Solution of the generalized Saint Venant problem
- 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, Vol. 189, Issue 12; Other Information: DOI: 10.1070/SM1998v189n12ABEH000362; Country of input: International Atomic Energy Agency (IAEA); ISSN 1064-5616
- Country of Publication:
- United States
- Language:
- English
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