We obtain bounds for the expected loss of torsional rigidity of a cylinder $Ω_L=(-L/2,L/2) \times Ω\subset \R^3$ of length $L$ due to a Brownian fracture that starts at a random point in $Ω_L,$ and runs until the first time it exits $Ω_L$. These bounds are expressed in terms of the geometry of the cross-section $Ω\subset \R^2$. It is shown that if $Ω$ 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.