Template:Inverse power law relationship

=Inverse Power Law (IPL)Relationship=

Introduction
The inverse power law (IPL) model (or relationship) is commonly used for non-thermal accelerated stresses and is given by:


 * $$L(V)=\frac{1}{K{{V}^{n}}}$$


 * where:

•	 $$L$$ represents a quantifiable life measure, such as mean life, characteristic life, median life,  $$B(x)$$  life, etc.

•	 $$V$$ represents the stress level.

•	 $$K$$ is one of the model parameters to be determined,  $$(K>0).$$

•	 $$n$$ is another model parameter to be determined.

The inverse power law appears as a straight line when plotted on a log-log paper. The equation of the line is given by:


 * $$\ln (L)=-\ln (K)-n\ln (V)$$

Plotting methods are widely used in estimating the parameters of the inverse power law relationship since obtaining $$K$$  and  $$n$$  is as simple as finding the slope and the intercept on Eqn. (log-inv).



A Look at the Parameter $$n$$
The parameter $$n$$  in the inverse power relationship is a measure of the effect of the stress on the life. As the absolute value of $$n$$  increases, the greater the effect of the stress. Negative values of $$n$$  indicate an increasing life with increasing stress. An absolute value of $$n$$  approaching zero indicates small effect of the stress on the life, with no effect (constant life with stress) when  $$n=0.$$

Example 1
Consider the following times-to-failure data at two different stress levels.



The data set was analyzed jointly and with a complete MLE solution over the entire data set using ReliaSoft's ALTA. The analysis yields:


 * $$\widehat{\beta }=2.61647$$


 * $$\widehat{K}=0.00102241$$


 * $$\widehat{n}=1.32729123$$

IPL and the Coffin-Manson Relationship
In accelerated life testing analysis, thermal cycling is commonly treated as a low-cycle fatigue problem, using the inverse power law relationship. Coffin and Manson suggested that the number of cycles-to-failure of a metal subjected to thermal cycling is given by [28]:


 * $$N=\frac{C}$$


 * where:

•	 $$N$$ is the number of cycles to failure. •	 $$C$$ is a constant, characteristic of the metal. •	 $$\gamma $$ is another constant, also characteristic of the metal. •	 $$\Delta T$$ is the range of the thermal cycle. This relationship is basically the inverse power law relationship, where instead of the stress, $$V,$$  the range  $$\Delta V$$  is substituted into Eqn. (inverse). This is an attempt to simplify the analysis of a time-varying stress test by using a constant stress model. It is a very commonly used methodology for thermal cycling and mechanical fatigue tests. However, by performing such a simplification, the following assumptions and shortcomings are inevitable. First, the acceleration effects due to the stress rate of change are ignored. In other words, it is assumed that the failures are accelerated by the stress difference and not by how rapidly this difference occurs. Secondly, the acceleration effects due to stress relaxation and creep are ignored.

Example
In this example the use of Eqn. (coffin) will be illustrated. This is a very simple example which can be repeated at any time. The reader is encouraged to perform this test.

Product:	ACME Paper Clip Model 4456 Reliability Target:	99% at a 90% confidence after 30 cycles of 45º

After consulting with our paper-clip engineers, the acceleration stress was determined to be the angle to which the clips are bent. Two bend stresses of 90º and 180º were used. A sample of six paper clips was tested to failure at both 90º and 180º bends with the following data obtained. The test was performed as shown in the next figures (a side-view of the paper-clip is shown).

Using the IPL-lognormal model, determine whether the reliability target was met.

Solution
By using the IPL relationship to analyze the data, we are actually using a constant stress model to analyze a cycling process. Caution must be exercised when performing the test. The rate of change in the angle must be constant and equal for both the 90 $$^{o}$$ and 180 $$^{o}$$  bends and constant and equal to the rate of change in the angle for the use life of 45 $$^{o}$$  bend. Rate effects are influencing the life of the paper clip. By keeping the rate constant and equal at all stress levels, we can then eliminate these rate effects from our analysis. Otherwise the analysis will not be valid.

The data were entered and analyzed using ReliaSoft's ALTA. The parameters of the IPL-lognormal model were estimated to be:


 * $$\begin{align}

& \sigma = & 0.198533 \\ & K= & 0.000012 \\ & n= & 1.856808 \end{align}$$

Using the QCP, the 90% lower 1-sided confidence bound on reliability after 30 cycles for a 45 $$^{o}$$ bend was estimated to be  $$99.6%$$, as shown below.

This meets the target reliability of 99%.