Weibull modulus
It represents the width of a probability density function (PDF) in which a higher modulus is a characteristic of a narrower distribution of values.
As m increases, the CDF distribution more closely resembles a step function at
, which correlates with a sharper peak in the probability density function (PDF) defined by:
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.
values and multiple reported moduli, m. The CDF for a bimodal Weibull distribution has the following form,[3] when applied to materials failure analysis:
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.
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:
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
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.
Standards organizations have created multiple standards for measuring and reporting values of Weibull parameters, along with other statistical analyses of strength data: When applying a Weibull distribution to a set of data the data points must first be put in ranked order.
The Weibull parameters, modulus and characteristic strength, can be obtained from fitting or using the linearization method detailed above.
[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] 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.
Much work has been done to describe brittle failure with the field of linear elastic fracture mechanics and specifically with the development of the ideas of the stress intensity factor and Griffith Criterion.
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.
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.
With careful manufacturing processes Weibull moduli of up to 98 have been seen for glass fibers tested in tension.
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[14] 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.
Variations in the modulus suggest differences in flaw populations between individual teeth, thought to 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[15] is determined to be 1.8, indicating a moderate degree of variability in strength among specimens.
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.
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.
A higher Weibull modulus allows for companies to more confidently predict the life of their product for use in determining warranty periods.
