In
solid mechanics
Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and ...
, it is common to analyze the properties of
beams with constant cross section. Saint-Venant's theorem states that the
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
cross section with maximal
torsional
rigidity is a circle.
[E. Makai, A proof of Saint-Venant's theorem on torsional rigidity, ]Acta Mathematica Hungarica
'' Acta Mathematica Hungarica'' is a peer-reviewed mathematics journal of the Hungarian Academy of Sciences, published by Akadémiai Kiadó and Springer Science+Business Media. The journal was established in 1950 and publishes articles on mathema ...
, Volume 17, Numbers 3–4 / September, 419–422,1966 It is named after the French mathematician
Adhémar Jean Claude Barré de Saint-Venant
Adhémar Jean Claude Barré de Saint-Venant (23 August 1797 – 6 January 1886) was a mechanician and mathematician who contributed to early stress analysis and also developed the unsteady open channel flow shallow water equations, also known as ...
.
Given a
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spa ...
domain ''D'' in the plane with area ''A'',
the radius and
the area of its greatest inscribed circle, the torsional rigidity ''P''
of ''D'' is defined by
:
Here the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
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 conjectured in 1856 that
of all domains ''D'' of equal area ''A'' the circular one has the greatest torsional rigidity, that is
:
A rigorous proof of this inequality was not given until 1948 by
Pólya. Another proof was given by
Davenport and reported in.
[G. Pólya and G. Szegő, Isoperimetric inequalities in Mathematical Physics (Princeton Univ.Press, 1951).] A more general proof and an estimate
:
is given by Makai.
Notes
Elasticity (physics)
Calculus of variations
Inequalities
Physics theorems