Zanaboni's treatment of Saint-Venant's principle

Abstract

Zanaboni's version of Saint-Venant's principle, which concerns an elastic body subjected to self-equilibrated loads distributed over a part of its surface, states that the stored energy tends to zero in regions increasingly remote from the load surface. Unlike other formulations, Zanaboni's version applies to bodies of arbitrary shape. Here, unnecessarily complicated aspects of the proof are simplified and rendered mathematically precise by appeal to a variant minimum principle. For cylindrically shaped bodies, a new decay estimate is derived that supplements Zanaboni's contributions and confirms that his approach provides a viable alternative to other studies of Saint-Venant's principle. In conclusion, various generalizations of the original results are briefly discussed, including an extension to nonlinear elasticity.

Keywords:

• elasticity: Saint-Venant's principle and energy decay
• general geometries

AMS Subject Classifications::

• Primary: 74G50
• Secondary: 74G65, 35A35