We have used the finite element method to estimate the Young's modulus of realistic random models of isotropic cellular solids. At low densities, the results can be described by the scaling relation E / Es = C ( ρ / ρs )n, where the parameters are reported in the text. At moderate to high densities, the results were described by Eq. (10). All our results were obtained using a solid Poisson's ratio of νs = 0.2. It has recently been shown [30] that the Young's modulus only varies by around 2 % for 0 < νs < 0.5 indicating that our results are valid for all usual values of the solid Poisson's ratio. 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 for closed-cell Voronoi tessellations were in general agreement with earlier studies on the tetrakaidecahedral foam, i.e., E / Es ≈ 1/3 ρ / ρ s ) as ( ρ / ρ s ) → 0 [4]. The actual exponent was n=1.19 which is greater than the value n=1 obtained for single-cell models and scaling arguments [Eq. (4), φ < 1]. If more than 70 % of of the cell faces are removed, the Young's modulus exponent increased to n=2, indicating that edge bending becomes the dominant mechanism of deformation.
The closed-cell random field model, with curved cell walls, showed a significantly greater density dependence than the Voronoi tessellation (with exponent n ≈ 1.5). Although the minimum density at which we were able to measure properties was ( ρ / ρs ) ≈ 0.15, the data showed no evidence of adopting a linear decay for ( ρ / ρ s ) → 0, as suggested by theory.
The semi-empirical closed-cell theory given in Eq. (4) was found to not be applicable to the models studied here, since the low edge fractions (φ < 0.5) caused the equation to exceed known upper bounds on the Young's modulus. Moreover, attempts to describe the deformation of closed or partially closed cellular materials in terms of a bending (exponent n=2) and plate stretching (exponent n=1) component were not successful. Instead, our results indicate that these mechanisms combine non-linearly and are best represented by a non-integer power law. Variational upper bounds, and other predictions for random closed cell foams, were found to significantly over-estimate the Young's moduli of the models. Therefore numerical simulation must be relied on for accurate predictions.
In this study, we have shown that it is important to consider large-scale (multi-cellular) models of random cellular solids in order to obtain realistic elastic properties. While the modulus of the closed cell Voronoi tessellation can be approximately described by a single cell of the tetrakaidecahedral model, it is not possible to model the effect of missing faces and irregular cells with curved walls using single-cell models. Our results are consistent with experimental data, and show a more complex density dependence than predicted by conventional theories based on scaling arguments and periodic cell models. Our results focus on the effect of multi-cellular disorder, rather than local characteristics (e.g., distribution of mass between cell edges and walls) 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.