Torsional rigidity for cylinders with a Brownian fracture

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