Printer Friendly

Nanomechanics of single-walled carbon nanotubes as composite reinforcement.


Carbon nanotubes [1] exhibit exceptional physical properties: small size, low density, high stiffness, high strength, and excellent electronic and thermal properties [2-7]. Of particular interest are their mechanical properties. Because of the high specific stiffness and strength, carbon nanotubes represent a very promising material as reinforcements in composite materials.

Elastic properties of both multi- and single-walled carbon nanotubes (SWCNT) have been investigated extensively through experimentation [8, 9]. SWCNTs are remarkably stiff and strong. The tensile modulus and strength of nanotubes have been reported [10, 11] to range from 0.27 TPa to 3.6 TPa and from 11 to 200 GPa, respectively. The theoretical approaches to predict properties can be classified into two categories, namely the "bottom up" approach based on quantum/molecular mechanics including classical molecular dynamics (MD) and ab initio methods and the "top down" approach based on continuum mechanics. In general, ab initio methods give more accurate results than MD, but they are also much more computationally expensive (only suitable for small systems containing at most hundreds of atoms). As indicated in Ref. 7, despite constant increases in available computational power and improvement in numerical algorithms, even classical MD computations are still limited to simulating on the order of [10.sup.6] - [10.sup.8] atoms for a few nanoseconds. The simulation of larger systems or longer times must currently be left to continuum methods. However, at the nanoscale, theories for describing continuum materials have reached their limit. The accuracy of using these continuum theories becomes questionable in many of the most interesting cases of nanomechanics. Thus, it would be very useful to have a nanomechanics theory that is seamless and generic to bridge the gap.

Some recent developments based on continuum mechanics have been reported for estimating elastic properties of nanotubes [12-16]. Among them, the analytical molecular mechanics model proposed in Ref. 13 is concise and capable of deriving closed form expressions for Young's moduli and Poisson's ratios as a function of the atomic structure of the nanotube. The force constants are determined by calibrating the model with experimental in-plane elastic constants of graphene sheet. The developed analytical molecular mechanics model has been extended by Xiao et al. [16] to incorporate the modified Morse potential function [17], to estimate elastic constants and stress-strain relationships of nanotubes under tensile and torsion loadings. Most of the available methods can only predict elastic constants like Young's moduli and/or Poisson's ratio of nanotubes because they were based on the force-field constants from harmonic energy potential functions. Only two continuum models [15, 16] are able to predict stress-strain relationships of SWCNTs. All of the above studies view SWCNT as a cylindrical shell with thickness. It is known that the thickness of SWCNTs is ill defined, and so is the stress. In the present study, the SWCNT is used as composite reinforcement and viewed as shell with no thickness (all nanotubes are assumed to have an infinite aspect ratio). Continuous fiber composite micromechanics for axial modulus is known to follow the rule of mixtures [18] derived from the assumption of constant strain in the constituents. The present study shows that one can obtain an equivalent result by defining tubes as solid nanofiber to have an identical volume fraction loading and equivalent elastic stiffness in the axial direction. We first present a simple analytical molecular structural mechanics model to estimate the elastic stiffness and load-strain relationships of nanotubes under tensile loadings, which is followed by prediction of the Young's modulus, Poisson's ratio, and strength of composites with different volume fractions based on the assumption of constant strain in the constituents, in which the force acting on nanotubes is calculated by the molecular structural mechanics model. Detailed derivations and predicted results are presented.


There are several different potential functions available [19-21] for describing C-C bond other than simple harmonic potential functions. Generally, the Tersoff-Brenner potential function is more accurate but it is complicated as presented in the atomistic-based analytical model [15]. Considering a SWCNT subjected to tension loadings, only bond stretching and angle variation terms are significant in the total system potential energy. The modified Morse potential function [17] is simple and is used in the present study. The modified Morse potential function was correlated to the Brenner potential function for strains below 10%. By using the simplified potential function in their molecular mechanics/dynamics models, Belytschko et al. [17] studied the fracture behavior of nanotubes, and excellent predictions have been achieved. The modified Morse potential function is given as follow:

E = [E.sub.stretch] + [E.sub.angle] (1)

[E.sub.stretch] = [D.sub.e]{[1 - [e.sup.-[beta]([DELTA]r)]][.sup.2] - 1} (2)

[E.sub.angle] = [1/2][k.sub.[theta]]([DELTA][theta])[.sup.2][1 + [k.sub.sextic]([DELTA][theta])[.sup.4]] (3)

where [E.sub.stretch] is the bond energy due to bond stretch [DELTA]r, and [E.sub.angle] is the bond energy due to bond angle variation [DELTA][theta]. The parameters calibrated by Belytschko, et al. [17] with the Brenner potential are as follows:

[D.sub.e] = 0.6031 nN x nm [beta] = 26.25 [nm.sup.-1] (4)

[k.sub.[theta]] = 1.42 nN x nm/rad[.sup.2] [k.sub.sextic] = 0.754 rad[.sup.-4]. (5)

This set of parameters corresponds with the Brenner potential for strain below 10% and a separation (dissociation) energy of 124 kcal/mol (5.62 eV/atom).

The validation of using this simple interatomic potential function has been well studied in Ref. 17, where it was reported that fracture is essentially independent of the separation energy and depends primarily on the inflection point of the interatomic potential. Consequently, the shape of the potential function after the inflection point is not important to fracture behavior. The stretch force and the angle-variation moment can be obtained from differentiations of Eqs. 2 and 3 as functions of bond stretch and bond angle variation, respectively:

F([DELTA]r) = 2[beta][D.sub.e](1 - [e.sup.-[beta][DELTA]r])[e.sup.-[beta][DELTA]r] (6)

M([DELTA][theta]) = [k.sub.[theta]][DELTA][theta][1 + 3[k.sub.sextic]([DELTA][theta])[.sup.4]]. (7)


SWCNT can be viewed as a hollow cylinder rolled from a graphene sheet. The nanotube, composed of carbon hexagons, is usually indexed by a pair of integers ([n.sub.1], [n.sub.2]) to represent its helicity. The diameter of the nanotube can be calculated as:

D = [[[square root of 3]a]/[pi]][square root of ([n.sub.1.sup.2] + [n.sub.2.sup.2] + [n.sub.1][n.sub.2])] (8)

where a = 0.142 nm is the C-C bond length.

Effective Diameter of SWCNT Fibers

When SWCNTs are used as reinforcement, one needs to determine the effective diameter of the reinforcement to take into account of the equilibrium separation distance (s) between SWCNTS and the polymer as shown in Fig. 1. The effective diameter is expressed as

[D.sub.f] = [[[square root of 3]a]/[pi]][square root of ([n.sub.1.sup.2] + [n.sub.2.sup.2] + [n.sub.1][n.sub.2])] + s. (9)

A value of s = 0.34 nm is used in the present study. Different surrounding media may give different separation distance. Only two types of nanotubes, armchair ([n.sub.1] = [n.sub.2]) and zigzag ([n.sub.2] = 0) nanotubes, subjected to tensile loading is studied in this article.


Tensile Behavior of Nanotubes

An armchair nanotube ([n.sub.1] = [n.sub.2]) subjected to a longitudinal tensile load [F.sub.T] is studied first. Figure 2 shows an equilibrium configuration of the tube and the associated forces and moments in three chemical bonds a, b, b, and three bond angles [alpha], [beta], [beta] resulting from a bond elongation [DELTA]a and two bond angle variances [DELTA][alpha] and [DELTA][beta]. The relationship between stress and the bond stretch and bond angle variation can be determined through equilibrium and geometry of the tube structure.

Similar to the idea of modeling the molecular structure as an effective "stick-spiral" system [13], we use a stick with Eq. 6 to model the force-stretch relationship of the C-C bond and a spiral spring with Eq. 7 to model the angle bending moment resulting from an angular variation of bond angle. The stick is assumed to have an infinite bending stiffness. Consider the force and moment acting on bond OA as shown in Fig. 2b. Force equilibrium to bond extension of stick OA and the moment equilibrium to bond OA lead to

fsin([alpha]/2) = F([DELTA]b) (10)

f[b/2]cos([alpha]/2) = M([DELTA][alpha]) + M([DELTA][beta])cos[phi] (11)

where. [phi] is the torsion angle between planes OA-OB and OA-OC, which is calculated as:

cos[phi] = -[tan([alpha]/2)/tan[beta]]. (12)

For the armchair nanotube, the geometry relationships satisfy

cos[beta] = cos([pi] - ([pi]/2[n.sub.1]))cos([alpha]/2) (13)

where [pi] - [[pi]/2[n.sub.1]] is the angle of the bond OC to the plane OA-OB as shown in Fig. 2c.

The total axial force [F.sub.T] acting on the armchair nanotube can be related to bond force f as [F.sub.T] = 2[n.sub.1]f, so the force density over tube circumference can be defined as

[T.sub.x] = [F.sub.T]/[pi]D = [2[n.sub.1]f]/[pi]D. (14)

The axial strain [[epsilon].sub.x] and circumferential strain [[epsilon].sub.[theta]] of armchair nanotube can be calculated as

[[epsilon].sub.x] = [[DELTA]b sin([alpha]/2) + [b/2]cos([alpha]/2)[DELTA][alpha]]/[bsin([alpha]/2)], [[epsilon].sub.[theta]] = [[DELTA]b cos([alpha]/2) - [b/2]sin([alpha]/2)[DELTA][alpha]]/[a + bcos([alpha]/2)]. (15)

Then, the Poisson's ratio of the SWCNT can be defined as v = -[[epsion].sub.[theta]]/[[epsilon].sub.x]. For any given bond stretch [DELTA]b, the axial force can be calculated by using Eq. 14 together with Eqs. 6 and 10, and an equilibrium configuration with [DELTA][alpha] and [DELTA][beta] corresponding to [DELTA]b can be uniquely determined through Eq. 11 together with Eqs. 7 and 13, so that the corresponding strain can be obtained by Eq. 15. The angle [alpha] and [beta] of armchair nanotubes have been found from ab initio calculations [22] where [alpha] [approximately equal to] 2[pi]/3 and [beta] = [pi] - arc cos[0.5cos([pi]/2n1)].

The load-strain relationship for a zigzag tube can be calculated in a similar manner to the armchair nanotube described earlier. More detailed derivations can be found in Ref. 16.




In-Plane Stiffness of SWCNTs

In this section, we present the predicted in-plane stiffness of nanotubes followed by their force-strain relationships. It is known that there is a large variation of the Young's moduli (E) of nanotubes among published data from both experimental and theoretical studies. Krishnan et al. [23] presented the experimental data as E = 1.3 -0.4/+ 0.6 TPa, where tubes were viewed as shell with thickness = 0.34 nm. Salvetat et al. [24] measured Young's modulus of nanotubes as 0.816 [+ or -] 0.41 TPa. Variation from experimental results may be due to the presence of defects in nanotube specimens and inherent limitations of current experimental techniques. Different theoretical values can result from using different definitions of the effective thickness of nanotube and by using different potential functions (force-constants) with different algorithms. For instance, Yakobson, et al. [25] used a thickness of 0.066 nm resulting in the graphite Young's modulus of 5.5 TPa. To avoid the confusion on thickness definition, one can use in-plane stiffness as the product of the conventional Young's modulus (E) with the tube thickness (t). As reviewed in Ref. 26, most quantum/molecular mechanics simulations resulted in similar graphite in-plane stiffness ([Y.sub.T] = Et) around 59 eV/atom = 360 J/[m.sup.2] though different effective thicknesses were assumed. Another interesting phenomena observed from theoretical investigations (such as the tight-binding results [27], lattice-dynamical results [28], atomistic-based continuum mechanics [15], structural mechanics [12], and analytical molecular mechanics [13]) is that the Young's moduli are size-dependent at small tube diameters. Again, because of the limitations of current experimental techniques, it is hard to validate/extract such dependence experimentally. More detail discussion can be found in Xiao et al. [16].

From the above molecular mechanics model, the closed-form solution of the in-plane stiffness ([Y.sub.T]) of the nanotube fiber can be given as:

[Y.sub.T] = [4[square root of 3]K]/[3[lambda]K[a.sup.2]/[k.sub.[theta]] + 9] (16)

where K = 2[[beta].sup.2][D.sub.e], [lambda] = [7 - cos([pi]/[n.sub.1])]/[34 + 2cos([pi]/[n.sub.1])] for armchair tubes, and [lambda] = [5 - 3cos([pi]/[n.sub.1])]/[14 - 2cos([pi]/[n.sub.1])] for zigzag tubes. Figure 3 gives the present predctions on the in-plane stiffness of nanotubes. It can be seen that the feature of the size-dependent in-plane stiffness at small diameter (< 2 nm) is captured by the present simple model. Same as the Young's moduli [16], the in-plane stiffness for both armchair and zigzag nanotubes decrease with decreasing tube diameter and approach the predicted graphite value when the tube diameter is increased. Some published data are also shown in Fig. 3 for comparison, and different values in Fig. 3 are normalized by their corresponding asymptotic value (i.e. predicted graphene value for large tube diameters). The present model gives almost the same trend as those of the tight-binding formulation by Goze et al. [27] and continuum structural mechanics [12], although there exists some difference at the plateau level.

Poisson's Ratios of SWCNTs

The dependence of the Poisson's ratio to the tube diameter is shown in Fig. 4. The present predicted Poisson's ratio for both armchair and zigzag tubes decrease with increasing tube diameter, approaching the limit value of 0.20 for graphene sheet. It is seen that the Poisson's ratio for zigzag tubes is more sensitive to the tube diameter than the armchair tubes. It should be noted that the present prediction for large diameter nanotubes (> 2 nm) and graphite is almost constant and is in excellent agreement with the theoretical value (0.21) in Ref. 28 based on a lattice-dynamics model.



Load-Strain Relationship of SWCNTs

The force-strain relationship of nanotubes is predicted using the above procedures up to the inflection point (i.e. the maximum of the interatomic force) only, though the procedure is able to give post failure response. However, the predicted post failure responses by the present model may not be reliable because the present model together with the simple interatomic potential function is not capable of describing the behaviors of the nanotube after the bonds are broken, such as formation of new bonds, rehybridization, and structural transformations. From the experimental [29] and theoretical studies [17] on the tensile behaviors of nanotubes, it was found that the stress exhibits a sudden drop to zero when stress reaches the tensile strength and the fracture is brittle.

Figure 5 shows the calculated force-strain relationships for armchair and zigzag nanotube fibers. Only four different types of nanotubes (i.e. (4, 4), (12, 12) armchairs and (4, 0), (20, 0) zigzags) are presented for illustration purpose. It should be noted that the present approach is developed based on the assumption of defect-free molecular structures. The present predictions on strength of nanotubes show agreement with the calculated results from the molecular mechanics/dynamics [17] where the maximum tensile force resultant of 38 GPa nm (converted from hollow tube) and failure strain of 18.7% were predicted for (12, 12) armchair nanotube and 32 GPa nm and 15.2% for (20, 0) zigzag nanotube. The predicted failure strains in the present study are 23.1% for armchair nanotubes and 15.6-17.5% for zigzag nanotubes. The predicted nonlinear behaviors of nanotubes are also very similar to those modeled by using MD [17].

Elastic Properties of Nanotube Composites

The micromechanics model for prediction aligned continuos nanofiber composites in the tube direction is based on the rule of mixtures approach, where constituents loaded in the tube direction are subjected to constant strain. For simplicity, one can consider the cross-section shown in Fig. 1 where the nanofiber is embedded in a polymer matrix. The polymer is taken to be a linear elastic isotropic material with a strain to failure greater than the nanofiber. Consequently, all nonlinearity in the stress-strain response will originate from the nanofiber and ultimate strength is defined by the strain to failure of the nanofiber. The volume fraction of the nanofiber in the unidirectional nanocomposite is defined as the area ratio of the fiber to the total area as shown in Fig. 1: [V.sub.f] = [D.sub.f.sup.2]/[D.sub.c.sup.2].

As discussed previously, both the longitudinal modulus [E.sub.c] and the major Poisson's ratio [v.sub.c] of fiber composites can be determined by considering a uniform axial strain [[member of].sub.0] applied to both the nanofiber and matrix. For a homogenous composite cylinder with diameter of [D.sub.c], the average stress is given by [[sigma].sub.x] = [E.sub.c][[epsilon].sub.0]. The total force acting on the cylinder in the x direction is

[F.sub.x] = [pi][D.sub.c.sup.2][[sigma].sub.x]/4 = [E.sub.c][[sigma].sub.0][pi][D.sub.c.sup.2]/4. (17)

Since the axial force on the fiber and matrix sum to the total force in the composite cylinders, one has

[F.sub.x] = [F.sub.T] + [pi]([D.sub.c.sup.2] - [D.sub.f.sup.2])[[sigma].sub.xm]/4 = [Y.sub.T][pi]D[[pi].sub.0] + [pi]([D.sub.c.sup.2] - [D.sub.f.sup.2])[[sigma].sub.xm]/4 (18)

where [F.sub.T] is the force in the nanotube and [[sigma].sub.xm] is the average stress in the matrix. Combining Eqs. 17 and 18 yields the expression for the composite modulus as a function of the constituent properties and volume fraction of nanotubes:

[E.sub.c] = [[4[Y.sub.T]D]/[D.sub.c.sup.2]] + (1 - [V.sub.f])[E.sub.m] = [[4D[Y.sub.T]]/[D.sub.f.sup.2]][V.sub.f] + [E.sub.m](1 - [V.sub.f]) (19)

in which [E.sub.m] is the Young's modulus of matrix (a typical value for the Young's modulus of matrix [E.sub.m] = 3.46 GPa (0.5 msi) is used in our parametric study [18]). The above approach views the tube as a shell without thickness. One can obtain the same expression [19] by treating the tube as a solid fiber with the effective diameter [D.sub.f]. In this case, the term [[4D[Y.sub.T]]/[D.sub.f.sup.2]] in Eq. 19 is the effective Young's modulus of the nanofiber. The difference between D and [D.sub.f] is the equilibrium distance (s) between the tube wall and the polymer. If one neglects the distance or the distance is much smaller than the tube diameter (s [much less than] D), one can define the effective Young's modulus of nanofiber as [E.sub.T] = [4[Y.sub.T]]/D. Clearly, the effective Young's modulus of the solid fiber is sensitive to the tube diameter.


It can be seen from Eq. 19 that the Young's modulus of composite depends on the tube diameter for a fixed volume fraction. Figure 6 shows such dependence graphically for the case of [V.sub.f] = 10%, where the Young's modulus increases by over 300% as the diameter is decreased from 5 nm to 1 nm. The solid line in Fig. 6 is calculated by using the in-plane stiffness of graphene sheet. The results indicate that there is no significant loss if one uses the asymptotic large diameter value (the in-plane stiffness of graphene sheet) to evaluate the Young's modulus of composites even at small diameters (< 1 nm). One concludes that the geometry effect of tube diameter has more impact on the Young's modulus of composites than the effect of the tube size dependent in-plane stiffness. It can also be seen that with 10% volume fraction of nanotubes, the axial Young's modulus of the composite is significantly greater than that of the polymer (3.46 GPa) with increases up to 30-100 GPa depending on what diameter of tubes used. The reason that higher composite tensile modulus is obtained with smaller diameter tubes is that one needs more small-diameter tubes to get the same volume fraction. For instance, if one makes composite with the same volume fraction [D.sub.f.sup.2]/[D.sub.c.sup.2] by replacing tube diameter [D.sub.f] with small tube diameter, like [D.sub.f]/2, one needs four times amount of such tubes ([D.sub.f]/2), which carries two times higher force compared with the tube diameter of [D.sub.f]. It is also noticed in Fig. 6, because of the size-dependent properties, particularly for zigzag tubes at smaller diameters (< 1 nm), one may obtain the smaller modulus with higher volume fraction using small diameter tubes than those using larger diameter tubes with higher volume fraction, which is clearly shown in Fig. 7. The moduli range in Fig. 7 is from 100 to 550 GPa when the volume fraction changes from 10 to 50%. The contour increment is 25 GPa in Fig. 7.

The effect of volume fraction on the predicted Young's moduli of composites reinforced with larger diameter tubes is presented in Fig. 8, in which three volume concentrations (2.5%, 5%, and 10%) were used, and tube diameters range from 5 to 50 nm. Both tube diameter and volume fraction play important role in nanocomposites.


Based on the rule of mixtures approach, one can also predict the major Poisson's ratio ([V.sub.c]) of the composite as a function of the nanofiber, the matrix and volume fraction:

[v.sub.c] = [v.sub.f][V.sub.f] + [v.sub.m](1 - [V.sub.f]). (20)

A typical value for the Poisson' ratio of matrix is taken as [v.sub.m] = 0.35 [18] in our calculations. The predicted Poisson's ratios of nanocomposites are shown in Fig. 9 with different volume fractions and different tube diameters. It can be seen that with a fixed volume fraction the Poisson's ratios of nanocomposites show a similar trend as nanotubes in that the Poisson's ratios decrease with increasing tube diameter, and when tube diameter is larger than 2 nm, one can use a constant value of the Poisson's ratio ([v.sub.f]) of graphene sheet. There is no geometry dependence for the Poisson's ratio of nanocomposites. Only fiber volume fractions are found to affects the Poisson's ratios of nanocomposites as shown in Fig. 10.



Analogous to the Young's modulus, the ultimate tensile strength of nanotube composites under uniform axial strain can be expressed as:

[[sigma].sub.c] = [[8[n.sub.1]f]/[[pi][D.sub.f.sup.2]]][V.sub.f] + [[sigma].sub.m](1 - [V.sub.f]) (21)

in which f = F([DELTA]b)/sin([alpha]/2)([alpha] [approximately equal to] 2[pi]/3) for armchair tubes and f = F([DELTA]b)/2 for zigzag tubes. As mentioned earlier, the matrix is assumed to be linear elastic and failure is defined at the failure load of the nanofiber.

For different matrix with different failure strains, one has to calculate the corresponding bond stretch [DELTA]b and bond force F([DELTA]b) at the same strains from the above molecular mechanics model because of the nonlinear tensile behavior of nanotubes. Figure 11 shows examples of tensile stress-strain curves of nanocomposites reinforced with two types of nanotubes and two different volume fractions. The modulus of matrix [E.sub.m] = 2 GPa is used for calculation. The present calculation is also compared with the MD simulation [30] in Fig. 12, where the reinforcement is (10, 10) CNT with volume fraction of 8% and the Brenner potential was used in the MD simulation. The modulus of matrix [E.sub.m] = 1.16 GPa extracted from the MD simulation [30] is used in the present calculation. It can be seen that the present prediction on stress-strain relationship of the composite has similar results as the MD simulation [30] at small deformation, and slightly overpredicts the MD simulation at larger deformation. The difference between two curves at larger deformation in Fig. 12 is partially due to different energy potential function used, and partially due to the interaction between CNT and polymer chains not taken into account in the present analytical approach.




If we simply assume the composite matrix and the nanofiber have the same ultimate failure strain, the ultimate tensile strength of nanotube composites can be easily predicted using the maximum bond stretch force F([DELTA]b)[.sub.max] = 7.92 nN as presented in Fig. 13 ([[sigma].sub.m] = 300 MPa is used as a typical value). Clearly, the tensile strengths of nanotube composites depend on both the diameter and nanofiber chirality. It should be noted this size-dependence is purely attributed to tube geometry. The effect of volume fraction on tensile strength of nanocomposites reinforced by zigzag tubes is presented in Fig. 14. Again, both the volume concentration and nanofiber diameter play important role on tensile strength of nanocomposites, which is not identified in conventional fiber-reinforced composites. The composite has higher composite tensile strength with smaller diameter tubes than those with larger diameters at the same volume fraction. It is the same reason as we described earlier for composite modulus that one needs more small-diameter tubes to get the same volume fraction of composite with larger diameter tubes.


By incorporating the modified Morse potential function into an analytical molecular structural mechanics model, the mechanical responses of armchair and zigzag nanotubes in tension are investigated. The present approach is capable of predicting in-plane stiffness, Poisson's ratios, and force-strain relationships of nanotubes. We have predicted the tensile force capacities and corresponding failure strains of SWCNT, and the same accuracy has been achieved when compared with classical molecular mechanics/dynamics simulations. The armchair nanotubes possess higher tensile capability and failure strain than zigzag nanotubes with the same diameters.

We further presented the elastic properties of nanocomposites reinforced by SWCNTs based on the rule of mixture. The Young's moduli of nanocomposites have been found to be sensitive to both the volume fraction and the reinforcement diameters, and Poisson ratios are influenced by the volume fraction only. We looked at the elastic properties of nanocomposites reinforced with both small and larger diameters and isolated the geometry effect from the effect of the size-dependent properties of nanotubes at small diameters. The tensile strength of nanocomposites is also studied for composites where matrix and nanofiber have similar failure strains. Similar to the behavior of Young's modulus, tensile strength of nanotube composites show dependence on both volume fraction and nanofiber diameter. Such behavior does not occur in conventional fiber-reinforced composites. It should be noted that the present study are looking at composites ideally reinforced with a unimodal distribution of tube diameters (i.e., all tubes in the nanocomposite are assumed identical). We understand that there is usually a distribution of nanotube diameters in a given sample of SWCNTs or MWCNTs. The effects of diameter distribution on elastic moduli can be found in Ref. 6. How to quantify such effect on stress-strain relationships and strengths of nanotube-composites will be studied further in the future.




1. S. Iijima, Nature, 354, 56 (1991).

2. M.S. Dresselhaus, G. Dresselhaus, and P.C. Eklund, Science of Fullerenes and Carbon Nanotubes, Academic Press, San Diego, (1996).

3. J. Che, T. Cagin, and W.A. Goddard, Nanotechnology, 11, 65 (2000).

4. M.B. Nardelli, J.L. Fattebert, D. Orlikowski, C. Roland, Q. Zhao, and J. Bernholc, Carbon, 38, 1703 (2000).

5. B.I. Yakobson and P. Avouris, "Mechanical Properties of Carbon Nanotubes," in Carbon Nanotubes. Topics in Applied Physics, Vol. 80, M.S. Dresselhaus, G. Dresselhaus, and P. Avouris, Editors, Springer Verlag, Berlin-Heidelberg, Germany, 287 (2001).

6. E.T. Thostenson and T.W. Chou, J. Phys. D: Appl. Phys., 36, 573 (2003).

7. D. Qian, G.J. Wagner, W.K. Liu, M.F. Yu, and R.S. Ruoff, Appl. Mech. Rev., 55, 495 (2002).

8. M.M.J. Treacy, T.W. Ebbesen, and J.M. Gibson, Nature, 381, 678 (1996).

9. E.W. Wong, P.E. Sheehan, and C.M. Lieber, Science, 277, 1971 (1997).

10. F. Li, B.S. Cheng, G. Su, and M.S. Dresselhaus, Appl. Phys. Lett., 77, 3161 (2000).

11. O. Lourie and H.D. Wagner, J. Mater. Res., 13, 2418 (1998).

12. C.Y. Li and T.S. Chou, Int. J. Solid. Struct., 40, 2487 (2003)

13. T. Chang and H. Gao, J. Mech. Phys. Solids, 51, 1059 (2003).

14. P. Zhang, Y. Huang, P.H. Geubelle, P.A. Klein, and K.C. Hwang, Int. J. Solid. Struct., 39, 3893 (2002).

15. H. Jiang, P. Zhang, B. Liu, Y. Huang, P.H. Geubelle, H. Gao, and K.C. Hwang, Comput. Meth. Appl. Mech. Eng., 193, 3419 (2003).

16. J.R. Xiao, B.A. Gama, and J.W. Gillespie Jr., Int. J. Solid. Struct., 42, 3075 (2005).

17. T. Belytschko, S.P. Xiao, G.C. Schatz, and R.S. Ruoff, Phys. Rev., B 65, 235430 (2002).

18. J.M. Whitney and R.L. McCullough, "Micromechanics Materials Modeling," in Delaware Composite Design Encyclopedia, Vol. 2, L.A. Carlsson and J.W. Gillespie, Editors, Technomic, Lancaster, USA, (1990).

19. G.C. Abell, Phys. Rev., B 31, 6184 (1985).

20. J. Tersoff, Phys. Rev. Lett., 61, 2872 (1988).

21. D.W. Brenner, Phys. Rev., B 42, 9458 (1990).

22. L.H. Ye, B.G. Liu, and D.S. Wang, Chin. Phys. Lett., 18, 1496 (2001).

23. A. Krishnan, E. Dujardin, T.W. Ebbesen, P.N. Yianilos, and M.M.J. Treacy, Phys. Rev., B 58, 14013 (1998).

24. J.P. Salvetat, J.M. Bonard, N.H. Thomson, A.J. Kulik, L. Forro, W. Benoit, and L. Zuppiroli, Appl. Phys., A 69, 255 (1999).

25. B.I. Yakobson, C.J. Brabec, and J. Bernholc, Phys. Rev. Lett., 76, 2511 (1996).

26. A. Pantano, D.M. Parks, and M.C. Boyce, J. Mech. Phys. Solids, 52, 789 (2004).

27. C. Goze, L. Vaccarini, L. Henrard, P. Bernier, E. Hernandez, and A. Rubio, Synth. Met., 103, 2500 (1999).

28. V.N. Popov, V.E. Van Doren, and M. Balkanski, Phys. Rev., B 61, 3078 (2000).

29. M.F. Yu, O. Lourie, M.J. Dyer, K. Moloni, T.F. Kelly, and R.S. Ruoff, Science, 287, 637 (2000).

30. S.J.V. Frankland, V.M. Harik, G.M. Odegard, D.W. Brenner, and T.S. Gates, Comp. Sci. Tech., 63, 1655 (2003).

Jia-Run Xiao, John W. Gillespie Jr.

Center for Composite Materials, University of Delaware, Newark, Delaware 19716

John W. Gillespie, Jr.

Department of Materials Science and Engineering, and Department of Civil and Environmental Engineering, University of Delaware, Newark, Delaware 19716

*Presented at Polymeric Nanocomposites 2005, Boucherville, Canada.

Correspondence to: Jia-Run Xiao;
COPYRIGHT 2006 Society of Plastics Engineers, Inc.
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2006 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Author:Xiao, Jia-Run; Gillespie, John W., Jr.
Publication:Polymer Engineering and Science
Date:Aug 1, 2006
Previous Article:Extensional flow mixer for polymer nanocomposites.
Next Article:PP-based nanocomposites with various intercalant types and intercalant coverages.

Related Articles
Superlong nanotubes can form a grid. (Nanoscale Networks).
Carbon nanotubes influence materials performance. (General Developments).
Nice threads: the golden secret behind spinning carbon-nanotube fibers.
Nanotech goes to new lengths: scientists create ultralong carbon nanotubes.
Nanotubes help broken bones mend.
Carbon tubes leave nano behind: colossal filaments could offer practical advantages.
Latest developments in carbon nanotubes based nanocomposites.
FE modelling of multi-walled carbon nanotubes/Mitmeseinalise susiniku nanotoru modelleerimine loplike elementide meetodil.
Carbon nanotubes; new research.
"We are done with metals".

Terms of use | Privacy policy | Copyright © 2022 Farlex, Inc. | Feedback | For webmasters |