Some Recent Results on Saint-Venant’s Principle

Abstract

Saint-Venant’s principle has been accepted by structural engineers, numerical analysts, and others as a reliable guide to how far edge and boundary effects penetrate into a body. Nevertheless, this conjecture by Saint-Venant in 1855, even though based upon plausible intuitive argument, has from the beginning provoked scepticism and the history of the subject has been characterized by attempts to formulate a precise mathematical definition supported by rigorous mathematical proof. Two main approaches may be discerned. One group investigates relevant properties of exact solutions: within this category are authors such as Boussinesq [4], Clebsch [5], Dougall [7], Synge [41], von Mises [32], Sternberg [39], and by slight extension, Mielke [30], [31] who has applied center manifold arguments. The other main area of study considers energy, and early contributors include Zanaboni [43], Goodier [23], [18], and Dou [6], the classic paper being by Toupin [42]. Later writers are, for example, Fichera [9], [10], [11], Oleinik et al. [24]–[28], [36], Payne and Horgan and their respective coworkers. The subject has been comprehensively surveyed by Gurtin [19], Maisonneuve [29], and Horgan and Knowles [20].