A Finite Element Approach for Estimation of Young's Modulus by sjb12334


									The Third Taiwan-Japan Workshop on Mechanical and Aerospace Engineering
Hualian, TAIWAN, R.O.C.
Nov. 28-29, 2005

           A Finite Element Approach for Estimation of Young’
               Modulus of Single-walled Carbon Nanotubes
                   Cheng-Wen Fan 1, Jhih-Hua Huang 2, Chyanbin Hwu 2 and Yu-Yang Liu2
           Department of Mechanical Engineering, Yung-Ta Institute of Technology & Commerce,
                                     Ping-Tung, Taiwan, R.O.C.
                   Institute of Aeronautics and Astronautics, National Cheng Kung University,
                                             Tainan, Taiwan, R.O.C.

    ABSTRACT: In this paper, the Young’ modulus of single-walled carbon nanotubes is
    estimated by a finite element approach. Individual carbon nanotube is simulated as a
    frame-like structure and the primary bonds between two nearest-neighboring atoms are treated
    as beam members. The beam properties for input into a finite element model are calculated
    via the concept of energy equivalence between molecular mechanics and structural mechanics.
    To verify this approach, the computed result of Young’ modulus of graphite sheets is
    examined, which shows good agreement with available literatures. Moreover, finite element
    models of both armchair and zigzag carbon nanotubes are established and the Young’moduli
    of these tubes are then effectively predicted. The relations of the obtained Young’moduli to
    the diameters of the nanotubes are also discussed. All the modeling and computing work in
    this paper are performed by the finite element commercial code ANSYS.

    KEYWORDS: carbon nanotubes, Young’modulus, finite element model


          Since Iijima’ discovery in 1991 (Iijima, 1991), carbon nanotubes (CNTs) have
    stimulated extensive research activities devoted to nanomechanics and their applications in
    nanoengineering. Due to their exceptional mechanical and electrical properties: small size,
    low density, high stiffness, high strength etc., CNTs represent a very promising material in
    many areas of science and industry.
          The elastic properties of multi- and singled-walled nanotubes (MWNTs and SWNTs)
    have been the subject of numerous research works in experimental, molecular dynamics, and
    elastic continuum modeling approaches. For examples, experimental investigations conducted
    by Treacy et al. (1996), who used TEM to measure the Young’modulus of MWNTs, reported
    a mean value of 1.8 Tpa with a variation from 0.40 to 4.15 Tpa; Krishnam et al. (1998) also
    used TEM to observe the vibration of a SWNT and obtained the Young’ modulus ranged
from 0.90 to 1.70 Tpa; Wang et al. (1997) conducted bending tests on cantilevered tubes using
atomic force microscopy and estimated a Young’modulus of 1.28 Tpa; Poncharal et al. (1999)
observed the static and dynamic mechanical deflections of cantilevered MWNTs, which is
induced by an electric field, and reported a modulus about 1.0 Tpa for small-diameter tubes.
      Besides the experimental works on CNTs, many researchers try to analyze the elastic
properties of CNTs by theoretical modeling techniques. These modeling approaches can be
generally classified into two categories. One is the molecular dynamics (MD) method (Iijima
et al., 1996; Gau et al., 1998; Zhang et al., 1998; Zhou et al., 2000; Belytschko et al., 2002),
which is based on the force field and total potential energy related to the interatomic
potentials for CNTs in a macroscopic sense. In this method, the bonding and nonbonding
potentials are represented in terms of the force constants and the deformation among the
atomic bonds, and then elastic moduli are determined by applying different small-strain
deformation modes. However, the computational expense of MD simulations limits the size of
CNTs that can be studied by this technique. The other approach is the continuum/finite
element method (Zhang et al., 2002a, 2002b; Jin and Yuan, 2003; Li and Chou, 2003). Since a
nanotube can be well described as a continuum solid beam or shell subject to tension, bending,
or torsional forces, it is reasonable to model the nanotube as a frame- or shell-like structure,
then the elastic properties of such a structure can be obtained by classical continuum
mechanics or finite element method. However, due to the uncertainty of the CNT wall
thickness and modulus for both of the above modeling techniques, the obtained elastic
properties of SWNTs or MWNTs have scattered values, for example, the results of axial
Young’ modulus ranged from about 1.0 Tpa to 5.5 Tpa can be found in the existing
      The purpose of this paper is to estimate the Young’ moduli of SWNTs by a finite
element (FE) approach. Since the carbon nanotube can be treated as a frame-like structure, the
primary bonds between two nearest-neighboring atoms can be modeled as beam elements in
view of the concept of finite element method. The beam properties for input into a finite
element model are calculated via the concept of energy equivalence between molecular
mechanics and structural mechanics. The Young’ moduli of both armchair and zigzag
nanotubes with different diameters will be calculated by this approach, and the relations of the
obtained moduli to the diameters of the nanotubes will be also discussed. For saving the effort
of program coding and making this approach more popular, all the modeling and computing
work are performed by the finite element commercial code ANSYS.

                             FINITE ELEMENT MODELING

     As mentioned above, CNTs can be treated as a frame-like structure with their bonds as
beam members and carbon atoms as joints. In (Li and Chou, 2003), the stiffness method was
used to simulate the mechanical behavior of nanotubes, they established the linkage between
the force constants in molecular mechanics and the element stiffness in structural mechanics
through the energy equivalence concept. The key point of this concept is that the simplest
harmonic forms of the various steric potential energy of nanotube bonds are adopted under the
assumption of small deformation, i.e.
                                     1                 1
                                U r  k r ( r  0 ) 2  kr (  ) 2 ,
                                               r              r                            (1)
                                     2                 2
                                    1             1
                                U  k( 0 ) 2  k(  )2 ,
                                                                                         (2)
                                    2             2
                               U      k( ) 2 ,
                                     U U                                             (3)
where U r   stands for bonded stretching energy, U θ for bonded angle bending energy, U 

is the combination of the dihedral angle torsion energy U  and the improper (out-of-plane)

torsion energy U ; kr , k and k are the bond stretching force constant, bond angle
bending force constant and torsional resistance respectively, and the symbols  ,  and
 represent the bond stretching increment, the bond angle change and the angle of bond
twisting, respectively. By comparing these energy forms to their counterparts in structural
mechanics, which are
                                 1 L N2 1 N 2 L 1 EA
                            U A   dL             ( ) 2 ,
                                                       L                                   (4)
                                 2 0 EA 2 EA    2 L
                                 1 LM2  2 EI 2 1 EI
                            U M   dL           (2 2 ,
                                                       )                                   (5)
                                 2 0 EI   L    2 L
                                  1 L T2        1 T 2 L 1 GJ
                            UT   dL                          
                                                               ( )2 ,                     (6)
                                  2 0 GJ        2 GJ      2 L
where U A , U M and U T     are the strain energy of a uniform beam of length L, cross-section
A and moment of inertia I under axial force, pure bending and pure torsion respectively; E and
G are Young’modulus and shear modulus;  is the total axial stretching deformation, 2α
             s                                 L
denotes the total relative rotation angle,  is the relative torsion angle and J the polar
moment of inertia.
    The concept of energy equivalence between the two systems implies that both U r and
U A represent the stretching energy, both U and U M the bending energy, both U  and
UT the torsional energy. Then it is reasonable to assume that  is equivalent to  , and
                                                               L                  r
                                        
2α equivalent to  , and  equivalent to  . Therefore, by comparing Eqs.(1)-(3)
                                                                       EA EI           GJ
with Eqs.(4)-(6), a direct relationship between the element stiffness     ,      and      and
                                                                        L    L          L
the force constants in molecular mechanics kr , k      and k can be obtained as follows
                                      EA       EI      GJ
                                         kr ,    k,    k.                             (7)
                                       L       L        L
Eq. (7) constructs the base of the stiffness method employed by (Li and Chou, 2003), and they
selected the values of the force constants kr , k and k according to the experience with
graphite sheets as
                                   mol  =6.53  -7 N/nm ,
                     kr = 938 kcal    1
                                       A        10

                     k= 126 kcal   /rad 2 =8.79  -10 N 
                                  mol 1             10     nm/rad 2 ,

and k is adopted as 40 kcal   /rad 2 =2.79  -10 N 
                             mol 1             10     nm/rad 2 , which is numerically

proven to have little influence on CNTs’            s
                                            Young’modulus in their paper. By self-developed
program, the computed elastic properties of the SWNTs are obtained and their dependence on
the tube diameter is also discussed.
      Note that the Poisson’ratio  which is one of the primary property of materials, does
                             s       ,
not appear explicitly in the formulation of Li and Chou briefly reviewed above. However, by
                                                               h o sn ao
                                                                e s s t
the selected values of the force constants, the restriction of t P i o’r i that should be a
positive number smaller than 0.5 is violated when calculating its value from Eq. (7) and the
well known relationship of E and G, i.e. G E / 2(1  ) , for an isotropic and uniform beam.
In fact, all the other available choices of the values of the force constants found in literatures
can’satisfy this restriction either. This is the main defect necessary to be solved or avoided
for theoretical completeness of this approach.
       In the present study, a finite element method implemented by the commercial code
ANSYS, instead of the stiffness method, is developed under the same theoretical base
described above. The major advantage of the FE approach proposed here is that no extra effort
is needed for program coding and consequently make the present approach more feasible for
mechanical analysis of nanotubes.
       In our finite element modeling work, the BEAM4 element in ANSYS is selected to
simulate the carbon bonds while the atoms are nodes. Fig.1 (modified from Tserpes and
Papanikos, 2005) depicts how the hexagonal lattice of the CNT is simulated as structure
elements of a space frame, where a c  stands for the initial length of C-C bond.
                       Fig. 1. Schematic finite element simulation of a CNT

       To determine the elastic properties of the BEAM4 element for input into the ANSYS
code, we assume that the cross sections of the beam elements are identical and circular, and
note that only the Young’modulus E, the Poisson’ratio  and the diameter of the circular
                           s                         s
cross section d are needed to be prepared for the properties of the BEAM4 element. From the
first two equations of Eq. (7), we have

                                          k          Lk 2
                                   d 4      ,     E r .                                  (8)
                                          kr         4 

By Eq. (8) and the values of kr and k given above, the numerical values of E and d are
calculated as 5.49 Tpa and 0.147 nm respectively. The element length is set to be equal to the
initial C-C bond length ac  of 0.142 nm. As to the Poisson’ratio  several values of 0.05,
                            c                               s       ,
0.1, 0.2 and 0.3 have been examined for the present FE model, and the tests prove that  has
little effect on the final result in our computation. Therefore, we set  = 0.3 as a
representative value in the FE model.
      Before we key in the input data of the BEAM4 element properties, the dimensions of the
parameters stated above should be further adjusted to avoid digits overflow/underflow error
during the computation performed by ANSYS. Thus, we adjust the dimensions as follows

                            Lin  10 L , Fin  20 F , Ein E ,
                                 10           10                                           (9)

where the original dimensions of length L, force F and modulus E are m, N and N/m2
respectively, and the subscript “ denotes the real input values. After such adjustment, the
numerical parts of the input data prepared for the BEAM4 element can be listed as follows
                               E in  .49  12 , Ain  .69 ,  0.3 ,
                                     5     10         1       in

where Ain  in / 4 is the area of the cross section, din = 1.47 is the diameter of the circular


      O N ’   D L S F R P IE H E A

     To verify the feasibility of the model described above, we first calculate the Young’     s
modulus of a graphite sheet, which can be rolled into a carbon nanotube. It is also expected to
provide useful information about the selection of those force constants from the calculation
result. By treating the graphite sheet as an elastic plane structure, a uniform tensile load is
applied at one end of the sheet and the other end is set to be fixed, as depicted in Fig.2. Then
the Young’modulus can be determined from classical elasticity theory, i.e.

                                       F / Ag   FH
                                  Yg             ,                                      (10)
                                        / H  tg
                                         H       HW

where Yg is the Young’modulus of the sheet, F is the total force applied on the nodes at one
end, Ag  t g is the cross-sectional area of the sheet with width W and thickness t g ,
         W                                                                                    H
and  are the initial length and the induced elongation respectively. The thickness t g is
taken as the interlayer spacing of graphite, 0.34 nm.

                               Fig. 2. FE model of the graphite sheet

     An energy form in elasticity theory to calculate the Young’modulus is also used as an
alternative approach, i.e.

                                           1 U N
                                                    1 U N
                                     Yg                    ,                             (11)
                                          Vg  2         
                                                   Wtg H  2

where Vg  g H is the total volume of the graphite sheet, and U N is the total strain
energy. In this approach, prescribed displacements instead of the tensile forces are applied at
the end nodes.
     Table 1 lists the computed Young’moduli of the graphite sheets by the ANSYS code for
different model sizes. From the results shown in Table 1, it can be seen that the obtained
Young’moduli of the graphite sheets are fairly closed to the commonly accepted value 1.025
Tpa (Kelly, 1981) and those presented by Li and Chou (2003). In addition, we can also
observe that the calculating result is weakly affected by model size.

      Table 1. The computed Young’moduli of the graphite sheets for different model sizes

      Width (nm)      Height (nm)                         s
                                                     Young’modulus (TPa)
                                         Eq. (10)         Eq. (11)      Li and Chou (2003)
         0.738           2.842            1.002             1.003               0.995
         0.985           3.126            1.011             1.070               1.002
         1.969           4.831            0.996             1.001               1.021
         1.969           9.847            1.048             1.051               1.024

                             YOUNG’ MODULUS OF A SWNT

     A successful verification of the present FE model is achieved in last section. In the
following, we will apply this method and employ the same parameters of the BEAM4 element
to estimate the Young’moduli of SWNTs. Two main types of SWNTs, zigzag and armchair,
are considered. Fig.3 depicts the typical FE model for calculating the Young’modulus of a
single SWNT. As shown in Fig.3, the prescribed forces (or displacements) are applied at one
end of the tube, while the other end is fixed. Similar to Eqs. (10) and (11), the equations used
                  on’ ou s
                       s       u
to calculate the Y ug m dl Y of a SWNT are slightly modified as follows
                                       F / At    FLt
                                    Y               ,                                   (12)
                                        t / Lt   t
                                         L        dt L
                                       1 U N
                                                1 U N ,
                                    Y                                                    (13)
                                      Vt  2    t 2
                                                dtL 
where At  dt is the cross-sectional area, t and d are respectively the initial wall thickness
and the diameter of the SWNT, Lt and  t are the initial tube length and the elongation in
the axial direction, Vt  dtLt denotes the total volume of the tube, U N is the total strain
energy. Note that different values of the wall thickness of the SWNTs, ranging from 0.064 to
0.69 nm, have been used in many researchersstudies. Moreover, they will result in widely
scattered values of the Young’ moduli of SWNTs. In majority of the studies, it has been
assumed that the wall thickness is equal to the interlayer spacing of graphite, i.e. 0.34 nm.
Thus, in order to make a reasonable comparison with existing literatures in which t = 0.34 nm
is specified, we also select this value to be the wall thickness in our FE model.
          Fig. 3. FE model for calculating the Young’modulus of a single SWNT

      Table 2 lists the computed Young’ moduli of both zigzag and armchair SWNTs with
different tube diameters by Eqs. (12) and (13), and these results are also displayed in Fig.4.
From Fig.4, it can be seen that the trend is similar for both zigzag and armchair SWNT, and
the effect of the tube chirality is not significant. It is also observed that the effect of the
diameter on the Young’moduli of both zigzag and armchair SWNTs is evident, especially for
diameters smaller than 0.8 nm. With increasing tube diameter, the Young’moduli of zigzag
SWNTs increase slightly faster than those of armchair SWNTs, and this increasing tendency
diminished gradually for both types as the diameters larger than 1.1 nm. As pointed out in Li
and Chou (2003), this increase for both zigzag and armchair SWNTs is due to the effect of
nanotube curvature. Higher curvature, which results in more significant distortion of C-C
bonds, leads to a smaller Young’modulus. In addition, the convergent values of the Young’     s
moduli estimated in this FE model are about 1.042 Tpa for zigzag SWNTs and 1.038 Tpa for
armchair SWNTs, which is in reasonable agreement with that obtained by Li and Chou (2003),
i.e. 1.04 Tpa for both types.
          Table 2. The computed Young’ moduli of both zigzag and armchair
                   SWNTs with different tube diameters

                                                                             Young’modulus (TPa)
                                       Tube type       Diameter (nm)
                                                                              Eq. (12)         Eq. (13)
                                         zigzag(5,0)         0.3917            1.0186          0.9903
                                       zigzag (10,0)         0.7830            1.0389          1.0175
                                       zigzag (15,0)         1.1750            1.0418          1.0381
                                       zigzag (20,0)         1.5670            1.0427          1.0405
                                       zigzag (25,0)         1.9590            1.0434          1.0422
                                      armchair (3,3)         0.4070            1.0042          1.0313
                                      armchair (6,6)         0.8140            1.0134          1.0346
                                      armchair (9,9)         1.2210            1.0162          1.0363
                    armchair (12,12)                         1.6280            1.0176          1.0368
                    armchair (15,15)                         2.0350            1.0185          1.0378

            Young's modulus (TPa) )

                                                                                               Zigzag [12]
                                                                                               Zigzag [13]
                                      1.05                                                     Arm-chair [12]
                                                                                               Arm-chair [13]



                                             0.2   0.5 0.8    1.1     1.4   1.7 2.0      2.3
                                                             Diameter (nm)

                                        Fig. 4. Young’moduli of SWNTs versus tube diameters


     A FE approach for estimating the Young’moduli of the zigzag and armchair SWNTs has
been developed and implemented by a commercial tools ANSYS. Based on the fact that the
nanotubes can be treated as a frame-like structure, a simple linkage between the force
constants in molecular mechanics and the elastic properties of the beam member in structural
mechanics is established through the energy equivalence concept. We obtain the Young’         s
moduli of about 1.042 Tpa for zigzag and 1.038 Tpa for armchair SWNTs at large tube
diameters. These results are comparable to those found in the existing literatures. Furthermore,
dependence of the Young’ moduli to tube diameters has been also investigated. The
investigation shows that the Young’ moduli increase with increasing diameter, and then
gradually approaches to a value close to that of graphite sheet when the tube diameter
becomes large. The main advantage of the proposed method is the significant saving of the
program coding time and the fair accuracy in estimating the Young’moduli of the SWNTs
when compared to the other numerical methodologies. It can be concluded that this approach
is a valuable tool for studying the mechanical behavior of carbon nanotubes.


     The support from National Science Council (NSC), R.O.C., through grant
NSC93-2212-E-006-025 is appreciated by the authors.


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