User:Matthew74205/Weibull modulus

Introduction
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:

$$F(x)=1-\exp(-(\frac{x-x_u}{x_0})^m)$$

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 PDF defined by:

$$F(x)=(\frac{m}{x_0})(\frac{x-x_u}{x_0})^{m-1}\exp(-(\frac{x-x_u}{x_0})^m)$$

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

$$F(\sigma)=1-\exp(-(\sigma/\sigma_0)^m)$$

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.

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The Weibull distribution can also be multi-modal, in which case there are multiple Weibull moduli and multiple σ0.

** INSERT CITATION ABOUT THIS**

something about using it to represent different failure modes in materials

Measurement
ASTM C1239-13: Standard Practice for Reporting Uniaxial Strength Data and Estimating Weibull Distribution Parameters for Advanced Ceramics

Calculations
Looking at the cumulative Weibull Distribution function

$$P=exp[-(\frac{\sigma}{\sigma_0})^m]$$

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

$$P=1-F$$

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

$$\ln(ln\left (\frac{1}{1-F} \right))=ln(\left ( \frac{\sigma}{\sigma_0} \right )^m)$$

This linear plot has a slope of the Weibull modulus and an x-intercept of $$ln(\sigma_0)$$.

Usage
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Brittle materials have a wide range of applications across various industries due to their unique properties. Their uses include, but are not limited to biomedical materials, filtration, catalyst supports, and Solid Oxide Fuel Cells (SOFC). An important characteristic to consider when selecting brittle materials is their mechanical reliability, which is quantified using the Weibull modulus. The importance of the Weibull modulus depends on the specific application context. For example, bone-filling ceramics used in orthopedic procedures require high mechanical reliability to ensure patient safety and efficacy.

Organic Materials
Studies examining organic brittle materials highlight the consistency and variability of the Weibull modulus within naturally occurring ceramics such as human dentin and abalone nacre. Research on human dentin samples indicates that the Weibull modulus remains stable across different depths or locations within the tooth, with an average value of approximately 4.5 and a range between 3 and 6. However, variations in the modulus suggest differences in flaw populations between individual teeth, which may be caused by random defects introduced during specimen preparation. Speculation exists regarding a potential decrease in the Weibull modulus with age due to changes in flaw distribution and stress sensitivity. Failure in dentin typically initiates at these flaws, which can be intrinsic or extrinsic in origin, arising from factors such as cavity preparation, wear, damage, or cyclic loading.

Studies on the abalone shell illustrate its unique structural adaptations, sacrificing tensile strength perpendicular to its structure to enhance strength parallel to the tile arrangement. The Weibull modulus of abalone nacre samples is determined to be 1.8, indicating a moderate degree of variability in strength among specimens.

Inorganic Materials
Some inorganic brittle materials, such as calcium phosphate ceramics (CPCs) are primarily used in non or moderate load-bearing areas due to mechanical instabilities. Research using Weibull statistics has examined the reliability of CPCs, which showed significant influences of factors such as compaction pressure and pore size on their mechanical properties. Macroporous CPCs demonstrate higher and more deterministic modulus values compared to their microporous counterparts, despite challenges in sample size requirements for reliability testing.

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Further data is needed for the range of P > 0.3, particularly relevant for applications involving porous brittle materials, such as bone substitutes. Studies have revealed a fundamental link between the behavior of the Weibull modulus and the evolution of porosity during sintering processes. Biaxial flexure testing demonstrated a U-shaped distribution of Weibull modulus as a function of porosity, with partially sintered Hydroxyapatite (HA) exhibiting higher moduli. Additionally, a decrease in mean fracture strength is observed with decreasing sintering temperature, indicating a relationship between microstructure and mechanical properties in the HA specimens. The interactions between microstructure, processing techniques, and mechanical behavior are highlighted, demonstrating the complexity of porous brittle materials' characteristics.

Quasi-brittle Materials
The Weibull modulus of quasi-brittle materials correlates with the decline in the slope of the energy barrier spectrum, as established in fracture mechanics models. This relationship allows for the determination of both the fracture energy barrier spectrum decline slope and the Weibull modulus while keeping factors like crack interaction and defect-induced degradation in consideration. Experimental results align with predictions, demonstrating the accuracy of the model. Temperature dependence and variations due to crack interactions or stress field interactions are observed in the Weibull modulus of quasi-brittle materials. Damage accumulation leads to a rapid decrease in the Weibull modulus, resulting in a right-shifted distribution with a smaller Weibull modulus as damage increases.