^{1}, Xu Huang

^{1}, Marino Arroyo

^{2}and Sulin Zhang

^{1,a)}

### Abstract

Under torsion and beyond the buckling point, multi-walled carbon nanotubes(MWCNTs) develop a periodic wave-like rippling morphology. Here, we show that torsional rippling deformations can be accurately described by a simple sinusoidal shape function. Combining this observation with the geometry optimization, we develop an effective coarse-grained model that reproduces the complex nonlinear mechanical responses of thick MWCNTs under torsion predicted by large-scale atomistic simulations. Furthermore, the model allows us to simulate super-thick tubes, inaccessible by other coarse-grained methods. With this effective coarse-grained model, we show from an energetic analysis that the rippling deformation is a result of in-plane strain energy relaxation, penalized by the increase in the interlayer van der Waals interaction energy. Our simulations reveal that the torsional response of MWCNTs with up to 100 layers approximately follows a simple bilinear law, and the ratio of the torsional rigidities in the pre- and post-buckling regimes is nearly a constant, independent of the tube radius. In contrast, the bifurcation torsional strain powerly scales with the tube radius. We also find that the wave number in the circumferential direction linearly increases with tube radius, while the wavelength monotonically increases with tube radius, and approaches a constant in the limit of bulk graphite. The bilinear constitutive relation, together with the scaling law of the bifurcation torsional strain, furnishes a simple nonlinear beam theory, which facilitates the analysis of MWCNT bundles and networks.

We gratefully acknowledge the grant support from the National Science Foundation grant under Awards No. 0600661 (0826841) and 0600642 (Clark V. Cooper, program manager). M. A. acknowledges the support of the European Commission (MIRG-CT-2005-029178) and the Ministerio de Ciencia e Innovacion (DPI2007-61054).

I. INTRODUCTION

II. METHODOLOGY

III. SIMULATION RESULTS

A. Model validation: A representative study

B. Size dependence and scaling law

IV. DISCUSSION AND CONCLUSIONS

### Key Topics

- Carbon nanotubes
- 90.0
- Elasticity
- 19.0
- Bifurcations
- 15.0
- Buckling
- 9.0
- Elasticity theory
- 8.0

## Figures

Twisted 37.67 nm long (30,30) nanotube: comparison between the atomistic model and the FCE model for two twisting angles. The atomistic system has 54 000 degrees of freedom while the continuum model has only 5070. The computational time with the continuum approach is seven times smaller than the full atomistic simulations, while the strain energy predicted by FCE calculations is only different from full atomistic simulations at 75° twisting and can be lower with a refined mesh. (a) Superimposed deformation configurations for atomistic (black spheres) and FCE (gray surface) calculations and (b) map of the strain energy density on the finite element computational mesh (red is high, blue is low).

Twisted 37.67 nm long (30,30) nanotube: comparison between the atomistic model and the FCE model for two twisting angles. The atomistic system has 54 000 degrees of freedom while the continuum model has only 5070. The computational time with the continuum approach is seven times smaller than the full atomistic simulations, while the strain energy predicted by FCE calculations is only different from full atomistic simulations at 75° twisting and can be lower with a refined mesh. (a) Superimposed deformation configurations for atomistic (black spheres) and FCE (gray surface) calculations and (b) map of the strain energy density on the finite element computational mesh (red is high, blue is low).

Rippling morphologies of a ten-walled MWCNT under torsion obtained by FCE simulations (Ref. 22) can be approximated by a simple sinusoidal function. (a) Longitudinal-section view of the deformation morphology. (b) Radial coordinates of the sample points (red symbols) in the outermost layer (50,50) as a function of axial coordinates fitted by a sinusoidal function (black curve). (c) Cross-sectional view. (d) Radial coordinates of the sample points (red symbols) as a function of the polar angle fitted by a sinusoidal function (black curve).

Rippling morphologies of a ten-walled MWCNT under torsion obtained by FCE simulations (Ref. 22) can be approximated by a simple sinusoidal function. (a) Longitudinal-section view of the deformation morphology. (b) Radial coordinates of the sample points (red symbols) in the outermost layer (50,50) as a function of axial coordinates fitted by a sinusoidal function (black curve). (c) Cross-sectional view. (d) Radial coordinates of the sample points (red symbols) as a function of the polar angle fitted by a sinusoidal function (black curve).

Evolution of rippling amplitude for the layer ( for the innermost and for the outermost) in a ten-walled MWCNT as a function of torsional deformation. The innermost two layers always have vanishing acting as the hard core in the entire torsional process due to the strong confinement by the outer layers. The insets depict the evolution of the cross sections from the initial circular shape to an oval and eventually to a hexagon with rounded corners.

Evolution of rippling amplitude for the layer ( for the innermost and for the outermost) in a ten-walled MWCNT as a function of torsional deformation. The innermost two layers always have vanishing acting as the hard core in the entire torsional process due to the strong confinement by the outer layers. The insets depict the evolution of the cross sections from the initial circular shape to an oval and eventually to a hexagon with rounded corners.

Rippling deformation of a ten-walled CNT under torsion with 34 nm in length and 3.4 nm in radius. (a) Longitudinal view. (b) Cross-sectional view. (c) Deformation map (green for ridges and blue for furrows). (d) Gaussian curvature map (white for zero, blue for negative, and red for positive Gaussian curvature). (e) Energy density map (red for higher energy state and blue for lower).

Rippling deformation of a ten-walled CNT under torsion with 34 nm in length and 3.4 nm in radius. (a) Longitudinal view. (b) Cross-sectional view. (c) Deformation map (green for ridges and blue for furrows). (d) Gaussian curvature map (white for zero, blue for negative, and red for positive Gaussian curvature). (e) Energy density map (red for higher energy state and blue for lower).

Energetics and mechanical responses of a ten-walled MWCNT under torsion calculated by the present model. For comparisons, an idealized case is also depicted, where the deformation of the MWCNT is constrained to the perfect cylindrical shape without rippling (by fixing for all the layers) throughout the entire loading process. (a) As compared with the idealized deformation mode, the rippling deformation (actual) beyond the bifurcation point releases the in-plane strain energy, penalized by the increase in the interlayer van der Waals energy. The undeformed configuration is taken as the reference energy state. In the idealized case, the interlayer van der Waals energy is nearly constant throughout the entire loading process, and thus the change in nonbonding energy (blue curve) almost coincides with the horizontal axis. (b) Applied torque as a function of the torsional deformation (torsional angle per unit length). The rippling deformation regime corresponds to a lower but nearly constant torsional rigidity than the idealized deformation mode . The torsion of bifurcation predicted by the present model is larger than that of the FCE model . The post-buckling torsional rigidity predicted by the present model is also slightly higher than that of the FCE calculation , owing to the over-constrained sinusoidal shape function.

Energetics and mechanical responses of a ten-walled MWCNT under torsion calculated by the present model. For comparisons, an idealized case is also depicted, where the deformation of the MWCNT is constrained to the perfect cylindrical shape without rippling (by fixing for all the layers) throughout the entire loading process. (a) As compared with the idealized deformation mode, the rippling deformation (actual) beyond the bifurcation point releases the in-plane strain energy, penalized by the increase in the interlayer van der Waals energy. The undeformed configuration is taken as the reference energy state. In the idealized case, the interlayer van der Waals energy is nearly constant throughout the entire loading process, and thus the change in nonbonding energy (blue curve) almost coincides with the horizontal axis. (b) Applied torque as a function of the torsional deformation (torsional angle per unit length). The rippling deformation regime corresponds to a lower but nearly constant torsional rigidity than the idealized deformation mode . The torsion of bifurcation predicted by the present model is larger than that of the FCE model . The post-buckling torsional rigidity predicted by the present model is also slightly higher than that of the FCE calculation , owing to the over-constrained sinusoidal shape function.

The scaling law of the twisted MWCNTs with up to 100 layers. Results from the FCE simulations are also presented for comparisons. (a) The scaled torsional rigidities in the pre- and post-buckling regimes as a function of number of layers in a MWCNT, respectively. Both the scaled torsional rigidities are nearly constants. (b) The ratio between the torsional rigidities in the pre- and post-buckling regimes is nearly a constant of . Results from the FCE model reveal that the ratio is . (c) The torsion of bifurcation scales with .

The scaling law of the twisted MWCNTs with up to 100 layers. Results from the FCE simulations are also presented for comparisons. (a) The scaled torsional rigidities in the pre- and post-buckling regimes as a function of number of layers in a MWCNT, respectively. Both the scaled torsional rigidities are nearly constants. (b) The ratio between the torsional rigidities in the pre- and post-buckling regimes is nearly a constant of . Results from the FCE model reveal that the ratio is . (c) The torsion of bifurcation scales with .

Rippling morphology of -walled CNTs under torsion with , respectively. (a) The cross-sectional deformation morphologies. (b) The circumferential wave number increases nearly linearly with the number of layers and tube radius . Correspondingly, the wavelength in circumferential direction monotonically increases with an upper limit of as the tube radius approaches infinity.

Rippling morphology of -walled CNTs under torsion with , respectively. (a) The cross-sectional deformation morphologies. (b) The circumferential wave number increases nearly linearly with the number of layers and tube radius . Correspondingly, the wavelength in circumferential direction monotonically increases with an upper limit of as the tube radius approaches infinity.

## Tables

Structural and mechanical properties of MWCNTs with different number of layers. is the number of layers in a MWCNT. is tube radius. is the torsion of bifurcation. and are torsional rigidities in the pre- and post-buckling regimes, respectively. Note that is nearly constant.

Structural and mechanical properties of MWCNTs with different number of layers. is the number of layers in a MWCNT. is tube radius. is the torsion of bifurcation. and are torsional rigidities in the pre- and post-buckling regimes, respectively. Note that is nearly constant.

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