Comparison of FEM results with experiment

To illustrate the utility of the FEM, we compare the computed results to experimental data (it should be remembered that the computed results have about a 10 % uncertainty, mainly due to digital resolution). 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 summarized [1,3].

Data for closed cell porous glass [31,32] (Fig. 11) agrees well with the FEM results obtained using the closed cell Voronoi tessellation. Micrographs of the glass studied by Zwissler and Adams [32] indicate a structure similar to that of the Voronoi tessellation shown in Fig. 2, indicating that the model is appropriate. Data for closed cell polymer foams is shown in Fig. 12. The data for expanded polystyrene [33] generally agree with the predictions of the closed-cell Gaussian random field model. The data for extruded polystyrene [34] decreases from the Voronoi tessellation towards the Gaussian random field result as the density decreases. Micrographs of polystyrene [13] indicate a cell structure similar to that of the Voronoi tessellation, but the cell walls show some curvature. This may explain why the results for the random field model (which contains curved cell walls) more closely matches the data.

Figure 11: Young's modulus of foamed glasses with closed cells. The data is from Morgan et al [31] ($\Box$), Zwissler and Adams [32] ($\circ$) (E_s=69 GPa [3]) and Walsh et al [35] ($\triangle$) (E_s=75 GPa). The solid line (---) corresponds to the closed-cell Voronoi tessellation. The closed-cell GRF model (- - -) and the conventional theory E / E_s = ( ρ / ρ_s ) ² ($\cdots$) are shown for comparison.

Figure 12: Young's modulus of closed cell polymer foams. The data is for extruded polystyrene [34] ($\circ$, E_s=1.4 GPa and ρ_s = 1050 Kg/m³), polystyrene beads [8] ($\triangle$, E_s=3.0 GPa and ρ_s = 1100 Kg/m³) expanded polystyrene [33] ($\Box$, E_s=2.65 GPa and ρ_s = 1020 Kg/m³ ) and for low-density polyethylene [36] ( $\bigtriangledown$). The solid (---) and dashed lines (- - -) correspond to the closed cell Voronoi tessellation and Gaussian random field models. The dotted line ($\cdots$) is the conventional theory for open cell foams E / E_s = ( ρ / ρ_s )² .