A higher-order Eringen model for Bernoulli-Euler nanobeams

Small-scale effects in carbon nanotubes are effectively assessed by resorting to the methods of nonlocal continuum mechanics. The crucial point of this approach consists in defining suitable constitutive laws which lead to reliable results. A nonlocal elastic law, diffusely adopted in literature, is that proposed by Eringen. According to this theory, the elastic equilibrium problem of a nonlocal nanostructure is equivalent to that of a corresponding local nanostructure subjected to suitable distortions simulating the nonlocality effect. Accordingly, transverse displacements and bending moments of a Bernoulli-Euler nonlocal nanobeam can be obtained by solving a corresponding linearly elastic (local) nanobeam, subjected to the same loading and kinematic constraint conditions of the nonlocal nanobeam, but with the prescription of suitable inelastic bending curvature fields. This observation leads naturally to the definition of a higher-order Eringen version for Bernoulli-Euler nanobeams, in which the elastic energy is assumed to be dependent on the total and inelastic bending curvatures and on their derivatives. Weak and strong formulations of elastic equilibrium of first-order gradient nanobeams are provided by a consistent thermodynamic approach. Exact solutions of fully clamped and cantilever nanobeams are given and compared with those of literature.