Modeling ductile to brittle fracture transition in steels - Micromechanical and physical challenges

International Journal of Fracture

Volumn 150, Issue 1-2, 2008, Pages 129-156

  Pineau, Andre ^a  

^a PSL RESEARCH UNIVERSITY   (France)

Author keywords

Cleavage; Ductile fracture; Ductile brittle transition; Fracture toughness; Local approach to fracture; Micromechanisms; Scatter

Indexed keywords

CHARPY IMPACT TESTING; CONSTITUTIVE EQUATIONS; DUCTILE FRACTURE; DUCTILITY; FINITE ELEMENT METHOD; FRACTURE TOUGHNESS; HOT ROLLING; NUMERICAL METHODS; PRESSURIZED WATER REACTORS; QUALITY CONTROL; SCATTERING; TEMPERATURE;

CLEAVAGE; DUCTILE BRITTLE TRANSITION; DUCTILE TO BRITTLE TRANSITIONS; FRACTURE TOUGHNESS TESTS; LOCAL APPROACH TO FRACTURE; METALLURGICAL PARAMETERS; MICROMECHANISMS; SCIENTISTS AND ENGINEERS;

BRITTLE FRACTURE;

EID: 50149105873     PISSN: 03769429     EISSN: 15732673     Source Type: Journal    
DOI: 10.1007/s10704-008-9232-4     Document Type: Article

References (132)

1. Aravas N, McMeeking RM (1985a) Finite element analysis of void growth near a blunting crack tip. J Mech Phys Solids 33: 25-37
2. Aravas N, McMeeking RM (1985b) Microvoid growth and failure in the ligament between a hole and a blunt crack tip. Int J Fract 29: 21-38
3. Amar E, Pineau A (1987) Application of a local approach to ductile-to-brittle transition in a low-alloyed steel. Nucl Eng Des 105: 85-96
4. ASTM E1921 (2002) Test method for the determination of reference temperature to for ferritic steels in the transition range. American Society for Testing and Materials, Philadelphia
5. Bakker A, Koers RWI (1991) Prediction of cleavage fracture events in the brittle-ductile transition region of a ferritic steel. In: Blauel JG, Schwalbe K-H (eds) Defect assessment in components - fundamentals and applications. ESIS/EG 9, Mechanical Engineering Publications, London, pp 613-632
6. Bauvineau L (1996) Approche locale de la rupture ductile: Applications à un acier Carbone-Manganèse. PhD thesis, Ecole des Mines de Paris
7. Benzerga A (2000) Rupture ductile des tôles anisotropes. PhD Thesis, Ecole des Mines de Paris
8. Benzerga A (2002) Micromechanics of coalescence in ductile fracture. J Mech Phys Solids 50: 1331-1362
9. Benzerga AA, Besson J, Batisse R, Pineau A (2002) Synergistic effects of plastic anisotropy and void coalescence on fracture mode in plane strain. Mod Simul Mater Sci Eng 10: 73-102
10. Beremin FM (1981) Cavity formation from inclusions in ductile fracture of A508 steel. Metall Trans A 12: 723-731
11. Beremin FM (1983) A local criterion for cleavage fracture of a nuclear pressure vessel steel. Metall Trans A 14A: 2277-2287
12. Berg A (1962) Proceedings of the motion of cracks in plane viscous deformation. In: Rosenberg RM (ed) 4th US national congress of applied mechanics. University of California, pp 885-892
13. Bernauer G, Brocks W, Schmitt W (1999) Modifications of the Beremin model for cleavage fracture in the transition region of a ferritic steel. Eng Fract Mech 64: 305-325
14. Besson J (2004) Local approach to fracture. In: Besson J (ed) Collective book. Presses de l'Ecole Nationale Supérieure des Mines de Paris
15. Bilat AS, Gourgues-Lorenzon AF, Besson J, Pineau A (2006) Brittle fracture in heat-affected zones of girth welds of modern line pipe steel (X 100) In: EFF 16 conference: failure analysis of nano and engineering materials and structures. Alexandroupolis, Greece, July 3-7
16. Bilby BA, Howard IC, Li ZH (1993) Prediction of the first spinning cylinder test using ductile damage theory. Fatigue Fract Eng Mater Struct 16: 1-20
17. Bordet SR, Karstensen AD, Knowles DM, Wiesner CS (2005a) A new statistical local criterion for cleavage fracture in steel. Part I - Model presentation. Eng Fract Mech 72: 435-452
18. Bordet SR, Karstensen AD, Knowles DM, Wiesner CS (2005b) A new statistical local criterion for cleavage fracture in steel. Part II - Application to an offshore structural steel. Eng Fract Mech 72: 453-474
19. Bouchet C, Tanguy B (2007) Private communication
20. Bouyne E (1999) Propagation et arrêt de fissures de clivage dans l'acier 2 1/4 Cr - 1 Mo. PhD Thesis, Ecole Nationale Supérieure des Mines de Paris
21. Bouyne E, Joly P, Houssin B, Wiesner C, Pineau A (2001) Mechanical and microstructural investigations into the crack arrest behaviour of a modern 2 1/4 Cr - 1 Mo pressure vessel steel. Fat Fract Eng Mater Struct 24: 105-116
22. Brocks W, Klingbeil D, Kunecke G, Sun D-Z (1995) Application of the Gurson model to ductile tearing resistance. In: Kirk M, Bakker A (eds) Constraint effects in fracture theory and applications: Second Volume, ASTM STP 1244. The American Society for Testing and Materials, Philadelphia, pp 232-252
23. Brocks W, Negre P, Scheider I, Schodel M, Steglich D, Zerbst U (2003) Structural integrity assessment by models of ductile crack extension in sheet metal. Steel Res Int 74: 504-513
24. Brückner A, Munz D (1984) Scatter of fracture toughness in the brittle-ductile transition region of a ferritic steel. In: Advances in probabilistic fracture mechanics - PVP. Am Soc Mech Eng NY 92:105-111
25. Buffière JY, Proudhon H, Ferrie E, Ludwig W, Maire E, Cloetens P (2005) Three dimensional imaging of damage in structural materials using high resolution micro-tomography. Nucl Instru Methods Phys Res B 238: 75-82
26. Busso E, Lei Y, O'Dowd NP, Webster GA (1998) Mechanistic prediction of fracture processes in ferritic steel welds within the transition temperature regime. J Eng Mater Technol Trans ASME 120: 328-337
27. Chen CR, Kolednik O (2005) Comparison of cohesive zone parameters and crack tip stress states between two different specimen types. Int J Fract 132: 135-152
28. f. Acta Metal Mater 38: 2527-2537
29. Chen JH, Wang GZ, Wang HJ (1996) A statistical model for cleavage fracture of low alloy steel. Acta Mater 44: 3979-3989
30. Chen JH, Wang Q, Wang GZ, Li Z (2003) Fracture behavior at crack tip - a new framework for cleavage mechanisms of steel. Acta Mater 51: 1841-1855
31. Chu C, Needleman A (1980) Void nucleation effects in biaxially stretched sheets. J Eng Mater Technol 102: 249-256
32. Cornec A, Scheider I, Schwalbe KH (2003) On the practical application of the cohesive model. Eng Fract Mech 70: 1963-1987
33. Curry DA, Knott JF (1979) Effect of microstructure on cleavage fracture toughness of quenched and tempered steels. Metal Science 13: 341-345
34. D'Escatha Y, Devaux JC (1979) Numerical study of initiation, stable crack growth and maximum load with a ductile fracture criterion based on the growth of holes. In: Landes JD, Begley JA, Clarke GA (eds) Elastic plastic fracture. ASTM STP, vol 668, pp 229-248
35. Decamp K, Bauvineau L, Besson J, Pineau A (1997) Size and geometry effect on ductile rupture of notched bars in a C-Mn steel: Experiments and modelling. Int J Fract 88: 1-18
36. Devillers-Guerville L, Besson J, Pineau A (1997) Notch fracture toughness of a cast duplex stainless steel: Modelling of experimental scatter and size effects. Nucl Eng Des 168: 211-225
37. Dodds RH, Ruggieri C, Koppenhoefer K (1997) 3D constraint effectcs on models for transferability of cleavage fracture toughness. In: Underwood JH, MacDonald BD, and Mitchell MR (eds) Fatigue and fracture mechanics: 28th Volume, ASTM STP 1321. American Society for Testing and Materials, pp 179-197
38. Faleskog J, Gao X, Shih C (1998) Cell model for nonlinear fracture analysis - I. Micromechanics calibration. Int J Fract 89: 355-373
39. François D, Pineau A (2002) From Charpy to present impact testing. ESIS Publication 30, Elsevier
40. Gao X, Dodds RH (2000) Constraint effects on the ductile-to-brittle transition temperature of ferritic steels: A Weibull stress model. Int J Fract 102: 43-69
41. Gao X, Faleskog J, Shih CF (1998a) Cell model for nonlinear fracture analysis - II. Fracture-process calibration and verification. Int J Fract 89: 374-386
42. Gao X, Ruggieri C, Dodds RH (1998b) Calibration of Weibull stress parameters using fracture toughness data. Int J Fract 92: 175-200
43. Gao X., Dodds RH, Tregoning RL, Joyce JA, Link RE (1999) A Weibull stress model to predict cleavage fracture in plates containing surface cracks. Fatigue Fract Eng Mater Struct 22: 481-493
44. Gao XS, Wang TH, Kim J (2005) On ductile fracture initiation toughness: effects of void volume fraction, void shape and void distribution. Int J Solids Struct 42: 5097-5117
45. Garrison WM, Moody NR (1987) Ductile rupture. J Phys Chem Solids 48: 1035-1074
46. Geers MGD (2004) Finite strain logarithmic hyperelasto-plasticity with softening: A strongly nonlocal implicit gradient framework. Comput Meth Appl Mech Eng 193: 3377-3401
47. Geers MGD, Ubachs RLJM, Engelen RAB (2003) Strongly nonlocal gradient-enhanced finite strain elastoplasticity. Int J Numer Meth Engng 56: 2039-2068
48. Ghosal AK, Narasimhan R (1996) Mixed-mode fracture initiation in a ductile material with a dual population of second-phase particles. Mat Sci Eng 211: 117-127
49. 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
50. Gologanu M, Leblond JB, Devaux J (1994) Approximate models for ductile metals containing non-spherical voids - case of axisymmetric oblate ellipsoidal cavities. Trans ASME J Eng Mat Tech 116: 290-294
51. Gullerud AS, Xiaoshheng Gao, Dodds RH, Haj-Ali R (2000) Simulation of ductile crack growth using computational cells: Numerical aspects. Eng Fract Mech 66: 65-92
52. Gurson A (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 Technol 99: 2-15
53. Hom CL, Mc Meeking RM (1989a) Void growth in elastic-plastic materials. J Appl Mech 56: 309-317
54. Hom CL, McMeeking RM (1989b) Three-dimensional void growth before a blunting crack tip. J Mech Phys Solids 37: 395-415
55. Keller K, Weihe S, Siegmund T, Kroplin B (1999) Generalized cohesive zone model: Incorporating triaxiality dependent failure mechanisms. Comput Mater Sci 16: 267-274
56. Kim J, Gao XS, Srivatsan TS (2003) Modeling of crack growth in ductile solids: A three-dimensional study. Int J Solids Struct 40: 7357-7374
57. Knott JF (1973) Fundamentals of fracture mechanics. Butterworths, London
58. Koers RWJ, Krom AHM, Bakker A (1995) Prediction of cleavage fracture in the brittle-to-brittle transition region of a ferritic steel. In: Kirk M, Bakker Ad (eds) Constraint effects in fracture theory and applications: Second volume, ASTM STP 1244. American Society for Testing and Materials, Philadelphia, pp 191-208
59. Koppenhoefer KC, RH Dodds (1998) Ductile crack growth in pre-cracked CVN specimens: Numerical studies. Nucl Eng Design 180: 221-241
60. Kotrechko SA (1995) Statistic model of brittle fracture in polycrystalline metals. Phys Metals 14: 1099-1120
61. Kotrechko S, Strnadel B, Dlouhy I (2007) Fracture toughness of cast ferritic steel applying local approach. Theor Appl Fract Mech 47: 171-181
62. Leblond J-B, Perrin G, Devaux J (1994) Bifurcation effects in ductile metals with nonlocal damage. J Appl Mech 61: 236-242
63. Lefèvre W, Barbier G, Masson R, Rousselier G (2002) A modified Beremin model to simulate the warm pre-stress effect. Nucl Eng Des 216: 27-42
64. Margolin BZ, Gulenko AG, Shvetsova VA (1998) Improved probabilistic model for fracture toughness prediction for nuclear pressure vessel steels. Int J Pres Ves Piping 75: 843-855
65. Martin-Meizoso A, Ocana-Arizcorreta I, Gil-Sevillano J, Fuentes-Pérez M (1994) Modelling cleavage fracture of bainitic steels. Acta Metall & Mater 42: 2057-2068
66. Mathur KK, Needleman A, Tvergaard V (1996) Three dimensional analysis of dynamic ductile crack growth in a thin plate. J Mech Phys Solids 44: 439-464
67. Mc Clintock FA (1968) A criterion for ductile fracture by the growth of holes. J App Mech 35: 363-371
68. McMeeking RM (1992) Crack blunting and void growth models for ductile fracture. In: Argon AS (ed) Topics in fracture and fatigue. Springer-Verlag, pp 179-198
69. Mediavilla J, Peerlings RHJ, Geers MGD (2006a) An integrated continuous-discontinuous approach towards damage engineering in sheet metal forming processes. Eng Fract Mech 73: 895-916
70. Mediavilla J, Peerlings RHJ, Geers MGD (2006b) Discrete crack modelling of ductile fracture driven by nonlocal softening plasticity. Int J Numer Meth Eng 66: 661-688
71. Mediavilla J, Peerling RHJ, Geers MGD (2006c) A nonlocal triaxiality-dependent ductile damage model for finite strain plasticity. Comput Meth Appl Mech Eng 195: 4617-4634
72. Minami F, Iida M, Takahara W, Konda N, Arimochi K (2002) Fracture mechanics analysis of Charpy test results based on the Weibull stress criterion. In: François D, Pineau A (eds) From Charpy to present impact testing. Elsevier and ESIS, pp 411-418
73. Moran PM, Shih CF (1998) Crack growth and cleavage in mismatched welds: a micromechanics study using a cell model. Int J Fract 92: 153-174
74. Morgeneyer TF, Starink MJ, Wang SC, Sinclair I. (2008) Quench sensitivity of toughness of an Al alloy l direct observation and analysis of failure initiation at the precipitate free zone. Acta Mater 56: 2872-2884
75. Mudry F, Di Rienzo F, Pineau A (1989) Numerical comparison of global and local fracture criteria in compact tension and center-crack panel specimens. In: Landes JD, Saxena A, Merkle JG (eds) Non-linear fracture mechanics: Volume II-elastic-plastic fracture, ASTM STP 995. American Society for Testing and Materials, Philadelphia, pp 24-39
76. Needleman A, Tvergaard V (1987) An analysis of ductile rupture modes at a crack tip. J Mech Phys Solids 35: 151-183
77. Needleman A, Tvergaard V (1991) A numerical study of void distribution effects on dynamic, ductile crack-growth. Eng Fract Mech 38: 157-173
78. Nilsson F, Faleskog J, Zaremba K, Oberg H (1992) Elastic-plastic fracture mechanics for pressure vessel design. Fatigue Fract Eng Mater Struct 15: 73-89
79. Pardoen T, Hutchinson J (2003) Micromechanics-based model for trends in toughness of ductile metals. Acta Mater 51: 133-148
80. u using the master curve. Eng Fract Mech 72: 91-120
81. Petti JP, Dodds RH (2005b) Ductile tearing and discrete void effects on cleavage fracture under small-scale yielding conditions. Int J Solids Struct 42: 3655-3676
82. Pineau A (1982) Review of fracture micromechanisms and a local approach to predicting crack resistance in low strength steels. In: François D (ed) Advances in fracture research, ICF5. Pergamon, Oxford, UK, pp 553-577
83. Pineau A (1992) Global and local approaches of fracture - transferability of laboratory test results to components. In: Argon AS(eds) Topics in fracture and fatigue. Springer, NewYork, pp 197-234
84. Pineau A (2003) Practical application of local approach methods. In: Ainsworth RA, Schwalbe K-H (eds) Comprehensive structural integrity. Elsevier, vol 7. pp 177-225
85. Pineau A (2006) Development of the local approach to fracture over the past 25 past years: Theory and applications. Int J Fract 138: 139-166
86. Pineau A (2007) Physical and statistical aspects of fracture. 11th symposium on continuum models and discrete systems CMDS 11, Paris, July 30-August 3
87. Pineau A, Pardoen T (2007) Failure mechanisms of metals. In: Ainsworth RA, Schwalbe K-H (eds) Comprehensive structural integrity. Elsevier, vol 2, chap 6, pp 686-797
88. Rice JR, Tracey DM (1969) On the ductile enlargement of voids in triaxial stress. J Mech Phys Solids 17: 201-217
89. Rivalin F, Besson J, Di Fant M, Pineau A (2001a) Ductile tearing of pipeline-steel wide plates: I. Dynamic and quasi static experiments. Eng Fract Mech 68: 329-345
90. Rivalin F, Besson J, Di Fant M, Pineau A (2001b) Ductile tearing of pipeline-steel wide plates: II. Modeling of in-plane crack propagation. Eng Fract Mech 68: 347-364
91. Rossol A, Berdin C, Forget P, Prioul C, Marini B (1999) Mechanical aspects of the Charpy impact test. Nucl Eng Des 188: 217-229
92. Rossol A, Berdin C, Prioul C (2002a) Charpy impact test modelling and local approach to fracture. In: François D, Pineau A (eds) From Charpy to present impact testing. Elsevier and ESIS, pp 445-452
93. Rossol A, Berdin C, Prioul C (2002b) Determination of the fracture toughness of a low alloy steel by the instrumented Charpy impact test. Int J Fract 115: 205-226
94. Rousselier G (1987) Ductile fracture models and their potential in local approach of fracture. Nucl Eng Des 105: 97-111
95. Rousselier G, Devaux J-C, Mottet G, Devesa G (1989) A methodology for ductile fracture analysis based on damage mechanics: An illustration of a local approach of fracture. In: Landes JD, Saxena A, Merkle JG (eds) Nonlinear fracture mechanics: Volume II - elastic-plastic fracture, ASTM STP 995. American Society for Testing and Materials, Philadelphia, pp 332-354
96. Roy YA, Dodds RH (2001) Simulation of ductile crack growth in thin aluminium panels using 3-D surface cohesive elements. Int J Fract 110: 21-45
97. Ruggieri C, Dodds RH Jr (1996) A transferability model for brittle fracture including constraint and ductile tearing effects: A probabilistic approach. Int J Fract 79: 309-340
98. Ruggieri C, Panontin TL, Dodds RH Jr (1996) Numerical modeling of ductile crack growth in 3-D using computational cell elements. Int J Fract 82: 67-95
99. Scheider I, Schodel M, Brocks W, Schonfeld W (2006) Crack propagation analyses with CTOA and cohesive model: Comparison and experimental validation. Eng Fract Mech 73: 252-263
100. Siegmund T, Brocks W (1999) Prediction of the work of separation and implications to modeling. Int J Fract 99: 97-116
101. Siegmund T, Brocks W (2000) A numerical study on the correlation between the work of separation and the dissipation rate in ductile fracture. Eng Fract Mech 67: 139-154
102. Tahar M (1998) Applications de l'approche locale de la rupture fragile à l'acier 16 MNDS: Corrélation reésilience-ténacité: Probabilité de rupture bimodale (clivage et intergranulaire). PhD Thesis, Ecole Nationale Supérieure des Mines de Paris
103. Tanguy B (2001) Modélisation de l'essai Charpy par l'approche locale de la rupture; Application au cas de l'acier 16 MND5 dans le domaine de la transition. PhD Thesis, Ecole Nationale Supérieure des Mines de Paris
104. Tanguy B, Besson J, Piques R, Pineau A (2002a) Ductile brittle transition in Charpy and CT tests/experiments and modelling. In: Neimitz A, Rokach IV, Kocanda D, Golos K (eds) ECF 14, Fracture mechanics beyond 2000's. EMAS Publishing, vol III. pp 391-398
105. Tanguy B, Besson J, Piques R, Pineau A (2002b) Numerical modeling of Charpy V-notch tests. In: François D, Pineau A (eds) From Charpy to present impact testing. Elsevier and ESIS, pp 461-468
106. Tanguy B, Besson J, Piques R, Pineau A (2005a) Ductile-to-brittle transition of an A 508 steel characterized by Charpy impact test - Part I: Experimental results. Eng Fract Mech 72: 49-72
107. Tanguy B, Besson J, Piques R, Pineau A (2005b) Ductile-to-brittle transition of an A 508 steel characterized by Charpy impact test - Part II: Modelling of the Charpy transition curve. Eng Fract Mech 72: 413-434
108. KIC·100 shifts due to irradiation for an A 508 pressure vessel steel. Eng Fract Mech 73: 191-206
109. Tanguy B (2007) Private communication
110. Thomason PF (1985a) Three-dimensional models for the plastic limit-load at incipient failure of the intervoid matrix in ductile porous solids. Acta Metall 33: 1079-1085
111. Thomason PF (1985b) A three-dimensional model for ductile fracture by the growth and coalescence of microvoids. Acta Metall 33: 1087-1095
112. Toyoda M, Minami F, Matsuo T, Hagiwara Y, Inoue T (1991) Effect of work hardening properties of high strength steels on cleavage/ductile fracture resistance. Natl Meeting Jpn Weld Soc 49: 112-113
113. Tvergaard V, Hutchinson JW (1992) The relationship between crack growth resistance and fracture process parameters in elastic plastic solids. J Mech Phys Solids 40: 1377-1397
114. Tvergaard V, Hutchinson JW (1996) Effect of strain-dependent cohesive zone model on predictions of crack growth resistance. Int J Solids Structures 33: 3297-3308
115. Tvergaard V, Hutchinson J (2002) Two mechanisms of ductile fracture: void by void growth versus multiple void interaction. Int J Solids Struct 39: 3581-3597
116. Tvergaard V, Needleman A (1984) Analysis of the cup and cone fracture in a round tensile bar. Acta Metall 32: 157-169
117. Tvergaard V, Needleman A (1988) An analysis of the temperature and rate dependence of Charpy V notch energies for a high nitrogen steel. Int J Fract 37: 197-215
118. Tvergaard V, Needleman A (1992) Effect of crack meandering on dynamic, ductile fracture. J Mech Phys Solids 40: 447-471
119. Wallin K (1989) The effect of ductile tearing on cleavage fracture probability in fracture toughness testing. Eng Fract Mech 32: 523-531
120. Wallin K (1991a) Fracture toughness transition curve shape for ferritic structural steels. In: Theoh SH, Lee KH(eds) Proceedings of the joint FEFG. ICF international conference on fracture of engineering materials and structures. Elsevier, London, pp 83-88
121. Wallin K (1991b) Statistical modelling of fracture in the ductile-to-brittle transition region. In: Blauel JG, Schwalbe K-H(eds) Defect assessment in components - fundamentals and applications ESIS/ECF 9. Mechanical Engineering Publications, London, pp 415-445
122. Wallin K (1993) Statistical aspects of constraint with emphasis on testing and analysis of laboratory specimens in the transition region. In: Hackett EM, Schwalbe KH, Dodds RH (eds) Constraint effects in fracture, ASTM STP 1171. Am. Soc. Test. Mater., Philadelphia, pp 264-288
123. Wallin K, Saario T, Torronen K (1984) Statistical model for carbide induced brittle fracture in steel. Metal Sci 18: 13-16
124. Weck A, Wilkinson DS, Toda H, Maire E (2006) 2D and 3D visualization of ductile fracture. Adv Eng Mater 8: 469-472
125. Xia L, Cheng L (1997) Transition from ductile tearing to cleavage fracture: A cell-model approach. Int J Fract 87: 289-306
126. Xia L, Shih CF (1995a) Ductile crack growth - I. A numerical study using computational cells with microstructurally based length scales. J Mech Phys Solids 43: 233-259
127. Xia L, Shih CF (1995b) Ductile crack growth - II. Void nucleation and geometry effects on macroscopic fracture behavior. J Mech Phys Solids 43: 1953-1981
128. Xia L, Shih CF (1996) Ductile crack growth: III. Transition to cleavage fracture incorporating statistics. J Mech Phys Solids 44: 603-639
129. Xia L, Shih C, Hutchinson J (1995) A computational approach to ductile crack growth under large scale yielding conditions. J Mech Phys Solids 43: 389-413
130. Yan C, Mai YW (1997) Effect of constraint on ductile crack growth and ductile-brittle fracture transition of a carbon steel. Int J Vessels Piping 73: 167-173
131. Yuan H, Chen J (2004) Comparison of computational predictions of material failure using nonlocal damage models. Int J Solids Struct 41: 1021-1037
132. Zhang ZL, Thaulow C, Odegard J (2000) A complete Gurson model approach for ductile fracture. Eng Fract Mech 67: 155-168