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Manufactured cellular materials have been developed for a range of applications [Gibson & Ashby, 1988] (e.g. insulation, light-weight reinforcement, and filtration), and their natural counterparts (e.g. bone, sponge and wood) have a cellular structure that optimises performance in a particular setting. 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 optimised for a given application. Many studies have focused on how local cell features, such as strut shape, affect properties. This approach is mainly useful for the study of single cells arranged in periodic arrays. Some theoretical results for such systems are generalisable to real materials, but some are not. 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 [Gibson & Ashby, 1988]:
n < 4, a wide range
of properties at
a given density. Experimental evidence suggests that n=2 for open cells.
The Poisson's ratio has been thought to be independent of
density [Gibson & Ashby, 1988].
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 optimise 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 struts or walls (e.g. curvature, and cross-sectional shape and uniformity). 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.
Most theoretical attention has been focused on simple cell structures with straight struts (or walls) arranged in periodic arrays at low densities. In this limit, explicit solution of the equations of elasticity can be avoided by using thin beam or plate theories. These results have elucidated some of the basic mechanisms of deformation, and their influence on the overall properties.
Consider periodic open-cell solids. If the material has 'straight-through' struts that traverse the extent of the sample, deformation occurs along the strut axis, and the moduli decreases in direct proportion to the density [Gent & Thomas, 1963,Ko, 1965,Christensen, 1986] (n=1). If the struts are finite, bending is activated at their intersection points and the Young's modulus can be shown to decrease quadratically with the density [Ko, 1965,Gibson & Ashby, 1982,Warren & Kraynik, 1997] (n=2) in agreement with experiment.
The 'tetrakaidecahedral' foam model, in particular, has been the subject of many recent studies [Zhu et al., 1997,Warren & Kraynik, 1997,Grenestedt, 1999]. 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. The foam has a relatively low anisotropy [Zhu et al., 1997] (the Young's modulus varies by less than 10 % with direction of loading), and is thought to be a good model of isotropic cellular solids. However, the model exhibits behaviour that is qualitatively different from that of real materials. Specifically, the Young's modulus deceases quadratically with density but the bulk modulus only decreases linearly [Zhu et al., 1997,Warren & Kraynik, 1997] (in disagreement with experiment). This corresponds to incompressible behaviour with a Poisson's ratio of 0.5 in the low-density limit, in contrast to the empirical value of around 0.33 [Gibson & Ashby, 1988].
There have been several attempts to incorporate isotropy in simple models of cellular structures. Christensen Christensen (1986) treated the case of straight through struts, which does not reproduce the quadratic density dependence of either the bulk or Young's modulus, while Warren and Kraynik (1988) examined the case of a rotationally averaged tetrahedral joint. The model predicts the same problematic linear dependence of the bulk modulus on density as the tetrakaidecahedral model. As Warren and Kraynik 1997) note, regular tetrahedra do not pack to fill space, and so the model cannot represent a real foam.
From the foregoing discussion it is clear that more complex, random models are necessary to improve predictions for cellular solids, since periodic models do not capture all the phenomena observable in real cellular solids. 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 emphasise that there are no exact analytical calculations available for general random materials, so that numerical methods become necessary.
Large scale computational methods [Garboczi & Day, 1995,Poutet et al., 1996] and sufficient computational power now exist for measuring the properties of complex digital microstructures with a reasonable degree of complexity. One route for property prediction is to directly image the porous structure, and then use a numerical method to predicts its moduli [Nieh et al., 1998]. This method can work well for a particular microstructure. However, it is also important to study how changes in microstructure affect properties (e.g., to guide optimisation). Statistical models, which allow variation of the density and structure, are ideally suited for this purpose.
There has been recent progress in this direction. Finite-element methods have been used to study the properties of random open-cell Voronoi tessellations [Van der Burg et al., 1997]. The similarities between the mathematical definition of the Voronoi tessellation and the physics of foam formation make the model a natural choice for cellular solids. A key question is whether the model can account for the properties of real cellular materials. As discussed above, the tetrakaidecahedral model, which is an example of the tessellation of a regular lattice, actually exhibits incompressible behaviour in the low density limit. It is interesting to see if this non-physical behaviour is due to the small anisotropy of the model. Since foaming is only one route to generating cellular solids [Gibson & Ashby, 1988], it is also interesting to investigate different types of cellular structures via alternate statistical models.
In this paper we use a finite element method (FEM) [Garboczi & Day, 1995] to estimate the elastic properties of four model cellular solids over a range of densities. The models are generated using tessellation methods [Stoyan et al., 1995], level-cut random field models [Berk, 1987], or by simply linking random nodes in space with struts. The models each have a unique microstructure, and are chosen to be broadly representative of the morphologies observed in real materials. The Young's moduli of the models can be described in terms of simple two parameter relations [e.g. equation (1) in the low density limit]. The results demonstrate the effect of microstructure, isotropic disorder, and finite density on the properties of cellular solids, including both Young's modulus and Poisson's ratio. Apart from the small numerical errors in the FEM, 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.