Transversely isotropic elastic properties of multiwalled carbon nanotubes

Five independent effective elastic moduli of a transversely isotropic multiwalled carbon nanotube (MWNT) are studied by analyzing its deformations under four loading conditions, i.e., axial tension, torsional moment, in-plane biaxial tension, and in-plane tension-compression stress. Two distributions of the tension loading on the outermost tube and on all tubes are considered, which correspond to the tensile and compressive Young's moduli. The general relations between the interwall stresses and strains are linearized due to the small strain condition, where the interwall stresses correspond to the variation of the interwall van der Waals forces. Three interwall elastic constants are used to characterize the linear relations associated with three basic interwall deformation modes, i.e., normal deformation in radial direction and two shear deformations in axial and circumferential directions. By taking each tube as a single-walled carbon nanotube, the analytical expressions for the interwall shear stress under the tensile loading on the outermost tube and five elastic moduli of a double-walled carbon nanotube are first obtained. Then, a replacement method is proposed to derive the corresponding expressions for the cases of more walls than two. These analytical expressions are plotted for the case of MWNT's composed of armchair tubes, where the interwall elastic constants are approximated as the corresponding ones of the graphite. The effect of the wall number, diameter, chirality, and length of the MWNT on the shear stress and five elastic moduli are displayed and discussed.