JOURNAL OF APPLIED PHYSICS 101, 064316 共2007兲
Hyperelastic behavior of single wall carbon nanotubes
Xianwu Linga兲 and S. N. Atluri
Center for Aerospace Research and Education, University of California at Irvine, 5251 California Ave.,
Suite 140, Irvine, California 92612
共Received 28 August 2006; accepted 13 November 2006; published online 28 March 2007兲
Single wall carbon nanotubes 共SWNTs兲 are shown to obey a hyperelastic constitutive model at
moderate strains and temperatures. The finite temperature is considered via the local harmonic
approach. The equilibrium configurations were obtained by minimizing the Helmholtz free energy
of a representative atom in an atom-based cell model. While the concept of strain-dependent tangent
modulus using linear elasticity was considered in prior literature, a constant ␮ for Ogden’s
hyperelastic model 关R. W. Ogden, Nonlinear Elastic Deformation 共Horwood, England, 1984兲兴 is
found in the current work for large tubes subjected to moderately large strains up to 900 K. © 2007
American Institute of Physics. 关DOI: 10.1063/1.2409646兴
I. INTRODUCTION
The carbon nanotubes 共CNTs兲 have aroused intense research interest due to their exceptional mechanical, thermal
and electrical properties. Mechanically, single wall carbon
nanotubes 共SWNTs兲 possess extremely high stiffness,
strength and resilience, and thus hold a great promise for use
in CNT-reinforced nanocomposites.
Knowledge of their mechanical properties is the first step
toward the rational application of CNTs as structural elements. Linear elasticity has been the tool used to derive the
Young’s modulus for the CNTs since their discovery. However, a very large scatter exists for the reported elastic
Young’s modulus 共Y兲, with a wide spread of data over one
order in magnitude. In Table I, we summarize a collection of
experimental and theoretical Young’s moduli, among many
others that cannot be listed here. Experimental measurements
are difficult and involve many uncertainties, such as the
structural uncertainty 共e.g., as caused by defect兲, the measurement uncertainty 共e.g., as caused by thermal vibration兲,
and the model uncertainty 共e.g., as caused by the elastic
beam or shell assumption兲. Hence, experimental errors are
generally very high. Computational modeling provides a
powerful tool to confirm, supplement and guide experimental
researches. Theoretical investigations have involved calculations from the first principle Hartree-Fock method through
the empirical potential molecular mechanics 共MM兲 and molecular dynamics 共MD兲 to the elastic continuum-based models. Nevertheless, no agreements in the predicted Young’s
moduli can be made.
Although the linear elasticity modeling of CNTs is ubiquitous in the literature, several authors also noticed the nonlinearity of the CNTs. Yao et al.10 indicated the strain effects
on the Young’s modulus in their MD simulations. Xiao
et al.15 fit the strain energy change of CNTs with a cubic
function and deduced a linearly strain-dependent in-plane
stiffness. Arroyo and Belytschko16 proposed a finite deformation nonlinear elastic theory based on the interatomic po-
tentials. Natsuki and Endo17 presented a MM-based mechanics approach and made similar observations as Xiao and
Liao.15
This paper aims to provide a hyperelastic constitutive
model to describe the mechanical behaviors of SWNTs. Although Xiao et al.15 and Natsuki et al.17 described the CNTs
stiffness with a linearly strain-dependent function, instead of
a singly strain-independent value prevalent in the literature,
they still implicitly assumed the linear elasticity model. In
this work, instead of a strain-dependent Young’s modulus
using linear elasticity, we show that in CNTs, the strain energy cannot be represented by a simple quadratic function of
the uniaxial strain. We then use Ogden’s hyperelasticity
model18 to obtain a constant material parameter ␮ for temperatures up to 900 K.
II. ATOM-BASED MODEL
The atom-based cell model is presented in Ling and
Atluri19 and illustrated in Fig. 1, where the representative
atom A is surrounded by three nearest neighbor atoms B , C,
and D, forming a lattice cell. The atoms interact through the
TABLE I. Reported Young’s modulus 共TPa兲.
Ref.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Method
Experimental 共TEM兲
Experimental 共AFM兲
M-R spectroscopy
Experimental 共TEM兲
Experimental 共AFM兲
MD
Quenched MD
Forced constant
Tight binding
Lattice dynamics
Hartree-Fock
LDA
Modified Morse
Molecular mechanics
Value
+1.65
1.85−1.45
0.69–1.87; 0.27–0.95
1.7–3.6
+.45
; 0.9
1.25−.35
0.06–1.3; 1.2
5.5; 0.64
0.2–2.0
0.97–1.11
1.24; 0.98
⬃1
0.72–1.12
0.764
0.94
0.99
Note
MWNTsa
MWNTs
CNTs
SWNTs
SWNTs
SWNTsb
共n , n兲
CNTs
CNTs
SWNTs
SWNTs
SWNTs
CNTs
CNTs
a
a兲
Electronic mail: xling@uci.edu
0021-8979/2007/101共6兲/064316/4/$23.00
First measurement.
Yakobson 共1996兲 used h = 0.066.
b
101, 064316-1
© 2007 American Institute of Physics
J. Appl. Phys. 101, 064316 共2007兲
064316-2
FIG. 1. Cell model of SWNT in the tubular structures.
Tersoff-Brenner potential.20 The polar coordinate system is
used to describe the position of the atoms, in which A is
positioned at 共r , 0 , 0兲 and ␸i = cos−1 xi / r for i = B , C , D.
The second nearest-neighbor atoms are determined using the
nearest-neighbor atom coordinates.
The finite temperature effect is accounted for by the
Helmholtz free energy H, which, via the local harmonic
approach,21 is expressed as
N
H = Utot + kBT 兺
3
兺
i=1 ␬=1
冋 冉 冊册
ln 2 sinh
h ␻ i␬
4kBT
,
共1兲
where Utot is the total potential energy and ␻i␬ are the atom
vibrating frequencies given by
冏
␻i2␬I 3⫻3 −
冏
1 ⳵2Utot
= 0, i = 1,2, . . . N.
mC ⳵ x i ⳵ x i
共2兲
For a SWNT, the frequencies ␻i␬ are independent of the
atom i and thus taken as ␻A␬.
For the Tersoff-Brenner formalism, the total potential energy Utot can be expressed as
Utot =
1
兺 兺 Vij = NUa ,
2 i j⫽i
共3兲
where Ua = 21 共VAB + VAC + VAD兲 is the potential energy per
atom. Therefore, the Helmholtz free energy per atom Ha is
obtained as
3
冋 冉 冊册
Ha = Ua + kBT 兺 ln 2 sinh
␬=1
h ␻ A␬
4kbT
共4兲
.
The equilibrium atom positions are obtained by minimizing
Ha, i.e.,
⳵ Ha
⳵ Ha
⳵ Ha
= 0,
= 0,
= 0, for i = B,C,D,
⳵r
⳵␸i
⳵ zi
共5兲
subjected to the nonlinear chirality constraint19
2
2
+ 关n共zD − zB兲 + m共zD − zC兲兴 ⬅ 0.
2
The uniaxial tension is imposed by enforcing that
ZB − ZC = ␭zl0, provided ZB ⱖ ZD ,
共6兲
共7兲
where ␧z and ␭z = exp共␧z兲 are, respectively, the axial strain
and stretch, and l0 = ZB0 − ZC0 is the initial z-axis length.
Equation 共7兲 provides a straightforward means to applying
the Cauchy-Born rule22 to the non-centrosymmetric atomic
structures.
III. RESULTS AND DISCUSSION
The strain energy E is stored in the form of the carbon
bond energy and atom vibration. Hence, we set E = ⌬Ha / V0,
where the change of the free energy ⌬Ha = Ha − Ha0 with Ha0
being the reference free energy at ␧z = 0, and V0 is the volume
per atom in the undeformed configuration. Figure 2 compares the strain energy versus the applied strains from several sources. Impressively, our computed ⌬Ha’s perfectly
match the MD results by Cornwell et al.7 and the structural
mechanics results by Natsuki et al.17 For small strains, the
present results also match very well the MD results by Robertson et al.23 and the MM results by Sears et al.14 The overall deviations are also small comparing to the MD results by
Xiao et al.15 Similar to Cornwell et al.,7 the tube radius is
seen to have a negligible effect on the strain energy. In our
calculations, the temperature effect on the strain energy is
slight, e.g., the strain energy at ␧ = 0.15 decreases by less
than 4% from T = 150 to T = 900 K, and for smaller strains,
even smaller changes incur. We mention that the results by
other researchers were reported only for absolute zero temperature.
As noticed by Xiao et al.,15 the strain energy can be
fitted by a cubic polynomial as
E = a2␧z2 + a3␧z3 ,
g = r 共关n共␸D − ␸B兲 + m共␸D − ␸C兲兴 − 4␲ 兲
2
FIG. 2. Energy change per atom under axial tension for 共10, 10兲 and 共5, 5兲
tubes at T = 300 K.
共8兲
where the coefficients of a2 , a3 are determined by a regression analysis. For the SWNT 共10, 10兲 at 300 K, we obtain
a2 = 20.10 and a3 = −3.38 eV/ atom, with a standard deviation
共␧兲 less than 3.28⫻ 10−4 for strain up to 0.15. Xiao et al.15
J. Appl. Phys. 101, 064316 共2007兲
064316-3
FIG. 3. Predicted tangent modulus for SWNTs at 300 K.
FIG. 4. Predicted ␮ for Ogden’s hyperelasticity model.
estimated a2 = 25.6, a3 = −48.2 in their simulations with ␧
= 0.005. Although the overall standard deviation is very
small, we note that the deviation is nearly totally contributed
from small strain data up to 0.04. Therefore, we divide the
fitting into two regions and obtain a2 = 19.45, a3 = 6.15 with
␧ = 2 ⫻ 10−6 for strains up to 0.04, and a2 = 20.12, a3 = −3.78
with ␧ = 3.53⫻ 10−4 for strains in between 0.04 and 0.15. The
initial stiffness estimations of a2 match quantitatively in all
the cases. However, the whole region regression in Xiao et
al.15 cannot capture the initial increase and the subsequent
decrease of the stiffness. The fact is that E is not a perfect
quadratic function in ␧z, which clearly indicates that the
SWNT is hyperelastic and cannot be simply described by
linear elasticity.
Under uniaxial tension, we define the tangent modulus
for uniaxial stiffness as
ure 3 shows that Y can be well characterized by a linear
function of ␧z before a critical strain. The strain dependence
of Y puts ambiguity into its conventional understanding as
the Young’s modulus. Nevertheless, we note that 共1兲 Y is of
the order of 1 TPA for the armchair CNTs, but nearly reduced
by half for the zigzag tubes; 共2兲 large tubes have greater
stiffness than the small tubes. No agreement on the size effect on the Young’s modulus has been observed 共cf. Lu,8
Hernández,9 Yao and Lordi10兲; and 共3兲 the critical strain at
which Y drops is nearly 4% − 6% strain for the armchair
tubes, but 10% − 20% strain for the zigzag tubes 共with the
larger tubes failing earlier兲.
While Eq. 共8兲 indicates the hyperelasticity behavior of
SWNTs, it does not provide a strain energy density function
under general deformation. Ogden18 derived the strain energy density function for a hyperelastic material as
Y=
⳵ 2E 1 ⳵ 2H a
=
.
⳵ ␧z2 V0 ⳵ ␧z2
共9兲
Y was generally regarded as the Young’s modulus in prior
literature, often with its single value evaluated at ␧z = 0. Fig-
E=
␮ 2
␮⬘
共␭z + ␭␪2 + ␭r2 − 3 − 2 ln J兲 + 共J − 1兲2 ,
2
2
共10兲
where J = ␭z␭␪␭r is the Jacobian of deformation 共with ␭␪ , ␭r
being, respectively, the stretches in the hoop and thickness
J. Appl. Phys. 101, 064316 共2007兲
064316-4
creasing temperatures, and the higher the temperature, the
greater the decrease. At high temperatures, no apparent elastic stage can be recognized, as even ␮ becomes strain dependent.
In sum, the hyperelasticity of SWNTs is explored using
Ogden’s model. A constant ␮ is obtained for large tubes up
to the critical strain. Small tubes reveal more nonlinearity.
Increasing temperature diminishes the elastic deformation.
ACKNOWLEDGMENT
This work is supported by the Army Research Laboratory and the Army Research Office.
1
FIG. 5. Temperature effect on ␮.
directions兲. Since the SWNT consists only a single layer of
atoms, the radial Kirchhoff stress ␴ˆ r = ␭r ⳵ E / ⳵␭r ⬅ 0, yielding ␮⬘ = 1 − ␭r2 / J共J − 1兲␮. Thus,
␮=
2E
␭z2
+
␭␪2
− 2 − 2 ln J
.
共11兲
We adopt a constant wall thickness h = 0.34 nm in our calculations.
Figure 4 shows the calculated ␮ for Ogden’s hyperelasticity model at 300 K. A nearly constant ␮ is observed for all
the tubes below the critical strains of 4% − 6% for armchair
tubes and of 10% − 20% for zigzag tubes, though each tube
has its own ␮ value. The nonlinearity becomes more obvious
for small tubes, as ␮ becomes more dependent on the applied
strains. This is understandable since in small tubes, the outof-plane ␲-bonds become more severely distorted. The critical strain can be clearly read from the curves for two large
armchair tubes 共20, 20兲 and 共10, 10兲, which take a sharp turn
at 0.065 and 0.07 strain, respectively. The deformation of an
SWNT was observed to be completely reversible 共i.e., elastic兲 subjected strains of more than 4%.2,3,6 Also in accordance with Nardelli et al. and Zhang et al.,24 the zigzag tubes
display a higher strain resilience. Hence, the critical strain
from our simulations can possibly indicate the elastic limit.
The present cell model cannot be used beyond the elastic
limit as it does not consider the Stone-Wales transformation.
The temperature effect on the predicted ␮ is plotted in
Fig. 5 for the armchair tube 共10, 10兲. ␮ decreases with in-
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