Effect of polydispersivity and porosity on the elastic properties of ...
Mechanics of Materials, 2010. 42(7): p. 726-739.
Effect of polydispersivity and porosity on the elastic properties of hollow particle filled composites M. Aurelia , M. Porfiri∗,a , N. Guptaa a Department of Mechanical and Aerospace Engineering Polytechnic Institute of New York University, Six MetroTech Center, Brooklyn, NY 11201, USA
Abstract Hollow particle filled composites are characterized by a variety of properties of interest to marine structural applications, including low density, high specific modulus, and low moisture absorption. Usually, dispersed microspheres exhibit widely different geometric characteristics, such as wall thickness and outer radius. In this paper, we develop a homogenization technique based on the differential scheme that accounts for polydispersivity in geometry of inclusion phases in hollow particle reinforced composites. We find manageable differential expressions to predict the composite effective moduli in terms of a variety of concurrent factors, including matrix and particle elastic constants, geometrical properties of hollow particles, volume fractions, and void content in the matrix. Theoretical findings are validated by experimental data and are compared with results of several published models. We find that polydispersion modeling allows for a closer agreement with experimental results than monodisperse schemes. In particular, qualitatively different predictions are found in syntactic foams containing thin walled particles by accounting for polydispersivity. Key words: homogenization, particle reinforced composites, polydispersivity, porosity, syntactic foams
1. Introduction Syntactic foams are fabricated by dispersing hollow microspheres in a matrix material with a twofold purpose: to embed closed cell porosity in the matrix, thus reducing the material density while controlling the size and distribution of the porosity; and to reinforce the matrix phase by using particles of a stiffer material than the matrix, see Narkis et al. (1982). Increasing interest towards understanding the properties of syntactic foams has resulted in extensive experimental characterization, see for example Gupta et al. (2001, 2004); Kishore et al. (2006). These studies have demonstrated that the properties of syntactic foams can be tailored by changing either the particle volume fraction or the wall thickness. Such approaches are also used in developing functionally graded syntactic foams, which show significantly higher energy absorption capabilities under compressive loading than the plain syntactic foams containing random distribution of particles, see for example Caeti et al. (2009). Through characterization of numerous compositions and material systems, these studies have provided insight into the structure-property correlations for syntactic foams under a variety of loading conditions, including tensile, compressive, flexural, and impact, see for example Kim and Mitchell (2003); Kishore et al. (2005); Gupta et al. (2009). These studies have highlighted the possibility of tailoring the mechanical, thermal, and electrical properties of syntactic foams, see for example Grosjean et al. (2009); Gupta et al. (2006); Shabde et al. (2006); Wouterson et al. (2004); Zhang and Ma (2009). The ability of tailoring syntactic foams has played an important role in enabling their applications in a diverse set of fields, including marine structures, aerospace structures, and sports equipment, see for example Ishai et al. (1995). Compared to the extensive experimental literature, relatively few modeling efforts on syntactic foams are available. In Huang and Gibson (1993), the elastic properties of syntactic foams are analyzed by studying the infinitely dilute hollow inclusion problem. In Marur (2005), the three phase homogenization scheme originally proposed in Christensen ∗ Corresponding
author. Tel.: +1-718-260-3681. Fax: +1-718-260-3532. Email addresses: [email protected] (M. Aureli), [email protected] (M. Porfiri), [email protected] (N. Gupta)
Preprint submitted to Mechanics of Materials
September 20, 2009
(a)
(b)
Figure 1: (a) Micrograph of a representative sample of commercially available microballoons used in syntactic foams. The particles’ outer radius distribution varies in the broad range, approximately 10–100 µm. (b) Schematic representation of polydispersivity in hollow spheres. In (b): (top row) polydispersivity in outer radius, with radius decreasing from left to right; and (bottom row) polydispersivity in wall thickness, with wall thickness increasing from left to right.
and Lo (1979) is used to study syntactic foams. In Porfiri and Gupta (2009), the differential scheme, see McLaughlin (1977); Norris (1985); Zimmerman (1991), is applied to model the elastic properties of syntactic foams, obtaining good agreement with experimental results in high inclusion volume fraction range. A common underlying theme in these models is to assume particles to be of same size and wall thickness. However, this assumption is not close to the reality and limits the usefulness of some of these approaches. Figure 1(a) shows a randomly selected sample of glass microspheres obtained from 3M, MN. These microballoons are commonly used in manufacturing syntactic foams, see for example Bardella and Genna (2001); Wouterson et al. (2004), and have an average particle diameter of 40 µm and density of 460 kg m−3 . The micrograph shows that the actual particle size varies over a wide range, from about 10 µm to 100 µm. In addition, the wall thickness of each particle can be different and can also exist over a large range. Figure 1(b) schematically represents such polydispersion in the size and wall thickness of these microballoons. The top row of particles in Fig. 1(b) illustrates inclusion size polydispersion, whereas the bottom row represents polydispersion in the inclusion wall thickness. A recent experimental study, see Gupta et al. (2009), has shown that the size and wall thickness values for a large batch of particles are indeed close to the nominal values supplied by the manufacturer. However, sieving these particles according to their size and measuring the true particle density of each fraction show that larger particles have lower density due to thinner walls, see Gupta et al. (2009). This implies that the larger size particles will be more compliant, thus indicating that inclusion of polydispersivity in models can provide better predictive capabilities. The problem of polydispersivity in solid particle filled composites has been originally addressed in Budiansky (1965); Hill (1965), where the first homogenization methods for polydisperse and multiphase particulate systems are discussed. In Huang et al. (1994), an extension of the three phase method to account for multiphase and polydisperse solid inclusion is presented. In Duan et al. (2007a,b), this formulation is generalized to include the presence of elastic interface effects. In Bardella and Genna (2001), the homogenization technique proposed in Herv´e and Pellegrini (1995) is adopted to account for filler particles gradation in syntactic foams. In Dai et al. (1998), a generalized self consistent Mori-Tanaka method, see Benveniste (1987), is proposed to determine the effective moduli of multiphase particulate composites. Modified Mori-Tanaka approaches have also been presented in Iwakuma and Koyama (2005); Peng et al. (2009) to study multiphase composites. Recently, in Zouari et al. (2008), a general framework for iterative homogenization of particle filled composites is proposed. It is therein assumed that the inclusion phase is progressively introduced into the matrix material in a number of successive steps, each step yielding the new effective media for the next addition step. The approach offers a more tractable and computationally oriented ground for analyzing polydispersivity than the morphological approach in Bardella and Genna (2001) without compromising the accuracy of the findings, see Zouari et al. (2008). 2
d
b
c d a c (a)
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Figure 2: (a) Micrograph and (b) schematic representation of the microstructure of a typical syntactic foam. In (b), the different phases are shown, including ‘a’ matrix, ‘b’ voids, ‘c’ particles, and ‘d’ porosity embedded in the particle shell.
The aim of this study is to develop a micromechanics-based model for the interplays of the polydispersivity in the hollow inclusion volume fraction and wall thickness on the properties of the resulting syntactic foams. We propose a method for determining the effective elastic properties of syntactic foams by combining the modeling results discussed in Porfiri and Gupta (2009) with a homogenization technique stemming from the limit for infinitely small inclusion volume fraction of the procedure developed in Zouari et al. (2008). By introducing distribution functions to describe the polydisperse nature of the composition of the inclusion phase in the matrix, we obtain an analytical differential formulation, in contrast to the algorithmic process discussed in Zouari et al. (2008), to evaluate the effective properties of the composite. The proposed model is applicable to study inclusion volume fractions up to the packing limit and to both continuous and discrete polydispersions. The model predictions are validated through comparison with published experimental results on vinyl ester and polyester-glass syntactic foams and with findings of several available models. 2. Problem Statement Figure 2(a) shows a representative syntactic foam microstructure, which is schematically redrawn in Fig. 2(b). These figures show that syntactic foams are three-phase materials with polydisperse inclusions. Matrix and microballoons are the two primary phases. The third phase is the air voids entrapped in the matrix during composite synthesis. Experimental studies have shown the presence of 5–10% voids in most foam compositions, see for example Tagliavia et al. (2009a). The air present inside hollow particles is sometimes counted as an additional phase. However, this phase is automatically accounted for by means of particle size and wall thickness and no further reference of this phase is required in the present work. In this work, matrix, particles, and voids are identified using subscripts m, p, and v, respectively. In addition, the term inclusion refers to both the particle phase and the void phase. We denote with Φ and Φm the volume fractions of the inclusion phase and of the matrix material in the composite, respectively, so that Φm + Φ = 1. The particle and void volume fractions are denoted with Φ p = φ p Φ and Φv = φv Φ, respectively, so that Φ p + Φv = Φ. In the limit Φ → 0, φ p and φv remain well-defined and finite. The inclusions are assumed to be of spherical shape and all particles are assumed to be made of the same material. However, this hypothesis is not essential in the development of the model, whose formulation allows for treatment of spherical particles of different materials. The ratio of the inner radius ri to the outer radius ro of the inclusion is called radius ratio and is denoted by η. Note that η = 0 corresponds to the case of a solid particle, whereas η = 1 denotes a spherical void embedded in the matrix. A large number of existing experimental studies on syntactic foams have used glass microspheres supplied by 3M. The average diameter and density of four types of microballoons provided by the manufacturer are presented in Table 1. The calculated values of wall thickness and radius ratio are also provided in Table 1, assuming the glass density 3
Table 1: Properties of hollow glass microspheres used in the experimental analysis.
Particle Typea
Average Average Density Diameter (kg/m3 ) (µm) S22 220 35 S32 320 40 S38 380 40 K46 460 40 a Manufacturer’s code
Calculated Wall Thickness (µm) 0.521 0.878 1.052 1.289
Radius Ratio η 0.970 0.956 0.947 0.936
of 2540 kg m−3 . Although most published studies refer to the average values of properties of these microballoons, they are actually polydispersed systems. The size and density distribution of these microballoons are presented in Figs. 3. The inclusion size is measured by sieving them in four size fractions using a sieve shaker and measuring the true particle density of each segment using a pycnometer to obtain the estimate of polydispersivity, see Gupta et al. (2009) for the detailed dataset. Figures 3 show that, in the four examined compositions, approximately 20 vol.% of the particles have radius ratio significantly lower than the average value. On the other hand, this implies that most particles have thinner walls than the average value. Therefore, wall thickness distribution should be considered to accurately predict the effective properties of the composite. 3. Model development 3.1. Description of inclusion polydispersivity The formulation of the polydispersivity problem can be achieved by following either a discrete or a continuous approach. More specifically, the radius ratio distribution is described by a discrete function in the former case, and by a continuous function in the latter case. In what follows, for clarity of presentation, we first consider the discrete approach and then adapt it to the continuous scenario through a limit process. We introduce a discrete distribution function by referring to N + 1 families of particles. The j-th inclusion family has radius ratio η j and the index j goes from 0 to N. The values ψ j express the volume fraction of the inclusion phase for the j-th family. The void volume fraction is associated to the family of inclusions with j = 0. Additionally, N X
N X
ψ j = 1;
j=0
ψ j = φp
(1)
j=1
Note that, according to this convention, the inclusion phase belonging to the j-th family, with volume V j , has a volume fraction given by V j /V = Φψ j ; ψ0 = V0 /V = Φv /Φ = φv (2) where V indicates the total volume of the composite material. In addition, we define a scaled version of the discrete distribution function ψ j , say χ j , defined for j = 1, . . . , N, so that φ p χ j = ψ j and χ j = V j /(Φ p V). Here, ψ j and χ j coincide when Φv = 0, that is, when there are no voids present in the matrix. The extension of these definitions to the continuous case is obtained by introducing a continuous distribution function ψ(η) such that ψ(η) dη represents the volume fraction of the inclusion for which the particles have a radius ratio in the interval [η, η + dη). The relations Z 0
Z
1
ψ(η) dη + φv = 1;
1 0
ψ(η) dη = φ p
(3)
represent the continuous counterpart of Eq. (1). We note that the distribution ψ(η) does not account explicitly for the presence of voids in the matrix, described by φv . In analogy with the discrete case, we introduce a scaled distribution R1 function φ p χ(η) = ψ(η), so that 0 χ(η) dη = 1. 4
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Figure 3: Experimental distribution of microballoon radius ratio η for (a) S22, (b) S32, (c) S38, and (d) K46 microballoons. The error bars correspond to one standard deviation. The gray dashed line indicates the average particle radius ratio calculated using the experimental data. The blue dash-dotted line indicates the nominal values reported in Table 1.
5
These distribution parameters are a suitable description of the inclusion polydisperse nature. In particular, using ψ j or ψ(η) allows to refer to the whole inclusion phase in the matrix, that is, to both the particles and the voids. On the other hand, χ j or χ(η) describe the composition of the sole particle phase, not accounting for the voids. The two descriptions are related by the inclusion composition parameters φ p and φv . 3.2. Differential scheme We model the composite as a linear elastic solid and assume it to be macroscopically homogeneous and isotropic. Hence, two effective elastic properties need to be determined to fully characterize the fourth order elasticity tensors that describe the effective constitutive behavior of the composite material, namely, the effective bulk modulus k and shear modulus µ or, equivalently, the effective Young’s modulus E and the Poisson’s ratio ν. For a monodisperse system of radius ratio η, the change in the composite’s elastic properties dE and dν due to an increment in the inclusion volume fraction dΦ is given by dE = fE (E p , ν p , E, ν, η)E dΦ dν = fν (E p , ν p , E, ν, η)ν dΦ
(4a) (4b)
where E p and ν p represent the inclusion elastic properties and Φ is the inclusion volume fraction. Thus, the effective elastic constants for a given inclusion volume fraction are expressed as the solution of the system of coupled ordinary differential equations given in Eqs. (4) with the initial conditions E = Em and ν = νm , for Φ = 0. The functions fE and fν are reported in the Appendix, specialized for the case of a solid or a hollow particle inclusion, fE(p) and fν(p) , or a void, fE(v) and fν(v) , see also Christensen (1979). We note that closed form solutions of Eqs. (4) may be derived in special cases, such as rigid particles or incompressible phases, see for example Pal (2005); Zimmerman (1991). To study polydisperse systems, we assume that the composition of any given volume fraction of the inclusion phase is described by the distribution function ψ j , thus explicitly and uniquely defining the homogenization path in the composite material construction, as introduced in Norris (1985). Hence, the increment in the composite’s modulus due to the replacement of an infinitesimal volume of the matrix material with an inclusion belonging to the j-th family, is obtained by substituting in Eqs. (4) η with η j and dΦ with ψ j dΦ, and extending the summation over j = 0, . . . , N, that is, N X (p) (v) (5a) dE = ψ j fE (E p , ν p , E, ν, η j ) + ψ0 fE (E, ν) E dΦ j=1 N X (p) (v) (5b) dν = ψ j fν (E p , ν p , E, ν, η j ) + ψ0 fν (E, ν) ν dΦ j=1
To account for the inclusion packing limit, the increment dΦ can be replaced with dΦ/(1 − Φ/ΦL ), see for example Pal (2005), where ΦL is the packing limit which is solely determined by the size polydispersivity. For a monodisperse system, ΦL = 0.637 is defined as the random close packing limit and, for widely polydisperse systems, ΦL can be over 0.9, see for example Torquato et al. (2000); Torquato (2001). Equations (5) can be rewritten in the form valid for a discrete distribution N X dE E (p) (v) φ p (6a) χ j fE (E p , ν p , E, ν, η j ) + φv fE (E, ν) = dΦ 1 − Φ/ΦL j=1 N X dν ν (p) (v) φ p (6b) = χ j fν (E p , ν p , E, ν, η j ) + φv fν (E, ν) dΦ 1 − Φ/ΦL j=1
where we also used ψ j = φ p χ j and ψ0 = φv . Equations (6) are solved with initial conditions E = Em and ν = νm , for Φ = 0. We note that, in contrast to the algorithmic procedure presented in Zouari et al. (2008) and based on an iterative homogenization technique, here the effective elastic properties are directly determined by solving the differential equations in Eqs. (6). 6
The approach proposed in Eqs. (6) is extended to the continuous polydispersion case by using Eq. (3) as " Z 1 # dE E (p) (v) = φp χ(η) fE (E p , ν p , E, ν, η) dη + φv fE (E, ν) dΦ 1 − Φ/ΦL 0 " Z 1 # dν ν = φp χ(η) fν(p) (E p , ν p , E, ν, η) dη + φv fν(v) (E, ν) dΦ 1 − Φ/ΦL 0
(7a) (7b)
Here, we choose ΦL = 1 to account for polydispersion and for the presence of the voids, see for example Christensen (1990). The continuous model in Eqs. (7) can not be obtained directly from the formulation developed in Zouari et al. (2008) through a limit process. In addition, we use a Beta probability distribution function, see for example Abramowitz and Stegun (1965), for the functional form of the function χ(η), that is, χ(η) =
Γ(β1 + β2 ) β1 −1 η (1 − η)β2 −1 Γ(β1 )Γ(β2 )
(8)
where Γ denotes Euler’s Gamma function, and the parameters β1 and β2 describe the shape of the distribution and its statistical moments. We note that the values of β1 and β2 can be potentially determined from experimental indirect measurements of the particle wall thickness. In addition, the Beta distribution has values in the interval [0, 1], that is, in the range of physical variation of η. The mean m1 , standard deviation m2 , and skewness m3 of the distribution χ(η) are defined in terms of β1 and β2 as follows p √ 2(β2 − β1 ) 1 + β1 + β2 β1 β1 β2 m1 = (9) ; m2 = ; m3 = √ p β1 + β2 β1 β2 (2 + β1 + β2 ) (β1 + β2 ) 1 + β1 + β2 Other continuous distribution models have been discussed in the scientific literature on polydispersivity, such as the log-normal distribution and the Schulz distribution, see for example Torquato (2001). These distributions have been used to model polydispersivity in the outer radius of spheres and generally admit values in the interval [0, ∞), that is, in a non-physical range of variation for the radius ratio. For this reason, the Beta distribution appears to be a more appropriate description for the present model. 3.3. Sensitivity analysis The proposed polydisperse homogenization schemes, see Eqs. (6) and (7), represent a higher order correction to the monodisperse differential scheme discussed in Porfiri and Gupta (2009). We perform a qualitative analysis of this correction by studying the following rearranged version of Eq. (6a), where we drop the dependence of fE(p) and fE(v) on E p , ν p , E, and ν to simplify the notation, X dE/E = φp χ j fE(p) (η j ) + φv fE(v) dΦ/[1 − Φ/ΦL ] j=1 N
(10)
We note that a similar argument can be adapted to Eq. (6b) and Eqs. (7). P By defining the mean value for the radius ratio as η¯ = Nj=1 χ j η j , and introducing the deviations from the mean value δ j = η j − η, ¯ the right hand side of Eq. (10) is expanded in series in the neighborhood of η¯ to obtain (p) N 1 ∂2 fE (η) X dE/E χ j δ2j ' φ p fE(p) (¯η) + φv fE(v) + φ p dΦ/[1 − Φ/ΦL ] 2 ∂η2 η=η¯ j=1
(11)
P P where we use the properties Nj=1 χ j = 1 and Nj=1 χ j δ j = 0. This approximate expression allows for a qualitative P analysis of the proposed model with the assumption that the distribution exhibits moderately low variance, Nj=1 χ j δ2j , or, in other words, that the contribution of order δ3j and higher is negligible. The first term on the right hand side of Eq. (11) corresponds to the monodisperse behavior, where φ p = 1, and the particle wall thickness is given by the mean value of the distribution. The second term, linear in φv = 1 − φ p , represents the contribution to the monodisperse 7
5 4.5
E (GPa)
4 3.5 3 2.5 2 1.5 1 0
Huang and Gibson (1993) Bardella and Genna (2001) Zouari et al. (2008) Monodisperse (no voids) Present model
0.05
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Figure 4: Comparison between experimental results from Huang and Gibson (1993) and different homogenization techniques. The dashed curve ‘Monodisperse’ reports the results of a homogenization procedure that does not account for the presence of voids inside the material.
behavior of embedded voids in the matrix. This effect results in general decrease in stiffness of the composite material, as it can be seen from the negative sign of the functions fE(v) , see also Eqs. (12a), (13c), and (13d) in the Appendix. The third term is the second order correction due to the inclusion polydispersivity. More specifically, the effect of inclusion polydispersivity on the composite’s modulus is related to the distribution’s variance and yields a decrease or an increase in stiffness, according to the sign of the second derivative of the function fE(p) evaluated at η¯ . The signs of these derivatives generally depend on the relative elastic properties of particle and matrix material and on the mean radius ratio η¯ . For a given distribution variance, stiffening depends on the magnitude of the second derivative of fE(p) evaluated at η, ¯ which can be regarded as a sensitivity function. We further note that the correction due to polydispersivity becomes less significant as the void content increases and the distribution variance decreases. 4. Model validation 4.1. Comparison with reference models The polydisperse model in its discrete formulation, see Eqs. (6), is applied to a test case, to assess its predictive capabilities against available schemes and experimental results reported in Huang and Gibson (1993). The experimental data are obtained on polyester-glass syntactic foams, synthesized by using K1 particles from 3M. The constituents’ elastic properties are provided in Huang and Gibson (1993); Bardella and Genna (2001) and are as follows: Em = 4.89 GPa, νm = 0.4, E p = 70.11 GPa, ν p = 0.23. Since no experimental data on the radius ratio distribution of the K1 particles is provided in the original experimental study, we characterize the particle dispersion by using only the reported mean value for the radius ratio, that is, η¯ = 0.983. Results of the present model are shown in Fig. 4 and are compared with findings from the cited literature. The predictions of the proposed model show very good agreement with experimental results and a close agreement with the findings reported in Bardella and Genna (2001), especially at low particle volume fraction. Figure 4 shows that the proposed homogenization technique is capable of capturing the effect of voids on the stiffness of the composite material. Results of a monodisperse homogenization scheme that does not account for voids, also reported in Fig. 4, are generally in greater deviation from the experimental findings, especially when the void content is not negligible. We note that all the models in Fig. 4 do not explicitly account for particle to particle interactions, that are studied for example in Ma et al. (2004); Sarvestani (2003); Segurado and LLorca (2006); Shodja and Sarvestani (2001). These models provide a greater insight in the microstructural aspects of particulate composite. However their formulation is more complex than the homogenization methods discussed in Fig. 4. 4.2. Experimental validation with vinyl ester-glass syntactic foams To further validate the proposed model, we compare its prediction to the experimental results obtained from three point bending tests on sixteen types of vinyl ester-glass syntactic foams reported in Tagliavia et al. (2009a). In this 8
(a)
(b)
Figure 5: Micrographs on neat vinyl ester samples. The arrows indicate spherical voids embedded in the matrix phase. The diameter of the void inclusions is in the range 5–50 µm.
study, the wall thickness distribution and the volume fraction of the particle phase are varied in syntactic foams. In Tagliavia et al. (2009a), only the mean and the standard deviation of modulus are reported for each syntactic foam type. However, we take the full data set to compare predictions for each specimen, to account for the composition and void content of that specimen. Void content for each specimen is determined by following the procedure described in Tagliavia et al. (2009a) using the density of neat vinyl ester resin as ρm = 1161 kg m−3 . In this work, since a more accurate description of the microballoons’ size and density is available, we use experimentally determined mean particle densities, see Gupta et al. (2009), to recalculate the void volume fraction. The constituents’ elastic properties are adapted from Tagliavia et al. (2009a), namely: Em = 3.52 GPa, νm = 0.3, E p = 60 GPa, and νm = 0.21. The value chosen for the Young’s modulus of the neat vinyl ester resin is the maximum value obtained in Tagliavia et al. (2009a). Indeed, scanning electron micrographs on neat resin samples show the presence of microvoids on the fracture surface of most specimens, see for example Figs. 5, which contribute to lowering the specimen modulus. Therefore, it is considered that the highest modulus value obtained is close to the actual modulus of the neat vinyl ester resin. Results of the present model with inclusion polydispersivity and voids are shown in Figs. 6–9 superimposed on experimental data. The theoretical results are in close agreement with experimental findings. The trends are correctly reproduced and the experimental points generally fall in a 10% error band from the model predictions. These results show that the present model is capable of predicting elastic properties of syntactic foams while accounting for the particle polydispersion and presence of voids. 5. Model analysis In this Section, we analyze some representative results of the proposed model applied to vinyl ester-glass syntactic foams. Throughout the Section, the constituents’ elastic properties are selected as in Section 4.2, that is: Em = 3.52 GPa, νm = 0.3, E p = 60 GPa, and ν p = 0.21. 5.1. Parametric study of vinyl ester-glass syntactic foams The proposed method is specialized to the particle radius ratio distributions illustrated in Figs. 3, when the void volume fraction is set to 0. Figures 10 compare the results of the proposed method with several monodispersion-based homogenization techniques. More specifically, model predictions (curve ‘Poly’ in Figs. 10) are compared to the results obtained by homogenizing the inclusion phase with monodisperse wall thickness equal to the average value of the polydispersion (curve ‘Mono’). Results of the second order correction in Eq. (11) are reported as well (curve ‘Corr’). Further, the averaging of multiple separate monodisperse behaviors (curve ‘Ave’) provides an additional means for 9
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Figure 6: Theoretical (squares) and experimental (circles) Young’s modulus for a vinyl ester-glass syntactic foam with S22 microballoons as inclusion. The particle volume fraction is (a) 30%, (b) 40%, (c) 50%, and (d) 60%. The particle wall thickness distribution is shown in Fig. 3(a).
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Figure 7: Theoretical (squares) and experimental (circles) Young’s modulus for a vinyl ester-glass syntactic foam with S32 microballoons as inclusion. The particle volume fraction is (a) 30%, (b) 40%, (c) 50%, and (d) 60%. The particle wall thickness distribution is shown in Fig. 3(b).
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1.5 640
800
Density (kg/m3)
Density (kg/m )
700
720
1.5 600
740
3
605
610
615
620
625
630
Density (kg/m3)
Density (kg/m ) (c)
(d)
Figure 8: Theoretical (squares) and experimental (circles) Young’s modulus for a vinyl ester-glass syntactic foam with S38 microballoons as inclusion. The particle volume fraction is (a) 30%, (b) 40%, (c) 50%, and (d) 60%. The particle wall thickness distribution is shown in Fig. 3(c).
12
4.5
4
4
3.5
3.5
E (GPa)
E (GPa)
4.5
3
3
2.5
2.5
2
2
1.5 895
900
905
910
915
1.5 760
920
770
780
3
4.5
4.5
4
4
3.5
3.5
3
2.5
2
2 810
810
820
830
660
665
670
3
2.5
805
800
(b)
E (GPa)
E (GPa)
(a)
1.5 800
790
Density (kg/m3)
Density (kg/m )
815
820
1.5 635
825
3
640
645
650
655
Density (kg/m3)
Density (kg/m ) (c)
(d)
Figure 9: Theoretical (squares) and experimental (circles) Young’s modulus for a vinyl ester-glass syntactic foam with K47 microballoons as inclusion. The particle volume fraction is (a) 30%, (b) 40%, (c) 50%, and (d) 60%. The particle wall thickness distribution is shown in Fig. 3(d).
13
1
1.05
0.9 0.8 0.7 0.6 0.5 0
E/Em
E/Em
1
Poly Mono Ave Corr Nom 0.2
0.95
0.9
0.4
Φ
0.6
0.8
0.85 0
1
Poly Mono Ave Corr Nom 0.2
(a)
0.6
0.8
1
0.6
0.8
1
(b)
Poly Mono Ave Corr Nom
1.4
E/Em
E/Em
1.15
Φ
1.5
1.25 1.2
0.4
1.1
1.3
Poly Mono Ave Corr Nom
1.2 1.05
1.1
1 0.95 0
0.2
0.4
Φ
0.6
0.8
1 0
1
(c)
0.2
0.4
Φ
(d)
Figure 10: Young’s modulus predictions for a syntactic foam with (a) S22, (b) S32, (c) S38, and (d) K46 microballoons. The void content is neglected, that is, Φv = 0. The curve ‘Poly’ represents the result of the homogenization with the polydisperse scheme with the experimentally determined wall thickness distribution. The curve ‘Mono’ is the result of the monodisperse homogenization with radius ratio corresponding to the average of the experimental polydispersion. The curve ‘Corr’ refers to the second order correction obtained with the variance of the distribution. The curve ‘Ave’ results from the averaging of the four monodisperse inclusions. The curve ‘Nom’ shows results obtained with the monodisperse homogenization method using the nominal microballoons wall thickness declared by the manufacturer, see Table 1. Note that the scale of the vertical axis is different in each case.
comparison. In other words, the curve shows the predictions obtained by solving Eqs. (4) with proper initial conditions for each inclusion type and averaging the results with the weights ψ j . Figures 10 shows that, especially for the case of thin walled particles such as S22 and S32 type, the prediction of the Young’s modulus of the composite material may vary significantly depending on the homogenization technique. Differences among the results generally tend to increase with the inclusion volume fraction. We note that this trend is consistent with the results in Christensen (1990), where different homogenization schemes are shown to yield approximately the same result for low inclusion volume fractions. We find that the same qualitative behavior is generally observed with different polydispersion compositions. In particular, the differences among the predicted values become more significant as the dispersion becomes broad and especially for thin walled particles. In addition, when compared to the monodisperse scheme, averaging provides results closer to the polydisperse scheme, especially for low to moderate inclusion volume fractions. A closer agreement is observed between the polydisperse scheme and the second order correction scheme in Eq. (11). In the four cases analyzed, the monodisperse homogenization based on average radius ratio values provides higher values than other techniques. We now use the proposed model to study the effect of void content in syntactic foams. Figures 11 present modulus predictions obtained from the homogenization scheme in Eqs. (6), using the polydispersion composition in Figs. 3. 14
0.2
1.3
0.2
1.3
1.2
1.2
1.1
0.15
1.1
0.15
1 v
0.9
0.1
Φ
Φv
1
0.8
0.9
0.1
0.8
0.7 0.05
0.7 0.05
0.6
0.6
0.5 0 0
0.1
0.2
0.3
Φp
0.4
0.5
0.6
0.7
0.5 0 0
0.4
0.1
0.2
(a)
0.3
Φp
0.4
0.5
0.6
0.7
(b)
0.2
1.3
0.2
1.3
1.2
1.2
1.1
0.15
1.1
0.15
1
0.1
v
0.9
Φ
Φv
1
0.8
0.9
0.1
0.8
0.7 0.05
0.7 0.05
0.6
0.6
0.5 0 0
0.1
0.2
0.3
Φp
0.4
0.4
0.5
0.6
0.7
0.5 0 0
0.4
(c)
0.1
0.2
0.3
Φp
0.4
0.5
0.6
0.7
0.4
(d)
Figure 11: Theoretical Young’s modulus for a composite material with (a) S22, (b) S32, (c) S38, and (d) K46 microballoons as inclusions, accounting for the matrix porosity. Note that, for Φ p = 0 and Φv = 0, E/Em = 1. Each line in the contour plot corresponds to a change of 5% in the value of the relative effective Young’s modulus of the composite.
Results for the different inclusion wall thickness distributions follow generally the same behavior discussed in Figs. 10. In particular, Fig. 11(a) shows that, for S22 and S32 particles, the relative Young’s modulus decreases as the inclusion volume fraction increases. For other particle types, the relative Young’s modulus increases with the inclusion volume fraction. This trend is more evident as the wall thickness increases. Furthermore, for the explored range of void content, that is, 0 ≤ Φv ≤ 0.2, the variations in the effective modulus scale approximately linearly with the matrix porosity. That is, in the contour plots in Figs. 11, the contour lines are approximately parallel. This simple scaling law can be attributed to the moderately low void volume fraction. Nevertheless, as the void content increases in the matrix material, the relative Young’s modulus tends to become less sensitive to the variation in the particle volume fraction, which is in line with the sensitivity analysis reported in Section 3.3. In other words, for a given Φv , the slope of the contour lines tends to decrease as Φ p increases. 5.2. Critical analysis of monodisperse homogenization for thin walled particle systems Syntactic foams characterized in most experimental studies use hollow particles with nominal value of η in the range [0.90, 0.98]. It is shown in Porfiri and Gupta (2009) that, in syntactic foams with constituents’ elastic constants similar to those explored in this study, the differential scheme predicts that the Young’s modulus increases with the inclusion volume fraction, for inclusion radius ratio below a critical value, say η∗ . In general, the value of η∗ depends on the elastic properties of the constituent materials and can be determined numerically by studying the functions fE(p) and fν(p) for each case of interest. For the material properties used in this work, we find that the critical value of the 15
14 12
1.1 1.05 1
8
E/Em
E/Em
10
1.15
Poly Mono Ave Corr
6
0.95 0.9
4
0.85
2 0 0
0.8
0.2
0.4
Φ
0.6
0.8
0.75 0
1
(a)
Poly Mono Ave Corr 0.2
0.4
Φ
0.6
0.8
1
(b)
Figure 12: Effective Young’s modulus for the composite material with polydisperse inclusion, according to different homogenization schemes. The inclusion phase composition in (a) is described by (η1 = 0.1, ψ1 = 0.25), (η2 = 0.2, ψ2 = 0.25), (η3 = 0.6, ψ3 = 0.25), and (η4 = 0.7, ψ4 = 0.25). In (b), the inclusion phase is described by the distribution (η1 = 0.90, ψ1 = 0.25), (η2 = 0.91, ψ2 = 0.25), (η3 = 0.98, ψ3 = 0.25), and (η4 = 0.99, ψ4 = 0.25).
radius ratio is approximately given by η∗ = 0.95. The value η∗ identifies a critical transition region rather than a fixed point in dependence of the effective modulus on the inclusion volume fraction. To highlight the main characteristics of the studied homogenization schemes, we consider two representative scenarios in which four families of inclusions with moderately different radius ratio are homogenized. The void content is neglected, Φv = 0, and the four families of inclusions are introduced in the matrix material with a constant volume ratio, that is, ψ j = 0.25, for j = 1, . . . , 4. Figures 12 show the influence of the selected homogenization scheme on the effective modulus of the composite material, as a function of the inclusion volume fraction Φ. A very close agreement among predictions from the four methods is observed for radius ratio in the range [0.1, 0.7], see Fig. 12(a). In this case, a polydispersion with average wall thickness value η¯ = 0.40 is considered. In particular, the second order correction scheme yields approximately the same results of the polydisperse homogenization. This hints that the modulus sensitivity on the distribution variance, as introduced in Eq. (11), dramatically decreases as thicker particles are considered in the homogenization scheme. In Fig. 12(b), we report the illustrative case where the polydisperse distribution has an average value of η¯ = 0.945, and the particle radius ratio values are in the interval [0.90, 0.99]. We find that, for the composite material under analysis, the prediction of the polydisperse homogenization scheme yields a decreasing Young’s modulus as the inclusion volume fraction increases, which is in contrast with the trend obtained for a monodisperse system. On the other hand, the second order correction to the monodisperse scheme presented in Eq. (11) is able to capture the decreasing trend in the composite’s Young’s modulus. Therefore, it is evident that the mean value of the radius ratio polydispersion may not be sufficient for a complete and accurate prediction of the effective Young’s modulus of the composite. The modulus is indeed extremely sensitive to the distribution variance, especially for thin walled particles that are relevant in practical applications. 5.3. Extension to continuous polydispersion In this Section, we study, for the continuous polydispersion case, the influence of the distribution statistical properties on the predictions of the model in Eqs. (9). To allow for a comparison with the results proposed in Fig. 12(b), we neglect porosity by selecting Φv = 0 and choose the distribution parameters β1 and β2 to obtain m1 = 0.9 or m1 = 0.945. Therefore, we determine the effect of polydispersivity through a parametric study, in which we vary the parameters of the distribution while keeping the mean constant. We also report standard deviation and skewness values obtained for each case analyzed. As shown in Figs. 3, the particle wall thickness distribution is generally not symmetric about the mean value, therefore the distribution’s skewness appears to be a significant feature of the description. Figures 13 illustrate the predictions of the continuous model for a set of Beta distributions with two 16
35
50 m = 0.065465, m = −1.1109
m = 0.04351, m = −1.4377
m2 = 0.039892, m3 = −0.69687
m2 = 0.03105, m3 = −1.044
2
2
40
m2 = 0.028333, m3 = −0.49925
25
Beta distribution
Beta distribution
30
3
m2 = 0.020079, m3 = −0.35538 m2 = 0.012716, m3 = −0.22567
20 15 10
3
m2 = 0.022058, m3 = −0.74843 m2 = 0.015634, m3 = −0.53292
30
m2 = 0.0099019, m3 = −0.33847
20 10
5 0 0.5
0.6
0.7
0.8
η (m1 = 0.9)
0.9
0 0.5
1
0.6
0.7
0.8
η (m1 = 0.945)
(a)
0.9
1
0.8
1
(b)
2.8
1.2 m = 0.065465, m = −1.1109 3
m2 = 0.039892, m3 = −0.69687
2.4
m2 = 0.028333, m3 = −0.49925
2.2
m2 = 0.020079, m3 = −0.35538
2
m2 = 0.012716, m3 = −0.22567
1
E/Em
E/Em
2.6
2
Monodisperse m = 0.9 1
1.8
0.8
m2 = 0.04351, m3 = −1.4377 m2 = 0.03105, m3 = −1.044 m2 = 0.022058, m3 = −0.74843
1.6 0.6
1.4
m2 = 0.015634, m3 = −0.53292 m = 0.0099019, m = −0.33847 2
1.2
3
Monodisperse m = 0.945 1
1 0
0.2
0.4
Φ
0.6
0.8
0.4 0
1
(c)
0.2
0.4
Φ
0.6
(d)
Figure 13: Beta distributions for χ(η) with (a) m1 = 0.9 and (b) m1 = 0.945, and varying standard deviation m2 and skewness m3 . Theoretical Young’s modulus for each distribution χ(η) with (c) m1 = 0.9 and (d) m1 = 0.945, compared with monodisperse results using mean values η = m1 (dashed line). The standard deviation and the absolute value of the skewness of the Beta distributions increase in the direction indicated by the arrow.
representative mean values, namely, m1 = 0.9 and m1 = 0.945, and decreasing standard deviation and skewness. The shape of the distribution is reported in Figs. 13(a) and 13(b), whereas the resulting moduli are shown in Figs. 13(c) and 13(d). These figures also compare the results obtained from the monodisperse homogenization scheme using only the mean radius ratio. Figures 13 show that significant discrepancies may arise among the homogenization techniques when the distribution involves moderately large standard deviations. In addition, the negative sign of the distribution’s skewness indicates that the shape of χ(η) is biased towards larger values of η, that is, a relevant percentage of the inclusion phase comprises particles with wall thickness significantly smaller than the mean value, see Figs. 13(a) and 13(b). The effect of thin walled particle inclusion is to lower the effective Young’s modulus of the composite, especially as the inclusion volume fraction increases and as the distribution is skewed towards η > η∗ . 6. Conclusions In this paper, we presented a homogenization method that accounts for polydispersivity in the inclusion phase in syntactic foams. The proposed method extends the monodisperse differential scheme for analysis of hollow spherical inclusion. The method provides a means to analyze the effects of particle wall thickness distribution, either discrete or 17
continuous, of different particle materials and the influence of voids on the elastic properties of syntactic foams. We derived manageable analytical expressions for the implementation of the technique, as well as simple descriptions of the qualitative dependence and sensitivity of the composite effective properties on the statistical characteristics of the included phases. We identified higher-order phenomena, that generally result in a decrease in the effective modulus, and their relation to the composition of the included phase. These features, while being pertinent to a more realistic description of the microstructure of typical syntactic foams, are generally discarded in available monodispersion studies. Theoretical results are compared with experimental work and are generally found to be in good agreement. The proposed approach may be extended to predicting plastic and viscoelastic response of composites by following the line of arguments in Kontou (2007); Remillat (2007); Tagliavia et al. (2009b). Acknowledgements This work was supported by the Office of Naval Research grant N00014-07-1-0419 with Dr. Y.D.S. Rajapakse as the program manager and by the National Science Foundation grant CMMI-0726723. Views expressed herein are those of authors, and not of the funding agency. The authors would like to gratefully acknowledge Mr. Gabriele Tagliavia for providing the experimental data on three point bending tests and some of the micrographs. A. Coefficients The functions introduced in Eqs. (5) are defined as follows, see Porfiri and Gupta (2009); Christensen (1979) " # 1 (i) 4 (i) (i) 2 fE = h (1 − 2ν) + hµ (1 + ν) (12a) E k 3 i 1 h (i) fν(i) = (−1 + ν + 2ν2 )(2h(i) (12b) µ (1 + ν) + 3hk (−1 + 2ν)) 3Eν where the superscript (i) can be specialized to the particle (p) and to the void (v) inclusion in the following form a0 + a3 η3 b0 + b3 η3 c0 + c3 η3 + c5 η5 + c7 η7 + c10 η10 h(p) µ (k p , µ p , k, µ, η) = d0 + d3 η3 + d5 η5 + d7 η7 + d10 η10 k(3k + 4µ) h(v) k (k, µ) = − 4µ 5µ(3k + 4µ) h(v) µ (k, µ) = − (9k + 8µ) h(p) k (k p , µ p , k, µ, η) =
(13a) (13b) (13c) (13d)
The coefficients ai , bi , ci , and di in Eqs. (13) are given by a0 = 4µ p (−k p + km )(−3km − 4µm )
(14a)
a3 = k p (3km + 4µ p )(−3km − 4µm )
(14b)
b0 = 4µ p (3k p + 4µm )
(14c)
b3 = −12k p (µ p − µm )
(14d)
c0 = 15(9λ p + 14µ p )(µ p − µm )µm (λm + 2µm )(14µ p (µ p + 4µm ) + λ p (19µ p + 16µm ))
(15a)
c3 = −375µm (λm + 2µm )(3(9µ2p − 10µm µ p + 8µ2m )λ2p + 4µ p (14µ2p − 9µm µ p + 16µ2m )λ p + 28(µ4p + 2µ2m µ2p )) c5 = 15120(λ p + µ p )2 (µ p − µm )2 µm (λm + 2µm ) c7 =
−375(27λ2p
28µ2p )(µ p
+ 56µ p λ p + 18
2
− µm ) µm (λm + 2µm )
(15b) (15c) (15d)
c10 = 15(19λ p + 14µ p )(µ p − µm )µm (λm + 2µm )(λ p (9µ p + 6µm ) + 2µ p (7µ p + 8µm ))
(15e)
d0 = (9λ p + 14µ p )(2µm (8µ p + 7µm ) + λm (6µ p + 9µm ))(14µ p (µ p + 4µm ) + λ p (19µ p + 16µm ))
(15f)
d3 = −50(3(9λm (3µ2p + µm µ p − 4µ2m ) − 2µm (−36µ2p + µm µ p + 28µ2m ))λ2p + 2µ p (3λm (28µ2p + 13µm µ p − 48µ2m ) + 14µm (16µ2p + 3µm µ p − 16µ2m ))λ p + 28µ2p (2µm (4µ2p + 3µm µ p − 7µ2m ) + 3λm (µ2p + µm µ p − 3µ2m )))
(15g)
d5 = 1008(λ p + µ p )2 (µ p − µm )(2µm (8µ p + 7µm ) + λm (6µ p + 9µm ))
(15h)
d7 = −25(27λ2p + 56µ p λ p + 28µ2p )(µ p − µm )(2µm (8µ p + 7µm ) + λm (6µ p + 9µm ))
(15i)
d10 = 2(19λ p + 14µ p )(µ p − µm )(3λ p (9λm (µ p − µm ) + 2(12µ p − 7µm )µm )+ 2µ p (3λm (7µ p − 12µm ) + 56(µ p − µm )µm ))
(15j)
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20
Effect of polydispersivity and porosity on the elastic properties of hollow particle filled composites M. Aurelia , M. Porfiri∗,a , N. Guptaa a Department of Mechanical and Aerospace Engineering Polytechnic Institute of New York University, Six MetroTech Center, Brooklyn, NY 11201, USA
Abstract Hollow particle filled composites are characterized by a variety of properties of interest to marine structural applications, including low density, high specific modulus, and low moisture absorption. Usually, dispersed microspheres exhibit widely different geometric characteristics, such as wall thickness and outer radius. In this paper, we develop a homogenization technique based on the differential scheme that accounts for polydispersivity in geometry of inclusion phases in hollow particle reinforced composites. We find manageable differential expressions to predict the composite effective moduli in terms of a variety of concurrent factors, including matrix and particle elastic constants, geometrical properties of hollow particles, volume fractions, and void content in the matrix. Theoretical findings are validated by experimental data and are compared with results of several published models. We find that polydispersion modeling allows for a closer agreement with experimental results than monodisperse schemes. In particular, qualitatively different predictions are found in syntactic foams containing thin walled particles by accounting for polydispersivity. Key words: homogenization, particle reinforced composites, polydispersivity, porosity, syntactic foams
1. Introduction Syntactic foams are fabricated by dispersing hollow microspheres in a matrix material with a twofold purpose: to embed closed cell porosity in the matrix, thus reducing the material density while controlling the size and distribution of the porosity; and to reinforce the matrix phase by using particles of a stiffer material than the matrix, see Narkis et al. (1982). Increasing interest towards understanding the properties of syntactic foams has resulted in extensive experimental characterization, see for example Gupta et al. (2001, 2004); Kishore et al. (2006). These studies have demonstrated that the properties of syntactic foams can be tailored by changing either the particle volume fraction or the wall thickness. Such approaches are also used in developing functionally graded syntactic foams, which show significantly higher energy absorption capabilities under compressive loading than the plain syntactic foams containing random distribution of particles, see for example Caeti et al. (2009). Through characterization of numerous compositions and material systems, these studies have provided insight into the structure-property correlations for syntactic foams under a variety of loading conditions, including tensile, compressive, flexural, and impact, see for example Kim and Mitchell (2003); Kishore et al. (2005); Gupta et al. (2009). These studies have highlighted the possibility of tailoring the mechanical, thermal, and electrical properties of syntactic foams, see for example Grosjean et al. (2009); Gupta et al. (2006); Shabde et al. (2006); Wouterson et al. (2004); Zhang and Ma (2009). The ability of tailoring syntactic foams has played an important role in enabling their applications in a diverse set of fields, including marine structures, aerospace structures, and sports equipment, see for example Ishai et al. (1995). Compared to the extensive experimental literature, relatively few modeling efforts on syntactic foams are available. In Huang and Gibson (1993), the elastic properties of syntactic foams are analyzed by studying the infinitely dilute hollow inclusion problem. In Marur (2005), the three phase homogenization scheme originally proposed in Christensen ∗ Corresponding
author. Tel.: +1-718-260-3681. Fax: +1-718-260-3532. Email addresses: [email protected] (M. Aureli), [email protected] (M. Porfiri), [email protected] (N. Gupta)
Preprint submitted to Mechanics of Materials
September 20, 2009
(a)
(b)
Figure 1: (a) Micrograph of a representative sample of commercially available microballoons used in syntactic foams. The particles’ outer radius distribution varies in the broad range, approximately 10–100 µm. (b) Schematic representation of polydispersivity in hollow spheres. In (b): (top row) polydispersivity in outer radius, with radius decreasing from left to right; and (bottom row) polydispersivity in wall thickness, with wall thickness increasing from left to right.
and Lo (1979) is used to study syntactic foams. In Porfiri and Gupta (2009), the differential scheme, see McLaughlin (1977); Norris (1985); Zimmerman (1991), is applied to model the elastic properties of syntactic foams, obtaining good agreement with experimental results in high inclusion volume fraction range. A common underlying theme in these models is to assume particles to be of same size and wall thickness. However, this assumption is not close to the reality and limits the usefulness of some of these approaches. Figure 1(a) shows a randomly selected sample of glass microspheres obtained from 3M, MN. These microballoons are commonly used in manufacturing syntactic foams, see for example Bardella and Genna (2001); Wouterson et al. (2004), and have an average particle diameter of 40 µm and density of 460 kg m−3 . The micrograph shows that the actual particle size varies over a wide range, from about 10 µm to 100 µm. In addition, the wall thickness of each particle can be different and can also exist over a large range. Figure 1(b) schematically represents such polydispersion in the size and wall thickness of these microballoons. The top row of particles in Fig. 1(b) illustrates inclusion size polydispersion, whereas the bottom row represents polydispersion in the inclusion wall thickness. A recent experimental study, see Gupta et al. (2009), has shown that the size and wall thickness values for a large batch of particles are indeed close to the nominal values supplied by the manufacturer. However, sieving these particles according to their size and measuring the true particle density of each fraction show that larger particles have lower density due to thinner walls, see Gupta et al. (2009). This implies that the larger size particles will be more compliant, thus indicating that inclusion of polydispersivity in models can provide better predictive capabilities. The problem of polydispersivity in solid particle filled composites has been originally addressed in Budiansky (1965); Hill (1965), where the first homogenization methods for polydisperse and multiphase particulate systems are discussed. In Huang et al. (1994), an extension of the three phase method to account for multiphase and polydisperse solid inclusion is presented. In Duan et al. (2007a,b), this formulation is generalized to include the presence of elastic interface effects. In Bardella and Genna (2001), the homogenization technique proposed in Herv´e and Pellegrini (1995) is adopted to account for filler particles gradation in syntactic foams. In Dai et al. (1998), a generalized self consistent Mori-Tanaka method, see Benveniste (1987), is proposed to determine the effective moduli of multiphase particulate composites. Modified Mori-Tanaka approaches have also been presented in Iwakuma and Koyama (2005); Peng et al. (2009) to study multiphase composites. Recently, in Zouari et al. (2008), a general framework for iterative homogenization of particle filled composites is proposed. It is therein assumed that the inclusion phase is progressively introduced into the matrix material in a number of successive steps, each step yielding the new effective media for the next addition step. The approach offers a more tractable and computationally oriented ground for analyzing polydispersivity than the morphological approach in Bardella and Genna (2001) without compromising the accuracy of the findings, see Zouari et al. (2008). 2
d
b
c d a c (a)
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Figure 2: (a) Micrograph and (b) schematic representation of the microstructure of a typical syntactic foam. In (b), the different phases are shown, including ‘a’ matrix, ‘b’ voids, ‘c’ particles, and ‘d’ porosity embedded in the particle shell.
The aim of this study is to develop a micromechanics-based model for the interplays of the polydispersivity in the hollow inclusion volume fraction and wall thickness on the properties of the resulting syntactic foams. We propose a method for determining the effective elastic properties of syntactic foams by combining the modeling results discussed in Porfiri and Gupta (2009) with a homogenization technique stemming from the limit for infinitely small inclusion volume fraction of the procedure developed in Zouari et al. (2008). By introducing distribution functions to describe the polydisperse nature of the composition of the inclusion phase in the matrix, we obtain an analytical differential formulation, in contrast to the algorithmic process discussed in Zouari et al. (2008), to evaluate the effective properties of the composite. The proposed model is applicable to study inclusion volume fractions up to the packing limit and to both continuous and discrete polydispersions. The model predictions are validated through comparison with published experimental results on vinyl ester and polyester-glass syntactic foams and with findings of several available models. 2. Problem Statement Figure 2(a) shows a representative syntactic foam microstructure, which is schematically redrawn in Fig. 2(b). These figures show that syntactic foams are three-phase materials with polydisperse inclusions. Matrix and microballoons are the two primary phases. The third phase is the air voids entrapped in the matrix during composite synthesis. Experimental studies have shown the presence of 5–10% voids in most foam compositions, see for example Tagliavia et al. (2009a). The air present inside hollow particles is sometimes counted as an additional phase. However, this phase is automatically accounted for by means of particle size and wall thickness and no further reference of this phase is required in the present work. In this work, matrix, particles, and voids are identified using subscripts m, p, and v, respectively. In addition, the term inclusion refers to both the particle phase and the void phase. We denote with Φ and Φm the volume fractions of the inclusion phase and of the matrix material in the composite, respectively, so that Φm + Φ = 1. The particle and void volume fractions are denoted with Φ p = φ p Φ and Φv = φv Φ, respectively, so that Φ p + Φv = Φ. In the limit Φ → 0, φ p and φv remain well-defined and finite. The inclusions are assumed to be of spherical shape and all particles are assumed to be made of the same material. However, this hypothesis is not essential in the development of the model, whose formulation allows for treatment of spherical particles of different materials. The ratio of the inner radius ri to the outer radius ro of the inclusion is called radius ratio and is denoted by η. Note that η = 0 corresponds to the case of a solid particle, whereas η = 1 denotes a spherical void embedded in the matrix. A large number of existing experimental studies on syntactic foams have used glass microspheres supplied by 3M. The average diameter and density of four types of microballoons provided by the manufacturer are presented in Table 1. The calculated values of wall thickness and radius ratio are also provided in Table 1, assuming the glass density 3
Table 1: Properties of hollow glass microspheres used in the experimental analysis.
Particle Typea
Average Average Density Diameter (kg/m3 ) (µm) S22 220 35 S32 320 40 S38 380 40 K46 460 40 a Manufacturer’s code
Calculated Wall Thickness (µm) 0.521 0.878 1.052 1.289
Radius Ratio η 0.970 0.956 0.947 0.936
of 2540 kg m−3 . Although most published studies refer to the average values of properties of these microballoons, they are actually polydispersed systems. The size and density distribution of these microballoons are presented in Figs. 3. The inclusion size is measured by sieving them in four size fractions using a sieve shaker and measuring the true particle density of each segment using a pycnometer to obtain the estimate of polydispersivity, see Gupta et al. (2009) for the detailed dataset. Figures 3 show that, in the four examined compositions, approximately 20 vol.% of the particles have radius ratio significantly lower than the average value. On the other hand, this implies that most particles have thinner walls than the average value. Therefore, wall thickness distribution should be considered to accurately predict the effective properties of the composite. 3. Model development 3.1. Description of inclusion polydispersivity The formulation of the polydispersivity problem can be achieved by following either a discrete or a continuous approach. More specifically, the radius ratio distribution is described by a discrete function in the former case, and by a continuous function in the latter case. In what follows, for clarity of presentation, we first consider the discrete approach and then adapt it to the continuous scenario through a limit process. We introduce a discrete distribution function by referring to N + 1 families of particles. The j-th inclusion family has radius ratio η j and the index j goes from 0 to N. The values ψ j express the volume fraction of the inclusion phase for the j-th family. The void volume fraction is associated to the family of inclusions with j = 0. Additionally, N X
N X
ψ j = 1;
j=0
ψ j = φp
(1)
j=1
Note that, according to this convention, the inclusion phase belonging to the j-th family, with volume V j , has a volume fraction given by V j /V = Φψ j ; ψ0 = V0 /V = Φv /Φ = φv (2) where V indicates the total volume of the composite material. In addition, we define a scaled version of the discrete distribution function ψ j , say χ j , defined for j = 1, . . . , N, so that φ p χ j = ψ j and χ j = V j /(Φ p V). Here, ψ j and χ j coincide when Φv = 0, that is, when there are no voids present in the matrix. The extension of these definitions to the continuous case is obtained by introducing a continuous distribution function ψ(η) such that ψ(η) dη represents the volume fraction of the inclusion for which the particles have a radius ratio in the interval [η, η + dη). The relations Z 0
Z
1
ψ(η) dη + φv = 1;
1 0
ψ(η) dη = φ p
(3)
represent the continuous counterpart of Eq. (1). We note that the distribution ψ(η) does not account explicitly for the presence of voids in the matrix, described by φv . In analogy with the discrete case, we introduce a scaled distribution R1 function φ p χ(η) = ψ(η), so that 0 χ(η) dη = 1. 4
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Figure 3: Experimental distribution of microballoon radius ratio η for (a) S22, (b) S32, (c) S38, and (d) K46 microballoons. The error bars correspond to one standard deviation. The gray dashed line indicates the average particle radius ratio calculated using the experimental data. The blue dash-dotted line indicates the nominal values reported in Table 1.
5
These distribution parameters are a suitable description of the inclusion polydisperse nature. In particular, using ψ j or ψ(η) allows to refer to the whole inclusion phase in the matrix, that is, to both the particles and the voids. On the other hand, χ j or χ(η) describe the composition of the sole particle phase, not accounting for the voids. The two descriptions are related by the inclusion composition parameters φ p and φv . 3.2. Differential scheme We model the composite as a linear elastic solid and assume it to be macroscopically homogeneous and isotropic. Hence, two effective elastic properties need to be determined to fully characterize the fourth order elasticity tensors that describe the effective constitutive behavior of the composite material, namely, the effective bulk modulus k and shear modulus µ or, equivalently, the effective Young’s modulus E and the Poisson’s ratio ν. For a monodisperse system of radius ratio η, the change in the composite’s elastic properties dE and dν due to an increment in the inclusion volume fraction dΦ is given by dE = fE (E p , ν p , E, ν, η)E dΦ dν = fν (E p , ν p , E, ν, η)ν dΦ
(4a) (4b)
where E p and ν p represent the inclusion elastic properties and Φ is the inclusion volume fraction. Thus, the effective elastic constants for a given inclusion volume fraction are expressed as the solution of the system of coupled ordinary differential equations given in Eqs. (4) with the initial conditions E = Em and ν = νm , for Φ = 0. The functions fE and fν are reported in the Appendix, specialized for the case of a solid or a hollow particle inclusion, fE(p) and fν(p) , or a void, fE(v) and fν(v) , see also Christensen (1979). We note that closed form solutions of Eqs. (4) may be derived in special cases, such as rigid particles or incompressible phases, see for example Pal (2005); Zimmerman (1991). To study polydisperse systems, we assume that the composition of any given volume fraction of the inclusion phase is described by the distribution function ψ j , thus explicitly and uniquely defining the homogenization path in the composite material construction, as introduced in Norris (1985). Hence, the increment in the composite’s modulus due to the replacement of an infinitesimal volume of the matrix material with an inclusion belonging to the j-th family, is obtained by substituting in Eqs. (4) η with η j and dΦ with ψ j dΦ, and extending the summation over j = 0, . . . , N, that is, N X (p) (v) (5a) dE = ψ j fE (E p , ν p , E, ν, η j ) + ψ0 fE (E, ν) E dΦ j=1 N X (p) (v) (5b) dν = ψ j fν (E p , ν p , E, ν, η j ) + ψ0 fν (E, ν) ν dΦ j=1
To account for the inclusion packing limit, the increment dΦ can be replaced with dΦ/(1 − Φ/ΦL ), see for example Pal (2005), where ΦL is the packing limit which is solely determined by the size polydispersivity. For a monodisperse system, ΦL = 0.637 is defined as the random close packing limit and, for widely polydisperse systems, ΦL can be over 0.9, see for example Torquato et al. (2000); Torquato (2001). Equations (5) can be rewritten in the form valid for a discrete distribution N X dE E (p) (v) φ p (6a) χ j fE (E p , ν p , E, ν, η j ) + φv fE (E, ν) = dΦ 1 − Φ/ΦL j=1 N X dν ν (p) (v) φ p (6b) = χ j fν (E p , ν p , E, ν, η j ) + φv fν (E, ν) dΦ 1 − Φ/ΦL j=1
where we also used ψ j = φ p χ j and ψ0 = φv . Equations (6) are solved with initial conditions E = Em and ν = νm , for Φ = 0. We note that, in contrast to the algorithmic procedure presented in Zouari et al. (2008) and based on an iterative homogenization technique, here the effective elastic properties are directly determined by solving the differential equations in Eqs. (6). 6
The approach proposed in Eqs. (6) is extended to the continuous polydispersion case by using Eq. (3) as " Z 1 # dE E (p) (v) = φp χ(η) fE (E p , ν p , E, ν, η) dη + φv fE (E, ν) dΦ 1 − Φ/ΦL 0 " Z 1 # dν ν = φp χ(η) fν(p) (E p , ν p , E, ν, η) dη + φv fν(v) (E, ν) dΦ 1 − Φ/ΦL 0
(7a) (7b)
Here, we choose ΦL = 1 to account for polydispersion and for the presence of the voids, see for example Christensen (1990). The continuous model in Eqs. (7) can not be obtained directly from the formulation developed in Zouari et al. (2008) through a limit process. In addition, we use a Beta probability distribution function, see for example Abramowitz and Stegun (1965), for the functional form of the function χ(η), that is, χ(η) =
Γ(β1 + β2 ) β1 −1 η (1 − η)β2 −1 Γ(β1 )Γ(β2 )
(8)
where Γ denotes Euler’s Gamma function, and the parameters β1 and β2 describe the shape of the distribution and its statistical moments. We note that the values of β1 and β2 can be potentially determined from experimental indirect measurements of the particle wall thickness. In addition, the Beta distribution has values in the interval [0, 1], that is, in the range of physical variation of η. The mean m1 , standard deviation m2 , and skewness m3 of the distribution χ(η) are defined in terms of β1 and β2 as follows p √ 2(β2 − β1 ) 1 + β1 + β2 β1 β1 β2 m1 = (9) ; m2 = ; m3 = √ p β1 + β2 β1 β2 (2 + β1 + β2 ) (β1 + β2 ) 1 + β1 + β2 Other continuous distribution models have been discussed in the scientific literature on polydispersivity, such as the log-normal distribution and the Schulz distribution, see for example Torquato (2001). These distributions have been used to model polydispersivity in the outer radius of spheres and generally admit values in the interval [0, ∞), that is, in a non-physical range of variation for the radius ratio. For this reason, the Beta distribution appears to be a more appropriate description for the present model. 3.3. Sensitivity analysis The proposed polydisperse homogenization schemes, see Eqs. (6) and (7), represent a higher order correction to the monodisperse differential scheme discussed in Porfiri and Gupta (2009). We perform a qualitative analysis of this correction by studying the following rearranged version of Eq. (6a), where we drop the dependence of fE(p) and fE(v) on E p , ν p , E, and ν to simplify the notation, X dE/E = φp χ j fE(p) (η j ) + φv fE(v) dΦ/[1 − Φ/ΦL ] j=1 N
(10)
We note that a similar argument can be adapted to Eq. (6b) and Eqs. (7). P By defining the mean value for the radius ratio as η¯ = Nj=1 χ j η j , and introducing the deviations from the mean value δ j = η j − η, ¯ the right hand side of Eq. (10) is expanded in series in the neighborhood of η¯ to obtain (p) N 1 ∂2 fE (η) X dE/E χ j δ2j ' φ p fE(p) (¯η) + φv fE(v) + φ p dΦ/[1 − Φ/ΦL ] 2 ∂η2 η=η¯ j=1
(11)
P P where we use the properties Nj=1 χ j = 1 and Nj=1 χ j δ j = 0. This approximate expression allows for a qualitative P analysis of the proposed model with the assumption that the distribution exhibits moderately low variance, Nj=1 χ j δ2j , or, in other words, that the contribution of order δ3j and higher is negligible. The first term on the right hand side of Eq. (11) corresponds to the monodisperse behavior, where φ p = 1, and the particle wall thickness is given by the mean value of the distribution. The second term, linear in φv = 1 − φ p , represents the contribution to the monodisperse 7
5 4.5
E (GPa)
4 3.5 3 2.5 2 1.5 1 0
Huang and Gibson (1993) Bardella and Genna (2001) Zouari et al. (2008) Monodisperse (no voids) Present model
0.05
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Figure 4: Comparison between experimental results from Huang and Gibson (1993) and different homogenization techniques. The dashed curve ‘Monodisperse’ reports the results of a homogenization procedure that does not account for the presence of voids inside the material.
behavior of embedded voids in the matrix. This effect results in general decrease in stiffness of the composite material, as it can be seen from the negative sign of the functions fE(v) , see also Eqs. (12a), (13c), and (13d) in the Appendix. The third term is the second order correction due to the inclusion polydispersivity. More specifically, the effect of inclusion polydispersivity on the composite’s modulus is related to the distribution’s variance and yields a decrease or an increase in stiffness, according to the sign of the second derivative of the function fE(p) evaluated at η¯ . The signs of these derivatives generally depend on the relative elastic properties of particle and matrix material and on the mean radius ratio η¯ . For a given distribution variance, stiffening depends on the magnitude of the second derivative of fE(p) evaluated at η, ¯ which can be regarded as a sensitivity function. We further note that the correction due to polydispersivity becomes less significant as the void content increases and the distribution variance decreases. 4. Model validation 4.1. Comparison with reference models The polydisperse model in its discrete formulation, see Eqs. (6), is applied to a test case, to assess its predictive capabilities against available schemes and experimental results reported in Huang and Gibson (1993). The experimental data are obtained on polyester-glass syntactic foams, synthesized by using K1 particles from 3M. The constituents’ elastic properties are provided in Huang and Gibson (1993); Bardella and Genna (2001) and are as follows: Em = 4.89 GPa, νm = 0.4, E p = 70.11 GPa, ν p = 0.23. Since no experimental data on the radius ratio distribution of the K1 particles is provided in the original experimental study, we characterize the particle dispersion by using only the reported mean value for the radius ratio, that is, η¯ = 0.983. Results of the present model are shown in Fig. 4 and are compared with findings from the cited literature. The predictions of the proposed model show very good agreement with experimental results and a close agreement with the findings reported in Bardella and Genna (2001), especially at low particle volume fraction. Figure 4 shows that the proposed homogenization technique is capable of capturing the effect of voids on the stiffness of the composite material. Results of a monodisperse homogenization scheme that does not account for voids, also reported in Fig. 4, are generally in greater deviation from the experimental findings, especially when the void content is not negligible. We note that all the models in Fig. 4 do not explicitly account for particle to particle interactions, that are studied for example in Ma et al. (2004); Sarvestani (2003); Segurado and LLorca (2006); Shodja and Sarvestani (2001). These models provide a greater insight in the microstructural aspects of particulate composite. However their formulation is more complex than the homogenization methods discussed in Fig. 4. 4.2. Experimental validation with vinyl ester-glass syntactic foams To further validate the proposed model, we compare its prediction to the experimental results obtained from three point bending tests on sixteen types of vinyl ester-glass syntactic foams reported in Tagliavia et al. (2009a). In this 8
(a)
(b)
Figure 5: Micrographs on neat vinyl ester samples. The arrows indicate spherical voids embedded in the matrix phase. The diameter of the void inclusions is in the range 5–50 µm.
study, the wall thickness distribution and the volume fraction of the particle phase are varied in syntactic foams. In Tagliavia et al. (2009a), only the mean and the standard deviation of modulus are reported for each syntactic foam type. However, we take the full data set to compare predictions for each specimen, to account for the composition and void content of that specimen. Void content for each specimen is determined by following the procedure described in Tagliavia et al. (2009a) using the density of neat vinyl ester resin as ρm = 1161 kg m−3 . In this work, since a more accurate description of the microballoons’ size and density is available, we use experimentally determined mean particle densities, see Gupta et al. (2009), to recalculate the void volume fraction. The constituents’ elastic properties are adapted from Tagliavia et al. (2009a), namely: Em = 3.52 GPa, νm = 0.3, E p = 60 GPa, and νm = 0.21. The value chosen for the Young’s modulus of the neat vinyl ester resin is the maximum value obtained in Tagliavia et al. (2009a). Indeed, scanning electron micrographs on neat resin samples show the presence of microvoids on the fracture surface of most specimens, see for example Figs. 5, which contribute to lowering the specimen modulus. Therefore, it is considered that the highest modulus value obtained is close to the actual modulus of the neat vinyl ester resin. Results of the present model with inclusion polydispersivity and voids are shown in Figs. 6–9 superimposed on experimental data. The theoretical results are in close agreement with experimental findings. The trends are correctly reproduced and the experimental points generally fall in a 10% error band from the model predictions. These results show that the present model is capable of predicting elastic properties of syntactic foams while accounting for the particle polydispersion and presence of voids. 5. Model analysis In this Section, we analyze some representative results of the proposed model applied to vinyl ester-glass syntactic foams. Throughout the Section, the constituents’ elastic properties are selected as in Section 4.2, that is: Em = 3.52 GPa, νm = 0.3, E p = 60 GPa, and ν p = 0.21. 5.1. Parametric study of vinyl ester-glass syntactic foams The proposed method is specialized to the particle radius ratio distributions illustrated in Figs. 3, when the void volume fraction is set to 0. Figures 10 compare the results of the proposed method with several monodispersion-based homogenization techniques. More specifically, model predictions (curve ‘Poly’ in Figs. 10) are compared to the results obtained by homogenizing the inclusion phase with monodisperse wall thickness equal to the average value of the polydispersion (curve ‘Mono’). Results of the second order correction in Eq. (11) are reported as well (curve ‘Corr’). Further, the averaging of multiple separate monodisperse behaviors (curve ‘Ave’) provides an additional means for 9
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Figure 6: Theoretical (squares) and experimental (circles) Young’s modulus for a vinyl ester-glass syntactic foam with S22 microballoons as inclusion. The particle volume fraction is (a) 30%, (b) 40%, (c) 50%, and (d) 60%. The particle wall thickness distribution is shown in Fig. 3(a).
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Figure 7: Theoretical (squares) and experimental (circles) Young’s modulus for a vinyl ester-glass syntactic foam with S32 microballoons as inclusion. The particle volume fraction is (a) 30%, (b) 40%, (c) 50%, and (d) 60%. The particle wall thickness distribution is shown in Fig. 3(b).
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610
615
620
625
630
Density (kg/m3)
Density (kg/m ) (c)
(d)
Figure 8: Theoretical (squares) and experimental (circles) Young’s modulus for a vinyl ester-glass syntactic foam with S38 microballoons as inclusion. The particle volume fraction is (a) 30%, (b) 40%, (c) 50%, and (d) 60%. The particle wall thickness distribution is shown in Fig. 3(c).
12
4.5
4
4
3.5
3.5
E (GPa)
E (GPa)
4.5
3
3
2.5
2.5
2
2
1.5 895
900
905
910
915
1.5 760
920
770
780
3
4.5
4.5
4
4
3.5
3.5
3
2.5
2
2 810
810
820
830
660
665
670
3
2.5
805
800
(b)
E (GPa)
E (GPa)
(a)
1.5 800
790
Density (kg/m3)
Density (kg/m )
815
820
1.5 635
825
3
640
645
650
655
Density (kg/m3)
Density (kg/m ) (c)
(d)
Figure 9: Theoretical (squares) and experimental (circles) Young’s modulus for a vinyl ester-glass syntactic foam with K47 microballoons as inclusion. The particle volume fraction is (a) 30%, (b) 40%, (c) 50%, and (d) 60%. The particle wall thickness distribution is shown in Fig. 3(d).
13
1
1.05
0.9 0.8 0.7 0.6 0.5 0
E/Em
E/Em
1
Poly Mono Ave Corr Nom 0.2
0.95
0.9
0.4
Φ
0.6
0.8
0.85 0
1
Poly Mono Ave Corr Nom 0.2
(a)
0.6
0.8
1
0.6
0.8
1
(b)
Poly Mono Ave Corr Nom
1.4
E/Em
E/Em
1.15
Φ
1.5
1.25 1.2
0.4
1.1
1.3
Poly Mono Ave Corr Nom
1.2 1.05
1.1
1 0.95 0
0.2
0.4
Φ
0.6
0.8
1 0
1
(c)
0.2
0.4
Φ
(d)
Figure 10: Young’s modulus predictions for a syntactic foam with (a) S22, (b) S32, (c) S38, and (d) K46 microballoons. The void content is neglected, that is, Φv = 0. The curve ‘Poly’ represents the result of the homogenization with the polydisperse scheme with the experimentally determined wall thickness distribution. The curve ‘Mono’ is the result of the monodisperse homogenization with radius ratio corresponding to the average of the experimental polydispersion. The curve ‘Corr’ refers to the second order correction obtained with the variance of the distribution. The curve ‘Ave’ results from the averaging of the four monodisperse inclusions. The curve ‘Nom’ shows results obtained with the monodisperse homogenization method using the nominal microballoons wall thickness declared by the manufacturer, see Table 1. Note that the scale of the vertical axis is different in each case.
comparison. In other words, the curve shows the predictions obtained by solving Eqs. (4) with proper initial conditions for each inclusion type and averaging the results with the weights ψ j . Figures 10 shows that, especially for the case of thin walled particles such as S22 and S32 type, the prediction of the Young’s modulus of the composite material may vary significantly depending on the homogenization technique. Differences among the results generally tend to increase with the inclusion volume fraction. We note that this trend is consistent with the results in Christensen (1990), where different homogenization schemes are shown to yield approximately the same result for low inclusion volume fractions. We find that the same qualitative behavior is generally observed with different polydispersion compositions. In particular, the differences among the predicted values become more significant as the dispersion becomes broad and especially for thin walled particles. In addition, when compared to the monodisperse scheme, averaging provides results closer to the polydisperse scheme, especially for low to moderate inclusion volume fractions. A closer agreement is observed between the polydisperse scheme and the second order correction scheme in Eq. (11). In the four cases analyzed, the monodisperse homogenization based on average radius ratio values provides higher values than other techniques. We now use the proposed model to study the effect of void content in syntactic foams. Figures 11 present modulus predictions obtained from the homogenization scheme in Eqs. (6), using the polydispersion composition in Figs. 3. 14
0.2
1.3
0.2
1.3
1.2
1.2
1.1
0.15
1.1
0.15
1 v
0.9
0.1
Φ
Φv
1
0.8
0.9
0.1
0.8
0.7 0.05
0.7 0.05
0.6
0.6
0.5 0 0
0.1
0.2
0.3
Φp
0.4
0.5
0.6
0.7
0.5 0 0
0.4
0.1
0.2
(a)
0.3
Φp
0.4
0.5
0.6
0.7
(b)
0.2
1.3
0.2
1.3
1.2
1.2
1.1
0.15
1.1
0.15
1
0.1
v
0.9
Φ
Φv
1
0.8
0.9
0.1
0.8
0.7 0.05
0.7 0.05
0.6
0.6
0.5 0 0
0.1
0.2
0.3
Φp
0.4
0.4
0.5
0.6
0.7
0.5 0 0
0.4
(c)
0.1
0.2
0.3
Φp
0.4
0.5
0.6
0.7
0.4
(d)
Figure 11: Theoretical Young’s modulus for a composite material with (a) S22, (b) S32, (c) S38, and (d) K46 microballoons as inclusions, accounting for the matrix porosity. Note that, for Φ p = 0 and Φv = 0, E/Em = 1. Each line in the contour plot corresponds to a change of 5% in the value of the relative effective Young’s modulus of the composite.
Results for the different inclusion wall thickness distributions follow generally the same behavior discussed in Figs. 10. In particular, Fig. 11(a) shows that, for S22 and S32 particles, the relative Young’s modulus decreases as the inclusion volume fraction increases. For other particle types, the relative Young’s modulus increases with the inclusion volume fraction. This trend is more evident as the wall thickness increases. Furthermore, for the explored range of void content, that is, 0 ≤ Φv ≤ 0.2, the variations in the effective modulus scale approximately linearly with the matrix porosity. That is, in the contour plots in Figs. 11, the contour lines are approximately parallel. This simple scaling law can be attributed to the moderately low void volume fraction. Nevertheless, as the void content increases in the matrix material, the relative Young’s modulus tends to become less sensitive to the variation in the particle volume fraction, which is in line with the sensitivity analysis reported in Section 3.3. In other words, for a given Φv , the slope of the contour lines tends to decrease as Φ p increases. 5.2. Critical analysis of monodisperse homogenization for thin walled particle systems Syntactic foams characterized in most experimental studies use hollow particles with nominal value of η in the range [0.90, 0.98]. It is shown in Porfiri and Gupta (2009) that, in syntactic foams with constituents’ elastic constants similar to those explored in this study, the differential scheme predicts that the Young’s modulus increases with the inclusion volume fraction, for inclusion radius ratio below a critical value, say η∗ . In general, the value of η∗ depends on the elastic properties of the constituent materials and can be determined numerically by studying the functions fE(p) and fν(p) for each case of interest. For the material properties used in this work, we find that the critical value of the 15
14 12
1.1 1.05 1
8
E/Em
E/Em
10
1.15
Poly Mono Ave Corr
6
0.95 0.9
4
0.85
2 0 0
0.8
0.2
0.4
Φ
0.6
0.8
0.75 0
1
(a)
Poly Mono Ave Corr 0.2
0.4
Φ
0.6
0.8
1
(b)
Figure 12: Effective Young’s modulus for the composite material with polydisperse inclusion, according to different homogenization schemes. The inclusion phase composition in (a) is described by (η1 = 0.1, ψ1 = 0.25), (η2 = 0.2, ψ2 = 0.25), (η3 = 0.6, ψ3 = 0.25), and (η4 = 0.7, ψ4 = 0.25). In (b), the inclusion phase is described by the distribution (η1 = 0.90, ψ1 = 0.25), (η2 = 0.91, ψ2 = 0.25), (η3 = 0.98, ψ3 = 0.25), and (η4 = 0.99, ψ4 = 0.25).
radius ratio is approximately given by η∗ = 0.95. The value η∗ identifies a critical transition region rather than a fixed point in dependence of the effective modulus on the inclusion volume fraction. To highlight the main characteristics of the studied homogenization schemes, we consider two representative scenarios in which four families of inclusions with moderately different radius ratio are homogenized. The void content is neglected, Φv = 0, and the four families of inclusions are introduced in the matrix material with a constant volume ratio, that is, ψ j = 0.25, for j = 1, . . . , 4. Figures 12 show the influence of the selected homogenization scheme on the effective modulus of the composite material, as a function of the inclusion volume fraction Φ. A very close agreement among predictions from the four methods is observed for radius ratio in the range [0.1, 0.7], see Fig. 12(a). In this case, a polydispersion with average wall thickness value η¯ = 0.40 is considered. In particular, the second order correction scheme yields approximately the same results of the polydisperse homogenization. This hints that the modulus sensitivity on the distribution variance, as introduced in Eq. (11), dramatically decreases as thicker particles are considered in the homogenization scheme. In Fig. 12(b), we report the illustrative case where the polydisperse distribution has an average value of η¯ = 0.945, and the particle radius ratio values are in the interval [0.90, 0.99]. We find that, for the composite material under analysis, the prediction of the polydisperse homogenization scheme yields a decreasing Young’s modulus as the inclusion volume fraction increases, which is in contrast with the trend obtained for a monodisperse system. On the other hand, the second order correction to the monodisperse scheme presented in Eq. (11) is able to capture the decreasing trend in the composite’s Young’s modulus. Therefore, it is evident that the mean value of the radius ratio polydispersion may not be sufficient for a complete and accurate prediction of the effective Young’s modulus of the composite. The modulus is indeed extremely sensitive to the distribution variance, especially for thin walled particles that are relevant in practical applications. 5.3. Extension to continuous polydispersion In this Section, we study, for the continuous polydispersion case, the influence of the distribution statistical properties on the predictions of the model in Eqs. (9). To allow for a comparison with the results proposed in Fig. 12(b), we neglect porosity by selecting Φv = 0 and choose the distribution parameters β1 and β2 to obtain m1 = 0.9 or m1 = 0.945. Therefore, we determine the effect of polydispersivity through a parametric study, in which we vary the parameters of the distribution while keeping the mean constant. We also report standard deviation and skewness values obtained for each case analyzed. As shown in Figs. 3, the particle wall thickness distribution is generally not symmetric about the mean value, therefore the distribution’s skewness appears to be a significant feature of the description. Figures 13 illustrate the predictions of the continuous model for a set of Beta distributions with two 16
35
50 m = 0.065465, m = −1.1109
m = 0.04351, m = −1.4377
m2 = 0.039892, m3 = −0.69687
m2 = 0.03105, m3 = −1.044
2
2
40
m2 = 0.028333, m3 = −0.49925
25
Beta distribution
Beta distribution
30
3
m2 = 0.020079, m3 = −0.35538 m2 = 0.012716, m3 = −0.22567
20 15 10
3
m2 = 0.022058, m3 = −0.74843 m2 = 0.015634, m3 = −0.53292
30
m2 = 0.0099019, m3 = −0.33847
20 10
5 0 0.5
0.6
0.7
0.8
η (m1 = 0.9)
0.9
0 0.5
1
0.6
0.7
0.8
η (m1 = 0.945)
(a)
0.9
1
0.8
1
(b)
2.8
1.2 m = 0.065465, m = −1.1109 3
m2 = 0.039892, m3 = −0.69687
2.4
m2 = 0.028333, m3 = −0.49925
2.2
m2 = 0.020079, m3 = −0.35538
2
m2 = 0.012716, m3 = −0.22567
1
E/Em
E/Em
2.6
2
Monodisperse m = 0.9 1
1.8
0.8
m2 = 0.04351, m3 = −1.4377 m2 = 0.03105, m3 = −1.044 m2 = 0.022058, m3 = −0.74843
1.6 0.6
1.4
m2 = 0.015634, m3 = −0.53292 m = 0.0099019, m = −0.33847 2
1.2
3
Monodisperse m = 0.945 1
1 0
0.2
0.4
Φ
0.6
0.8
0.4 0
1
(c)
0.2
0.4
Φ
0.6
(d)
Figure 13: Beta distributions for χ(η) with (a) m1 = 0.9 and (b) m1 = 0.945, and varying standard deviation m2 and skewness m3 . Theoretical Young’s modulus for each distribution χ(η) with (c) m1 = 0.9 and (d) m1 = 0.945, compared with monodisperse results using mean values η = m1 (dashed line). The standard deviation and the absolute value of the skewness of the Beta distributions increase in the direction indicated by the arrow.
representative mean values, namely, m1 = 0.9 and m1 = 0.945, and decreasing standard deviation and skewness. The shape of the distribution is reported in Figs. 13(a) and 13(b), whereas the resulting moduli are shown in Figs. 13(c) and 13(d). These figures also compare the results obtained from the monodisperse homogenization scheme using only the mean radius ratio. Figures 13 show that significant discrepancies may arise among the homogenization techniques when the distribution involves moderately large standard deviations. In addition, the negative sign of the distribution’s skewness indicates that the shape of χ(η) is biased towards larger values of η, that is, a relevant percentage of the inclusion phase comprises particles with wall thickness significantly smaller than the mean value, see Figs. 13(a) and 13(b). The effect of thin walled particle inclusion is to lower the effective Young’s modulus of the composite, especially as the inclusion volume fraction increases and as the distribution is skewed towards η > η∗ . 6. Conclusions In this paper, we presented a homogenization method that accounts for polydispersivity in the inclusion phase in syntactic foams. The proposed method extends the monodisperse differential scheme for analysis of hollow spherical inclusion. The method provides a means to analyze the effects of particle wall thickness distribution, either discrete or 17
continuous, of different particle materials and the influence of voids on the elastic properties of syntactic foams. We derived manageable analytical expressions for the implementation of the technique, as well as simple descriptions of the qualitative dependence and sensitivity of the composite effective properties on the statistical characteristics of the included phases. We identified higher-order phenomena, that generally result in a decrease in the effective modulus, and their relation to the composition of the included phase. These features, while being pertinent to a more realistic description of the microstructure of typical syntactic foams, are generally discarded in available monodispersion studies. Theoretical results are compared with experimental work and are generally found to be in good agreement. The proposed approach may be extended to predicting plastic and viscoelastic response of composites by following the line of arguments in Kontou (2007); Remillat (2007); Tagliavia et al. (2009b). Acknowledgements This work was supported by the Office of Naval Research grant N00014-07-1-0419 with Dr. Y.D.S. Rajapakse as the program manager and by the National Science Foundation grant CMMI-0726723. Views expressed herein are those of authors, and not of the funding agency. The authors would like to gratefully acknowledge Mr. Gabriele Tagliavia for providing the experimental data on three point bending tests and some of the micrographs. A. Coefficients The functions introduced in Eqs. (5) are defined as follows, see Porfiri and Gupta (2009); Christensen (1979) " # 1 (i) 4 (i) (i) 2 fE = h (1 − 2ν) + hµ (1 + ν) (12a) E k 3 i 1 h (i) fν(i) = (−1 + ν + 2ν2 )(2h(i) (12b) µ (1 + ν) + 3hk (−1 + 2ν)) 3Eν where the superscript (i) can be specialized to the particle (p) and to the void (v) inclusion in the following form a0 + a3 η3 b0 + b3 η3 c0 + c3 η3 + c5 η5 + c7 η7 + c10 η10 h(p) µ (k p , µ p , k, µ, η) = d0 + d3 η3 + d5 η5 + d7 η7 + d10 η10 k(3k + 4µ) h(v) k (k, µ) = − 4µ 5µ(3k + 4µ) h(v) µ (k, µ) = − (9k + 8µ) h(p) k (k p , µ p , k, µ, η) =
(13a) (13b) (13c) (13d)
The coefficients ai , bi , ci , and di in Eqs. (13) are given by a0 = 4µ p (−k p + km )(−3km − 4µm )
(14a)
a3 = k p (3km + 4µ p )(−3km − 4µm )
(14b)
b0 = 4µ p (3k p + 4µm )
(14c)
b3 = −12k p (µ p − µm )
(14d)
c0 = 15(9λ p + 14µ p )(µ p − µm )µm (λm + 2µm )(14µ p (µ p + 4µm ) + λ p (19µ p + 16µm ))
(15a)
c3 = −375µm (λm + 2µm )(3(9µ2p − 10µm µ p + 8µ2m )λ2p + 4µ p (14µ2p − 9µm µ p + 16µ2m )λ p + 28(µ4p + 2µ2m µ2p )) c5 = 15120(λ p + µ p )2 (µ p − µm )2 µm (λm + 2µm ) c7 =
−375(27λ2p
28µ2p )(µ p
+ 56µ p λ p + 18
2
− µm ) µm (λm + 2µm )
(15b) (15c) (15d)
c10 = 15(19λ p + 14µ p )(µ p − µm )µm (λm + 2µm )(λ p (9µ p + 6µm ) + 2µ p (7µ p + 8µm ))
(15e)
d0 = (9λ p + 14µ p )(2µm (8µ p + 7µm ) + λm (6µ p + 9µm ))(14µ p (µ p + 4µm ) + λ p (19µ p + 16µm ))
(15f)
d3 = −50(3(9λm (3µ2p + µm µ p − 4µ2m ) − 2µm (−36µ2p + µm µ p + 28µ2m ))λ2p + 2µ p (3λm (28µ2p + 13µm µ p − 48µ2m ) + 14µm (16µ2p + 3µm µ p − 16µ2m ))λ p + 28µ2p (2µm (4µ2p + 3µm µ p − 7µ2m ) + 3λm (µ2p + µm µ p − 3µ2m )))
(15g)
d5 = 1008(λ p + µ p )2 (µ p − µm )(2µm (8µ p + 7µm ) + λm (6µ p + 9µm ))
(15h)
d7 = −25(27λ2p + 56µ p λ p + 28µ2p )(µ p − µm )(2µm (8µ p + 7µm ) + λm (6µ p + 9µm ))
(15i)
d10 = 2(19λ p + 14µ p )(µ p − µm )(3λ p (9λm (µ p − µm ) + 2(12µ p − 7µm )µm )+ 2µ p (3λm (7µ p − 12µm ) + 56(µ p − µm )µm ))
(15j)
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