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Torsional rigidity for cylinders with a Brownian fracture
Author(s) -
Berg Michiel van den,
den Hollander Frank
Publication year - 2018
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12138
Subject(s) - mathematics , brownian motion , ball (mathematics) , rigidity (electromagnetism) , geometry , mathematical analysis , newtonian fluid , classical mechanics , physics , quantum mechanics , statistics
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 5for 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 inR 3 with radius 1, and runs until the first time it exits this ball.

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