St. Petersburg Paradox and Failure Probability

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.

Figure 1

Schematic of the tensometer and force evolution on a fiber. (a) The tensile load was applied and measured on the fiber samples as a function of length and time using a tensometer. (b) A representative graph of the measured force as a function of time from a polyester fiber (Gutermann white sewing thread, filament $\text{diameter}=0.025\mathrm{mm}$, fiber $\text{diameter}=0.25\mathrm{mm}$). The top left inset shows a microscope image of a polyester fiber sample and the bottom shows a polyamide fiber sample (Eagle Claw 6 lb nylon monofilament fishing line, $\text{diameter}=0.22\mathrm{mm}$). The scale bars in both images are equal to 0.1 mm.

Figure 2

Normalized fracture force for different length fibers. Normalized fracture force for the (a) polyester ($L_0=1.2\mathrm{mm}$, $F_0=17.84N$) and (b) polyamide ($L_0=1.0\mathrm{mm}$, $F_0=42.60N$) fibers as a function of fiber length from 1 mm to 1 km. The datasets were fit using the two parameter $(\alpha ,\beta )$ St. Petersburg, Weibull, and mean-field models and are summarized in the tables above each graph. ${\overline{R}}^2$ was used as a figure of merit.

Figure 3

Fracture force for fibers at different strain rates. The force to fracture a polyester and polyamide fiber as a function of strain rate. The data were fit using a linear regression.