A Sharp Upper Bound for the¶Torsional Rigidity of Rods¶by Means of Web Functions |
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Authors: | Graziano Crasta Ilaria Fragalà Filippo Gazzola |
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Institution: | 1.Dipartimento di Matematica Pura e Applicata?via Campi 213/B?41100 Modena, Italy,IT;2.Dipartimento di Matematica del Politecnico?piazza L. da Vinci 32?20133 Milano, Italy,IT;3.Dipartimento di Scienze T.A.?via Cavour 84?15100 Alessandria, Italy,IT |
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Abstract: | Using web functions, we approximate the Dirichlet integral which represents the torsional rigidity of a cylindrical rod with planar convex cross-section Ω. 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 Ω varies, we show that the ratio between these two values is always larger than ¾; 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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