Torsional rigidity for cylinders with a Brownian fracture

We obtain bounds for the expected loss of torsional rigidity of a cylinder CL of length L and planar cross-section Ω due to a Brownian fracture that starts at a random point in CL and runs until the first time it exits CL. These bounds are expressed in terms of the geometry of the cross-section Ω⊂R2. It is shown that if Ω is a disc with radius R, then in the limit as L→∞ the expected loss of torsional rigidity equals cR5 for some c∈(0,∞). 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 R3 with radius 1, and runs until the first time it exits this ball.