How to Define a Complicated Stress Profile

In accelerated life tests, constant and step stresses are commonly used. It is not difficult to get the profile functions for them because of their simplicity. However, in some applications, the stress profile is very complicated. Obtaining the function for such a profile is not a simple task. In this article, we will discuss how to use Weibull++ and ALTA to define a complicated profile function. Parabolic functions are used as an example. However, the method described is a general one and can be extended to any stress profile function.

Example
An engineer is conducting an accelerated life test. A stress is applied to the system. The test lasts 65 minutes, and the stress is measured by a sensor every 5 minutes during the test. It is known that the stress profile is a curve consisting of several segments. The duration and the shape of each segment are given in Table 1.

Segment	Shape	Start Time	End Time
S1	Parabolic Curve	0	10
S2	Parabolic Curve	10	20
S3	Parabolic Curve	20	35
S4	Constant Line	35	45
S5	Parabolic Curve	45	60
S6	Linear Ramp	60	65

Segment	Time	Value
S1	0	9.38
	5	60.81
	10	210.21
S2	10	210.21
	15	60.20
	20	9.86
S3	20	9.86
	25	187.81
	30	187.61
	35	9.87
S4	35	9.87
	40	10.85
	45	10.07
S5	45	10.07
	50	5.49
	55	6.10
	60	9.73
S6	60	9.73
S6	65	100

Because of the measurement error, the readings can only approximate the true stress value (which is unknown) for each measurement. For example, although the stress in segment S4 is known to be constant, the readings are not constant. They are values around the nominal value of 10.

In the following section, we will discuss how to fit parabolic functions to the stress profile segments and how to enter the function into ALTA to create a stress profile. The general equation for a parabolic function is:

The above function applies to a parabolic function with a vertical axis (time axis) of symmetry. For horizontal axis of symmetry, it is necessary to switch the positions of t and y.

From the above equation, we can see that in order to estimate the three parameters, we need at least three different measurements. Examination of Table 2 shows that we can meet this requirement. For segments S1 and S2, we have exactly three points. Therefore, using these three points, we can set up three equations. For example, for S1, we have:

By solving the above equations simultaneously, the solutions for a, b and c can be obtained.

For segments S3 and S5, four measurements are available. However, we have only three parameters to estimate. Therefore, the least square error method should be used to estimate the function parameters. Weibull++'s Non-Linear Equation Fit Solver can be used to obtain the parameter values. We will use S3 to illustrate how this tool works. First the data points are entered, as shown next. Note that the X column contains time values and the Y column is the value of the stress profile at each time.

The parabolic equation is then entered into the Control Panel on the right side of the utility and the user clicks the Calculate icon. The results of the calculation are shown next.

Using the same process, the parabolic equations for S1, S2 and S5 also can be obtained as:

S4 is supposed to be a constant. Thus, we take the average of all the measurements and get the average value of:

S6 is a linear curve. The inputs and the results from the Non-Linear Equation Fit Solver are shown next.

Once the functions for all segments have been obtained, we can put them in ALTA, as shown next.

Further Discussion
In the above example, we illustrated how to create a complicated stress profile with parabolic segments from discrete measurement data. However, in general, the curve shape of each segment may not be known. In this case, different functions could be used to fit the measurement points by trial and error. As the number of points in the profile increases, it becomes easier to identify an appropriate function. Usually, a simple scatter plot can help a lot. For example, consider the scatter plot shown next.

You can see from the plot that an exponential function might describe this stress profile well. Weibull++ has a degradation tool that includes linear, exponential, power, logarithmic, Gompertz and Lloyd-Lipow functions. The degradation tool can be used to estimate the function parameters. For the data in the scatter plot, we can get an exponential curve like: