Weibull modulus

The Weibull modulus is a dimensionless parameter of the Weibull distribution. It represents the width of a probability density function (PDF) in which a higher modulus correlates with a sharper peak. Use case examples include biological and brittle material failure analysis, where modulus is reported after preforming a fit to the Weibull distribution.

Definition

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

CDF of Weibull distribution for the example of predicting failure in materials, σ₀= 50 MPa

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

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:

PDF of Weibull distribution for the example of predicting failure in materials, σ₀= 50 MPa

${\displaystyle F(\sigma )=1-\exp(-(\sigma /\sigma _{0})^{m})}$

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.

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 bi-modal 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 set which fail by the first mode. The corresponding PDF is defined by:

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

Calculations

Looking at the cumulative Weibull Distribution function

${\displaystyle P=\exp(-(\sigma /\sigma _{0})^{m})}$

where m is the Weibull modulus. If the probability is plotted vs the stress, we find that the graph is sigmoidal. However if the equation is rearranged via

${\displaystyle F=1-P}$

and the fact that e is the base of the natural log. A linear plot can be created via the equation being rearranged to

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

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

This linear plot has a slope of the Weibull modulus and an x-intercept of ${\displaystyle ln(\sigma _{0})}$.

Use Cases

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.

Test procedures for determining the Weibull modulus are specified in DIN EN 843-5 and DIN 51 110-3.

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.

References

1. Weibull, Waloddi (1951). "A Statistical Distribution Function of Wide Applicability". Journal of Applied Mechanics.
2. Meyers, Marc; Chawla, Krishan (2009). Mechanical Behavior of Materials (2nd ed.). Cambridge University Press. ISBN   978-0-511-45557-5.
3. Loidl, Dieter; Paris, Oskar; Rennhofer, H.; Müller, Martin; Peterlik, Herwig (November 2007). "Skin-core structure and bimodal Weibull distribution of the strength of carbon fibers". Carbon. 45 (14): 2801–2805. doi:10.1016/j.carbon.2007.09.011. ISSN   0008-6223.