Bending of nanobeams in finite elasticity

Motivated by the need to have a fully nonlinear beam model usable at the nanoscale, in this paper, the equilibrium problem of inflexed nanobeams in the context of nonlocal finite elasticity is investigated. Considering both deformations and displacements large, a three-dimensional kinematic model has been proposed. Extending the linear nonlocal Eringen theory, a constitutive law in integral form for the nonlocal Cauchy stress tensor has been defined. Finally, by imposing the equilibrium conditions, the governing equations are obtained. These take the form of a coupled system of three equations in integral form, which is solved numerically. Explicit formulae for displacements, stretches and stresses in every point of the nanobeam are derived. By way of example, a simply supported nanobeam, which is inflexed under nonlinear conditions, has been considered. The nonlocal effects on the deformation and internal actions are shown through some graphs and discussed in detail.