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3.4.1 Estimation of Stress For the Weibull Inverse Power Law

Charles Hagwood
Statistical Engineering Division, ITL

Roger Clough and Richard Fields
Metallurgy Division, MSEL

The 1986 amendments to the Safe Drinking Water Act mandated that solder used in potable water systems be free of lead. A partnership between NIST and the Copper Development Association was created to study various lead-free solders. The statistical problem is to estimate the failure time models of these lead-free alternatives.

Failure time data, (stress, tf), on stress rupture of copper piping joints made with certain lead free solders suggest that specimens under stress below a certain threshold run indefinitely without failure. A commonly used model for this type of data is the Weibull inverse power law that includes a threshold. The log of the failure times, z=ln(tf), satisfy the model

\begin{displaymath}z=\left\{ \begin{array}{ll}
a+blog(x-x_0)+\sigma \epsilon& x > x_0\\
\infty & x \leq x_0
\end{displaymath} (3.1)

x=stress, x0=threshold, $\epsilon$ has the standard extreme value distribution and for the parameter space $x_0,\sigma > 0$, $-\infty < a,b < \infty$.

If the threshold is unknown, this estimation problem presents several difficulties for statistical treatment. The largest problem being that as the threshold approaches the minimum of the data (stresses) the likelihood approaches infinity, thus there is no global maximum. The threshold is crucial for establishing permissible stresses for copper pressure fitting codes.

A modified maximum likelihood approach is taken where first the threshold is estimated in terms of the other parameters by using a moment estimator. This estimator of the threshold is then substituted into the likelihood equations. Having removed the threshold parameter, the problem is reduced to determining two parameters, and is solvable. Censoring creates another obstacle, because if the threshold is unknown, it can not be determined whether a censored observation will eventually fail or not. Thus, the likelihood will depend on how many censored observations have associated stresses above the threshold or not, which results in a partitioning of the likelihood into several cases. The case with the highest likelihood is chosen as the best model.

Shown in TABLE 1 are the soldered joint shear stresses and their corresponding times to failure. These are data for Sb5 (95% tin and 5% antimony) solder tested in a heating chamber at 250oC. The best fitting model based on the above approach appears in Figure 1. A manuscript reporting our analysis has been accepted for publication in the IEEE Transactions on Reliability.

Stress-Failure Time Data for Solder

Stresses Failure Times
(MPa) (hours)


37.38, 9.88, 13.45, 39.28
x2=4.82269504 59.37, 107.59, 92.18
x3=4.53900709 74.72, 85.42, 122.37, 93.05, 79.38, 88.77
x4=3.97163121 79.61, 110.13
x5=3.54609929 140.06, 145.10, 256.28
x6=2.83687943 1968.71, 582.68, 328.50
x7=0.70921986 8585.24*

* censored data.


Figure 24: Fitted Curve Using TABLE 1 Data with 95% Confidence Bands.

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Date created: 7/20/2001
Last updated: 7/20/2001
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