A theory of plasticity for carbon nanotube reinforced composites

International Journal of Plasticity 27 (2011) 539–559

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A theory of plasticity for carbon nanotube reinforced composites Pallab Barai, George J. Weng ⇑ Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, NJ 08903, United States

a r t i c l e

i n f o

Article history: Received 23 June 2010 Received in final revised form 11 August 2010 Available online 20 August 2010 Keywords: Two-scale homogenization Plasticity Carbon nanotubes CNT agglomeration and interface effects Nanocomposites

a b s t r a c t Carbon nanotubes (CNTs) possess exceptional mechanical properties, and when introduced into a metal matrix, it could significantly improve the elastic stiffness and plastic strength of the nanocomposite. But current processing techniques often lead to an agglomerated state for the CNTs, and the pristine CNT surface may not be able to fully transfer the load at the interface. These two conditions could have a significant impact on its strengthening capability. In this article we develop a two-scale micromechanical model to analyze the effect of CNT agglomeration and interface condition on the plastic strength of CNT/metal composites. The large scale involves the CNT-free matrix and the clustered CNT/matrix inclusions, and the small scale addresses the property of these clustered inclusions, each containing the randomly oriented, transversely isotropic CNTs and the matrix. In this development the concept of secant moduli and a field fluctuation technique have been adopted. The outcome is an explicit set of formulae that allows one to calculate the overall stress–strain relations of the CNT nanocomposite. It is shown that CNTs are indeed a very effective strengthening agent, but CNT agglomeration and imperfect interface condition can seriously reduce the effective stiffness and elastoplastic strength. The developed theory has also been applied to examine the size (diameter) effect of CNTs on the elastic and elastoplastic response of the composites, and it was found that, with a perfect interface contact, decreasing the CNT radius would enhance the overall stiffness and plastic strength, but with an imperfect interface the size effect is reversed. A comparison of the theory with some experiments on the CNT/Cu nanocomposite serves to verify the applicability of the theory, and it also points to the urgent need of eliminating all CNT agglomeration and improving the interface condition if the full potential of CNT reinforcement is to be realized. Ó 2010 Elsevier Ltd. All rights reserved.

1. Introduction Carbon nanotubes (CNTs) are known to possess exceptional mechanical stiffness and strength. The axial Young’s modulus of both single-walled and multi-walled CNTs can be as high as 1–1.2 TPa (Treacy et al., 1996; Krishnan et al., 1998; Salvetat et al., 1999; Shen and Li, 2004, 2005a,b), as compared to diamond at about 1.2 TPa, steel at 200 GPa, and copper at 100 GPa. Their tensile strength is about 100–200 GPa, as compared to annealed steel at about 700 MPa and annealed copper at about 200 MPa. CNTs also have very superior thermal and electrical conductivities (Dai and Lieber, 1996; Hone et al., 1999; Hone, 2004), mostly because of their nearly perfect atomic structures on the surface. Theoretical calculation of CNT thermal conductivity suggests that it can be as high as 6600 W/mK for an isolated (10, 10) nanotube at room temperature (Berber et al., 2000), while experimental measurement at 3000 W/mK for an individual multi-walled nanotube has also been reported (Kim et al., 2001). This is much higher than the thermal conductivity of copper, at about 400 W/mK. Single-walled CNTs also ⇑ Corresponding author. E-mail address: [email protected] (G.J. Weng). 0749-6419/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijplas.2010.08.006

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can carry super electric current (Kasumov et al., 1999; Jarillo-Herrero et al., 2006), and have displayed a current-carrying capacity of about 1000 times of copper wire (Collins and Avouris, 2000). The addition of CNTs into metals, polymers, or ceramics (CNT-reinforced composites) can also enhance the thermal and electrical properties of the matrix material (Ounaies et al., 2003), which in turn have significant impact to the industry of electronic devices. While the benefits of utilizing CNTs in a base material can be seen in many ways, our concern here is on the mechanical properties, with a special reference to plasticity. Over the past decade considerable works have been done on the stiffness of singe-walled (SWCNTs) and multi-walled CNTs (MWCNTs), as well as the elastic moduli of CNT/polymer composites. A meaningful review on both lines of studies is beyond the scope of this article, but it can be mentioned that, for both SWCNTs and MWCNTs the molecular mechanics approach has been able to blend both atomistic and continuum ideas successfully to deliver their transversely isotropic elastic moduli (Li and Chou, 2003a; Chang and Gao, 2003; Shen and Li, 2004, 2005a,b). For the CNT/polymer composites, Odegard et al. (2002, 2003), Frankland et al. (2003), Li and Chou (2003b,c), Fisher et al. (2003), Shi et al. (2004), Seidel and Lagoudas (2006), Anumandla and Gibson (2006), Ashrafi and Hubert (2006), Eitan et al. (2006), Luo et al. (2007), Selmi et al. (2007), Pantano et al. (2008), Liu and Brinson (2008) and Shao et al. (2009), among others, have developed various multi-scale models to elucidate the influence of CNT volume concentration, alignment, waviness, and interfacial conditions on the overall elastic properties of the nanocomposites. A different approach based on the percolation theory and micromechanical homogenization technique has also been developed to estimate the elastic moduli of long fiber networks which have potential application to CNT-reinforced composites (Chatterjee and Prokhorova, 2007, 2009; Chatterjee, 2008a,b). Experimentally it has also been demonstrated that, even with a small load of CNTs, the overall stiffness of the composites can be markedly improved (Schadler et al., 1998; Shaffer and Windle, 1999; Qian et al., 2000,2002; Gong et al., 2000; Andrews et al., 2002). These studies have shed significant insights into the elastic behavior of CNTs and CNT/polymer composites, but none has touched upon the issue of plasticity. It appears that the benefit of utilizing CNTs to improve the yield strength of a material has not been widely recognized, but its low density, high stiffness, and high tensile strength can make it a truly attractive reinforcing agent not just for the elastic stiffness. Motivated by this observation, and in line with our recent pursuit for high yield strength through grain-size refinement in nanocrystalline materials (Jiang and Weng, 2003, 2004a,b; Li and Weng, 2007; Barai and Weng, 2008a,b, 2009; Weng, 2009; see also Khan and Zhang, 2000; Khan et al. 2000, 2006; Farrokh and Kahn, 2009), we decided to look into this issue more closely. Several synthesizing techniques for both single-walled and multi-walled CNT-reinforced composites have been reported in the literature. One of the common features of CNT morphology is the formation of agglomeration in the matrix. Such an agglomerated state is highly undesirable, and various processing techniques have been introduced to reduce such an undesirable state. For instance, a melt compounding process was introduced to fabricate a MWCNT-reinforced composite with polypropylene (PP) matrix (Prashantha et al., 2009). In this process a PP/MWCNT master batch was first diluted with a twin screw extruder. It was shown that, by proper application of shearing, the formation of CNT agglomerates could be effectively reduced at low load of CNTs. Another procedure made use of the high energy ball milling and sintering of nano sized Al or Cu powder (used as the matrix) and powders of multi-walled CNTs (Cha et al., 2005; Kim et al., 2006). Enhanced hardness of the composite was observed, but agglomerated CNTs still remain in the microstructure. In addition, the load transfer condition at the CNT/matrix interface remains an issue. At present uniform distribution of CNTs within the matrix and improved load transfer through surface functionalization remains two challenging tasks in the fabrication process. For this reason any theoretical model must be developed with the capability of predicting these two effects on the overall properties of the composite. To address the issue of agglomeration, a two-scale micromechanics model will be developed to study the plasticity of a CNT-reinforced metal matrix composite. The small scale considers the agglomerated CNT/matrix as a composite. This smallscale composite in turn serves as inclusions in the non-CNT containing pure matrix to form the final CNT composite on the large scale. The interface condition will be modeled through an imperfect interface, spring-like, model in the small-scale composite. This formulation automatically includes the size (diameter) and aspect-ratio effects of CNTs. In this way the influence of CNT agglomeration, interface condition, and CNT size can all be assessed. One novel feature of the theory is that the developed model will be purely analytical. As such, it can be applied by others to study the elasticity and plasticity problems of the CNT/matrix composite with these three important microstructural features. 2. Morphology of the CNT-reinforced composite A typical morphology of CNT-reinforced composite is shown in Fig. 1, where the appearance of CNT agglomeration is quite conspicuous. To reflect such an agglomerated state, we have adopted the morphology of Fig. 2 in our model. In this model the entire composite is divided into two domains, the CNT-free matrix denoted as phase O (volume concentration cO) and the spherical inclusions that contain the agglomerated CNTs and some matrix material denoted as phase I (volume concentration cI). It follows that:

cI þ cO ¼ 1:

ð1Þ

Phase I is itself a composite with uniform dispersion of the CNT fibers. The volume fraction of the matrix inside this ðIÞ ðIÞ small-scale composite will be denoted by c0 and that of the CNTs denoted by c1 , so that

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Fig. 1. Agglomeration of multi-walled carbon nanotubes in polypropylene composites (Prashantha et al., 2009).

Fig. 2. The two-scale morphology of the CNT-reinforced composite. The large-scale composite is comprised of the pure matrix and the CNT-containing spherical inclusions, each representing a small-scale composite consisting of the randomly oriented CNTs and the metal matrix.

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ðIÞ

c1 þ c0 ¼ 1:

ð2Þ

The volume concentration of CNTs in the entire composite will be denoted by c1 and that of the matrix by c0. The foregoing relations lead to ðIÞ

c1 ¼ c 1  c I ;

ðIÞ

c0 ¼ cO þ c0  cI :

ð3Þ

So we have phase I and O as the inclusions and matrix, respectively, in the large-scale problem, and phase 1 and 0 in the small-scale one. The agglomerated state is defined through cI. If all the CNTs are uniformly dispersed, then cI = 1 and ðIÞ c1 ¼ c1 , but due to agglomeration this kind of distribution is rarely observed. 3. Constitutive relations of the CNTs, interface, and matrix Carbon nanotubes will be taken to deform only elastically. The hexagonal distribution of carbon atoms on the tube surface makes the surface property isotropic, but in the lateral direction the stiffness is substantially softer due to the hollow nature of the tube. As a result the overall elastic properties of CNTs are transversely isotropic, with five independent elastic constants. Based on such an observation Shen and Li (2004, 2005a) have applied the interatomic potentials to derive the five independent elastic constants of single-walled and multi-walled CNTs. To describe the transversely isotropic elastic response, we shall adopt Hill’s (1964) notation so that the stress–strain relation can be written as

r ¼ Le; with L ¼ ð2k; l; n; 2m; 2pÞ;

ð4Þ

where L is the elastic stiffness tensor, and k, l, n, m and p are, respectively, its plane-strain bulk modulus, cross modulus, axial modulus under an axial strain, transverse shear modulus, and axial shear modulus. By taking direction 1 to be symmetric and plane 2–3 isotropic, Eq. (4) carries the explicit form

ðr22 þ r33 Þ ¼ 2kðe22 þ e33 Þ þ 2le11 ;

r11 ¼ lðe22 þ e33 Þ þ ne11 ; ðr22  r33 Þ ¼ 2mðe22  e33 Þ; r23 ¼ 2me23 ; r12 ¼ 2pe12 ; r13 ¼ 2pe13 :

ð4aÞ

One advantage of adopting such a notation is that Walpole’s scheme (1969, 1981) for the manipulation of transversely isotropic tensors, such as the inner product and inverse, can be easily called upon to obtain the explicit results. The interface condition will be modeled by the linear spring layer model with a continuous traction but a displacement jump, in the form (e.g. Benveniste, 1985; Hashin, 1990, 1991; Qu, 1993a,b)

Drij nj ¼ ½rij ðxÞnj ¼ 0; Dui ¼ ½ui ðxÞ ¼ gij rjk nk ;

ð5Þ

where the brackets [] represent the jump of the said quantity at the interface with an outward normal, ni, from the inclusion to the matrix, and gij represents the compliance of the interface layer. A particularly useful form of this interface parameter is

gij ¼ cdij þ ðb  cÞni nj :

ð6Þ

When b = 0, the interface is allowed to slide but without normal separation or interpenetration. This condition will be assumed in this study. So our concern is the influence of c on the overall property of the composite. The ductile matrix will be taken to be isotropic, with its stress–strain relation described by the modified Ludwik’s equation

re ¼ ry þ h  ðepe Þn ; where re and

re

ð7Þ

p e

e are the usual Mises’ effective stress and effective plastic strain, defined as

 1=2 3 0 0 ¼ rij rij ; 2

epe ¼

 1=2 2 p p eij eij 3

ð8Þ

in terms of the deviatoric stress and plastic strain components, where ry, h and n are, respectively, the yield stress, strength coefficient, and strain-hardening exponent of the matrix. Its dilatational response is taken to be plastically incompressible. Most experimental evidence such as shown in Fig. 1 indicates that CNTs are randomly oriented within the agglomeration which results in an isotropic behavior for phase I, the spherical inclusions. We shall take these inclusions to be also homogeneously dispersed in the pure matrix, phase O, so that the overall composite is isotropic. 4. The elastic state of the two-scale model Before initial yielding, the agglomerated CNT composite responds elastically, and the overall elastic moduli depends on the elastic properties of the matrix and CNTs, as well as c1 and cI. For both the initial response and the subsequent

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543

elastoplastic deformation, we first recall the elastic principles of a dual-phase material. For the large-scale problem involving spherical inclusions, many micromechanical models have been developed, but to keep the inclusion/matrix morphology in perspective, the generalized self-consistent model (GSC, Christensen and Lo, 1979) and the Mori–Tanaka model (M–T Mori and Tanaka, 1973; see also Weng, 1984, 1990a, 1992 and Benveniste, 1987 for the elastic moduli) are likely to be the two most suitable ones. The field-fluctuation method to be used in the calculation of the effective stress of the ductile phase further requires that the expression of the effective moduli be as simple as possible. For this reason, and also for the reason that both the large-scale and the small-scale problems could be treated with the same micromechanical principle, we shall adopt the Mori–Tanaka approach here. This will provide the needed consistency in the exposition of both homogenization schemes. The advantage of using the M–T approach was also recognized by Tan et al. (2005) in their study on the effect of nonlinear interface debonding in composites. 4.1. The large-scale problem involving spherical inclusions For the elastic response of the larger-scale problem with spherical inclusions, the effective bulk and shear moduli of the composite can be written as (Weng, 1984)



j ¼ j0 1 þ

 cI ðjI  j0 Þ ; cO a0 ðjI  j0 Þ þ j0



l ¼ l0 1 þ

 cI ðlI  l0 Þ ; cO b0 ðlI  l0 Þ þ l0

ð9Þ

where cI and cO are the volume concentrations of domain I (the CNT- containing agglomerates) and domain O (the CNT-free matrix), j0 and l0 are the bulk and shear moduli of the matrix, and jI and lI are the bulk and shear moduli of the CNT-containing domain I, respectively. In addition, a0 and b0 are the dilatational and deviatoric components of Eshelby’s (1957) S-tensor for a spherical inclusion, given by

a0 ¼

3j0 ; 3j0 þ 4l0

b0 ¼

6 j0 þ 2l0 : 5 3j0 þ 4l0

ð10Þ

For later development of the small-scale homogenization scheme, it is useful to record the average hydrostatic and deviatoric stress of the agglomerated inclusions, the phase I, in terms of the external stress rij (Weng, 1984)

jI

ðIÞ rkk ¼

ðcI þ cO a0 ÞðjI  j0 Þ þ j0

rkk ; r0ðIÞ ij ¼

lI ðcI þ cO b0 ÞðlI  l0 Þ þ l0

r0ij :

ð11Þ

ðIÞ

This pair of rij then serves as the externally applied stress for the treatment of the effective properties of phase I, the smallscale composite that consists of the matrix and the randomly oriented CNTs. The corresponding values for the matrix phase (phase O), are given by ðOÞ rkk ¼

a0 ðjI  j0 Þ þ j0 rkk ; ðcI þ cO a0 ÞðjI  j0 Þ þ j0

r0ðOÞ ¼ ij

b0 ðlI  l0 Þ þ l0 r0 : ðcI þ cO b0 ÞðlI  l0 Þ þ l0 ij

ð12Þ

4.2. The small-scale problem involving randomly oriented carbon nanotubes with a perfect interface ðIÞ

ðIÞ

In this case the relevant volume concentrations are c0 and c1 . To signify CNTs as phase 1 in this small-scale composite, its transversely isotropic moduli tensor in Eq. (4) is rewritten as

L1 ¼ ð2k1 ; l1 ; n1 ; 2m1 ; 2p1 Þ;

ð13Þ

which is equivalent to

  L1 ¼ 2j23 ; C 12 ; C 11 ; 2l23 ; 2l12

ð14Þ

in terms of the traditional engineering constants, with direction 1 as the axial direction and plane 2–3 isotropic. The longitudinal Young’s modulus and major Poisson ratio in turn are given by E11 = n  l2/k and m12 = l/2k. Based on the general formulae of a two-phase composite containing randomly oriented ellipsoidal inclusion (Weng, 1984, 1990a), Qiu and Weng (1990) have derived the effective bulk and shear moduli of a two-phase composite with an isotropic matrix and randomly oriented, transversely isotropic, spheroids, as (see their Eq. (5.8)) ðIÞ

jI ¼

c0

LA j0 þ cðIÞ 1 n

ðIÞ c0

þ

ðIÞ 3c1 nA

ðIÞ

;

lI ¼

c0

LA l0 þ cðIÞ 1 g ; ðIÞ A þ 2c1 g

ðIÞ c0

ð15Þ

where we have added the subscript I to both j and l to signify that the effective bulk and shear moduli refer to phase I, the small-scale composite. These moduli are to be used in Eqs. (9)–(12) in the large-scale problem. In addition, parameters ð3nLA ; 2gLA Þ are the hydrostatic and deviatoric components of the orientational average of the tensor product L1 and A1, denoted as {L1 A1}, and ð3nA ; 2gA Þ comes from the orientational average of {A1} in their original paper. In these expressions the subscript 1 refers to phase 1, the inclusion phase, and L1 and A1 are its transversely isotropic moduli and strain concentration tensor embedded in the infinitely extended isotropic matrix, respectively. The four parameters, nLA ; nA ; gLA and gA,

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depend on the elastic properties j0 and l0 of the matrix and k, l, n, m and p of the CNTs, as well as the CNT aspect ratio and ðIÞ volume concentration, c1 . Detailed derivation for these four parameters can be found in Qiu and Weng (1990). For convenience their expressions are listed in Appendix. Since Eshelby’s S-tensor is involved in these expressions, the aspect ratio of the CNTs is accounted for, but for all practical purpose CNTS could be treated with an aspect ratio approaching infinity. In that case the results of Shi et al. (2004) are recovered. We note in passing that, for simplicity, the agglomerated domains are assumed to be spherical so that its effective moduli are calculated by Eq. (9), but domains with a spheroidal shape can be accounted for by applying the formula Eq. (13) to replace Eq. (9), in which the inclusion phase is just isotropic, not transversely isotropic. But an explicit formula for such case can be found directly in Tandon and Weng (1986). 4.3. The small-scale randomly oriented carbon nanotubes with an interfacial sliding effect The two-scale theory developed in the last two subsections for the elastic response of the CNT-reinforced nanocomposite assumes that the carbon nanotubes are perfectly attached to the matrix material. No discontinuity between the CNT fibers and the adjacent matrix in terms of traction or displacement has been considered. Since the surface of a pristine carbon nanotubes fiber is usually quite smooth, continuity in displacement at the CNT-matrix interface cannot always be assured. Based on the concept of slightly weakened interfaces (Qu, 1993a,b), the effect of displacement discontinuity given by Eqs. (5) and (6) can be incorporated here to address  the interface issue of CNT in the small-scale composite. From Eq. (2.13) of Qu (1993a,b), the modified Eshelby S-tensor SM is written as ijkl

SM ijkl ¼ Sijkl þ ðI ijpq  Sijpq Þ  H pqrs  Lrsmn  ðImnkl  Smnkl Þ:

ð16Þ

Here Sijkl is the regular Eshelby S-tensor, Lrsmn is the stiffness tensor of the matrix material, Iijpq is the identity tensor and Hpqrs represents the imperfection compliance tensor. The expression for each entry in the H-tensor is provided in Appendix B. In parallel to Eq. (15) but including the additional contribution from the displacement jump at the interface, the effective stiffness of the 3-D CNT-reinforced composite can now be expressed as ðIÞ

jI ¼

c0 ðIÞ c0

þ

LA j0 þ cðIÞ 1 n

ðIÞ 3c1 nA

þ

ðIÞ 3c1 nHLA

ðIÞ

;

lI ¼

LA l0 þ cðIÞ 1 g : ðIÞ A ðIÞ þ 2c1 g þ 2c1 gHLA

c0

ðIÞ c0

ð17Þ

Because of the effect of the imperfect interface tensor, H, the only extra terms that got into Eq. (17) are nHLA and gHLA . These two terms are the hydrostatic and deviatoric components of the orientational average of the tensor,fH1 L1 A1 g, respectively. Here H1 is the imperfect interface compliance tensor for the spheroidal inclusions, L1 is the stiffness tensor of the transversely isotropic CNT fibers, and A1 is the strain concentration tensor for the inclusions using the modified Eshelby S-tensor (SM) The expression of A1 is given in equation (4.16) of Qu (1993a,b) in terms of the stiffness of the inclusion (L1) and the matrix (L0) as, 1 A1 ¼ ½1 þ SM : L1 0 : ðL1  L0 Þ :

ð18Þ HLA

HLA

HLA

They are written as an isotropic tensor with the dilatational and deviatoric components, (3n , 2g ). The values of n and gHLA are computed using Walpole’s (1981) mathematical scheme developed by Qiu and Weng (1990) in their Section 2. 5. A two-scale homogenization scheme for the plasticity of CNT-reinforced composite After yielding, the overall composite enters into the elastoplastic state. To treat this problem an explicit, two-scale homogenization scheme based on the concept of secant moduli will be developed. The secant-moduli approach makes use of a linear elastic comparison composite as the starting point, and then replaces the elastic moduli of the constituent phases by their respective secant moduli. A key step in this latter process is the determination of the effective stress of the ductile phase. To this end a field-fluctuation approach will be adopted. The use of a linear comparison composite to determine the nonlinear behavior of a heterogeneous material has a long history. It started from Talbot and Willis (1985) in their construction of nonlinear bounds. In the elastoplastic context, Tandon and Weng (1988), Weng (1990b), Ponte Castañeda (1991), Qiu and Weng (1992), Suquet (1995) and Hu (1996, 1997) have made related contributions. There are of course other nonlinear approaches for the heterogeneous solids (e.g. Hill, 1965; Hutchinson, 1976; Berveiller and Zaoui, 1979; Weng, 1982; Dvorak, 1992; Fotiu and Nemat-Nasser, 1996; Masson et al., 2000; Doghri and Tinel, 2005; Pierard and Doghri, 2006; Love and Batra, 2006; Berbenni et al., 2007; Mercier and Molinari, 2009), but for the problem at hand, we found the secant-moduli approach to be the most convenient one to use. As with the elastic case we first outline the procedure for the large-scale problem and then go down to the small-scale one. 5.1. The large-scale dual-phase plasticity involving spherical inclusions Based on a linear comparison composite with an identical microgeometry, the secant-moduli approach allows one to obtain the effective secant moduli of the composite from Eq. (9), as

P. Barai, G.J. Weng / International Journal of Plasticity 27 (2011) 539–559



js ¼ j0 1 þ

 cI ðjsI  j0 Þ ; cO asO ðjsI  j0 Þ þ j0



ls ¼ ls0 1 þ

l l l l

cI ð sI  sO Þ cO bsO ð sI  sO Þ þ

 ; s

lO

545

ð19Þ

where

asO ¼

3j0 ; 3j0 þ 4lsO

bsO ¼

6 j0 þ 2lsO : 5 3j0 þ 4lsO

ð20Þ

It should be noted that capital letter O has been used in lsO (we deserve ls0 for the secant shear modulus of the matrix in the small-scale problem). The pair jsI and lsI are the effective secant moduli of the CNT-containing spherical domain, phase I, given by Eqs. (15) and (17) with a perfect or an imperfect interface, respectively, but with l0 replaced by ls0 . It is evident that even though the ductile matrix is plastically incompressible, both the large and the small-scale composites are plastically compressible. In general the secant shear modulus ls0 is related to the secant Young’s modulus Es0 which in turn depends on the plastic strain epe of the matrix. That is,

Es0 ¼

1 ; 1=E0 þ epe =re

ls0 ¼

3j0 Es0 : 9j0  Es0

ð21Þ

As plastic deformation continues, the secant modulus of the ductile matrix continues to decrease. It follows that, once ls0 at a given state of external stress (or strain) is known, this pair of effective secant moduli will allow us to determine the elastoplastic behavior of the large-scale composite. Since epe depends on re from the constitutive Eq. (7), a key step is to determine the effective stress of the ductile matrix at a given stage of deformation. This could be accomplished through the field fluctuation approach, originally proposed by Bobeth and Diener (1986) and Kreher and Pompe (1989) in the elastic context, and extended to plasticity by Suquet (1995) and Hu (1996, 1997). The novelty of this approach is that, under the same boundary condition, a change in a material parameter of a constituent phase will result in a field fluctuation that gives rise to a new overall energy. Then by taking a partial derivative of this energy with respect to this material parameter, a relation between the stress (or strain, or strain-rate) of the individual phase and the overall stress (or strain, or strain-rate) of the composite will result. By choosing ls0 and j0 as the parameters to vary, this approach will give the effective and hydrostatic stresses of the ductile phase (phase O), respectively, as (Hu, 1996, 1997)

( ðOÞ e

r

"   #)1  s 2 2 1 1 lsO @ js 2 lO @ ls 2 2 r þ r ; cO 3 js @ lsO kk ls @ lsO e

¼

ð22Þ

ðOÞ

where r2e ¼ ð3=2Þr0ij r0ij . This re then can be used in Eq. (4) to find epe , and both can be used in Eq. (21) and determine the current state of lsO at the current level of applied stress, rij . ðOÞ 0ðOÞ It should be recognized that this re includes contributions from both the average stress rij in Eq. (12) and the locally perturbed stress field in the heterogeneously deforming matrix. This was done through the energy equivalence principle in the field-fluctuation approach. As a consequence its value is greater than that calculated from the mean deviatoric stress alone, and the estimated overall elastoplastic composite will also be softer than that based on the direct mean-stress approach. These two approaches are known as the second-moment and the first-moment approach, respectively, in literature. Another distinct difference is that, under a pure dilatational loading, the second-moment approach could deliver the desirable nonlinear behavior leading to plastic compressibility, but the first-moment approach always gives rise to a linear response. Further details on these and other issues can be found in Qiu and Weng (1992). 5.2. Plasticity of the small-scale composite with randomly oriented CNTs in the ductile matrix On the smaller scale, a parallel line of analysis allows us to calculate the effective elastoplastic behavior of the two-phase composite containing the randomly oriented CNTs and the ductile matrix. The corresponding js and ls in the large-scale ðIÞ ðIÞ problem are now the js and ls , which follow from jI and lI given in Eq. (15) and (17), with the l0 contained there replaced s by l0 , as ðIÞ

jsI ¼

c0

LA j0 þ cðIÞ 1 ns

ðIÞ c0

þ

ðIÞ 3c1 nAs

ðIÞ

jsI ¼

c0 ðIÞ c0

þ

ðIÞ

;

lsI ¼

LA j0 þ cðIÞ 1 ns

ðIÞ 3c1 nAs

þ

ðIÞ 3c1 nHLA s

c0

LA ls0 þ cðIÞ 1 gs ; ðIÞ A þ 2c1 gs

ðIÞ c0

ðIÞ

;

lsI ¼

ls0 þ c1ðIÞ gLA s ; ðIÞ ðIÞ þ 2c1 gAs 2c1 gHLA s

ð23Þ

c0 ðIÞ c0

ð24Þ

A HLA A HLA where nLA ; gLA are the secant expressions of their elastic counterparts, in which l0 has been replaced by s ; gs and gs s ; ns ; ns ls0 . ðIÞ The level of the applied stress, rij , that corresponds to the level of the external rij in the large-scale problem is given by Eq. (11), with l0, a0 and b0 replaced by their secant counterparts. That is,

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P. Barai, G.J. Weng / International Journal of Plasticity 27 (2011) 539–559 ðIÞ rkk ¼

jsI ðcI þ cO asO ÞðjsI  j0 Þ þ j0

rkk ; r0ðIÞ ij ¼

lsI

ðcI þ cO bsO ÞðlsI  lsO Þ þ lsO

r0ij :

ð25Þ

To find the effective stress of the ductile phase 0 inside the agglomerate phase I, the same field fluctuation approach leads to

( ð0Þ e

r ¼ ðIÞ2

"   #)1  s 2 2 1 ls0 @ jsI ðIÞ2 l0 @ lsI ðIÞ2 2 r þ s r ; s ðIÞ @ ls0 kk lI @ ls0 e c0 3 jI 1

0ðIÞ

ð26Þ

0ðIÞ

where re ¼ ð3=2Þrij rij . As with the large-scale composite, this effective stress allows us to determine the secant shear modulus, ls0 of the matrix, and this in turn gives rise to the effective secant-moduli for the small composite through Eq. (18). Then the overall stress–strain relation of the small-scale composite can be evaluated. 6. A computational procedure There are many ways to carry out the computation for the developed two-scale theory. Even though the theory has been developed from top down, we find it easier to go from bottom up. We first evaluate the elastic response of the composite from Eqs. (9)–(13) to find the relevant elastic state, and Eqs. (23) and (25) to find the effective stress of phase O and phase ðOÞ ð0Þ 0 inside inclusion I. Then check the yield criterion to see if re > ry and re > ry for the plastic deformation to commence in phase O and phase 0. Plastic deformation usually starts from phase 0 inside phase I. Once the yield condition has been reached, we continue with some assumed value for ls0 (slightly lower than the elastic value) of the matrix phase in the ð0Þ small-scale composite, and use it to compute its effective stress re from its constitutive equations (Eqs. (21) and (7)), s s and the secant moduli jI and lI of this small-scale composite from Eqs. 23 and 24. This effective stress is also the value ð0Þ ðIÞ of re in Eq. (26). To determine the value of rij , the external boundary condition must be specified. For instance under a ðIÞ ðIÞ 0 pure tension r11 ; rkk ¼ r11 andr11 ¼ ð2=3Þr11 , etc., and thus rkk andrij can be specified from Eq. (19) in terms of r11 and s the yet-unknown lO of the non-CNT containing matrix (phase O). After substituting these into Eq. (26), the effective stress ð0Þ s s rð0Þ e is now also related to r11 and lO . Since, re is already known from the assumed l0 , Eq. (26) provides a relation that links s the desired r11 with the yet unknown lO , that is, r11 is now given in terms of lsO of the matrix in domain O. So, once this lsO is determined, the problem is solved. To determine this lsO we now go to the large-scale problem. From Eq. (19), the overall secant moduli js and ls depend on ðOÞ s lO , and then from Eq. (22), the effective stress rðOÞ now solely depends on lsO . By setting this re into the constitutive equae tion, Eq. (4), the current state of lsO can be found. Once this lsO is known, the desired r11 is obtained. sðIÞ This step can be repeated with a new (lower) value of l0 . In this way the entire stress–strain curve of the two-scale CNTreinforced composite can be calculated. To assist the computation, a flow chart is given in Fig. 3. This computational procedure is always convergent and stable with a sufficiently small strain increment (we set it at 0.002%). 7. Results and discussion The foregoing theory is applicable to both elastic and elastoplastic response of CNT-reinforced composite, without or with CNT agglomeration. In this section we first apply it to calculate the elastic properties of a CNT/polymer composite, and then move on to examine the nonlinear stress–strain relation of a CNT/metal system. In this process the role of CNT agglomeration will be highlighted. 7.1. Elastic response of a CNT/polymer composite Our calculation on the effective Young’s modulus of a CNT-reinforced composite was directed to the CNT/polyimide system tested by Odegard et al. (2003). We used the transversely isotropic properties of single-walled CNTs given by Shen and Li (2004) and the matrix properties given by Odegard et al. (2003) in the calculations. Free-standing single-walled CNTs have been considered as hollow thin shells. Along this line Wu et al. (2008) and Peng et al. (2008) have been able to link the tension and bending rigidities of single-walled CNTS in terms of the interatomic potentials. But in this study CNTs are embedded in the matrix, and will be treated as long rods. The elastic properties of CNTs and polyimide are listed in Table 1. We first used these properties in Eqs. (15) and (17) to calculate the effective bulk and shear moduli of the small-scale composite, and then in Eq. (9) to calculate those of the large-scale one. This two-scale methodology in turn gives rise to the effective Young’s modulus, E, of the composite as a function of CNT concentration, c1, at five levels of CNT agglomerations: cI = 0.2, 0.4, 0.6, 0.8, and 1, where at cI = 1, the CNTS are uniformly distributed and at 0.2, the CNTs are clustered in 20% of the matrix, leading to a highly agglomerated state. With a perfect interface the calculated results are shown in Fig. 4a at these five levels of agglomeration, along with the test data. It is seen that uniform distribution of the randomly oriented CNTs results in the best reinforcement, while an agglomerated state would lower the overall stiffness. The experimental data is seen to lie close to the agglomerated state of cI = 0.2–0.3. This comparison reflects the inevitability of CNT agglomeration, so the assumption of uniform distribution could result in a significant overestimate for the elastic stiffness.

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547

Fig. 3. Flow chart for the computation of elastoplastic stress–strain curves of CNT/metal composites.

Despite the agglomeration, a very encouraging sign shown here is that the Young’s modulus could still increase by about 75% with only 1vol% of CNTs. Such a remarkable effect is almost unmatched by any types of reinforcement. The corresponding results with an imperfect interface are shown in Fig. 4b. This calculation made use of the formulae given in Eq. (17) for the small-scale composite, with the sliding parameter, c = 5  108 nm/MPa. It is seen that the experimental result can be reproduced with the level of agglomeration, cI = 0.4. The Young’s modulus at the same level of agglomeration in Fig. 4a is seen to be slightly stiffer than the experimental result. Introduction of the interface imperfection makes the response softer, as expected. To investigate the effect of the sliding parameter, c on the elastic response of the two-scale CNT nanocomposite further, the Young’s modulus, shear modulus, and bulk modulus are plotted in Figs. 5a–c, respectively, for different values of the interface imperfection. In Figs. 5a and 5b for very small values of c (<108 nm/MPa), the interface acts as almost a perfect one and the curves for both Young’s and shear moduli almost overlap on each other. As the sliding parameter increases,

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P. Barai, G.J. Weng / International Journal of Plasticity 27 (2011) 539–559 Table 1 Parameters used in the calculation of elastic properties of CNT-reinforced polymer composite (matrix properties taken from Odegard et al. (2003) and SWCNT properties from Shen and Li (2004)). Material properties

Matrix phase

CNT

Isotropic Young’s modulus (E) Poisson’s ratio (m) Axial Young’s modulus (E11) Transverse bulk modulus (K23) Transverse shear modulus (l23) In plane shear modulus (l12) In plane Poisson’s ratio (m12)

0.85 GPa 0.4 – – – – –

– – 1.06 TPa 271 GPa 17 GPa 442 GPa 0.162

Fig. 4a. Comparison of the Young’s moduli obtained using the two-scale model at different level of agglomeration with the experimental data of Odegard et al. (2003) on CNT/polymer composite.

Fig. 4b. Comparison of the Young’s moduli obtained using the two-scale model with imperfect interface effect at different level of agglomeration with the experimental data of Odegard et al. (2003) on CNT/polymer composite.

the response becomes softer. When the imperfection becomes as large as 106 nm/MPa, the load transfer between CNT and the matrix is almost negligible and CNTs effectively act as cylindrical voids. As a result softening is observed in both Young’s

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Fig. 5a. The effect of imperfect interface on the Young’s moduli of the carbon nanotubes reinforced composite. The plots are generated for a particular radius of the SWNT assuming a constant level of agglomeration. As the imperfection at the interface increases, the effective Young’s modulus of the composite material decreases, and CNTs with very weak interfaces behaves as voids.

Fig. 5b. The effect of imperfect interface on the shear moduli of the carbon nanotubes reinforced composite display similar behavior as that of the Young’s moduli. The magnitude of the shear moduli of the entire composite material decreases with increasing interface imperfection, and for very weak interfaces the CNTs behave as voids leading to softening of the composite with increasing volume fraction of the carbon nanotubes fibers.

and shear moduli. On the other hand, the bulk modulus shown in Fig. 5c remains totally unchanged with the increase of c This is a result that only the tangential interface imperfection has been considered (b = 0). The level of agglomeration in all the figures is assumed to be cI = 0.6 in these computations. It is also of interest to see the size effect of CNTs on the Young’s moduli of the overall CNT nanocomposite. Fig. 6 illustrates such an effect as the average CNTs radius increases from 0.341 nm to 3.39 nm. In these computations, the increasing elastic moduli of the SWNT fibers with decreasing radius have been taken from the molecular mechanics calculations of Shen and Li (2004), and then implemented into our two-scale composite model. To eliminate the interference from c on this size effect, no imperfection was assumed, and the same agglomeration state of cI = 0.6 was also taken. 7.2. Plasticity of a CNT/metal composite Our calculations were directed to a CNT/Cu composite that has been tested by Kim et al. (2006). Their scanning electron micrograph, reproduced in Fig. 7, indicated the two-domain morphology as envisioned in our model: one with the Cu matrix devoid of CNTs and the other containing both CNTs and Cu. We have used the two-scale model to calculate the stress–strain curves of such CNT composite. In this calculation the material constants for the transversely isotropic CNTs were taken from Shen and Li (2005a) for multi-walled CNTs. These constants and those of Cu are listed in Table 2.

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Fig. 5c. The effect of imperfect interface on the bulk moduli of the carbon nanotubes reinforced composite clearly portrays that the bulk moduli remains unaffected by the change in interface imperfection. It is because in the formulation of the problem only the tangential imperfection has been considered and the normal component has been assumed to be zero (see Appendix B).

Fig. 6. Increase in Young’s moduli of the entire CNT-reinforced nanocomposite with increasing volume fraction of the carbon nanotubes for different radius of the SWNT fibers. The values of elastic moduli of the SWNT fibers with decreasing radius are taken from Shen and Li (2004). The aspect ratio of the fibers is assumed to be very large with almost zero interface imperfection. The level of agglomeration assumed in this figure is cI = 0.6.

Assuming a perfect interface, the calculated results at 5% and 10% CNT concentrations are shown in Fig. 8a, along with the test data of Kim et al. (2006). The 0% CNT corresponds to the base Cu. It is seen that the nonlinear stress–strain relations of the CNT/Cu composite are effectively strengthened by addition of CNTs. These calculations have been made with the agglomerated state of cI = 0.2. The effect of an interface imperfection on the stress–strain response of the CNT/Cu nanocomposite is shown in Fig. 8b. The same experimental result by Kim et al. (2006) is also reproduced here for comparison. In general the non-zero value of the imperfection parameter softens the overall response of the composite, but increasing the level of morphology uniformity, cI, makes it stiffer. In Fig. 8b the sliding parameter is assumed to be c = 108 nm/MPa and the agglomerated state was taken to be cI = 0. 29, as opposed to 0.2. The elastic moduli and plastic constants used in both Figs. 8a and b are the same, as given in Table 2. To reflect the influence of agglomeration on the nonlinear stress–strain behavior more clearly, we have made another round of calculations based on the assumption of uniform distribution (i.e. cI = 1). The results with 5% and 10% CNTs are shown as solid lines in Fig. 9, where the dashed lines are for the agglomerated state taken from Fig. 8a. The results under perfect uniform distribution are seen to be substantially higher than the latter. One is led to conclude that, as in Young’s

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551

Fig. 7. SEM micrographs showing the CNT/Cu agglomerated region and a pure Cu region in a 5 vol% CNT/Cu nanocomposite (from Kim et al. (2006)).

Table 2 Parameters used in the calculation of elastoplastic properties of CNTreinforced metal composite (matrix properties taken from Kim et al. (2006) and MWCNT properties from Shen and Li (2005a)). Elastic and plastic constituent properties

Metal matrix (phase 0)

CNTs (phase 1)

Isotropic Young’s modulus (E) Poisson’s ratio (m) Axial Young’s modulus (E11) Transverse bulk modulus (K23) Transverse shear modulus (l23) In plane shear modulus (l12) In plane Poisson’s ratio (m12) Work hardening exponent (n) Initial yield stress (ry) Strength coefficient (h)

128 GPa

–

0.36 – –

– 1.17 TPa 130 GPa

–

5.98 GPa

– – 0.2

277 GPa 0.139 –

70 MPa 200 MPa

– –

Fig. 8a. Comparison of the elastoplastic response as obtained using our two-scale model with the experimental results of Kim et al. (2006) on CNT/Cu composite. No interface imperfection has been assumed here. The level of agglomeration assumed in this figure is cI = 0.2.

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Fig. 8b. Comparison of the elastoplastic response as obtained using our two-scale model including imperfect interface effect with the experimental results of Kim et al. (2006) on CNT/Cu composite. Different level of agglomeration has been assumed in this figure (cI = 0.29) as compared to the previous one without any interface imperfection.

Fig. 9. Comparison between the plastic response with and without agglomeration at two different CNT volume fractions. The stress–strain results for the non-uniform distribution of the CNT are the same theoretical estimations as shown in Fig. 8a. No imperfection at the MWNT–matrix interface has been assumed here.

modulus, the elastoplastic behavior of CNT nanocomposites could be greatly overestimated without considering the CNT agglomeration. Likewise to reflect the effect of interface imperfection on the nonlinear stress–strain relations of the composite more clearly, we have made another set of calculations with various c values. The results are shown in Fig. 10. The elastoplastic behavior of the entire composite is observed to soften with increasing magnitude of the sliding parameter, c. Fig. 11 shows the calculated results with various aspect ratio (length-to-diameter ratio), a, of MWNTs, ranging from 2 all the way to 104. For high values of a(a > 103) the MWNTs behave as infinitely long fibers and their response overlap. As a decreases to lower values, softening is observed. This reduction in strength continues until a = 10 but there is a reversal at a = 2. Here a non-zero value of the interface imperfection (c = 106 nm/MPa) has been assumed, along with a constant value of the carbon nanotube volume fraction (c1 = 5%). To analyze the size (radius) effect of MWNTs on the nonlinear elastoplastic response of the composite, we have made two separate calculations, one with a nearly perfect interface and the other with an imperfect interface, with c = 109 nm/MPa and 106 nm/MPa. The results are given in Figs. 12a and 12b, respectively. Here the dependence of the five elastic constants of the MWNTs has been taken from Shen and Li (2005a), who provided a list showing how the elastic parameters increase with decreasing MWNT radius. Fig. 12a shows that, with decreasing MWNT radius the overall elastoplastic response becomes stiffer. But with a strong interfacial imperfection as in Fig. 12b, the trend is reversed. The overall response of the composite material shows a softening effect with decreasing MWNT radius. This happens because of the fact that as the

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Fig. 10. The effect of MWNT-matrix interface imperfection on the overall stress–strain response of the two-phase composite material is shown here. With increasing values of the imperfection, c, the composite material becomes softer. A constant volume fraction of the CNT fibers under a constant level of agglomeration is assumed. The radius of the MWNT has been kept constant at 3.39 nm. The fibers are assumed to be infinitely long.

Fig. 11. The effect of different aspect ratio on the stress–strain behavior of the CNT-reinforced metal matrix nanocomposite is shown here. Fibers with very large aspect ratio, a > 103, behave as long fibers and their response overlap. As a is decreased, the response becomes softer, but again for a < 10 some hardening is visible. A constant volume fraction of the CNT fibers under a constant level of agglomeration is assumed. The radius of the MWNT has been kept constant at 3.39 nm.

radius of the MWNT decreases, the same value of the sliding parameter has a more pronounced effect on softening the overall response of the composite material. For small values of the MWNT radius, the effect of interface imperfection overcomes the hardening effect due to decrease in radius. Finally to illustrate the effect of CNT agglomeration on the yield strength of the nanocomposite further, we vary the volume concentration of the spherical domain, cI, from 0.2 to 1, while keeping the CNT volume concentration fixed, to calculate the flow stress at 0.2% proof strain. The results for the same two volume concentrations, c1 = 5% and 10% discussed above, are given in Fig. 13. In each case the yield strength of the CNT nanocomposite is seen to increase steadily with the increasing degree of uniformity. With 5% CNTs, the yield stress can change from 114 MPa to 147 MPa, and with 10% CNTs, it can increase from 117 MPa to nearly 180 MPa. This again shows the strong effect of the CNT agglomeration on the plastic strength of CNT-reinforced composites. 8. Concluding remarks Determination of the effective properties of CNT-reinforced composite is a highly complicated issue, for there are many factors that could affect the overall response. These include the interfacial load transfer condition, surface functionalization

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Fig. 12a. Under very small values of interface imperfection (which effectively leads to zero imperfection) with decreasing radius of the MWNT, the entire CNT-reinforced metal matrix nanocomposite is observed to harden. This happens because the modulus of the MWNT increases as the radius decreases (see Shen and Li, 2005a). A constant volume fraction of the CNT fibers under a constant level of agglomeration is assumed.

Fig. 12b. The stress–strain response of the CNT-reinforced composite for different values of the MWNT fiber radius under a non-zero interface imperfection. The composite softens with decreasing radius of the MWNT fibers. Even though the moduli of the CNT fibers increase with decreasing radius, the effect of the imperfection at the CNT–matrix interface is much stronger which leads to an overall softening of the entire composite material.

to improve the load transfer and CNT dispersion, and CNT waviness and agglomeration, among others. Depending on the processing conditions, some factors are more dominant than the others. In this article we have developed an idealized two-scale model to capture the overall elastoplastic response of carbon nanotube reinforced composites in which CNT agglomeration and interface conditions can play the key role. The large-scale model involves the pure matrix and the homogeneously distributed spherical domains, while the small-scale model considers this spherical domain that is comprised of the randomly oriented CNTs and matrix. The CNTs are modeled as transversely isotropic long fibers, with an aspect ratio that tends to infinity. The matrix phase is taken to be isotropic, both elastic and plastic. The developed model is explicit, and can be readily applied to examine the influence of CNT concentration, agglomeration, and interface imperfection. All of these factors are shown to have significant effect on both the elastic and plastic properties of CNT nanocomposites. The calculated results indicate that the formation of CNT agglomeration and an imperfect interface can seriously reduce the elastic stiffness and yield strength of the nanocomposite. To realize the full potential of CNT-reinforcement, improved processing that could lead to a more uniform distribution of CNTs and a better interface load transfer condition are the top priority. In retrospect it must be recognized that the developed theory is mainly intended for the plasticity of CNT/metal composites, and that idealizations have been adopted in this development. In reality CNT clustering in the fine scale can exhibit highly complex structure (such as CNT ropes) and CNTs are not well-separated straight rods, and in the coarse scale the

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555

Fig. 13. The increase in yield strength as the distribution of the CNTs within the composite goes from highly agglomerated to uniformly distributed configuration, for carbon nanotubes volume fraction of 5% and 10%. Zero interface imperfection is assumed here.

homogenized clusters are not exactly spherical and uniformly dispersed in the matrix. The adoption of Qu’s imperfect interface model, which can be considered as a simple case of a cohesive zone model, is another idealization to a finite-thickness interphase. The developed model is based on a continuum micromechanics approach and it is desirable to tailor it more specifically to CNT-reinforced materials. Incorporation of the interatomic potential – though involving a higher degree of computational complexity – is another desirable route to model the interface and interphase effects (e.g. Odegard et al., 2002, 2003; Jiang et al., 2006; Tan et al., 2007). Such an approach could link the linear elastic modulus and nonlinear behavior to parameters in the interatomic potentials (Zhang et al., 2002, 2004). Under large deformation defects in the form of Stone-Wales transformation could also occur, and this could eventually lead to fracture of CNTs (Jiang et al., 2004; Song et al., 2006). Implementation of these and other nanoscale characteristics could provide additional physical insights. These and other factors deserve to be implemented in a future theory. Acknowledgment This work was supported by the NSF Division of Civil, Mechanical and Manufacturing Innovation, Mechanics and Structure of Materials Program, under Grant CMS-0510409. Appendix A. Constants nLA, nA, gLA and gA in Eq. (13) for the CNT-containing composite This list was taken from Qiu and Weng (1990). In doing so, the original notations have been preserved for ease of cross reference. Note that k1, l1, n1, m1 and p1 are the transversely isotropic moduli of the carbon nanotubes.

1 h

nLA ¼

nA ¼

9l

1 h ð1Þ

9l

gLA ¼

gA ¼ ð1Þ

ð1Þ

ð1Þ

2d h

1 15l

ð1Þ

4ðk1 d

ð1Þ

1 30l

ð1Þ

ð1Þ

ð1Þ

i ð1Þ ð1Þ  n1 g ð1Þ þ l1 cð1Þ  2k1 h Þ þ ðn1 cð1Þ  2l1 h Þ ;

i ð1Þ  2ðg ð1Þ þ h Þ þ cð1Þ ;

ðk1 d

½d ð1Þ

ð1Þ

 l1 g ð1Þ Þ þ 2ðl1 d

ð1Þ

 l1 g ð1Þ Þ  ðl1 d

ð1Þ

 i 2 m p1 ð1Þ ð1Þ 1 ;  n1 g ð1Þ þ l1 cð1Þ  2k1 h Þ þ ðn1 cð1Þ  2l1 h Þ þ þ 5 eð1Þ f ð1Þ

þ 2ðg ð1Þ þ h Þ þ 2cð1Þ  ð1Þ

  1 1 1 ; þ 5 eð1Þ f ð1Þ

ðA:1Þ

where l ¼ cð1Þ d  2g ð1Þ h . In terms of the matrix moduli, the five elastic constants of the CNTs, and Eshelby’s S-tensor for a spheroidal inclusion (aligned along direction 1), the six transversely isotropic parameters that appear in Walpole’s notations are given by

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cð1Þ ¼ 1 þ ð1Þ

d

¼1þ

2ðk1  k0 Þ 2ðl1  l0 Þ ½ð1  v 0 ÞðS2222 þ S2233 Þ  2m0 S2211  þ ½ð1  v 0 ÞS1122  2m0 S1111 ; E0 E0 ðn1  n0 Þ 2ðl1  l0 Þ ½S2222  2m0 S1122  þ ½ð1  v 0 ÞS1122  m0 S1111 ; E0 E0

eð1Þ ¼ 1 þ

2ðm1  m0 Þ S2323 ; m0

f ð1Þ ¼ 1 þ

2ðp1  p0 Þ S1212 ; p0

g ð1Þ ¼

ð1Þ

h

¼

2ðk1  k0 Þ ðl1  l0 Þ ½ð1  v 0 ÞS1122  m0 S2211  þ ½S1111  2m0 S1122 ; E0 E0

ðn1  n0 Þ ðl1  l0 Þ ½S2211  m0 ðS2222 þ S2233 Þ þ ½ð1  m0 ÞðS2222 þ S2233 Þ  2m0 S2211 ; E0 E0

ðA:2Þ

where k0, l0, n0, m0 and p0 are the counterparts of k1, l1, n1, m1 and p1 for the isotropic matrix, given by k0 = j0 + l0/3, l0 = j0  2l0/3, and n0 = j0 + 4l0/3, and m0 = p0 = l0. With the S-tenor of an infinitely long circular fiber that is suitable for CNTs, we further have

cð1Þ ¼ 1 þ

k1  k0 ; k0 þ m0

g ð1Þ ¼ 0;

h

ð1Þ

¼

ð1Þ

d

l1  l0 ; 2ðk0 þ m0 Þ

ðk0 þ 2m0 Þðm1  m0 Þ ; 2m0 ðk0 þ m0 Þ k1  k0 ¼1þ : k0 þ m0

eð1Þ ¼ 1 þ

¼ 1; l

ð1Þ

f ð1Þ ¼ 1 þ

p1  p0 ; 2p0

ðA:3Þ

Appendix B. Explicit evaluation of Qu’s H-tensor The formula to derive the expression for the components of the H-tensor is defined in Appendix C of Qu (1993b). Considering only the tangential imperfection, c and making the normal imperfection, b zero, one can write the imperfection compliance tensor H as,

Hijkl ¼ cðPijkl  Q ijkl Þ;

ðA:4Þ

where

Pijkl ¼

Q ijkl with, n ¼

3 16p

Z p  Z 2p 0

 ^j n ^ l þ djk n ^i n ^ l þ dil n ^k n ^ j þ djl n ^k n ^ i Þ  n1 dh sin /d/; ðdik n

0

 Z p  Z 2p 3 ^j n ^k n ^ l Þ  n3 dh sin /d/ ^i n ¼ ðn 4p 0 0

ðA:5Þ

  cos / sin /  cosh sin /  sin h ; : ; a1 a a

ðA:6Þ

pffiffiffiffiffiffiffiffi ^ i n^i ; n

^i ¼ n

Aspect ratio, a is defined as the length-to-diameter ratio (a1/a), which signifies that for long fibers a approaches infinity and for spheres it is equal to 1. Using this definition the value of n is computed as

n¼

1 aa

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 cos2 / þ a2 sin /:

The nonzero terms of Pijkl and Qijkl can be found to be

P1111

" # pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 1 a a2  1 1 1  ¼  : : pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin 2a a2  1 a a a2  1

P2222 ¼

" # pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 a a2  2 a2  1 a 1 : þ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin 2a a 2ða2  1Þ a2  1 2ða2  1Þ

ðA:7Þ

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P1212 ¼

" ( )# pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 a a2  1 a2  2 a a2  1 1 1 1 p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin ;  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin   8a 2ða2  1Þ a a a a2  1 a2  1

P3333 ¼ P2222 ; P2323 ¼

Q 1111

557

1 P2222 2

and P1313 ¼ P1212 ;

ðA:8Þ

" pffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 3 2a2 þ 1 3 a2  1 1  ¼  sin ; 2a aða2  1Þ2 ða2  1Þ5=2 a

Q 2222 ¼

" pffiffiffiffiffiffiffiffiffiffiffiffiffiffi# 9 a3 ð2a2  5Þ a3 ða2  4Þ a2  1 1 ;  a þ sin 2 5=2 2 8a a 2ða  1Þ 2ða  1Þ

Q 1122 ¼

" # pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 aða2 þ 2Þ a2  1 aða2  4Þ 1 ;  sin þ 4a ða2  1Þ5=2 a ða2  1Þ2

Q 3333 ¼ Q 2222 ;

Q 1212 ¼ Q 1133 ¼ Q 1313 ¼ Q 1122

and Q 2233 ¼ Q 2323 ¼

1 : Q 3 2222

ðA:9Þ

These expressions are useful to the application of Qu’s theory with an imperfect interface. References Andrews, R., Jacques, D., Minot, M., Rantell, T., 2002. Fabrication of carbon multiwalled nanotube/polymer composites by shear mixing. Macromol. Mater. Eng. 287, 395–403. Anumandla, V., Gibson, R.F., 2006. A comprehensive closed form micromechanics model for estimating the elastic modulus of nanotube-reinforced composites. Compos. Part A 37, 2178–2185. Ashrafi, B., Hubert, P., 2006. Modeling the elastic properties of carbon nanotube array/polymer composites. Compos. Sci. Tech. 66, 387–396. Barai, P., Weng, G.J., 2008a. Mechanics of creep resistance in nanocrystalline solids. Acta Mech. 195, 327–348. Barai, P., Weng, G.J., 2008b. The competition of grain size and porosity in the viscoplastic response of nanocrystalline solids. Int. J. Plasticity 24, 1380–1410. Barai, P., Weng, G.J., 2009. 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