This paper presents a mesh-free numerical modeling approach for carbon nanotubes (CNTs) subjected to bending loads. The higher-order Cauchy-Born rule was employed to construct the higher-order continuum constitutive model. An initial equilibrium single-walled CNT (SWCNT) was viewed as been formed by rolling up a graphite sheet into a cylindrical shape. The deformation from an original SWCNT to the current configuration was approximated with the moving least-square (MLS) approximation, and the mesh-free computational framework was established in the theoretical scheme of higher-order gradient continuum. Mesh-free numerical simulations were carried out for SWCNTs, and the accuracy and convergence were discussed in comparison with the results of atomistic simulation. The buckling behavior was studied for various types of SWCNTs upon bending, and the buckling mechanism was investigated in virtue of the continuum variables, which showed that the maximum axial compressive strain played a vital role in the development of kinking.