Nonlocal nonlinear model of Bernoulli–Euler nanobeam with small initial curvature and its application to single-walled carbon nanotubes

Although the small-scale effect and initial curvature have significant influence on the nonlinear mechanical property of the nanobeam, there are few researches to consider both factors. On the basis of the nonlocal differential constitutive relation, this paper, for the first time, presents a nonlinear partial differential–integral equation model of Bernoulli–Euler nanobeam with a small initial curvature. Then the model is applied to research the static bending and the frequency of free vibration for the single-walled carbon nanotubes (SWCNTs). This study indicates that the static deformations are relate to the load direction due to the initial curvature. In addition, the initial curvature complicates the relation between the free vibration frequencies and vibration amplitudes.