St. Petersburg Paradox and Failure Probability and Extreme Value Statistics of brittle failure

Jake Fontana and coauthor found in PRL (See attached)

The St. Petersburg paradox provides a simple paradigm for systems that show sensitivity to rare events. Here, we demonstrate a physical realization of this paradox using tensile fracture, experimentally verifying for six decades of spatial and temporal data and two different materials that the fracture force depends  logarithmically on the length of the fiber. The St. Petersburg model may be useful in a variety fields where failure and reliability are critical.

However, Taloni and Zapperi show that the result,however, was derived assuming that "the force required to fracture the fiber is a linear function of the defect size" [1], which is in glaring contrast with fracture mechanics. Here we address the problem combining extreme value theory (EVT) [2] with the Griffith's stability crack crite-rion [3]. According to the Griffith's assumption, the fail-ure stress should be inversely proportional to the square root of the largest defect size. We also show that in the asymptotic limit, the wire strength follows the Gumbel's distribution, in full agreement with the data reported in [1], as we demonstrate using the maximum likelihood   method. We thus conclude that the load carrying capac-ity of the wires studied in [1] follows EVT, in agreement with previous observations for different materials [2].

Papers attached.

What do you think?  I would be pleased to discuss this with you, since I had previous work on brittle fracture and Griffith theory (from power law distribution of defects) see

Is Weibull’s modulus really a material constant? Example case with interacting collinear cracks

L Afferrante M.Ciavarella E.Valenza

https://doi.org/10.1016/j.ijsolstr.2005.08.002Get rights and content

Under an Elsevier user licenseopen archive

Abstract

The Weibull distribution is widely used to describe the scatter of the strength in brittle (but also quasi-brittle) materials, often assuming that the Weibull modulus is a “material constant”. One possible motivation of this perhaps comes from the classical Freudenthal’s interpretation of Weibull modulus depending on the crack size distribution, which however assumes the cracks to be at large distance one from the other. It is here found with simple numerical experiments with collinear cracks that Weibull distributions tend to be obtained also with interaction taken into account, but the Weibull modulus depends on both the crack size distribution and the distribution of ligaments. Hence, Weibull modulus should not be considered a “material constant” or to correspond to an “intrinsic” microstructure of the material, as assumed in many industrial applications and commercial postprocessors of FEM softwares, even in the case of a varying stress fields. In the limit case of a crack or sharp notch this leads to paradoxically a zero scale parameter (and the usual Weibull modulus). Hence, in the case of a blunt notch, we suggest the Weibull modulus would vary depending on the distribution of cracks, their distances, and the interaction with the geometry and stress field. Only numerical simulations where the distribution of cracks is directly included in the geometry under consideration can provide the correct scale factor and Weibull modulus.