Life-Stress Relationship Plotting

Once the parameters of the life distribution have been obtained using probability plotting methods, a second plot is created in which life is plotted versus stress. To do this, a life characteristic must be chosen to be plotted. The life characteristic can be any percentile, such as B(x) life, the scale parameter, mean life, etc. The plotting paper used is a special type of paper that linearizes the life-stress relationship. For example, a log-log paper linearizes the inverse power law relationship, and a log-reciprocal paper linearizes the Arrhenius relationship. The parameters of the model are then estimated by solving for the slope and the intercept of the line. This methodology is illustrated in Example 1.

Example 1

Consider the following times-to-failure data at three different stress levels.

Estimate the parameters for a Weibull assumed life distribution and for the inverse power law life-stress relationship.

Solution

First the parameters of the Weibull distribution need to be determined. The data is individually analyzed (for each stress level) using the probability plotting method, or software such as ReliaSoft's Weibull++, with the following results:

where:

, are the parameters of the 393 psi data.
, are the parameters of the 408 psi data.
, are the parameters of the 423 psi data.

Since the shape parameter, β is not common for the three stress levels, the average value is estimated:

Averaging the betas is one of many simple approaches available. One can also use a weighted average, since the uncertainty on beta is greater for smaller sample sizes. In most practical applications the value of will vary (even though it is assumed constant) due to sampling error, etc. The variability in the value of is a source of error when performing analysis by averaging the betas. MLE analysis, which uses a common , is not susceptible to this error. MLE analysis is the method of parameter estimation used in ALTA and it is explained in the next section.

Redraw each line with a = 4 and estimate the new eta values as follows.

= 6650
= 5745
= 4774.

The IPL relationship is given by:

where:

L represents a quantifiable life measure (η in the Weibull case), V represents the stress level, K is one of the parameters and n is another model parameter. The relationship is linearized by taking the logarithm of both sides, which yields:

where L = η, (-lnK) is the intercept, and (-n) is the slope of the line.

The values of η obtained previously are now plotted on a log-linear scale, yielding the following plot:

The slope of the line is the η parameter, which is obtained from the plot:

Thus:

Solving the inverse power law equation with respect to K yields:

Substituting V = 403, the corresponding L (from the plot), L = 6,00 and the previously estimated n: