Nonlocal second-order strain gradient elasticity model and its application in wave propagating in carbon nanotubes

Recent experimental studies indicate that Young’s modulus of carbon nanotubes increases steeply with tube diameter decreasing. The consideration of this effect is of great importance for the fabrication and exploitation of nano-electromechanical devices. Nevertheless, the rapid stiffness enhancement effect noticed from experimental observation maybe unable to be predicted by using size-dependent elasticity models available in literatures. It is strongly necessary to further shed light on the size-dependent mechanical mechanism and characterize the rapid strengthening effect of stiffness for nano-sized materials. To achieve this goal, the nonlocal second-order strain gradient elasticity model is established by introducing the second-order strain gradient field with nonlocal effect into the stored energy function of nonlocal first-order strain gradient elasticity theory. With the aids of the laws of thermodynamics, the constitutive relations are obtained. The Hamilton principle is used to derive the governing equations of equilibrium and boundary conditions. The proposed model is applied to investigate the problem of wave propagating in carbon nanotubes. The new dispersion relations derived are presented for evaluating the influences of size-dependent parameters on the characteristics of wave propagation. The results show that present model can predict the rapid increasing effect of carbon nanotubes with the decrease of tube size.