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The effect of inclusion waviness and waviness distribution on elastic properties of fiber-reinforced composites
Composites: Part B 42 (2011) 62–70
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Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
The effect of inclusion waviness and waviness distribution on elastic properties of fiber-reinforced composites Chao-hsi Tsai a, Chuck Zhang a,⇑, David A. Jack b, Richard Liang a, Ben Wang a a b
Department of Industrial and Manufacturing Engineering, and High Performance Material Institute (HPMI), Florida State University, Tallahassee, FL 32310, USA Department of Mechanical Engineering, School of Engineering and Computer Science, Baylor University, Waco, TX 76798, USA
a r t i c l e
i n f o
Article history: Received 25 April 2010 Accepted 16 September 2010 Available online 18 September 2010 Keywords: A. Polymer–matrix composites (PMCs) B. Mechanical properties C. Micro-mechanics C. Statistical properties/methods
a b s t r a c t In this study we investigated the effects of inclusion waviness and its distribution to the effective composite stiffness. Different waviness conditions were analyzed: uniform waviness with variable inclusion orientation or aspect ratio, and uniform aspect ratio with variable waviness. The inclusion waviness was found to have a greater effect on tensile moduli and shear modulus for unidirectional composites; however, if the inclusions are either randomly dispersed or partially aligned, the degree of waviness effect was smaller. The elastic moduli were also over-estimated if inclusion aspect ratio or waviness followed symmetric distributions. In addition, the waviness distribution effect was larger when larger inclusion waviness was introduced in a composite. The lack of fiber waviness distribution assumption was found to lead to inaccurate composite stiffness predictions, especially for composite with higher fiber volume fraction. We also demonstrated the inclusion waviness effects on the tensile modulus of carbon nanotube-reinforced composites. The results showed that the disparity between theoretical predictions and experimental data could be due to different inclusion waviness conditions. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction The widespread use of composites in industrial applications is largely due to their exceptional performances and high propertyto-weight ratios. Among all types of composites, a popular one is made from fibrous inclusions with polymer matrix, also known as fiber-reinforced composites (FRC). The overall properties of FRC depend on the properties of the constituent materials, as well as the microstructure therein. With respect to the stiffness of composites, several microstructure parameters have been found to be influential on composite stiffness: the inclusion aspect ratio (ratio of inclusion length and diameter) [1–3], their orientation [2,4–6], waviness [7–11], and the overall inclusion volume fraction [1–3]. To compute composite stiffness via constituent properties and corresponding microstructures is known as composite micromechanics. Although the theory of micro-mechanics has been well-studied, most existing models are originally derived for composites having unidirectional inclusions with constant inclusion structure and properties [1]. In other words, those models were deterministic in nature, and the statistical variations of the input variables were ignored. This may bias the results when using these models to predict composite stiffness, especially for fiber-reinforced composites ⇑ Corresponding author. Tel.: +1 850 410 6355; fax: +1 850 410 6342. E-mail address: [email protected] (C. Zhang). 1359-8368/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2010.09.004
with each inclusion having different aspect ratio, orientation and waviness. Researchers have been trying to solve these problems over the past decades. For instance, some researches focused on investigating the effects of misaligned inclusions to the stiffness of short-fiber-reinforced composites [2,4–6]. Many researchers also studied the variations of inclusion aspect ratio [2–3]. However, limited information on studies are available with respect to the inclusion waviness variation. Some of the existing studies focused on deriving analytical expressions for effective composite stiffness with wavy inclusions (e.g. Hsiao and Daniel [7], Chan and Wang [8]). Some others applied finite element analysis to construct an effective wavy inclusion element, and combined with traditional micromechanical models to predict the overall effective stiffness of composites [9–11]. In the studies of Anumandla and Gibson [10], the inclusion waviness in the composites was assumed homogeneous (i.e. all inclusions possess identical waviness), and thus the waviness distribution effects were ignored. Fisher et al. [9] and Bradshaw et al. [11] extended Mori–Tanaka method to composites consisting of more than one type of inclusion type via the method introduced by Tendon and Weng [12], so that the distribution of CNT waviness can be considered. They also demonstrated the effect of CNT waviness distribution by introducing a discrete probability distribution. The objective of this study was to conduct a comprehensive analysis focusing specifically on the stochastic effect of inclusion waviness to the stiffness of fiber-reinforced composites. The
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effective composite stiffness were computed through micromechanical models, specifically, Mori–Tanaka method [13] for unidirectional composites, Hsiao-Daniel model [7] for composite with wavy inclusions, and Advani–Tucker’s orientation averaging [4] for composite with misaligned inclusions. The Voigt method of composite homogenization and numerical Monte-Carlo simulation were conducted in conjunction of micromechanical models to compute average composite stiffness through composite microstructural (i.e. fiber aspect ratio, waviness and orientation) distributions. Continuous inclusion waviness and aspect ratio distribution with both symmetric and asymmetric shapes were considered, and the four composite elastic constants (elongation and transversely tensile moduli E11 and E22, shear modulus G12, and Poisson’s ratio m12) were computed and compared under different conditions. We found that the distribution of inclusion waviness is influential to the composite stiffness in most cases, especially when asymmetric distribution was presented. More details of the simulation results will be discussed in Section 3. We also introduced a carbon nanotube-reinforced composite in Section 4 to demonstrate the waviness distribution effect on nanocomposite system. The simulated composite tensile moduli were compared to the physical experimental data conducted by Dynamic Mechanical Analysis (DMA). Several modeling limitations are listed here. As micro-mechanics were used as the basis of computation, standard micromechanics assumptions (see, e.g. Tucker and Liang [1]) hold. It is also noted that different micromechanical models could result in different waviness effects because of the underlying assumption differences. Our results are only held under the models applied. 2. Technical approach 2.1. Micromechanical modeling for composite stiffness with wavy inclusions The properties of composites were determined by the constituent materials, their structure and distributions. With respect to the stiffness prediction of fiber-reinforced composites, the composite micro-mechanics is a very efficient technique and is well-accepted by many researchers. A popular model is the mean field method developed by Mori and Tanaka [13], which is based on Eshelby’s strain-concentration tensor around an ellipsoidal inclusion in the matrix [14]: Eshelby
A
m
f
m
1
¼ ½I þ ES ðC C Þ
ð1Þ
Eshelby
where A is Eshelby’s strain-concentration tensor, I is the identity tensor, and E is the Eshelby tensor, which depends only on the inclusion aspect ratio and the matrix elastic constants [14]. S and C are composite compliance tensor and stiffness tensor, with the subscripts f and m designating the fiber and matrix, respectively. Based on Eshelby’s theory, Mori and Tanaka [13] proposed the concept of effective field theory, which assumes that when many identical inclusions are introduced in a concentrated composite, the strain over each inclusion is equal to the average strain in the matrix, and thus provided results applicable to higher inclusion volume fractions [13]:
AMT ¼ AEshelby ½ð1 V f ÞI þ V f AEshelby 1
63
As a physical consideration, we recognize that the inclusions within the composites usually possess certain degree of curvature, especially for flexible inclusions with larger aspect ratio. A good example for long flexible inclusion is the carbon nanotubes (CNTs). Fig. 1 shows a scanning electron microscopic (SEM) image of a twodimensional CNT texture surface. This kind of CNT textures (also known as CNT buckypapers) can be used as preforms for CNT-reinforce composites [15]. Fig. 1 clearly shows that most of the CNT bundles possess certain degree of curvature, indicating that the curvature effect must be taken into account for predicting the stiffness of this kind of composites. Several researchers introduced the concept of inclusion waviness to model the curvature effect [7–11]. Hsiao and Daniel [7] derived analytical expressions for unidirectional composite stiffness with uniform fiber waviness. A complete set of twelve elastic constants were derived with respect to the defined waviness parameters. Their work is used to introduce the concept of inclusion waviness and its effect to the composite stiffness. In the definition of inclusion waviness, it is assumed that each inclusion forms sinusoidal wave shape, as shown in Fig. 2, where A is the amplitude and L is the wavelength. For uniform waviness, it is sufficient to consider a representative volume element containing one period of the waviness. Assume axis X1 being the direction along the aligned inclusion and applied load. The elastic properties of the composites are determined from the average strain obtained in the representative volume, which is the integration of the strain of every infinitesimal thickness over one wavelength in the loading (X1) direction. To get the effective stiffness of the uniform-waviness model, a waviness parameter w is introduced as a function of amplitude and wavelength as w = A/L. Twelve elastic constants, three axial moduli, three shear moduli, and six Poisson’s ratios, were derived with respect to the defined waviness parameter (see Hsiao and Daniel [7] for detailed discussions). Note that in the presented paper, we considered inclusion waviness occurring in the 1–2 plane, while in Hsiao–Daniel paper [7] the waviness was in x–z plane. Therefore a swap of y and z in their notation is required (i.e. E22 in the presented paper is Ez in Hsiao-Daniel paper [7]). It is also worthwhile to emphasize that the waviness effect in Hsiao–Daniel paper [7] only depends on waviness parameter (i.e. the ratio of wavelength and amplitude). Therefore the actual magnitude of wavelength and amplitude (and their relationship to fiber diameter) is irrelevant. 2.2. Inclusion waviness distribution and modeling Before the inclusion waviness effects on the overall properties of composites can be analyzed, its distribution function and
ð2Þ
where Vf is the composite volume fraction. The Mori–Tanaka strainconcentration tensor AMT can then be used to compute the overall composite stiffness tensor C as [1]:
C ¼ C m þ V f ðC f C m ÞAMT
ð3Þ
It is noted that the original form of Mori–Tanaka method can only work for composites with straight and unidirectional inclusions.
Fig. 1. Surface image of a single-walled carbon nanotube texture taken by scanning electron microscopy (SEM).
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orientation sampled from the corresponding distributions, and the same concentration as the original RVE (as depicted in the right section of Fig. 3). The effective stiffness tensor of the ith unit element, labeled Ci, defined as:
X2
ei ¼ C i ri
ð6Þ
can thus be computed through corresponding inclusion aspect ratio and waviness using micromechanical models, where ei and ri are stress tensor and strain tensor for the ith element, respectively. From the Voigt method, a constant mean strain over each unit element was assumed,
X1
ei ¼ e Fig. 2. Illustration of lamina with fiber waviness.
modeling techniques must be defined. As a physical consideration, a composite may contain inclusions having various structure parameters (e.g. aspect ratio and waviness). Although the variations can be statistically represented by their mean and variance, it is often insufficient if one does not consider the whole probability distributions. For instance, a symmetric distribution and a nonsymmetric distribution can have totally different behavior, even if they have the same mean and variance. Probability distribution can be roughly grouped into two types: normal (or symmetric) distributions, and non-normal (or asymmetric) distributions [16]. A normal distribution function with random variable X has the form of [16]:
" 2 # 1 1 X l PðXÞ ¼ pffiffiffiffiffiffiffi exp 2 r r 2p
ð4Þ
where l and r are mean and variance of the distribution, respectively. However, many real distributions fall into the category of asymmetric distribution, such as Gamma distribution and exponential distribution. One of the objectives of this research is to study the distribution effects of the inclusion waviness to the effective composite stiffness. We introduce a log-normal distribution as the representation of asymmetric waviness distribution, which has the function form of [16]:
" 2 # 1 1 ln X k pðXÞ ¼ pffiffiffiffiffiffiffi exp 2b2 b 2p X
ð5Þ
where k and b are distribution parameters and both have positive values. The mean and variance of a log-normal distribution are exp(k + b2/2) and exp(2k + b2)(exp(b2) 1), respectively [16]. As mentioned in the previous section, the micromechanical models (e.g. Mori–Tanaka method and Hsiao–Daniel model) can only deal with uniform inclusion properties and microstructures because of their deterministic nature. To address this issue, the Voigt method of composite homogenization were often applied (see, e.g. Jack and Smith [6]), and the concepts are described as follows. We first considered a simplified composite representation volume element (RVE) with misaligned inclusions having variable aspect ratio and waviness, as shown in the left section of Fig. 3. The inclusion aspect ratio, waviness and orientation within the RVE are assumed to follow certain distributions; therefore, the effective stiffness tensor of the RVE, hCRVEi, cannot be computed directly through micromechanical models because of their deterministic limitations (i.e. the inputted microstructural parameters are not constants). The composite homogenization process is thus applied to decompose the whole RVE into several independent unit elements. Each unit element is assumed having unidirectional inclusions with uniform (but unique) waviness, aspect ratio and
ð7Þ
therefore the stiffness of the RVE is simply the expectation of the total unit elements combined. If the number of total unit elements is N, the expectation for the effective RVE stiffness tensor can be computed as:
hC RVE i ¼
N 1X Ci N i¼1
ð8Þ
Here we introduced a semi-numerical approach to compute the stiffness expectation. The computational steps are summarized as follows. (1) Define probability distribution functions for inclusion aspect ratio, waviness and orientation. (2) Apply Monte-Carlo sampling to draw aspect ratio and waviness samples from the corresponding aspect ratio and waviness distribution. (3) Compute effective composite stiffness Ci through inclusion aspect ratio and waviness sampled from the previous step (Mori–Tanaka method and Hsiao–Daniel model). Store the result. (4) Repeat step (3) for N times. (5) Compute average stiffness tensor (i.e. expectation) using Eq. (8). (6) Apply orientation averaging (Advani and Tucker [4]) to compute effective composite stiffness through defined orientation distribution function. Note that we assumed that inclusion aspect ratio, waviness and orientation are decoupled; therefore they can be sampled independently from the corresponding probability distributions using Monte-Carlo sampling. It is worthwhile to mention that the expectation of composite stiffness with misaligned inclusions having uniform aspect ratio and waviness has been analytically solved (see, e.g. Advani and Tucker [4], Jack and Smith [6]). However, for conjunction to non-uniform inclusion aspect ratio and waviness, the analytical form of the composite stiffness expectation has yet to be developed, and needs to be computed numerically through Monte-Carlo method or similar approaches.
3. Discussion In this section we conduct a comprehensive analysis on the effects of inclusion waviness and its distribution to the effective stiffness of composites. Different waviness conditions were considered. First, we consider a uniform inclusion waviness state with either variable inclusion orientation or aspect ratio. Then the variation of inclusion waviness was analyzed under uniform aspect ratio and unidirectional inclusions. Finally, the waviness influence on a carbon nanotube-reinforce composite is discussed.
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Fig. 3. Illustration of composite homogenization.
3.1. Effect of inclusion waviness to composite stiffness under different orientation states The inclusion orientation problems have been studied intensively by many researchers over the past decades (see, e.g. Refs. [4–6]). We followed the orientation representation introduced by Advani and Tucker [4], but similar works can also be found elsewhere. Three different fiber orientation states are discussed in this Section: (1) perfect fiber alignment (unidirectional), (2) planar fiber orientation state (2-D orientation), and (3) three-dimensional orientation state. The inclusion waviness under different orientation conditions were discussed. Note that for the later two orientation states, the fiber can either be randomly distributed or partially aligned. To model inclusion orientation, it is convenient to introduce a three-dimensional orientation function defined as: 2m
wðh; /Þ ¼ k sin
h cos2n /
ð9Þ
where n and m are positive integers related to the degree of fiber alignment, and higher the m, n values, the higher the degree of fiber alignment along X1 corresponding to (h, /)=(p/2, 0) and (h, /) = (p/ 2, p). k is a constant determined by satisfying the normalized condition of W. h and / are Euler’s angles defining the orientation for each individual inclusion, as shown in Fig. 4. Advani and Tucker [4] proposed a orientation averaging process to compute the effective composite stiffness through the defined orientation function. This technique was employed in this study to compute the effective stiffness of composites under different orientation states (see, e.g. Advani and Tucker [4] for the complete discussion of orientation averaging). Note that for unidirectional fiber with constant fiber aspect ratio and waviness, the composite stiffness tensor was computed directly through the Mori–Tanaka method, as outlined in Eqs. (1)– (3). However, different fiber orientation distribution functions were chosen to compute effective composite stiffness with misaligned fiber states. For 2-D plane orientation state, the out-of-plane angle h was p/2, which eliminates the cosine term in Eq. (9), and the orientation distribution function was reduced to w(/) = kcos2n /. This reduced form is the 2-D orientation distribution function,
X3
θ
X2
φ X1 Fig. 4. Coordinate system defining the fiber orientation in 3-D space.
and we selected n = 0 and n = 1 to describe 2-D random and 2-D partially aligned state, respectively. 3-D orientation states are represented by Eq. (9), where n = m = 0 and n = m = 1 were used to describe 3-D random and 3-D partially aligned states, respectively. The material properties used in the simulation were typical of fiber-reinforced engineering thermoplastics adopted from Tucker and Liang [1]. Specifically, the fibers were assumed isotropic with Young’s modulus and Poisson’s ratio equating 30 GPa and 0.2, respectively, and the matrix were 1 GPa and 0.38. The fiber volume fraction was 20%. A plane stress state was assumed; therefore, the elastic behavior of the composite can be described by four elastic constants: axial and transverse tensile moduli E11 and E22, shear modulus G12, and Poisson ratio m12. The sample size N used for Monte-Carlo simulation is set 107. The effects of orientation distribution on composites with constant fiber aspect ratio and waviness are demonstrated in Fig. 5. The fiber aspect ratio was set constant at 10. Fig. 5a shows that fiber waviness has dramatic effect on E11 for unidirectional composite, but the degree of effects are smaller if the fibers are somewhat misaligned. For 3-D random orientation state, the inclusion
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Fig. 5. Fiber orientation distribution effects on composite stiffness: (a) E11/Em, (b) E22/Em, (c) G12/Gm, (d)
waviness had a minimal effect on E11. Fiber waviness also had significant effects on E22 and m12 for most of the fiber orientated states, except for 3-D random composite, whose E22 and m12 stay nearly constants for different fiber waviness levels. The increase of fiber waviness was observed to lead to an increase of composite E22, if the fiber orientation is partially aligned or unidirectional. The fiber waviness also affects the shear modulus of composites with different fiber orientation states, but the effects were only significant when the fibers are either 2-D random or unidirectional. Also note that under 3-D random and 2-D aligned conditions, the resulting composites have identical shear modulus and therefore the results are overlapped in Fig. 5c.
3.2. Effect of inclusion aspect ratio distribution with uniform fiber waviness In this section, we discuss the effects of different fiber aspect ratio (denoted as a) distributions on unidirectional composites with constant fiber waviness. Three different fiber aspect ratio conditions: (1) non-symmetric distribution (represented by log-normal distribution), (2) symmetric distribution (represented by normal distribution), and (3) constant aspect ratio, were considered. For all the three conditions, the average fiber aspect ratio was set to be either 10 or 100, with the standard deviation equaling 50% of the mean based on the experimental observation conducted by Yeh [17].
v12/vm.
Fig. 6 illustrates the effects of different fiber aspect ratio distribution on the four composite elastic constants. It can be seen in most cases, different aspect ratio distributions lead to different composite property predictions. When the fiber aspect ratio is relatively large (i.e. a = 100), the prediction results for composites with Lognormal aspect ratio distribution and the constant one is very similar. However, the properties of composites would be over-estimated if the fiber aspect ratio follows normal distribution. With respect to the composites with smaller fiber aspect ratio (i.e. a = 10), similar prediction results can also be seen. If the aspect ratio follows normal distribution, the composite E11, E22, and G12, will be over-estimated. On the contrary, the three of the four elastic properties will be under-estimated if the composite has lognormal fiber aspect ratio distribution. In summary, the distribution of fiber aspect ratio has significant effects on composite property prediction. This also implies the lack of prediction accuracy if one considers only the average fiber aspect ratio to predict the composite stiffness using micromechanical models.
3.3. Effect of fiber waviness distribution with constant fiber aspect ratio The effect of different fiber waviness distributions on unidirectional composites stiffness is also discussed. For the constant fiber aspect ratios, three different waviness distributions were considered: non-symmetric distribution (represented by log-normal dis-
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Fig. 6. Fiber aspect ratio distribution effects on composite stiffness a (a) E11/Em, (b) E22/Em, (c) G12/Gm, (d)
tribution), symmetric distribution (represented by normal distribution), and constant fiber waviness. Similar to the previous section, two different waviness means, 0.1 (high) and 0.01 (low), were assumed. The standard deviation of the waviness distribution set were 50% of the mean. Fig. 7 illustrates the effective composite properties with different fiber waviness distributions. It was observed that at smaller fiber waviness conditions, the predicted results for composite with lognormal waviness distribution were nearly identical to the one with constant waviness. On the other hand, under the current modeling assumption, if the composite possesses normal distributed fiber waviness, it was very likely that three of the four elastic constants (E11, E22, G12) would be over-estimated. With respect to the composites with larger fiber waviness, the three different waviness distributions lead to different predictions for composites E11, E22 and G12. Composites with Lognormal fiber waviness distribution had the highest predictions on E11 and E22. On the other hand, composite with constant fiber waviness had the highest predictions on G12 and m12. Normal distributed fiber waviness appeared to have the tendency to result in lower composite property predictions. Waviness distribution effects on composite stiffness at different volume fraction levels were also investigated. Similar to the previous analysis, three different fiber waviness distributions were assumed. The analysis was conducted based on different fiber aspect ratio and waviness combinations; therefore, four different
v12/vm.
conditions were considered. Fig. 8 demonstrated the simulation results, and different fiber aspect ratio and waviness combination are represented by different colors. It was observed that the distribution of fiber waviness had a significant effect on composite properties. In most cases, the lack of fiber waviness distribution assumption would lead to inaccurate composite stiffness predictions, especially for composite with higher fiber volume fraction. 3.4. Inclusion waviness effect on carbon nanotube-reinforced composites In this section, we demonstrate the inclusion waviness effect on the tensile modulus of carbon nanotube-reinforced composites. A unique manufacturing technique was employed in this research to produce multi-layered CNT-reinforced composite laminates. This process was first developed by the researchers at High Performance Material Institute (HPMI) [15]. In their process, carbon nanotube structure and dispersion were controlled by fabricating them into thin films, also known as the CNT buckypapers (BPs), using a multi-step sonication and filtration process [15]. Buckypapers were used as preforms to make multi-layered buckypaperpolymer composite laminates, which were found to have good mechanical properties and high nanotube loading [15]. It is accepted by many researchers that CNT-reinforced composites can be considered as fiber-reinforced composites, so that their
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Fig. 7. Fiber waviness distribution effects on composite stiffness: (a) E11/Em, (b) E22/Em, (c) G12/Gm, (d)
mechanical properties can be predicted using micro-mechanics (see, e.g. Ref. [18]). In this study, an eight-layered buckypaperpolymer composite was produced using randomly dispersed single-walled CNT buckypapers and Epon 862 polymer matrix. The composite was cured at 177 °C under vacuum to remove possible void formation. Five samples were produced using the same process. The mechanical properties of the BP-polymer composites were measured using Dynamic Mechanical Analysis (DMA). For each composite sample, the DMA test was conducted three times to get the average tensile modulus. It is worthwhile to mention that the buckypaper is a two-dimensional CNT texture; therefore, the CNT orientation distribution can be represented by two-dimensional random orientation distribution function introduced above. In addition, based on some prior studies, the CNT length and diameter within the buckypapers were measured using image processing techniques [17]. Specifically, both CNT length and diameter followed the Weibull distribution, with length follows Weibull (1.89 lm, 1.93), and diameter follows Weibull (8.225 nm, 1.63) [17]. The average CNT aspect ratio was 470 [17]. Also, as a final physical consideration, we recognize that the stiffness of a nanotube bundle will be a function of the effective diameter of the bundle. Liu et al. [19] presented a hybrid atom/continuum model to study the bulk elastic properties of SWNT bundles and
v12/vm.
developed a complete set of the five independent elastic constants from molecular dynamics (MD) simulations. They found that the five elastic constants were all directly related to bundle diameter. Therefore, for each Monte-Carlo sampling, one nanotube bundle diameter was generated from the corresponding Weibull distribution, and thus the five elastic constants for the specific nanotube bundle were determined. The bundle’s effective transversely isotropic stiffness tensor, was then formed for use with the Mori–Tanaka method as outlined in Eqs. (1)–(3). With respect to the properties of the surrounding epoxy resin matrix, we used the data in Yamini and Young’s paper [20], with an elastic modulus of Em = 2.5 GPa, shear modulus Gm = 1.2 GPa, and Poisson’s ratio mm = 0.3 thus completely defining the matrix’s isotropic stiffness tensor Cm expressed in Eq. (3). The waviness distribution of the CNT bundle within the buckypapers, however, is not recognized yet. From the SEM image of buckypaper surface illustrated in Fig. 1, only slight bundle waviness was observed. Therefore, a normal waviness distribution is assumed, with the mean ranges from 0.01 to 0.05, and the standard deviation equals 50% of the mean. Fig. 9 shows the tensile modulus comparison of experimental data and micromechanical simulation predictions. The trend of DMA results seem to correspond with the simulation quite well. With the assumption of normal CNT bundle waviness distribution with the mean equal to 0.05, the micromechanical simulation
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Fig. 8. Fiber waviness distribution effects on composite stiffness with different volume fraction (a) E11/Em, (b) E22/Em, (c) G12/Gm, (d)
v12/vm.
results were very close to the experimental data. However, as the CNT waviness distribution used here were arbitrarily selection, no clear conclusion can be made from the results of Fig. 9. The purpose of this analysis is rather to demonstrate the possible CNT bundle waviness effects on the resulting BPP composite modulus. More accurate simulation predictions can be made once the true CNT waviness distribution within the buckypapers is recognized.
4. Conclusions In this study we investigated the effect of inclusion waviness and its distribution to the effective composite stiffness. Different waviness conditions: uniform waviness with variable inclusion orientation or aspect ratio, uniform aspect ratio and variable waviness, were analyzed. The inclusion waviness was found to have a great effect on tensile moduli and shear modulus for unidirectional composites. However, if the inclusions were either randomly dispersed or only partially aligned, the degree of waviness effect was smaller. With respect to unidirectional composites with variable aspect ratio and uniform waviness, the composite properties tended to be over-estimated if inclusion aspect ratio follows sym-
Fig. 9. Experimental data and simulation results comparison.
metric distribution. The elastic moduli would also be over-estimated if variable inclusion waviness follows symmetric
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distribution. In addition, the waviness distribution effect is larger when larger inclusion waviness was introduced in the composite. The lack of fiber waviness distribution assumption was found to lead to inaccurate composite stiffness predictions, especially for composite with higher fiber volume fractions. We also demonstrated the inclusion waviness effect to the tensile modulus of carbon nanotube-reinforced composites. The results show that the disparity between theoretical predictions and experimental data could be due to different inclusion waviness conditions. References [1] Tucker CL, Liang E. Stiffness predictions for unidirectional short-fiber composites review and evaluation. Compos Sci Technol 1999;59:655–71. [2] Jiang B, Liu C, Zhang C, Wang B, Wang Z. The effect of non-symmetric distribution of fiber orientation and aspect ratio on elastic properties of composites. Compos Part B – Eng 2007;38:24–34. [3] Hine PJ, Lusti HR, Gusev AA. Numerical simulation of the effects of volume fraction, aspect ratio and fibre length distribution on the elastic and thermoelastic properties of short fibre composites. Compos Sci Technol 2002;62:1445–53. [4] Advani SG, Tucker CL. The use of tensors to describe and predict fiber orientation in short fiber composites. J Rheol 1987;31:751–84. [5] Chen CH, Cheng CH. Effective elastic moduli of misoriented short-fiber composites. Int J Solids Struct 1996;33:2519–39. [6] Jack DA, Smith DE. Elastic properties of short-fiber polymer composites, derivation and demonstration of analytical forms for expectation and variance from orientation tensors. J Compos Mater 2008;42:277–308. [7] Hsiao HM, Daniel IM. Elastic properties of composites with fiber waviness. Compos Part A – Appl Sci Manuf 1996;27:931–41. [8] Chan WS, Wang JS. Influence of fiber waviness on the structural response of composite laminates. J Thermoplast Compos Mater 1994;7:243–60.
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Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
The effect of inclusion waviness and waviness distribution on elastic properties of fiber-reinforced composites Chao-hsi Tsai a, Chuck Zhang a,⇑, David A. Jack b, Richard Liang a, Ben Wang a a b
Department of Industrial and Manufacturing Engineering, and High Performance Material Institute (HPMI), Florida State University, Tallahassee, FL 32310, USA Department of Mechanical Engineering, School of Engineering and Computer Science, Baylor University, Waco, TX 76798, USA
a r t i c l e
i n f o
Article history: Received 25 April 2010 Accepted 16 September 2010 Available online 18 September 2010 Keywords: A. Polymer–matrix composites (PMCs) B. Mechanical properties C. Micro-mechanics C. Statistical properties/methods
a b s t r a c t In this study we investigated the effects of inclusion waviness and its distribution to the effective composite stiffness. Different waviness conditions were analyzed: uniform waviness with variable inclusion orientation or aspect ratio, and uniform aspect ratio with variable waviness. The inclusion waviness was found to have a greater effect on tensile moduli and shear modulus for unidirectional composites; however, if the inclusions are either randomly dispersed or partially aligned, the degree of waviness effect was smaller. The elastic moduli were also over-estimated if inclusion aspect ratio or waviness followed symmetric distributions. In addition, the waviness distribution effect was larger when larger inclusion waviness was introduced in a composite. The lack of fiber waviness distribution assumption was found to lead to inaccurate composite stiffness predictions, especially for composite with higher fiber volume fraction. We also demonstrated the inclusion waviness effects on the tensile modulus of carbon nanotube-reinforced composites. The results showed that the disparity between theoretical predictions and experimental data could be due to different inclusion waviness conditions. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction The widespread use of composites in industrial applications is largely due to their exceptional performances and high propertyto-weight ratios. Among all types of composites, a popular one is made from fibrous inclusions with polymer matrix, also known as fiber-reinforced composites (FRC). The overall properties of FRC depend on the properties of the constituent materials, as well as the microstructure therein. With respect to the stiffness of composites, several microstructure parameters have been found to be influential on composite stiffness: the inclusion aspect ratio (ratio of inclusion length and diameter) [1–3], their orientation [2,4–6], waviness [7–11], and the overall inclusion volume fraction [1–3]. To compute composite stiffness via constituent properties and corresponding microstructures is known as composite micromechanics. Although the theory of micro-mechanics has been well-studied, most existing models are originally derived for composites having unidirectional inclusions with constant inclusion structure and properties [1]. In other words, those models were deterministic in nature, and the statistical variations of the input variables were ignored. This may bias the results when using these models to predict composite stiffness, especially for fiber-reinforced composites ⇑ Corresponding author. Tel.: +1 850 410 6355; fax: +1 850 410 6342. E-mail address: [email protected] (C. Zhang). 1359-8368/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2010.09.004
with each inclusion having different aspect ratio, orientation and waviness. Researchers have been trying to solve these problems over the past decades. For instance, some researches focused on investigating the effects of misaligned inclusions to the stiffness of short-fiber-reinforced composites [2,4–6]. Many researchers also studied the variations of inclusion aspect ratio [2–3]. However, limited information on studies are available with respect to the inclusion waviness variation. Some of the existing studies focused on deriving analytical expressions for effective composite stiffness with wavy inclusions (e.g. Hsiao and Daniel [7], Chan and Wang [8]). Some others applied finite element analysis to construct an effective wavy inclusion element, and combined with traditional micromechanical models to predict the overall effective stiffness of composites [9–11]. In the studies of Anumandla and Gibson [10], the inclusion waviness in the composites was assumed homogeneous (i.e. all inclusions possess identical waviness), and thus the waviness distribution effects were ignored. Fisher et al. [9] and Bradshaw et al. [11] extended Mori–Tanaka method to composites consisting of more than one type of inclusion type via the method introduced by Tendon and Weng [12], so that the distribution of CNT waviness can be considered. They also demonstrated the effect of CNT waviness distribution by introducing a discrete probability distribution. The objective of this study was to conduct a comprehensive analysis focusing specifically on the stochastic effect of inclusion waviness to the stiffness of fiber-reinforced composites. The
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effective composite stiffness were computed through micromechanical models, specifically, Mori–Tanaka method [13] for unidirectional composites, Hsiao-Daniel model [7] for composite with wavy inclusions, and Advani–Tucker’s orientation averaging [4] for composite with misaligned inclusions. The Voigt method of composite homogenization and numerical Monte-Carlo simulation were conducted in conjunction of micromechanical models to compute average composite stiffness through composite microstructural (i.e. fiber aspect ratio, waviness and orientation) distributions. Continuous inclusion waviness and aspect ratio distribution with both symmetric and asymmetric shapes were considered, and the four composite elastic constants (elongation and transversely tensile moduli E11 and E22, shear modulus G12, and Poisson’s ratio m12) were computed and compared under different conditions. We found that the distribution of inclusion waviness is influential to the composite stiffness in most cases, especially when asymmetric distribution was presented. More details of the simulation results will be discussed in Section 3. We also introduced a carbon nanotube-reinforced composite in Section 4 to demonstrate the waviness distribution effect on nanocomposite system. The simulated composite tensile moduli were compared to the physical experimental data conducted by Dynamic Mechanical Analysis (DMA). Several modeling limitations are listed here. As micro-mechanics were used as the basis of computation, standard micromechanics assumptions (see, e.g. Tucker and Liang [1]) hold. It is also noted that different micromechanical models could result in different waviness effects because of the underlying assumption differences. Our results are only held under the models applied. 2. Technical approach 2.1. Micromechanical modeling for composite stiffness with wavy inclusions The properties of composites were determined by the constituent materials, their structure and distributions. With respect to the stiffness prediction of fiber-reinforced composites, the composite micro-mechanics is a very efficient technique and is well-accepted by many researchers. A popular model is the mean field method developed by Mori and Tanaka [13], which is based on Eshelby’s strain-concentration tensor around an ellipsoidal inclusion in the matrix [14]: Eshelby
A
m
f
m
1
¼ ½I þ ES ðC C Þ
ð1Þ
Eshelby
where A is Eshelby’s strain-concentration tensor, I is the identity tensor, and E is the Eshelby tensor, which depends only on the inclusion aspect ratio and the matrix elastic constants [14]. S and C are composite compliance tensor and stiffness tensor, with the subscripts f and m designating the fiber and matrix, respectively. Based on Eshelby’s theory, Mori and Tanaka [13] proposed the concept of effective field theory, which assumes that when many identical inclusions are introduced in a concentrated composite, the strain over each inclusion is equal to the average strain in the matrix, and thus provided results applicable to higher inclusion volume fractions [13]:
AMT ¼ AEshelby ½ð1 V f ÞI þ V f AEshelby 1
63
As a physical consideration, we recognize that the inclusions within the composites usually possess certain degree of curvature, especially for flexible inclusions with larger aspect ratio. A good example for long flexible inclusion is the carbon nanotubes (CNTs). Fig. 1 shows a scanning electron microscopic (SEM) image of a twodimensional CNT texture surface. This kind of CNT textures (also known as CNT buckypapers) can be used as preforms for CNT-reinforce composites [15]. Fig. 1 clearly shows that most of the CNT bundles possess certain degree of curvature, indicating that the curvature effect must be taken into account for predicting the stiffness of this kind of composites. Several researchers introduced the concept of inclusion waviness to model the curvature effect [7–11]. Hsiao and Daniel [7] derived analytical expressions for unidirectional composite stiffness with uniform fiber waviness. A complete set of twelve elastic constants were derived with respect to the defined waviness parameters. Their work is used to introduce the concept of inclusion waviness and its effect to the composite stiffness. In the definition of inclusion waviness, it is assumed that each inclusion forms sinusoidal wave shape, as shown in Fig. 2, where A is the amplitude and L is the wavelength. For uniform waviness, it is sufficient to consider a representative volume element containing one period of the waviness. Assume axis X1 being the direction along the aligned inclusion and applied load. The elastic properties of the composites are determined from the average strain obtained in the representative volume, which is the integration of the strain of every infinitesimal thickness over one wavelength in the loading (X1) direction. To get the effective stiffness of the uniform-waviness model, a waviness parameter w is introduced as a function of amplitude and wavelength as w = A/L. Twelve elastic constants, three axial moduli, three shear moduli, and six Poisson’s ratios, were derived with respect to the defined waviness parameter (see Hsiao and Daniel [7] for detailed discussions). Note that in the presented paper, we considered inclusion waviness occurring in the 1–2 plane, while in Hsiao–Daniel paper [7] the waviness was in x–z plane. Therefore a swap of y and z in their notation is required (i.e. E22 in the presented paper is Ez in Hsiao-Daniel paper [7]). It is also worthwhile to emphasize that the waviness effect in Hsiao–Daniel paper [7] only depends on waviness parameter (i.e. the ratio of wavelength and amplitude). Therefore the actual magnitude of wavelength and amplitude (and their relationship to fiber diameter) is irrelevant. 2.2. Inclusion waviness distribution and modeling Before the inclusion waviness effects on the overall properties of composites can be analyzed, its distribution function and
ð2Þ
where Vf is the composite volume fraction. The Mori–Tanaka strainconcentration tensor AMT can then be used to compute the overall composite stiffness tensor C as [1]:
C ¼ C m þ V f ðC f C m ÞAMT
ð3Þ
It is noted that the original form of Mori–Tanaka method can only work for composites with straight and unidirectional inclusions.
Fig. 1. Surface image of a single-walled carbon nanotube texture taken by scanning electron microscopy (SEM).
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orientation sampled from the corresponding distributions, and the same concentration as the original RVE (as depicted in the right section of Fig. 3). The effective stiffness tensor of the ith unit element, labeled Ci, defined as:
X2
ei ¼ C i ri
ð6Þ
can thus be computed through corresponding inclusion aspect ratio and waviness using micromechanical models, where ei and ri are stress tensor and strain tensor for the ith element, respectively. From the Voigt method, a constant mean strain over each unit element was assumed,
X1
ei ¼ e Fig. 2. Illustration of lamina with fiber waviness.
modeling techniques must be defined. As a physical consideration, a composite may contain inclusions having various structure parameters (e.g. aspect ratio and waviness). Although the variations can be statistically represented by their mean and variance, it is often insufficient if one does not consider the whole probability distributions. For instance, a symmetric distribution and a nonsymmetric distribution can have totally different behavior, even if they have the same mean and variance. Probability distribution can be roughly grouped into two types: normal (or symmetric) distributions, and non-normal (or asymmetric) distributions [16]. A normal distribution function with random variable X has the form of [16]:
" 2 # 1 1 X l PðXÞ ¼ pffiffiffiffiffiffiffi exp 2 r r 2p
ð4Þ
where l and r are mean and variance of the distribution, respectively. However, many real distributions fall into the category of asymmetric distribution, such as Gamma distribution and exponential distribution. One of the objectives of this research is to study the distribution effects of the inclusion waviness to the effective composite stiffness. We introduce a log-normal distribution as the representation of asymmetric waviness distribution, which has the function form of [16]:
" 2 # 1 1 ln X k pðXÞ ¼ pffiffiffiffiffiffiffi exp 2b2 b 2p X
ð5Þ
where k and b are distribution parameters and both have positive values. The mean and variance of a log-normal distribution are exp(k + b2/2) and exp(2k + b2)(exp(b2) 1), respectively [16]. As mentioned in the previous section, the micromechanical models (e.g. Mori–Tanaka method and Hsiao–Daniel model) can only deal with uniform inclusion properties and microstructures because of their deterministic nature. To address this issue, the Voigt method of composite homogenization were often applied (see, e.g. Jack and Smith [6]), and the concepts are described as follows. We first considered a simplified composite representation volume element (RVE) with misaligned inclusions having variable aspect ratio and waviness, as shown in the left section of Fig. 3. The inclusion aspect ratio, waviness and orientation within the RVE are assumed to follow certain distributions; therefore, the effective stiffness tensor of the RVE, hCRVEi, cannot be computed directly through micromechanical models because of their deterministic limitations (i.e. the inputted microstructural parameters are not constants). The composite homogenization process is thus applied to decompose the whole RVE into several independent unit elements. Each unit element is assumed having unidirectional inclusions with uniform (but unique) waviness, aspect ratio and
ð7Þ
therefore the stiffness of the RVE is simply the expectation of the total unit elements combined. If the number of total unit elements is N, the expectation for the effective RVE stiffness tensor can be computed as:
hC RVE i ¼
N 1X Ci N i¼1
ð8Þ
Here we introduced a semi-numerical approach to compute the stiffness expectation. The computational steps are summarized as follows. (1) Define probability distribution functions for inclusion aspect ratio, waviness and orientation. (2) Apply Monte-Carlo sampling to draw aspect ratio and waviness samples from the corresponding aspect ratio and waviness distribution. (3) Compute effective composite stiffness Ci through inclusion aspect ratio and waviness sampled from the previous step (Mori–Tanaka method and Hsiao–Daniel model). Store the result. (4) Repeat step (3) for N times. (5) Compute average stiffness tensor (i.e. expectation) using Eq. (8). (6) Apply orientation averaging (Advani and Tucker [4]) to compute effective composite stiffness through defined orientation distribution function. Note that we assumed that inclusion aspect ratio, waviness and orientation are decoupled; therefore they can be sampled independently from the corresponding probability distributions using Monte-Carlo sampling. It is worthwhile to mention that the expectation of composite stiffness with misaligned inclusions having uniform aspect ratio and waviness has been analytically solved (see, e.g. Advani and Tucker [4], Jack and Smith [6]). However, for conjunction to non-uniform inclusion aspect ratio and waviness, the analytical form of the composite stiffness expectation has yet to be developed, and needs to be computed numerically through Monte-Carlo method or similar approaches.
3. Discussion In this section we conduct a comprehensive analysis on the effects of inclusion waviness and its distribution to the effective stiffness of composites. Different waviness conditions were considered. First, we consider a uniform inclusion waviness state with either variable inclusion orientation or aspect ratio. Then the variation of inclusion waviness was analyzed under uniform aspect ratio and unidirectional inclusions. Finally, the waviness influence on a carbon nanotube-reinforce composite is discussed.
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Fig. 3. Illustration of composite homogenization.
3.1. Effect of inclusion waviness to composite stiffness under different orientation states The inclusion orientation problems have been studied intensively by many researchers over the past decades (see, e.g. Refs. [4–6]). We followed the orientation representation introduced by Advani and Tucker [4], but similar works can also be found elsewhere. Three different fiber orientation states are discussed in this Section: (1) perfect fiber alignment (unidirectional), (2) planar fiber orientation state (2-D orientation), and (3) three-dimensional orientation state. The inclusion waviness under different orientation conditions were discussed. Note that for the later two orientation states, the fiber can either be randomly distributed or partially aligned. To model inclusion orientation, it is convenient to introduce a three-dimensional orientation function defined as: 2m
wðh; /Þ ¼ k sin
h cos2n /
ð9Þ
where n and m are positive integers related to the degree of fiber alignment, and higher the m, n values, the higher the degree of fiber alignment along X1 corresponding to (h, /)=(p/2, 0) and (h, /) = (p/ 2, p). k is a constant determined by satisfying the normalized condition of W. h and / are Euler’s angles defining the orientation for each individual inclusion, as shown in Fig. 4. Advani and Tucker [4] proposed a orientation averaging process to compute the effective composite stiffness through the defined orientation function. This technique was employed in this study to compute the effective stiffness of composites under different orientation states (see, e.g. Advani and Tucker [4] for the complete discussion of orientation averaging). Note that for unidirectional fiber with constant fiber aspect ratio and waviness, the composite stiffness tensor was computed directly through the Mori–Tanaka method, as outlined in Eqs. (1)– (3). However, different fiber orientation distribution functions were chosen to compute effective composite stiffness with misaligned fiber states. For 2-D plane orientation state, the out-of-plane angle h was p/2, which eliminates the cosine term in Eq. (9), and the orientation distribution function was reduced to w(/) = kcos2n /. This reduced form is the 2-D orientation distribution function,
X3
θ
X2
φ X1 Fig. 4. Coordinate system defining the fiber orientation in 3-D space.
and we selected n = 0 and n = 1 to describe 2-D random and 2-D partially aligned state, respectively. 3-D orientation states are represented by Eq. (9), where n = m = 0 and n = m = 1 were used to describe 3-D random and 3-D partially aligned states, respectively. The material properties used in the simulation were typical of fiber-reinforced engineering thermoplastics adopted from Tucker and Liang [1]. Specifically, the fibers were assumed isotropic with Young’s modulus and Poisson’s ratio equating 30 GPa and 0.2, respectively, and the matrix were 1 GPa and 0.38. The fiber volume fraction was 20%. A plane stress state was assumed; therefore, the elastic behavior of the composite can be described by four elastic constants: axial and transverse tensile moduli E11 and E22, shear modulus G12, and Poisson ratio m12. The sample size N used for Monte-Carlo simulation is set 107. The effects of orientation distribution on composites with constant fiber aspect ratio and waviness are demonstrated in Fig. 5. The fiber aspect ratio was set constant at 10. Fig. 5a shows that fiber waviness has dramatic effect on E11 for unidirectional composite, but the degree of effects are smaller if the fibers are somewhat misaligned. For 3-D random orientation state, the inclusion
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Fig. 5. Fiber orientation distribution effects on composite stiffness: (a) E11/Em, (b) E22/Em, (c) G12/Gm, (d)
waviness had a minimal effect on E11. Fiber waviness also had significant effects on E22 and m12 for most of the fiber orientated states, except for 3-D random composite, whose E22 and m12 stay nearly constants for different fiber waviness levels. The increase of fiber waviness was observed to lead to an increase of composite E22, if the fiber orientation is partially aligned or unidirectional. The fiber waviness also affects the shear modulus of composites with different fiber orientation states, but the effects were only significant when the fibers are either 2-D random or unidirectional. Also note that under 3-D random and 2-D aligned conditions, the resulting composites have identical shear modulus and therefore the results are overlapped in Fig. 5c.
3.2. Effect of inclusion aspect ratio distribution with uniform fiber waviness In this section, we discuss the effects of different fiber aspect ratio (denoted as a) distributions on unidirectional composites with constant fiber waviness. Three different fiber aspect ratio conditions: (1) non-symmetric distribution (represented by log-normal distribution), (2) symmetric distribution (represented by normal distribution), and (3) constant aspect ratio, were considered. For all the three conditions, the average fiber aspect ratio was set to be either 10 or 100, with the standard deviation equaling 50% of the mean based on the experimental observation conducted by Yeh [17].
v12/vm.
Fig. 6 illustrates the effects of different fiber aspect ratio distribution on the four composite elastic constants. It can be seen in most cases, different aspect ratio distributions lead to different composite property predictions. When the fiber aspect ratio is relatively large (i.e. a = 100), the prediction results for composites with Lognormal aspect ratio distribution and the constant one is very similar. However, the properties of composites would be over-estimated if the fiber aspect ratio follows normal distribution. With respect to the composites with smaller fiber aspect ratio (i.e. a = 10), similar prediction results can also be seen. If the aspect ratio follows normal distribution, the composite E11, E22, and G12, will be over-estimated. On the contrary, the three of the four elastic properties will be under-estimated if the composite has lognormal fiber aspect ratio distribution. In summary, the distribution of fiber aspect ratio has significant effects on composite property prediction. This also implies the lack of prediction accuracy if one considers only the average fiber aspect ratio to predict the composite stiffness using micromechanical models.
3.3. Effect of fiber waviness distribution with constant fiber aspect ratio The effect of different fiber waviness distributions on unidirectional composites stiffness is also discussed. For the constant fiber aspect ratios, three different waviness distributions were considered: non-symmetric distribution (represented by log-normal dis-
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Fig. 6. Fiber aspect ratio distribution effects on composite stiffness a (a) E11/Em, (b) E22/Em, (c) G12/Gm, (d)
tribution), symmetric distribution (represented by normal distribution), and constant fiber waviness. Similar to the previous section, two different waviness means, 0.1 (high) and 0.01 (low), were assumed. The standard deviation of the waviness distribution set were 50% of the mean. Fig. 7 illustrates the effective composite properties with different fiber waviness distributions. It was observed that at smaller fiber waviness conditions, the predicted results for composite with lognormal waviness distribution were nearly identical to the one with constant waviness. On the other hand, under the current modeling assumption, if the composite possesses normal distributed fiber waviness, it was very likely that three of the four elastic constants (E11, E22, G12) would be over-estimated. With respect to the composites with larger fiber waviness, the three different waviness distributions lead to different predictions for composites E11, E22 and G12. Composites with Lognormal fiber waviness distribution had the highest predictions on E11 and E22. On the other hand, composite with constant fiber waviness had the highest predictions on G12 and m12. Normal distributed fiber waviness appeared to have the tendency to result in lower composite property predictions. Waviness distribution effects on composite stiffness at different volume fraction levels were also investigated. Similar to the previous analysis, three different fiber waviness distributions were assumed. The analysis was conducted based on different fiber aspect ratio and waviness combinations; therefore, four different
v12/vm.
conditions were considered. Fig. 8 demonstrated the simulation results, and different fiber aspect ratio and waviness combination are represented by different colors. It was observed that the distribution of fiber waviness had a significant effect on composite properties. In most cases, the lack of fiber waviness distribution assumption would lead to inaccurate composite stiffness predictions, especially for composite with higher fiber volume fraction. 3.4. Inclusion waviness effect on carbon nanotube-reinforced composites In this section, we demonstrate the inclusion waviness effect on the tensile modulus of carbon nanotube-reinforced composites. A unique manufacturing technique was employed in this research to produce multi-layered CNT-reinforced composite laminates. This process was first developed by the researchers at High Performance Material Institute (HPMI) [15]. In their process, carbon nanotube structure and dispersion were controlled by fabricating them into thin films, also known as the CNT buckypapers (BPs), using a multi-step sonication and filtration process [15]. Buckypapers were used as preforms to make multi-layered buckypaperpolymer composite laminates, which were found to have good mechanical properties and high nanotube loading [15]. It is accepted by many researchers that CNT-reinforced composites can be considered as fiber-reinforced composites, so that their
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Fig. 7. Fiber waviness distribution effects on composite stiffness: (a) E11/Em, (b) E22/Em, (c) G12/Gm, (d)
mechanical properties can be predicted using micro-mechanics (see, e.g. Ref. [18]). In this study, an eight-layered buckypaperpolymer composite was produced using randomly dispersed single-walled CNT buckypapers and Epon 862 polymer matrix. The composite was cured at 177 °C under vacuum to remove possible void formation. Five samples were produced using the same process. The mechanical properties of the BP-polymer composites were measured using Dynamic Mechanical Analysis (DMA). For each composite sample, the DMA test was conducted three times to get the average tensile modulus. It is worthwhile to mention that the buckypaper is a two-dimensional CNT texture; therefore, the CNT orientation distribution can be represented by two-dimensional random orientation distribution function introduced above. In addition, based on some prior studies, the CNT length and diameter within the buckypapers were measured using image processing techniques [17]. Specifically, both CNT length and diameter followed the Weibull distribution, with length follows Weibull (1.89 lm, 1.93), and diameter follows Weibull (8.225 nm, 1.63) [17]. The average CNT aspect ratio was 470 [17]. Also, as a final physical consideration, we recognize that the stiffness of a nanotube bundle will be a function of the effective diameter of the bundle. Liu et al. [19] presented a hybrid atom/continuum model to study the bulk elastic properties of SWNT bundles and
v12/vm.
developed a complete set of the five independent elastic constants from molecular dynamics (MD) simulations. They found that the five elastic constants were all directly related to bundle diameter. Therefore, for each Monte-Carlo sampling, one nanotube bundle diameter was generated from the corresponding Weibull distribution, and thus the five elastic constants for the specific nanotube bundle were determined. The bundle’s effective transversely isotropic stiffness tensor, was then formed for use with the Mori–Tanaka method as outlined in Eqs. (1)–(3). With respect to the properties of the surrounding epoxy resin matrix, we used the data in Yamini and Young’s paper [20], with an elastic modulus of Em = 2.5 GPa, shear modulus Gm = 1.2 GPa, and Poisson’s ratio mm = 0.3 thus completely defining the matrix’s isotropic stiffness tensor Cm expressed in Eq. (3). The waviness distribution of the CNT bundle within the buckypapers, however, is not recognized yet. From the SEM image of buckypaper surface illustrated in Fig. 1, only slight bundle waviness was observed. Therefore, a normal waviness distribution is assumed, with the mean ranges from 0.01 to 0.05, and the standard deviation equals 50% of the mean. Fig. 9 shows the tensile modulus comparison of experimental data and micromechanical simulation predictions. The trend of DMA results seem to correspond with the simulation quite well. With the assumption of normal CNT bundle waviness distribution with the mean equal to 0.05, the micromechanical simulation
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Fig. 8. Fiber waviness distribution effects on composite stiffness with different volume fraction (a) E11/Em, (b) E22/Em, (c) G12/Gm, (d)
v12/vm.
results were very close to the experimental data. However, as the CNT waviness distribution used here were arbitrarily selection, no clear conclusion can be made from the results of Fig. 9. The purpose of this analysis is rather to demonstrate the possible CNT bundle waviness effects on the resulting BPP composite modulus. More accurate simulation predictions can be made once the true CNT waviness distribution within the buckypapers is recognized.
4. Conclusions In this study we investigated the effect of inclusion waviness and its distribution to the effective composite stiffness. Different waviness conditions: uniform waviness with variable inclusion orientation or aspect ratio, uniform aspect ratio and variable waviness, were analyzed. The inclusion waviness was found to have a great effect on tensile moduli and shear modulus for unidirectional composites. However, if the inclusions were either randomly dispersed or only partially aligned, the degree of waviness effect was smaller. With respect to unidirectional composites with variable aspect ratio and uniform waviness, the composite properties tended to be over-estimated if inclusion aspect ratio follows sym-
Fig. 9. Experimental data and simulation results comparison.
metric distribution. The elastic moduli would also be over-estimated if variable inclusion waviness follows symmetric
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distribution. In addition, the waviness distribution effect is larger when larger inclusion waviness was introduced in the composite. The lack of fiber waviness distribution assumption was found to lead to inaccurate composite stiffness predictions, especially for composite with higher fiber volume fractions. We also demonstrated the inclusion waviness effect to the tensile modulus of carbon nanotube-reinforced composites. The results show that the disparity between theoretical predictions and experimental data could be due to different inclusion waviness conditions. References [1] Tucker CL, Liang E. Stiffness predictions for unidirectional short-fiber composites review and evaluation. Compos Sci Technol 1999;59:655–71. [2] Jiang B, Liu C, Zhang C, Wang B, Wang Z. The effect of non-symmetric distribution of fiber orientation and aspect ratio on elastic properties of composites. Compos Part B – Eng 2007;38:24–34. [3] Hine PJ, Lusti HR, Gusev AA. Numerical simulation of the effects of volume fraction, aspect ratio and fibre length distribution on the elastic and thermoelastic properties of short fibre composites. Compos Sci Technol 2002;62:1445–53. [4] Advani SG, Tucker CL. The use of tensors to describe and predict fiber orientation in short fiber composites. J Rheol 1987;31:751–84. [5] Chen CH, Cheng CH. Effective elastic moduli of misoriented short-fiber composites. Int J Solids Struct 1996;33:2519–39. [6] Jack DA, Smith DE. Elastic properties of short-fiber polymer composites, derivation and demonstration of analytical forms for expectation and variance from orientation tensors. J Compos Mater 2008;42:277–308. [7] Hsiao HM, Daniel IM. Elastic properties of composites with fiber waviness. Compos Part A – Appl Sci Manuf 1996;27:931–41. [8] Chan WS, Wang JS. Influence of fiber waviness on the structural response of composite laminates. J Thermoplast Compos Mater 1994;7:243–60.
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