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The International Conference on Experiment
al Solid Mechanics and Dynamics

(
X
-
Mech
-
2012
)

March 6
-
7, 2012, Tehran, Iran

Center of Excellence in Experimental Solid Mechanics and Dynam
ics,

School of Mechanical Engineering, Iran University of Science and Technology



1

Development of a Full Range Multi
-
scale Modeling to Obtain Elastic
Properties of CNT/Polymer


1
M. M. Shokrieh
*

and
2
R. Rafiee


1
Professor,
2
PhD Student, Composites Research Laboratory, Center of

Excellence in Solid Mechanics
and
Dynamics, Department of Mec
hanical Engineering, Iran University of Science & Technology, Narmak,
1684613114, Tehran, Iran,


*
(Tel: (98) 21 77208127, e
-
mail: Shokrieh@iust.ac.ir)



Abstract


The main goal of this research is to develop a full range multi
-
scale modeling approach to e
xtract
Young’s modulus and Poisson’s ratio of carbon nanotube reinforced polymer (CNTRP) covering all
nano, micro, meso and macro scales. The developed model consists of two different phases as top
-
down scanning and bottom
-
up modeling. At the first stage,
the material region will scanned from the
macro level downward to the nano scale. Effective parameters associated with each and every scale
will be identified through this scanning procedure. Accounting for identified effective parameters of
each specific
scale, suitable representative volume elements (RVE) will be defined, separately for all
nano, micro, meso and macro scales. In the second stage of modeling a hierarchical multi
-
scale
modeling approach is developed. This modeling strategy will process the
material at each scale and
feed the obtained information to proceeding scale as input information. It has been shown that the
developed modeling procedure provides a clear insight to the properties of CNTRP and it is very
efficient tool in simulation of CN
TRP.


Keywords:

Carbon Nanotube; Composites; Multi
-
scale Modeling; Young's modulus



1. Introduction


The supreme mechanical properties of carbon nanotubes (CNTs) are rendered them as a potential candidate of
reinforcing agents for new generation of poly
meric composites [1
-
4]. There are some evidences in literature [5
-
7] mentioning the significant enhancement in mechanical properties of polymeric resins by utilizing small
portion of CNTs. Consequently, prediction of mechanical properties of carbon nanotub
e
-
based composites plays
an important role in understanding their behavior and can pave the road toward their industrial application
.

CNT is a reinforcing agent at nano
-
scale, while mechanical properties of carbon nanotube reinforced polymer
(CNTRP) are c
haracterized at micro
-
scale. Different involved length scales are schematically depicted in Fig. 1
on the basis of top
-
down scanning method. The diversity of scales requires a full
-
range multi
-
scale modeling
approach to cover all scales of Nano, Micro, Mes
o and Macro.

Micromechanics rules can not be directly used to obtain properties of CNTRP, since they treat a reinforcing
agent as a continuum medium which is not pertinent to the lattice structure of CNT. On the other side,
micromechanics equations can not

take into account the scale of nano, while the first scale of modeling process
in these equations is micro.

Due to formation of local aggregates in the form of bundled CNTs diminishing efficiency of CNTs in
reinforcement, it is not recommended to use hom
ogenization methods at macro scale. Localized homogenous
regions should be distinguished at lower scales prior to application of these techniques and they should be
employed at those specific regions.

2

In addition to the aforementioned critical issues in m
odeling of CNTRP, there are some uncertainties in CNTRP
attributed to the processing methods which should be taken into account in modeling process. CNTs are
intending to be oriented in different directions. Dispersion of CNTs in different points of the ma
terial region
does not follow a uniform pattern as a result of agglomeration of CNTs. TEM pictures of embedded CNTs in
polymeric resin have revealed that CNTs rarely remains straight in matrix [5, 8]. This phenomenon is arisen
from the very large aspect ra
tios and low bending stiffness of CNTRP.





Figure 1: TEM pictures implying on non
-
straight CNTs [5, 8]


These process
-
induced uncertainties will affect the performance of CNT in reinforcement of CNTRP and should
be considered in the modeling
procedure. The existence of abovementioned uncertainties necessitates
implementation of stochastic modeling, while deterministic approaches simply overlook them. The main reason
of reported discrepancies between experimental observations and theoretical m
odeling [4] can be routed to the
ignorance of uncertainties using deterministic approaches.

The main objective of this research is to develop an appropriate multi
-
scale modeling to predict Young’s
modulus and Poisson’s ratio of CNTRP on the basis of botto
m
-
up modeling approach. Prior the development of
the modeling procedure, a top
-
down scanning will be carried out to identify involved parameters and their
effective scales, accordingly.



2. Development of Multi
-
Scale Modeling



The modeling procedure cons
tructed on the basis of bottom
-
up modeling starts from the nano
-
scale and lasting
in macro
-
scale, passing the in
-
between scales as micro and meso. Since the developed multi
-
scale modeling is a
full
-
range multi
-
scale modeling covering all scales of Nano, Mi
cro, Meso and Macro, it will be called as N3M
multi
-
scale modeling. The modeling technique required an accurate and careful definition of representative
volume elements (RVE) at each level which are identified on the basis of top
-
down scanning method. Diff
erent
stages of developed multi
-
scale modeling on the basis of hierarchical approach are presented as it follows.


2.1 Nano
-
Scale Modeling


Proper RVE at the nano scale should simulate the lattice structure of CNT, diameter and its chirality. In this
study

an armchair CNT in the form of (10, 10) is selected. The finite element model of the lattice structure of
the CNT is constructed on the basis of developed nano
-
scale continuum mechanics approach developed by Li
and Chou [9]. In their method each Carbon
-
Ca
rbon (C
-
C) bond is replaced with an equivalent structural beam
element using a linkage between inter
-
atomic potential energies of the molecular mechanics and strain energies
of the structural mechanics. The method is also verified by a closed
-
from solution

developed by authors in
another research [10].


2.2 Micro
-
Scale Modeling


At this level, interaction between CNT and the surrounding polymer and load transfer from the matrix to CNT
have to be studied. For this purpose, a concurrent multi
-
scale finite el
ement modeling approach is employed
wherein CNT is treated at its nano
-
scale while the resin is modeled at the micro
-
scale. Solid element is used to
simulate the surrounding polymer as a continuum medium. The size of each element is selected as small as th
e
length of hexagon rings on the lattice structure of the CNT. The inter
-
phase region is treated as non
-
bonded van
der Waals (vdW) interactions. The vdW interactions between carbon atoms of CNT and the nodes of inner
3

surface of the surrounding resin is mod
eled using three dimensional non
-
linear spring elements and
corresponding data of the non
-
linear curve of vdW force [11] which was obtained from Lennard
-
Jones “6
-
12”
potential is fed into the software.

The model is subjected to three different loadings to

obtain longitudinal, transverse and shear moduli. A non
-
linear analysis is performed due to nature of vdW interactions. The non
-
linear analysis is performed using the
full Newton
-
Raphson iterative method due to the inherently non
-
linear behavior of the vd
W interaction. In each
sub
-
step, the vdW interactions are rearranged and updated based on their new corresponding arrangements. In
other word, in each sub
-
step of the non
-
linear analysis, some previously active vdW interactions will be
deactivated as their

distances exceed the cut
-
off length and can not transfer any load. In the meantime, some
new vdW interactions will be formed and activated in accordance with updated situation of the model. This
procedure called adaptive vdW interaction (AVI) is completel
y explained in the previous research published by
present authors [12].

Comparing the results of developed AVI method with the rule of mixture [13], it was observed that direct
application of micromechanics rules will lead to overestimated results. This di
fficulty was also reported by other
investigators [14
-
16]. The main reason of this draw back stems from the fact that micromechanics rule have
been developed for a continuum fiber which is perfectly bonded to the surrounding resin; while CNT consists of
a
lattice structure interacting with surrounding polymer through vdW Interactions. So, micromechanics
equations are not able to simulate the scale difference between nano and micro. In order to overcome this
difficulty, CNT and its inter
-
phase are converted
to a continuum equivalent fiber. The mechanical properties of
the developed equivalent fiber are obtained using developed AVI method. Developed equivalent fiber shows a
transversely
-
isotropic behavior [12].

Wang, et al. [17] reported that the length of CNT

is varying from 50 nm to 1700 nm when single walled carbon
nano
-
tubes are dispersed in the resin. Since the reinforcement efficiency of CNT highly depends on the length of
CNT, a full range of analyses is performed on different length of CNT on mentioned
range and corresponding
properties of the developed equivalent fibers with different length are obtained and reported in table 1.


Table 1: Longitudinal effective modulus of developed equivalent fiber for different lengths


CNT Length [nm]

E [GPa]

CNT Leng
th [nm]

E [GPa]

CNT Length [nm]

E [GPa]

50



㔵5

㌶3

㄰㔰

㔱5

㄰1

ㄲ1

㘰6

㌸3

ㄱ〰

㔲5

ㄵ1

ㄶ1

㘵6

㐰4

ㄱ㔰

㔴5

㈰2

㈰2

㜰7

㐱4

ㄲ㔰

㔶5

㈵2

㈳2

㜵7

㐳4

ㄳ〰

㔷5

㌰3

㈶2

㠰8

㐴4

ㄳ㔰

㔷5

㌵3

㈸2

㠵8

㐶4

ㄴ㔰

㔹5

㐰4

㌱3

㤰9

㐷4

ㄵ㔰

㘰6

㐵4

㌳3

9


㐹4



㔰5

㌴3

㄰〰

㔰5




2.3 Meso
-
Scale Modeling


At the meso
-
scale, a RVE consists of fully dispersed CNTs with different length which was converted to
equivalent fibers with different properties in the preceding scale. The effective parameters of t
his scale are
characterized as orientation and volume fraction of CNTs. Since an equivalent fiber was developed at the micro
-
scale, it is permissible to use micromechanics equation at the level of meso. Mori
-
Tanaka technique is selected
at this scale and i
ts modified form which accounts for random orientation of the cylindrical inclusion is
employed [16]. The main advantages of Mori
-
Tanaka technique can be divided into two major issues. Firstly, it
takes into account the interaction between inclusions; and
secondly it can be expanded to consider multiple
inclusions. So, it is appropriate for the case of the developed equivalent fiber with different properties.

The RVE of meso scale is presented in Fig. 2. Embedded equivalent fibers in presented block of Fig
. 2 are
oriented in random directions and they can exist in both straight and curved forms. For sake of simplicity, it is
assumed here that all CNTs will remain straight. They can be either concentrated in local aggregates or
dispersed in some other areas.

It is assumed that aggregates will be appeared in the form of spherical regions.
All other equivalent fibers located out of the spherical regions are considered to be fully dispersed.

4

Exhibiting isotropic behavior due to random orientations of CNT, Young’
s modulus and Poisson’s ratio of the
constitutive block of Fig. 2 can be calculated using improved Mori
-
Tanaka model presented by Shi, et al. [18].
Each and every length of CNT which leads to a different effective stiffness of developed equivalent fiber wi
ll be
considered as a new phase in multiple
-
phase Mori
-
Tanaka equations. Using the volume fraction of each block,
the mechanical properties of each constitutive block will be calculated.


2.4

Macro
-
scale Modeling


The material region consists of several blocks

as depicted in Fig. 2. In order to obtain the overall properties of
investigated material region, average of properties associated with each constitutive block is obtained using
Voigt model [13].




Figure 2: Tessellated mate
rials regions at macro scale with constitutive blocks at meso scale


Due to the random orientation of CNT in matrix, the material region can

be assumed as an isotropic material and
two properties out of three properties (E, G and ν) can fully describe the material properties.



3. Stochastic Multi
-
scale Modeling


In this section, the developed N3M multi
-
scale modeling in preceding section will

be employed on the basis of a
stochastic analysis. First of all, the investigated material region is partitioned into smaller square blocks and
random volume fractions are assigned to each block. This represents the random dispersion of CNTs in the
matrix
. A convergence study is performed to obtain appropriate mesh density for tessellated region.

Gaussian distribution is selected for the volume fractions with mean value of 5% and different standard
deviations as 0.5%, 1%, 1.5%, 2% and 2.5%.

Wang, et al.
[17] performed a statistical study on the length of CNTs using experimental observation and
presented the probability density function of the CNT length. Using probability distribution of CNT length by
Wang, et al. [17], effective stiffness of equivalent f
iber with different length accompanied by their probability of
existence is fed into the simulation process of each block. The simulation is performed 500 times in order to
achieve Young’s modulus with 0.01% coefficient of variations. As a case study, a ty
pical Epoxy resin available
in the market with Young’s modulus of 10 GPa and Poisson’s ratio of 0.3 is selected. As a pertinent assumption,
the selected resin is considered as an isotropic material.



4. Results and Discussion


The results of simulation pr
ocedure explained in previous section are presented in Fig. 5 for the case of Young’s
modulus and Poisson’s ratio versus different pattern of volume fractions. Young’s modulus of simulated
material region is varying from 15.21 GPa to 15.26 GPa representing

0.3% fluctuation in result. On the other
hand, Poisson’ ratio can be considered 0.28 for all cases when the accuracy is considered as two order of
magnitudes. So, it is inferred that one can simply replace the random volume fraction with the mean value of

the
volume fraction regardless of its deviations.

The possibility of replacing random lengths of CNT by corresponding mean value is also examined and the
result is presented in Fig. 3. It is worth mentioning that for this case the mean value of CNT length

is obtained
using probability density function presented in Fig. 3 and then effective stiffness of equivalent fiber with the
mean length is obtained using explained developed technique in the previous section.

5

As it can be seen in Fig. 4, the results of t
wo different approached (random length versus mean length) are very
close. The maximum difference is reported 0.07%, while the runtime of the developed modeling technique
decrease significantly. As a consequent, the length parameter can also be taken into
account as a deterministic
value using its average.




Figure 3: Young’s modulus and Poisson’s ratio versus volume fraction


Comparing the result of the simulation taking into account mean values for volume fraction and length of CNT
with the case of max
imum length of CNT, shows that the former reports Young’s modulus 21% lower than the
later. It implies on the important role of CNT length and its subsequent effect on the load transfer efficiency.

A sensitivity analysis has been performed and it was reve
aled that the results are highly sensitive to
agglomeration comparing with other random parameters. A comparison has been done between agglomerated
CNTs and fully dispersed CNTs in matrix and it was shown in Fig. 5. The results show a considerable
differen
ce between these two cases implying on importance of agglomeration in stiffness reduction of CNTRP.




Figure 4: Comparison between utilized random and mean length approaches


6



Figure 5: Comparison between agglomerated and fully dispersed CNTs in matrix


5. Conclusion


A stochastic multi
-
scale modeling is developed in this study to predict the mechanical properties of carbon
nano
-
tube reinforced polymers. The developed modeling technique scans the whole scales starting from nano
and lasting at macro
-
scal
e on the basis of bottom
-
up modeling. In each scale, effective parameters are identified
and studied and the results are fed into the immediate upper level. Three main parameters as orientations, length
and volume fraction of the CNTs accounts for random b
ehavior of the material region. It was shown that
volume fraction and length of CNTs can be simply replaced by their associated mean values with a very good
approximation. It was also observed that the length of CNT has a great influence on the reinforceme
nt efficiency
due to this fact that the load transfer depends on the effective length of CNT. The results show that Young’s
modulus increased about 52% by adding 5% of the CNT. All random parameters including volume fraction,
length, orientation and agglom
eration of CNTs are considered here except waviness. Taking into account the
waviness of CNTs is the subject of another study by the present authors.


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