A gradient model for Timoshenko nanobeams

The elastostatic problem of a Timoshenko nanobeam is formulated by a new constitutive behaviour of gradient-type. Unlike previous approaches which directly substitute the expression of the nonlocal stress into the classical equilibrium equations, the proposed model starts from a nonlocal thermodynamic formulation. A suitable definition of the expression for the internal energy provides the variational formulation of Timoshenko nanobeams in terms of rotations and transverse displacements so that a higher-order system of ordinary differential equations is consistently obtained with the corresponding boundary conditions. Different from other nonlocal models, the solutions indicate that the stiffness of nanobeams is significantly increased at smaller scales due to size effects. The solutions corresponding to local models are perfectly obtained as a special case of the nonlocal ones.