  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.
