Comparison of FEM results with experiment

To illustrate the utility of the FEM we compare the computed results to experimental data. Since real foams can have densities lower than those we are currently able to computationally study, we use the formula E / E_s = C( ρ / ρ_s ) ⁿ to extrapolate the results. This is justified by the fact that the low density FEM data appear to fall on a straight line when plotted against log-log axes. Accurate comparison of theoretical and experimental results is hindered by the imprecision involved in estimating the properties of the solid skeleton E_s and ρ_s. We report E_s and ρ_s when they have been given, but some data sets are reported only in terms of E/E_s and ρ_s. Some of the data sets we have taken from the literature have been previously summarised [Gibson & Ashby, 1988,Green, 1985].

Data for open-cell foams are compared with the open-cell FEM derived theories in Figure 10. The data for rubber latex foam lies above the line E / E_s = ( ρ / ρ _s ) ² and agree reasonably well with the FEM result for high-coordination number node-bond models. If the estimated value of E_s is correct, this suggests that the co-ordination number of the foam is quite high. The single data point ($\diamond$) obtained for a carbon foam [Christensen, 1986] falls on the same line. A micrograph in the reference indicates that the struts were tetrahedrally coordinated, unlike the model. Note that the normalisation constant E_s=6.9 GPa used by Christensen seems low compared to the value K_s=24 GPa (E_s =K_s if ν = 1/3) adopted for carbonised aerogels [Pekala et al., 1990]. Indeed, if E_s=24 GPa is assumed, the data point falls close to the line E / E_s = ( ρ / ρ_s ) ² .

Figure 10: Young's moduls of open-cell foams. The data is for alumina [Hagiwara & Green, 1987] ($\circ$, E_s=380 GPa, ρ_s =3970 kg/cm³), rubber latex obtained by Lederman [Lederman, 1971] ($\Box$) and Gent and Thomas [Gent & Thomas, 1963] ($\triangle$), open-cell foams [Gibson & Ashby, 1982] ( $\bigtriangledown$), and reticulated vitreous carbon [Christensen, 1986] ($\diamond$ , E_s=6.9 GPa). The lines corresponds to the four open-cell FEM theories derived in this paper; high- (- $\cdot$ -) and low (- - -) coordination number foams, open-cell Voronoi tesselation (---) and the open cell Gaussian random field model ($\cdots$).

The data for porous alumina agree reasonably well with the predictions of the open-cell GRF model. However, micrographs of the structure indicate a structure closer to that of the open-cell Voronoi tessellation (with occasional closed faces) so the agreement seems fortuitous. The fact that the data lies below the results for the open-cell Voronoi tessellation may be attributable to dead mass in the closed faces (i.e., if the mass in these faces was eliminated from the relative density the data would shift to the left [Gross et al., 1997], and possibly agree with open-cell Voronoi tessellation results). Data for the open-cell materials considered by Gibson and Ashby [Gibson & Ashby, 1982] is seen to agree well with the FEM results for the low-coordination number node-bond model and open-cell Voronoi tessellation. In general the data agree with the conventional theory, and hence our results for the open-cell Voronoi tessellation and low-coordination number node-bond model.

In a prior paper [Roberts, 1997] it was suggested that open-cell GRF's provide useful models of organic aerogels. It was shown that the models could reproduce the scattering intensities and predict the contribution of the solid network to the overall thermal conductivity of these low density materials. These prior results provide evidence that the model is reasonable, but it is also important to compare the elastic properties with experimental data. Data for the bulk modulus of open-cell organic aerogels is compared with the FEM results in Figure 11. We have assumed that K/K_s=E/E_s, which corresponds to the assumption that Poisson's ratio is constant with density [ ν ( ρ / ρ_s ) = ν_s], which we have shown to be approximately true at low density. The FEM results over-estimate the data for `polymeric' aerogels [Pekala et al., 1990] by factors of 2.4 and 1.5 for samples before and after carbonisation, respectively. However, the decay of modulus with density is reasonable, indicating that the basic structure of the model is correct, but that there is more elastically inefficient mass in the real materials (such as dangling ends, or struts of non-uniform width). The microstructure (and elastic properties) of aerogels are highly variable, and the data shown is for the stiffest structures. For example, data is also shown for a 'colloidal' aerogel in the figure. The struts of colloidal aerogels tend to be granular, with the narrow inter-particle necks decreasing the overall stiffness. Note that the random-field model can be modified to mimic this type of structure by shifting the position of the level-cuts [Roberts, 1997].

Figure 11: Bulk modulus of open-cell aerogel foams. The data is for carbonized ($\Box$, K_s=24 GPa, $\rho _s$_s=1500 kg/cm ³) and uncarbonized organic aerogels ($\triangle$, K_s=3.5 GPa, $\rho _s$_s =1300 kg/cm ³) [Pekala et al., 1990]. The circles correspond to data from Gross et al [Gross & Fricke, 1992,Gross et al.] for an uncarbonized organic prepared under different conditions. The lines corresponds to the open-cell Gaussian random field (---) and the conventional theory K / K_s = ( / _s ) ² (- - -).