NONLINEAR FREE VIBRATION OF NANOBEAMS BASED ON NONLOCAL STRAIN GRADIENT THEORY WITH THE CONSIDERATION OF THICKNESS-DEPENDENT SIZE EFFECT

Although the strain gradient and stress gradient parameters have been widely considered in the frame of nonlocal strain gradient theory, the literature concerned with the additional effect of slender ratio parameter in nonlocal strain gradient beam models is limited. In this paper, a nonlinear dynamical model for nonlocal strain gradient beams is developed and its nonlinear free vibration is analyzed. In the proposed dynamical model, the size-dependent properties associated not only with the nonlocal strain gradient and nonlocal stress gradient parameters but also with the slender ratio parameter are discussed. The effect of slender ratio parameter, which may be also interpreted as the thickness-dependent size effect, is caused by the stress on account of the thickness-direction strain gradient. Based on nonlocal strain gradient theory, the nonlinear governing equation of boundary conditions of the nanobeam are derived first. Then the nonlinear governing equation is simplified for special symmetric boundary conditions and external loadings. In the nonlinear free vibration analysis, an analytical solution for predicting the nonlinear free vibration frequencies is derived via the homotopy analysis method. It is shown that the nonlinear frequencies of the nanobeam display significant size-dependent phenomena for large values of slender ratio parameter and either stiffness-softening or stiffness-hardening behavior may occur. Our results also demonstrate that, besides conventional strain gradient and stress gradient effects, the thickness dependent size effect can be significant for slender nanobeams and cannot be ignored in many cases.