Stress/Strain Induced Void?


Fracture micro mechanisms of ductile porous solids are substantially researched worldwide since last 60 years through different experiments, theories, thermodynamics and computer models. It is still attracting immense interests to the scientists/engineers as evidenced by a slew of many interesting and innovative techniques, different perceptions and philosophies to elucidate ductile fracture micro mechanisms of materials. The damage accumulation (i.e., void volume fraction, fv) inside a ductile material under tensile deformation is strongly dependent on many engineering/metallurgical variables with their complex and unknown interactions. The role of micro void nucleation and growth during ductile fracture of materials under different environments and loading conditions is well established and documented, but the details of some micro mechanisms governing this fracture process are not still clearly understood. Such as, the coalescence of micro voids has not been clarified clearly, since it is an unstable and rapidly occurred phenomenon in materials. A comprehensive and exhaustive literature review has been performed to realize these facts completely. This article also critically reviews the standard computational methods often widely used for fracture mechanics analysis, which have been proposed/developed by eminent scientists to simulate the ductile fracture of materials. Many studies monitoring the damage accumulation during tensile deformation of different ductile alloys which are, in principle, affected by the imposed stress triaxiality, applied stress and the resulting plastic strains, are already available in the open published literatures. But it is still not clear in these circumstances, whether this damage accumulation is stress assisted or strain induced. In the current investigation, it has been demonstrated through experiments, modeling and reviewing from existing literature that damage accumulation inside a material can be effectively explained by imposed stress triaxiality. This article would be truly being a gift to the structural materials and solid mechanics communities as a whole.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20



Advanced high strength steels


Artificial neural network


American iron and steel institute

bcc :

Body centered cubic


Bayesian neural network

C :

Micro void coalescence


Continuum damage mechanics

CP :

Crystal plasticity


Crystal plasticity-finite element modeling


Deformation induced martensite

DP :

Dual phase


Dynamic recrystallization


Electron back scatter diffraction


Embedded-atom method

EL :


F :



Finite element modeling


Finite element analysis

fcc :

Face centered cubic

GT :

Gurson–Tvergaard model

G :

Micro void growth


Geometrically necessary dislocations

GB :

Grain boundary


Grain boundary triple points

GS :

Grain size


Gurson–Tvergaard–Needleman model


Golaganu–Leblond–Devaux model

HT :

High temperature


High strength low alloy steels

hcp :

Hexagonal closed packed

IP :

Image processing

J :


K :

Strain hardening co-efficient

LT :

Low temperature


Log predictive error

MD :

Molecular dynamics


Metal matrix composites

N :

Micro void nucleation

NL :

Notch length

L :


M :

Strain rate sensitivity


Nearest neighboring distance


Optical light microscope


Oxygen-free high thermal conductivity


Poly (methyl methacrylate)


Representative Volume Element

RT :

Room temperature

RA :

Reduction in area

SR :

Strain rate


Scanning electron microscope


Stacking fault energy

SD :

Standard deviation


Sretch zone width

T :


TB :

Twin boundary


Transformation induced plasticity


Transmission electron microscope


Ultimate tensile strength

YS :

Yield strength


X-ray diffraction


X-ray computer tomography

ΔG c :

Critical free energy change

E :

Young’s modulus

ε :


ε c :

Critical strain

ε f :

Fracture strain

ε eff :

Effective strain

ε eq :

Equivalent strain

ε p :

Plastic strain

f 0 :

Initial void volume fraction

f c :

Critical void volume fraction

f v :

Void volume fraction

γ :

Shear stress

l 0 :

Intervoid spacing

l c :

Critical intervoid ligament distance

n :

Strain hardening exponent

P :


r 0 :

Initial void size

ρ d :

Dislocation density

σ :

Applied stress

σ N :

Normal stress

σ h :

Hydrostatic tensile stress

σ r :

Residual stress

σ c :

Critical stress

σ f :

Fracture stress

σ m :

Mean stress

σ eq :

Equivalent stress

σ eff :

Effective stress

σ υ :

Sigma noise

T e :

Test error

t r :

Rupture time

τ :

Stress triaxiality ratio


  1. 1.

    Gurson AL (1977) Continuum theory of ductile rupture by void nucleation and growth: part I: yield criteria and flow rules for porous ductile media. J Eng Mater Tech 99:2

    Article  Google Scholar 

  2. 2.

    Knott JF (1973) Fundamentals of fracture mechanics. Gruppo Italiano Frattura, Butterworth, London

    Google Scholar 

  3. 3.

    Benzerga AA, Leblond JB (2010) Ductile fracture by void growth to coalescence. Book Chapter: Advances in Applied Mechanics 44:169–305 (Edited by Hassan Aref, Erik van der Giessen Volume 44, Pages 1–308 (2010))

  4. 4.

    Pineau A, Benzerga AA, Pardoen T (2016) Failure of metals I: brittle and ductile fracture. Acta Mater 107:424–483

    Article  Google Scholar 

  5. 5.

    Garrison WM Jr, Moody NR (1987) Ductile fracture. J Phys Chem Sol 48:1035–1074

    Article  Google Scholar 

  6. 6.

    Goods SH, Brown LM (1979) Overview No. 1: the nucleation of cavities by plastic deformation. Acta Metall 27:1–15

    Article  Google Scholar 

  7. 7.

    Rice JR, Johnson MA (1970) Inelastic behavior of solids. McGraw-Hill, New York, p 641

    Google Scholar 

  8. 8.

    Levitas VI, Altukhova NS (2011) Thermodynamics and kinetics of nanovoid nucleation inside elastoplastic material. Acta Mater 59:7051–7059

    Article  Google Scholar 

  9. 9.

    Das A (2017) Fracture complexity of pressure vessel steels. Philos Mag 97:3084–3141

    Article  Google Scholar 

  10. 10.

    Ruggieri C, Dodds RH (1996) A transferability model for brittle fracture including constraint and ductile tearing effects: a probabilistic approach. Int J Fract 79:309–340

    Article  Google Scholar 

  11. 11.

    Davies PW, Dennison JP (1958) Void nucleation sites in creep of metals. Nature 182:131–132

    Article  Google Scholar 

  12. 12.

    Hull D, Rimmer DE (1959) The growth of grain-boundary voids under stress. Philos Mag 4:673–687

    Article  Google Scholar 

  13. 13.

    Puttick KE (1959) Ductile fracture in metals. Philos Mag 4:964–969

    Article  Google Scholar 

  14. 14.

    Greenwood JN, Miller DR, Suiter JW (1954) Intergranular cavitation in stressed metals. Acta Metall 2:250–258

    Article  Google Scholar 

  15. 15.

    Kear BH, Piearcey BJ (1967) Tensile and creep properties of single crystals of the nickel–base super alloy Mar-M 200 (Nickel-base single crystal super alloy tensile and creep properties, comparing single and multiple slip orientations). AIME Trans 239:1209–1215

    Google Scholar 

  16. 16.

    Rice JR, Tracey DM (1969) On the ductile enlargement of voids in triaxial stress fields. J Mech Phys Sol 17:201–217

    Article  Google Scholar 

  17. 17.

    McClintock FA (1968) A criterion for ductile fracture by the growth of holes. J Appl Mech 35:363–371

    Article  Google Scholar 

  18. 18.

    Calhoun CD, Stoloff NS (1970) The effects of particles on fracture processes in magnesium alloys. Metall Trans 1:997–1006

    Google Scholar 

  19. 19.

    Kirman I (1971) The relation between microstructure and toughness in 7075 aluminum alloy. Metall Trans 2:1761–1770

    Google Scholar 

  20. 20.

    Greenfield MA, Margolin H (1972) The mechanism of void formation, void growth and tensile fracture in an alloy consisting of two ductile phases. Metall Trans 3:2649–2659

    Article  Google Scholar 

  21. 21.

    Cox TB, Low JR Jr. (1972) NASA Technical Report No. 4 on Research Grant NGR-39-087-003, Carnegie-Mellon University, USA

  22. 22.

    Broek D (1973) The role of inclusions in ductile fracture and fracture toughness. Eng Fract Mech 5:55IN157–6IN666

    Google Scholar 

  23. 23.

    Raj R, Ashby MF (1975) Intergranular fracture at elevated temperature. Acta Metall 23:653–666

    Article  Google Scholar 

  24. 24.

    Dyson BF, Loveday MS, Rodgers MJ (1976) Grain boundary cavitation under various states of applied stress. Proc R Soc Lond A 349:245–259

    Article  Google Scholar 

  25. 25.

    Okamoto S, Terasaki F, Kunitake T (1977) Influence of MnS inclusion on ductility and ductile fracture process of high strength steels. Tetsu-to-Hagane 63:1878–1886

    Article  Google Scholar 

  26. 26.

    Van Stone RH, Low JR, Shannon JL (1978) Investigation of the fracture mechanism of Ti–5Al–2.5 Sn at cryogenic temperatures. Metall Trans A 9:539–552

    Article  Google Scholar 

  27. 27.

    Koss DA, Chan KS (1980) Fracture along planar slip bands. Acta Metall 28:1245–1252

    Article  Google Scholar 

  28. 28.

    Nieh TG, Nix WD (1980) A comparison of the dimple spacing on intergranular creep fracture surfaces with the slip band spacing for copper. Scripta Metall 14:365–368

    Article  Google Scholar 

  29. 29.

    Tvergaard V, Needleman A, Lo KK (1981) Flow localization in the plane strain tensile test. J Mech Phys Sol 29:115–142

    MATH  Article  Google Scholar 

  30. 30.

    Beremin FM (1981) Cavity formation from inclusions in ductile fracture of A508 steel. Metall Trans A 12:723–731

    Article  Google Scholar 

  31. 31.

    Kim NJ, Thomas G (1981) Effects of morphology on the mechanical behavior of a dual phase Fe–2Si–0.1C steel. Metall Trans A 12:483–489

    Article  Google Scholar 

  32. 32.

    Speich GR, Spitzig WA (1982) Effect of volume fraction and shape of sulfide inclusions on through-thickness ductility and impact energy of high-strength 4340 plate steels. Metall Trans A 13:2239–2258

    Article  Google Scholar 

  33. 33.

    Garber R, Bernstein IM, Thompson AW (1981) Hydrogen assisted ductile fracture of spheroidized carbon steels. Metall Trans A 12:225–234

    Article  Google Scholar 

  34. 34.

    Shi YW, Barnby JT (1984) Void nucleation during tensile deformation of a C-Mn structural steel. Int J Fract 25:143–151

    Article  Google Scholar 

  35. 35.

    Tweed JH, Knott JF (1987) Micromechanisms of failure in C–Mn weld metals. Acta Metall 35:1401–1414

    Article  Google Scholar 

  36. 36.

    Thomson RD, Hancock JW (1984) Ductile failure by void nucleation, growth and coalescence. Int J Fract 26:99–112

    Article  Google Scholar 

  37. 37.

    Vijayaraghavan TV, Margolin H (1988) The effect of matrix strength on void nucleation and growth in an alpha-beta titanium alloy, CORONA-5. Metall Trans A 19:591–601

    Article  Google Scholar 

  38. 38.

    Bowen P, Druce SG, Knott JF (1986) Effects of microstructure on cleavage fracture in pressure vessel steel. Acta Metall 34:1121–1131

    Article  Google Scholar 

  39. 39.

    Maloney JL, Garrison WM (1989) Comparison of void nucleation and growth at MnS and Ti2CS inclusions in HY180 steel. Scripta Metall 23:2097–2100

    Article  Google Scholar 

  40. 40.

    Shi YW, Barnby JT, Nadkarni AS (1991) Void growth at ductile crack initiation of a structural steel. Eng Fract Mech 39:37–44

    Article  Google Scholar 

  41. 41.

    Bray JW, Handerhan KJ, Garrison WM, Thompson AW (1992) Fracture toughness and the extents of primary void growth. Metall Trans A 23:485–496

    Article  Google Scholar 

  42. 42.

    Kosco JB, Koss DA (1993) Work hardening of mechanically alloyed Fe2-Y2O3 alloys. Mater Sci Eng A 169:1–7

    Article  Google Scholar 

  43. 43.

    Kwon D, Asaro RJ (1990) A study of void nucleation, growth and coalescence in spheroidized 1518 steel. Metall Trans A 21:117–134

    Article  Google Scholar 

  44. 44.

    Helbert AL, Feaugas X, Clavel M (1996) The influence of stress trlaxiality on the damage mechanisms in an equiaxed & #x03B1;/β Ti–6Al–4V alloy. Metall Mater Trans A 27:3043–3058

    Article  Google Scholar 

  45. 45.

    Wang ZP (1994) Void growth and compaction relations for ductile porous materials under intense dynamic general loading conditions. Int J Sol Str 31:2139–2150

    MATH  Article  Google Scholar 

  46. 46.

    Leblond JB, Perrin G, Devaux J (1994) Bifurcation effects in ductile metals with nonlocal damage. J Appl Mech 61:236–242

    MATH  Article  Google Scholar 

  47. 47.

    Geltmacher AB, Matic P, Harvey DP (1996) An experimental study evaluating macroscopic cavity interactions as a model of microvoid ductility. J Eng Mater Tech 118:515–521

    Article  Google Scholar 

  48. 48.

    Atkins AG (1996) Fracture in forming. J Mater Proc Tech 56:609–618

    Article  Google Scholar 

  49. 49.

    Miyata T, Tagawa T, Aihara S (1997) Influence of pre-strain on fracture toughness and stable crack growth in low carbon steels. In: Fatigue and fracture mechanics: 28th Volume. ASTM International

  50. 50.

    Pardoen T, Delannay F (1998) Assessment of void growth models from porosity measurements in cold-drawn copper bars. Metall Mater Trans A 29:1895–1909

    Article  Google Scholar 

  51. 51.

    Goto DM, Koss DA, Jablokov V (1999) The influence of tensile stress states on the failure of HY-100 steel. Metall Mater Trans A 30:2835–2842

    Article  Google Scholar 

  52. 52.

    Chan KS, Davidson DL (1999) Evidence of void nucleation and growth on planar slip bands in a Nb–Cr–Ti alloy. Metall Mater Trans A 30:579–585

    Article  Google Scholar 

  53. 53.

    Ishikawa N, Parks DM, Kurihara M (2000) Micromechanism of ductile crack initiation in structural steels based on void nucleation and growth. ISIJ Int 40:519–527

    Article  Google Scholar 

  54. 54.

    Zhang ZL, Thaulow C, Odegard J (2000) A complete Gurson model approach for ductile fracture. Eng Fract Mech 67:155–168

    Article  Google Scholar 

  55. 55.

    Bandstra JP, Koss DA (2001) Modeling the ductile fracture process of void coalescence by void-sheet formation. Mater Sci Eng A 319:490–495

    Article  Google Scholar 

  56. 56.

    Nagumo M, Yoshida H, Shimomura Y, Kadokura T (2001) Ductile crack growth resistance in hydrogen-charged steels. Mater Trans 42:132–137

    Article  Google Scholar 

  57. 57.

    Erdogan M, Tekeli S (2002) The effect of martensite particle size on tensile fracture of surface-carburised AISI 8620 steel with dual phase core microstructure. Mater Des 23:597–604

    Article  Google Scholar 

  58. 58.

    Xue Q, Meyers MA, Nesterenko VF (2002) Self-organization of shear bands in titanium and Ti–6Al–4V alloy. Acta Mater 50:575–596

    Article  Google Scholar 

  59. 59.

    Agarwal H, Gokhale AM, Graham S, Horstemeyer MF (2003) Void growth in 6061-aluminum alloy under triaxial stress state. Mater Sci Eng A 341:35–42

    Article  Google Scholar 

  60. 60.

    Kumar KS, Suresh S, Chisholm MF, Horton JA, Wang P (2003) Deformation of electrodeposited nanocrystalline nickel. Acta Mater 51:387–405

    Article  Google Scholar 

  61. 61.

    Park KT, Myung SH, Shin DH, Lee CS (2004) Size and distribution of particles and voids pre-existing in equal channel angular pressed 5083 Al alloy: their effect on cavitation during low-temperature superplastic deformation. Mater Sci Eng A 371:178–186

    Article  Google Scholar 

  62. 62.

    Chae D, Koss DA (2004) Damage accumulation and failure of HSLA-100 steel. Mater Sci Eng A 366:299–309

    Article  Google Scholar 

  63. 63.

    Shabrov MN, Briant CL, Needleman A, Kim S, Sylven E, Sherman DH, Chuzhoy L (2004) Void nucleation by inclusion cracking. Metall Mater Trans A 35:1745–1755

    Article  Google Scholar 

  64. 64.

    Benzerga AA, Besson J, Pineau A (2004) Anisotropic ductile fracture: part I: experiments. Acta Mater 52:4623–4638

    Article  Google Scholar 

  65. 65.

    Molinari A, Wright TW (2005) A physical model for nucleation and early growth of voids in ductile materials under dynamic loading. J Mech Phys Sol 53:1476–1504

    MATH  Article  Google Scholar 

  66. 66.

    Potirniche GP, Hearndon JL, Horstemeyer MF, Ling XW (2006) Lattice orientation effects on void growth and coalescence in fcc single crystals. Int J Plast 22:921–942

    MATH  Article  Google Scholar 

  67. 67.

    Gallais C, Simar A, Fabregue D, Denquin A, Lapasset G, De Meester B, Brechet Y, Pardoen T (2007) Multiscale analysis of the strength and ductility of AA 6056 aluminum friction stir welds. Metall Mater Trans A 38:964–981

    Article  Google Scholar 

  68. 68.

    Das A, Tarafder S (2009) Experimental investigation on martensitic transformation and fracture morphologies of austenitic stainless steel. Int J Plast 25:2222–2247

    Article  Google Scholar 

  69. 69.

    Das A, Das SK, Tarafder S (2009) Correlation of fractographic features with mechanical properties in systematically varied microstructures of Cu-strengthened high-strength low-alloy steel. Metall Mater Trans A 40:3138–3145

    Article  Google Scholar 

  70. 70.

    Lillo T, Cole J, Frary M, Schlegel S (2009) Influence of grain boundary character on creep void formation in alloy 617. Metall Mater Trans A 40:2803–2811

    Article  Google Scholar 

  71. 71.

    Avramovic-Cingara G, Saleh CA, Jain MK, Wilkinson DS (2009) Void nucleation and growth in dual-phase steel 600 during uniaxial tensile testing. Metall Mater Trans A 40:3117

    Article  Google Scholar 

  72. 72.

    Avramovic-Cingara G, Ososkov Y, Jain MK, Wilkinson DS (2009) Effect of martensite distribution on damage behaviour in DP600 dual phase steels. Mater Sci Eng A 516:7–16

    Article  Google Scholar 

  73. 73.

    Bringa EM, Traiviratana S, Meyers MA (2010) Void initiation in fcc metals: effect of loading orientation and nanocrystalline effects. Acta Mater 58:4458–4477

    Article  Google Scholar 

  74. 74.

    Landron C, Bouaziz O, Maire E, Adrien J (2010) Characterization and modeling of void nucleation by interface decohesion in dual phase steels. Scripta Mater 63:973–976

    Article  Google Scholar 

  75. 75.

    Landron C, Maire E, Bouaziz O, Adrien J, Lecarme L, Bareggi A (2011) Validation of void growth models using x-ray microtomography characterization of damage in dual phase steels. Acta Mater 59:7564–7573

    Article  Google Scholar 

  76. 76.

    Nielsen KL, Tvergaard V (2011) Failure by void coalescence in metallic materials containing primary and secondary voids subject to intense shearing. Int J Sol Str 48:1255–1267

    MATH  Article  Google Scholar 

  77. 77.

    Hsu CY, Lee BJ, Mear ME (2009) Constitutive models for power-law viscous solids containing spherical voids. Int J Plast 25:134–160

    MATH  Article  Google Scholar 

  78. 78.

    Ghahremaninezhad A, Ravi-Chandar K (2012) Ductile failure behavior of polycrystalline Al 6061-T6. Int J Fract 174:177–202

    Article  Google Scholar 

  79. 79.

    Xue W, Pan QG, Ren YY, Shang W, Zeng HQ, Liu H (2012) Microstructure and type IV cracking behavior of HAZ in P92 steel weldment. Mater Sci Eng A 552:493–501

    Article  Google Scholar 

  80. 80.

    Fabrègue D, Landron C, Bouaziz O, Maire E (2013) Damage evolution in TWIP and standard austenitic steel by means of 3D X ray tomography. Mater Sci Eng A 579:92–98

    Article  Google Scholar 

  81. 81.

    Yerra SK, Martin G, Veron M, Brechet Y, Mithieux JD, Delannay L, Pardoen T (2013) Ductile fracture initiated by interface nucleation in two-phase elastoplastic systems. Eng Fract Mech 102:77–100

    Article  Google Scholar 

  82. 82.

    Berdin C, Wang H (2013) Local approach to ductile fracture and dynamic strain aging. Int J Fract 182:39–51

    Article  Google Scholar 

  83. 83.

    Yang ZN, Zhang FC, Qu L, Yan ZG, Xiao YY, Liu RP, Zhang XY (2014) Formation of duplex microstructure in Zr-2.3 Nb alloy and its plastic behaviour at various strain rates. Int J Plast 54:163–177

    Article  Google Scholar 

  84. 84.

    Zhang H, Ponge D, Raabe D (2014) Superplastic Mn–Si–Cr–C duplex and triplex steels: interaction of microstructureand void formation. Mater Sci Eng A 610:355–369

    Article  Google Scholar 

  85. 85.

    Komori K (2014) Proposal and use of a void model for inclusion cracking for simulating inner fracture defects in drawing of ferrite-pearlite steels. Mech Mater 77:98–109

    Article  Google Scholar 

  86. 86.

    Das A, Roy N, Ray AK (2014) Stress induced creep cavity. Mater Sci Engg A 598:28–33

    Article  Google Scholar 

  87. 87.

    Brown AD, Wayne L, Pham Q, Krishnan K, Peralta P, Luo SN, Patterson BM, Greenfield S, Byler D, McClellan KJ, Koskelo A (2015) Microstructural effects on damage nucleation in shock-loaded polycrystalline copper. Metall Mater Trans A 46:4539–4547

    Article  Google Scholar 

  88. 88.

    Saeidi N, Ashrafizadeh F, Niroumand B, Forouzan MR, Barlat F (2016) Examination and modeling of void growth kinetics in modern high strength dual phase steels during uniaxial tensile deformation. Mater Chem Phys 172:54–61

    Article  Google Scholar 

  89. 89.

    Nemcko MJ, Wilkinson DS (2016) Impact of microstructure on void growth and linkage in pure magnesium. Int J Fract 200:31–47

    Article  Google Scholar 

  90. 90.

    Gariboldi E, Naumenko K, Ozhoga-Maslovskaja O, Zappa E (2016) Analysis of anisotropic damage in forged Al–Cu–Mg–Si alloy based on creep tests, micrographs of fractured specimen and digital image correlations. Mater Sci Eng A 652:175–185

    Article  Google Scholar 

  91. 91.

    Zhang Y, Jiang S, Zhu X, Zhao Y (2017) Influence of void density on dislocation mechanisms of void shrinkage in nickel single crystal based on molecular dynamics simulation. Phys E 90:90–97

    Article  Google Scholar 

  92. 92.

    Cooper AJ, Tuck OCG, Burnett TL, Sherry AH (2018) A statistical assessment of ductile damage in 304L stainless steel resolved using X-ray computed tomography. Mater Sci Eng A 728:218–230

    Article  Google Scholar 

  93. 93.

    Fu W, Wang R, Zhang J, Wu K, Liu G, Sun J (2018) The effect of precipitates on voiding, twinning, and fracture behaviors in Mg alloys. Mater Sci Eng A 720:98–109

    Article  Google Scholar 

  94. 94.

    Fisher JR, Gurland J (1981) Void nucleation in spheroidized carbon steels part 1: experimental. Metal Sci 15:185–192

    Article  Google Scholar 

  95. 95.

    Fisher JR, Gurland J (1981) Void nucleation in spheroidized carbon steels Part 2: model. Metal Sci 15:193–202

    Article  Google Scholar 

  96. 96.

    Teirlinck D, Embury JD, Ashby MF (1984) Damage accumulation during ductile rupture and the development of failure maps. Fracture 84:105–125, Pergamon

  97. 97.

    Zhang X, Mai YW (2006) A simple damage-accumulation model for constraint effects on ductile fracture. Int J Frac 141:135–145

    MATH  Article  Google Scholar 

  98. 98.

    Wu SX, Cotterell B, Mai YW (1991) On the relationship between crack tip opening displacement at the initiation of a ductile tear in low carbon steel, hydrostatic stress, and void growth. Int J Frac 51:207–218

    Google Scholar 

  99. 99.

    Tracey DM (1971) Strain-hardening and interaction effects on the growth of voids in ductile fracture. Eng Frac Mech 3:301–315

    Article  Google Scholar 

  100. 100.

    Cox TB, Low JR (1974) An investigation of the plastic fracture of AISI 4340 and 18 Nickel-200 grade maraging steels. Metall Trans 5:1457–1470

    Article  Google Scholar 

  101. 101.

    Andersson H (1974) Finite element treatment of a uniformly moving elastic-plastic crack tip. J Mech Phys Sol 22:285–308

    MATH  Article  Google Scholar 

  102. 102.

    Argon AS, Im J, Safoglu R (1975) Cavity formation from inclusions in ductile fracture. Metall Trans A 6:825

    Article  Google Scholar 

  103. 103.

    Cocks ACF, Ashby MF (1982) On creep fracture by void growth. Prog Mater Sci 27:189–244

    Article  Google Scholar 

  104. 104.

    Budiansky B, Hutchinson JW, Slutsky S (1982) Void growth and collapse in viscous solids. In: Mechanics of solids: the Rodney Hill 60th Anniversary, vol 13–45

  105. 105.

    Kim KH, Kim DW (1983) The effect of void growth on the limit strains of steel sheets. Int J Mech Sci 25:293–300

    Article  Google Scholar 

  106. 106.

    Hancock JW, Brown DK (1983) On the role of strain and stress state in ductile failure. J Mech Phys Sol 31:1–24

    Article  Google Scholar 

  107. 107.

    Yoon HS, Taya M (1984) Prediction of the void growth at its early stage in a viscous two-phase material. Int J Eng Sci 22:1035–1040

    Article  Google Scholar 

  108. 108.

    Marino B, Mudry F, Pineau A (1985) Experimental study of cavity growth in ductile rupture. Eng Fract Mech 22:989–996

    Article  Google Scholar 

  109. 109.

    Fleck NA, Hutchinson JW (1986) Void growth in shear. Proc R Soc Lond A 407:435–458

    Article  Google Scholar 

  110. 110.

    Koplik J, Needleman A (1988) Void growth and coalescence in porous plastic solids (Doctoral dissertation, Brown University)

  111. 111.

    Magnusen PE, Dubensky EM, Koss DA (1988) The effect of void arrays on void linking during ductile fracture. Acta Metall 36:1503–1509

    Article  Google Scholar 

  112. 112.

    Hom CL, McMeeking RM (1989) Void growth in elastic-plastic materials. J Appl Mech 56:309–317

    Article  Google Scholar 

  113. 113.

    Shi YW (1989) Critical void growth for ductile rupture of steel welds. Eng Fract Mech 34:901–907

    Article  Google Scholar 

  114. 114.

    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

    Article  Google Scholar 

  115. 115.

    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 Sol 41:1723–1754

    MATH  Article  Google Scholar 

  116. 116.

    Benson DJ (1993) An analysis of void distribution effects on the dynamic growth and coalescence of voids in ductile metals. J Mech Phys Sol 41:1285–1308

    Article  Google Scholar 

  117. 117.

    Steglich D, Brocks W (1997) Micromechanical modelling of the behaviour of ductile materials including particles. Comput Mater Sci 9:7–17

    Article  Google Scholar 

  118. 118.

    Pardoen T, Doghri I, Delannay F (1998) Experimental and numerical comparison of void growth models and void coalescence criteria for the prediction of ductile fracture in copper bars. Acta Mater 46:541–552

    Article  Google Scholar 

  119. 119.

    Pardoen T, Delannay F (1998) The coalescence of voids in prestrained notched round copper bars. Fat Fract Eng Mater Str 21:1459

    Article  Google Scholar 

  120. 120.

    Horstemeyer MF, Gokhale AM (1999) A void-crack nucleation model for ductile metals. Int J Sol Str 36:5029–5055

    MATH  Article  Google Scholar 

  121. 121.

    Pardoen T, Hutchinson JW (2000) An extended model for void growth and coalescence. J Mech Phys Sol 48:2467–2512

    MATH  Article  Google Scholar 

  122. 122.

    Tvergaard V, Hutchinson JW (2002) Two mechanisms of ductile fracture: void by void growth versus multiple void interaction. Int J Sol Str 39:3581–3597

    MATH  Article  Google Scholar 

  123. 123.

    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

    MATH  Article  Google Scholar 

  124. 124.

    Tvergaard V, Needleman A (2006) Three dimensional microstructural effects on plane strain ductile crack growth. Int J Sol Str 43:6165–6179

    MATH  Article  Google Scholar 

  125. 125.

    Liu WH, Zhang XM, Tang JG, Du YX (2007) Simulation of void growth and coalescence behavior with 3D crystal plasticity theory. Comput Mater Sci 40:130–139

    Article  Google Scholar 

  126. 126.

    Hammi Y, Horstemeyer MF (2007) A physically motivated anisotropic tensorial representation of damage with separate functions for void nucleation, growth, and coalescence. Int J Plast 23:1641–1678

    MATH  Article  Google Scholar 

  127. 127.

    Bandstra JP, Koss DA (2008) On the influence of void clusters on void growth and coalescence during ductile fracture. Acta Mater 56:4429–4439

    Article  Google Scholar 

  128. 128.

    Cvijovic Z, Vratnica M, Rakin M, Cvijovi-Alagi I (2008) Micromechanical model for fracture toughness prediction in Al–Zn–Mg–Cu alloy forgings. Philos Mag 88:3153–3179

    Article  Google Scholar 

  129. 129.

    Fabregue D, Pardoen T (2008) A constitutive model for elastoplastic solids containing primary and secondary voids. J Mech Phys Sol 56:719–741

    MATH  Article  Google Scholar 

  130. 130.

    Benzerga AA, Leblond JB (2010) Ductile fracture by void growth to coalescence. Adv App Mech 44:169–305

    Article  Google Scholar 

  131. 131.

    Landron C, Maire E, Adrien J, Bouaziz O, Di Michiel M, Cloetens P, Suhonen H (2012) Resolution effect on the study of ductile damage using synchrotron X-ray tomography. Nucl Instr Metho Phys Res Sec B: Beam Inter Mater Atom 284:15–18

    Article  Google Scholar 

  132. 132.

    Monchiet V, Bonnet G (2013) A Gurson-type model accounting for void size effects. Int J Sol Str 50:320–327

    Article  Google Scholar 

  133. 133.

    He J, Cui Z, Chen F, Xiao Y, Ruan L (2013) The new ductile fracture criterion for 30Cr2Ni4MoV ultra-super-critical rotor steel at elevated temperatures. Mater Des 52:547–555

    Article  Google Scholar 

  134. 134.

    Tekoglu C, Hutchinson JW, Pardoen T (2015) On localization and void coalescence as a precursor to ductile fracture. Philos Trans R Soc A 373:20140121

    Article  Google Scholar 

  135. 135.

    Tekoglu C (2015) Void coalescence in ductile solids containing two populations of voids. Eng Fract Mech 147:418–430

    Article  Google Scholar 

  136. 136.

    Morin L, Leblond JB, Kondo D (2015) A Gurson-type criterion for plastically anisotropic solids containing arbitrary ellipsoidal voids. Int J Sol Str 77:86–101

    Article  Google Scholar 

  137. 137.

    Morin L, Kondo D, Leblond JB (2015) Numerical assessment, implementation and application of an extended Gurson model accounting for void size effects. Eur J Mech A/Sol 51:183–192

    MathSciNet  MATH  Article  Google Scholar 

  138. 138.

    Pushkareva M, Adrien J, Maire E, Segurado J, Llorca J, Weck A (2016) Three-dimensional investigation of grain orientation effects on void growth in commercially pure titanium. Mater Sci Eng A 671:221–232

    Article  Google Scholar 

  139. 139.

    Komori K (2017) Predicting ductile fracture in ferrous materials during tensile tests using an ellipsoidal void model. Mech Mater 113:24–43

    Article  Google Scholar 

  140. 140.

    Needleman A, Triantafyllidis N (1978) Void growth and local necking in biaxially stretched sheets. J Eng Mater Tech 100:164–169

    Article  Google Scholar 

  141. 141.

    Burke MA, Nix WD (1979) A numerical study of necking in the plane tension test. Int J Sol Str 15:379–393

    MATH  Article  Google Scholar 

  142. 142.

    Barnby JA, Shi YW, Nadkarni AS (1984) On the void growth in C-Mn structural steel during plastic deformation. Int J Fract 25:273–283

    Article  Google Scholar 

  143. 143.

    Thompson HE, Knott JF (1986) Effects of crack length and pre-strain on ductile fracture. In: ECF4, Amsterdam

  144. 144.

    Hori M, Nemat-Nasser S (1988) Mechanics of void growth and void collapse in crystals. Mech Mater 7:1–13

    Article  Google Scholar 

  145. 145.

    Hori M, Nemat-Nasser S (1988) Dynamic response of crystalline solids with microcavities. J Appl Phys 64:856–863

    Article  Google Scholar 

  146. 146.

    Becker R, Needleman A, Richmond O, Tvergaard V (1988) Void growth and failure in notched bars. J Mech Phys Sol 36:317–351

    Article  Google Scholar 

  147. 147.

    Lee BJ, Mear ME (1992) Axisymmetric deformation of power-law solids containing a dilute concentration of aligned spheroidal voids. J Mech Phys Sol 40:1805–1836

    Article  Google Scholar 

  148. 148.

    Lee BJ, Mear ME (1992) Effective properties of power-law solids containing elliptical inhomogeneities. part I: rigid inclusions. Mech Mater 13:313–335

    Article  Google Scholar 

  149. 149.

    Lee BJ, Mear ME (1992) Effective properties of power-law solids containing elliptical inhomogeneities. part II: voids. Mech Mater 13:337–356

    Article  Google Scholar 

  150. 150.

    Margolin BZ, Karzov GP, Shvetsova VA, Kostylev VI (1998) Modelling for transcrystalline and intercrystalline fracture by void nucleation and growth. Fat Fract Eng Mater Str 21:123–137

    Article  Google Scholar 

  151. 151.

    Tasan CC, Hoefnagels JMP, Peerlings RHJ, Geers MGD, Ten Horn CHLJ, Vegter H (2007) Ductile damage evolution and strain path dependency. In: AIP conference proceedings, AIP, vol 907, No 1, pp 187–192

  152. 152.

    Khan IA, Bhasin V (2017) On the role of secondary voids and their distribution in the mechanism of void growth and coalescence in porous plastic solids. Int J Sol Str 108:203–215

    Article  Google Scholar 

  153. 153.

    Nielsen KL, Andersen RG, Tvergaard V (2018) Void coalescence mechanism for combined tension and large amplitude cyclic shearing. Eng Fract Mech 189:164–174

    Article  Google Scholar 

  154. 154.

    Perrin G, Leblond JB (2000) Accelerated void growth in porous ductile solids containing two populations of cavities. Int J Plast 16:91–120

    MATH  Article  Google Scholar 

  155. 155.

    Tvergaard V (1998) Interaction of very small voids with larger voids. Int J Sol Str 35:3989–4000

    MathSciNet  MATH  Article  Google Scholar 

  156. 156.

    Ohno N, Hutchinson JW (1984) Plastic flow localization due to non-uniform void distribution. J Mech Phys Sol 32:63–85

    MATH  Article  Google Scholar 

  157. 157.

    Becker R (1987) The effect of porosity distribution on ductile failure. J Mech Phys Sol 35:577–599

    Article  Google Scholar 

  158. 158.

    Brown LM, Embury JD (1973) In Proc. Conf. on Microstructure and Design of Alloys, Institute of Metals and Iron and Steel Insitute, London. 1, (33), pp 164–169

  159. 159.

    Ritchie RO, Server WL, Wullaert RA (1979) Critical fracture stress and fracture strain models for the prediction of lower and upper shelf toughness in nuclear pressure vessel steels. Metall Trans A 10:1557–1570

    Article  Google Scholar 

  160. 160.

    Thomason PF (1990) Ductile fracture of metals. Pergamon Press plc, Ductile Fracture of Metals (UK), 1990, 1990:219

  161. 161.

    Gurland J, Plateau J (1963) The mechanism of ductile rupture of metals containing inclusions (No. TID-17914). Brown Univ., Providence; Institut de Recherches de la Siderugie, St.-Germain-en-Laye, France

  162. 162.

    Hancock JW, Mackenzie AC (1976) On the mechanisms of ductile failure in high-strength steels subjected to multi-axial stress-states. J Mech Phys Sol 24:147–160

    Article  Google Scholar 

  163. 163.

    Rice JR (1976) Localization of plastic deformation (No. COO-3084/43; CONF-760835-1). Brown Univ., Providence, RI (USA). Div. of Engineering

  164. 164.

    Needleman A, Rice JR (1978) Limits to ductility set by plastic flow localization. In: Koistinen DP, Wang NM (eds) Mechanics of sheet metal forming. Springer, Boston, MA. Print ISBN: 978-1-4613-2882-7

  165. 165.

    Cocks ACF, Ashby MF (1980) Intergranular fracture during power-law creep under multiaxial stresses. Metal Sci 14:395–402

    Article  Google Scholar 

  166. 166.

    Thomason PF (1981) Ductile fracture and the stability of incompressible plasticity in the presence of microvoids. Acta Metall 29:763–777

    Article  Google Scholar 

  167. 167.

    Chen IW, Argon AS (1981) Diffusive growth of grain-boundary cavities. Acta Metall 29:1759–1768

    Article  Google Scholar 

  168. 168.

    Tvergaard V (1982) Material failure by void coalescence in localized shear bands. Int J Sol Str 18:659–672

    MATH  Article  Google Scholar 

  169. 169.

    Tvergaard V, Needleman A (1984) Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 32:157–169

    Article  Google Scholar 

  170. 170.

    Needleman A, Tvergaard V (1984) An analysis of ductile rupture in notched bars. J Mech Phys Sol 32:461–490

    Article  Google Scholar 

  171. 171.

    Aravas N, McMeeking RM (1985) Finite element analysis of void growth near a blunting crack tip. J Mech Phys Sol 33:25–49

    Article  Google Scholar 

  172. 172.

    Johnson GR, Cook WH (1985) Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng Fract Mech 21:31–48

    Article  Google Scholar 

  173. 173.

    Rousselier G (1987) Ductile fracture models and their potential in local approach of fracture. Nucl Eng Des 105:97–111

    Article  Google Scholar 

  174. 174.

    Dubensky EM, Koss DA (1987) Void/pore distributions and ductile fracture. Metall Trans A 18:1887–1895

    Article  Google Scholar 

  175. 175.

    McMeeking RM, Hom CL (1990) Finite element analysis of void growth in elastic-plastic materials. Int J Fract 42:1–19

    Article  Google Scholar 

  176. 176.

    Magnusen PE, Srolovitz DJ, Koss DA (1990) A simulation of void linking during ductile microvoid fracture. Acta Metall Mater 38:1013–1022

    Article  Google Scholar 

  177. 177.

    Barton DC, Waheed M, Mirza MS, Church P (1995) A numerical study of ductile void growth under dynamic loading conditions. Int J Fract 73:325–343

    Article  Google Scholar 

  178. 178.

    Gologanu M, Leblond JB, Perrin G, Devaux J (1997) Recent extensions of Gurson’s model for porous ductile metals. In: Suquet P (eds) Continuum micromechanics. International centre for mechanical sciences (Courses and Lectures), vol 377. Springer, Vienna. Print ISBN: 978-3-211-82902-8

  179. 179.

    Thomason PF (1999) Ductile spallation fracture and the mechanics of void growth and coalescence under shock-loading conditions. Acta Mater 47:3633–3646

    Article  Google Scholar 

  180. 180.

    Leblond JB, Perrin G (1999) A self-consistent approach to coalescence of cavities in inhomogeneously voided ductile solids. J Mech Phys Sol 47:1823–1841

    MATH  Article  Google Scholar 

  181. 181.

    Gologanu M, Leblond JB, Perrin G, Devaux J (2001) Theoretical models for void coalescence in porous ductile solids. I. Coalescence, “in layers”. Int J Sol Str 38:5581–5594

    MATH  Article  Google Scholar 

  182. 182.

    Gologanu M, Leblond JB, Devaux J (2001) Theoretical models for void coalescence in porous ductile solids. II. Coalescence “in columns”. Int J Sol Str 38:5595–5604

    MATH  Article  Google Scholar 

  183. 183.

    Guo TF, Cheng L (2002) Modeling vapor pressure effects on void rupture and crack growth resistance. Acta Mater 50:3487–3500

    Article  Google Scholar 

  184. 184.

    Benzerga AA (2002) Micromechanics of coalescence in ductile fracture. J Mech Phys Sol 50:1331–1362

    MATH  Article  Google Scholar 

  185. 185.

    Besson J, Guillemer-Neel C (2003) An extension of the Green and Gurson models to kinematic hardening. Mech Mater 35:1–18

    Article  Google Scholar 

  186. 186.

    Klocker H, Tvergaard V (2003) Growth and coalescence of non-spherical voids in metals deformed at elevated temperature. Int J Mech Sci 45:1283–1308

    MATH  Article  Google Scholar 

  187. 187.

    Siruguet K, Leblond JB (2004) Effect of void locking by inclusions upon the plastic behavior of porous ductile solids-I: theoretical modeling and numerical study of void growth. Int J Plast 20:225–254

    MATH  Article  Google Scholar 

  188. 188.

    Lemaitre J, Desmorat R (2005) Engineering damage mechanics: ductile, creep, fatigue and brittle failures. Springer, Berlin

    Google Scholar 

  189. 189.

    Lassance D, Scheyvaerts F, Pardoen T (2006) Growth and coalescence of penny-shaped voids in metallic alloys. Eng Fract Mech 73:1009–1034

    Article  Google Scholar 

  190. 190.

    Barsoum I, Faleskog J (2007) Rupture mechanisms in combined tension and shear-Experiments. Int J Sol Str 44:1768–1786

    MATH  Article  Google Scholar 

  191. 191.

    Xue L (2008) Constitutive modeling of void shearing effect in ductile fracture of porous materials. Eng Fract Mech 75:3343–3366

    Article  Google Scholar 

  192. 192.

    Nahshon K, Hutchinson JW (2008) Modification of the Gurson model for shear failure. Eur J Mech A/Sol 27:1–17

    MATH  Article  Google Scholar 

  193. 193.

    Weck A, Wilkinson DS, Maire E, Toda H (2008) Visualization by X-ray tomography of void growth and coalescence leading to fracture in model materials. Acta Mater 56:2919–2928

    Article  Google Scholar 

  194. 194.

    Weck A, Wilkinson DS, Maire E (2008) Observation of void nucleation, growth and coalescence in a model metal matrix composite using X-ray tomography. Mater Sci Eng A 488:435–445

    Article  Google Scholar 

  195. 195.

    Tvergaard V, Hutchinson JW (2008) Mode III effects on interface delamination. J Mech Phys Sol 56:215–229

    MathSciNet  MATH  Article  Google Scholar 

  196. 196.

    Nahshon K, Xue Z (2009) A modified Gurson model and its application to punch-out experiments. Eng Fract Mech 76:997–1009

    Article  Google Scholar 

  197. 197.

    Besson J (2010) Continuum models of ductile fracture: a review. Int J Dam Mech 19:3–52

    Article  Google Scholar 

  198. 198.

    Ha S, Kim K (2010) Void growth and coalescence in fcc single crystals. Int J Mech Sci 52:863–873

    Article  Google Scholar 

  199. 199.

    Jacques N, Mercier S, Molinari A (2012) Void coalescence in a porous solid under dynamic loading conditions. Int J Fract 173:203–213

    Article  Google Scholar 

  200. 200.

    Tekoglu C, Leblond JB, Pardoen T (2012) A criterion for the onset of void coalescence under combined tension and shear. J Mech Phys Sol 60:1363–1381

    Article  Google Scholar 

  201. 201.

    Khan AS, Liu H (2012) A new approach for ductile fracture prediction on Al 2024-T351 alloy. Int J Plast 35:1–12

    Article  Google Scholar 

  202. 202.

    Ma D, Chen D, Wu S, Wang H, Cai C, Deng G (2012) Dynamic experimental verification of void coalescence criteria. Mater Sci Eng A 533:96–106

    Article  Google Scholar 

  203. 203.

    Srivastava A, Needleman A (2013) Void growth versus void collapse in a creeping single crystal. J Mech Phys Sol 61:1169–1184

    MathSciNet  Article  Google Scholar 

  204. 204.

    Besson J, McCowan CN, Drexler ES (2013) Modeling flat to slant fracture transition using the computational cell methodology. Eng Fract Mech 104:80–95

    Article  Google Scholar 

  205. 205.

    Benzerga AA, Leblond JB (2014) Effective yield criterion accounting for microvoid coalescence. J Appl Mech 81:031009

    Article  Google Scholar 

  206. 206.

    Cao TS, Gachet JM, Montmitonnet P, Bouchard PO (2014) A Lode-dependent enhanced Lemaitre model for ductile fracture prediction at low stress triaxiality. Eng Fract Mech 124:80–96

    Article  Google Scholar 

  207. 207.

    Malcher L, Pires FA, De Sa JC (2014) An extended GTN model for ductile fracture under high and low stress triaxiality. Int J Plast 54:193–228

    Article  Google Scholar 

  208. 208.

    Molinari A, Jacques N, Mercier S, Leblond JB, Benzerga AA (2015) A micromechanical model for the dynamic behavior of porous media in the void coalescence stage. Int J Sol Str 71:1–18

    Article  Google Scholar 

  209. 209.

    Jiang W, Li Y, Su J (2016) Modified GTN model for a broad range of stress states and application to ductile fracture. Eur J Mech A/Sol 57:132–148

    MathSciNet  MATH  Article  Google Scholar 

  210. 210.

    Gatea S, Ou H, Lu B, McCartney G (2017) Modelling of ductile fracture in single point incremental forming using a modified GTN model. Eng Fract Mech 186:59–79

    Article  Google Scholar 

  211. 211.

    Jablokov V, Goto DM, Koss DA (2001) Damage accumulation and failure of HY-100 steel. Metall Mater Trans A 32:2985–2994

    Article  Google Scholar 

  212. 212.

    Jablokov V, Goto DM, Koss DA, McKirgan JB (2001) Temperature, strain rate, stress state and the failure of HY-100 steel. Mater Sci Eng A 302:197–205

    Article  Google Scholar 

  213. 213.

    Chae D, Young CJ, Goto DM, Koss DA (2001) Failure behavior of heat-affected zones within HSLA-100 and HY-100 steel weldments. Metall Mater Trans A 32:2229–2237

    Article  Google Scholar 

  214. 214.

    Chae D, Bandstra JP, Koss DA (2000) The effect of pre-strain and strain-path changes on ductile fracture: experiment and computational modeling. Mater Sci Eng A 285:165–171

    Article  Google Scholar 

  215. 215.

    Goto DM, Garrett RK, Bingert JF, Chen SR, Gray GT (2000) The mechanical threshold stress constitutive-strength model description of HY-100 steel. Metall Mater Trans A 31:1985–1996

    Article  Google Scholar 

  216. 216.

    Berg CA (1962) The motion of cracks in plane viscous deformation. In: Proceedings of Fourth US national congress of applied mechanics, Berkeley, vol 2, pp 885–892

  217. 217.

    Thomason PF (1968) A theory for ductile fracture by internal necking of cavities. J Inst Metals 96:360

    Google Scholar 

  218. 218.

    Rice JR, Tracey DM (1973) Computational fracture mechanics. In: Fenves SJ, Perrone N, Robinson A, Schnobrich WC (eds) Numerical and computer methods in structural mechanics. Academic Press, New York, pp 585–623

  219. 219.

    Needleman A (1972) Void growth in an elastic-plastic medium. J Appl Mech 39:964–970

    Article  Google Scholar 

  220. 220.

    Needleman A (1972) A numerical study of necking in circular cylindrical bar. J Mech Phys Sol 20:111–127

    MATH  Article  Google Scholar 

  221. 221.

    Green G, Knott JF (1975) Effects of side grooves on initiation and propagation of ductile fracture. Metals Tech 2:422–427

    Article  Google Scholar 

  222. 222.

    McMeeking RM (1976) Finite deformation analysis of crack tip opening in elastic-plastic materials and implications for fracture initiation (No. COO-3084/44). Brown Univ., Providence, RI (USA). Div. of Engineering

  223. 223.

    Mackenzie AC, Hancock JW, Brown DK (1977) On the influence of state of stress on ductile failure initiation in high strength steels. Eng Fract Mech 9:167–188

    Article  Google Scholar 

  224. 224.

    Goldenberg T, Lee TD, Hirth JP (1978) Ductile fracture of U-notched bend specimens of spheroidized AISI 1095 steel. Metall Trans A 9:1663–1671

    Article  Google Scholar 

  225. 225.

    Chu CC, Needleman A (1980) Void nucleation effects in biaxially stretched sheets. ASME J Eng Mater Technol 102:249–256

    Article  Google Scholar 

  226. 226.

    Wilkinson DS, Vitek V (1983) The propagation of cracks by cavitation: a general theory. In: Perspectives in creep fracture. Pergamon, pp 161–170

  227. 227.

    Needleman A, Rice JR (1983) Plastic creep flow effects in the diffusive cavitation of grain boundaries. In: Perspectives in creep fracture, pp 107–124

  228. 228.

    Lemaitre J (1985) A continuous damage mechanics model for ductile fracture. J Eng Mater Tech 107:83–89

    Article  Google Scholar 

  229. 229.

    Tai WH, Yang BX (1986) A new microvoid-damage model for ductile fracture. Eng Fract Mech 25:377–384

    Article  Google Scholar 

  230. 230.

    Thompson AW (1987) Modeling of local strains in ductile fracture. Metall Trans A 18:1877–1886

    Article  Google Scholar 

  231. 231.

    Hutchinson JW, Tvergaard V (1989) Softening due to void nucleation in metals. In: Fracture mechanics: perspectives and directions (Twentieth Symposium) 1989 Jan. ASTM International

  232. 232.

    Needleman A, Tvergaard V (1991) An analysis of dynamic, ductile crack growth in a double edge cracked specimen. Int J Fract 49:41–67

    Article  Google Scholar 

  233. 233.

    Tvergaard V, Hutchinson JW (1992) The relation between crack growth resistance and fracture process parameters in elastic-plastic solids. J Mech Phys Sol 40:1377–1397

    MATH  Article  Google Scholar 

  234. 234.

    Sun LZ, Huang ZP (1992) Dynamic void growth in rate-sensitive plastic solids. Int J Plast 8:903–924

    MATH  Article  Google Scholar 

  235. 235.

    Ruggieri C, Panontin TL, Dodds RH (1996) Numerical modeling of ductile crack growth in 3-D using computational cell elements. Int J Fract 82:67–95

    Article  Google Scholar 

  236. 236.

    Ghosal AK, Narasimhan R (1997) A finite element study of the effect of void initiation and growth on mixed-mode ductile fracture. Mech Mater 25:113–127

    Article  Google Scholar 

  237. 237.

    Bandstra JP, Goto DM, Koss DA (1998) Ductile failure as a result of a void-sheet instability: experiment and computational modeling. Mater Sci Eng A 249:46–54

    Article  Google Scholar 

  238. 238.

    Ragab AR, Saleh CA (2005) Evaluation of bendability of sheet metals using void coalescence models. Mater Sci Eng A 395:102–109

    Article  Google Scholar 

  239. 239.

    Komori K (1999) Proposal and use of a void model for the simulation of ductile fracture behavior. Acta Metar 47:3069–3077

    Article  Google Scholar 

  240. 240.

    Klocker H, Tvergaard V (2000) Void growth and coalescence in metals deformed at elevated temperature. Int J Fract 106:259–276

    MATH  Article  Google Scholar 

  241. 241.

    Horstemeyer MF, Ramaswamy S (2000) On factors affecting localization and void growth in ductile metals: a parametric study. Int J Dam Mech 9:5–28

    Article  Google Scholar 

  242. 242.

    Ragab AR, Saleh CA (2000) Effect of void growth on predicting forming limit strains for planar isotropic sheet metals. Mech Mater 32:71–84

    Article  Google Scholar 

  243. 243.

    Ragab AR (2004) Application of an extended void growth model with strain hardening and void shape evolution to ductile fracture under axisymmetric tension. Eng Fract Mech 71:1515–1534

    Article  Google Scholar 

  244. 244.

    Huber G, Brechet Y, Pardoen T (2005) Predictive model for void nucleation and void growth controlled ductility in quasieutectic cast aluminium alloys. Acta Mater 53:2739–2749

    Article  Google Scholar 

  245. 245.

    Tvergaard V (2007) Discrete modelling of ductile crack growth by void growth to coalescence. Int J Fract 148:1–12

    MATH  Article  Google Scholar 

  246. 246.

    Ragab AR (2008) Prediction of fracture limit curves in sheet metals using a void growth and coalescence model. J Mater Proc Tech 199:206–213

    Article  Google Scholar 

  247. 247.

    Wright TW, Ramesh KT (2009) Statistically informed dynamics of void growth in rate dependent materials. Int J Imp Eng 36:1242–1249

    Article  Google Scholar 

  248. 248.

    Bareggi A, Maire E, Bouaziz O, Di Michiel M (2012) Damage in dual phase steels and its constituents studied by X-ray tomography. Int J Fract 174:217–227

    Article  Google Scholar 

  249. 249.

    Hosokawa A, Wilkinson DS, Kang J, Maire E (2013) Onset of void coalescence in uniaxial tension studied by continuous X-ray tomography. Acta Mater 61:1021–1036

    Article  Google Scholar 

  250. 250.

    Cao TS, Mazière M, Danas K, Besson J (2015) A model for ductile damage prediction at low stress triaxialities incorporating void shape change and void rotation. Int J Sol Str 63:240–263

    Article  Google Scholar 

  251. 251.

    Wong WH, Guo TF (2015) On the energetics of tensile and shear void coalescences. J Mech Phys Sol 82:259–286

    MathSciNet  Article  Google Scholar 

  252. 252.

    Chang HJ, Segurado J, Lorca LJ (2015) Three-dimensional dislocation dynamics analysis of size effects on void growth. Scripta Mater 95:11–14

    Article  Google Scholar 

  253. 253.

    Margolin H, Mahajan Y (1978) Void formation, void growth and tensile fracture in Ti-6AI-4V. Metall Trans A 9:781–791

    Article  Google Scholar 

  254. 254.

    Le Roy G, Embury JD, Edwards G, Ashby MF (1981) A model of ductile fracture based on the nucleation and growth of voids. Acta Metall 29:1509–1522

    Article  Google Scholar 

  255. 255.

    Sauter AI, Nix WD (1992) A study of stress-driven diffusive growth of voids in encapsulated interconnect lines. J Mater Res 7:1133–1143

    Article  Google Scholar 

  256. 256.

    Everett RK, Geltmacher AB, Simmonds KE (2002) 3D image-based modeling of void interactions in HY 100 steel. Plasticity, Damage, and Fracture at Macro, Micro, and Nano Scales, 699–701

  257. 257.

    Qiu H, Mori H, Enoki M, Kishi T (1999) Evaluation of ductile fracture of structural steels by microvoid model. ISIJ Int 39:358–364

    Article  Google Scholar 

  258. 258.

    Fowler JP, Worswick MJ, Pilkey AK, Nahme H (2000) Damage leading to ductile fracture under high strain-rate conditions. Metall Mater Trans A31:831–844

    Google Scholar 

  259. 259.

    Everett RK, Simmonds KE, Geltmacher AB (2001) Spatial distribution of voids in HY-100 steel by X-ray tomography. Scripta Mater 44:165–169

    Article  Google Scholar 

  260. 260.

    Brown KR, Creton C (2002) Nucleation and growth of cavities in soft viscoelastic layers under tensile stress. Eur Phy J E 9:35–40

    Article  Google Scholar 

  261. 261.

    Morgeneyer TF, Starink MJ, Sinclair I (2008) Evolution of voids during ductile crack propagation in an aluminium alloy sheet toughness test studied by synchrotron radiation computed tomography. Acta Mater 56:1671–1679

    Article  Google Scholar 

  262. 262.

    Velmanirajan K, Thaheer ASA, Narayanasamy R, Madhavan R, Suwas S (2013) Effect of annealing temperature in Al 1145 alloy sheets on formability, void coalescence, and texture analysis. J Mater Eng Perf 22:1091–1107

    Article  Google Scholar 

  263. 263.

    Seo D, Toda H, Kobayashi M, Uesugi K, Takeuchi A, Suzuki Y (2015) In situ observation of void nucleation and growth in a steel using X-ray tomography. ISIJ Int 55:1474–1482

    Article  Google Scholar 

  264. 264.

    Bhadeshia HKDH (1999) Neural networks in materials science. ISIJ Int 39:966–979

    Article  Google Scholar 

  265. 265.

    Keralavarma SM, Chockalingam S (2016) A criterion for void coalescence in anisotropic ductile materials. Int J Plast 82:159–176

    Article  Google Scholar 

  266. 266.

    Tipper CF (1949) The fracture of metals. Metallurgia 39:133–137

    Google Scholar 

  267. 267.

    Hahn GT, Rosenfield AR (1975) Metallurgical factors affecting fracture toughness of aluminum alloys. Metall Trans A 6(4):653–668

    Article  Google Scholar 

  268. 268.

    Walsh JA, Jata KV, Starke EA (1989) The influence of Mn dispersoid content and stress state on ductile fracture of 2134 type Al alloys. Acta Metall 37:2861–2871

    Article  Google Scholar 

  269. 269.

    Qiu H, Enoki M, Mori H, Takeda N, Kishi T (1999) Effect of strain rate and plastic pre-strain on the ductility of structural steels. ISIJ Int 39:955–960

    Article  Google Scholar 

  270. 270.

    French IE, Weinrich PF (1974) The influence of hydrostatic pressure on the tensile deformation of a spheroidised 0.5% steel. Scripta Metall 8:87–90

    Article  Google Scholar 

  271. 271.

    Jonas JJ, Baudelet B (1977) Effect of crack and cavity generation on tensile stability. Acta Metall 25:43–50

    Article  Google Scholar 

  272. 272.

    Nicolaou PD, Semiatin SL, Ghosh AK (2000) An analysis of the effect of cavity nucleation rate and cavity coalescence on the tensile behavior of superplastic materials. Metall Mater Trans A 31:1425

    Article  Google Scholar 

  273. 273.

    Senior BA, Noble FW, Eyre BL (1986) The nucleation and growth of voids at carbides in 9Cr–1Mo steel. Acta Metall 34:1321–1327

    Article  Google Scholar 

  274. 274.

    Budiansky B (1983) Micromechanics. Comp Str 16:3–12

    MATH  Article  Google Scholar 

  275. 275.

    Bridgeman PW (1952) Studies in large plastic flow and fracture: with special emphasis on the effects of hydrostatic pressure. In: Irwin GR (ed) Science, vol 115. McGraw-Hill, New York, 362 pp

  276. 276.

    Benzerga AA, Keralavarma SM (2009) Finite element analyses of combined void shape and plastic anisotropy effects in ductile fracture: with special emphasis on the effects of hydrostatic pressure. In: Irwin GR (ed) Science, vol 115. McGraw-Hill, New York, 362 pp

  277. 277.

    Pilling J, Ridley N (1927–1989) Superplasticity in crystalline solids. Institute of Metals, London

  278. 278.

    Kiran R, Khandelwal K (2015) A coupled microvoid elongation and dilation based ductile fracture model for structural steels. Eng Fract Mech 145:15–42

    Article  Google Scholar 

  279. 279.

    Brünig M, Brenner D, Gerke S (2015) Stress state dependence of ductile damage and fracture behavior: experiments and numerical simulations. Eng Fract Mech 141:152–169

    Article  Google Scholar 

  280. 280.

    Van der Giessen E, Needleman A (1995) Discrete dislocation plasticity: a simple planar model. Modell Sim Mater Sci Eng 3:689

    Article  Google Scholar 

  281. 281.

    Faleskog J, Shih CF (1997) Micromechanics of coalescence-I. Synergistic effects of elasticity, plastic yielding and multi-size-scale voids. J Mech Phys Sol 45:21–50

    Article  Google Scholar 

  282. 282.

    Needleman A (1987) Continuum model for void nucleation by inclusion debonding. J Appl Mech 54:525–531

    MATH  Article  Google Scholar 

  283. 283.

    Tvergaard V (1982) On localization in ductile materials containing spherical voids. Int J Frac 18:237–252

    Google Scholar 

  284. 284.

    Thomason PF (1985) A three-dimensional model for ductile fracture by the growth and coalescence of microvoids. Acta Metall 33:1087–1095

    Article  Google Scholar 

  285. 285.

    Beremin FM (1981) IUTAM Symp. on Three-Dimensional Constitutive Relations and Ductile Fracture, North-Holland Publishing Com- pany, Dourdan, France

  286. 286.

    Bauer RW, Wilsdorf HG (1974) Void Initiation in Ductile Fracture (No. MS-3533-107-74). Virginia Univ Charlottesville Research Labs for the Engineering Sciences

  287. 287.

    Budiansky B, Amazigo JC, Evans AG (1988) Small-scale crack bridging and the fracture toughness of particulate-reinforced ceramics. J Mech Phys Sol 36:167–187

    Article  Google Scholar 

  288. 288.

    Rose LF (1987) Crack reinforcement by distributed springs. J Mech Phys Sol 35:383–405

    MathSciNet  MATH  Article  Google Scholar 

  289. 289.

    Gologanu M, Leblond JB, Devaux J (1994) Approximate models for ductile metals containing non-spherical voids-case of axisymmetric oblate ellipsoidal cavities. J Eng Mat Tech 116:290–297

    MATH  Article  Google Scholar 

  290. 290.

    Benzerga AA, Leblond JB, Needleman A, Tvergaard V (2016) Ductile failure modeling. Int J Fract 201:29–80

    Article  Google Scholar 

  291. 291.

    Das A, Chowdhury T, Tarafder S (2014) Ductile fracture micro-mechanisms of high strength low alloy steels. Mater Des 54:1002–1009

    Article  Google Scholar 

  292. 292.

    Das A, Sivaprasad S, Tarafder M, Das SK, Tarafder S (2013) Estimation of damage in high strength steels. Appl Soft Comp 13:1033–1041

    Article  Google Scholar 

  293. 293.

    MacKay DJC (1992) A practical Bayesian framework for backpropagation networks. Neural Comp 4:448–472

    Article  Google Scholar 

  294. 294.

    Bhadeshia HKDH, MacKay DJC, Svensson LE (1995) Impact toughness of C-Mn steel arc welds–Bayesian neural network analysis. Mater Sci Tech 11:1046–1051

    Article  Google Scholar 

  295. 295.

    Das A (2016) Revisiting stacking fault energy of steels. Metall Mater Trans A 47:748–768

    Article  Google Scholar 

  296. 296.

    Das A (2018) Calculation of ductility from pearlite microstructure. Mater Sci Tech 34:1046–1063

    Article  Google Scholar 

  297. 297.

    Das A (2018) An artificial intelligence paradigm in heuristic search of tensile behaviour of titanium alloys. Int J Mater Res 109:979–1004

    Google Scholar 

  298. 298.

    Qu R, Zhang P, Zhang Z (2014) Notch effect of materials: strengthening or weakening? J Mater Sci Tech 30:599–608

    Article  Google Scholar 

  299. 299.

    Liu B, Huang Y, Li M, Hwang KC, Liu C (2005) A study of the void size effect based on the Taylor dislocation model. Int J Plast 21:2107–2122

    MATH  Article  Google Scholar 

  300. 300.

    Siruguet K (2000) Rupture ductile à basse triaxialité: effet des inclusions sur la croissance et la coalescence des cavités (Doctoral dissertation, Paris 6)

  301. 301.

    Edelson BI, Baldwin WM Jr (1962) The effect of second phases on the mechanical properties of alloys. Trans ASM 55:230

    Google Scholar 

  302. 302.

    Kerlins V, Phillips A (1987) Modes of fracture. ASM Handb 12:12–71

    Google Scholar 

  303. 303.

    Le May I (1978) Failure mechanisms and metallography: a review. In: McCall JL, French PM (eds) Metallography in failure analysis. Springer, Boston, MA, pp 1–31

    Google Scholar 

  304. 304.

    Das A (2016) Intervention of martensite variants on the spatial aspect of microvoids. Mater Res Exp 3:066501

    Article  Google Scholar 

  305. 305.

    Cocks ACF (1985) The nucleation and growth of voids in a material containing a distribution of grain-boundary particles. Acta Metall 33:129–137

    Article  Google Scholar 

  306. 306.

    Bao Y, Wierzbicki T (2004) On fracture locus in the equivalent strain and stress triaxiality space. Int J Mech Sci 46:81–98

    Article  Google Scholar 

Download references


All the experiments and simulations (FEM through ABAQUS 6.9) were carried out at CSIR-National Metallurgical Laboratory, Jamshedpur when the author was previously employed as a Scientist during 2004–2014. I am extremely grateful to Professor Sir H.K.D.H. Bhadeshia, Phase Transformation and Complex Properties Research Group, Department of Materials Science and Metallurgy, University of Cambridge, UK for the provision of Neuromat Neural Network software for the current analysis. I would also like to thank Professor Eugenio Onate (The Editor in Chief - Archives of Computational Methods in Engineering) for the provision of facilities for the revision of the manuscript. The insightful suggestions, comments and strong recommendations about the manuscript by the anonymous reviewers are also gratefully appreciated.

Author information



Corresponding author

Correspondence to Arpan Das.

Ethics declarations

Conflict of interest

No potential conflict of interest was reported by the author.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Das, A. Stress/Strain Induced Void?. Arch Computat Methods Eng (2020).

Download citation