The Weibull distribution, represented as a cumulative distribution function (CDF), is defined by:

${\displaystyle F(x)=1-\exp(-({\frac {x-x_{u}}{x_{0}}})^{m})}$ 

in which m is the Weibull modulus.^[1] ${\displaystyle x_{0}}$  is a parameter found during the fit of data to the Weibull distribution and represents an input value for which ~67% of the data is encompassed. As m increases, the CDF distribution more closely resembles a step function at ${\displaystyle x_{0}}$ , which correlates with a sharper peak in the probability density function (PDF) defined by:

${\displaystyle f(x)=({\frac {m}{x_{0}}})({\frac {x-x_{u}}{x_{0}}})^{m-1}\exp(-({\frac {x-x_{u}}{x_{0}}})^{m})}$ 

Failure analysis often uses this distribution^[2], as a CDF of the probability of failure F of a sample, as a function of applied stress σ, in the form:

${\displaystyle F(\sigma )=1-\exp \left[-({\frac {\sigma }{\sigma _{0}}})^{m}\right]}$ 

Failure stress of the sample, σ, is substituted for the ${\displaystyle x}$  property in the above equation. The initial property ${\displaystyle x_{u}}$  is assumed to be 0, an unstressed, equilibrium state of the material.

In the plotted figure of the Weibull CDF, it is worth noting that the plotted functions all intersect at a stress value of 50 MPa, the characteristic strength for the distributions, even though the value of the Weibull moduli vary. It is also worth noting in the plotted figure of the Weibull PDF that a higher Weibull modulus results in a

The Weibull distribution can also be multi-modal, in which there would be multiple reported ${\displaystyle x_{0}}$  values and multiple reported moduli, m. The CDF for a bimodal Weibull distribution has the following form,^[3] when applied to materials failure analysis:

${\displaystyle F(\sigma )=1-\phi \exp[(-({\frac {\sigma }{\sigma _{01}}})^{m_{1}}]-(1-\phi )\exp[-({\frac {\sigma }{\sigma _{02}}})^{m_{2}}]}$ 

This represents a material which fails by two different modes. In this equation m₁ is the modulus for the first mode, and m₂ is the modulus for the second mode. Φ is the fraction of materials from the sample which fail by the first mode. The corresponding PDF is defined by:${\displaystyle f(\sigma )=\phi ({\frac {m_{1}}{\sigma _{01}}})({\frac {\sigma }{\sigma _{01}}})^{m_{1}-1}\exp[-({\frac {\sigma }{\sigma _{01}}})^{m_{1}}]+(1-\phi )({\frac {m_{2}}{\sigma _{02}}})({\frac {\sigma }{\sigma _{02}}})^{m_{2}-1}\exp[-({\frac {\sigma }{\sigma _{02}}})^{m_{2}}]}$ 

Examples of a bimodal Weibull PDF and CDF are plotted in the figures of this article with values of the characteristic strength being 40 and 120 MPa, the Weibull moduli being 4 and 10, and the value of Φ is 0.5, corresponding to 50% of the specimens failing by each failure mode.

The compliment of the cumulative Weibull distribution function can be expressed as:

${\displaystyle P=1-F}$ 

Where P corresponds to the probability of survival of a specimen for a given stress value. Thus, it follows that:

${\displaystyle P(\sigma )=1-\left[1-\exp \left[-({\frac {\sigma }{\sigma _{0}}})^{m}\right]\right]=\exp \left[-({\frac {\sigma }{\sigma _{0}}})^{m}\right]}$ 

where m is the Weibull modulus. Here, P corresponds to the probability of survival of a specimen for a given stress value. If the probability is plotted vs the stress, we find that the graph is sigmoidal, as shown in the figure above. Taking advantage of the fact that the exponential is the base of the natural logarithm, the above equation can be rearranged to:

${\displaystyle \ln \left[\ln \left({\frac {1}{1-F}}\right)\right]=\ln \left[\left({\frac {\sigma }{\sigma _{0}}}\right)^{m}\right]}$ 

Which using the properties of logarithms can be expressed as:

${\displaystyle \ln \left[\ln \left({\frac {1}{1-F}}\right)\right]=m\ln(\sigma )-m\ln(\sigma _{0})}$ 

When the LHS of this equation is plotted as a function of the natural logarithm of stress, a linear plot can be created which has a slope of the Weibull modulus, m, and an x-intercept of ${\displaystyle \ln(\sigma _{0})}$ .

Looking at the plotted linearizations of the CDFs from above it can be seen that all of the lines intersect the x-axis at the same point because all of the functions have the same value of the characteristic strength. The slopes vary because of the differing values of the Weibull moduli.

Standards organizations have created multiple standards for measuring and reporting values of Weibull parameters, along with other statistical analyses of strength data:

• ASTM C1239-13: Standard Practice for Reporting Uniaxial Strength Data and Estimating Weibull Distribution Parameters for Advanced Ceramics^[4]
• ASTM D7846-21: Standard Practice for Reporting Uniaxial Strength Data and Estimating Weibull Distribution Parameters for Advanced Graphites ^[5]
• ISO 20501:2019 Fine Ceramics (Advanced Ceramics, Advanced Technical Ceramics) - Weibull Statistics for Strength Data^[6]
• ANSI DIN EN 843-5:2007 Advanced Technical Ceramics - Mechanical Properties of Monolithic Ceramics at Room Temperature - Part 5: Statistical Analysis^[7]

When applying a Weibull distribution to a set of data the data points must first be put in ranked order. For the use case of failure analysis specimens' failure strengths are ranked in ascending order, i.e. from lowest to greatest strength.

Weibull statistics are often used for ceramics and other brittle materials^[8][9]. They have also been applied to other fields as well such as meteorology where wind speeds are often described using Weibull statistics^[10][11][12].

A table is provided with the Weibull moduli for several common materials. However, it is important to note that the Weibull Modulus is a fitting parameter from strength data, and therefore the reported value may vary from source to source.

For ceramics and other brittle materials, the maximum stress that a sample can be measured to withstand before failure may vary from specimen to specimen, even under identical testing conditions. This is related to the distribution of physical flaws present in the surface or body of the brittle specimen, since brittle failure processes originate at these weak points. When flaws are consistent and evenly distributed, samples will behave more uniformly than when flaws are clustered inconsistently. This must be taken into account when describing the strength of the material, so strength is best represented as a distribution of values rather than as one specific value. The Weibull modulus is a shape parameter for the Weibull distribution model which, in this case, maps the probability of failure of a component at varying stresses.

Consider strength measurements made on many small samples of a brittle ceramic material. If the measurements show little variation from sample to sample, the calculated Weibull modulus will be high and a single strength value would serve as a good description of the sample-to-sample performance. It may be concluded that its physical flaws, whether inherent to the material itself or resulting from the manufacturing process, are distributed uniformly throughout the material. If the measurements show high variation, the calculated Weibull modulus will be low; this reveals that flaws are clustered inconsistently and the measured strength will be generally weak and variable. Products made from components of low Weibull modulus will exhibit low reliability and their strengths will be broadly distributed.

A further method to determine the strength of brittle materials has been described by the Wikibook contribution Weakest link determination by use of three parameter Weibull statistics.