Non-local continuum modeling of carbon nanotubes; computation of spatial kernal using helical symmetry lattice dynamics

The longitudinal, transverse and torsional wave dispersion curves in single walled carbon nanotubes (SWCNT) are used to estimate the non-local kernel for use in continuum elasticity models of nanotubes. The dispersion data for an armchair (10,10) SWCNT was obtained using lattice dynamics of SWNTs while accounting for the helical symmetry of the tubes. In our approach, the Fourier transformed kernel of non-local linear elastic theory is directly estimated by matching the atomistic data to the dispersion curves predicted from non local beam theory and axisymmetric shell theory. The distribution of kernel weights in the Fourier space indicate that a Gaussian kernel can o er a better prediction for wave dispersion in CNTs than the non-local kernel from gradient theory. Reconstructions of these kernels are provided in this paper and the dispersion data obtained from such kernels are directly compared with gradient theories. The numerically computed kernels obtained from this study will help in development of improved and e cient continuum models for predicting the mechanical response of CNTs.