The International Conference on Experimental Solid Mechanics and Dynamics
(X-Mech-2012)
March 6-7, 2012, Tehran, Iran
Center of Excellence in Experimental Solid Mechanics and Dynamics,
School of Mechanical Engineering, Iran University of Science and Technology
1
*
2
1 Professor, 2 PhD Student, Composites Research Laboratory, Center of Excellence in Solid Mechanics and
Dynamics, Department of Mechanical Engineering, Iran University of Science & Technology, Narmak,
1684613114, Tehran, Iran,
* (Tel: (98) 21 77208127, e-mail: Shokrieh@iust.ac.ir)
The main goal of this research is to develop a full range multi-scale modeling approach to extract
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 topdown 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 CNTRP.
Carbon Nanotube; Composites; Multi-scale Modeling; Young's modulus
The supreme mechanical properties of carbon nanotubes (CNTs) are rendered them as a potential candidate of reinforcing agents for new generation of polymeric 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 nanotube-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 characterized 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, Meso 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 homogenization 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.
1

In addition to the aforementioned critical issues in modeling 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 material 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 ratios 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 modeling [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 bottom-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.
The modeling procedure constructed 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, Micro, 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. Different 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-Carbon (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 element 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 the 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
2

surface of the surrounding resin is modeled 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 nonlinear 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 vdW 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 completely 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 difficulty 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 Length [nm] E [GPa] CNT Length [nm] E [GPa]
50 58 550 366 1050 516
100
150
129
164
600
650
384
400
1100
1150
528
540
200
250
300
350
400
450
500
203
232
261
287
310
330
348
700
750
800
850
900
950
1000
416
432
448
462
476
490
504
1250
1300
1350
1450
1550
561
570
578
591
600
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 this scale are characterized as orientation and volume fraction of CNTs. Since an equivalent fiber was developed at the microscale, it is permissible to use micromechanics equation at the level of meso. Mori-Tanaka technique is selected at this scale and its 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.
3

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 will 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 materials 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.
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 fiber 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 typical 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.
The results of simulation procedure 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.
4

As it can be seen in Fig. 4, the results of two 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.
Young's Modulus
15.5
15.4
15.3
15.2
15.1
15
Max
Min
Avg
5% 5%±0.5% 5%±1% 5%±1.5%
Volume Fraction
5%±2% 5%±2.5%
Poisson's Ratio
0.288
0.2875
0.287
0.2865
Max
Min
Avg
0.286
0.2855
0.285
5% 5%±0.5% 5%±1% 5%±1.5% 5%±2% 5%±2.5%
Volume Fraction
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 maximum 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 revealed 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 difference between these two cases implying on importance of agglomeration in stiffness reduction of CNTRP.
Mean Values for Young's Modulus
16.1
16.08
16.06
16.04
16.02
16
15.98
15.96
15.94
15.92
Variable Length
Mean Length
5% 5%±0.5% 5%±1% 5%±1.5% 5%±2% 5%±2.5%
Volume Fraction
Figure 4: Comparison between utilized random and mean length approaches
5

Young's Modulus
16.4
16.2
16
15.8
15.6
15.4
15.2
15
Mean-Agglomerated
Mean-Dispersed
5% 5%±0.5% 5%±1% 5%±1.5% 5%±2% 5%±2.5%
Volume Fraction
Figure 5: Comparison between agglomerated and fully dispersed CNTs in matrix
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-scale 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 behavior 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 reinforcement 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 agglomeration of CNTs are considered here except waviness. Taking into account the waviness of CNTs is the subject of another study by the present authors.
[1] Dai, H. (2002), Surface Science 500 , 218–241.
[2] Salvetat-Delmotte and J.P., Rubio, A. (2002), Carbon 40 , 1729–1734.
[3] Lau, K.T. et al. (2006), Composites: Part B 37 , 425–436.
[4] Shokrieh, M. M. and Rafiee, R. (2010), Journal of Mechanics of Composite Materials 37(2) , 235-240.
[5] Qian, D. et al. (2000), Applied Physics Letter 76(20) , 2868-2870.
[6] Schadler, L. (1998), Applied Physics Letter 73(26) , 3842–3844.
[7] Zhu, J. et al. (2004), Advanced Functional Materials 14(7) , 643–648.
[8] Shaffer, M.S.P. (1999), Advanced Materials 11(11) : 937-941.
[9] Li, C. and Chou, T.W. (2004), International Journal of Solids and Structures 40 , 2487–2499.
[10] Shokrieh, M. M. and Rafiee, R. (2010), Journal of Materials and Design 32 , 790-795.
[11] Kalamkarov, A.L. (2006), International Journal of Solids and Structures 43 , 6832-6854.
[12] Shokrieh, M. M. and Rafiee, R. (2010), Journal of Composite Structures 92 , 647-652.
[13] Tsai, S.W. and Hoa, C.V. and Gay, D. (2003), Composite Materials, Design and Applications , CRC Press
[14] Odegard, G.M. et al. (2003), Composites Science and Technology 63 , 1671–1687.
[15] Frankland, S.J.V. (2003), Composites Science and Technology 63 , 1655–1661.
[16] Villoria, R.G. and Miravete, A. (2007), Acta Materialia 55 : 3025–3031.
[17] Wang, S. et al. (2006), Nanotechnology 17 , 634-639.
[18] Shi, D.L., et al. (2004), Engineering Materials and Technology 126 , 250-257.
6