Solution of the generalized Saint Venant problem

Journal Article · Thu Dec 31 04:00:00 EST 1998 · Sbornik. Mathematics

DOI:https://doi.org/10.1070/SM1998V189N12ABEH000362; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)· OSTI ID:21202823

Avkhadiev, F G ^[1]

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1. Institute of Mathematics and Mechanics, Kazan State University (Kazakhstan)

A well-known problem in the mathematical theory of elasticity about the torsional rigidity P({omega}) of a bar whose cross-section is an arbitrary simply connected domain {omega} is considered. It is shown that P({omega}) is equivalent to the moment of inertia of the domain relative to its boundary. Thus, a new interpretation of the well-known Coulomb's formula is suggested, and on this basis the following problem, which has its origins in works of Cauchy and Saint Venant, is solved: find a geometric parameter equivalent to the torsional rigidity coefficient of elastic bars with simply connected cross-sections. The proof is based on the definition of the torsional rigidity as the norm of a certain embedding operator in a Sobolev space and on the theory of conformal maps. In particular, some conformally invariant inequalities are established.

• OSTI ID: 21202823
• Journal Information: Sbornik. Mathematics, Journal Name: Sbornik. Mathematics Journal Issue: 12 Vol. 189; ISSN 1064-5616
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
CROSS SECTIONS
ELASTICITY
MAPS
MATHEMATICAL SOLUTIONS
MOMENT OF INERTIA