10.24423/EngTrans.1172.20210607
Analysis of Estimation of Parameters in 3P-Weibull KJc Distribution: Sample Size Effect
The minimum sample size for a good estimation of the parameters in both three-parameter Weibull KJc distribution (3P-W) and ASTM E1921 methods was analyzed. Additionally, the estimations provided by maximum likelihood (ML) and linear regression (LR) were compared. Fracture toughness sets with different sample sizes were randomly generated following a 3P-W with parameters corresponding to experimental datasets from the Euro round robin fracture toughness test. Then, LR and ML were applied to the sets and the parameters were estimated. Standard deviation (SD) and interquartile range (IQR) were employed to analyze the goodness of fit. The results of this paper were consistent with the necessity of large sample sizes (over 30) to find a representative value of the threshold and shape parameters. However, the scale parameter showed a lower scatter and can be estimated with a smaller sample size (around six samples), as used in the standard ASTM E1921-19b.
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References
Landes J.D., Shaffer D.H., Statistical characterization of fracture in the transition region, [in:] STP 700 Fracture Mechanics: 12th Conference, West Conshohocken, PA: ASTM International, Paris P. [Ed.], pp. 368–382, 1980.
Landes J.D., McCabe D.E., Effect of section size on transition temperature behavior of structural steels, [in:] STP 833 Fracture Mechanics: 15th Symposium, West Conshohocken, PA: ASTM International, Sanford R. [Ed.], pp. 378–392, 1984.
Wallin K., Saario T., Törrönen K., Statistical model for carbide induced brittle fracture in steel, Metal Science, 18(1):13–18, 1984, doi: 10.1179/030634584790420384.
Wallin K., The scatter in KIC-results, Engineering Fracture Mechanics, 19(6): 1085–1093, 1984, doi: 10.1016/0013-7944(84)90153-X.
Wallin K., Fracture toughness transition curve shape for ferritic structural steels, Proceedings of International Conference on Fracture of Engineering Materials and Structures, Singapore, August 6–8, 1991, Teoh S.H., Lee K.H. [Eds.], pp. 83–88, 1991.
Kirk M.T., The technical basis for application of the master curve to the assessment of nuclear reactor pressure vessel integrity, United States Nuclear Regulatory Commission, ADAMS ML093540004, 2009.
Wallin K., Fracture Toughness of Engineering Materials: Estimation and Application, EMAS Publishing, Warrington 2011.
ASTM E1921-19b, Standard Test Method for Determination of Reference Temperature, To, for Ferritic Steels in the Transition Range, ASTM International, West Conshohocken, PA, 2019, doi: 10.1520/E1921-19B, www.astm.org.
He J., Lian J., Golisch G., Jie X., Münstermann S., A generalized Orowan model for cleavage fracture, Engineering Fracture Mechanics, 186: 105–118, 2017, doi: 10.1016/j.engfracmech.2017.09.022.
Wallin K., Karjalainen-Roikonen P., Suikkanen P., Sub-sized CVN specimen conversion methodology, Procedia Structural Integrity, 2: 3735–3742, 2016, doi: 10.1016/j.prostr.2016.06.464.
Barbosa V.S., Ruggieri C., Fracture toughness testing using non-standard bend specimens – Part II: Experiments and evaluation of T0 reference temperature for a low alloy structural steel, Engineering Fracture Mechanics, 195: 297–312, 2018, doi: 10.1016/j.engfracmech.2018.03.028.
Ruggieri C., Savioli R.G., Dodds R.H., Comments on W.S. Lei’s 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”, Engineering Fracture Mechanics, 178: 535–540, 2017, doi: 10.1016/j.engfracmech.2016.03.050.
Barbosa V.S., Ruggieri C., Fracture toughness testing using non-standard bend specimens – Part I: Constraint effects and development of test procedure, Engineering Fracture Mechanics., 195: 279–296, 2018, doi: 10.1016/j.engfracmech.2018.03.029.
Ishihara K., Hamada T., Meshii T., T-scaling method for stress distribution scaling under small-scale yielding and its application to the prediction of fracture toughness temperature dependence, Theoretical and Applied Fracture Mechanics, 90: 182–192, 2017, doi: 10.1016/j.tafmec.2017.04.008.
Mu M.Y., Wang G.Z., Xuan F.Z., Tu S.-T., Unified correlation of wide range of in-plane and out-of-plane constraints with cleavage fracture toughness, Procedia Engineering., 130: 803–819, 2015. doi: 10.1016/j.proeng.2015.12.199.
Meshii T., Characterization of fracture toughness based on yield stress and successful application to construct a lower-bound fracture toughness master curve, Engineering Failure Analysis., 116: 104713, 2020, doi: 10.1016/j.engfailanal.2020.104713.
Meshii T., Failure of the ASTM E 1921 master curve to characterize the fracture toughness temperature dependence of ferritic steel and successful application of the stress distribution T-scaling method, Theoretical and Applied Fracture Mechanics, 100: 354–361, 2019, doi: 10.1016/j.tafmec.2019.01.027.
Wallin K.R.W., Objective assessment of scatter and size effects in the Euro fracture toughness data set, Procedia Engineering, 10: 833–838, 2011, doi: 10.1016/j.proeng.2011.04.137.
Ipiña J.E.P., Berejnoi C., Size effects in the transition region and the beginning of the upper shelf for ferritic steels, Fatigue & Fracture of Engineering Materials & Structures, 33(3): 195–202, 2010, doi: 10.1111/j.1460-2695.2009.01432.x.
Larrainzar C., Berejnoi C., Ipiña J.E.P., Comparison of 3P-Weibull parameters based onJC and KJC values, Fatigue & Fracture of Engineering Materials & Structure, 34(6): 408–422, 2011, doi: 10.1111/j.1460-2695.2010.01533.x.
Weibull W., A statistical distribution function of wide applicability, Journal of Applied Mechanics, 18(3): 293–297, 1951, doi: 10.1115/1.4010337.
Prabhakar Murthy D.N., Xie M., Jiang R., Weibull Models, John Wiley & Sons, Hoboken, New Jersey, 2004.
Dodson B., The Weibull Analysis Handbook, ASQ Quality Press, 2006.
Sandon F., "Mathematics of Statistics. II" by J.F. Kenney. Pp. ix, 202. 15s. 1939; rep. 1947 (Van Nostrand, New York; Macmillan, London), The Mathematical Gazette, 33(305): 228–229, 1949, doi: 10.2307/3611432.
Heerens J., Hellmann D., Development of the Euro fracture toughness dataset, Engineering Fracture Mechanics, 69 (4): 421–449, 2002, doi: 10.1016/S0013-7944(01)00067-4.
Knuth D.E., The Art of Computer Programming, Vol. 2. Seminumerical Algorithms (3rd ed.), Addison-Wesley Longman Publishing Co., Inc., 1997.
Pobočíková I., Sedliačková Z., Comparison of four methods for estimating the Weibull distribution parameters, Applied Mathematical Sciences, 8(83): 4137–4149, 2014, doi: 10.12988/ams.2014.45389.
Ferreira L.A., Silva J.L., Parameter estimation for Weibull distribution with right censored data using EM algorithm, Eksploatacja i Niezawodność – Maintenance and Reliability, 19(2): 310–315, 2017, doi: 10.17531/ein.2017.2.20.
Ng H.K.T., Luo L., Hu Y., Duan F., Parameter estimation of three-parameter Weibull distribution based on progressively Type-II censored samples, Journal of Statistical Computation and Simulation, 82(11): 1661–1678, 2012, doi: 10.1080/00949655.2011.591797.
Kenney J.F., Mathematics of Statistics, Chapman & Hall LTD, London, 1947.
Wallin K., Master curve Analysis of the “Euro” fracture toughness dataset, Engineering Fracture Mechanics, 69(4), 451–481, 2002, doi: 10.1016/S0013-7944(01)00071-6.
Ipiña J.E.P., Centurion S.M.C., Asta E.P., Minimum number of specimens to characterize fracture toughness in the ductile-to-brittle transition region, Engineering Fracture Mechanics, 47(3): 457–463, 1994, doi: 10.1016/0013-7944(94)90102-3.
McCabe D.E., Zerbst U., Heerens J., Development of Test Practice Requirements for a Standard Method on Fracture Toughness Testing in the Transition Regime (Report GKSS–93/E/81), Germany, 1993.
Miglin M. , Oberjohn L. , Van Der Sluys W., Analysis of results from the MPC/JSPS round robin testing program in the ductile-to-brittle transition region, [in:] Fracture Mechanics: Twenty-Fourth Volume, J. Landes, D. McCabe, J. Boulet [Eds.], West Conshohocken, PA: ASTM International, pp. 342–354, 1994, doi: 10.1520/stp13713s.
Van Der Sluys W., Miglin M., Results of MPC/JSPS cooperative testing program in the brittle-to-ductile transition region, [in:] Fracture Mechanics: Twenty-Fourth Volume, J. Landes, D. McCabe, J. Boulet [Eds.], West Conshohocken, PA: ASTM International, pp. 308–324, 1994, doi: 10.1520/STP13711S.
DOI: 10.24423/EngTrans.1172.20210607