Torsional rigidity for cylinders with a Brownian fracture.

Abstract: We obtain bounds for the expected loss of torsional rigidity of a cylinder C L of length L and planar cross‐section Ω due to a Brownian fracture that starts at a random point in C L and runs until the first time it exits C L. These bounds are expressed in terms of the geometry of the cross‐section Ω ⊂ R 2. 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 c R 5 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 R 3 with radius 1, and runs until the first time it exits this ball. [ABSTRACT FROM AUTHOR]