Atomistic-based continuum representation of the effective proper

Atomistic-based continuum representation of the effective properties of nano

-reinforced epoxies

S.A. Meguid ^,^a^,, J.M. Wernik ^a and Z.Q. Cheng ^a

^a Mechanics and Aerospace Design Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, 5 King’s College Road, Toronto, Ontario, Canada M5S 3G8

Received 2 September 2009;

revised 5 March 2010.

Available online 16 March 2010.

Abstract

In this paper, an atomistic-based representative volume element (RVE) is developed to characterize the behavior of carbon nanotube (CNT) reinforced amorphous epoxies. The RVE consists of the carbon nanotube, the surrounding epoxy matrix, and the CNT/epoxy interface. An atomistic-based continuum representation is adopted throughout all the components of the RVE. By equating the associated strain energies under identical loading conditions, we were able to homogenize the RVE into a representative fiber. The homogenized RVE was then employed in a micromechanical analysis to predict the effective properties of the newly developed CNT-reinforced amorphous epoxy. Numerical examples show that the effect of volume fraction, orientation, and aspect ratio of the continuous fibres on the properties of the CNT-reinforced epoxy adhesives can be significant. These results have a direct bearing on the design and development of nano-tailored adhesives for use in structural adhesive bonds.

Keywords: Atomistic-based continuum; Nano-reinforced epoxies; Representative volume element

Nomenclature

U: potential energy

V_r: energy due to bond stretching

V_θ: energy due to bond bending

V_t: energy due to bond dihedral angle torsion

V_e: energy due to out-of-plane torsion

V_v: energy due to non-bonded van der Waals interaction

V_s: energy due to sum of torsional contributions

k_r: bond stretching force constant

k_θ: bond bending force constant

k_τ: torsional resistance force constant

r: deformed length of C–C bond

r₀: undeformed length of C–C bond

Δr: bond stretching increment

θ: deformed C–C bond angle

θ₀: undeformed C–C bond angle

Δθ: change in bond angle

Δ: change in angle of bond twisting

EA: tensile resistance

EI: flexural rigidity

GJ: torsional stiffness

l: length of beam

Δl: bond stretching increment

2α: bond angle change

Δβ: angle change of bond twisting

α_i: non-dimensional parameters

t: wall thickness of CNT

ψ: hard sphere radius of the atom

γ: potential well depth

U_t: strain energy of a truss rod

A_v: cross-sectional area

E_v: Young’s modulus of truss rod

R_v: undeformed length of truss rod

ε_ij: strain components

σ_ij: stress components

E^f: Young’s modulus of the representative fiber

ν^f: Poisson’s ratio of the representative fiber

G^f: shear modulus of the representative fiber

n_j: components of the outward normal vector

u_i: displacement components

x_j: cartesian coordinates

U_f: strain energy of representative fibre

V: volume of the representative fibre

d: diameter of representative fibre

L: length of representative fibre

e: applied strain

E₁₁: longitudinal Young’s modulus

E₂₂: transverse Young’s modulus

μ₁₂: longitudinal shear modulus

μ₂₃: transverse shear modulus

E: Young’s modulus

G: shear modulus

Article Outline

Nomenclature

Introduction

Representative volume element: 2.1. Carbon nanotube representation; 2.2. Epoxy matrix representation; 2.3. CNT/epoxy interface representation

Representative fiber

Micromechanical analysis

Results and discussion

Conclusion

Acknowledgements

References

1. Introduction

Since their discovery by Iijima (1991), it has been theoretically and experimentally confirmed that carbon nanotubes (CNTs) possess exceptional high stiffness and strength. These properties, amongst others, suggest that CNTs show great promise as reinforcing agents in composite materials (Endo et al., 2004). In recent years,

nano

-reinforced epoxy composites have attracted great interest from the materials and mechanics communities partly because of their exceptional electro-thermo-mechanical properties and partly because of their varied length/time scales. In the past few years researchers have shown that the dispersion of just a few weight percentages of nanofillers into composites could lead to dramatic changes in their mechanical ([Schadler et al., 1998], [Qian et al., 2004], [Kim et al., 2008] and [Zhai et al., 2008]), thermal ([Salehi-Khojin et al., 2007] and [Huang, 2007]), and electrical (Qinghua and Jianhua, 2007) properties with added functionalities. For example, Qian et al. (2000) have demonstrated that with the weight addition of 1% multi-walled carbon nanotubes (MWCNTs) in a polystyrene matrix, the stiffness of the composite film can increase by up to 42% and the tensile strength by up to 25%. Schadler et al. (1998) have also demonstrated the effectiveness of using carbon nanotubes as reinforcing agents in adhesive materials. They found that with 5% weight addition of CNTs in an epoxy resin, the stiffness can be improved by as much as 20% in tension and 25% in compression. However, since experimentation at the nanoscale is still developing, it is important to develop accurate and efficient models to predict the material properties of CNT-reinforced structures. Since the length scales of interest in CNT-reinforced epoxy composites vary from the nanoscale to macroscale, the development of an appropriate model to accurately and efficiently predict the physical and mechanical properties at different length scales is one of the main issues in the simulation of these classes of problems.

For the analysis of nanostructured materials, atomistic simulation methods such as first-principle quantum-mechanical methods (Ding, 2005), molecular dynamics (MD) ([Liew et al., 2004] and [Unnikrishnan et al., 2008]) and Monte Carlo (Zhou et al., 2006) simulations have been routinely adopted. However, these methods are computationally intensive and limited by the realistic system size that they can represent because of the enormous number of degrees of freedom involved. Even the use of state-of-the-art parallel supercomputers can only handle a limited number of atoms ( not, vert, similar 10⁹), corresponding to less than one cubic micron ([Rudd, 2001] and [Abraham et al., 2002]). For a detailed description of the different techniques adopted in atomistic simulations and coupled multiscale methods, refer to the recent reviews by (Wernik and Meguid, 2009), (Ghoniem et al., 2003) and (Vvedensky, 2004). On the other hand, continuum mechanics may not be directly applicable to nanostructures. At the nanoscale, traditional continuum mechanical concepts do not maintain their validity (Chang et al., 2006) and gross oversimplifications can arise from the use of a purely continuum model. For the case of nano-reinforced adhesives, these models cannot accurately describe the influence of the nanofillers upon the mechanical properties, bond formation/breakage and their interactions in the composite systems because they lack the appropriate constitutive relations that govern material behaviour at this scale. Another modeling approach is the atomistic-based continuum technique. It has the unique advantage of describing the atomistic structure–property relations in a continuum framework, thereby reducing the computational demand while employing the appropriate atomistic constitutive relations. In the case of nano-reinforced epoxy systems, it is important to consider the atomic bonding and interaction between two phases: the nanofiller(s) and the epoxy matrix. This is typically carried out at the atomistic scale using appropriate interatomic potentials. For computational simplicity, and to adequately address scale-up issues, it is also desirable to couple atomistic-based continuum models of nanotube-reinforced epoxy composites with established micromechanical models to describe their mechanical behavior on a macroscopic scale.

In this paper, we develop an atomistic-based representative volume element (RVE) for the study of CNT-reinforced epoxy composites. This method allows for the molecular properties obtained through molecular mechanics to be used directly in determining the corresponding bulk properties of the material at the macroscopic scale. The RVE consists of the carbon nanotube, the surrounding epoxy matrix, and the CNT/epoxy interface. The RVE is then reduced into a homogenized continuous representative fiber. Adopting micromechanical analysis techniques, we were able to predict the effective mechanical properties of the macroscopic CNT-reinforced epoxy composite. Fig. 1 provides a schematic illustration of analysis process and clearly identifies all the components used in the development of the RVE. The results from these analyses are compared with published findings to confirm the validity of the model. Numerical examples are given to show the effect of CNT length, volume fractions, orientation, and the aspect ratio of the representative fibre on the properties of the CNT-reinforced epoxy composite.

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Fig. 1.

Schematic illustration of the analysis procedure and the components used in the development of the representative volume element.

2. Representative volume element

Let us consider an epoxy matrix reinforced by single-walled CNTs. The single-walled CNT is a armchair (10, 10) nanotube of radius 6.78 Å and length 4.3 nm and is modeled using a space frame structure. The epoxy matrix immediately adjacent to the CNT is represented as individual epoxy chains aligned in the axial direction thereby maintaining the atomistic representation. As an approximation, it is assumed that the epoxy maintains its structure throughout the entire material and that the CNT is directly incorporated in the epoxy. The CNT/epoxy interface is represented by a truss rod model whereby each van der Waals interaction is simulated using a truss rod. This description implies the assumption of a non-bonded interfacial region. A detailed description of all the components used in the development of the RVE is presented below.

2.1. Carbon nanotube representation

From the viewpoint of molecular mechanics, carbon nanotubes may be treated as a large array of molecules consisting of carbon atoms. The general expression of the molecular mechanics force field or potential energy among atoms, when omitting the electrostatic interaction, can be expressed as follows (Cornell et al., 1995);

(1)U=∑V_r+∑V_θ+∑V_t+∑V_e+∑V_v,where V_r is the energy due to bond stretching, V_θ the energy due to bending (bond angle variation), V_t the energy due to dihedral angle torsion, V_e the energy due to out-of-plane torsion, and V_v is the energy due to non-bonded van der Waals interaction. The atomic interaction mechanisms are depicted in Fig. 2; excluding the electrostatic and hydrogen bonding forces.

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Fig. 2.

Interatomic interactions in molecular mechanics.

In general, for covalent systems, the main contribution to the total energy comes from the first four terms of Eq. (1). Under the assumption of small deformation, harmonic approximations are adequate for describing the different energy contributions. For the sake of simplicity and merging dihedral angle torsion and out-of-plane torsion into a single equivalent term, we can arrive at the following expressions for the bond stretching, bond angle bending, and bond torsion potentials (Li and Chou, 2003)

(2)

(3)

(4)where , and k_τ are the bond stretching, bond bending, and torsional resistance force constants, respectively, while , and Δ represent the respective bond stretching increment, the bond angle change, and the angle change of bond twisting, respectively.

According to classical structural mechanics, the strain energy of a uniform beam of length l can be expressed as

(5)where , and Δβ are the axial stretching deformation, the rotational angle at the end of the beam, and the relative rotation between the ends of the beam, respectively. The three terms in the above expression represent the energy associated with stretching, bending, and torsion, respectively. It is reasonable to assume that the rotation 2α is equivalent to the total change Δθ of the bond angle, Δl is equivalent to Δr, and Δβ is equivalent Δ. Equating (2)–(4)(2), (3) and (4) with the individual terms in Eq. (5), the following direct relationships between the structural mechanics parameters EA,EI, and GJ and the molecular mechanics force constants k_r,k_θ, and k_τ are obtained, viz.

(6)Eq. (6) establishes the foundation of applying the theory of structural mechanics to the modeling of carbon nanotube structures. As long as the force constants k_r,k_θ, and k_τ are known, the sectional stiffness parameters EA,EI, and GJ can be determined and the deformation and elastic behavior of carbon nanotubes at the atomistic scale can be simulated. In the present paper, , and as taken from Li and Chou (2003).

2.2. Epoxy matrix representation

The carbon nanotube used in this study is assumed to be dispersed inside a generalized amorphous epoxy matrix, which is represented by covalently bonded beads of CH₂ united atoms. Each epoxy chain consists of approximately 35 units. The entire surrounding epoxy matrix was modeled using a total of 16 united atom epoxy chains equally spaced at a distance of 0.3816 nm apart from each other and aligned in the axial direction as depicted in Fig. 3. This separation distance corresponds to the equilibrium van der Waals separation distance which will be given more detail in the coming section. The ‘CH₂’ units of the same chain are connected by covalent bonds of length 0.153 nm. As we have done for the CNT, the covalent bonds in the epoxy chains can also be simulated by using a similar space frame model, which is shown in Fig. 4. Consequently, the interaction of the ‘CH₂’ units in the epoxy chains was defined in terms of bond stretching, bond angle variation, and van der Waals contributions. The force constants and geometrical parameters for the epoxy chains are taken from Cornell et al. (1995) and are as follows: bond length l=0.1526 nm, bond angle and .

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Fig. 3.

The epoxy chain configuration surrounding the carbon nanotube.

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Fig. 4.

Epoxy chain representation.

2.3. CNT/epoxy interface representation

Of importance to the development of an accurate and viable RVE is the appropriate representation of the interface between the CNT and the epoxy. Different approaches can be adopted in characterizing the mechanisms and magnitudes of load transfer between a nanotube and the polymer matrix. The interfacial characteristics between the CNTs and polymer matrix remain unclear and researchers have reported a large range of interfacial shear stresses. Four approaches are possible. First, it can be assumed that no chemical bonding exists between the nanotubes and polymer matrix. In this case, van der Waals forces dominate the solution. To avoid weak interfacial strength, some researchers proposed that the chain of the polymer wrap around the nanotube in a helical fashion to enhance the non-bonded nanotube–polymer interaction, which has been observed experimentally and theoretically (Lordi and Yao, 2000). The second approach is to assume that there exist strong chemical bonds. In this case, C–C covalent bonds are included between the nanotube and polymer, which increases the interfacial strength significantly. The third consideration assumes that covalent cross-links form between the nanotube and polymer matrix. In this case, only a small percentage of covalent bonds form from the introduction of multifunctional amines, which act as intermediary bonding sites between the nanotube and polymer chains. However, it is possible that the chemical bonding in the form of functionalization may compromise the properties of the nanotube by introducing structural changes in the graphitic layers of the nanotube (Fiedler et al., 2006). Finally, the load transfer can be attributed to the mechanical interlocking of the polymer and nanotube as a result of geometrical inconsistencies in the structure of the nanotube. However, the carbon atoms on CNT walls are chemically stable because of the aromatic nature of the bonding. As a result, the reinforcing CNTs interact with the surrounding matrix mainly through van der Waals interactions (Hu et al., 2006). Therefore, in this study, we investigate the non-bonded configuration which implies that only van der Waals interactions are considered.

A number of approaches have been considered to account for the interfacial properties. These depend on the type of bonding and load transfer mechanisms, hence, the interfacial thickness has not yet been unambiguously defined. Several different values have been used in both atomistic and continuum simulations. Hu et al. (2005) simulated the helical wrapping of one polystyrene chain around a carbon nanotube considering only van der Waals interactions via molecular dynamics. The equilibrium distance between the hydrogen atoms in the polymer and carbon atoms in the nanotube ranged from 0.2851 to 0.5445 nm. However, only one polymer chain was considered when in practical cases there may be other chains which also wrap around the nanotube. In comparison, Li and Chou (2006) studied the compressive behaviour of carbon nanotube/epoxy composites and assumed that the inside surface of the epoxy matrix was located at the same position as the outside surface of the nanotube giving an interfacial thickness equal to 0.17 nm or half the thickness of the nanotube itself. Given the above variance, it was reasonable to assume an interfacial thickness of 0.3816 nm in our simulation. This value corresponds to the equilibrium separation distance of the Lennard–Jones potential. This same value was used by Montazeri and Naghdabadi (2008) in their molecular structural mechanics model of SWCNT–epoxy composites.

In order to simulate the van der Waals interactions, we used truss elements whereby each interaction was represented by one truss rod. Each rod extends out from a carbon atom in the CNT structure to a united atom in the epoxy matrix. van der Waals interactions have most commonly been described using the Lennard–Jones pair potential because of its simplicity and sole dependence on the atomic separation distance. The Lennard–Jones potential is defined as

(7)where γ is the potential well depth, ψ is the hard sphere radius of the atom or the distance at which U_v(r) is zero, and r is the distance between the two atoms. The proposed truss rod model was also used to simulate the van der Waals interactions between united atoms in the same epoxy chain as well as the interactions between united atoms of different epoxy chains. All the interactions that were considered in this model are depicted in Fig. 5 along with their respective Lennard–Jones parameters as taken from (Binder, 1995) and are summarized in Table 1.

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Fig. 5.

van der Waals truss rod connectivity (a) CNT–epoxy interface, (b) united atoms in same epoxy chain, and (c) united atoms in different epoxy chains.

Table 1. The Lennard–Jones interatomic potential parameters used for simulating van der Waals interactions in the RVE.

Interaction	Potential well depth (γ) (KJ/mol)	Hard sphere radius (ψ) (nm)
CNT–polymer chain	0.4492	0.340
CH₂–CH₂	0.4742	0.352

In this study, the number of truss rods or van der Waals interactions is governed by the separation distance between two interacting carbon atoms or united CH₂ atom units. Only atoms within the usual Lennard–Jones cut-off distance of 2.5ψ were considered. At this distance the forces acting between interacting atoms is negligible and can be neglected. If the distance between the atoms is greater than the equilibrium distance of the potential (0.3816 nm for interacting carbon units) the truss rod was assigned an initial strain corresponding to this separation distance.

From Eq. (7), it is clear that the energy associated with van der Waals forces is highly non-linear. It can be fairly complicated to determine Young’s modulus of the truss rods that represent van der Waals forces through consideration of the non-linear behavior and the large range of values for the separation of the interacting units in an equilibrium configuration. The energy associated with the van der Waals force is given in Eq. (7), and the classical continuum strain energy of a truss rod can be defined as follows

(8)where , and R_v are Young’s modulus, the cross-sectional area, the deformed length, and the undeformed length of truss element, respectively. By equating Eqs. (7) and (8), the Young’s modulus for every van der Waals interaction may be approximated as being

(9)

It should be noted that the truss model used to simulate the interaction between the CNT and the epoxy has previously been used by other researchers. Li and Chou (2006) have also used a truss model to simulate the interaction between the CNT and epoxy matrix. They treated the epoxy matrix as a continuous solid and in doing so neglected the nanoscale effect of the epoxy chains. Thus, the number of van der Waals interactions included in their model was arbitrarily defined. In our study, we consider the nanoscale effect of both the CNT and the epoxy lending for a direct determination of the number of interactions. Thus, the present RVE model is a more realistic approximation.

3. Representative fiber

After developing the atomistic-based continuum structure, we homogenized the RVE into a representative fiber. The resulting fiber was assumed to be linear elastic, homogeneous, and continuous, and has the same cylindrical geometry as the atomistic-based continuum structure. Consequently, the mechanical properties of the cylinder can be determined by equating the total strain energies of the atomistic-based continuum structure and representative fiber under identical loading conditions. In this paper, it is further assumed that the representative fiber is isotropic. Therefore, only two independent elastic constants exist. The elastic constants can be determined by applying a single boundary condition to both the fiber and atomistic-based continuum structure.

The RVE is treated as the representative fiber throughout the remainder of the paper. The governing constitutive law of the representative fiber can be assumed as follows

(10a)

(10b)

(10c)where are the strain components, are stress components, are Young’s modulus, shear modulus, and Poisson’s ratio, respectively. Two sets of boundary conditions were chosen to apply at the boundaries of the RVE to determine the two independent elastic constants, as given below by the respective displacement and traction expressions

(11)where B is the boundary, x_j is the coordinate axes, and n_j are the components of the outward normal vector to B. The total strain energy of the representative fiber is given by

(12)where V, d, L are the respective volume, diameter, and length of the fiber.

To determine Young’s modulus, a strain was prescribed along the x₁ axis, ε₁₁=e, with all of the shear strain components set to zero. From Eq. (11), the boundary conditions are

(13)The strain energy is:

(14)Given that all the parameters in Eq. (14) were known, the Young’s modulus can be determined.

The shear modulus is another elastic parameter to be determined. By prescribing a pure shear strain in the x₁–x₂ plane, with all other strain components equal to zero, the boundary conditions applied at the surface can be described as

(15)

The resulting strain energy of the representative fiber is then reduced to

(16)which will allow for the direct determination of the shear modulus.

The displacements and tractions addressed above were applied to each node at the boundary of the atomistic-based continuum structure, and the total strain energies were obtained by summing the strain energies of each finite element in the corresponding structure. The Young’s modulus and shear modulus of the representative fiber were determined to be 528.4 and 161.7 GPa, respectively, for a representative fiber diameter, d, of 2.2 nm, length, L, of 4.3 nm, and applied strain, e, of 0.1%.

4. Micromechanical analysis

Having determined the effective properties of the representative fiber, the effective material properties of the macroscopic CNT-reinforced epoxy composites can be determined. Again, the representative fiber radius and length were chosen to be 1.1 and 4.3 nm, respectively. The fiber accurately accounts for the structure–property relationship at the nanoscale and provides a bridge to the continuum model. The constitutive relations of the CNT-reinforced epoxy composite are constructed using micromechanics. In this paper, the bulk, amorphous epoxy matrix was assumed to be isotropic, with a representative Young’s modulus of 0.9 GPa and Poisson’s ratio of 0.3 which are typically representative general epoxy properties. We have also assumed the case of perfect bonding between the bulk epoxy and representative fiber.

The Mori–Tanaka method (Mori and Tanaka, 1973) is widely regarded as a powerful micromechanical model for conventional micro-particle reinforced polymers ([Benevensite, 1987] and [Qui and Weng, 1990]) and has effectively been utilized in modeling nanocomposites with transversely isotropic or orthotropic material properties (Liu et al., 2008). In this paper, we used a form of the Mori–Tanaka method presented by Tandon and Weng (1984) in which case the nanofiber and matrix are both assumed to be linearly elastic, homogeneous and isotropic. The details of this method are not presented here but can be found in the publication by Tandon and Weng (1984). The cases of both uniformly aligned and randomly oriented representative fibers were examined in the present analysis. The representative fibers were assumed to be spheroidal in geometry for the Eshelby tensor and both the CNT and the representative fibers have the same length. It was also assumed that the CNT volume fraction was defined as the total space occupied by the CNT, including half of the interfacial region. From this, it was determined that the CNT volume fraction was 63% of the total representative fiber’s volume fraction.

5. Results and discussion

Using the atomistic-based continuum mechanics approach and the micromechanics method, the elastic effective properties of the CNT-reinforced epoxy composites can be determined. In the following, we investigate the sensitivity of the effective properties of the CNT-reinforced composites on the CNT length, volume fraction, orientation and aspect ratio of the representative fiber. A CNT volume fraction of 1% is implied for all cases where the effect of CNT length on the respective modulus has been investigated. Likewise, a CNT length of 100 nm is used when investigating the effect of CNT volume fraction, unless otherwise specified.

The variation of longitudinal Young’s modulus, E₁₁, of the aligned and randomly oriented CNT-reinforced epoxy composites is plotted against the CNT volume fraction and CNT length in [Fig. 6] and [Fig. 7], respectively. It can be seen that E₁₁ of both the aligned and randomly oriented CNT-reinforced epoxy composites are sensitive to both parameters. It can also be observed that E₁₁ increases dramatically with the increase of CNT volume fraction and seems to level of at a constant value with the variation in CNT length. Furthermore, E₁₁ of aligned CNT composites is significantly larger than E₁₁ of randomly oriented CNT composites for both cases. The data in Fig. 7 indicates that further increases in CNT length beyond 400 nm result in relatively small increases in longitudinal Young’s modulus for a given CNT volume fraction. It should also be noted that the CNT volume fraction has been extended up to a maximum of 5% in Fig. 6. CNT concentrations above this magnitude are not normally realized. The attractive van der Waals interactions between carbon nanotubes coupled with their high aspect ratio leads to considerable agglomeration and aggregation at high concentrations. The resulting agglomerates act as defect sites rather than reinforcements which would ultimately lead to a subsequent degradation of the nanocomposite properties (Sun and Meguid, 2004). An efficient utilization of the nanotube properties in polymeric materials is therefore related to their homogenous dispersion in the matrix. The present study uses an idealized model which assumes a perfect dispersion of the nanophase particles even at these high concentrations. Therefore, we can expect that the results would indicate a positive influence of the nanotubes on the elastic properties even at higher CNT volume fractions. Fig. 6 demonstrates that anomaly in the inset. In that inset, we extend the results to a 45% CNT volume fraction to illustrate the unrealistic positive reinforcement effect at high concentrations.

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Fig. 6.

Effect of CNT volume fraction on the longitudinal Young’s modulus, E₁₁, of the CNT-reinforced epoxy composite for both aligned and random orientations with different lengths.

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Fig. 7.

Influence of CNT length on the longitudinal Young’s modulus, E₁₁, of aligned and randomly orientated CNT-reinforced epoxy composites for a CNT volume fraction of 1%.

We present a comparison of the above results for the case of aligned fibers of aspect ratio a=100 with the findings of (Liu and Brinson, 2008) and (Odegard et al., 2005) for CNT volume fractions up to 5%. The variation of longitudinal Young’s modulus, E₁₁, is presented in Fig. 8 against the CNT volume fraction for all models. It can be observed that E₁₁ of all three models increases dramatically with the increase of CNT volume fraction. The results of the present analysis show better agreement with the results of Odegard et al. (2005) than those of Liu and Brinson (2008). This is partly because both the present model and the model developed by Odegard incorporate a nanoscale representation of all the components used in the RVE, while Liu and Brinson directly applied the Mori–Tanaka method without giving consideration to the development of an RVE from atomistic principles. The discrepancy can also be attributed to the use of different polymer systems in both studies.

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Fig. 8.

Comparison of the variation of longitudinal Young’s modulus, E₁₁, of the aligned CNT-reinforced epoxy composite for the present model and those of (Liu and Brinson, 2008) and (Odegard et al., 2005).

The longitudinal shear modulus μ₁₂ of the aligned and randomly oriented CNT-reinforced epoxy composites is plotted against the nanotube volume fraction and length in [Fig. 9] and [Fig. 10], respectively. It can be observed that μ₁₂ of randomly oriented CNT-reinforced epoxy composites is more sensitive to the variation of the volume fraction and length of CNT when compared to the perfectly aligned configuration. The longitudinal shear modulus of the aligned CNT-reinforced epoxy composites showed no dependence on CNT length and only a small variation at CNT volume fractions above 35%. It can also be seen that μ₁₂ of randomly oriented CNT-reinforced epoxy composites is much larger than μ₁₂ of aligned CNT-reinforced epoxy composites at comparable CNT volume fractions and lengths

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Fig. 9.

Influence of CNT volume fraction on longitudinal shear modulus, μ₁₂, of the aligned and randomly orientated CNT-reinforced epoxy composites for a CNT length of 100 nm.

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Fig. 10.

Longitudinal shear modulus, μ₁₂, of the CNT-reinforced epoxy composite for a 1% CNT volume fraction vs. CNT length.

The sensitivity of the transverse Young’s modulus, E₂₂, and transverse shear modulus, μ₂₃, of the aligned CNT-reinforced epoxy composites on the CNT volume fraction for the different CNT lengths are shown in [Fig. 11] and [Fig. 12], respectively. It can be observed that both E₂₂ and μ₂₃ increase with the subsequent increase of CNT volume fraction. The data in [Fig. 11] and [Fig. 12] indicate that further increases in CNT volume fraction beyond 30% result in relative large increases in E₂₂ and μ₂₃ for a given CNT length as evidenced by the slope of in the data curve. The results of [Fig. 11] and [Fig. 12] also show that increasing the CNT length results in a decrease in both moduli while further increases in CNT length beyond 20 nm result in very small changes in E₂₂ and μ₂₃ for a given CNT volume fraction. It can be concluded that the CNT length has a small influence on E₂₂ and μ₂₃ of the aligned CNT-reinforced epoxy composites.

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Fig. 11.

Effect of CNT volume fraction on transverse Young’s modulus, E₂₂, of the CNT-reinforced epoxy composites with different CNT length.

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Fig. 12.

Variation of transverse shear modulus, μ₂₃, of the CNT-reinforced epoxy composite with CNT volume fraction for different CNT lengths.

As we have considered the RVE as a representative fiber the important parameter to investigate is the influence of the aspect ratio on the properties of the CNT-reinforced epoxy composites. Plotted in [Fig. 13], [Fig. 14], [Fig. 15] and [Fig. 16] are the variations of the effective properties of the aligned CNT-reinforced epoxy composites with the CNT volume fraction and aspect ratio,a, of the representative fiber. [Fig. 17] and [Fig. 18] illustrate the sensitivity of Young’s modulus and shear modulus of the randomly oriented case, respectively. It can be seen that E₁₁ of both the aligned and randomly oriented CNT-reinforced epoxy composites, and μ₁₂ of the randomly oriented CNT-reinforced epoxy composites are significantly more sensitive to the aspect ratio than other moduli. It can also be observed that both E₁₁ and μ₁₂ of both the aligned and randomly oriented CNT-reinforced epoxy composites increase with increasing aspect ratio, whereas E₂₂ and μ₂₃ show a relative decrease. It should be noted that the value of μ₁₂, as shown in Fig. 16, for an aspect ratio of 1.5 is larger than when compared to an aspect ratio of 100 or even 400. This same phenomenon has been observed by Tandon and Weng (1984) in their micromechanical analysis of glass-fiber reinforced composites. From further calculation, it can be determined that the value of μ₁₂ increases initially, with increasing aspect ratio and then begins to decrease. This variation is usually small. It is also worth noting that we can consider the representative fiber with aspect ratio beyond 100 as being a continuous fiber.

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Fig. 13.

Influence of CNT volume fraction on longitudinal Young’s modulus, E₁₁, of the CNT-reinforced epoxy composite vs. CNT volume fraction for different aspect ratios.

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Fig. 14.

Transverse Young’s modulus, E₂₂, of the CNT-reinforced epoxy composite vs. CNT volume fraction for different aspect ratios.

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Fig. 15.

Effect of CNT volume fraction on transverse shear modulus, μ₂₃, of the CNT-reinforced epoxy composites with different aspect ratios.

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Fig. 16.

Variation of CNT volume fraction with longitudinal shear modulus, μ₁₂, of the CNT-reinforced epoxy composite with different aspect ratios.

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Fig. 17.

Influence of CNT volume fraction on longitudinal Young’s modulus, E, of the randomly oriented CNT-reinforced epoxy composite with different aspect ratios.

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Fig. 18.

Effect of CNT volume fraction on longitudinal shear modulus G of the randomly oriented CNT-reinforced epoxy composites with different aspect ratio.

6. Conclusion

In this paper, an atomistic-based continuum model had been developed to study CNT-reinforced epoxy composites. In this model, a representative volume element (RVE) which consists of a carbon nanotube, the surrounding epoxy matrix, and CNT/epoxy interface has been simulated using the finite element method. Through equating the associated strain energies, the RVE was homogenized and studied as a continuous representative fiber. The fiber was then used in a micromechanical analysis of the macroscopic CNT-reinforced epoxy composite system. A form of the Mori–Tanaka method applicable to linear elastic, homogeneous, isotropic fibers and polymeric matrices was used to predict the effective elastic properties of the macroscopic CNT-reinforced epoxy composite. The major advantages of our model include the simplicity of the structure, the nanoscale effects and the improved computational efficiency for predicting the effective properties of the CNT-reinforced composites. Numerical results show that the CNT length, volume fraction, orientation and the aspect ratio of the representative fibers have significant effects on the effective properties of the CNT-reinforced composites.

Acknowledgement

The authors acknowledge the financial support provided by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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