A reciprocity theorem in linear gradient elasticity and the corresponding Saint-Venant principle

In many practical applications of nanotechnology and in microelectromechanical devices, typical structural components are in the form of beams, plates, shells and membranes. When the scale of such components is very small, the material microstructural lengths become important and strain gradient elasticity can provide useful material modelling. In addition, small scale beams and bars can be used as test specimens for measuring the lengths that enter the constitutive equations of gradient elasticity. It is then useful to be able to apply approximate solutions for the extension, shear and flexure of slender bodies. Such approach requires the existence of some form of the Saint-Venant principle. The present work presents a statement of the Saint-Venant principle in the context of linear strain gradient elasticity. A reciprocity theorem analogous to Betti’s theorem in classic elasticity is provided first, together with necessary restrictions on the constitutive equations and the body forces. It is shown that the order of magnitude of displacements are in accord with the Sternberg’s statement of the Saint-Venant principle. The cases of stretching, shearing and bending of a beam were examined in detail, using two-dimensional finite elements. The numerical examples confirmed the theoretical results.