Consistent Euler – Bernoulli beam theories in statics for classical and explicit gradient elasticities

Abstract

The Euler – Bernoulli beam bending theory in elementary (engineering) mechanics uses two fundamental assumptions, first that the material behaviour is isotropic elastic and secondly that plane cross sections remain plane, rigid and perpendicular to the beam axis. It is well – known, that this theory suffers from the inconsistency that, e.g., the shear strain is always vanishing, whereas the shear stress is supposed to be not vanishing. In the present paper we remove this inconsistency in engineering mechanics by postulating the material response to be a limiting case of anisotropic elasticity. We show that all results within the “inconsistent” isotropic elasticity remain valid. Then, the anisotropic elasticity approach is extended to model bending of Euler – Bernoulli beam in explicit gradient elasticity. Especially, attention is focused on deriving distributions of stress components and discussing limiting responses in dependence of a material length parameter inherent in the elasticity law.