dormieux L, Kondo D, Vairo G (2016) A computational insight into void-size effects on strength properties of nanoporous materials. Mech Mater 101:102–117
Brach S, Dormieux L, Kondo D, Vairo G (2017) Nanoporous materials with a general isotropic plastic matrix: exact limit state under isotropic loadings. Int J Plast 89:1–28
Brach S, Anoukou K, Kondo D, Vairo G (2018) Limit analysis and homogenization of nanoporous materials with a general isotropic plastic matrix. Int J Plast 105:24–61
Dormieux L, Kondo D (2010) An extension of Gurson model incorporating interface stresses effects. Int J Eng Sci 48:575–581
Espinosa HD, Prorok BC, Peng B (2004) Plasticity size effects in free-standing submicron polycrystalline FCC films subjected to pure tension. J Mech Phys Solids 52:667–689
Fleck NA, Hutchinson JW (1993) A phenomenological theory for strain gradient effects in plasticity. J Mech Phys Solids 41:1825–1857
Fleck NA, Hutchinson JW (1997) Strain gradient plasticity. Adv Appl Mech 33:295–361
Fleck NA, Hutchinson JW (2001) A reformulation of strain gradient plasticity. J Mech Phys Solids 49:2245–2271
Fleck NA, Willis JR (2009) A mathematical basis for strain-gradient plasticity theory-Part I: scalar plastic multiplier. J Mech Phys Solids 57:161–177
Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: theory and experiment. Acta Metall Mater 42:475–487
Gao H, Huang Y (2001) Taylor-based nonlocal theory of plasticity. Int J Solids Struct 38:2615–2637
Gao H, Huang Y, Nix WD, Hutchinson JW (1999) Mechanism-based strain gradient plasticity-I. Theory. J Mech Phys Solids 47:1239–1263
Gologanu M, Leblond JB, Devaux J (1993) Approximate models for ductile metals containing non-spherical voids-case of axisymmetric prolate ellipsoidal cavities. J Mech Phys Solids 41:1723–1754
Gurson AL (1977) Continuum theory of ductile rupture by void nucleation and growth: Part I-yield criteria and flow rules for porous ductile media. ASME J Eng Mater Technol 99:2–15
Holte I, Niordson CF, Nielsen KL, Tvergaard V (2019) Investigation of a gradient enriched Gurson–Tvergaard model for porous strain hardening materials. Eur J Mech A 75:472–484
Huang Y, Gao H, Nix WD, Hutchinson JW (2000) Mechanism-based strain gradient plasticity-II. Analysis. J Mech Phys Solids 48:99–128
Huang Y, Qu S, Hwang KC, Li M, Gao H (2004) A conventional theory of mechanism-based strain gradient plasticity. Int J Plast 20:753–782
Hure J, Barrioz PO, Tanguy B (2020) Assessing size effects on the deformation of nanovoids in metallic materials. Scr Mater 177:54–57
Hutchinson JW (2000) Plasticity at the micron scale. Int J Solids Struct 37:225–238
Keralavarma S, Benzerga A (2010) A constitutive model for plastically anisotropic solids with non-spherical voids. J Mech Phys Solids 58:874–901
Lacroix R, Leblond JB, Perrin G (2016) Numerical study and theoretical modelling of void growth in porous ductile materials subjected to cyclic loadings. Eur J Mech A 55:100–109
Leblond J, Perrin G, Devaux J (1995) An improved Gurson-type model for hardenable ductile metals. Eur J Mech A 14:499–527
Leblond JB, Kondo D, Morin L, Remmal A (2018) Classical and sequential limit analysis revisited. Comptes Rendus Mécanique 346:336–349
Ling C, Forest S, Besson J, Tanguy B, Latourte F (2018) A reduced micromorphic single crystal plasticity model at finite deformations. Application to strain localization and void growth in ductile metals. Int J Solids Struct 134:43–69
Liu D, He Y, Dunstan DJ, Zhang B, Gan Z, Hu P, Ding H (2013) Toward a further understanding of size effects in the torsion of thin metal wires: an experimental and theoretical assessment. Int J Plast 41:30–52
Madou K, Leblond JB (2012a) A Gurson-type criterion for porous ductile solids containing arbitrary ellipsoidal voids-I: Limit-analysis of some representative cell. J Mech Phys Solids 60:1020–1036
Madou K, Leblond JB (2012b) A Gurson-type criterion for porous ductile solids containing arbitrary ellipsoidal voids-II: determination of yield criterion parameters. J Mech Phys Solids 60:1037–1058
Mi C, Buttry DA, Sharma P, Kouris DA (2011) Atomistic insights into dislocation-based mechanisms of void growth and coalescence. J Mech Phys Solids 59:1858–1871
Monchiet V, Bonnet G (2010) Interfacial models in viscoplastic composites materials. International Journal of Engineering Science 48:1762–1768
Monchiet V, Bonnet G (2013) A Gurson-type model accounting for void size effects. Int J Solids Struct 50:320–327
Monchiet V, Kondo D (2013) Combined voids size and shape effects on the macroscopic criterion of ductile nanoporous materials. Int J Plast 43:20–41
Monchiet V, Cazacu O, Charkaluk E, Kondo D (2008) Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids. Int J Plast 24:1158–1189
Morin L, Kondo D, Leblond JB (2015a) Numerical assessment, implementation and application of an extended Gurson model accounting for void size effects. Eur J Mech A 51:183–192
Morin L, Leblond JB, Benzerga AA (2015b) Coalescence of voids by internal necking: theoretical estimates and numerical results. J Mech Phys Solids 75:140–158
Morin L, Leblond JB, Kondo D (2015c) A Gurson-type criterion for plastically anisotropic solids containing arbitrary ellipsoidal voids. Int J Solids Struct 77:86–101
Morin L, Leblond JB, Tvergaard V (2016) Application of a model of plastic porous materials including void shape effects to the prediction of ductile failure under shear-dominated loadings. J Mech Phys Solids 94:148–166
Morin L, Michel JC, Leblond JB (2017) A Gurson-type layer model for ductile porous solids with isotropic and kinematic hardening. Int J Solids Struct 118:167–178
Niordson CF (2008) Void growth to coalescence in a non-local material. Eur J Mech A 27:222–233
Niordson CF, Tvergaard V (2019) A homogenized model for size-effects in porous metals. J Mech Phys Solids 123:222–233
Nix WD, Gao H (1998) Indentation size effects in crystalline materials: a law for strain gradient plasticity. J Mech Phys Solids 46:411–425
Perrin G, Leblond JB (1990) Analytical study of a hollow sphere made of plastic porous material and subjected to hydrostatic tension-application to some problems in ductile fracture of metals. Int J Plast 6:677–699
Pineau A, Benzerga AA, Pardoen T (2016) Failure of metals I - Brittle and ductile fracture. Acta Mater 107:424–483
Scherer JM, Hure J (2019) A size-dependent ductile fracture model: constitutive equations, numerical implementation and validation. Eur J Mech A 76:135–145
Segurado J, Llorca J (2009) An analysis of the size effect on void growth in single crystals using discrete dislocation dynamics. Acta Mater 57:1427–1436
Segurado J, LLorca J (2010) Discrete dislocation dynamics analysis of the effect of lattice orientation on void growth in single crystals. Int J Plast 26:806–819
Stelmashenko NA, Walls MG, Brown LM, Milman YV (1993) Microindentations on W and Mo oriented single crystals: an STM study. Acta Metall Mater 41:2855–2865
Tang FL, Cai HM, Bao HW, Xue HT, Lu WJ, Zhu L, Rui ZY (2014) Molecular dynamics simulations of void growth in \(\gamma \)-TiAl single crystal. Comput Mater Sci 84:232–237
Taylor GI (1934) The mechanism of plastic deformation of crystals Part I. Theoretical containing papers of a mathematical and physical character. Proc R Soc Lond Ser A 145:362–387
Torki ME, Tekoglu C, Leblond JB, Benzerga AA (2017) Theoretical and numerical analysis of void coalescence in porous ductile solids under arbitrary loadings. Int J Plast 91:160–181
Traiviratana S, Bringa EM, Benson DJ, Meyers MA (2008) Void growth in metals: atomistic calculations. Acta Mater 56:3874–3886
Tvergaard V (1981) Influence of voids on shear band instabilities under plane strain conditions. Int J Fract 17:389–407
Valiev RZ, Islamgaliev RK, Alexandrov IV (2000) Bulk nanostructured materials from severe plastic deformation. Progr Mater Sci 45:103–189
Wen J, Huang Y, Hwang KC, Liu C, Li M (2005) The modified Gurson model accounting for the void size effect. Int J Plast 21:381–395
Zong Z, Lou J, Adewoye OO, Elmustafa AA, Hammad F, Soboyejo WO (2006) Indentation size effects in the nano- and micro-hardness of fcc single crystal metals. Mater Sci Eng A 434:178–187
Acknowledgements
The authors are indebted to Jean-Baptiste Leblond, whose work on ductile fracture has been a continuous source of inspiration.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Morin, L., Kondo, D. An interphase approach of size effects in ductile porous materials. Int J Fract 230, 71–82 (2021). https://doi.org/10.1007/s10704-020-00507-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10704-020-00507-6