Fatigue Life is a Statistical Quantity

Introduction to the Weibull distribution

Consider 2 competing Spring Designs, called A and B.

10 Samples of each spring are tested in fatigue

The number of cycles to failure are recorded.

Spring A:

726000

615000

508000

808000

755000

849000

384000

667000

515000

483000

529000

730000

651000

446000

343000

960000

730000

730000

973000

258000

Spring B

Objective

You need a design where 90% of the springs last (at least) to 400 000 cycles.

First Primitive Test:

Average Lifetime

Design A Design B

726000 615000 508000 808000 755000 849000 384000 667000 515000 483000

631000

529000 730000 651000 446000 343000 960000 730000 730000 973000 258000

635000 Average

Design B looks better…..

Next test, a bit more sophisticated:

We plot fraction of springs failed vs number of cycles. We use this to get estimates of reliability of Design A and Design B as a function of cyles

Example: If first failure in design A is at 200 000 cycles, reliability above 200 000 cycles is reduced from 100% to 90%

We now fit a smooth curve (2nd order power) to the data and to find F (400000)

We repeat the same for Design B

The question is :

Is our method to decide the best there is ?

(best meaning “standing up in court”)

Sadly no

But it is not a bad method to make a crude estimate of what the Weibull distribution predicts.

The Weibull distribution was found by Weibull strictly by trial and error.

He tried to model the distribution of failure strength of steels and derive probabilities for a high reliability (such as 99.9 %) from a limited set of test data.

After settled on this distribution:

X is the variable (here the number of cycles to failure) and and are the Weibull parameters ( is the shape parameter also known to material scientists as the Weibull modulus and is the scale or length parameter.

Note that has a physical meaning. If a tensile specimen is twice as long, the probability for a flaw terminating its fatigue life is twice as high.

Size matters.

There is nothing god given about the Weibull distribution. There are other reliability distributions - all invented before Weibull.

But it does fit, experimentally, a very wide variety of phenomena.

Weibull wrote a famous paper demonstrating it fit the size distribution of beans, the height distribution of the population on an island (I forgot which one) and so on, i.e. Biological phenomena as well as steels, with a total of 10 examples

The paper is a classic - I will put it on the website.

The reason is that depending on how the modulus is picked it fits both infant mortality and wear out phenomena.

And fatigue failure is a kind of wear out phenomena.

MOST IMPORTANTLY IT HAS BECOME A DE FACTO STANDARD FOR ENGINEERS.

TO PREVAIL IN COURT, YOU BETTER SHOW YOU USED IT – PROPERLY !

From Wiki

The Weibull distribution is used

•In survival analysis[6]•In reliability engineering and failure analysis•In industrial engineering to represent manufacturing and delivery times•In extreme value theory•In weather forecasting (To describe wind speed distributions, as the natural distribution often matches the Weibull shape[7] Fitted cumulative Weibull distribution to maximum one-day rainfalls)• In communications systems engineering (In radar systems to model the dispersion of the received signals level produced by some types of clutters. To model fading channels in wireless communications, as the Weibull fading model seems to exhibit good fit to experimental fading channel measurements)•In General insurance to model the size of Reinsurance claims, and the cumulative development of Asbestosis losses•In forecasting technological change (also known as the Sharif-Islam model)[citation needed]

• In hydrology the Weibull distribution is applied to extreme events such as annual maximum one-day rainfalls and river discharges. • In describing the size of particles generated by grinding, milling and crushing operations, the 2-Parameter Weibull distribution is used, and in these applications it is sometimes known as the Rosin-Rammler distribution. (In this context it predicts fewer fine particles than the Log-normal distribution and it is generally most accurate for narrow particle size distributions).[

Stolen From Wiki

So we need to learn to do the proper Weibull analysis

This involves three steps

“Massaging” the data point . (we simply took the first spring to fail as the 10% probability for reliability –not a good idea )

Plotting the massaged data point in a “double ln” plot to get the Weibull parameters

Learning on how to put confidence limits on the results.

Massaging the data point

The data are plotted in increasing sequence (ranking low to high) The equivalent of our failure percentage becomes approximately

This will do for most engineering problems. One can do this better by looking up the F distribution.

There is a nifty website; “teach yourself statistics” which might come in handy when you are in industry

(Modern industry runs on statistics)

Once you have done the ranking, it is just plotting :

And, from the double ln plot to extract the values for K and

So here is the plot :

Spring A Spring B

beta = 4.39 2.604167

alpha= 686938 714973

Reliabiliy 0.911098 0.802222

From which you get the following results:

This does not look all that earth shattering different from our primitive estimate which yielded

0.8807 0. 74866

Which shows you that it is a good idea to make a primitive estimate first before setting out to do a “state of the art Weibull with an F distribution)

Now, we should put confidence limits on our answer…

But as this is not a statistics course, I will leave it at this.

But before you use a statistics package to analyze your data it is a good idea to make the primitive plot we started out and decide what other curves might “reasonably” be drawn to the data.

This will give you a rough idea as to the confidence limits you can put on the data.

Moving on to the three parameter Weibull Distribution

The three parameter Weibull distribution

The three parameter Weibull distribution uses an additional parameter to shift the distribution “sideways” along the horizontal axis (number of cycles to failure, time to failure ….)

Physical meaning in brittle fracture is generally “density of flaws per unit length.

Fiber Optics

For example, if a Corning glass fiber has one defect per 10”, and you test 10 specimens, each one inch long, then 9 will have high strength and only one low strength. The average “fracture strength” will be high.

On the other hand, if you test 10 meter long (~ 40 inches) long section, the chances are that you 98% of the time will have a specimen with a flaw. The “average fracture strength” will be low.

Electronics:

The classic example is the break down strength of DRAM capacitors (DRAM cells). The break down depends on both the intrinsic breakdown strength of the oxide and the “defects” (known as weak spots” in the oxide*)

If you have two oxides A and B, where flawless A has a break down voltage of 8 MV/cm and flawless B 7 MV/cm, then A is the better oxide.

On the other hand, if A has 1 defect per 20 square μm ( square micron) and B has one defect per 1 cm2 and the presence of a defect lowers the break down voltage by 20% AND you measure the break down strength on test capacitors made 10x10 micron, (because the research lab does not have state of the art immersion steppers), than your conclusion will be opposite !

Cause you 10x10 test capacitors made in A will have, on average, 5 defects, and therefore now return a breakdown voltage of about 6.4 MV/cm.

Whereas the test capacitors made with the B oxide, will on average, have near zero defects, you will measure an average breakdown of 7 MV/cm THE RESEARCH LAB WILL RECOMMEND TO USE OXIDE A

Electronics continued:

But the production capacitors are 0.5 x 0.5 micron….. An area 40 times smaller than your research lab test specimens !!!

On average, they will not contain a defect, if made either in A or B. Hence the production people will report that capacitors made with the A oxide have a higher breakdown voltage, around 8 MV/cm and those with B will have a breakdown field, on average, of 7 MV/cvm

THE PRODUCTION PEOPLE WILL RECOMMEND OXIDE A

Conflict resolution:

In general, you have no idea about the “length scale of the defect population”. Perhaps the difference between the research lab and the production line is that the production line uses Plasma Processing that uses higher electric fields than the research lab to etch the metallization. A long metal line, acting as an antenna, can “partly blow out a gate oxide” i.e. damage it, by stressing it too much.

So, a priory it is not clear what is going on…………………

Approach A for the research lab

You plot the Weibull distribution of the breakdown voltage of your 10x10 with a two parameter Weibull. If the distribution is curved, chances are you have a length scale parameter .

Approach B for the research lab

You make test capacitors of different size. From 1x1 cm down to 2x2 which is the best your equipment can do with enough geometric precision to know the area within +/- 10%.

You then investigate the dependence of the 2 parameter Weibull on the capacitor size.

If it does depend, you have a length scale problem and you can extract the length scale on which the defect occurs.

The three parameter Weibull, accumulated failure function

))exp((1)(

ttF

Failure probability density function

There are various methods of determining . The most simple one is to plot the data first as a two parameter Weibull (i.e ) . If that one is curved, then one adds/subtracts values of until the line is reasonably straight.

Of course, one can also do this with linear regression. To see how, visit http://www.weibull.com/LifeDataWeb/estimation_of_the_weibull_parameter.htm

Weibull.com is a wonderfully informative website which will teach you everything you ever wanted to know about the Weibull distribution. Here is (stolen from the Website) an example on how to adjust gamma

The curved original data a very familiar to glass fiber guys.

And to capacitor testing guys… math is math.

And to solar cell guys, that test conversion efficiency vs cell size…

And so on…. Math is math

As a good experimenter (good experimenters are those who a paranoid) you now investigate design A and B for “hidden curvature.Is there a hidden length effect ???