Nonlinear stick-spiral model for predicting mechanical behavior of single-walled carbon nanotubes

Based on a molecular mechanics concept, a nonlinear stick-spiral model is developed to investigate the mechanical behavior of single-walled carbon nanotubes (SWCNTs). The model is capable of predicting not only the initial elastic properties (e.g., Young’s modulus) but also the stress-strain relations of a SWCNT under axial, radial, and torsion conditions. The elastic properties, ultimate stress, and failure strain under various loading conditions are discussed and special attention has been paid to the effects of the tube chirality and tube size. Some unique mechanical behaviors of chiral SWCNTs, such as axial strain-induced torsion, circumferential strain-induced torsion, and shear strain-induced extension are also studied. The predicted results from the present model are in good agreement with existing data, but very little computational cost is needed to yield them.

Figure 1

(Color online) Chirality dependent axial stress-strain relationships for SWCNTs.

Figure 2

Chirality and size dependent initial axial Young’s modulus of SWCNTs.

Figure 3

(Color online) Dependence of the axial secant modulus on the axial strain.

Figure 4

Chirality and size dependent axial Poisson’s ratio of SWCNTs.

Figure 5

(a) Axial tensile strength versus tube diameter. (b) Axial failure strain versus tube diameter.

Figure 6

(Color online) Variation of axial tensile strength and failure strain against tube chiral angle. Note that although the tensile strength is increases with increasing chiral angle, the variation of failure strain is not monotonic.

Figure 7

(Color online) Axial strain-induced torsion angle of chiral SWCNTs with approximately the same diameter. The maximum untwisting angle occurs for SWCNTs with a chiral angle of about $\pi ∕12$. Note that a chiral SWCNT will twist in the same direction under a relatively large compressive strain and under a tensile strain.

Figure 8

(Color online) Chirality dependent circumferetial stress-strain relationships for SWCNTs.

Figure 9

Chirality and size dependent initial circumferential Young’s modulus of SWCNTs.

Figure 10

Chirality and size dependent circumferential Poisson’s ratio of SWCNTs.

Figure 11

(Color online) Dependence of the circumferential secant modulus on the circumferential strain.

Figure 12

(a) Circumferential tensile strength versus tube diameter. (b) Circumferential failure strain versus tube diameter.

Figure 13

(Color online) Circumferential tensile strength and failure strain versus tube chirality.

Figure 14

(Color online) Circumferential strain-induced torsion angle of chiral SWCNTs with approximately the same diameter. The maximum untwisting angle occurs for SWCNTs with a chiral angle of about $\pi ∕12$. Note that a relatively large tensile strain will result in a torsion in the same direction as under a compressive strain.

Figure 15

(Color online) Shear stress-strain curves for SWCNTs with different chiralities. Note that the torsion response of a chiral SWCNT is significantly dependent on the twisting direction because of their geometrical asymmetry.

Figure 16

(Color online) Non-monotonous variations of ideal shear strength and failure strain versus tube chiralty. Note that the shear strength and failure strain of a chiral SWCNT in the twisting and untwisting directions are quite different.

Figure 17

Chirality and size dependent initial shear modulus of SWCNTs.

Figure 18

(Color online) Dependence of the secant shear modulus on the shear strain.

Figure 19

(Color online) Shear strain-induced axial extension of chiral SWCNTs with approximately the same diameter. Note that a chiral SWCNT can even be elongated by a torsion in the twisting direction.

Figure 20

Schematic illustration of a graphene sheet and definitions of geometrical parameters used to describe a nanotube. A (3, 1) tube would be formed by rolling up the graphene sheet bounded by the two dashed lines. The unit cell of the nanotube is shown in gray.

Figure 21

(Color online) (a) Global structure of a chiral nanotube. (b) Front view of local structure and schematic of the stick-spiral model. In the model, a stick with infinite bending stiffness is used to model force-stretch relationship of the carbon-carbon bond, and a spiral spring is used to describe the twisting moment resulting from an angular distortion of the bond angle. (c) Top view of local structure.