2 Effective Elastic Properties of a Novel Continuous Fuzzy Fiber-Reinforced Composite with Wavy Carbon Nanotubes
M. C. Ray and S. I. Kundalwal
ABSTRACT
This chapter deals with the investigation of the effect of carbon nanotube (CNT) waviness on the effective elastic properties of a novel continuous fuzzy fiber-reinforced com-posite (FFRC). Effective elastic properties of this composite estimated by the mechanics of materials (MOM) approach have been compared with those predicted by the Mori–Tanaka (MT) method. The interphase between a CNT and the polymer matrix which models the nonbonded van der Waals interaction between them is also considered for estimating the effective elastic properties of the FFRC. The effect of CNT waviness on the effective elastic properties of the FFRC is investigated when wavy CNTs are coplanar with either of the two mutually orthogonal planes. Development of wavy CNTs and the micromechanics models are presented in detail.
2.1 INTRODUCTION
Graphene, a monolayer of sp2-hybridized carbon atoms arranged in a two-dimensional lattice, has emerged as an exotic material in recent years owing to its exceptional charge transport, thermal, electrical, optical, and mechani-cal properties. Graphene and its derivatives are being stud-ied in nearly every field of science and engineering. Recent
progress has shown that graphene-based materials can have a profound impact on electronic and optoelectronic devices, chemical sensors, nanocomposites, and energy storage (Singh et al., 2011). However, until 2004, single-layer graphene was believed to be thermodynamically unstable under ambi-ent conditions (Mermin, 1968; Landau and Lifshitz, 1980). Novoselov et al. (2004) successfully identified single layers of graphene in a simple tabletop experiment. This revolutionary discovery has added a new dimension of research in the fields of physics, chemistry, biotechnology, and materials science. The “thinnest” known material, graphene exhibits excellent multifunctional properties (Novoselov et al., 2004; Huang et al., 2006; Balandin et al., 2008; Bao et al., 2009; Blake et al., 2008; Ghosh et al., 2008; Li et al., 2008; Wang et al., 2008; Jiang et al., 2009a,b; Lee et al., 2009; Timo et al., 2009; Tsoukleri et al., 2009; Wang et al., 2009; Dreyer et al., 2010; Seol et al., 2010; Milowska et al., 2012). Table 2.1 summarizes the mechanical, thermal, and electrical properties of single and multilayer graphene. Recently, graphene has been used as an alternative carbon-based reinforcement in the preparation of polymer nanocomposites demonstrating improved multi-functional properties (Stankovich et al., 2006; Ramanathan et al., 2008; Ansari and Giannelis, 2009; Bose et al., 2010; Fan et al., 2010; Lv et al., 2010; Zhang et al., 2010; Zhao et al., 2010; Kuila et al., 2011; Potts et al., 2011; Song et al., 2014;
CONTENTS
Abstract ....................................................................................................................................................................................... 172.1 Introduction ....................................................................................................................................................................... 172.2 Architecture of the FFRC Containing Wavy CNTs .......................................................................................................... 202.3 Models of Wavy CNTs ...................................................................................................................................................... 222.4 Effective Elastic Properties of the Continuous FFRC Containing Wavy CNTs ............................................................... 22
2.4.1 MOM Approach ..................................................................................................................................................... 222.4.1.1 Effective Elastic Properties of the PMNC .............................................................................................. 222.4.1.2 Effective Elastic Properties of the CFF .................................................................................................. 262.4.1.3 Effective Elastic Properties of the FFRC................................................................................................ 27
Yang et al., 2014). Graphene is expected to play an important role in the fabrication of sensor, fuel cell, nanoelectronic, and bioelectronic devices in the near future (Geim and Novoselov, 2007; Kauffman and Star, 2010). Graphene is also capable of replacing metal conductors in electronic and electrical devices due to its excellent electrical conductivity and mechanical flexibility (Kim et al., 2009; Li et al., 2009; Bae et al., 2010). Ongoing research shows that graphene can replace brittle and chemically unstable indium in oxide, in flexible displays, and touch screens. Therefore, graphene, as an electronic circuit material, is considered to be potentially superior to other car-bon-based nanofillers (Huang et al., 2007; Wu et al., 2008; Park et al., 2010).
The research on the synthesis of molecular carbon struc-ture by an arc-discharge method for evaporation of carbon led to the discovery of an extremely thin needle-like graphitic car-bon molecule (Iijima, 1991). While examining under an elec-tron microscope at the level of atomic resolution, Iijima (1991) observed that such needle-like carbon materials are seamless coaxial tubes made of carbon atom sheets. The thickness of the carbon atom sheet was less than a nanometer while the separation between the walls of the tubes was found to be 0.34 nm. The outer diameter of such needle-like material is in the range of a few nanometers. Iijima (1991) named such nee-dle-like carbon materials as multiwalled carbon nanotubes
(MWCNT). Within a couple of years, Iijima and Ichihashi (1993) discovered the synthesis of single-walled carbon nano-tube (SWCNT). The other manufacturing processes, such as laser ablation, chemical vapor deposition (CVD) and plasma-enhanced CVD are being employed to synthesize CNTs on a large scale.
Researchers probably thought that CNTs may be useful as nanoscale fibers for developing novel nanocomposites, and this conjecture motivated them to accurately predict the prop-erties of CNTs. Hence, since the discovery of CNTs, research-ers have been carrying out extensive studies to estimate their elastic properties. Both experimental measurements and theo-retical studies confirmed that SWCNTs have axial Young’s modulus in the terapascal range (Treacy et al., 1996; Lu, 1997; Krishnan et al., 1998; Lourie and Wagner, 1998; Popov et al., 2000; Jin and Yuan, 2003; Shen and Li, 2004, 2005; Liu et al., 2005; Tsai et al., 2010). Table 2.2 summarizes the elastic prop-erties of different types of CNTs in which the symbols have their usual meaning. The quest for utilizing such exceptional mechanical properties of CNTs and their high aspect ratio led to the opening of an emerging area of research on the devel-opment of CNT-reinforced composites (Odegard et al., 2003; Gao and Li, 2005; Seidel and Lagoudas, 2006; Jiang et al., 2009a,b; Meguid et al., 2010; Tsai et al., 2010; Wernik and Meguid, 2010; Wernik et al., 2012).
TABLE 2.1Mechanical, Thermal, and Electrical Properties of Graphene and Graphene Oxide-Based Materials
References Method Material Properties
Huang et al. (2006) Molecular mechanics (using second-generation Brenner potential)
Single-layer graphene, T = 0.0574 nm E = 4.23 TPa and ν = 0.397 (uniaxial tension)
Single-layer graphene, T = 0.0678 nm E = 3.58 TPa and ν = 0.397 (uniaxial stretching)
Single-layer graphene, T = 0.0811 nm E = 2.99 TPa and ν = 0.397 (equibiaxial stretching)
Balandin et al. (2008) Confocal micro-Raman spectroscopy Single-layer graphene (4.84 ± 0.44) × 103 to (5.30 ± 0.48) × 103 W/mK at RT
Ghosh et al. (2008) Confocal micro-Raman spectroscopy Suspended graphene flake ~3080–5150 W/mK at RT
Lee et al. (2008) AFM Monolayer graphene E = 1 ± 0.1 TPa, σint = 130 ± 10 GPa at εint = 0.25
Bao et al. (2009) Experimental Single-layer graphene CTE = −7 × 10−6 K−1 at 300 K
Jiang et al. (2009a,b) Nonequilibrium Green’s function approach
Single-layer graphene CTE = −6 × 10−6 K−1 at 300 K
Lee et al. (2009) AFM Monolayer graphene E = 1.02 TPa and σ = 130 GPa
Bilayer graphene E = 1.04 TPa and σ = 126 GPa
Trilayer graphene E = 0.98 TPa and σ = 101 GPa
Tsoukleri et al. (2009) Raman Graphene Strain ~1.3% in tension;strain ~0.7% in compression
Timo et al. (2009) Electrical four-point measurement Reduced graphene oxide flake 0.14–2.87 W/mK
6.2 × 102 to 6.2 × 103 Ω−1 m−1
Seol et al. (2010) Thermal measurement Single-layer (suspended) 3000–5000 W/mK at RT (suspended)
Single-layer (on SiO2 support) 6000 W/mK at RT (on a silicon dioxide support)
Milowska et al. (2012)
DFT Monolayer graphene E = 1.05 TPa, ν = 0.11, K = 0.47 TPa, and G = 0.46 TPa
Note: AFM, atomic force microscopy; DFT, density functional theory.
19Effective Elastic Properties of a Novel Continuous Fuzzy Fiber-Reinforced Composite
It has been experimentally observed that CNTs are actu-ally curved cylindrical tubes with a relatively high aspect ratio (Shaffer and Windle, 1999; Qian et al., 2000; Vigolo et al., 2000; Berhan et al., 2004; Tsai et al., 2011; Tyson et al., 2011). Images analyzed by Qian et al. (2000) and Berhan et al. (2004) have been illustrated in Figure 2.1a and b, which clearly demonstrate that the embedded CNTs remains highly curved when dispersed in the polymer matrix. It is hypothe-sized that their affinity to become curved is due to their high aspect ratio and the associated low bending stiffness. Many research investigations revealed that CNT curvature reduces the effective elastic properties of the CNT-reinforced com-posite (Fisher et al., 2002; Berhan et al., 2004; Shi et al., 2004; Anumandla and Gibson, 2006). The effect of CNT curvature on the polymer matrix nanocomposite (PMNC) stiffness has been investigated by Pantano and Cappello (2008). They concluded that in the presence of weak bond-ing, the enhancement of the nanocomposite stiffness can be achieved through the bending energy of CNTs rather than through the axial stiffness of CNTs. Tsai et al. (2011) studied the effects of CNT waviness and its distribution on the effec-tive nanocomposite stiffness. In their study, elastic moduli were overestimated when CNT aspect ratio or waviness fol-lowed a symmetric distribution. Farsadi et al. (2012) devel-oped a three-dimensional finite element model to investigate
the influence of CNT waviness on the elastic moduli of the CNT-reinforced composite.
Additionally, with CNT waviness, the manufacturing of unidirectional continuous CNT-reinforced composites in large scale has to encounter some challenging difficulties, such as the agglomeration of CNTs, the misalignment, and the difficulty in manufacturing very long CNTs. Further research on improving the out-of-plane properties of com-posites and the better application of short CNTs led to the growth of CNTs on the surface of advanced fibers. For exam-ple, Bower et al. (2000) have grown aligned CNTs on the substrate surface using high-frequency microwave plasma-enhanced CVD. Veedu et al. (2006) demonstrated that the remarkable improvements in the interlaminar fracture tough-ness, hardness, delamination resistance, in-plane mechani-cal properties, damping, and thermoelastic behavior of the laminated composite can be obtained by growing MWCNTs of about 60 µm long on the surface of fibers. Qiu et al. (2007) presented a method to fabricate multifunctional multiscale composites through an effective infiltration-based vacuum-assisted resin transfer-molding process. Ray (2010) devel-oped a shear lag model of a novel hybrid smart composite to analyze the load transferred to the coated piezoelectric fibers from the CNT-reinforced matrix in the absence and the presence of the electric field. Such a fiber augmented with
TABLE 2.2Comparison of Engineering Constants for Different Types of CNTs
radially grown CNTs on its circumferential surface is being called “fuzzy fiber” (Garcia et al., 2008; Mathur et al., 2008; Chatzigeorgiou et al., 2011), and the resulting composite is called fuzzy fiber-reinforced composite (FFRC). Recently, a novel continuous FFRC reinforced with zigzag SWCNTs and carbon fibers was proposed by the authors (Kundalwal and Ray, 2011).
The distinct constructional feature of the unidirectional continuous FFRC studied by the authors is that the uni-formly spaced straight CNTs are radially grown on the cir-cumferential surface of carbon fibers. However, in practice, such radially grown CNTs may be wavy. Since CNT wavi-ness influences the effective elastic properties of the CNT-reinforced composite, CNT waviness may also affect the effective elastic properties of the FFRC. Such an effect cannot be predicted from the above-mentioned existing studies as the constructional feature of the FFRC is distinct and not simi-lar to CNT-reinforced composites studied elsewhere. Thus, a critical issue that is yet to be addressed is to examine the effect of waviness of CNTs on the effective elastic properties of the FFRC. In this chapter (Kundalwal and Ray, 2013), an endeavor has been made to investigate the effect of waviness of radially grown CNTs on the effective elastic properties of the continuous FFRC.
2.2 ARCHITECTURE OF THE FFRC CONTAINING WAVY CNTs
Figure 2.2 shows a schematic of a lamina of the FFRC containing wavy CNTs being studied here. The novel constructional feature of the FFRC is that wavy CNTs of equal length are radially and uniformly grown on the
circumferential surface of carbon fiber. Such a resulting fuzzy fiber is shown in Figure 2.3. Here, wavy CNTs are modeled as sinusoidal solid CNTs (Fisher et al., 2002; Berhan et al., 2004; Anumandla and Gibson, 2006; Pantano and Cappello, 2008; Tsai et al., 2011), while at any location along the length of CNT, it is considered as transversely isotropic (Jin and Yuan, 2003; Shen and Li, 2004; Shi et al., 2004; Liu et al., 2005; Li and Guo, 2008; Song et al., 2009; Tsai et al., 2010) with CNT axis being the axis of symmetry. Also, the CNTs are grown on the circumferential surface of carbon fiber in such a way that their axes of transverse isotropy at their roots are normal to the circumferential surface of carbon fiber. The radially
2A
T/2
100 nm Mag = 50.00 K X300 nm
1.21.4
10.80.6
0 0.2 0.4 0.6Length scale (μm)
Nan
otub
e wt%
0.8 1 1.2 1.4
(a) (b)
FIGURE 2.1 Images of CNT-reinforced polymers showing that the embedded CNTs exhibit significant curvature with the polymer. (a) Transmission electron microscopy image of MWCNT (1 wt%) in polystyrene. (Reproduced from Qian, D. et al. 2000. Applied Physics Letters 76(20):2868–70. With permission.) (b) Scanning electron microscopy image of CNT showing straight and sinusoidal segments. (Reproduced from Berhan, L., Yi, Y. B., and Sastry, A. M. 2004. Journal of Applied Physics 95(9):5027–5034. With permission.)
2, z
3, y
Carbon fiber Polymer Wavy CNTs CFF
1, x
FIGURE 2.2 Schematic diagram of a lamina made of the FFRC containing wavy CNTs. (Reproduced from Kundalwal, S. I. and Ray, M. C. 2013. ASME Journal of Applied Mechanics 80:021010. With permission.)
21Effective Elastic Properties of a Novel Continuous Fuzzy Fiber-Reinforced Composite
grown wavy CNTs eventually reinforce the polymer matrix surrounding the carbon fiber along the direction transverse to the length of carbon fiber. Thus, the combination of the fuzzy fiber with wavy CNTs and the polymer matrix can be viewed as a composite fuzzy fiber (CFF) in which a carbon fiber is embedded in wavy CNT-reinforced PMNC. It may be noted that the variations of the constructional feature of the CFF can be such that the planes of wavy CNTs are
coplanar with the 2–3 (2′–3′) plane or the 1–3 (1′–3′) plane as shown in Figure 2.4a and b, respectively. Note that the plane passing through the axis of carbon fiber is the 1–3 or 1′–3′ plane, while the plane transverse to the carbon fiber is the 2–3 or 2′–3′ plane. For the various steps involved in the micromechanical modeling of the continuous FFRC containing wavy CNTs, the schematic shown in Figure 2.5 will be followed.
Wavy CNTs
Carbon fiber
FIGURE 2.3 Fuzzy fiber with wavy CNTs radially grown on its circumferential surface.
Polymer CNTs Carbon fiber
PMNC
CFF
FFRC
FIGURE 2.5 Modeling of the FFRC and its phases. (Reproduced from Kundalwal, S. I. and Ray, M. C. 2013. ASME Journal of Applied Mechanics 80:021010. With permission.)
2
2R
2′
3′
2R
2′
2
3 3
1, 1′
1, 1′
3′
3
ϕ
ϕ
Carbon fiberCarbon fiber
Wavy CNTs
Wavy CNTs
Wavy CNTs
Polymer
Polymer
Transverse cross-section
Transverse cross-section
Longitudinal cross-sections
Longitudinal cross-section
Polymer
Polymer
Wavy CNTs
2
2L
2L
2a
2a
Carbon fiber
(a)
(b)Carbon fiber
FIGURE 2.4 Transverse and longitudinal cross sections of the CFF in which wavy CNTs are coplanar with either the 2–3 (2′–3′) or the 1–3 (1′–3′) plane. (Reproduced from Kundalwal, S. I. and Ray, M. C. 2013. ASME Journal of Applied Mechanics 80:021010. With permission.)
As mentioned in Section 2.2, a CNT wave is assumed to have a sinusoidal shape. Considering a carbon fiber and an unwound PMNC lamina while this lamina is composed of sinusoidally wavy CNTs which are coplanar with either the 2–3 (2′–3′) plane or the 1–3 (1′–3′) plane as shown in Figure 2.6, the CFF can be viewed to be formed by wrapping the carbon fiber with the unwound PMNC lamina as shown in Figure 2.7. As shown in Figure 2.8, the representative vol-ume element (RVE) is divided into infinitesimally thin slices
of thickness dy. Averaging the effective elastic properties of these slices over the length (Ln) of the RVE (i.e., the thickness of the unwound PMNC lamina), the homogenized effective elastic properties of the unwound PMNC can be estimated. Each slice can be treated as an off-axis unidirectional PMNC lamina, and its elastic coefficients can be determined by trans-forming the elastic coefficients of the corresponding specially orthotropic lamina.
Now, these wavy CNTs are characterized by
z = = =A y) or x A
nLn
sin( sin ( y);ω ω ω π
(2.1)
corresponding to the plane of CNT waviness being copla-nar with the 2–3 plane and the 1–3 plane, respectively. In Equation 2.1, A and Ln are the amplitude of CNT wave and the linear distance between the CNT ends, respectively, and n represents the number of CNT waves. The running length of CNT (Lnr) can be expressed in the following form:
L A y) dynr2
Ln
= +∫ 1 2 2
0
ω ωcos (
(2.2)
in which the angle θ shown in Figure 2.8 is given by
tan cos ( y)θ
ωω θ
ωω=
= =dz
dy Acos ( y) or tan =
dxdy A
corresponding to the plane of CNT waviness being copla-nar with the 2–3 plane and the 1–3 plane, respectively. Note that for a particular value of ω, the value of θ varies with the amplitude of CNT wave.
2.4 EFFECTIVE ELASTIC PROPERTIES OF THE CONTINUOUS FFRC CONTAINING WAVY CNTs
This section deals with the procedures for employing the two micromechanics methods; namely, the mechanics of materi-als (MOM) approach and the Mori–Tanaka (MT) method for estimating the effective elastic properties of the FFRC con-taining wavy CNTs.
2.4.1 MoM aPProach
This section presents the derivation of analytical microme-chanics models based on the MOM approach for estimating the effective elastic properties of the PMNC material, CFF, and FFRC.
2.4.1.1 Effective Elastic Properties of the PMNCIn order to estimate the effective elastic properties of the PMNC material surrounding the carbon fiber, the consideration of the nonbonded van der Waals interaction between an atom of a CNT and an atom of the polymer matrix is an important issue.
Carbon fiber Carbon fiberPolymer
Polymer
Wavy CNTs
Unwound PMNC lamina
Wavy CNTs
3
2′ 3′
ϕ
3
2
1
2
1
Carbon fiber(b)
(a)
Carbon fiberPolymer
Polymer
Wavy CNTs
Unwound PMNC lamina
3
2′ 3′
ϕ
3
2
1
2
1
FIGURE 2.6 Unwound lamina of the PMNC containing wavy CNTs.
InterphaseInterphase
Carbon fiber
Unwound PMNC laminaInterphase
Carbon filter
Wavy CNTs
Wavy CNTs2
2
3
2′
3′
3
Polymer
ϕ
FIGURE 2.7 Transverse cross sections of the CFF with unwound and wound PMNC. (Reproduced from Kundalwal, S. I. and Ray, M. C. 2013. ASME Journal of Applied Mechanics 80:021010. With permission.)
23Effective Elastic Properties of a Novel Continuous Fuzzy Fiber-Reinforced Composite
The atomic structure of CNTs consisting of sp2-hybridized carbon atoms hinders the formation of the strong covalent bonds between the carbon atoms of a CNT and the surround-ing polymer matrix. Naturally, the bonding between a CNT and the surrounding polymer takes place through van der Waals noncovalent interactions which are obviously weaker than covalent bonding between carbon–carbon bonds (Li and Chou, 2003). Following the previous research (Odegard et al., 2003; Seidel and Lagoudas, 2006; Tsai et al., 2010; Li et al., 2011a,b), an interphase may be considered between a CNT and the polymer matrix which characterizes the nonbonded van der Waals interaction, and researchers treated this inter-phase as an equivalent solid continuum. The effective elastic properties of such interphase resembling a solid continuum can be determined by molecular dynamics simulation and are readily available in the open literature (Tsai et al., 2010). Thus, because of the consideration of this interphase between a CNT and the polymer matrix, the micromechanical model of the unwound PMNC by the MOM approach turns out to be a three-phase MOM model.
It may be noted that the effective elastic properties at any point in the unwound PMNC lamina containing sinusoidally
wavy CNTs where CNT axis makes an angle θ with the 3 (3′)–axis can be approximated by transforming the effective elastic properties of the unwound lamina of PMNC contain-ing straight CNTs. Figure 2.9 illustrates the cross sections of the RVE of the unwound lamina with straight CNTs. The three-phase micromechanics model based on the MOM approach derived by Ray (2010) has been modified to predict
2(a) Interphase
Interphase
θ
CNT3
dy
A3, y
Polymer
Carbon fiberLn
λ
2, z
1(b) Interphase
Interphase
θ
CNT3
dy
A3, y
Polymer
Carbon fiber
Ln
λ
1, x
FIGURE 2.8 RVEs of the unwound PMNC material containing a wavy CNT radially grown on the carbon fiber. (a) CNT is coplaner with yz plane, (b) CNT is coplaner with xy plane. (Reproduced from Kundalwal, S. I. and Ray, M. C. 2013. ASME Journal of Applied Mechanics 80:021010. With permission.)
2 Polymer matrix
Polymer matrix
2CNT fiber
CNT fiber
Interphase
Interphase
13
FIGURE 2.9 Cross sections of the RVE of the unwound PMNC material. (Reproduced from Kundalwal, S. I. and Ray, M. C. 2011. International Journal of Mechanics and Materials in Design 7(2):149–166. With permission.)
the effective elastic constant matrix [Cnc] of the unwound PMNC material with straight CNTs. On the basis of the prin-cipal material coordinate (1–2–3) axes shown in Figure 2.2, the constitutive relations for the constituent phases of the PMNC are given by
{ } [ ]{ }; rrσ = =C n, i, and pr r� (2.3)
where the state of stress vector, the state of strain vector and the elastic coefficient matrix of the rth phase are
{ } [ ] ,σ σ σ σ σ σ σr r r r r r r T= 1 2 3 23 13 12 { } [ ]� � � � � � �r r r r r r r T= 1 2 3 23 13 12 and
[C ]r
r r r
r r r
r r r
r
C C C
C C C
C C C
C=
11 12 13
12 22 23
13 23 33
44
0 0 0
0 0 0
0 0 0
0 0 0 0 00
0 0 0 0 0
0 0 0 0 055
66
C
C
r
r
.
In Equation 2.3, the superscripts n, i, and p denote, respec-tively, CNT, interphase, and polymer matrix. For the con-stituent phase denoted by r, σ1
r , σ2r , and σ3
r are the normal stresses along the principal material coordinate axes 1, 2, and 3, respectively; �1
r , �2r , and �3
r are the corresponding normal strains; σ12
r , σ13r , and σ23
r are the shear stresses; �12r , �13
r , and �23r
are the shear strains; and Cijr (i, j = 1, 2, 3,…, 6) are the elastic
coefficients.Several researchers (Gao and Li, 2005; Li and Chou,
2003, 2009; Odegard et al., 2003; Thostenson and Chou, 2003; Seidel and Lagoudas, 2006; Jiang et al., 2009a,b; Li et al., 2011b) assumed a perfectly bonding condition between a CNT and the polymer matrix. Esteva and Spanos (2009) emphatically reported that the imperfect bonding does not affect the effective longitudinal Young’s modulus of the CNT-reinforced polymer matrix composite and marginally affects the transverse elastic properties of the composite for high-volume fraction (>0.8) of CNTs. In their extensive research, Tsai et al. (2010) investigated the effect of the inter-phase between a CNT and the polymer matrix formed due to the nonbonded van der Waals interaction for estimating the effective elastic properties of CNT-reinforced polymer matrix composite. They found that the interphase margin-ally enhances the effective Young’s modulus of the com-posite transverse to CNT fiber over that of the composite without the consideration of the interphase. Hence, in the MOM approach being presented here, it is assumed that CNT is perfectly bonded to the neighboring phases. The iso-field conditions and the rules of mixture for satisfying the perfect bonding conditions between a CNT and the neigh-boring phases can be expressed as (Smith and Auld, 1991; Benveniste and Dvorak, 1992; Ray, 2010).
σσ
σσσ
σσ
σσ
1
2
3
23
13
12
1
2
3
23
n
n
n
n
n
n
i
i
i
i
� �
=
113
12
1
2
3
23
13
12
i
i
p
p
p
p
p
pσ
σσ
σσσ
=
�
=
σσ
σσσ
1
2
3
23
13
12
nc
nc
nc
nc
nc
nc
�
(2.4)
and
v vn
n
n
n
n
n
n
i
i
i
i
�
�
�
�
�
�
�
�
1
2
3
23
13
12
1
2
3σ σ
+223
13
12
1
2
3
23
13
12
i
i
i
p
p
p
p
p
p
p
v
�
�
�
�
�
�
�
+σ
=
�
�
�
�
�
1
2
3
23
13
12
nc
nc
nc
nc
nc
nc
σ
(2.5)
In Equation 2.5, vn, vi, and vp represent the volume frac-tions of CNT fiber, interphase, and polymer material, respec-tively, present in the RVE of the PMNC, and are determined in Section 2.5. Also, the superscript nc represents the unwound PMNC material containing straight CNTs. Substituting Equation 2.3 into Equations 2.4 and 2.5, the stress and strain vectors in the unwound PMNC material containing straight CNTs can be expressed in terms of the corresponding stress and strain vectors of the constituent phases as follows:
25Effective Elastic Properties of a Novel Continuous Fuzzy Fiber-Reinforced Composite
[ ]C vC C C
3 p
p p p
=
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
13 23 33
=
,
[C ]4
11 12 13
12 22 23
0 0 0
0 0 0
0 0 1 0 0 0
0
C C C
C C C
n n n
n n n
00 0 0 0
0 0 0 0 0
0 0 0 0 0
44
55
66
5
11
C
C
C
C C
n
n
n
i
=
,
[C ]
112 13
12 22 23
44
55
66
0 0 0
0 0 0
0 0 1 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
i i
i i i
i
i
C
C C C
C
C
Cii
p p p
p p p
C C C
C C C
=
,
[C ]6
11 12 13
12 22 23
0 0 0
0 0 0
00 0 1 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
44
55
66
1
C
C
C
p
p
p
,
[V ]] =
v
v
v
v
v
n
n
n
n
n
0 0 0 0 0
0 0 0 0 0
0 0 1 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
=
,
[V ]2
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
v
v
v
v
v
i
i
i
1
1
=
and
v
v
v
v
p
p
p
p
[V ]3
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 00
0 0 0 0 0 vp
(2.10)
Using Equations 2.7 through 2.9, the local strain vectors { }�n and { }�p can be expressed in terms of the composite strain { }�nc and subsequently, using them in Equation 2.6, the following constitutive relation between the states of stresses and strains at any point in the unwound PMNC material con-taining straight CNTs is obtained:
{ } [C ]{ }nc ncσ = �nc (2.11)
where the effective elastic coefficient matrix [Cnc] of the lam-ina of the unwound PMNC containing straight CNTs is given by
[C ] [C ][V ] [C ][ ]nc = +− −1 5
17
1V6 (2.12)
in which
[ ] [C ] [C ][C ] [ ],
[V ] [V ] [V ][C ] [C ],
[V ] [V
C C7 6= +
= +
=
−
−
3 2 51
4 3 2 51
6
5 1]] [V ][C ] [C ], and
[V ] [V ] [ ][ ] [C ]
+
= +
−
−
4 61
4
6 41
6V C1 4
(2.13)
Once [Cnc] is computed by Equation 2.12, the effective elastic coefficients (Cij
NC) at any point in the unwound PMNC lamina where CNT is inclined at an angle θ with the 3 (3′)–axis can be derived in a straightforward manner by employing the appropriate transformation law. Thus, if the plane of wavy CNTs is coplanar with the 2–3 (2′–3′) plane, the effective elastic coefficients (Cij
NC) at any point in the unwound PMNC lamina are given by
Similarly, if the plane of wavy CNTs is coplanar with the 1–3 (1′–3′) plane, then the effective elastic coefficients I(cIij
NC) at any point in the unwound PMNC lamina where CNT is inclined at an angle θ with the 3 (3′)–axis are given by
C C k C C C k
C C k C
NC nc 4 nc nc nc 2
NC nc 2
11 11 334
13 552
12 12 2
1 2 2 1= + + +
= +
( ) ,
332
13 11 33 552
134
22
1
4 1 1
nc
NC nc nc nc 2 nc 4
NC
C C C C k C k
C
,
( ) ( ),
C
= + − + +
= 222 23 122
23
33 114
33 13
1
1 2
nc NC nc nc 2
NC nc nc 4 nc
C C C k
C C C k C
, ,
(
= +
= + + + 22 1
1
2
552
44 44 662
55 11 33 13
C k
C C k C
C C C C
nc 2
NC nc 2 nc
NC nc nc
) ,
,
(
= +
= + − nnc nc 2 nc
NC nc nc 2
C k C k
and C C C k
− + +
= +
2 1 1
1
552
554 4
66 442
66
) ( ),
(2.15)
It is now obvious that the effective elastic properties of the unwound PMNC lamina with wavy CNTs vary along the length of CNT as the value of θ varies over the length of CNT. The average effective elastic coefficient matrix [C ]NC of the lamina of such unwound PMNC material containing wavy CNTs can be obtained by averaging the transformed elastic coefficients (C )ij
NC over the linear distance between CNT ends as follows (Hsiao and Daniel, 1996):
[C ] [C ]NC NC= ∫1
0L
dyn
Ln
(2.16)
It may also be noted that when the carbon fiber is viewed to be wrapped by such unwound PMNC lamina, the matrix [C ]NC provides the effective elastic properties at a point located in the PMNC where CNT axis (3′–axis) is oriented at an angle ф with the 3–axis in the 2–3 plane. Hence, at any point in the PMNC surrounding the carbon fiber, the effective elastic coefficient matrix [C ]PMNC of the PMNC with respect to the 1–2–3 coordinate system turns out to be location depen-dant and can be determined by the following transformations:
[C ] [ ] [ ][T]PMNC = − −T CT NC 1 (2.17)
where
[T] =−
− −−
1 0 0 0 0 0
0 0 0
0 0 0
0 2 2 0 0
0 0 0 0
0 0 0 0
2 2
2
2 2
m n mn
n m mn
mn mn m n
m n
n m
2
with m = cosф and n = sinф.
From Equation 2.17, it is obvious that the effective elastic properties at any point of the PMNC surrounding the carbon fiber with respect to the principal material coordinate (1–2–3) axes of the FFRC vary over an annular cross section of the PMNC phase of the RVE of the CFF. The volume average of these effective elastic properties [C ]PMNC over the volume of the PMNC can be treated as the homogenized effective elastic properties of the PMNC with sinusoidally wavy CNTs sur-rounding the carbon fiber. However, without loss of general-ity, it may be considered that the volume averages of these effective elastic properties [C ]PMNC over the volume of the PMNC can be treated as the homogenized effective elastic properties [CPMNC] of the PMNC material surrounding the carbon fiber with respect to the 1–2–3 coordinate axes of the FFRC and are given by
[ ]( )
[C ]CR a
r dr dPMNC2 2
PMNC
a
R
=− ∫∫1
0
2
πθ
π
(2.18)
Thus, the effective constitutive relations for the PMNC material with respect to the principal material coordinate (1–2–3) axes of the FFRC can be expressed as
{ } [C ]{ }PMNC PMNCσ = �PMNC (2.19)
2.4.1.2 Effective Elastic Properties of the CFFThe effective elastic properties of the CFF can be predicted by estimating the effective elastic properties of a lamina of continuous unidirectional fiber reinforced composite in which the carbon fiber is the reinforcement and the matrix phase is the PMNC material. The cross sections of the RVE of such lamina have been illustrated in Figure 2.10. Here, the length of carbon fiber aligns with the 1–direction. The MOM approach derived in Section 2.4.1.1 is augmented to estimate the effective elastic properties of the CFF. Similar to Equations 2.4 and 2.5, the iso-field conditions and the rules of mixture for the RVE shown in Figure 2.10 can be written as
2 PMNC
PMNC
Carbon fiber
Carbon fiber
2
1 3
FIGURE 2.10 Cross sections of the RVE of the CFF. (Reproduced from Kundalwal, S. I. and Ray, M. C. 2011. International Journal of Mechanics and Materials in Design 7(2):149–166. With permission.)
27Effective Elastic Properties of a Novel Continuous Fuzzy Fiber-Reinforced Composite
� �1
2
3
23
13
12
1
2
3
f
f
f
f
f
f
PMNC
PMNCσσσσσ
σσ
=PPMNC
PMNC
PMNC
PMNC
CFF
CF
σσσ
σ
23
13
12
1
2
=
�FF
CFF
CFF
CFF
CFF
σσσσ
3
23
13
12
(2.20)
and
v vf
f
f
f
f
f
f
PMNC
PMNCσ σ1
2
3
23
13
12
1
�
�
�
�
�
�
+
22
3
23
13
12
1
PMNC
PMNC
PMNC
PMNC
PMNC
�
�
�
�
=
σCCFF
CFF
CFF
CFF
CFF
CFF
�
�
�
�
�
2
3
23
13
12
(2.21)
In Equation 2.21, vf and vPMNC are the volume fractions of the carbon fiber and the PMNC material, respectively, with respect to the volume of the RVE of the CFF, and have been determined in Section 2.5. Using Equations 2.19 through 2.21, and following the procedure for deriving Equation 2.12, the constitutive relations of the CFF can be obtained as follows:
{ } [C ]{ }CFF CFFσ = �CFF (2.22)
in which the effective elastic coefficient matrix [CCFF] of the CFF is given by
[C ] [C ][V ] [C ][ ]CFF = +− −8 9
19
1V10 (2.23)
The various matrices appearing in Equation 2.23 are
[V ] [V ] [V ][C ] [ ],
[ ] [ ] [V ][ ] [ ],
[C
9 7 8 111
71
8
= +
= +
−
−
C
V V C C
10
10 8 10 11
]] vf=
C C Cf f f11 12 13 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
=
,
[C ]10
12 22 23
13 23 33
1 0 0 0 0 0
0 0 0
0 0 0
0 0
C C C
C C C
f f f
f f f
00 0 0
0 0 0 0 0
0 0 0 0 0
44
55
66
C
C
C
f
f
f
,
[C ]9
11 12 13
12 22 23
=
v C v C v C
C C CPMNC
PMNCPMNC
PMNCPMNC
PMNC
PMNC PMNC PPMNC
PMNC PMNC PMNC
P
C C C
C
13 23 33
44
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
MMNC
PMNC
PMNC
PMNC P
C
C
C C
0 0
0 0
0 0
1 0 0 0 0 0
55
66
11
12 22
=
,
[C ]
MMNC PMNC
PMNC PMNC PMNC
PMNC
P
C
C C C
C
C
23
13 23 33
44
55
0 0 0
0 0 0
0 0 0 0 0
0 0 0 0 MMNC
PMNC
f
C
v
v
0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0
66
7
=
,
[V ]ff
f
f
f
v
v
v
and0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
1 0 0 0
8
=[V ]
00 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
v
v
v
v
v
PMNC
PMNC
PMNC
PMNC
PMNC
(2.24)
2.4.1.3 Effective Elastic Properties of the FFRCIt may be reiterated that the RVE of the FFRC lamina can be viewed as being comprised of a CFF and the polymer material. The cross sections of such an RVE are shown in Figure 2.11. In order to satisfy the perfectly bonding situation between the CFF and the polymer, the iso-field conditions and the rules of mixture appropriate for this RVE are given by
In Equation 2.26, vCFF and vP are the volume fractions of the CFF and the polymer matrix, respectively, with respect to the volume of the FFRC, and have been determined in Section 2.5. It may be noted that unlike the constituent phases, the symbols denoting the states of stresses and strains in the FFRC are written without using the superscript. Using Equations 2.22, 2.25, and 2.26, and following the procedure for deriving Equation 2.12, the constitutive relations for the FFRC are derived as follows:
{ } [C]{ }σ = � (2.27)
where the effective elastic coefficient matrix [C] of the FFRC is given by
[C] [C ][V ] [C ][ ]= +− −12 13
113
1V14 (2.28)
with
[V ] [V ] [V ][C ] [C ],
[V ] [V ] [V ][C ] [C
13 11 12 151
14
14 12 11 141
15
= +
= +
−
− ]],
[V ]11
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
=
v
v
v
v
v
CFF
CFF
CFF
CFF
CFFF
,
[V ]12
1 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
=
v
v
v
v
v
P
P
P
P
P
=[C ] vCFF12
11 12 13 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
C C CCFF CFF CFF
00
0 0 0 0 0 0
0 0 0 0 0 0
13
11 12 12
=
,
[C ]
v C v C v CpP
pP
pPP
P P P
P P P
P
P
C C C
C C C
C
C
0 0 0
0 0 0
0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
12 11 12
12 12 11
44
44
CC
C C C
P
CFF CFF CFF
44
14
12 22 23
1 0 0 0 0 0
0 0
=
,
[C ]
00
0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
13 23 33
44
55
66
C C C
C
C
C
CFF CFF CFF
CFF
CFF
CFF
and
C C C
C C C
P P P
P P P
1 0 0 0 0 0
0 0 0
0 0 012 11 12
12 12 11
00 0 0 0 0
0 0 0 0 0
0 0 0 0 0
44
44
44
C
C
C
P
P
P
(2.29)
2.4.2 Mt Method
The micromechanics model (i.e., the MOM approach) of the FFRC, derived in the preceding section, is based on the assumptions delineated by Equations 2.4 and 2.5, respec-tively, which imply the perfect bonding condition between a CNT fiber and the neighboring phases. However, it is required to justify the validity of these assumptions for modeling the perfect bonding conditions of the MOM approach. For this purpose, another micromechanics model based on the MT method which does not require to satisfy the relations similar to Equations 2.4 and 2.5 will be presented here. If the inter-phase between a CNT and the polymer matrix is considered, then the micromechanical model of the unwound PMNC with
Polymer matrix
Polymer matrix
2 2CFF
CFF
31
FIGURE 2.11 Cross sections of the RVE of the FFRC. (Reproduced from Kundalwal, S. I. and Ray, M. C. 2011. International Journal of Mechanics and Materials in Design 7(2):149–166. With permission.)
29Effective Elastic Properties of a Novel Continuous Fuzzy Fiber-Reinforced Composite
straight CNTs will be a three-phase MT model. Utilizing the elastic properties of CNT, interphase, and polymer matrix properties, the three-phase MT method can be derived for the unwound PMNC material with straight CNTs. The explicit formulation of such three-phase MT method can be derived as (Dunn and Ledbetter, 1995)
[C ] [C ][I] [C ][A ] v [ ][A ]
[I] v [A ] v [
ncp
pn
nn i
ii
p n n i
v v C
v
= + +
× + + AA ]11
−
(2.30)
The concentration tensors [An] and [Ai] appearing in Equation 2.30 are given by
[A ] [I] [S ]{([C ]) ([C ] [C ])}n n
p n p= + − − −1 1
(2.31)
[A ] [I] [S ]{([C ]) ([C ] [C ])}i i
p i p= + − − −1 1
(2.32)
Also, in the above matrices, [Sn] and [Si] indicate the Eshelby tensors for the domains denoted by n and i, respec-tively, and [I] is an identity matrix. As assumed in Section 2.4.1.1, a CNT may be treated as a solid circular cylinder. Thus, the specific form of the Eshelby tensor for the cylindri-cal inclusion given by Qui and Weng (1990) is utilized to com-pute the matrices [ n]S↓ and [Si]. The elements of the Eshelby tensors for the cylindrical CNT reinforcement in the isotropic interphase and the cylindrical interphase in the isotropic poly-mer matrix are explicitly written as follows (Qui and Weng, 1990):
[S ]n
n n n
n n n
n
S S S
S S S
S S=
1111 1122 1133
2211 2222 2233
3311 3322
0 0 0
0 0 0nn n
n
n
n
S
S
S
S
3333
2323
1313
1212
0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
=
and
[S ]
S S S
S S S
Si
i i i
i i i
1111 1122 1133
2211 2222 2233
0 0 0
0 0 0
33311 3322 3333
2323
1313
1212
0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
i i i
i
i
i
S S
S
S
S
(2.33)
in which
S S S S Sn n
in n n
i
1111 2222 3333 1122 22115 4
8 10
4 18 1
= = −−
= = = −νν
ν( )
, ,(i −− νi )
,
S S
)S S S Sn n
i
pn n n n
1133 2233 3311 3322 1313 23232 1
014
= =−
= = = =νν(
, ,
S S S Sn
i
ii i
P
1212 1111 2222 333313 4
8 15 4
8 10= −
−= = −
−=ν
ννν( )
,( )
, ,P
S S S
S
i ip
pi i
p
p1122 2211 1133 2233
33111
4 18 1 2 1
= = −−
= =−
=
νν
νν( )
, S( )
,
SS33221 0= ,
S S and S
vi i ip
1313 2323 121214
3 48 1
= = = −−( v )p
where νi and νp denote the Poisson’s ratios of the interphase, and the polymer matrix, respectively. Once [Cnc] is computed from Equation 2.30, Equations 2.14 through 2.18 are used to estimate the average effective elastic coefficient matrix [CPMNC] of the PMNC material surrounding the carbon fiber containing wavy CNTs.
The effective elastic properties of the CFF can be predicted by estimating the effective elastic properties of a composite in which the carbon fiber is the reinforcement and the matrix phase is the PMNC material. Thus, according to the two-phase model by the MT method (Mori and Tanaka, 1973), the effective elastic coefficient matrix for the CFF is given by
[C ] [C ] v ([C ] [C ])[A ]CFF PMNCf
f PMNC= + − 2 (2.34)
in which the matrices of the strain concentration factors are as follows:
[A ] [A ] [I] v [A ]f2 2 2
1= +
−� �vPMNC and
[A ] [I] [S ]([C ]) ([C ] [C ])�
21 1
= + − − −
fPMNC f PMNC
(2.35)
where the Eshelby tensor [Sf] is computed based on the prop-erties of the PMNC matrix and the shape of the carbon fiber. It is worth noting that the PMNC matrix is transversely isotro-pic and, consequently, the Eshelby tensor (Li and Dunn, 1998) corresponding to transversely isotropic material is utilized for computing [Sf] while the inclusion is a circular cylindrical fiber. The elements of the Eshelby tensor for the cylindrical carbon fiber embedded in the transversely isotropic PMNC material are explicitly given by (Li and Dunn, 1998)
Finally, considering the CFF as the cylindrical inclusion embedded in the isotropic polymer matrix, the effective elas-tic properties [C] of the FFRC can be determined by utiliz-ing the two-phase MT method (Mori and Tanaka, 1973) as follows:
[ ] [ ] ([ ] [ ])[A ]C C v C CpCFF
CFF p= + − 3 (2.37)
in which the matrices of the strain concentration factors are given by
[A ] [A ][v [I] [A ]]
[I] [S ](C ) ([C
p
CFFp C
3 3 31
1
= +
= +
−
−
� �
�
v and
[A ]
CFF
3FFF p] [C ])−
−1
in which the Eshelby tensor [SCFF] is computed based on the properties of the polymer matrix and the shape of the CFF. The elements of the Eshelby tensor for the cylindrical CFF reinforcement in the isotropic polymer matrix are explicitly given by (Qui and Weng, 1990)
[S ]CFF
S S S
S S S
S S S=
1111 1122 1133
2211 2222 2233
3311 3322 3333
0 0 0
0 0 0
00 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
2323
1313
1212
S
S
S
(2.38)
in which
S S S S1111 2222 3333
p
p 3311
p
p= = = −−
= =−
05 4
8 1 2 12211,
( ), S
( ),
νν
νν
S S
vv
S S S S2233 3322
p
p 1122 1133 1313 1212= = −−
= = = =4 18 1
014( )
, ,
and
S2323
p
p= −−
3 48 1
νν( )
2.5 RESULTS AND DISCUSSION
In this section, numerical results for the effective elastic prop-erties of the FFRC containing wavy CNTs are evaluated using the two different micromechanical models derived in Sections 2.4.1 and 2.4.2. Zigzag SWCNTs, carbon fiber, and polyimide material are used for evaluating the numerical results. Their material properties are summarized in Table 2.3. The effec-tive elastic properties and the thickness of the hollow circular cylindrical continuum representing the interphase between a CNT and the polyimide matrix are also listed in Table 2.3. Volume fraction of CNTs (VCNT) in the FFRC depends on CNT diameter, running length of CNT, carbon fiber diam-eter, and surface-to-surface distance between two adjacent radially aligned CNTs at their roots. If there are no other
TABLE 2.3Material Properties of the Constituent Phases of the FFRC
31Effective Elastic Properties of a Novel Continuous Fuzzy Fiber-Reinforced Composite
phases or materials in between CNTs, it is reported (Jiang et al., 2009a,b) that the minimum surface-to-surface distance between two adjacent CNTs is the equilibrium van der Waals distance, which is about 0.34 nm. Since in the PMNC mate-rial, polymer molecules fill the gap between CNTs and the formation of the interphase is also considered, the surface-to-surface distance between the two adjacent CNTs at their roots is considered as 1.7 nm. Recall that the FFRC lamina can be viewed as being comprised of CFFs and the polymer matrix (see Figure 2.2). For fibers with circular cross section, it is a well-known fact that the hexagonal packing array is the opti-mal packing of fibers and the corresponding maximum fiber volume fraction is 0.9069. Hence, in the FFRC, the hexagonal packing array of CFFs is considered as shown in Figure 2.12 for evaluating the numerical results. It is also assumed that the radially grown CNTs are uniformly spaced on the circumfer-ential surface of carbon fiber. Note that although the volume fraction of the CFF is 0.9069, the volume fraction of carbon fiber with respect to the volume of FFRC is much less than 0.9069.
The determination of VCNT in the FFRC containing wavy CNTs is an important issue. It is obvious that the construc-tional feature of the FFRC imposes a constraint on the maxi-mum value of VCNT as the number of wavy CNTs grown on the circumferential surface of a particular carbon fiber diameter is limited. Since the radially grown wavy CNTs are uniformly spaced on the circumferential surface of carbon fiber, the maximum number of CNTs grown on the circum-ferential surface of carbon fiber of particular diameter can be determined. Referring to Figure 2.12, the cross section of the RVE of the FFRC can be considered as an equilateral triangle. The volume fractions of different constituent phases of the PMNC, CFF, and FFRC can be determined from this equilateral triangle. First, the volume of the RVE of the FFRC (VFFRC) is determined from this equilateral triangle as follows:
V D LFFRC = 3
42
(2.39)
where D = 2R. The volume (Vf) of the carbon fiber is
V d Lf = π
82
(2.40)
where d = 2a. Thus, the carbon fiber volume fraction in the FFRC (νf) can be expressed as
v
VV
dD
f
f
FFRC= = π2 3
2
2
(2.41)
Using Equation 2.41, the carbon fiber volume fraction (vf ) and the PMNC volume fraction (vPMNC) in the CFF can be derived as
vd L
D Lv and v vf f PMNC f= = = −
π
π π8
8
2 31
2
2
(2.42)
The maximum number (NCNT)max of radially grown aligned wavy CNTs on the circumferential surface of carbon fiber is given by
(N )
(d . )maxCNT
n
dL=+
π2 1 7 2
(2.43)
The volume (VCNT) of wavy CNTs is
V d LCNT
n nr CNT= π4
2 (N )max
(2.44)
and the volume (Vi) of the CNT/polymer interphase is
Vi
i n CNT= − −π4
2 2[d d ](R a)(N )max
(2.45)
where di is the outer diameter of the RVE of the hollow cylin-drical CNT/polymer interphase.
Thus, the maximum CNT volume fraction (VCNT)max with respect to the volume of the FFRC containing wavy CNTs can be determined as follows:
(V )maxCNT
CNT
FFRCn nr
nf
VV
d Ld(d +1.7)
v= = π 2
2
(2.46)
Since the lamina of the FFRC has been made of the hex-agonal packing array of CFFs and the polymer matrix, the volume fractions of the CFF (vCFF) and the polymer matrix (vp) with respect to the volume of the FFRC are given by
v and v vCFF CFF= ≅ = −π
2 30 9069 1. p
(2.47)
CFFs Wavy CNTs
Carbon fiber Polymer
FIGURE 2.12 Hexagonal packing array comprised of CFFs. (Reproduced from Kundalwal, S. I. and Ray, M. C. 2013. ASME Journal of Applied Mechanics 80:021010. With permission.)
The maximum CNT volume fraction with respect to the volume of the PMNC (vn)max can be determined in terms of (VCNT)max as follows:
(v )
( v )(V )max maxn
CNT
PMNCf
CNTV
V= =
−2 3
2 3π (2.48)
Finally, the volume fractions of the CNT/polymer matrix interphase (vi) and the polymer matrix (vp) with respect to the volume of the PMNC can be determined in terms of (Vn)max as follows:
v
VV
and vi
i
PMNCi n
n np n i= = − = − −(d d )
(d )(v )(v ) (v )
maxmax
2 2
2 1
(2.49)
The carbon fiber volume fraction can vary according to the manufacturing route, typically from 0.3 to 0.7. In this study, the value of the diameter of carbon fiber is assumed as 2a = 10 µm, and its volume fraction (vf) is considered as 0.6. The diameter of the CFF and the length of straight CNT become 12.2943 µm and 1.1472 µm, respectively, when the value of vf is 0.6. The degree of CNT waviness is charac-terized by waviness factor, A/Ln. It should be noted that for straight CNT, the value of waviness factor is zero. Unless oth-erwise mentioned, amplitude variations of CNT waviness in the 1–3 and 2–3 planes have been considered for a fixed value of CNT wave frequency ( )ω π= 12 /Ln . Also, unless other-wise mentioned, the maximum amplitude for CNT is consid-ered as A = 100dn based on the diameter of a zigzag (10, 0) CNT which yields the upper limit for the value of waviness factor as A/Ln = 0 068. . It is evident from Equations 2.2 and 2.46 that the increase in the amplitude of CNT increases the running length of CNT which eventually increases the maxi-mum CNT volume fraction (VCNT)max in the FFRC. Figure 2.13 illustrates the variation of the maximum value of CNT volume fraction in the FFRC with waviness factor for the three types of zigzag CNTs. It may be observed from this figure that the maximum value of VCNT increases with the increase in the
value of A/Ln. It may also be noted from Figure 2.13 that for a particular value of waviness factor (A/Ln), the value of VCNT is increased with the increase in the value of integer designat-ing a zigzag CNT. This is due to the fact that as the integer for designating a zigzag CNT increases, the diameter of CNT increases.
It may be noted that the micromechanics models derived in Sections 2.4.1 and 2.4.2 are applicable for estimating the effective elastic properties of any continuous CNT-reinforced composite. Hence, in order to verify the validity of these micromechanics models, CNT and the matrix material of the nanocomposite studied by Liu and Chen (2003) are consid-ered for the constituents of the unwound PMNC material with straight CNTs. The engineering constants of this unwound PMNC material computed by these micromechanics mod-els are compared with those of the same predicted by Liu and Chen (2003) using the finite element model. Table 2.4 illustrates this comparison and it may be observed that the three sets of results are in excellent agreement validating the
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
0.02
0.04
0.06
0.08
0.1
0.12
A/Ln
CNT
volu
me f
ract
ion
(VCN
T)
CNT (10, 0) CNT (14, 0) CNT (18, 0)
FIGURE 2.13 Variation of the maximum CNT volume fraction in the FFRC with the waviness factor (ω = 12π/Ln).
TABLE 2.4Comparison of the Engineering Constants of the Unwound PMNC Material with Straight CNTs
Note: En and Ep are the Young’s moduli of CNT and the polymer matrix, respectively; E1 and E2 are the axial Young’s modulus and the transverse Young’s modulus of the unwound PMNC, respectively; νn and νp are the Poisson’s ratios of CNT and the polymer matrix, respectively; ν12 and ν23 are the axial Poisson’s ratio and the transverse Poisson’s ratio of the unwound PMNC, respectively.
a En = 1000 GPa, νn = 0.3, νp = 0.3, and CNT volume fraction, νn = 0.04871.b Liu and Chen (2003).
33Effective Elastic Properties of a Novel Continuous Fuzzy Fiber-Reinforced Composite
micromechanics models derived in this chapter. This agree-ment also ensures the validity of the assumptions adopted for the MOM approach.
First, the effective elastic properties of the PMNC are computed by employing the MOM approach and the MT method. It should be noted that the models by the MOM approach and the MT method estimate the effective elastic properties of the PMNC with the presence of the CNT/poly-mer interphase. Subsequently, the estimated effective elastic properties of the PMNC are used to compute the effective elastic properties of the CFF in which the carbon fiber is the reinforcement and the matrix phase is the PMNC mate-rial. However, for the sake of brevity, the effective elastic properties of the PMNC and the CFF are not presented here. Figure 2.14 illustrates the variation of the effective elastic coefficient C11 of the FFRC with waviness factor. It may be observed that the effective values of C11 predicted by the MT method excellently agree with those predicted by the MOM approach while wavy CNTs are coplanar with both the 1–3 (1′–3′) and the 2–3 (2′–3′) planes. It may also be observed from Figure 2.14 that the effective values of C11 of the FFRC are not affected by the variations of the amplitude of wavy CNTs in the 2–3 plane. This is attributed to the fact that when wavy CNTs are coplanar with the 2–3 plane, the value
of CNC11 of the PMNC is not affected by CNT waviness as
given by Equation 2.14. When wavy CNTs are coplanar with the 1–3 plane, Equation 2.15 reveals that the transformed
value of CNC11 of the PMNC varies with the amplitude of CNT
waves. Hence, the effective value of the elastic coefficient C11 of the FFRC varies with the amplitude of CNT waves being coplanar with the 1–3 plane as shown in Figure 2.14. Both the MOM approach and the MT method yield identi-cal estimates for the value of C12 as shown in Figure 2.15. It may be observed that CNT waviness causes significant increase in the effective value of C12 when wavy CNTs are
coplanar with the 1–3 plane. The constructional feature of the FFRC reveals that the FFRC would be a transversely isotropic material. This is corroborated by the prediction of C13 which is identical to that of C12 as shown in Figure 2.16. Figure 2.17 demonstrates that the increase in the value of waviness factor decreases the value of C22 when CNT wavi-ness is coplanar with the 1–3 plane, whereas the value of C22 enhances for the higher values of waviness factor when CNT waviness is coplanar with the 2–3 plane. Also, it may be noted that the MT method slightly overestimates the value of C22 as compared to that estimated by the MOM approach. Since the FFRC is a transversely isotropic mate-rial, identical results are also obtained for the effective elastic coefficient C33 but are not presented here. Similar trend of results has been obtained for the effective elastic coefficient
C23 as shown in Figure 2.18. Figure 2.19 demonstrates that the increase in the value of waviness factor increases the value of C55 when CNT waviness is coplanar with the 1–3 plane, whereas the value of C55 is slightly improved for the higher values of waviness factor when CNT waviness is coplanar with the 2–3 plane. Also, it may be noted that the MT method overestimates the values of C55 as com-pared to those estimated by the MOM approach. Although not shown here, identical results are also obtained for the effective elastic coefficient C66.
It is important to note from Figures 2.14 through 2.19 that if wavy CNTs are coplanar with the 1–3 (1′–3′) plane, then the effective values of C11, C12, C13, and C55 of the FFRC are significantly improved over their values with straight CNTs (ω = 0) for the higher values of waviness factor. The pre-dicted elastic coefficients of the FFRC containing sinusoi-dally wavy CNTs in the present study are found to be coherent with the previously reported results (Pantano and Cappello, 2008; Tsai et al., 2011; Farsadi et al., 2012) for the CNT-reinforced composite containing sinusoidally wavy CNTs with symmetric distributions. When a wavy CNT is copla-nar with the 1–3 (1′–3′) plane, the amplitudes of CNT waves become parallel to the 1–axis. This results into the aligning of the projections of parts of CNT lengths with the 1–axis leading to the stiffening of the polymer matrix along the 1–axis. The enhancement of the effective elastic coefficients of the FFRC presented in Figures 2.14, 2.15, 2.16, and 2.19 for the values of C11, C12, C13, and C55, respectively, is attributed to such increase in the axial stiffness along the 1–direction when CNT waves are coplanar with the 1–3 or the (1′–3′) plane. On the other hand, if wavy CNTs are coplanar with the 2–3 (2′–3′) plane then the transverse elastic coefficients of the FFRC are improved over their values with straight CNTs (ω = 0), and the reverse is true when wavy CNTs are coplanar with the 1–3 (1′–3′) plane.
Comparison of the model by the MOM approach with the MT method reveals that the MOM approach yields con-servative estimates for the effective elastic coefficients of the FFRC. Hence, predictions by the MOM approach have been considered in the subsequent results for investigat-ing the effect of CNT diameter, presence of CNT/polymer matrix interphase and higher values of CNT wave frequen-cies on the effective elastic properties of the FFRC con-sidering wavy CNTs to be coplanar with the 1–3 (1′–3′) plane. Figures 2.20 and 2.21 illustrate the comparisons of the effective elastic coefficients C11 and C22 of the FFRC for different values of CNT diameter, respectively. Figure 2.20 reveals that CNT diameter does not influence the effective elastic coefficient C11 of the FFRC, and the same is true for the elastic coefficients C12 and C55 but are not presented here. Figure 2.21 depicts that the increase in CNT diameter marginally increases the value of C22 for the higher values of waviness factor. Although not presented here, the same is true for the effective elastic coefficients C23 and C44. This may be attributed to the fact that as the diameter of CNT increases the value of CNT volume fraction in the FFRC increases as depicted in Figure 2.13 which results in the
35Effective Elastic Properties of a Novel Continuous Fuzzy Fiber-Reinforced Composite
increase in the values of the effective elastic coefficients of the PMNC.
Next, the effective elastic properties of the FFRC con-taining wavy CNTs are computed by employing the MOM approach with the presence and absence of the CNT/polymer interphase as illustrated in Figures 2.22 through 2.24. To investigate the effect of the interphase on the effective elastic properties of the FFRC, zigzag (10, 0) and (18, 0) CNTs are considered to be coplanar with the 1–3 (1′–3′) plane. From Figures 2.22 and 2.24, it may be observed that the models by the two- and three-phase MOM approach predict almost identical estimates for the effective values of C11 and C55, respectively. From Figure 2.23 it may be observed that the three-phase MOM approach provides a slightly enhanced estimate for the value of C22 as compared to that predicted by
the two-phase MOM approach for the higher values of wavi-ness factor. Although not presented here, similar predictions have also been obtained for the effective elastic coefficients C23 and C44.
So far, in this chapter, the effect of wavy CNTs on the effective elastic properties of the FFRC have been studied by considering CNT wave frequency (ω) as 12π /Ln for a par-ticular value of waviness factor (A/Ln). To this end, however, the variation of CNT wave frequency for a particular value of A/Ln would be an important study. For this, the discrete val-ues of wave frequency of zigzag (18, 0) CNT are considered as 5π /Ln 10π /Ln 15π /Ln and 20π /Ln. Figures 2.25 through 2.27 illustrate the comparisons of the effective elastic coef-ficients of the FFRC for different values of CNT wave fre-quencies. Figure 2.25 illustrates that the increase in the value
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
20
40
60
80
100
120
140
160
180
A/Ln
C 11 (G
Pa)
CNT (10, 0) CNT (14, 0) CNT (18, 0)
FIGURE 2.20 Variation of the effective elastic coefficient C11 of the FFRC with the waviness factor while the wavy CNTs are copla-nar with the 1–3 (1′–3′) plane (ω = 12π/Ln).
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
5
10
15
20
25
A/Ln
C22
(GPa
)
CNT (10, 0) CNT (14, 0) CNT (18, 0)
FIGURE 2.21 Variation of the effective elastic coefficient C22 of the FFRC with the waviness factor while the wavy CNTs are copla-nar with the 1–3 (1′–3′) plane (ω = 12π/Ln).
FIGURE 2.22 Variation of the effective elastic coefficient C11 of the FFRC with the waviness factor while the wavy CNTs are copla-nar with the 1–3 (1′–3′) plane (ω = 12π/Ln).
FIGURE 2.23 Variation of the effective elastic coefficient C22 of the FFRC with the waviness factor while the wavy CNTs are copla-nar with the 1–3 (1′–3′) plane (ω = 12π/Ln).
of CNT wave frequency significantly enhances the value of C11. This is attributed to the fact that as CNT wave frequency increases the value of CNT volume fraction and the aligned projections of parts of CNT lengths with the 1–axis increases. The more the value of CNT wave frequency, the more will be such projections, and hence the axial stiffness increases with the increase in CNT wave frequency. Similar predictions for the effective elastic coefficient C12 have also been obtained but are not presented here. Figure 2.26 reveals that, in general, the value of C22 of the FFRC decreases with increase in the values of CNT wave frequency and waviness factor. Although not shown here, the same is true for the effective elastic coef-ficients C23 and C44. Figure 2.27 depicts that the value of C55
of the FFRC is significantly enhanced initially and then stabi-lized for the higher values of waviness factor.
2.6 CONCLUSIONS
In this chapter, the effect of CNT waviness on the effective elastic properties of a novel continuous FFRC has been stud-ied considering the sinusoidally wavy CNTs to be coplanar with either of the two mutually orthogonal planes. Analytical micromechanics models based on the MOM approach and the MT method are employed to determine the effective elastic properties of this novel composite. The effective elastic prop-erties of the FFRC are estimated in the presence and absence
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
25
50
75
100
125
150
175
200
225
A/Ln
C 11 (G
Pa)
ω = 5π/Ln
ω = 10π/Lnω = 15π/Ln
ω = 20π/Ln
FIGURE 2.25 Variation of the effective elastic coefficient C11 of the FFRC with the waviness factor while the wavy CNTs are copla-nar with the 1–3 (1′–3′) plane ((18, 0) CNT).
FIGURE 2.24 Variation of the effective elastic coefficient C55 of the FFRC with the waviness factor while the wavy CNTs are copla-nar with the 1–3 (1′–3′) plane (ω = 12π/Ln).
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
5
10
15
20
25
A/Ln
C 22 (G
Pa)
ω = 5π/Ln
ω = 10π/Lnω = 15π/Ln
ω = 20π/Ln
FIGURE 2.26 Variation of the effective elastic coefficient C22 of the FFRC with the waviness factor while the wavy CNTs are copla-nar with the 1–3 (1′–3′) plane ((18, 0) CNT).
0 0.01 0.02 0.03 0.04 0.05 0.06 0.070
2
4
6
8
10
12
A/Ln
C55
(GPa
)
ω = 5π/Ln
ω = 10π/Ln
ω = 15π/Ln
ω = 20π/Ln
FIGURE 2.27 Variation of the effective elastic coefficient C55 of the FFRC with the waviness factor while the wavy CNTs are copla-nar with the 1–3 (1′–3′) plane ((18, 0) CNT).
37Effective Elastic Properties of a Novel Continuous Fuzzy Fiber-Reinforced Composite
of an interphase between a CNT and the polymer matrix. The following important conclusions are drawn from the study carried out in this chapter.
1. The effective values of the axial elastic coefficients of the FFRC containing wavy CNTs being coplanar with the 1–3 plane are significantly improved over their values with straight CNTs (ω = 0) for the higher values of waviness factor and CNT wave frequency. The effect of wavy CNTs being coplanar with the 2–3 plane is found to be marginal on the effective axial elastic coefficients of the FFRC.
2. CNT waviness in the 2–3 plane causes improvement in the effective transverse elastic coefficients of the FFRC compared to that of the composite with straight CNTs (ω = 0). The effective transverse elastic coeffi-cients are significantly improved for the higher values of waviness factor and CNT wave frequency when wavy CNTs are coplanar with the 2–3 plane.
3. The diameter of zigzag CNT negligibly affects the effective elastic coefficients of the FFRC containing wavy CNTs.
4. The nonbonded van der Waals interaction between a CNT and the polymer matrix negligibly affects the effective elastic coefficients of the FFRC containing wavy CNTs. Thus, for conservative and intuitive esti-mates, one may neglect the nonbonded van der Waals interaction between a CNT and the polymer matrix, and consider the perfect bonding between them.
The present study reveals that wavy CNTs can be properly used to construct novel advanced nanocomposites with supe-rior elastic properties.
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