Strength of Materials/Weakest link determination by use of three parameter Weibull statistics

When for a particular property a material shows a large scatter in measured data (e.g. strength measurement results of brittle materials, or of threads with important variations in thickness as listed in reference [1]), a characteristic value for, or the weakest link of such a property can be obtained by analyzing the measured data on the basis of three parameter Weibull statistics.

Introduction
When a material is subject for strength determination, series of strength measurements will show a scatter. When the scatter is low compared with the average value of the series, a minimum strength value $$f_{min}$$ may be defined by:

(1) $$ f_{min} = f_a-u.s $$

in which $$f_a$$ is the average strength, $$s$$ is the standard deviation of the measurement series, and $$u$$ is the factor related to the probability of not failing (e.g. relates to a fail probability of 0.0001)

However, a number of materials, e.g. threads with important variations in thickness, brittle materials as annealed glass, will show a large, or even a very large scatter compared with the average value of the series. And even the scatter of the average values of a number of series of measurements may show such an important scatter compared with the overall average value, that applying expression (1) leads to a nonsensical minimum strength.

In those cases, strength measurements series may be described by two parameter Weibull distributions that indicate that the minimum value approaches to zero, or:

(2) $$F(f;\alpha,\beta)=1-\exp \left[ - \left( \frac{f}{\beta}\right)^\alpha \right]$$

in which $$F$$ is the cumulated probability of failing, $$f$$ the measured strength value, $$\alpha$$ the first or shape parameter, and $$\beta$$ the second or scale parameter.

The minimum applicable strength value $$f_{min}$$ can be determined by the following expression:

(3) $$f_{min} = \beta \left[ \ln \frac{1}{1-F} \right]^\frac{1}{\alpha}$$

in which the cumulated probability $$F$$ of failing has a low value, e.g. 0.0001.

When the scatter of strength measurements is extremely important, or the minimum strength value is certainly larger than zero, straight application of the two parameter Weibull distribution is of no meaning and should be replaced by a more complex method leading to the use of three parameter Weibull distributions, expressed as:

(4) $$F(f;\alpha,\beta) = 1 - \exp \left[ -\left( \frac{f-\gamma}{\beta} \right)^\alpha \right]$$

in which $$\gamma$$ is the third parameter, or the minimum value of the strength. Or in other words, the weakest link.

It should also be taken into account that the parameters shall be considered as distributions $$\mathcal {D}$$ of random variables with a mean value and a variance:

(5) 1st, or shape parameter:   $$\alpha = \mathcal{D}( \alpha_m,Var_\alpha)$$

(6) 2nd, or scale parameter:   $$\beta = \mathcal{D}( \beta_m,Var_\beta)$$

(7) 3rd, or location parameter: $$\gamma = \mathcal{D}( \gamma_m,Var_\gamma)$$

Quantification method
In order to quantify the three parameters, and especially the third one, the weakest link, a relative large number of data should be available, grouped in series. Those series have to be obtained from measurement pieces drawn from lots, where each lot should be considered as more or less homogeneous. The estimation of the parameters then may follow the following iterative procedure:

step i: Use the three parameter Weibull distribution, however declare the third parameter $$\gamma_{min} = 0$$. Put each series of data into that distribution by means of a best fitted method. This gives for each series the parameters $$\alpha$$ and $$\beta$$. For a best fitting method, see reference [2], and possibly the Wikipedia contribution "Weibull modulus".

step ii: With those parameters $$\alpha$$ and $$\beta$$, calculate a specific value and range, e.g. the average $$f_a$$ value and the range $$R_{p2-p1}=f_{p2}-f_{p1}$$ with possibly $$p1=0.05$$ and $$p2=0.95$$.

 step iii: Place all pairs of specific range/specific value, here as example the $$R_{p2-p1}$$/$$f_a$$ pairs, into a Cartesian diagram with as abscise the specific range and as ordinate the specific value. The result should be a stretched cloud pointing to the ordinate.

step iv: Calculate the trend, or regression line and calculate the specific value when the specific range equals to zero. The value obtained, $$f_{a,R=0}$$, is also the average third parameter $$\gamma_{a,R=0}$$.

step v: Calculate the best estimation of the residual standard deviation $$s_r$$, needed for the estimation of the lower value of the third parameter, the real weakest link $$\gamma_{min}$$, with the expressions:

(8) $$s_r = \sqrt{K_f \frac{1-r^2}{n-2}}$$ and (9) $$K_f = \sum_{i=1}^n f_a^2 - \frac{1}{n} \left(\sum_{i=1}^n f_a\right)^2$$

where $$K_f$$ is the square sum of the average $$f_a$$ values, $$n$$ is the number of pairs, and $$r$$ is the correlation factor.

step vi: Calculate the lower value of the third parameter by means of

(10) $$\gamma_{min} = \gamma_{a,R=0} - u.s_r$$

where $$u$$ is the eccentricity of the normal distribution, e.g. $$u=1.64$$ for a probability of 0.05.

step vii: Replace the used third parameter in step i by the $$\gamma_{min}$$ value estimated in step vi. Repeat steps i to vii until the difference between last and fore last value is sufficient small.

Conditions for representation
In order to select a most representative number of strength measurement series of the entire, more or less inhomogeneous population, but ensuring within each lot a most representative homogeneous number of measurement pieces, some conditions should be taken into account.

a)	The number of series should represent a sufficient time interval of production of the concerned material. A population consisting out of up to five series is certainly too small, a population from six to ten series may lead to doubtful results.

b)	The number of measurement pieces per series should be recommended to have at least 10 pieces. The test pieces for one series should be selected randomly from one lot manufactured within a production interval that may be assumed to be as homogeneous as possible.

c)	The measurement pieces per series should be of one size: the measurement pieces of different series should differ in size.

Examples of possible minimum numbers of measurement series:

Threads: From each of e.g. four production intervals or lots, spread over a production period of one month, where each production of threads is as homogeneous as possible, a first series of tensile measurements should consist out of measurement pieces of length $$l$$, a second series should consist out of measurement pieces of 3 times $$l$$ and a third series should consist out of measurement pieces of 10 times $$l$$. So in total 12 series of tensile measurement pieces.

Annealed flat glass: From each of e.g. four production intervals or lots, spread over a production period of some months, where each production of annealed flat glass is as homogeneous as possible (e.g. series could be taken from one very large sheet of glass), a first series should consist out of measurement pieces of surface $$A_1$$, a second series should consist out of measurement pieces of surface $$A_2$$, and a third series should consist out of measurement pieces of surface $$A_3$$. For various surfaces, refer to the series of international standards CEN ISO 1288 (reference [3]). So in total 12 series

Computer generated example


An example with computer generated data is given below.

Measurement pieces:

- 4 lots, where each lot contains 3348 strength numbers;

- each lot deliver 3 series of measurement pieces, where the pieces contain 121, 25 or 9 strength numbers, in respectively the first, second and third series;

- each series consists out of 12 measurement pieces;

- total 4 x 3 x 12 = 144 measurement pieces;

- strength numbers $$f_i$$ are randomly distributed following the expression

(11) $$f_i = 20+4 \mu_0 + \left[\left(40+40 \mu_0 \right) \left(\mu_1+\mu_2 \right) \right]$$

where $$\mu_0$$ is a random variable related to lots with values between 0 and 1, and $$\mu_1$$ and $$\mu_2$$ are random variables related to any stress number, also with values between 0 and 1;

Third parameter estimation:

The steps i to vii have been performed two times. The result is graphically given in figure 1. The stability of the method has been investigated by repeating the whole process 20 times. The averaged minimum third parameter value $$\gamma_{min}$$ was 16.2 by a standard deviation $$s_{\gamma_{min}}$$=2.2.

Symbols
$$ f $$	= occurrence, e.g. breaking stress

$$ m $$	= subscript that indicates mean value

$$ A $$	= a dimension, e.g. length, surface, ..

$$ F $$	= probability that an occurrence occurs

$$ K $$	= intensity of irregularities that influences, most negatively, an occurrence

$$ R $$	= range, or value of difference between two occurrences

$$Var $$	= variance of within a series of values

$$ \alpha $$	= first parameter of Weibull distributions

$$ \beta $$	= second or scale parameter of Weibull distributions

$$ \gamma $$	= third parameter of Weibull distributions

Weibull distribution with parameters subject of random variation
The following sections give the information for the development from expressions in the “Introduction” to those in step v and step vi in the “Quantification method”.

From expression (4) the stress $$f_{F}$$ belonging to a certain occurrence probability F is

(i) $$f_{F} = \gamma + \beta \left( \ln \frac{1}{1-F} \right)^\frac{1}{\alpha}$$

From expression (i) and expressions (5) to (7) it can be concluded that

 at first  the variance of the stress with a determined probability to occur, is

(ii) $$Var_{F} = Var_{ \gamma} + Var_{ \beta + \alpha}$$ where + means "combined with".

When the probability $$ F $$ is selected in such a way that $$ \ln \frac{1}{1-F} = 1$$, then expression (i) will be simplified to

(iii) $$f_{F} = \gamma + \beta$$ which can be detailed as

(iv.a)	mean value $$f_{F;m} = \gamma_{m} + \beta_{m}$$ and

(iv.b)	variance $$Var_F = Var_{\gamma} + Var_{\beta}$$

 and secondly  the value of the stress $$f_{F}$$ with a determined probability $$ F $$ to occur varies linear with the value of the second parameter $$ \beta $$. When the value of $$ \beta $$ can be reduced to zero, the remaining expression will be

(v)	$$f_F = \gamma $$ which can be detailed as

(vi.a)	mean value $$f_{F;m} = \gamma_m $$ and

(vi.b)	variance $$Var_F = Var_{\gamma} $$

Properties of the second or scale parameter $$ \beta $$
Failing of an object under stress of inhomogeneous objects such as threads, or brittle materials such as annealed glass, depends on the size $$ A $$ of the objects as well as the intensity $$ K $$ of those irregularities in or on the object from where failing starts. For instance for threads the size is characterized by the length, the intensity of the irregularities by a.o. variations in thickness. In the case of annealed glass the size is characterized by the surface of the object, the intensity of the irregularities by the presence and depth of flaws on that surface.

This dependency can be expressed as

(vii) $$ \frac{ \beta_{A_1}}{ \beta_{A_2}} = K \left ( \frac{A_2}{A_1} \right )^ \frac{1}{a} $$

so that the expression (6) can be more detailed by

(viii) $$ \beta _{A_1} = \mathcal {D} \left [ K \left ( \frac{A_2}{A_1} \right )^\frac{1}{a} \beta_{A_{2;m}}, K \left ( \frac{A_2}{A_1} \right )^ \frac{1}{a} Var_{A_{2;m}} \right ] = K \left ( \frac{A_2}{A_1} \right )^ \frac{1}{a} \mathcal {D} \left ( \beta_{A_{2;m}} , Var_{A_{2;m}} \right ) $$

This expression demonstrates that when the size $$ A_{1} $$ approaches the infinitive and, or the irregularities are such that $$ K $$ approaches zero, the value of the second parameter $$ \beta $$ approaches zero.

Reduction of second parameter $$ \beta $$ to zero
For a certain measurement series, the occurrence $$f_{F_1} $$ occurring with a probability $$F_1 $$, and the occurrence $$f_{F_2} $$ occurring with a probability $$ F_2 $$, are

(ix) $$ f_{F_1} = \gamma + \beta \left( \ln \frac {1}{1-F_1} \right)^ \frac {1}{\alpha} $$ and

(x) $$ f_{F_2} = \gamma + \beta \left( \ln \frac {1}{1-F_2} \right)^ \frac {1}{\alpha} $$

When stating that the range $$ R_{F_1-F_2} $$ for that particular measurement series is

(xi) $$ R_{F_1-F_2} = f_{F_1} - f_{F_2} = \beta \left [ \left ( \ln \frac {1}{1-F_1}\right )^ \frac {1}{\alpha} - \left ( \ln \frac {1}{1-F_2}\right )^ \frac {1}{\alpha} \right ] $$

then the second parameter $$ \beta $$ can be expressed as

(xii) $$ \beta = \frac {R_{F_1-F_2}}{ \left ( \ln \frac {1}{1-F_1} \right )^ \frac {1}{\alpha} - \left ( \ln \frac {1}{1-F_2}\right )^ \frac {1}{\alpha}} $$

When substituting $$ \beta $$ in expression (ix) by expression (xii), the result is that

(xiii) $$ f_{F_1} = \gamma + \frac {R_{ \left ( F_1-F_2 \right )} \ln \left( \frac {1}{1-F_1} \right ) ^ \frac {1}{\alpha}}{ \left ( \ln \frac {1}{1-F_1}\right )^ \frac {1}{\alpha} - \left ( \ln \frac {1}{1-F_2} \right )^ \frac {1}{\alpha}} $$

Further, when taking into account the variability of $$ \beta $$ by using expression (viii), it can be concluded that the occurrence $$ f $$ with a probability of $$ F_1 $$ is

(xiv) $$ f_{F_1} = \gamma + K \left ( \frac{A_2}{A_1} \right )^ \frac{1}{\alpha} \frac {R_{ \left ( F_1-F_2 \right )} \ln \left( \frac {1}{1-F_1} \right ) ^ \frac {1}{\alpha}}{ \left ( \ln \frac {1}{1-F_1}\right )^ \frac {1}{\alpha} - \left ( \ln \frac {1}{1-F_2} \right )^ \frac {1}{\alpha}} $$

This means that when the size $$ A_1 $$ becomes an infinitive large, or when the intensity of the irregularities are so important that $$ K $$ approaches zero, the occurrence equals to the value of the third parameter, or

(xv) $$ f_{F_1} = \gamma $$

to be detailed as

(xvi.a) mean value: $$ f_{F_{1;m}} = \gamma_m $$ (xvi.b) and variance: $$ Var_{F_1} = Var_{ \gamma } $$

Conclusion
On the basis of (xiii), the measured values of pairs $$ R_{F_1-F_2} $$ and $$ f_{F_1} $$ can be placed in a Cartesian diagram, which should give a scatter of points where the scatter is determined by the distributions of $$ \alpha $$, $$ \beta $$ and $$ \gamma $$. When the specimens per measurement series do not differ in dimension and, or when the series are drawn from lots with no significant difference in the intensity of irregularities, the scatter will be an undetermined cloud of points. No conclusion can be drawn. When on the other hand these differences have been respected, the scatter allows to draw a regression line of $$ f_{F_1} $$ as function of the range $$ R_{F_1-F_2} $$, and there where the range $$ R_{F_1-F_2} $$ equals to zero, to apply expression (xv), and thus the quantification method as exposed above.