Equivalent Formulations of Euler–Bernoulli Beam Theory for a Simple Gradient Elasticity Law

Existing Euler-Bernoulli beam theories in classical elastostatics suffer from the inconsistency that either the elasticity law or the equilibrium equations are not satisfied in local form. It has recently been shown that by assuming elastic anisotropy subject to internal constraints, it is possible to make the theory consistent. This has been proved to be true also for a simple gradient elasticity law. Usually, bending of beams is viewed as a one-dimensional problem. We consider in this paper two known one-dimensional formulations for Euler-Bernoulli beam and gradient elastic material behavior. The two formulations seem to be different, as the free energy functional of the one includes the cross-sectional area of the beam, whereas the other does not. The aim is, by using consistent Euler-Bernoulli beam theory, to derive the two one-dimensional formulations as special cases of a three-dimensional simple gradient elasticity model and to show that these are equivalent to each other.