Buckling of single-walled carbon nanotubes upon bending: Molecular dynamics simulations and finite element method

The bending buckling behaviors of single-walled carbon nanotubes (SWCNTs) are systematically investigated by using both molecular dynamics (MD) simulation and finite element method (FEM), to analyze the relationships between critical bending buckling curvature, critical buckling strain and nanotube geometry parameters (e.g., tube diameter, length and chirality). The postbuckling shape of SWCNT and the effect of loading boundary conditions are also discussed. The comparison between MD and FEM simulations shows that the continuum shell model provides some useful insights into the bending buckling mechanisms, yet it cannot quantitatively reproduce the bending buckling behavior of SWCNTs, since the continuum model does not account for the geometrical imperfections in the atomic system that are critical to the onset of buckling. Improvements of continuum models are suggested based on the findings.

Figure 1

(Color online) The relationships between critical bending buckling curvature and nanotube diameter. Results computed from MD and FEM analyses are compared with the model by Yakobson and co-workers.[15] The tube length is fixed at $24\mathrm{nm}$.

Figure 2

(Color online) The comparison between the critical buckling strains of bending and axial compression of SWCNTs with same length $\left(24\mathrm{nm}\right)$. Both MD and FEM results are presented and compared with the axishell theory for compression.

Figure 3

The length dependence of critical bending buckling curvature of (5,5), (9,0) and (10,10) tubes.

Figure 4

(Color online) The relationship between critical bending buckling curvature and nanotube geometrical parameters established from FEM analysis.

Figure 5

The geometry of (9,0) tubes at the onset of bending buckling with different lengths: (a) $L=6.11\mathrm{nm}$, (b) $L=7.36\mathrm{nm}$, (c) $L=17.78\mathrm{nm}$, and (d) $L=34.86\mathrm{nm}$.

Figure 6

The geometry of zigzag tubes at the onset of bending buckling with fixed length $(L=24\mathrm{nm})$ and different radii: (a) (35,0), (b) (26,0), (c) (17,0), and (d) (9,0).

Figure 7

(Color online) The effect of boundary condition simulated by FEM. (a) The ideal distribution of axial strain of pure bending. For (10,10) tube with $L=6.2\mathrm{nm}$ and $t=0.066\mathrm{nm}$: (b) The contour plot of axial strain ${\epsilon }_{22}$ just before buckling, and (c) the subsequent buckled geometry. For (10,10) tube with $L=12.5\mathrm{nm}$ and $t=0.066\mathrm{nm}$: (d) The contour plot of axial strain ${\epsilon }_{22}$ just before buckling, and (e) the subsequent buckled geometry.

Figure 8

(Color online) The effect of boundary condition simulated by FEM. For (20,20) tube with $L=25\mathrm{nm}$ and $t=0.066\mathrm{nm}$: (a) The contour plot of axial strain ${\epsilon }_{22}$ just before buckling, and (b) the subsequent buckled geometry. For (20,20) tube with $L=50\mathrm{nm}$ and $t=0.066\mathrm{nm}$: (c) The contour plot of axial strain ${\epsilon }_{22}$ just before buckling, and (d) the subsequent buckled geometry.