Size-dependent vibration analysis of nanobeams based on the nonlocal strain gradient theory

A size-dependent sinusoidal shear deformation beam model is developed to investigate the free vibration of nanobeams based on the nonlocal strain gradient theory. The new model contains a nonlocal parameter and a material length scale parameter which can capture the size effect. The governing equations and boundary conditions are derived by employing Hamilton's principle. Navier's method is utilized to obtain analytical solutions for natural frequencies of simply supported nanobeams. The results are compared with other beam models and other classical and non-classical theories. Several numerical examples are presented to illustrate the effects of nonlocal parameter, material length scale parameter, slenderness ratio and shear deformation on the free vibration of nanobeams. It is found that natural frequencies predicted by the nonlocal strain gradient theory are higher than those predicted by nonlocal theory and lower than those obtained by strain gradient theory. When the length scale parameter is smaller than the nonlocal parameter, the nanobeam exerts a stiffness-softening effect. When the length scale parameter is larger than the nonlocal parameter, the nanobeam exerts a stiffness-hardening effect. Moreover, it is observed that the effect of shear deformation becomes more significant for nanobeams with lower values of slenderness ratios and for higher modes.