Flexure mechanics of nonlocal modified gradient nano-beams

Two frameworks of the nonlocal integral elasticity and the modified strain gradient theory are consistently merged to conceive the nonlocal modified gradient theory. The established augmented continuum theory is applied to a Timoshenko-Ehrenfest beam model. Nanoscopic effects of the dilatation, the deviatoric stretch, and the symmetric rotation gradients together with the nonlocality are suitably accommodated. The integral convolutions of the constitutive law are restored with the equivalent differential model subject to the nonclassical boundary conditions. Both the elastostatic and elastodynamic flexural responses of the nano-sized beam are rigorously investigated and the well posedness of the nonlocal modified gradient problems on bounded structural domains is confirmed. The analytical solution of the phase velocity of flexural waves and the deflection and the rotation fields of the nano-beam is detected and numerically illustrated. The transverse wave propagation in carbon nanotubes is furthermore reconstructed and validated by the molecular dynamics simulation data. Being accomplished in revealing both the stiffening and softening structural responses at nano-scale, the proposed nonlocal modified gradient theory can be beneficially implemented for nanoscopic examination of the static and dynamic behaviors of stubby nano-sized elastic beams.