Torsional rigidity for cylinders with a Brownian fracture

TL;DR

This paper derives bounds on the expected reduction in torsional rigidity of cylindrical structures caused by a Brownian fracture, linking the loss to the geometry of the cross-section and analyzing the asymptotic behavior for large lengths.

Contribution

It provides new bounds for the expected torsional rigidity loss due to Brownian fractures, connecting geometric properties and probabilistic analysis, especially for cylindrical shapes.

Findings

Expected loss of torsional rigidity scales as R^5 for large length L.

Bounds are expressed in terms of the cross-section's geometry.

Asymptotic behavior characterized for cylindrical shapes with circular cross-section.

Abstract

We obtain bounds for the expected loss of torsional rigidity of a cylinder $\Omega_L=(-L/2,L/2) \times \Omega\subset \R^3$ of length $L$ due to a Brownian fracture that starts at a random point in $\Omega_L,$ and runs until the first time it exits $\Omega_L$. These bounds are expressed in terms of the geometry of the cross-section $\Omega \subset \R^2$. It is shown that if $\Omega$ is a disc with radius $R$, then in the limit as $L \rightarrow \infty$ the expected loss of torsional rigidity equals $cR^5$ for some $c\in (0,\infty)$. We derive bounds for $c$ in terms of the expected Newtonian capacity of the trace of a Brownian path that starts at the centre of a ball in $\R^3$ with radius $1,$ and runs until the first time it exits this ball.