Introduction

Manufactured cellular materials have been developed for a range of applications [1] (e.g., insulation, light-weight reinforcement), and their natural counterparts (e.g. wood) have a cellular structure that optimizes performance for their particular requirements. The useful properties of cellular solids depend on the material from which they are made, their relative density, and their internal geometrical structure. It is important to link the physical properties of cellular solids to their density and complex microstructure, in order to understand how such properties can be optimized for a given application. Many studies have focussed on the elastic response of periodic materials. Equally important is the effect of disorder (e.g., isotropy), and the interaction between cells on a mesoscopic scale, as most real cellular solids are not periodic. In this paper we study model isotropic cellular solids at scales (≈ 100 cells) where these effects can be probed.

At low densities, experimental results indicate that the Young's modulus (E) of cellular solids is related to their density ( ρ ) through the relation [1]:

$$\frac E{E_s}= C \left( \frac{\rho}{\rho_s} \right)^n = C p^n$$ (1)

where E_s and ρ_s are the Young's modulus and density of the solid skeleton and p = ρ / ρ_s is the reduced density. The constants C and n depend on the microstructure of the solid material. The value of n generally lies in the range 1 < n < 4, giving a wide range of properties at a given density. For closed-cell foams, experimental studies indicate that 1 < n < 2. The complex dependence of C and n on microstructure is not well understood, and this remains a crucial problem in the ability to predict and optimize the elastic properties of cellular solids. At the local or cellular scale, important variables include the cell character (e.g. open or closed), the geometrical arrangement of the cell elements (e.g. angle of intersection), and the shape of the cell walls (e.g. curvature). At a larger scale, the geometrical arrangement of the cells is also crucial. The values of both C and n will depend on whether the material is periodic or disordered.

Analysis of simple models shows that three basic mechanisms of deformation are important for closed-cellular solids. If the cell walls are much thinner than the cell edges, the deformation is governed by edge-bending. In this case, E varies quadratically with density (n=2), and can be described by results for open cellular solids [1]. If cell-wall bending is the mechanism of deformation, Gibson and Ashby [2] have shown that Eshould vary cubicly (n=3) with density. However, the fact that 1 < n < 2 indicates that cell-wall stretching (n=1) is actually the dominant behaviour [3,1].

The "tetrakaidecahedral" foam model, in particular, has been the subject of many recent studies [4,5,6,7,8]. The cells of the model uniformly partition space, and are defined by truncating the corners of a cube giving eight hexagonal and six square faces (Fig. 1). The foam has a relatively low anisotropy [5] (E varies by less than 10 % with direction of loading), and is thought to be a good model of isotropic cellular solids. In all cases, E was found to decrease linearly with density (n=1). However, real materials exhibit a larger dependence of E on density (n>1), indicating that periodic models do not capture salient features of foam microstructure. It is likely that the random disorder is responsible, and it is important to study its influence on the properties of cellular solids.

Figure 1: The tetrakaidecahedral model. The model corresponds to the Voronoi tessellation of a body-centered cubic lattice.

There have been several recent studies of the effect of disorder in cellular solids. For two-dimensional models, variation in cell-shape leads to a variation of 4 % to 9 % in elastic properties [9], while deletion of 5 % of cell struts decreased the modulus by 35 % [10]. Similar effects of 'imperfections' were seen in spring lattices, which have some similarities to foams [11]. In three-dimensions, Grenestedt [6] showed that disorder decreased the Young's modulus of the tetrakaidecahedral foam (with 16 cells) by 10 %. However, only the pre-factor of Eq. (1) was affected; the scaling exponent remained constant (n=1). Grenestedt [12] has also estimated the effect of "wavy-imperfections" on the stiffness of a cube with closed cell walls. If the wave-amplitude was five times the cell-wall thickness, the stiffness decreased by 40 % compared to the case of flat faces.

From the foregoing discussion it is clear that more complex, three-dimensional random models are necessary to improve predictions for cellular solids [13]. There are two main problems in studying random models. First, a sufficiently accurate model of the microstructure must be developed. And second, the properties of the model must be accurately evaluated. We emphasize that there are no exact analytical calculations available for general random materials, so that numerical methods become necessary.

In this paper we use a finite element method (FEM) [14] to estimate the elastic properties of model cellular solids over a range of densities. The models are generated using tessellation methods [15] and level-cut random field models [16]. The Young's moduli of the models can be described in terms of simple two parameter relations [e.g., Eq. (1) in the low density limit]. The results demonstrate the effect of microstructure, isotropic disorder, and finite density on the elastic properties of cellular solids, including both Young's modulus and Poisson's ratio. Apart from the small numerical errors in the finite element method, 10 % or less, the results are exact for each of the models. Therefore, the results can be used to predict the properties of cellular solids if their structure is similar to one of the models, or be used to accurately interpret experimental data.