Effect of polydispersivity and porosity on the elastic properties of hollow particle filled composites

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 accounting for polydispersion in the model allows for a closer agreement with experimental results than monodisperse schemes. In particular, qualitatively different predictions are observed in syntactic foams containing thin walled particles when accounting for polydispersivity.

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 studies (see for example Gupta et al., 2001, Gupta et al., 2004, Kishore et al., 2010). 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 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., 2010). 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 and Lo (1979) is adapted 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. Fig. 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. Fig. 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., 2010) 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., 2010). This implies that the larger size particles will be more compliant, thus indicating that modeling polydispersivity 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, Duan et al., 2007b, this formulation is generalized to include the presence of elastic interface effects. In Bardella and Genna (2001), the homogenization technique proposed in Hervé 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).

The aim of this study is to develop a micromechanics-based model for predicting the effects 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 extending the classical differential scheme sequentially developed in McLaughlin, 1977, Norris, 1985, Zimmerman, 1991 with a homogenization technique stemming from the limit for infinitely small inclusion volume fraction of the procedure presented 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.