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.

Figure 1

Four types of loading conditions: (a) axial tension, (b) torsional moment, (c) in-plane biaxial tension, and (d) tension-compression stress.

Figure 2

Schematic diagrams for the stresses and deformations of a small segment of two tubes with MWNT’s under tension loading condition, (a) for the axial analysis and (b) for the radial analysis.

Figure 3

Variations of the outermost interwall shear stress for various wall numbers.

Figure 4

(a) The dependence of the effective longitudinal tensile Young’s modulus on the MWNT length for various wall numbers, which is normalized by that of the outermost tube; (b) the dependence of the stable value of the tensile Young’s modulus on the wall number and diameter of MWNT’s.

Figure 5

(a) The dependence of the major Poisson’s ratio on the MWNT length for various wall numbers; (b) the dependence of the major Poisson’s ratio on the wall number and diameter of MWNT’s.

Figure 6

A schematic diagram to show the stresses and deformations of two adjacent tubes under torsional moment loading.

Figure 7

(a) The dependence of the longitudinal shear modulus on the MWNT length for various wall numbers, (b) the dependence of the longitudinal shear modulus on the wall number and diameter of MWNT’s.

Figure 8

The dependence of the plane-strain bulk modulus on the wall number and diameter of MWNT’s.

Figure 9

A schematic diagram to show the shape changes from circles to ellipse of the outer and inner tubes.

Figure 10

The dependence of the in-plane shear modulus on the wall number and diameter of MWNT’s.