Load symmetry and energy minimization in Saint-Venant's problem
Saint-Venant's solution, if regarded as the solution of an elastic three- dimensional problem, is not necessarily unique, since tractions on the bases are not locally specified, but only their resultants are given. To overcome this indeterminateness Saint-Venant himself imposed the further condition that three stress components \((\sigma_{xx},\sigma_{yy},\sigma_{xy})\) are identically zero. Since this condition may appear artificial, it has been tried, in more recent times, to justify this assumption more rationally. \textit{J. K. Knowles} and \textit{E. Sternberg} [Arch. Ration. Mech. Anal. 22, 100-120 (1966; Zbl 0149.434)] proved that Saint-Venant's solutions have the peculiar property of minimizing the strain energy when tractions on the bases satisfy a certain local condition of equilibrium, but this does not happen when, for instance, the characteristics of displacement, instead of stress, on the bases are prescribed. It is thus convenient to enlarge the class introduced by Knowles and Sternberg, and the suitable choice is that of elastic states obeying a certain condition of global equilibrium at the bases.
scientific article; zbMATH DE number 4074590