On comparing the shape parameters of two Weibull distributions

doi:10.1016/j.msea.2011.07.013

Materials Science and Engineering: A

Volume 528, Issue 27, 15 October 2011, Pages 8028-8030

https://doi.org/10.1016/j.msea.2011.07.013 Get rights and content

Abstract

Probability estimators developed previously by the authors have been used to compare unbiased estimates of the shape parameters (Weibull modulus) from two distributions by the linear regression method for thirteen sample sizes ranging from 10 to 100. Percentiles of the ratio of two estimated Weibull moduli for various confidence levels have been developed. The use of these percentiles for hypothesis testing, i.e., comparison of the two Weibull moduli is demonstrated by using datasets from the literature.

Highlights

► Estimating Weibull modulus by using linear regression and unbiased probability estimators. ► Hypothesis testing on two estimated Weibull moduli. ► Percentile tables are generated for this hypothesis test.

Introduction

Weibull statistics are widely used to model the variability in the fracture properties of ceramics and metals, where the concept of weakest link applies. For the two-parameter Weibull distribution, the cumulative probability, P, that a part will fracture at a given stress, σ, or below can be predicted as [1]; $P = 1 - exp [- {(\frac{σ}{σ_{0}})}^{m}]$ where σ₀ is the scale parameter and m is the Weibull modulus, alternatively referred to as the shape parameter. The Weibull distribution has been successfully applied to materials in which fracture properties are determined overridingly by the defect distributions. For instance, Green and Campbell [2] showed that the tensile strength (S_T) of A356 castings alloys follows a 2-parameter Weibull distribution and that the filling system design has a strong effect on the Weibull modulus. Since the results of Green and Campbell, the 2-parameter Weibull distribution has been used extensively in the casting literature to characterize fracture-related mechanical properties such as tensile strength (S_T) [3], elongation [4] and fatigue life (N_f) [5], [6], [7]. According to Campbell [8], Weibull modulus, m, is often between 1 and 10 for pressure die castings, and between 10 and 30 for many gravity-filled castings. For good quality aerospace castings, m is expected to be between 50 and 100. Hence Weibull modulus has been used as one of the measures of the casting quality and reliability.

There are several methods available in the literature to estimate the Weibull parameters such as linear regression, maximum likelihood method and method of moments. Among these methods, the most popular method is linear regression mainly because of its simplicity. After taking the logarithm of Eq. (1) twice yields: $ln [- ln (1 - P)] = m ln (σ) - m ln (σ_{0})$ Note that Eq. (2) has a linear form when the left-hand side of the equation is plotted versus ln(σ) with a slope of m and an intercept of −m ln(σ₀). Hence, the estimates of σ₀ and m, ${\hat{σ}}_{0}$ and $\hat{m}$ , respectively, are determined from the slope and intercept in Eq. (2).

Processing engineers working on improving the quality of material need tools to measure the degree of effectiveness of their efforts. When the Weibull modulus is used as one of the measures of quality and reliability, the effect of process changes can then be evaluated by comparing the Weibull moduli of samples taken prior to and after the implementation of changes. Alternatively, Weibull moduli from two different processes and/or supplier may need to be compared. Therefore there is a need to develop a method to compare Weibull moduli from two separate samples. This study is motivated by this need.

Section snippets

Research methodology

Monte Carlo simulations were used to generate two samples from a Weibull distribution with σ₀ = 1 and m = 3. The probability estimators (plotting position formulas) that yield unbiased estimates of the Weibull parameters reported by the authors [9] were used to assign each data point a probability. These estimators have the form, $P = \frac{i - a}{n + b}$ where i is the rank of the data point in the sample in ascending order, n represents the sample size, and a and b are numbers, such that 0 ≤ a ≤ 0.5 and 0 ≤ b ≤ 1.0. The

Results and discussion

Recently, the authors analyzed the distribution of normalized ${\hat{σ}}_{0}$ and $\hat{m}$ and found that (i) the distribution of normalized ${\hat{σ}}_{0}$ is normal [9], and (ii) the distribution of normalized $\hat{m}$ is neither normal, lognormal, 3-parameter Weibull nor 3-parameter log-Weibull [10]. Therefore it was expected that the distribution of ${\hat{m}}_{1} / {\hat{m}}_{2}$ would not follow these distribution. Analysis of several combinations of sample sizes confirmed this expectation. As a result, it is necessary to generate tables for the

Conclusions

The distributions of the ratio of Weibull moduli estimated from two samples in this paper are neither normal, lognormal, 3-parameter Weibull nor 3-parameter log-Weibull.

Therefore percentiles of the distribution of the ratio of two Weibull moduli have been developed.

The use of these percentiles for comparing two Weibull moduli has been demonstrated by using datasets from the literature.

References (11)

N.R. Green et al.
Mater. Sci. Eng. A
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D. Casellas et al.
Mater. Sci. Eng. A
(2005)
Q.G. Wang et al.
J. Light Metals
(2001)
W. Weibull:
J. Appl. Mech.
(1951)
G.E. Byczynski et al.

There are more references available in the full text version of this article.

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Rapid communicationOn comparing the shape parameters of two Weibull distributions

Abstract

Highlights

Introduction

Section snippets

Research methodology

Results and discussion

Conclusions

Mater. Sci. Eng. A

Mater. Sci. Eng. A

J. Light Metals

J. Appl. Mech.

Rapid communication
On comparing the shape parameters of two Weibull distributions