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TECHNICAL PAPERS

# An Equivalent Orthotropic Representation of the Nonlinear Elastic Behavior of Multiwalled Carbon Nanotubes

[+] Author and Article Information
M. Garg

Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139-4307

A. Pantano

Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139-4307; Department of Mechanical Engineering, Università degli Studi di Palermo, Viale delle Scienze, 90128 Palermo, Italy

M. C. Boyce1

Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139-4307mcboyce@mit.edu

Since our target is to model MWCNTs of increasingly larger diameter, we neglect the initial internal stress distribution resulting due to the wall curvature (which is found to be influential for small diameter tubes, but negligible for larger diameter tubes).

As shown in Appendix , for the case of MWCNTs with less than $∼10$ walls, the properties for the concentric layers vary with radial location; the differences in mechanical properties become negligible as the number of walls increases in an MWCNT.

1

Corresponding author.

J. Eng. Mater. Technol 129(3), 431-439 (Dec 02, 2006) (9 pages) doi:10.1115/1.2744408 History: Received May 29, 2006; Revised December 02, 2006

## Abstract

An equivalent orthotropic representation (EOR) of the nonlinear elastic behavior of multiwalled carbon nanotubes (MWCNTs) was developed based on a nested shell structural representation of MWCNTs. The EOR model was used together with the finite element method to simulate the large deformation of MWCNTs under bending, axial compression and radial compression. Results were compared with those of the nested shell model for four-, eight-, nine-, 14-, and 19-walled carbon nanotubes. The EOR model provides a dramatic improvement in computational efficiency and successfully quantitatively replicates the overall deformation behavior including the initial linear elastic behavior, the onset of local buckling, and the post-buckling compliance. The proposed EOR model together with the finite element method offers a computationally efficient method for simulating large and complex systems of MWCNTs.

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## Figures

Figure 1

Finite element bending simulation of a 14-walled CNT using the nested structural shell representation. The rippling mode is first: (a) observed to initiate at a wavelength predicted from shell theory based on the outer wall diameter and outer wall thickness; and (b) to then transition to a longer wavelength consistent with experimental observations (TEM image of a buckled MWCNT in CNT-polymer composite (17)) (c).

Figure 3

Radial modulus versus compressive strain for the s layer derived from LJ potential

Figure 4

Comparison of EOR results to NSSR results (13) of bending of a 14-walled CNT of L=15nm and outer radius of Ro=4.76nm

Figure 5

EOR and NSSR 14-walled CNT bending simulation results depicting deformed meshes at different stages of deformation, showing the influence of element size on the EOR model predictions

Figure 6

Comparison of the EOR model results with the NSSR model results for bending of four-, eight-, 14-, and 19-walled CNT of L=15nm(13)

Figure 7

Axial compression of 14-walled CNT of L=15nm as predicted using EOR model (with the different element sizes) and NSSR. EOR is properly predictive of NSSR when the element size is set to that predicted by shell theory (12)

Figure 8

Compression simulations of 14-walled CNT using EOR model with different element size compared to NSSR, showing the initial configuration, buckling initiation, and final configuration of the MWCNT

Figure 9

Comparison of the axial compression results from the EOR model and the NSSR for four-, nine-, and 14-walled CNT (12)

Figure 10

Deformed mesh configurations at different stages of deformation for a ten-walled CNT subjected to lateral compression: EOR-based MWCNT with element size=0.14nm(a) and (b), =0.24nm(c) and (d), and =0.34nm(e) and (f); NSSR based MWCNT with element size =0.24nm(g) and (h)

Figure 11

Lateral compression force-displacement behavior of ten-walled CNT: (a) comparing EOR results (with element sizes 0.14nm, 0.24nm, and 0.34nm) with NSSR results; and (b) comparing EOR results with fixed tangential element length of 0.24nm and varied radial element length as well as fixed radial element length of 0.24nm and varied tangential element length

Figure 12

Variation of the volume fraction of graphene (fg) with number of layers in an (n,n) MWCNT (tg=0.075nm, ts=0.34nm)

Figure 13

Variation of the volume fraction (fg) of graphene with radial position in an (n,n) MWCNT (tg=0.075nm and ts=0.34nm)

Figure 2

N-walled CNT: (a) with top-view (b) and depicted cylindrical coordinate system; the equivalent hollow cylinder model of the MWNT (c). Graphene layers and the interlayer spacing that constitute a material point RVE depicted in cylindrical coordinate system (d); the unwrapped N-layered RVE at a material point used in the analysis (e); and the equivalent orthotropic representation (EOR) RVE (f).

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