FACTA UNIVERSITATIS
Series: Mechanical Engineering Vol. 18, No 4, 2020, pp. 525 - 536

https://doi.org/10.22190/FUME201005043R

Original scientific paper

EVOLUTION OF THE CARBON NANOTUBE BUNDLE

STRUCTURE UNDER BIAXIAL AND SHEAR STRAINS

Leysan Kh. Rysaeva1, Dmitry V. Bachurin2, Ramil T. Murzaev1,

Dina U. Abdullina1, Elena A. Korznikova1,3, Radik R. Mulyukov1,3,

Sergey V. Dmitriev3,4

1Institute for Metals Superplasticity Problems of the Russian Academy of Sciences, Russia
2Institute for Applied Materials, Karlsruhe Institute of Technology, Germany

3Ufa State Petroleum Technological University, Russia
4National Research Tomsk State University, Russia

Abstract. Close packed carbon nanotube bundles are materials with highly deformable

elements, for which unusual deformation mechanisms are expected. Structural evolution

of the zigzag carbon nanotube bundle subjected to biaxial lateral compression with the

subsequent shear straining is studied under plane strain conditions using the chain model

with a reduced number of degrees of freedom. Biaxial compression results in bending of

carbon nanotubes walls and formation of the characteristic pattern, when nanotube cross-

sections are inclined in the opposite directions alternatively in the parallel close-packed

rows. Subsequent shearing up to a certain shear strain leads to an appearance of shear

bands and vortex-like displacements. Stress components and potential energy as the

functions of shear strain for different values of the biaxial volumetric strain are analyzed

in detail. A new mechanism of carbon nanotube bundle shear deformation through

cooperative, vortex-like displacements of nanotube cross sections is reported.

Key Words: Carbon Nanotube Bundle, Plane Strain Conditions, Lateral Compression,

Shear Deformation, Deformation Mechanisms

Received October 05, 2020 / Accepted November 11, 2020

Corresponding author: Sergey V. Dmitriev

Ufa State Petroleum Technological University, Kosmonavtov St. 1, Ufa 450062, Russia. National Research
Tomsk State University, Lenin Ave. 36, Tomsk 634050, Russia

E-mail: dmitriev.sergey.v@gmail.com

526 L.KH. RYSAEVA, D.V. BACHURIN, R.T. MURZAEV, D.U. ABDULLINA,...

1. INTRODUCTION

In nature, there are many allotropic modifications of carbon with different physical

properties. One of such modifications is carbon nanotube (CNT), which is but a rolled sheet

of graphene. The peculiarity of CNTs is that they interact rather weakly with each other and

can form CNT bundles [1-4]. The interest in CNTs is primarily due to their unique

mechanical properties such as very high tensile strength, high Young's modulus, and

ultimate fracture strain [5-8]. CNTs are flexible and lightweight and have good thermal and

electric conductivity. Mechanical applications of CNTs include high-strength ropes [2,9],

fibers [10-14], polymer- and metal-matrix composites [15-17], solid lubricants [17-20], and

many others.

Tensile [2,9-14] and compressive [21-28] behavior of vertically aligned CNT brushes

and forests have been most extensively studied experimentally. Straining of vertically

aligned CNTs implies deformation along their axis, while straining of horizontally

aligned CNTs means deformation normal to their axis. Lateral compression of isolated

CNTs or CNT bundles has not been widely studied yet [29-33]. It should be noted that

the methods of winding, drawing, micromechanical rolling, and shear pressing [34-37]

can be applied to obtaining horizontally aligned CNT bundles from the vertically aligned

ones. It has been revealed that CNT bundles can deform elastically up to hydrostatic

pressure of 1.5 GPa, and the hydrostatic deformation of CNT lattice is reversible up to

4 GPa [38]. Karmakar and co-authors have demonstrated that deformation of CNT bundles

under non-hydrostatic pressure is reversible (elastic) at stresses below 5 GPa [39].

In the last two decades, considerable attention has been paid to computer modeling of

the mechanical properties of CNT bundles for a better understanding of their properties

and deformation mechanisms. Since a nanotube bundle can be represented as a material

with highly deformable elements, new deformation mechanisms different from those

typical for conventional materials can be expected. In particular, transformation of vertically

aligned CNT forest into a horizontally aligned one under pressure has been studied via

mesoscopic modeling [40,41]. Different morphological patterns of CNTs subjected to large

deformations have been revealed by means of continuum shell model [42]. The rigidity of

CNT crystal does not decrease with increasing CNT diameter [1]. Authors [43-45] have found

that CNTs can exist in two stable configurations (circular and collapsed ones) depending on

their diameter. Nonlinear coarse-grained potentials specially developed for CNTs have

allowed studying the mechanical response and failure of CNT bundles [46]. The chain

model developed in Ref. [47] has been applied successfully to the investigation of structure

and properties of carbon nanoribbons [47-51] and dynamics of surface ripplocations [52].

Recently, the chain model has been used for simulation of evolution of CNT bundle

structure under both lateral uniaxial and biaxial compression [53-55].

The present work is devoted to a study of structural evolution of zigzag CNT bundle

under biaxial compression with the subsequent shear straining using the chain model

[47,53]. The latter uses a reduced number of degrees of freedom but at the same time

gives at least one order of magnitude acceleration of computations without loss of

accuracy, in comparison with full atomistic modeling under plane strain conditions.

Despite the fact that the chain model [47,53] can be applied for various CNTs, here we

restrict ourselves to the consideration of only single-walled CNTs of equal diameter.

Evolution of Carbon Nanotube Bundle Structure under Biaxial and Shear Strains 527

2. MODEL AND COMPUTATIONAL DETAILS

The computational cell of the modeled bundle has a parallelogram shape and contains

1080 zigzag CNTs (30 horizontal layers and 36 vertical layers) of equal diameter aligned

along the z-axis. For clarity, Fig. 1 represents only a part of the computational cell (two

horizontal layers and two vertical layers). Each CNT consists of 60 carbon atoms. Total

number of atoms in the cell is N=60×30×36=64800. Periodic boundary conditions are

applied along the two (x,y) Cartesian directions, i.e., infinitely elongated along the z-axis

CNTs are considered.

Lateral compression of the CNT bundle is applied using plane strain conditions,

namely when each carbon atom stays rigidly in “its own” atomic row along the z-axis, but

can move freely along the (x,y) plane. Thus, each atom has only two degrees of freedom,

which allows us to reduce the dimensionality of the problem from three-dimensional to

two-dimensional one.

Fig. 1 Geometry of the computational cell. Only two horizontal layers and two vertical

layers are presented for clarity, while in simulation 30 horizontal layers and 36

vertical layers are used. Carbon atoms in the upper row are indicated by large

green circles, and those in the lower row are indicated by small circles. Each CNT

contains 60 carbon atoms. The CNT cross-sections represent a triangular lattice

The following geometrical parameters for the model of CNT bundle are chosen. The

valence bond length in graphene is =1.418 Å. The distance between neighboring atomic
rows in the zigzag CNT is a=1.228 Å. The CNT diameter is D=23.46 Å and the

equilibrium distance between the neighboring CNT walls is d=3.30 Å. Thus, the centers

of neighboring CNTs are at the distance of A=D+d=26.76 Å.

The Hamiltonian describing the interaction between carbon atoms includes the four

terms, namely

H=K+UB+UA+UVdW. (1)

Here the first term, K, gives the kinetic energy of the carbon atoms, UB stands for the

energy of valence bonds, UA is the energy of valence angles, and UVdW is the energy of

van der Waals interactions between CNTs. The model parameters were calculated based

528 L.KH. RYSAEVA, D.V. BACHURIN, R.T. MURZAEV, D.U. ABDULLINA,...

on the interatomic potential developed for sp2-carbon by Savin et al. [56] and further

applied for investigation of various phenomena [56-61].

The initially constructed CNT bundle is subjected to relaxation in order to obtain

minimum energy configuration. A biaxial compression (with xx<0 and yy<0) is applied

by a strain increment of xx=yy= −0.0025, which is followed by giving to carbon
atoms small random displacements in the range from −10-6 to 10-6 Å along the x- and y-

axes and further minimization of potential energy U=UB+UA+UVdW. The relaxation of the

system is stopped, when the absolute value of the maximal force acting on atoms

becomes less than 10-10 eV/Å. This biaxially strained structure is deformed again by

applying subsequently a shear strain up to =0.30 with the step of =0.01. The strain

state of deformed system is thus characterized by two parameters: the absolute value of

the volumetric strain,

| |=|xx+yy|, (2)

and shear strain, . In the present study, no thermal effects are taken into consideration.

For further details related to the simulation setup and construction of the Hamiltonian of

the chain model, we refer the reader to our previous publications [53-55].

3. SIMULATION RESULTS AND DISCUSSION

Firstly, structural evolution of the CNT bundle under simple biaxial compression is

considered, and, secondly, biaxial compression followed by a shear strain is analyzed.

Fig. 2 Structural evolution of CNT bundle at different values of biaxial volumetric strain,

 (presented under each structure). Translation cells of the structures are shown
by red lines and includes four CNTs marked for clarity with red, blue, black and

light green colors. For a more detailed visualization, only a part (lower left corner)

of the computational cell is shown

Evolution of Carbon Nanotube Bundle Structure under Biaxial and Shear Strains 529

3.1. Biaxial compression

Fig. 2 demonstrates the evolution of CNT bundle structure subjected to biaxial

volumetric strain, , along the x- and y-axes. At strains of |θ | < 0.02, CNTs preserve
mainly their circular cross-sections and only small distortions of CNTs, barely visible to

the naked eye in the figure, are observed.

An increase of biaxial strain up to |θ | = 0.03 results in an appearance of elliptical

cross-sections. At that, as clearly seen in Fig. 2, in the horizontal close-packed rows, the

elliptic cross-sections are inclined alternatively in the opposite directions. The latter is the

necessary condition to maintain an equilibrium state within the bundle. The straining up

to |θ | = 0.1 leads to a further compression of CNTs within the bundle and no collapsed

cross-sections are observed. Note that the deformation is performed up to the

compressive biaxial volumetric strain |θ | ≤ 0.1 to avoid the first-order phase transition

related to a formation of the collapsed CNTs, as revealed in Ref. [55].

Fig. 3 Structural evolution of CNT bundle at different values of biaxial volumetric strain,

|θ | (horizontally) and subsequent shear strain, γ (vertically). For a more detailed

visualization, only a part (lower left corner) of the computational cell is shown

3.2. Shearing of biaxially pre-strained CNT bundle

The structural evolution of the CNT bundle under biaxial volumetric strain, ,
followed by shear straining is presented in Fig. 3. At low biaxial pre-strains in the range

of |θ | ≤ 0.02, the deformation is almost uniform, and only small distortions (will be

discussed later in more detail) in the form of shear bands in a triangular lattice of CNT

bundle are observed at  = 0.3. It should be noted that all nanotubes in the computational
cell are deformed approximately in the same way, and their cross-sections have the

elliptical non-collapsed shape inclined towards the direction of the applied shear strain.

An increase of the biaxial pre-strain up to |θ |=0.03 results in an appearance of

longitudinal-transverse shear bands and the nuclei of vortex-like displacements.

530 L.KH. RYSAEVA, D.V. BACHURIN, R.T. MURZAEV, D.U. ABDULLINA,...

=0.03 =0.15 =0.30

| |=0.02

| |=0.03

| |=0.05

| |=0.07

| |=0.10

Fig. 4 Displacement field of the centers of mass for each CNT in the computational cell

at different values of biaxial volumetric strain, |θ | (vertically), and shear strain, γ

(horizontally)

Evolution of Carbon Nanotube Bundle Structure under Biaxial and Shear Strains 531

Moreover, the compression of the horizontal CNT layers becomes inhomogeneous:

some layers turn out to be less deformed in contrast to the elements of the neighboring

layers, where the deformation is much more pronounced (best seen for |θ |=0.03 and

 = 0.3). The alternation of such layers with different compression values is to maintain
an equilibrium state, which is seen in Fig. 3 for |θ |=0.03. Further increase in biaxial

deformation from |θ |=0.05 to 0.1 and subsequent shear result in disordering of the CNT

bundle structure. The fraction of collapsed CNTs gradually increases with an increase in

biaxial deformation and subsequent shear strain and is maximal at |θ | = 0.1 and  = 0.3.
Fig. 4 illustrates the true displacement of the centers of mass of CNTs with respect to

the uniformly deformed state. At that, the uniform biaxial compression and shear

deformations were subtracted for clarity. At pure shear without preliminary biaxial

deformation (| | = 0.0), no formation of shear bands or vortices in the structure occurs

(not shown in Fig. 4). The same can be said about the case of | | = 0.02, as seen, at low
biaxial strains, no significant distortions in the structure of CNT bundle are observed.

With an increase in the preliminary biaxial deformation, shear bands and vortices begin

to appear, which become wider with an increase in the shear strain. The vortices are best

seen in Fig. 4 in the structure at | | = 0.1 and  = 0.30.
Since the average size of vortices is significantly smaller than the size of the

computational cell, it can be argued that in this work we consider a representative volume

of a CNT bundle.

Fig. 5 Dependence of stress components σxx (a) and σxy (b) on shear strain, γ at different

values of biaxial volumetric strain, | |, as indicated in the legends

The structural evolution can be also well seen on the stress-strain curves shown in

Fig. 5. The behavior of stress components xx and yy is very similar, which is expected due

to the hexagonal symmetry of the structure, that is why only xx is shown in (a). An increase

in biaxial compression (| | > 0.03) leads to the first increase of both xx and yy components

and saturation at  = 0.03, so that xx and yy practically do not change with increasing shear

strain. For all values of | |, pressure p=-(xx + yy)/2 decreases with increasing . Note that

for | | =0 for  > 0 pressure becomes negative. This means that the CNT bungle with no

pre-stress shrinks volumetrically under shear deformation. The behavior of xy component

is shown in Fig. 5(b). For the structures with | | > 0.03, the stresses linearly decrease with

532 L.KH. RYSAEVA, D.V. BACHURIN, R.T. MURZAEV, D.U. ABDULLINA,...

increasing strain as shown in Fig. 5(c). This occurs because the structure of nanotubes

initially has a rearranged structure that noticeably differs from the initial one, and due to

this, the nucleation of vortex segments occurs already at the compressive state, and the

following shear deformation leads to their growth and increase in number. Shear stress for

| | =0 has the opposite sign as compared to the other values of volumetric strain.
Fig. 6 demonstrates potential energy of the CNT bundle per atom as the function of

shear strain for different values of the compressive biaxial volumetric strain (indicated for

each curve). In Figs. 6(b) to 6(d), the three components of the potential energy

U=UB+UA+UVdW are given separately, i.e., the energy of van-der-Waals interactions (UVdW),

the energy of valence bonds (UB) and the energy of valence angles (UA), respectively.

Fig. 6 Potential energy of the CNT bundle per atom as the function of shear strain, γ,

calculated for different values of the compressive volumetric strain, | | (indicated for
each curve). Total potential energy, U=UB+UA+UVdW, is shown in (a). Its three

components are presented separately: the energy of van-der-Waals interactions, UVdW

(b), the energy of valence bonds, UB (c), and the energy of valence angles, UA (d)

The energy of the van-der-Waals interactions is negative since it is described by the

Lennard-Jones potential with the zero energy level corresponding to well-separated

atoms. The creation of the van-der-Waals bonds leads to the reduction of energy below

zero. Energies UB and UA are positive because for them the zero energy level corresponds

to the unstrained flat graphene so that the bending deformation to create a CNT results in

an increase of potential energy above zero.

Evolution of Carbon Nanotube Bundle Structure under Biaxial and Shear Strains 533

As seen in Fig. 6(b), at | | ≤ 0.03 the energy of van-der-Waals interactions, UVdW, shows
the tendency of a very slow reduction with increasing shear strain. This can be explained by

elliptization of CNTs and, as a consequence, an increase in the contact area between

adjacent CNT walls, which effectively results in the formation of new van-der-Waals bonds

and, therefore, to the reduction of the corresponding energy. At | |=0.02, in the range of

 < 0.1, UVdW slightly increases. This can be a result of the interplay between CNT
elliptization, which leads to a decrease of UVdW and pore opening between CNTs due to a

better packing under not very high compressive strain. At | | ≥ 0.05, UVdW decreases with
increasing shear strain first very rapidly and then more slowly. This is obviously due to the

collapse of CNTs, which leads to the formation of new van-der-Waals bonds between the

inner walls of collapsed CNTs.

Fig. 6(c) reveals that the energy stored by valence bonds, UB, is for two orders of

magnitude smaller than the corresponding values of both UVdW and UA. This is due to very

high rigidity of the valence bonds with respect to their tension/compression. One can

conclude that the deformation of the CNT bundle occurs mainly due to the bending of the

CNT walls and van-der-Waals interactions between CNTs with a negligible contribution

from the CNT walls compression or tension.

Fig. 6(d) clearly shows that the energy stored by valence angles, UA, increases with ,

which is due to elliptization of CNTs growing with shear strain. At | | ≥ 0.05, a reduction

of UA is seen, when  < 0.05. In this initial range of shear strain, collapse of CNTs takes
place which frees up the space for non-collapsed CNTs and their cross-sections become

closer to circles leading to the net reduction of UA.

4. CONCLUSIONS

In summary, for the first time, structural evolution of zigzag CNT bundle with the

nanotubes of the same diameter under biaxial lateral compression with the subsequent

shearing was investigated in the frames of the chain model. Nanotube bundle is a material

with highly deformable elements, and, therefore, the deformation mechanisms in them

can differ significantly from those typical for conventional polycrystalline or amorphous

materials. Simulations revealed that simple biaxial compression results in elliptization of

CNT cross sections and formation of the characteristic pattern where CNTs in the

horizontal close-packed rows are inclined in the opposite directions. Formation of shear

bands and vortex-like displacements in CNT bundle subjected to biaxial pre-straining

with subsequent shearing were found. The analysis of the potential energy during shear

showed that the energy stored by the valence bonds is for two orders of magnitude

smaller than both the energy of van-der-Waals interactions and the energy stored by the

valence angles. Thus, the deformation of CNT bundle occurs mainly due to the bending

of the CNT walls and van-der-Waals interactions between CNTs play a negligible role.

It should be emphasized that the formation of shear bands is very common for

conventional crystalline and amorphous materials, while the formation of vortex-type

displacement patterns observed in the present study can be regarded as a new deformation

mechanism which can be realized in the material with highly deformable structural units.

Investigation of the effect of the CNT diameter and multi-walled CNTs nanotubes on

the deformation behavior and structural evolution may become a natural continuation of

the present work. In addition, investigation of CNT bundle with armchair orientation (in

534 L.KH. RYSAEVA, D.V. BACHURIN, R.T. MURZAEV, D.U. ABDULLINA,...

contrast to zigzag orientation studied here) are also of interest for future research. Test

calculations show that the chain model used in this work reproduces well the stiffness of

the CNT wall in bending and compression, as well as van der Waals interactions;

nevertheless, it is important to compare the results reported here with the results of full-

atomic modeling, which is planned to be done in one of the future works.

Acknowledgements: The work of E.A.K. (design of the research and simulations) was supported

by the Russian Foundation for Basic Research, Grant no. 18-29-19135. This work was partly

supported by the State Assignment of IMSP RAS No. AAAA-A17-117041310220-8.

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