Comptes Rendus
Computational metallurgy and changes of scale / Métallurgie numérique et changements d'échelle
Multiscale modeling of ductile failure in metallic alloys
Comptes Rendus. Physique, Volume 11 (2010) no. 3-4, pp. 326-345.

Micromechanical models for ductile failure have been developed in the 1970s and 1980s essentially to address cracking in structural applications and complement the fracture mechanics approach. Later, this approach has become attractive for physical metallurgists interested by the prediction of failure during forming operations and as a guide for the design of more ductile and/or high-toughness microstructures. Nowadays, a realistic treatment of damage evolution in complex metallic microstructures is becoming feasible when sufficiently sophisticated constitutive laws are used within the context of a multilevel modelling strategy. The current understanding and the state of the art models for the nucleation, growth and coalescence of voids are reviewed with a focus on the underlying physics. Considerations are made about the introduction of the different length scales associated with the microstructure and damage process. Two applications of the methodology are then described to illustrate the potential of the current models. The first application concerns the competition between intergranular and transgranular ductile fracture in aluminum alloys involving soft precipitate free zones along the grain boundaries. The second application concerns the modeling of ductile failure in friction stir welded joints, a problem which also involves soft and hard zones, albeit at a larger scale.

Les modèles micromécaniques pour la rupture ductile ont été développés essentiellement au cours des années 1970 et 1980 avec comme but de traiter des problèmes de rupture dans des applications structurales et de compléter l'approche par la mécanique de la rupture. Plus tard, cette approche micromécanique a attiré les spécialistes de la métallurgie physique soucieux de prédire les problèmes de rupture pendant les opérations de mise en forme ainsi que d'être guidés pour l'élaboration de microstructures plus ductiles et/ou plus tenaces. Aujourd'hui, il est devenu possible d'aborder de façon réaliste l'évolution du dommage dans des microstructures métalliques complexes à condition d'utiliser des lois constitutives suffisamment sophistiquées dans le contexte d'une stratégie de modélisation multi-échelle. La compréhension actuelle et l'état de l'art dans la modélisation des phénomènes de germination, croissance et coalescence de cavités sont passés en revue en portant une attention particulière à la physique sous-jacente. La prise en compte des différentes longueurs caractéristiques associées à la microstructure et au processus d'endommagement est discutée. Deux applications de la méthodologie sont décrites pour illustrer le potentiel des modèles. La première application concerne la compétition entre rupture ductile intergranulaire et transgranulaire dans des alliages d'aluminium impliquant des zones molles sans précipité le long des joints de grain. La seconde application concerne la modélisation de la rupture ductile de joints soudés par friction malaxage, un problème qui implique également des zones dures et molles, mais à une échelle plus grande.

Published online:
DOI: 10.1016/j.crhy.2010.07.012
Keywords: Ductile fracture, Multiscale modelling, Ductility, Fracture toughness, Damage, Voids
Mot clés : Rupture ductile, Modélisation multiéchelle, Ductilité, Ténacité, Endommagement, Cavités

Thomas Pardoen 1; Florence Scheyvaerts 1; Aude Simar 1; Cihan Tekoğlu 1; Patrick R. Onck 2

1 Institute of Mechanics, Materials and Civil Engineering, Université catholique de Louvain, place Sainte Barbe 2, B-1348 Louvain-la-Neuve, Belgium
2 Micromechanics of Materials, Zernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, NL-9747 AG Groningen, The Netherlands
@article{CRPHYS_2010__11_3-4_326_0,
     author = {Thomas Pardoen and Florence Scheyvaerts and Aude Simar and Cihan Teko\u{g}lu and Patrick R. Onck},
     title = {Multiscale modeling of ductile failure in metallic alloys},
     journal = {Comptes Rendus. Physique},
     pages = {326--345},
     publisher = {Elsevier},
     volume = {11},
     number = {3-4},
     year = {2010},
     doi = {10.1016/j.crhy.2010.07.012},
     language = {en},
}
TY  - JOUR
AU  - Thomas Pardoen
AU  - Florence Scheyvaerts
AU  - Aude Simar
AU  - Cihan Tekoğlu
AU  - Patrick R. Onck
TI  - Multiscale modeling of ductile failure in metallic alloys
JO  - Comptes Rendus. Physique
PY  - 2010
SP  - 326
EP  - 345
VL  - 11
IS  - 3-4
PB  - Elsevier
DO  - 10.1016/j.crhy.2010.07.012
LA  - en
ID  - CRPHYS_2010__11_3-4_326_0
ER  - 
%0 Journal Article
%A Thomas Pardoen
%A Florence Scheyvaerts
%A Aude Simar
%A Cihan Tekoğlu
%A Patrick R. Onck
%T Multiscale modeling of ductile failure in metallic alloys
%J Comptes Rendus. Physique
%D 2010
%P 326-345
%V 11
%N 3-4
%I Elsevier
%R 10.1016/j.crhy.2010.07.012
%G en
%F CRPHYS_2010__11_3-4_326_0
Thomas Pardoen; Florence Scheyvaerts; Aude Simar; Cihan Tekoğlu; Patrick R. Onck. Multiscale modeling of ductile failure in metallic alloys. Comptes Rendus. Physique, Volume 11 (2010) no. 3-4, pp. 326-345. doi : 10.1016/j.crhy.2010.07.012. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2010.07.012/

[1] A. Pineau Review of fracture micromechanisms and a local approach to predicting crack resistance in low strength steels (D. François et al., eds.), ICF 5, Advances in Fracture Research, vol. 2, Pergamon, 1981, pp. 553-577

[2] V. Tvergaard Material failure by void growth to coalescence, Adv. Appl. Mech., Volume 27 (1990), pp. 83-151

[3] A. Pineau; T. Pardoen Failure mechanisms of metals, Comprehensive Structural Integrity Encyclopedia, vol. 2, Elsevier, 2007 (Chap. 6, on-line)

[4] A.A. Benzerga, J.-B. Leblond, Ductile fracture by void growth to coalescence, Adv. Appl. Mech. (2010), in press.

[5] C.F. Tipper The fracture of metals, Metallurgica, Volume 39 (1949), pp. 133-137

[6] K.E. Puttick Ductile fracture in metals, Philos. Mag., Volume 4 (1959), pp. 964-969

[7] H.C. Rogers Tensile fracture of ductile metals, Trans. Metall. Soc. AIME, Volume 218 (1960), pp. 498-506

[8] C.D. Beachem An electron fractographic study of the influence of plastic strain conditions upon ductile rupture processes in metals, Trans. ASM, Volume 56 (1963), p. 318

[9] J. Gurland; J. Plateau The mechanism of ductile rupture of metals containing inclusions, Trans. ASM, Volume 56 (1963), pp. 442-454

[10] F.A. McClintock A criterion for ductile fracture by the growth of holes, J. Appl. Mech., Volume 35 (1968), pp. 363-371

[11] J.R. Rice; D.M. Tracey On the ductile enlargement of voids in triaxial stress, J. Mech. Phys. Solids, Volume 17 (1969), pp. 201-217

[12] Y. D'Escatha; J.C. Devaux Numerical study of initiation, stable crack growth and maximum load with a ductile fracture criterion based on the growth of holes (J.D. Landes; J.A. Begley; G.A. Clarke, eds.), Elastic Plastic Fracture, ASTM STP, vol. 668, American Society for Testing and Materials, Philadelphia, 1979, pp. 229-248

[13] A. Needleman; V. Tvergaard An analysis of ductile rupture modes at a crack tip, J. Mech. Phys. Solids, Volume 35 (1987), pp. 151-183

[14] L. Xia; C.F. Shih; J. Hutchinson A computational approach to ductile crack growth under large scale yielding conditions, J. Mech. Phys. Solids, Volume 43 (1995), pp. 389-413

[15] W. Brocks; D. Klingbeil; G. Kunecke; D.-Z. Sun Application of the Gurson model to ductile tearing resistance (M. Kirk; A. Bakker, eds.), The Constraint Effects in Fracture Theory and Applications, vol. 2, ASTM STP, vol. 1244, American Society for Testing and Materials, Philadelphia, 1995, p. 232

[16] K.C. Koppenhoefer; R.H. Dodds Ductile crack growth in pre-cracked CVN specimens: numerical studies, Nuclear Eng. Design, Volume 180 (1998), pp. 221-241

[17] R.M. McMeeking Finite deformation analysis of crack-tip opening in elastic-plastic materials and implication for fracture, J. Mech. Phys. Solids, Volume 25 (1977), pp. 357-381

[18] M. Brunet; F. Morestin; H. Walter Damage identification for anisotropic sheet-metals using a non-local damage model, Int. J. Damage Mech., Volume 13 (2004), pp. 35-57

[19] D. Lassance; D. Fabrègue; F. Delannay; T. Pardoen Micromechanics of room and high temperature fracture in 6xxx Al alloys, Prog. Mater. Sci., Volume 52 (2007), pp. 62-129

[20] T. Pardoen Numerical simulation of low stress triaxiality ductile fracture, Comput. Struct., Volume 84 (2006), pp. 1641-1650

[21] B. Chéhab; Y. Bréchet; M. Véron; P.J. Jacques; G. Parry; J.-C. Glez; J.-D. Mithieux; T. Pardoen Micromechanics of the high temperature damage in a dual phase stainless steel, Acta Mater., Volume 58 (2010), pp. 626-637

[22] L. Devillers-Guerville; J. Besson; A. Pineau Notch fracture toughness of a cast duplex stainless steel: modelling of experimental scatter and size effects, Nuclear Eng. Design, Volume 168 (1997), pp. 211-225

[23] S. Bugat; J. Besson; A. Pineau Micromechanical modeling of the behavior of duplex stainless steels, Comput. Mater. Sci., Volume 16 (1999), pp. 158-166

[24] F. Rivalin; J. Besson; M. Di Fant; A. Pineau Ductile tearing of pipeline-steel wide plates: II. Modeling of in-plane crack propagation, Eng. Fract. Mech., Volume 68 (2001), pp. 347-364

[25] T. Pardoen; D. Dumont; A. Deschamps; Y. Bréchet Grain boundary versus transgranular ductile failure, J. Mech. Phys. Solids, Volume 51 (2003), pp. 637-665

[26] M. Gologanu; J.-B. Leblond; G. Perrin; J. Devaux Recent extensions of Gurson's model for porous ductile metals (P. Suquet, ed.), Continuum Micromechanics, CISM Lectures Series, Springer, New York, 1997, p. 61

[27] T. Pardoen; Y. Marchal; F. Delannay Thickness dependence of cracking initiation criteria in thin aluminum plates, J. Mech. Phys. Solids, Volume 47 (1999), pp. 2093-2123

[28] T. Pardoen; F. Hachez; B. Marchioni; H. Blyth; A.G. Atkins Mode I fracture of sheet metal, J. Mech. Phys. Solids, Volume 52 (2004), pp. 423-452

[29] G. Lacroix; T. Pardoen; P.J. Jacques On the fracture toughness of TRIP-assisted multiphase steels, Acta Mater., Volume 56 (2008), pp. 3900-3913

[30] N. Clément, Phase transformations and mechanical properties of the Ti-5553 β-metastable titanium alloy, Ph.D. thesis, Université catholique de Louvain, 2009.

[31] A.S. Argon; J. Im; R. Safoglu Cavity formation from inclusions in ductile fracture, Metall. Trans. A, Volume 6 (1975), pp. 825-837

[32] M.N. Shabrov; E. Sylven; S. Kim; D.H. Sherman; L. Chuzhoy; C.L. Briant; A. Needleman Void nucleation by inclusion cracking, Metall. Mater. Trans. A, Volume 35 (2004), pp. 1745-1755

[33] B. Budiansky; J.W. Hutchinson; S. Slutsky Void growth and collapse in viscous solids (H.G. Hopkins; M.J. Sewell, eds.), Mechanics of Solids. The Rodney Hill 60th Anniversary Volume, Pergamon Press, Oxford, 1982, p. 13

[34] G. Huber; Y. Brechet; T. Pardoen Void growth and void nucleation controlled ductility in quasi eutectic cast aluminium alloys, Acta Mater., Volume 53 (2005), pp. 2739-2749

[35] D. Lassance; F. Scheyvaerts; T. Pardoen Growth and coalescence of penny-shaped voids in metallic alloys, Eng. Fract. Mech., Volume 73 (2006), pp. 1009-1034

[36] K.C. Liao; J. Pan; S.C. Tang Approximate yield criteria for anisotropic porous ductile sheet metals, Mech. Mater., Volume 26 (1997), pp. 213-226

[37] F. Bron; J. Besson; A. Pineau Ductile rupture in thin sheets of two grades of 2024 aluminum alloy, Mater. Sci. Eng. A, Volume 380 (2004), pp. 356-364

[38] A. A Benzerga; J. Besson; A. Pineau Anisotropic ductile fracture — Part I: experiments, Acta Mater., Volume 52 (2004), pp. 4623-4638

[39] A.A. Benzerga; J. Besson; A. Pineau Anisotropic ductile fracture — Part II: theory, Acta Mater., Volume 52 (2004), pp. 4639-4650

[40] A. Needleman; A.S. Kushner An analysis of void distribution effects on plastic flow in porous solids, Eur. J. Mech. A/Solids, Volume 9 (1990), pp. 193-206

[41] V. Tvergaard Interaction of very small voids with larger voids, Int. J. Solids Struct., Volume 35 (1998), pp. 3989-4000

[42] P.F. Thomason Ductile Fracture of Metals, Pergamon Press, Oxford, 1990

[43] M.J. Worswick; Z.T. Chen; A.K. Pilkey; D. Lloyd; S. Court Damage characterization and damage percolation modeling in aluminum alloy sheet, Acta Mater., Volume 49 (2001), pp. 2791-2803

[44] C. Tekoğlu; T. Pardoen A micromechanics based damage model for composite materials, Int. J. Plast., Volume 26 (2010), pp. 549-569

[45] T.B. Cox; J.R. Low An investigation of the plastic fracture of AISI 4340 and 18 nickel-200 grade maraging steels, Metall. Trans. A, Volume 5 (1974), pp. 1457-1470

[46] G. Perrin; J.B. Leblond Accelerated void growth in porous ductile solids containing two populations of cavities, Int. J. Plast., Volume 16 (2000), pp. 91-120

[47] J. Faleskog; C.F. Shih Micromechanics of coalescence – I. Synergistic effects of elasticity, plastic yielding and multi-size-scale voids, J. Mech. Phys. Solids, Volume 45 (1997), pp. 21-45

[48] D. Fabrègue; T. Pardoen A constitutive model for elastoplastic solids containing primary and secondary voids, J. Mech. Phys. Solids, Volume 56 (2008), pp. 719-741 (Corrigendum to “A constitutive model for elastoplastic solids containing primary and secondary voids” J. Mech. Phys. Solids, 57, 2009, pp. 869-870)

[49] P.F. Thomason Three-dimensional models for the plastic limit-load at incipient failure of the intervoid matrix in ductile porous solids, Acta Metall., Volume 33 (1985), pp. 1079-1085

[50] T. Pardoen; I. Doghri; F. Delannay Experimental and numerical comparison of void growth models and void coalescence criteria for the prediction of ductile fracture in copper bars, Acta Mater., Volume 46 (1998), pp. 541-552

[51] F. Scheyvaerts; P.R. Onck; T. Pardoen A new model for void coalescence by internal necking, Int. J. Damage Mech., Volume 19 (2010), pp. 95-126

[52] J.W. Hancock; A.C. Mackenzie On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states, J. Mech. Phys. Solids, Volume 14 (1977), pp. 147-169

[53] B. Marini; F. Mudry; A. Pineau Experimental study of cavity growth in ductile rupture, Eng. Fract. Mech., Volume 6 (1985), pp. 989-996

[54] A. Simar; K.L. Nielsen; B. de Meester; V. Tvergaard; T. Pardoen Micro-mechanical modelling of ductile failure in 6005A aluminium using a physics based strain hardening law including stage IV, Eng. Fract. Mech., Volume 77 (2010), pp. 2491-2503

[55] V. Tvergaard; J.W. Hutchinson Two mechanisms of ductile fracture: void by void growth versus multiple void interaction, Int. J. Solids Struct., Volume 39 (2002), pp. 3581-3597

[56] A. Needleman; V. Tvergaard An analysis of ductile rupture in notched bars, J. Mech. Phys. Solids, Volume 32 (1984), pp. 461-490

[57] J. Besson; D. Steglich; W. Brocks Modelling of plane strain ductile rupture, Int. J. Plast., Volume 19 (2003), pp. 1517-1541

[58] T. Pardoen; J.W. Hutchinson Micromechanics-based model for trends in toughness of ductile metals, Acta Mater., Volume 51 (2003), pp. 133-148

[59] A. 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., Volume 99 (1977), pp. 2-15

[60] M. Gologanu; J.B. Leblond; J. Devaux Approximate models for ductile metals containing non-spherical voids – case of axisymmetric prolate ellipsoidal cavities, J. Mech. Phys. Solids, Volume 41 (1993), pp. 1723-1754

[61] M. Gologanu; J.B. Leblond; J. Devaux Approximate models for ductile metals containing non-spherical voids – case of axisymmetric oblate ellipsoidal cavities, J. Eng. Mater. Technol., Volume 116 (1994), pp. 290-297

[62] V. Monchiet; O. Cazacu; E. Charkaluk; D. Kondo Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids, Int. J. Plast., Volume 24 (2008), pp. 1158-1189

[63] T. Pardoen; J.W. Hutchinson An extended model for void growth and coalescence, J. Mech. Phys. Solids, Volume 48 (2000), pp. 2467-2512

[64] O. Pierard; I. Doghri A study of various estimates of the macroscopic tangent operator in the incremental homogenization of elasto-plastic composites, Int. J. Multiscale Comput. Eng., Volume 4 (2006), pp. 521-543

[65] I. Doghri; C. Friebel Effective elasto-plastic properties of inclusion-reinforced composites study of shape, orientation and cyclic response, Mech. Mater., Volume 37 (2005), pp. 45-68

[66] I. Doghri; A. Ouaar Homogenization of two-phase elasto-plastic composite materials and structures — study of tangent operators, cyclic plasticity and numerical algorithms, Int. J. Solids Struct., Volume 40 (2003), pp. 1681-1712

[67] M. Bornert; T. Bretheau; P. Gilormini Homogénéisation en mécanique des matériaux, 1. Matériaux aléatoires élastiques et milieux périodiques, HERMES Science, Paris, 2001

[68] C. Chu; A. Needleman Void nucleation effects in biaxially stretched sheets, J. Eng. Mater. Technol., Volume 102 (1980), pp. 249-256

[69] F.M. Beremin Cavity formation from inclusions in ductile fracture of A508 steel, Metall. Trans. A, Volume 12 (1981), pp. 723-731

[70] J.B. Leblond; G. Perrin; J. Devaux An improved Gurson-type model for hardenable ductile metals, Eur. J. Mech. A/Solids, Volume 14 (1995), pp. 499-527

[71] H. Klöcker; V. Tvergaard Growth and coalescence of non-spherical voids in metals deformed at elevated temperature, Int. J. Mech. Sci., Volume 25 (2003), pp. 1283-1308

[72] M. Grange; J. Besson; E. Andrieu An anisotropic Gurson model to represent the ductile rupture of hydrided Zircaloy–4 sheets, Int. J. Fract., Volume 105 (2000), pp. 273-293

[73] A.A. Benzerga; J. Besson; R. Batisse; A. Pineau Synergistic effects of plastic anisotropy and void coalescence on fracture mode in plane strain, Mod. Simul. Mater. Sci. Eng., Volume 10 (2002), pp. 73-102

[74] S.M. Keralavarma; A. Benzerga An approximate yield criterion for anisotropic porous media, C. R. Mécanique, Volume 336 (2008), pp. 685-692

[75] M. Mear; J.W. Hutchinson Influence of yield surface curvature on flow localization in dilatant plasticity, Mech. Mater., Volume 4 (1985), pp. 395-407

[76] J. Besson; C. Guillemer-Neel An extension of the Green and Gurson models to kinematic hardening, Mech. Mater., Volume 35 (2003), pp. 1-18

[77] U. Mühlich; W. Brocks On the numerical integration of a class of pressure-dependent plasticity models including kinematic hardening, Comput. Mech., Volume 31 (2003), pp. 479-488

[78] K. Nahshon; J.W. Hutchinson Modification of the Gurson Model for shear failure, Eur. J. Mech. A/Solids, Volume 27 (2008), pp. 1-17

[79] P.F. Thomason A theory for ductile fracture by internal necking of cavities, J. Institute Metals, Volume 96 (1968), pp. 360-365

[80] P.F. Thomason A three-dimensional model for ductile fracture by the growth and coalescence of microvoids, Acta Metall., Volume 33 (1985), pp. 1087-1095

[81] F. Scheyvaerts, P.R. Onck, T. Pardoen, Multiscale modelling of ductile fracture in heterogeneous metallic alloys, Ph.D. thesis, Universite catholique de Louvain, 2009.

[82] M. Kaisalam; P. Ponte-Castañeda A general constitutive theory for linear and nonlinear particulate media with microstructure evolution, J. Mech. Phys. Solids, Volume 46 (1998), pp. 427-465

[83] S.K. Yerra; C. Tekoğlu; F. Scheyvaerts; L. Delannay; A. Van Bael; P. Van Houtte; T. Pardoen Void growth and coalescence in single crystals, Int. J. Solids Struct., Volume 47 (2010), pp. 1016-1029

[84] W. Brocks; D.Z. Sun; A. Hönig Verification of the transferability of micromechanical parameters by cell model calculations with viscoplastic materials, Int. J. Plast., Volume 11 (1995), pp. 971-989

[85] X. Gao; J. Kim Modelling of ductile fracture: significance of void coalescence, Int. J. Solids Struct., Volume 43 (2006), pp. 6277-6293

[86] J. Koplik; A. Needleman Void growth and coalescence in porous plastic solids, Int. J. Solids Struct., Volume 24 (1988), pp. 835-853

[87] A.A. Benzerga Micromechanics of coalescence in ductile fracture, J. Mech. Phys. Solids, Volume 50 (2002), pp. 1331-1362

[88] V. Tvergaard Studies of void growth in a thin ductile layer between ceramics, Comput. Mech., Volume 20 (1997), pp. 186-191

[89] J.B. Leblond; G. Mottet A theoretical approach of strain localization within thin planar bands in porous ductile materials, C. R. Mécanique, Volume 336 (2008), pp. 176-189

[90] N.A. Fleck; J.W. Hutchinson Strain gradient plasticity, Adv. Appl. Mech., Volume 33 (1997), pp. 295-361

[91] J. Wen; Y. Huang; K.C. Hwang; C. Liu; M. Li The modified Gurson model accounting for the void size effect, Int. J. Plast., Volume 21 (2005), pp. 381-395

[92] C.F. Niordson Strain gradient plasticity effects in whisker-reinforced metals, J. Mech. Phys. Solids, Volume 51 (2003), pp. 1863-1883

[93] J.R. Rice The localization of plastic deformation (W.T. Koiter, ed.), Theoretical and Applied Mechanics, North-Holland Publ. Co., 1976, pp. 207-220

[94] A. Needleman; J.R. Rice Limits to ductility set by plastic flow localization (D.P. Koistinen; N.M. Wang, eds.), Mechanics of Sheet Metal Forming, Plenum Press, New York, 1978, p. 237

[95] A. Needleman; V. Tvergaard Analyses of plastic flow localization in metals, Appl. Mech. Rev., Volume 45 (1992), p. S3-S18

[96] H. Yamamoto Conditions for shear localization in the ductile fracture of void-containing materials, Int. J. Fract., Volume 14 (1978), pp. 347-365

[97] L.M. Brown, J.D. Embury, The initiation and growth of voids at second phase particles, in: Proc. Third Int. Conf. on the Strength of Metals and Alloys, ICSMA 3, Cambridge, England, 1973, p. 164.

[98] J.-B. Leblond; G. Perrin; J. Devaux Bifurcation effects in ductile metals with nonlocal damage, J. Appl. Mech., Volume 61 (1994), pp. 236-242

[99] V. Tvergaard; A. Needleman Effects of non local damage in porous plastic solids, Int. J. Solids Struct., Volume 32 (1995), pp. 1063-1077

[100] F. Reusch; B. Svendsen; D. Klingbeil Local and non-local Gurson-based ductile damage and failure modelling at large deformation, Eur. J. Mech. A/Solids, Volume 22 (2003), pp. 770-792

[101] M.G.D. Geers Finite strain logarithmic hyperelasto-plasticity with softening: a strongly nonlocal implicit gradient framework, Comput. Methods Appl. Mech. Eng., Volume 193 (2004), pp. 3377-3401

[102] J. Mediavilla; R.H.J. Peerlings; M.G.D. Geers A nonlocal triaxiality-dependent ductile damage model for finite strain plasticity, Comput. Methods Appl. Mech. Eng., Volume 66 (2006), pp. 661-688

[103] A.E. Huespe; A. Needleman; J. Oliver; P.J. Sánchez A finite thickness band method for ductile fracture analysis, Int. J. Plast., Volume 25 (2009), pp. 2349-2365

[104] V. Tvergaard; J.W. Hutchinson The relationship between crack growth resistance and fracture process parameters in elastic plastic solids, J. Mech. Phys. Solids, Volume 40 (1992), pp. 1377-1397

[105] D. Steglich; W. Brocks; J. Heerens; T. Pardoen Anisotropic ductile damage modelling of Al2024 alloys, Eng. Fract. Mech., Volume 75 (2008), pp. 3692-3706

[106] D. Dumont; A. Deschamps; Y. Bréchet A model for predicting fracture mode and toughness in 7000 series aluminium alloys, Acta Mater., Volume 52 (2004), pp. 2529-2540

[107] T.F. Morgeneyer; M.J. Starink; S.C. Wang; I. Sinclair Quench sensitivity of toughness in an Al alloy: direct observation and analysis of failure initiation at the precipitate-free zone, Acta Mater., Volume 56 (2008), pp. 2872-2884

[108] T. Krol; D. Baither; E. Nembach The formation of precipitate free zones along grain boundaries in a superalloy and the ensuing effects on its plastic deformation, Acta Mater., Volume 52 (2004), pp. 2095-2108

[109] T. Pardoen; Y. Bréchet Influence of microstructure driven strain localisation on ductile fracture of metallic alloys, Philos. Mag., Volume 84 (2004) no. 3–5, pp. 269-298

[110] R.S. Mishra; Z.Y. Ma Friction stir welding and processing, Mater. Sci. Eng. R, Volume 50 (2005), pp. 1-78

[111] R. Nandan; T. DebRoy; H.K.D.H. Bhadeshia Recent advances in friction-stir welding – process, weldment structure and properties, Prog. Mater. Sci., Volume 53 (2008), pp. 980-1023

[112] P.L. Threadgill; A.J. Leonard; H.R. Shercliff; P.J. Withers Friction stir welding of aluminium alloys, Int. Mater. Rev., Volume 54 (2009), pp. 49-93

[113] A. Simar; J. Lecomte-Beckers; T. Pardoen; B. de Meester Effect of boundary conditions and heat source distribution on temperature distribution in friction stir welding, Sci. Technol. Weld. Joining, Volume 11 (2006), pp. 170-176

[114] A. Simar; T. Pardoen; B. de Meester Effect of rotational material flow on temperature distribution in friction stir welds, Sci. Technol. Weld. Joining, Volume 12 (2007) no. 4, pp. 324-332

[115] A. Simar; Y. Bréchet; B. de Meester; A. Denquin; T. Pardoen Microstructure, local and global mechanical properties of friction stir welds in aluminium alloy 6005A-T6, Mater. Sci. Eng. A, Volume 486 (2008), pp. 85-95

[116] A. Simar; Y. Bréchet; B. de Meester; A. Denquin; T. Pardoen Sequential modelling of local precipitation, strength and strain hardening in friction stir welds of an aluminium alloy 6005A-T6, Acta Mater., Volume 55 (2007), pp. 6133-6143

[117] K.L. Nielsen; T. Pardoen; V. Tvergaard; B. de Meester; A. Simar Modelling of plastic flow localisation and damage development in friction stir welded 6005A aluminium alloy using physics based strain hardening law, Int. J. Solids Struct., Volume 47 (2010), pp. 2359-2370

[118] C. Gallais; A. Simar; D. Fabrègue; A. Denquin; G. Lapasset; B. de Meester; Y. Bréchet; T. Pardoen Multiscale analysis of the strength and ductility of 6056 aluminium friction stir welds, Metall. Mater. Trans. A, Volume 38 (2007), pp. 964-981

Cited by Sources:

Comments - Policy