Calculus of Variations and Geometric Measure Theory
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G. Crasta - I. FragalĂ  - F. Gazzola

A sharp upper bound for the torsional rigidity of rods by means of web functions

created on 03 Oct 2001
modified on 10 Dec 2003


Published Paper

Inserted: 3 oct 2001
Last Updated: 10 dec 2003

Journal: Arch. Rational Mech. Anal.
Volume: 164
Number: 3
Pages: 453-469
Year: 2002


Using web functions, we approximate the Dirichlet integral which represents the torsional ri\-gi\-di\-ty of a cylindrical rod with planar convex cross section $\Omega$. To this end, we use a suitably defined piercing function, which enables us to obtain bounds for both the approximate and the exact value of the torsional rigidity. When $\Omega$ varies, we show that the ratio between these two values is always larger than $3/4$; we prove that this is a sharp lower bound and that it is not attained. Several extremal cases are also analyzed and studied by numerical methods.

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