Weibull: What’s that?

Aplication of Weibull statistics in fatigue experiments

The Weibull statistics (probability of survival) is mainly used in pseudo static strength tests in which the applied stress is increased until the specimen breaks. The stress at which the specimen breaks is called the "strength" and the phenomenon of breaking is called the "event". In order to use this method in fatigue tests some modifications and assumptions have to be made. In a fatigue test the fatigue life N1 is determined for a given (constant) strain or stress amplitude

One way for applying the probability of survival model is possible if we are able to obtain a lot of the same fatigue lives N (e.g. 100.000 cycles) for different strain (or stress) amplitudes. So, one considers the fatigue life N=100.000 as the event. Note that a measurement of N unequal to 100.000 can not be used. The applied strain amplitudes of the measurements (each giving N=100.000 cycles) are called the ("measured") strength values. However, this will of course never be reached or be possible in practice.

The other way is to define the measured fatigue life as the strength, the point at which the change in dissipated energy per cycle happens as the event and the applied strain amplitude as the procesparameter. The last one is than comparable to the rate in which the stress is increased in the pseudo-static tests. However, for a good determination of the Weibull modulus m one needs at least 15 measurements at one strain level. In practice normally only three repetitions are carried out.

So, there is a problem in the determination of the Weibull modulus.However the following way of "thinking" might help.We start with the probability of survival S1of an unit volume for N1 cycles.

m1

1o

- - R1

1

= = S e e = "Risk of Rupture"R

εεεεεεεε

Note we can only use measurements at which the fatigue life is just N1 cycles for a given strain amplitude.If we raise the "reference" level N1 to N2 the probability of survival this higher fatigue life at the same strain or stress amplitude should decrease. Or we can start with the probability of survival of an unit volume for a given strain amplitude. After all we do have measurements (repetitions) at the same strain amplitude. So, the measured fatigue lives N1 represents already a reflection of the survival.If we raise the strain amplitude, the probability of survival for the same fatigue life will decrease

The problem is how to incorperate all the fatigue measurements in one "survival equation".

One way to incorporate this is by taking into account the Wöhler curve N.εb = A .

You see the Wöhler curve as a "transfer" function. If the "reference" strain level ε1(at which the mean value N1 is measured) is changed to a level ε2, the Wöhler equation will predict the (mean) value of the fatigue life N2 at the new strain level. This can be seen as the points at which the chances to fail (e.g. 50 %) are equal.

If the reference level ε is zero the "risk of rupture" will be zero and the probability of survival will be one.

In a popular way this formula can be seen as the probability of survival with regard to the Wöhler curve. The following step is to incorperate the volume effect. For an uniaxial test this becomes:

m b

b

N . -

AS = eεεεε

Determination of the Weibullmodulus m is possible by plotting the logarithm of the logarithm of 1/S as a function of the logarithm of the parameter N.εb.

m b

b

N . - Vol.

AS = eεεεε