Longitudinal vibration of size-dependent rods via nonlocal strain gradient theory

The longitudinal vibration analysis of small-scaled rods is studied in the framework of the nonlocal strain gradient theory. The equations of motion and boundary conditions for the vibration analysis of small-scaled rods are derived by employing the Hamilton principle. The model contains a nonlocal parameter considering the significance of nonlocal elastic stress field and a material length scale parameter considering the significance of strain gradient stress field. The analytical solutions of predicting the natural frequencies and mode shapes of the rods with some specified boundary conditions are derived. A finite element method is developed and can be used to calculate the vibration problem by arbitrarily applying classical and non-classical boundary conditions. It is shown that the nonlocal strain gradient rod model exerts a stiffness-softening effect when the nonlocal parameter is larger than the material length scale parameter, and exerts a stiffness-hardening effect when the nonlocal parameter is smaller than the material length scale parameter. The higher-order frequencies are more sensitive to the non-classical boundary conditions in comparison with the lower-order frequencies, and the type of non-classical boundary conditions has a little effect on mode shapes.