A framework for statistical modelling of plastic yielding initiated cleavage fracture of structural steels

References

• ASTM E 1921-13, Standard test method for determination of reference temperature, T₀, for ferritic steels in the transition range, 2013. Available at https://www.astm.org/Standard/standards-and-publications.html.
   Google Scholar
• W. Böhme, Experience with instrumented Charpy tests obtained by a DVM round-robin and further development, in Evaluating Material Properties by Dynamic Testing, ESIS Publication 20, E. Van Walle, ed., Mechanical Engineering Publications, London, 1996, pp. 1–23.
   Google Scholar
• C.M. Moura, J.J. Vilela, E.G. Rabello, G. de Paula Martins, and J.R.G. Carneiro, Evaluation of the ductile-to-brittle transition temperature in low carbon steel, in International Nuclear Atlantic Conference (INAC 2009), Rio de Janeiro, RJ, Brazil, Sept. 27 Oct 2, 2009. Available at http://www.iaea.org/inis/collection/NCLCollectionStore/_Public/40/106/40106396.pdf.
   Google Scholar
• P.P. Puzak, E.W. Eschbacher, and W.S. Pellini, Initiation and propagation of brittle fracture in structural steels, Weld. J. 31 (1954), pp. 561-s–581-s.
   Google Scholar
• ASTM E208-06, Standard Test Method for Conducting Drop-Weight Test to Determine Nil-Ductility Transition Temperature of Ferritic Steels, 2012. Available at https://www.astm.org/Standard/standards-and-publications.html.
   Google Scholar
• P.P. Puzak, M.E. Schuster, and W.S. Pellini, Crack starter tests of ship fracture and project steels, Weld. J. 33 (1954), pp. 481-s–496-s.
   Google Scholar
• P.P. Puzak, A.J. Babecki, and W.S. Pellini, Correlations of brittle fracture service failures with laboratory notch-ductility tests, Weld. J. 37 (1958), pp. 391-s–410-s.
   Google Scholar
• W.S. Pellini and P.P. Puzak, Fracture analysis diagram procedures for the fracture-safe engineering design of steel structures, Weld. Res. Counc. Bull. No. 88 (1963), 28 pages.
   Google Scholar
• A. Cosham, D.G. Jones, R. Eiber, and P. Hopkins, Don’t drop the drop weight tear test, J. Pipeline Eng. 9 (2010), pp. 69–84.
   Google Scholar
• R.J. Eiber, Correlation of full scale tests with laboratory tests in the 3rd Symposium on Line Pipe Research. American Gas Association (AGA) Catalogue No. L30000e, Pipeline Research Council International (PRCI), Inc. Report PRCI NG-18, Dalias, TX, 1965, pp. 83–118. Available at http://www.techstreet.com/aga/standards/prci-ng-18?product_id=1702965#product.
   Google Scholar
• R.J. Eiber and B.N. Leis, Fracture control technology for pipelines-ca. 2000, Final Report on Project PR-3-00108 to the Line Pipe Research Supervisory Committee of the Pipeline Research Council International, PRCI Report PR-3-00108, Battelle, Ohio, 2002.
   Google Scholar
• Y.Y. Wu, H.L. Yu, C. Lu, K. Tieu, A.R. Godbole, and G. Michal, Transition of ductile and brittle fracture during DWTT by FEM, in 13th International Conference on Fracture (ICF13), Curran Associates, Inc., China, 2013, pp. 1648–1655.
   Google Scholar
• ANSI/API RP5L3, Recommended Practice for Conducting Drop-Weight Tear Tests on Line Pipe, 3rd ed., 1996. Available at http://www.api.org/~/media/files/publications/whats%20new/5l3_e4%20pa.pdf.
   Google Scholar
• ASTM E436-03, Standard Test Method for Drop-Weight Tear Tests of Ferritic Steels, 2014, Available at https://www.astm.org/Standard/standards-and-publications.html.
   Google Scholar
• C.S. Wiesner, Predicting structural crack arrest behaviour using small-scale material characterization tests, Int. J. Press. Vessels Pip. 69 (1996), pp. 185–196.10.1016/0308-0161(95)00131-X
   Web of Science ®Google Scholar
• C. Gallo, J.A. Álvarez, F. Gutiérrez-Solana, and J.A. Polanco, Predicting crack arrest behaviour of structural steels using small-scale material characterization tests, in From Charpy to Present Impact Testing, ESIS Publication 30, D. François and A. Pineau, eds., Elsevier, Poitiers, 2002, pp. 271–278.10.1016/S1566-1369(02)80030-9
   Google Scholar
• T.L. Anderson, Fracture Mechanics: Fundamentals and Applications, 3rd ed., CRC Press, Boca Raton, FL, 2005.
   Google Scholar
• N.P. O’Dowd and C.F. Shih, Family of crack-tip fields characterized by a triaxiality parameter: Part I-Structure of fields, J. Mech. Phys. Solids 39 (1991), pp. 989–1015.10.1016/0022-5096(91)90049-T
   Web of Science ®Google Scholar
• N.P. O’Dowd and C.F. Shih, Family of crack-tip fields characterized by a triaxiality parameter: Part II-Fracture applications, J. Mech. Phys. Solids 40 (1992), pp. 939–963.10.1016/0022-5096(92)90057-9
   Web of Science ®Google Scholar
• X.K. Chao and Y.J. Chao, J-A2 characterization of crack-tip fields: Extent of J-A2 dominance and size requirements, Int. J. Fract 89 (1998), pp. 285–307.10.1023/A:1007487911376
   Web of Science ®Google Scholar
• E. Orowan, Classical and Dislocation Theories of Brittle Fracture, in Fracture, B.L. Averbach, D.K. Felbeck, G.T. Hahn, and D.A. Thmas, eds., Wiley, New York, 1959, pp. 147–160.
   Google Scholar
• W. Weibull, A statistical theory of the strength of materials, Ingeniorsvetenskapakademiens. Handlinger 151 (1939), pp. 1–45.
   Google Scholar
• P. Kittl and G. Diaz, Weibull’s fracture statistics, or probabilistic strength of materials: State of the art, Res Mech. 24 (1988), pp. 99–207.
   Google Scholar
• R. Danzer, P. Supancic, J. Pascual, and T. Lube, Fracture statistics of ceramics – Weibull statistics and deviations from Weibull statistics, Eng. Fract. Mech. 74 (2007), pp. 2919–2932.10.1016/j.engfracmech.2006.05.028
   Web of Science ®Google Scholar
• J.D. Landes and D.H. Shaffer, Statistical characterization of fracture in the transition region, in Fracture Mechanics: 12th Conference, ASTM STP 700, ASTM, Philadelphia, 1980, pp.368–382.
   Google Scholar
• F.M. Beremin, A local approach for cleavage fracture of a nuclear pressure vessel steel, Metall. Trans. A 14 (1983), pp. 2277–2287.10.1007/BF02663302
   Web of Science ®Google Scholar
• D.A. Curry, Cleavage micromechanisms of crack extensions in steels, Met. Sci. 14 (1980), pp. 319–326.10.1179/msc.1980.14.8-9.319
   Web of Science ®Google Scholar
• J.H. Chen and R. Cao, Micromechanism of Cleavage Fracture of Metals, 1st ed., Butterworth-Heinemann, Oxford, 2014.
   Google Scholar
• A. Pineau, Development of the local approach to fracture over the past 25 years: Theory and applications, Int. J. Fract. 138 (2006), pp. 139–166.
   Web of Science ®Google Scholar
• C. Ruggieri and R.H. Dodds, An engineering methodology for constraint corrections of elastic-plastic fracture toughness – Part I: A review on probabilistic models and exploration of plastic strain effects, Eng. Fract. Mech. 134 (2015), pp. 368–390.10.1016/j.engfracmech.2014.12.015
   Web of Science ®Google Scholar
• C. Zener, Fracture of metals, in 29th National Metal Congress and Exposition, American Society for Metals, Cleveland, OH, 1948, pp. 3–31.
   Google Scholar
• A.N. Stroh, The formation of cracks as a result of plastic flow, Proc. Roy. Soc. London A 223 (1954), pp. 404–414.10.1098/rspa.1954.0124
   Web of Science ®Google Scholar
• A.H. Cottrell, Theory of brittle facture in steel and similar metals, Trans. Metall. Soc. AIME 212 (1958), pp. 192–203.
   Google Scholar
• J. Friedel, Dislocation, Pergamon Press Editions, New York, 1964.
   Google Scholar
• E. Smith, The nucleation and growth of cleavage microcracks in mild steel, in Proceedings of Conference on Physical Basis of Yield and Fracture, Institute of Physics and Physical Society, Oxford, 1966, pp. 36–46.
   Google Scholar
• A. Lambert-Perlade, A.F. Gourgues, J. Besson, T. Sturel, and A. Pineau, Mechanisms and modeling of cleavage fracture in simulated heat-affected zone microstructures of a high-strength low alloy steel, Metall. Mater. Trans. A 35 (2004), pp. 1039–1053.10.1007/s11661-004-1007-6
   Web of Science ®Google Scholar
• A. Echeverria and J.M. Rodriguez-Ibabe, Cleavage micromechanisms on microalloyed steels: Evolution with temperature of some critical parameters, Scripta Mater. 50 (2004), pp. 307–312.10.1016/j.scriptamat.2003.09.003
   Web of Science ®Google Scholar
• B.Z. Margolin, V.A. Shvetsova, and G.P. Karzov, Brittle fracture of nuclear pressure vessel steels – I. Local criterion for cleavage fracture, Int. J. Press. Vessels Pip.. 72 (1997), pp. 73–87.10.1016/S0308-0161(97)00012-4
   Google Scholar
• B.Z. Margolin, V.A. Shvetsova, and A. Gulenko, Radiation embrittlement modelling in multi-scale approach to brittle fracture of RPV steels, Int. J. Fract. 179 (2013), pp. 87–108.10.1007/s10704-012-9775-2
   Web of Science ®Google Scholar
• J.H. Chen and G.Z. Wang, Study of mechanism of cleavage fracture at low temperature, Metall. Trans. A 23 (1992), pp. 509–517.10.1007/BF02801168
   Google Scholar
• M. Kroon and J. Faleskog, Micromechanics of cleavage fracture initiation in ferritic steels by carbide cracking, J. Mech. Phys. Solids 53 (2005), pp. 171–196.10.1016/j.jmps.2004.05.008
   Web of Science ®Google Scholar
• M. Kroon and J. Faleskog, Influence of crack deflection into the carbide/ferrite interface on cleavage fracture initiation in ferritic steels, Mech. Mater. 40 (2008), pp. 695–707.10.1016/j.mechmat.2008.03.006
   Web of Science ®Google Scholar
• I. Kawata, K. Sugimoto, and S. Aihara, Simulation of cleavage crack propagation in steels with anisotropic crystal orientation distribution, Procedia Mater. Sci. 3 (2014), pp. 531–536.10.1016/j.mspro.2014.06.088
   Google Scholar
• R.O. Ritchie, J.F. Knott, and J.R. Rice, On the relationship between critical tensile stress and fracture toughness in mild steel, J. Mech. Phys. Solids 21 (1973), pp. 395–410.10.1016/0022-5096(73)90008-2
   Web of Science ®Google Scholar
• D.A. Curry and J.F. Knott, Effects of microstructure on cleavage fracture stress in steel, Met. Sci. 12 (1978), pp. 511–514.10.1179/msc.1978.12.11.511
   Google Scholar
• S. Nohut, A. Usbeck, H. Özcoban, D. Krause, and G.A. Schneider, Determination of the multiaxial failure criteria for alumina ceramics under tension-torsion test, J. Eur. Ceram. Soc. 30 (2010), pp. 3339–3349.10.1016/j.jeurceramsoc.2010.08.008
   Web of Science ®Google Scholar
• P.T. Williams, B.R. Bass, and W.J. McAfee, Shallow flaws under biaxial loading conditions, Part II. Application of a Weibull stress analysis of the cruciform bend specimen using a hydrostatic stress criterion, ASME J. Pres. Ves. Tech. 123 (2000), pp. 25–31.
   Web of Science ®Google Scholar
• W.-S. Lei, A cumulative failure probability model for cleavage fracture in ferritic steels, Mech. Mater. 93 (2016), pp. 184–198.10.1016/j.mechmat.2015.11.001
   Web of Science ®Google Scholar
• G. Pluvinage, Fracture criteria; Global or local?, Struct. Int. Life 11 (2011), pp. 173–176.
   Google Scholar
• J.R. Griffiths and D.R.J. Owen, An elastic-plastic stress analysis for a notched bar in plane strain bending, J. Mech. Phys. Solids 19 (1971), pp. 419–431.10.1016/0022-5096(71)90009-3
   Web of Science ®Google Scholar
• H. Duennewaldarfmann, M. Twickler, R. Twickler, and W. Dahl, Investigations on cleavage fracture below and at general yield, in Advances in Fracture Research, K. Salama, K. Ravi-Chandar, D.M.R. Taplin, and P.R. Rao, eds., Vol. 1, Pergamon Press, Houston, TX, 1989, pp. 230–238.
   Google Scholar
• C. Juede-Esser, F. Grimpe, and W. Dahl, Investigations on microscopic cleavage fracture stress, in Proceedings of the Tenth European Conference of Fracture – Structural Integrity: Experiments, Models and Applications, K.H. Schwalbe and C. Berger eds., Engineering Materials Advisory Services, Berlin, 1994, pp. 1017–1023.
   Google Scholar
• F. Grimpe, J. Heyer, and W. Dahl, Influence of temperature, strain rate and specimen geometry on the microscopic cleavage fracture stress, Nucl. Eng. Des. 188 (1999), pp. 155–160.10.1016/S0029-5493(99)00012-6
   Web of Science ®Google Scholar
• X.X. Xu, Q.G. Cai, and Y. Su, A tensile stress criterion for cleavage in precracked, nothed and smooth bars, Int. J. Fract. 50 (1991), pp. 51–65.
   Web of Science ®Google Scholar
• D.A. Curry and J.F. Knott, Effect of microstructure on cleavage fracture toughness of quenched and tempered steels, Met. Sci. 13 (1979), pp. 341–345.10.1179/msc.1979.13.6.341
   Google Scholar
• A.A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. R. Soc. London, A 221 (1921), pp. 163–198.10.1098/rsta.1921.0006
   Google Scholar
• T. Lin, A.G. Evans, and R.O. Ritchie, A statistical model of brittle fracture by transgranular cleavage, J. Mech. Phys. Solids 34 (1986), pp. 477–497.10.1016/0022-5096(86)90013-X
   Web of Science ®Google Scholar
• A.S. Eldin and S.C. Collins, Fracture and yield stress of 1020 steel at low temperatures, J. Appl. Phys. 22 (1951), pp. 1296–1297.
   Web of Science ®Google Scholar
• J.R. Matthews, F.A. Mcclintock, and W. Shack, Statistical determination of surface flaw density in brittle materials, J. Am. Ceram. Soc. 59 (1976), pp. 304–308.10.1111/jace.1976.59.issue-7-8
   Web of Science ®Google Scholar
• A.G. Evans, Evaluation of a fundamental approach for the statistical analysis of fracture, J. Am. Ceram. Soc. 61 (1978), pp. 156–160.10.1111/jace.1978.61.issue-3-4
   Web of Science ®Google Scholar
• S.B. Bartdorf and H.L. Heinisch Jr, Weakest link theory reformulated for arbitrary fracture criterion, J. Am. Ceram. Soc. 61 (1978), pp. 355–358.10.1111/jace.1978.61.issue-7-8
   Web of Science ®Google Scholar
• J. Lamon, Statistical approaches to failure for ceramic reliability assessment, J. Am. Ceram. Soc. 71 (1988), pp. 106–112.10.1111/jace.1988.71.issue-2
   Web of Science ®Google Scholar
• M. Kroon and J. Faleskog, A probabilistic model for cleavage fracture with a length scale-influence of material parameters and constraint, Int. J. Fract. 118 (2002), pp. 99–118.10.1023/A:1022864513654
   Web of Science ®Google Scholar
• P. Novak, R. Yuan, B.P. Somerday, P. Sofronis, and R.O. Ritchie, A statistical, physical-based, micro-mechanical model of hydrogen-induced intergranular fracture in steel, J. Mech. Phys. Solids 58 (2010), pp. 206–226.10.1016/j.jmps.2009.10.005
   Web of Science ®Google Scholar
• D. Jeulin, Multi-scale random sets: From morphology to effective properties and to fracture statistics, J. Phys.: Conf. Ser. 319 (2011), pp. 1–11.
   Google Scholar
• C.S. Wiesner and M.R. Goldthorpe, The effect of temperature and specimen geometry on the parameters of the “Local Approach” to cleavage fracture, J. Phys. IV 6 (1996), pp. 295–304.
   Google Scholar
• G. Bernauer, W. Brocks, and W. Schmitt, Modifications of the Beremin model for cleavage fracture in the transition region of a ferritic steel, Eng. Fract. Mech. 64 (1999), pp. 305–325.10.1016/S0013-7944(99)00076-4
   Web of Science ®Google Scholar
• P.P. Milella and N. Bonora, On the dependence of the Weibull exponent on geometry and loading conditions and its implications on the fracture toughness probability curve using a local approach criterion, Int. J. Fract. 104 (2000), pp. 71–87.10.1023/A:1007680819217
   Web of Science ®Google Scholar
• B. Wasiluk, J.P. Petti, and R.H. Dodds, Temperature dependence of Weibull stress parameters: Studies using the Euro-material, Eng. Fract. Mech. 73 (2006), pp. 1046–1069.10.1016/j.engfracmech.2005.11.006
   Web of Science ®Google Scholar
• J.P. Mathieu, K. Inal, S. Berveiller, and O. Diard, A micromechanical interpretation of the temperature dependence of Beremin model parameters for French RPV steel, J. Nucl. Mater. 406 (2010), pp. 97–112.10.1016/j.jnucmat.2010.02.025
   Web of Science ®Google Scholar
• A. Bakker and R.W.J. Koers, Prediction of cleavage fracture events in the brittle-ductile transition region of a ferritic steel, in Defect Assessment in Components Fundamentals and Applications, ESIS/EGF9, J.G. Blauel and K.H. Schwalbe, eds., Mechanical Engineering Publications, London, 1991, pp. 613–632.
   Google Scholar
• L. Xia and C.F. Shih, Ductile crack growth – III. Transition to cleavage fracture incorporating statistics, J. Mech. Phys. Solids 44 (1996), pp. 603–639.10.1016/0022-5096(95)00086-0
   Web of Science ®Google Scholar
• P. Haušild, C. Berdin, and P. Bompard, Prediction of cleavage fracture for a low-alloy steel in the ductile-to-brittle transition temperature range, Mater. Sci. Eng., A 391 (2005), pp. 188–197.10.1016/j.msea.2004.08.067
   Web of Science ®Google Scholar
• T. Lin, A.G. Evans, and R.O. Ritchie, Stochastic modeling of the independent roles of particle size and grain size in transgranular cleavage fracture, Metall. Trans. A 18 (1987), pp. 641–651.10.1007/BF02649480
   Web of Science ®Google Scholar
• S. Hadidi-Moud, A. Mirzaee-Sisan, C.E. Truman, and D.J. Smith, A local approach to cleavage fracture in ferritic steels following warm pre-stressing, Fatigue Fract. Eng. Mater. Struct. 27 (2004), pp. 931–942.10.1111/ffe.2004.27.issue-10
   Web of Science ®Google Scholar
• X. Gao, G. Zhang, and T.S. Srivatsan, A probabilistic model for prediction of cleavage fracture in ferritic steels: A modified Weibull stress model, Mater. Sci. Eng., A 394 (2005), pp. 210–219.10.1016/j.msea.2004.11.035
   Web of Science ®Google Scholar
• M. Moattari, I. Sattari-Far, I. Persechino, and N. Bonora, Prediction of fracture toughness in ductile-to-brittle transition region using combined CDM and Beremin models, Mater. Sci. Eng., A 657 (2016), pp. 161–172.10.1016/j.msea.2015.12.090
   Web of Science ®Google Scholar
• J.P. Petti and R.H. Dodds, Calibration of the Weibull stress scale parameter, σu, using the Master Curve, Eng. Fract. Mech. 72 (2005), pp. 91–120.10.1016/j.engfracmech.2004.03.009
   Google Scholar
• K. Wallin, T. Saario, and K. Toerroenen, Fracture of brittle particles in a ductile matrix, Int. J. Fract. 32 (1987), pp. 201–209.
   Web of Science ®Google Scholar
• K. Wallin and A. Laukkanen, New developments of the Wallin, Saario, Toerroenen cleavage fracture model, Eng. Fract. Mech. 75 (2008), pp. 3367–3377.10.1016/j.engfracmech.2007.07.018
   Web of Science ®Google Scholar
• J. Faleskog, M. Kroon, and H. Öberg, A probabilistic model for cleavage fracture with a length scale-parameter estimations of stationary crack experiments, Eng. Fract. Mech. 71 (2004), pp. 57–79.10.1016/S0013-7944(03)00066-3
   Web of Science ®Google Scholar
• S.R. Bordet, A.D. Karstensen, D.M. Knowles, and C.S. Wiesner, A new statistical local criterion for cleavage fracture in steel. Part I: Model presentation, Eng. Fract. Mech. 72 (2005), pp. 435–452.10.1016/j.engfracmech.2004.02.009
   Web of Science ®Google Scholar
• S.R. Bordet, B. Tanguy, J. Besson, D. Moinereau, and A. Pineau, Cleavage fracture of a RPV steel following warm pre-stressing: Microstructural analysis and interpretation through a new model, Fatigue Fract. Eng. Mater. Struct. 29 (2006), pp. 799–816.10.1111/ffe.2006.29.issue-9-10
   Web of Science ®Google Scholar
• W.-S. Lei, A statistical model of cleavage fracture in structural steels with power-law distribution of microcrack size, Philos. Mag. Lett. 96 (2016), pp. 101–111.10.1080/09500839.2016.1158425
   Web of Science ®Google Scholar
• W.–S. Lei, A discussion of “An engineering methodology for constraint corrections of elastic–plastic fracture toughness – Part II: Effects of specimen geometry and plastic strain on cleavage fracture predictions” by C. Ruggieri, R.G. Savioli, R.H. Dodds, Eng. Fract. Mech. 146 (2015):185–209). Eng. Fract. Mech. 2016, In press. http://dx.doi.org/10.1016/j.engfracmech.2016.02.052.
   Google Scholar
• Z.P. Bazant, Y. Xi, and S. Reid, Statistical size effect in quasi-brittle structures: I. Is Weibull theory applicable?, J. Eng. Mech. 117 (1991), pp. 2609–2622.10.1061/(ASCE)0733-9399(1991)117:11(2609)
   Web of Science ®Google Scholar
• W.-S. Lei, Statistics of single-fiber tensile strength incorporating spatial distribution of microcracks, Fatigue Fract. Eng. Mater. Struct. 2016, In press. http://dx.doi.org/10.1111/ffe.12445.
   Google Scholar
• H. Riesch-Oppermann, K. Heiermann, and S. Roudi, Advanced probabilistic tools for the uncertainty assessment in component reliability predictions, in Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS), P. Neittaanmaeki, T. Rossi, S. Korotov, E. Onate, J. Periaux, and D. Knoerzer, eds., Jyväskylä, 2004, pp. 1–20.
   Google Scholar
• C. Ruggieri and R.H. Dodds Jr., Numerical evaluation of probabilistic fracture parameters using WSTRESS, Eng. Comput. 15 (1998), pp. 49–73.10.1108/02644409810200686
   Web of Science ®Google Scholar
• S. Kotrechko, The key problems of local approach to cleavage fracture, J. Theor. Appl. Mech. 51 (2013), pp. 75–89.
   Web of Science ®Google Scholar
• S.B. Batdorf and J.G. Crose, A statistical theory for the fracture of brittle structures subjected to nonuniform polyaxial stresses, J. Appl. Mech. 41 (1974), pp. 459–464.10.1115/1.3423310
   Web of Science ®Google Scholar
• S. Lee, S. Kim, B. Hwang, B. Lee, and C.G. Lee, Effect of carbide distribution on the fracture toughness in the transition temperature region of an SA 508 steel, Acta Mater. 50 (2002), pp. 4755–4762.10.1016/S1359-6454(02)00313-0
   Web of Science ®Google Scholar
• A. Pineau and B. Tanguy, Advances in clavage fracture modelling in steels: Micromechanical, numerical and multiscale aspects, C.R. Phys. 11 (2010), pp. 316–325.10.1016/j.crhy.2010.07.013
   Web of Science ®Google Scholar
• A.G. Evans, Statistical aspects of cleavage fracture in steel, Metall. Trans. A 14A (1993), pp. 1349–1355.
   Google Scholar
• F. Minami, A. Brueckner-Foit, D. Munz, and B. Trolldenier, Estimation procedure for the Weibull parameters used in the local approach, Int. J. Fract. 54 (1992), pp. 197–210.
   Web of Science ®Google Scholar
• ESIS P6-98, Procedure to Measure and Calculate Materials Parameters for the Local Approach to Fracture using Notch Tensile Specimens, ESIS Publication, 1998. Available at http://www.lavoisier.eu/books/other/esis-procedure-p6-98-procedure-to-measure-et-calculate-material-parameters-for-the-local-approach-to-fracture-using-notched-tensile-specimens/description_2272290.
   Google Scholar
• K. Wallin, T. Saario, and K. Toerroenen, Statistical model for carbide induced brittle fracture in steel, Met. Sci. 18 (1984), pp. 13–16.10.1179/030634584790420384
   Web of Science ®Google Scholar
• X.X. Xu, Q.G. Cai, Y. Su, C.X.Hou, and. W.D. Ma, Statistical model of cleavage fracture in granular bainite in simulated weld heat affected zone, Mater. Sci. Technol.. 5 (1989), pp. 913–917.10.1179/mst.1989.5.9.913
   Web of Science ®Google Scholar
• S. Ortner, J. Duff, and D. Beardsmore, Characterization of Euro A reference steel for application of EOH model of brittle fracture, Technical report SA/EIG/15234/R003 Project PERFECT, SERCO Assurance, 2005.
   Google Scholar
• P.M. James, M. Ford, and A.P. Jivkov, A novel particle failure criterion for cleavage fracture modelling allowing measured brittle particle distributions, Eng. Fract. Mech. 121–122 (2014), pp. 98–115.10.1016/j.engfracmech.2014.03.005
   Web of Science ®Google Scholar
• B. Tanguy, J. Besson, and A. Pineau, Comment on effect of carbide distribution on the fracture toughness in the transition region of an SA 508 steel, Scripta Mater. 49 (2003), pp. 191–197.10.1016/S1359-6462(03)00239-2
   Web of Science ®Google Scholar
• L. Vincent, M. Libert, B. Marini, and C. Rey, Towards a modeling of RPV steel brittle fracture using crystal plasticity computations on polycrystalline aggregrates, J. Nucl. Mater. 406 (2010), pp. 91–96.10.1016/j.jnucmat.2010.07.022
   Web of Science ®Google Scholar
• M. Capurro, P. Brozzo, and P. Cirillo, Statistical aspects of cleavage resistance of high strength microalloyed structural steels, Mater. Sci. Technol. 13 (1997), pp. 645–649.10.1179/mst.1997.13.8.645
   Web of Science ®Google Scholar
• W.-S. Lei, Evaluation of the basic formulations for the cumulative probability of brittle fracture with two different spatial distributions of microcracks, Fatigue Fract. Eng. Mater. Struct. 39 (2016), pp. 611–623.
   Web of Science ®Google Scholar