On the equilibrium of a cylindrical elastic solid: Comparisons with St. Venant's theory
The author considers a linearly elastic cylinder of length 2h submitted to end tractions. In a first part, after establishing some identities on stress means, he derives a quadratic functional of the stress that has the property of being minimized by the stress of the Saint Venant solution. This functional is used to compute explicit lower bounds on the stored energy in terms of applied forces in the cases of extension, uniform flexion and torsion. In a second part, a general notion of ``overall strain'' is proposed, which is the least square rigid motion interpolate of the relative displacements of the ends of the cylinder. Explicit formulae for this overall strain in terms of applied forces are given. A comparison with Saint Venant's theory is effected. It shows that, in some instances, these formulae agree with the latter theory when \(h\to \infty\).
minimized by the stress of the Saint Venant solution
scientific article; zbMATH DE number 4094916
scientific article