A Statistical Analysis on Modeling Uncertainty Through Crack Initiation Tests

  • Jae Phil Park
  • Chanseok Park
  • Chi Bum Bahn
Conference paper
Part of the The Minerals, Metals & Materials Series book series (MMMS)


Because a large time spread in most crack initiation tests makes it a daunting task to predict the initiation time of cracking, a probabilistic model, such as the Weibull distribution, has been usually employed to model it. In this case, although it might be anticipated to develop a more reliable cracking model under ideal cracking test conditions (e.g., large number of specimen, narrow censoring interval, etc.), it is not straightforward to quantitatively assess the effects of these experimental conditions on model estimation uncertainty . Therefore, we studied the effects of some key experimental conditions on estimation uncertainties of the Weibull parameters through the Monte Carlo simulations. Simulation results suggested that the estimated scale parameter would be more reliable than the estimated shape parameter from the tests. It was also shown that increasing the number of specimen would be more efficient to reduce the uncertainty of estimators than the more frequent censoring.


Weibull distribution Estimation Monte carlo simulation 



Cumulative distribution function

\( F\left( \cdot \right) \)

Cumulative distribution function of Weibull distribution


End cracking fraction

\( \hat{\eta } \)

Estimator of Weibull scale parameter

\( \hat{\beta } \)

Estimator of Weibull shape parameter


Extreme value distribution


Extreme value distribution for minima


Generalized extreme value distribution


Independent and identically distributed

\( s_{i} \)

Last censoring time of ith suspended specimen


Length of censoring interval

\( L\left( \cdot \right) \)

Likelihood function

\( \mu \)

Location parameter of generalized extreme value distribution

\( l\left( \cdot \right) \)

Log-likelihood function


Lower bound

\( c_{{j_{L} }} \)

Lower bound time of censoring interval for jth cracking


Maximum likelihood estimation


Number of interval-censored cracked specimens


Number of suspended specimens


Probability density function

\( g\left( \cdot \right) \)

Probability density function of generalized extreme value distribution

\( {\text{RE}}\left( \cdot \right) \)

Relative error

\( {\text{RE}}_{50\% } \)

Relative error of median estimates

\( {\text{RLCI}}_{90\% } \)

Relative length of 90% confidence interval


Relative test duration

\( \sigma \)

Scale parameter of generalized extreme value distribution

\( \eta \)

Scale parameter of Weibull distribution

\( \xi \)

Shape parameter of generalized extreme value distribution

\( \beta \)

Shape parameter of Weibull distribution


Stress corrosion cracking



\( \eta_{\text{true}} \)

True Weibull scale parameter

\( \beta_{\text{true}} \)

True Weibull shape parameter


Upper bound

\( c_{{j_{U} }} \)

Upper bound time of censoring interval for jth cracking

\( x \)

Variable of generalized extreme value distribution



This work was supported by the Nuclear Safety Research Program through the Korea Foundation of Nuclear Safety (KOFONS) granted financial resource from the Nuclear Safety and Security Commission (NSSC), Republic of Korea (No. 1403006), and was supported by “Human Resources Program in Energy Technology” of the Korea Institute of Energy Technology Evaluation and Planning (KETEP), who granted the financial resources from the Ministry of Trade, Industry & Energy, Korea. (No. 20164010201000).


  1. 1.
    W. Lunceford, T. DeWees, P. Scott, EPRI Materials Degradation Matrix, Rev. 3. EPRI, Palo Alto, CA, USA 3002000628, 2013Google Scholar
  2. 2.
    P. Scott, M.-C. Meunier, Materials Reliability Program: Review of Stress Corrosion Cracking of Alloys 182 and 82 in PWR Primary Water Service (MRP-220). EPRI, Palo Alto, CA, USA Rep. No. 1015427, 2007Google Scholar
  3. 3.
    K.J. Kim and E.S. Do, Technical Report: Inspection of Bottom Mounted Instrumentation Nozzle (in Korean). Korea Institute of Nuclear Safety (KINS), Daejeon, Korea KINS/RR-1360, 2015Google Scholar
  4. 4.
    C. Amzallag, S.L. Hong, C. Pages, A. Gelpi, Stress Corrosion Life Assessment of Alloy 600 PWR Components. in 9th International Symposium on Environmental Degradation of Materials in Nuclear Power Systems—Water Reactors, 1999, pp. 243–250Google Scholar
  5. 5.
    Y.S. Garud, Stress Corrosion Cracking Initiation Model for Stainless Steel and Nickel Alloys. EPRI, Palo Alto, CA, USA 1019032, 2009Google Scholar
  6. 6.
    M. Erickson, F. Ammirato, B. Brust, D. Dedhia, E. Focht, M. Kirk, et al., Models and Inputs Selected for Use in the xLPR Pilot Study. EPRI, Palo Alto, CA, USA 1022528, 2011Google Scholar
  7. 7.
    G. Troyer, S. Fyfitch, K. Schmitt, G. White, C. Harrington, Dissimilar Metal Weld PWSCC Initiation Model Refinement for XLPR Part I: A Survey of Alloy 82/182/132 Crack Initiation Literature. in 17th International Conference on Environmental Degradation of Materials in Nuclear Power Systems—Water Reactors, Ottawa, Ontario, Canada, August 9–13, 2015Google Scholar
  8. 8.
    W. Weibull, A statistical distribution function of wide applicability. J. Appl. Mech. 18, 293–297 (1951)Google Scholar
  9. 9.
    E. Eason, Materials Reliability Program: Effects of Hydrogen, pH, Lithium and Boron on Primary Water Stress Corrosion Crack Initiation in Alloy 600 for Temperatures in the Range 320–330°C (MRP-147). EPRI, Palo Alto, CA, USA 1012145, 2005Google Scholar
  10. 10.
    I.S. Hwang, S.U. Kwon, J.H. Kim, S.G. Lee, An intraspecimen method for the statistical characterization of stress corrosion crack initiation behavior. Corrosion 57, 787–793 (2001)CrossRefGoogle Scholar
  11. 11.
    J. McCool, Using the Weibull distribution: reliability, modeling, and inference (Wiley, Hoboken, N.J., USA, 2012)CrossRefGoogle Scholar
  12. 12.
    S.M. Ross, Introduction to Probability and Statistics for Engineers and Scientists, 4th edn. (Elsevier Academic Press, USA, 2009)Google Scholar
  13. 13.
    R.A. Fisher L.H.C. Tippett, Limiting Forms of the Frequency Distribution of the Largest or Smallest Member of a Sample. in Mathematical Proceedings of the Cambridge Philosophical Society, 1928, pp. 180–190CrossRefGoogle Scholar
  14. 14.
    R.L. Wolpert, Extremes, Available online:, 2014
  15. 15.
    J.P. Park, C. Park, J. Cho, C.B. Bahn, Effects of cracking test conditions on estimation uncertainty for Weibull parameters considering time-dependent censoring interval. Materials 10, 3 (2016)CrossRefGoogle Scholar
  16. 16.
    D. McFadden,Modeling the Choice of Residential Location. Transportation Research Record 1978Google Scholar
  17. 17.
    U. Genschel, W.Q. Meeker, A comparison of maximum likelihood and median-rank regression for Weibull estimation. Q. Eng. 22, 236–255 (2010)CrossRefGoogle Scholar
  18. 18.
    J.P. Park, C.B. Bahn, Uncertainty evaluation of Weibull estimators through Monte Carlo simulation: applications for crack initiation testing. Materials 9, 521 (2016)CrossRefGoogle Scholar
  19. 19.
    F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw, W.A. Stahel, Robust statistics: the approach based on influence functions vol 114, Wiley, 2011Google Scholar
  20. 20.
    J.D. Hong, C. Jang, T.S. Kim, PFM application for the PWSCC integrity of Ni-Base alloy welds—development and application of PINEP-PWSCC. Nuclear Eng. Technol. 44, 961–970 (2012)CrossRefGoogle Scholar
  21. 21.
    K. Dozaki, D. Akutagawa, N. Nagata, H. Takihuchi, K. Norring, Effects of dissolved hydrogen condtent in PWR primary water on PWSCC initiation property. E-J. Adv. Main. 2, 65–76 (2010)Google Scholar
  22. 22.
    Y.S. Garud, SCC initiation model and its implementation for probabilistic assessment. in ASME Pressure Vessels & Piping Division, July 18–22, 2010Google Scholar

Copyright information

© The Minerals, Metals & Materials Society 2019

Authors and Affiliations

  • Jae Phil Park
    • 1
  • Chanseok Park
    • 2
  • Chi Bum Bahn
    • 1
  1. 1.School of Mechanical EngineeringPusan National UniversityBusanRepublic of Korea
  2. 2.Department of Industrial EngineeringPusan National UniversityBusanRepublic of Korea

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