Discussion and Conclusion

We have used the finite element method to estimate the Young's modulus of four realistic random models of isotropic cellular solids. The open-cell Voronoi tessellations have a microstructure similar to that observed in foams. The node-bond and level-cut Gaussian models were considered representative of cellular solids generated by other (non-foaming) processes. At low densities, the results could be described by the scaling relation E / E_s = C( / _s )ⁿ , where C and n are given in the text. At moderate to high densities the results could be described by (equation 9) using the parameters reported in the text. The fitting relations we have derived can be used to predict the properties of cellular materials that have a microstructure similar to one of the models, and can be useful for interpreting experimental data.

Our results have theoretical implications for predicting the elastic properties of cellular solids. The most widely used formula for the Young's modulus of open cell materials is E / E_s = ( / _s ) ² , with a Poisson's ratio of  = 0.33 (independent of density and solid Poisson's ratio $\nu_s$_s). While this result is broadly applicable, we have shown that the properties of open-cell materials are more complex. The highly coordinated ($\bar z=12.5$ = 12.5) node-bond model has an exponent of n=1.3 due to spatial 'locking' of the nodes, so that the major mechanism of deformation is axial tension (or compression) rather than bending. The open-cell random field model, which is not based on an underlying polyhedral structure, had an exponent of n=3.0. This was attributed to enhanced deformation in the struts, which are bent and have non-uniform thickness. Generally the Poisson's ratio showed a weak dependence on density. For three of the four open-cell models the Poisson's ratio converged to $\nu\approx 0.25$ 0.25 at low densities, in agreement with several independent theories. The fact that the solid Poisson's ratio was taken to be 0.2 meant that the overall Poisson's ratio was nearly independent of density. If the solid Poisson's ratio was much different, the overall Poisson's ratio would be a much stronger function of density, because of its convergence or 'flow diagram' behaviour [Garboczi & Roberts, 2000].

One of the most surprising results was that isotropic open-cell Voronoi tessellations were nearly incompressible (Poisson's ratio $\approx 0.5$ 0.5) at low densities. This was directly related to the fact that the model foam is much stiffer under uniform compression than axial (or shear) deformation (K while E, G ² ). At low densities, the properties of the model foam are actually very well predicted by Warren and Kraynik's theoretical result for an isotropically oriented tetrahedral joint. This provides an explanation of the unusual behaviour. In a perfect tetrahedral joint under uniform compression, the forces are balanced so that the central node is locked in position. Therefore the deformation is only along the strut directions (i.e. K $K\propto \rho$ ). Even though the struts in the random model are not perfectly tetrahedral, our results indicate the same node locking occurs on average.

The Poisson's ratio of the tesselation is significantly higher than observed in experiments ( $\nu_{\rm expt} \approx 0.33$_expt 0.33). The most likely explantion indicated by our results is that real foams contain broken bonds (around 15 %). Bent bonds may also play a role, although a recent calculation of Grenestedt [Grenestedt, 1998] indicated that this does not influence the Poisson's ratio (the bulk and Young's modulus decrease by a common factor). A more sophisticated model may show otherwise. Our results for a poly-disperse model (with maximum to minimum cell diameter ratio of around 4, see Figure 7) showed that the Poisson's ratio decreased by about 10 %. Since real foams can have greater polydispersity, this provides another possible explanation of the discrepancy between the model and experimental results. We will consider these questions in future work. Note also that it is difficult to measure the Poisson's ratio experimentally. Indeed the data presented by Gibson and Ashby falls in the range 0.1 < < 0.45 (with $\langle \nu \rangle=0.33$). Since there is no independent evidence that as many as 15 % of the bonds are broken in foams, and the Voronoi tesselation appears to be a reasonable microstructure model, it is also possible that more precise experiments will in fact show a higher Poisson ratio.

In this study, we have shown that it is absolutely necessary to consider large-scale (multi-cellular) models of random cellular solids in order to obtain realistic elastic properties. Our results are consistent with experimental data, and show a more complex density dependence than predicted by conventional theories based on periodic cell models. Our results focus on the global (e.g. connectivity and geometrical cell arrangement), rather than local characteristics (e.g. strut cross-sectional shape or curvature) of cellular materials, for the following reasons. First, it is difficult to simultaneously model the local and global variables with finite computational power, and second, study of single cell models probably provides a more fruitful route to understanding the influence of local cell-character on the overall properties. We believe that the results of both approaches may be beneficially combined.

Acknowledgements

A.P.R. thanks the Fulbright Foundation and the Australian Research Council for financial support. We also thank the Partnership for High Performance Concrete Technology program of the National Institute of Standards and Technology for partial support of this work.