Fatigue Life is a Statistical Quantity

8/12/2019 Fatigue Life is a Statistical Quantity

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Fatigue Life is a Statistical

Quantity

Introduction to the Weibull

distribution


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Objective

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


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First Primitive Test:

Average Lifetime

Design A Design B

726000

615000

508000

808000

755000

849000

384000

667000

515000

483000

631000

529000

730000

651000

446000343000

960000

730000

730000

973000

258000

635000Average

Design B looks better..


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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%

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70

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90

100

0 100000 200000 300000 400000 500000 600000 700000 800000 900000

Survival Probability vs Cycles Design A



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

y = -7.3352E-11x2- 1.0216E-04x + 1.4026E+020

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0 100000 200000 300000 400000 500000 600000 700000 800000 900000

Chart Title



Poly. (Survival Probability vs Cycles Design A)

Y = -6.0918E-11 N2 -1.1619E-4N +144.3

This fit predicts that

the number of

permissible cycles, for

90% reliability, is

386637, for design A.

For 400000, our goal,

the probability is88.07% for Design A


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We repeat the same for Design B

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0 200000 400000 600000 800000 1000000 1200000

Survival Plot Design B



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y = -5.5949E-12x2- 1.1626E-04x + 1.2137E+02

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0 200000 400000 600000 800000 1000000 1200000



Poly. (Survival Plot Design B)

This plot predicts

that the reliability of

Design B at 400000

cycles is74.866 %


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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 testdata.

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.


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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 ITPROPERLY !


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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 cumulativedevelopment of Asbestosis losses

In forecasting technological change (also known as the Sharif-Islam model)[citation

needed]


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In hydrology the Weibull

distribution is applied to

extreme events such as

annual maximum one-

day rainfalls and riverdischarges.

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 sizedistributions).[

Stolen From Wiki


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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)


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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 :


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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)


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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


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The three parameter Weibull distribution

The three parameter Weibull distribution uses an additional parameter

to shift the distribution sideways along the horizontal axis (number ofcycles 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.


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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 theoxide*)

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 m ( square micron) andB 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, 5defects, 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


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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


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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 enoughgeometric 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 canextract the length scale on which the defect occurs.


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The three parameter Weibull, accumulated failure function

))exp((1)(

t

tF

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_param

eter.htm


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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 mathis math.

And to solar cell

guys, that test

conversion

efficiency vs cellsize

And so on. Math

is math


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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 ???

y = 0.0092x2+ 0.2435x + 13.441

y = -0.0193x2+ 0.3536x + 13.496

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-4 -3 -2 -1 0 1 2

To do so, you make a usual two

parameter Weibull plot, but instead

of fitting a straight line, you fit apower series.

See left. The quadratic term gives

you the curvature.

The curvature is very small, and ofopposite sign for A and B. So it

seems a wash, given the scatter of

the data.

And so we can conclude (without

Doing a Sigma Analysis) that there

is no length scale or threshold effect

in the spring test data.

QED

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Fatigue Life is a Statistical Quantity