Recent Developments Concerning Saint-Venant's Principle

This chapter provides an overview of the recent developments concerning Saint-Venant's principle. The task of determining, within the framework of the linear theory of elasticity, the stresses and displacements in an elastic cylinder in equilibrium, under the action of loads that arise solely from tractions applied to its plane ends has come to be called Saint- Venant's problem. Saint-Venant's construction does not permit the arbitrary preassignment of the point-by-point variation of the end tractions giving rise to these forces and moments; indeed, this variation is essentially determined as a consequence of the special assumptions made in connection with his so-called semi-inverse procedure. The early work of Saint-Venant and Boussinesq furnished the seeds from which grew a large number of more general assertions, most referring to elastic solids of arbitrary shape and many being rather imprecise, concerning the effect on stresses within the body of replacing the tractions acting over a portion of its surface by statically equivalent ones. Such propositions usually went by the name of Saint-Venunt's principle, despite the fact that Saint-Venant's original conjecture was intended to apply only to cylinders. This chapter discusses in detail about flow in a cylinder, a representation for the exact solution, and energy decay for other linear elliptic second-order problem. Linear elastostatic problems are also stated in the chapter