Prior results

In this section we discuss prior results for closed-cell foams. The results illustrate, and attempt to quantify, the basic mechanisms of deformation. We compare the results to our FEM results in subsequent sections to illustrate the effect of disorder in multi-cellular models.

Christensen [17] has derived a result for a closed cell material comprised of randomly located and isotropically oriented large intersecting thin plates. The results are,

$$\frac EE_s= \frac{2 (7-5\nu_s)}{3(1-\nu_s)(9+5\nu_s)} \frac \rho\rho_s;\;\; \nu= \frac{1+5\nu_s}{9+5\nu_s}.$$ (2)

where the subscript "s" indicates the solid phase. The linear dependence (n=1) of modulus on density is typical for cellular materials with straight-through' elements. In this case, cell-wall stretching is the only mechanism of deformation.

Analysis of more complex closed cell foams is very difficult, but computational results  [6,7,8,4,5] have been obtained for the closed cell tetrakaidecahedral foam shown in Fig. 1. Simone and Gibson [5] recently found that the Young's modulus is nearly equal (within 10 %) for loading in the $\langle100\rangle$, $\langle111\rangle$ and $\langle110\rangle$ directions. For the density range 0.05 < ρ / ρ_s < 0.20, their results for the $\langle100\rangle$ direction were fitted with the formula

$$\frac {E_{100}}{E_s} \approx 0.315 \left( \frac\rho\rho_s \right) + 0.209 \left( \frac\rho\rho_s \right)^2$$ (3)

which compares well with the earlier result E₁₀₀/E_s ≈ 0.33 ( ρ / ρ_s )  [4]. For the case where the face thickness is 5 % of the edge thickness, Mills and Zhu [8] found E₁₀₀ / E_s ≈ 0.06( ρ / ρ_s ) ^1.06 in the density range 0.015 < ρ / ρ_s < 0.1.

Gibson and Ashby have proposed the semi-empirical formulae

$$\frac EE_s \approx \phi^2 \left( \frac\rho\rho_s \right)^2 + (1-\phi) \frac\rho\rho_s; \;\; \nu \approx \frac 13$$ (4)

where φ is the fraction of solid mass contained in the cell-edges (the remaining fraction 1 − φ is in the cell faces). Gas trapped in the cells can also increase the stiffness, but this effect is usually negligible [1]. The first term of Eq. (4) accounts for deformation in the cell edges. Note that the case φ = 1 corresponds to a commonly used semi-empirical formula for open-cell solids, i.e. Eq. (1) with C=1 and n=2. The second term corresponds to stretching deformation in the cell faces. The result provides good agreement with data for closed-cell foams when 0.6 φ 0.8 [1]. In part, the implied relatively high cell-edge fractions can be attributed to the fact that surface tension forces drive mass out of the cell walls into the edges. However, in some foams, the cell faces are relatively thick, giving φ = 0.01 - 0.07 [8] (note that these authors report the fraction of total mass in the faces which corresponds to 1 − φ) and we expect Eq. (4) to overestimate the measured values.

There are also several kinds of exact bounds that have been derived for the elastic properties of composite materials [18]. If the properties of each phase in a composite are not too dissimilar, the bounds can be quite restrictive. For porous materials, however, the bounds on Poisson's ratio are no more restrictive than the range guaranteed by the non-negativity of K and G for isotropic materials ( -1 ν 0.5 ), and the lower bound on E reduces to zero. Nevertheless, the upper bound on Eis sometimes found to provide a reasonable approximation of the actual property.

The most commonly applied bounds for isotropic composites are due to Hashin-Shtrikman [19]. These bounds can be evaluated if the elastic properties and volume fraction of each phase are available. The upper bound E_u is

$$\frac{E_u}{E_s} \frac{p}{1+C_H(1-p)};\;\; \nu_u=0.5$$ = (5)

$$\frac{(1+\nu_s)(13-15\nu_s)}{2(7-5\nu_s)}.$$ C_H = (6)

Note that C_H (ν_s = 0.2) = 1, and 11/12 C_H < 1.006 for ν_s > 0 (the maximum occurring near ν_s = 0.27). Therefore as ( ρ / ρ_s ) → 0 E/E_s ≈ ½ ( φ / φ_s ). In order to improve the bound, it is necessary to know the N-point correlation functions of the composite [18,20]. These functions are generally only available for certain models to order N=3. In this case, the bounds are referred to as 3-point bounds.

It is interesting to compare the bound with the formulae reported above. It is simple to show that Eqs. (2) and  (6) are identical as ρ / ρ_s → 0. This indicates that, to the accuracy of Christensen's approximation, randomly oriented straight-through plates provide an optimally stiff microstructure. Also note that the semi-empirical formula given in Eq. (4) actually violates the bound if more that half the solid material resides in the cell faces ( $\phi > \frac12$ > ½).