The Physical Meaning of Astatic Equilibrium in Saint-Venant’s Principle for Linear Elasticity

Abstract

When a boundary loading which is not only self-equilibrated but has the additional property that the loading system remains self-equilibrated when all the forces are rotated through an arbitrary angle about their points of application (astatic equilibrium), is applied to a small region of the surface of a linear elastic body, the long range stress field produced by the loading is in general of smaller order (with respect to the radius of the loaded segment of the boundary) than would be the long range stress field produced by a loading system which was merely self-equilibrating but which would not continue to be self-equilibrating if each force were rotated (von Mises [3], Sternberg [6]). The physical distinctions between astatic equilibrium loadings and merely self-equilibrated loadings, and the physical reasons why astatic equilibrium loadings produce smaller long range stresses, are examined. It is pointed out that astatic equilibrium loadings always produce zero mean deformation in a linear elastic body and that, therefore, if a small volume element, in the neighborhood of a small patch of the boundary surface subject to astatic equilibrium loading were considered as an isolated body, this small volume would undergo no mean deformation and would be easier to fit back into the main body than if it had been subject to merely self-equilibrated loading which would have caused mean deformation.