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 is a characteristic of a narrower distribution of values. Use case examples include biological and brittle material failure analysis, where modulus is used to describe the variability of failure strength for materials.

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 is a characteristic of a narrower distribution of values. Use case examples include biological and brittle material failure analysis, where modulus is used to describe the variability of failure strength for materials.

Definition

The Weibull distribution, represented as a cumulative distribution function (CDF), is defined by: F(x)=1-\exp\left(-\left(\frac{x-x_u}{x_0}\right)^m\right) in which m is the Weibull modulus. 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 x_0 , which correlates with a sharper peak in the probability density function (PDF) defined by: f(x)=\left(\frac{m}{x_0}\right)\left(\frac{x-x_u}{x_0}\right)^{m-1}\exp\left(-\left(\frac{x-x_u}{x_0}\right)^m\right) Failure analysis often uses this distribution, as a CDF of the probability of failure F of a sample, as a function of applied stress σ, in the form: F(\sigma)=1-\exp\left[-\left(\frac{\sigma}{\sigma_0}\right)^m\right] Failure stress of the sample, σ, is substituted for the x property in the above equation. The initial property 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 steeper slope within the plot. The Weibull distribution can also be multi-modal, in which there would be multiple reported x_0 values and multiple reported moduli, m. The CDF for a bimodal Weibull distribution has the following form, when applied to materials failure analysis: F(\sigma)=1-\phi\exp\left[-\left(\frac{\sigma}{\sigma_{01}}\right)^{m_1}\right] -(1-\phi)\exp\left[-\left(\frac{\sigma}{\sigma_{02}}\right)^{m_2}\right] This represents a material which fails by two different modes. In this equation m 1 is the modulus for the first mode, and m 2 is the modulus for the second mode. Φ is the fraction of the sample set which fail by the first mode. The corresponding PDF is defined by: f(\sigma)=\phi\left(\frac{m_1}{\sigma_{01}}\right)\left(\frac{\sigma}{\sigma_{01}}\right)^{m_1-1}\exp\left[-\left(\frac{\sigma}{\sigma_{01}}\right)^{m_1}\right]+(1-\phi)\left(\frac{m_2}{\sigma_{02}}\right)\left(\frac{\sigma}{\sigma_{02}}\right)^{m_2-1}\exp\left[-\left(\frac{\sigma}{\sigma_{02}}\right)^{m_2}\right] 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. P=1-F Where P corresponds to the probability of survival of a specimen for a given stress value. Thus, it follows that: 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. 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: \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 also be expressed as: \ln \left[\ln\left (\frac{1}{1-F} \right)\right]= m\ln(\sigma)-m\ln(\sigma_0) When the left side 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 \ln(\sigma_0) . Looking at the plotted linearization 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. F(\sigma)=\frac{i-0.5}{N} where i is the specimen number as ranked and N is the total number of specimens in the sample. From there F can be plotted against failure strength to obtain a Weibull CDF. The Weibull parameters, modulus and characteristic strength, can be obtained from fitting or using the linearization method detailed above.

References

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