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.
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