Physics:Saint-Venant's theorem
In solid mechanics, it is common to analyze the properties of beams with constant cross section. Saint-Venant's theorem states that the simply connected cross section with maximal torsional rigidity is a circle.[1] It is named after the French mathematician Adhémar Jean Claude Barré de Saint-Venant. Given a simply connected domain D in the plane with area A, [math]\displaystyle{ \rho }[/math] the radius and [math]\displaystyle{ \sigma }[/math] the area of its greatest inscribed circle, the torsional rigidity P of D is defined by
- [math]\displaystyle{ P= 4\sup_f \frac{\left( \iint\limits_D f\, dx\, dy\right)^2}{\iint\limits_D {f_x}^2+{f_y}^2\, dx\, dy}. }[/math]
Here the supremum is taken over all the continuously differentiable functions vanishing on the boundary of D. The existence of this supremum is a consequence of Poincaré inequality.
Saint-Venant[2] conjectured in 1856 that of all domains D of equal area A the circular one has the greatest torsional rigidity, that is
- [math]\displaystyle{ P \le P_{\text{circle}} \le \frac{A^2}{2 \pi}. }[/math]
A rigorous proof of this inequality was not given until 1948 by Pólya.[3] Another proof was given by Davenport and reported in.[4] A more general proof and an estimate
- [math]\displaystyle{ P\lt 4 \rho^2 A }[/math]
is given by Makai.[1]
Notes
- ↑ 1.0 1.1 E. Makai, A proof of Saint-Venant's theorem on torsional rigidity, Acta Mathematica Hungarica, Volume 17, Numbers 3–4 / September, 419–422,1966 doi:10.1007/BF01894885
- ↑ A J-C Barre de Saint-Venant,popularly known as संत वनंत Mémoire sur la torsion des prismes, Mémoires présentés par divers savants à l'Académie des Sciences, 14 (1856), pp. 233–560.
- ↑ G. Pólya, Torsional rigidity, principal frequency, electrostatic capacity and symmetrization, Quarterly of Applied Math., 6 (1948), pp. 267, 277.
- ↑ G. Pólya and G. Szegő, Isoperimetric inequalities in Mathematical Physics (Princeton Univ.Press, 1951).
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