Generalization of Saint-Venant's principle to micropolar continua

Abstract

The mathematical formulation and proof of Saint-Venant's principle as given by Toupin for non-polar solids is generalized to the case of micropolar elasticity. On one end of a micropolar cylinder of arbitrary length and cross-section we apply a system of self-equilibrated stresses and couple stresses. We first prove that the norms of the stress and couple stress tensors are bounded by the energy density. By means of Rayleigh's principle for the lowest natural eigenfrequency for a slice of the cylinder we then prove that the energy, stored in the cylinder beyond a certain distance from the loaded end, has an exponential decrease with this distance, thus establishing Saint-Venant's principle for the system.