Free vibrations of FG elastic Timoshenko nano-beams by strain gradient and stress-driven nonlocal models

Size-dependent vibrational behavior of functionally graded (FG) Timoshenko nano-beams is investigated by strain gradient and stress-driven nonlocal integral theories of elasticity. Hellinger-Reissner's variational principle is preliminarily exploited to establish the equations governing the elastodynamic problem of FG strain gradient Timoshenko nano-beams. Differential and boundary conditions of dynamical equilibrium of FG Timoshenko nano-beams, with nonlocal behavior described by the stress-driven integral theory, are formulated. Free vibrational responses of simple structures of technical interest, associated with nonlocal stress-driven and strain gradient strategies, are analytically evaluated and compared in detail. The stress-driven nonlocal model for FG Timoshenko nano-beams provides an effective tool for dynamical analyses of stubby composite parts of Nano-Electro-Mechanical Systems.