Recent Developments Concerning Saint-Venant’s Principle: An Update

Abstract

The simplifications arising in elasticity theory from consideration of resultant boundary conditions instead of mathematically exact pointwise conditions have been the key to widespread application of the subject. Thus, for example, theories for strength of materials, plates, and shells rely on such relaxed boundary conditions for their development. The justification of this approximation is usually based on some form of the celebrated Saint-Venant’s principle. A comprehensive survey of contemporary research concerning Saint-Venant’s principle (covering primarily the period 1965–1981) was given by Horgan and Knowles (1983). Since that time, several developments have taken place demonstrating continued interest in understanding the ramifications of Saint-Venant’s principle from both a physical and mathematical point of view. In this article we review these developments, thus providing an update on contributions to this fundamental engineering principle.

Spatial decay of transient end effects in functionally graded heat conducting materials

“…For the example (7.3), if we further assume that cp = c0poexp(-/3x2//i), (7.5) where ( is a thermal diffusivity constant. Thus the decay factor (6.11) for the case a > /3 is simply "p (-sh) • (7)(8) while for a < (3, the decay factor is e(a-/3)02 oxpr~iM~J' (7)(8)(9) which yields a decay rate smaller than that in (7.8). These results reflect the interplay between the degree of inhomogeneity parameters a and /?, i.e., the measures of inhomogeneity of K(x2) and c{x2)p(x2) respectively.…”

“…For homogeneous heat conducting solids, this question was first raised by Boley (see, e.g., Boley [3], Boley and Weiner [4]) in the context of a Saint-Venant principle for heat conduction. Saint-Venant's principle in the equilibrium theory of linear elasticity has a long history (see, e.g., the reviews by Horgan and Knowles [13], Horgan [9,10]) and is a consequence of the elliptic character of the governing partial differential equations. The study of related issues for parabolic equations is more recent.…”

Spatial evolution of harmonic vibrations in linear elasticity

“…The standard procedure used in engineering practice to determine the extent of local stresses or edge effects is based on some form of the celebrated Saint Venant principle. A comprehensive surveys of contemporary research concerning Saint Venant principle can be found in [Horgan and Knowles 1983;Horgan 1989;1996].…”

Spatial behavior for a fourth-order dispersive equation

“…The history and development of these questions was extensively surveyed by Horgan and Knowles [11] in a work periodically updated by Horgan [9,10]. Moreover, the books by Ames and Straughan [1], Flavin and Rionero [6] and Antontsev et al [2] are devoted to such matters.…”