Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Abbreviations
- CDF:
-
Cumulative distribution function
- \( F\left( \cdot \right) \) :
-
Cumulative distribution function of Weibull distribution
- ECF:
-
End cracking fraction
- \( \hat{\eta } \) :
-
Estimator of Weibull scale parameter
- \( \hat{\beta } \) :
-
Estimator of Weibull shape parameter
- EVD:
-
Extreme value distribution
- EVDm :
-
Extreme value distribution for minima
- GEVD:
-
Generalized extreme value distribution
- iid:
-
Independent and identically distributed
- \( s_{i} \) :
-
Last censoring time of i th suspended specimen
- LCI:
-
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
- LB:
-
Lower bound
- \( c_{{j_{L} }} \) :
-
Lower bound time of censoring interval for j th cracking
- MLE:
-
Maximum likelihood estimation
- C:
-
Number of interval-censored cracked specimens
- S:
-
Number of suspended specimens
- PDF:
-
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
- RTD:
-
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
- SCC:
-
Stress corrosion cracking
- t :
-
Time
- \( \eta_{\text{true}} \) :
-
True Weibull scale parameter
- \( \beta_{\text{true}} \) :
-
True Weibull shape parameter
- UB:
-
Upper bound
- \( c_{{j_{U} }} \) :
-
Upper bound time of censoring interval for jth cracking
- \( x \) :
-
Variable of generalized extreme value distribution
References
W. Lunceford, T. DeWees, P. Scott, EPRI Materials Degradation Matrix, Rev. 3. EPRI, Palo Alto, CA, USA 3002000628, 2013
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, 2007
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, 2015
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. 243–250
Y.S. Garud, Stress Corrosion Cracking Initiation Model for Stainless Steel and Nickel Alloys. EPRI, Palo Alto, CA, USA 1019032, 2009
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, 2011
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, 2015
W. Weibull, A statistical distribution function of wide applicability. J. Appl. Mech. 18, 293–297 (1951)
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, 2005
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)
J. McCool, Using the Weibull distribution: reliability, modeling, and inference (Wiley, Hoboken, N.J., USA, 2012)
S.M. Ross, Introduction to Probability and Statistics for Engineers and Scientists, 4th edn. (Elsevier Academic Press, USA, 2009)
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. 180–190
R.L. Wolpert, Extremes, Available online: https://www2.stat.duke.edu/courses/Fall15/sta711/lec/topics/extremes.pdf, 2014
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)
D. McFadden,Modeling the Choice of Residential Location. Transportation Research Record 1978
U. Genschel, W.Q. Meeker, A comparison of maximum likelihood and median-rank regression for Weibull estimation. Q. Eng. 22, 236–255 (2010)
J.P. Park, C.B. Bahn, Uncertainty evaluation of Weibull estimators through Monte Carlo simulation: applications for crack initiation testing. Materials 9, 521 (2016)
F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw, W.A. Stahel, Robust statistics: the approach based on influence functions vol 114, Wiley, 2011
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)
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)
Y.S. Garud, SCC initiation model and its implementation for probabilistic assessment. in ASME Pressure Vessels & Piping Division, July 18–22, 2010
Acknowledgements
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).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 The Minerals, Metals & Materials Society
About this paper
Cite this paper
Park, J.P., Park, C., Bahn, C.B. (2019). A Statistical Analysis on Modeling Uncertainty Through Crack Initiation Tests. In: Jackson, J., Paraventi, D., Wright, M. (eds) Proceedings of the 18th International Conference on Environmental Degradation of Materials in Nuclear Power Systems – Water Reactors. The Minerals, Metals & Materials Series. Springer, Cham. https://doi.org/10.1007/978-3-030-04639-2_135
Download citation
DOI: https://doi.org/10.1007/978-3-030-04639-2_135
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-04638-5
Online ISBN: 978-3-030-04639-2
eBook Packages: Chemistry and Materials ScienceChemistry and Material Science (R0)