Postbuckling prediction of axially loaded double-walled carbon nanotubes with temperature dependent properties and initial defects

Buckling and postbuckling analysis is presented for a double-walled carbon nanotube subjected to axial compression in thermal environments. The analysis is based on a continuum mechanics model in which each tube of a double-walled carbon nanotube is described as an individual orthotropic shell with the presence of van der Waals interaction forces, and the interlayer friction is negligible between the inner and outer tubes. The governing equations are based on higher order shear deformation shell theory with a von Kármán-Donnell type of kinematic nonlinearity, and they include thermal effects. Temperature-dependent material properties, which come from molecular dynamics simulations, and initial point defect, which is simulated as a dimple on the tube wall, are both taken into account. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling response of perfect and imperfect, axially loaded armchair, and zigzag carbon nanotubes under different sets of thermal environments. The results reveal that temperature change has a significant effect on the postbuckling behavior of the single-walled carbon nanotube, but it has a small effect on the postbuckling behavior of the double-walled carbon nanotube. The single-walled carbon nanotube has an unstable postbuckling path, and the structure is imperfection sensitive. In contrast, the double-walled carbon nanotube has a very weak “snap-through” postbuckling path, and the structure is virtually imperfection insensitive.

Figure 1

Molecular structure of a double-walled carbon nanotube with an initial point defect.

Figure 2

An elastic shell model for a double-walled carbon nanotube under axial compression: (a) geometry and loading case; (b) coordinate system and point defect.

Figure 3

Comparisons of postbuckling behavior for a (10,10) single-walled carbon nanotube under axial compression ($L=9.262\mathrm{nm}$, $R=0.682\mathrm{nm}$; 1: $E=5.5\mathrm{TPa}$, $\nu =0.19$, $h=0.066\mathrm{nm}$; 2: $E=4.84\mathrm{TPa}$, $\nu =0.19$, $h=0.075\mathrm{nm}$; 3: $E=5.1\mathrm{TPa}$, $\nu =0.24$, $h=0.074\mathrm{nm}$; 4: $E=1.28\mathrm{TPa}$, $\nu =0.25$, $h=0.154\mathrm{nm}$; 5: MD simulation results).

Figure 4

Comparisons of postbuckling behavior for single-walled carbon nanotubes (SWCTN) under axial compression in thermal environments: (a) armchair (10,10) SWCTN; (b) zigzag (17,0) SWCTN.

Figure 5

The effect of initial point defects on the postbuckling behavior of a (10,10) single-walled carbon nanotube under axial compression in thermal environmental condition $T=700\mathrm{K}$: (a) load shortening; (b) load deflection.

Figure 6

The effect of temperature rise on the postbuckling behavior for an armchair double-walled carbon nanotube under axial compression: (a) load shortening; (b) load deflection.

Figure 7

The effect of initial point defects on the postbuckling behavior of an armchair double-walled carbon nanotube under axial compression in environmental condition $T=700\mathrm{K}$: (a) load shortening; (b) load deflection.

Figure 8

Comparisons of postbuckling behavior for armchair and zigzag double-walled carbon nanotubes under axial compression in thermal environments: (a) load shortening; (b) load deflection (1: $L=9.262\mathrm{nm}$, $R_{\mathit{I}}=1.021\mathrm{nm}$, $R_{\mathit{II}}=0.681\mathrm{nm}$, $h=0.065\mathrm{nm}$; 2: $L=9.116\mathrm{nm}$, $R_{\mathit{I}}=1.009\mathrm{nm}$, $R_{\mathit{II}}=0.669\mathrm{nm}$, $h=0.076\mathrm{nm}$).