Elastic properties and pressure-induced structural transitions of single-walled carbon nanotubes

The mechanical properties and pressure-induced structural transitions of single-walled carbon nanotubes (SWCNTs) are studied in this paper in the theoretical scheme of the higher-order gradient continuum. Both the first- and second-order deformation gradients are involved in the approximation of the deformations of microscale lattice vectors, and the continuum elastic potential is obtained by equating the deformation energy of a representative cell with that of an equivalent volume of the continuum, and thus, the established continuum constitutive model is more reasonable. By setting an appropriate equation as the deformation map from a graphite sheet to an initial undeformed SWCNT, the elastic constants of SWCNTs are obtained by minimizing the energy of a representative cell. The response of SWCNTs under axis-symmetrical loadings can also be obtained by changing the values of the set geometrical parameters. Based on the derived constitutive model, a computational method is developed to study the response of SWCNTs under hydrostatic pressure.

Figure 1

(Color online) SWCNT formed by rolling an equilibrium graphite sheet into a cylindrical shape through three steps; the change of the representative cell is also depicted.

Figure 2

(Color online) $\mathbf{F}\left({\mathbf{X}}^{\prime }\right)\cdot \mathbf{R}$ maps the vector $\mathbf{R}$ onto the tangent plane as ${\mathbf{r}}^{″}$; the enhanced term $\mathbf{G}\left({\mathbf{X}}^{\prime }\right):\left(\mathbf{R}\otimes \mathbf{R}\right)/2$ pulls ${\mathbf{r}}^{″}$ to $\mathbf{r}$ that is closer to the real case.

Figure 3

(Color online) Representative cell composed of bonds $I-J\left(J=1,2,3\right)$.

Figure 4

Axial and circumferential Young’s moduli versus tube radius for $\left(n,0\right)$, $\left(n,n\right)$, and $\left(3n,2n\right)$ CNTs: the solid line for axial Young’s modulus and the dashed line for circumferential Young’s modulus.

Figure 5

Radial Young’s modulus versus tube radius for $\left(n,0\right)$, $\left(n,n\right)$, and $\left(3n,2n\right)$ CNTs.

Figure 6

Poisson’s ratio versus tube radius for $\left(n,0\right)$, $\left(n,n\right)$, and $\left(3n,2n\right)$ CNTs.

Figure 7

Shear Young’s modulus versus tube radius for $\left(n,0\right)$, $\left(n,n\right)$, and $\left(3n,2n\right)$ CNTs.

Figure 8

Hydrostatic pressure-radial strain relationship curve that is obtained with an analytical-numerical method.

Figure 9

(Color online) (a) SWCNT simplified as an elastic ring (b) that can be mapped from a line segment with length ${\lambda }_2\Gamma$. With increasing hydrostatic pressure, the shape of the cross section transforms from a circle to an ellipse, then to a peanut shape (c).

Figure 10

Variation of lengths of the long and short axes with pressure for (10,10) CNT.

Figure 11

Critical pressure of the first structural transition versus tube radius for $\left(n,0\right)$, $\left(n,n\right)$, and $\left(3n,2n\right)$ CNTs.