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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)

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.

Keywords

Weibull distribution Estimation Monte carlo simulation 

Nomenclature

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 ith 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 jth 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

Notes

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).

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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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