Effect of deleting faces

Depending on the physical conditions for foam formation, it is possible for the final foam to contain both open and closed cells. It is relatively easy to delete cell-walls from the Voronoi tessellation, which allows us to quantitatively investigate how the presence of partially open cells degrades the foam stiffness. In the Voronoi tessellation, a cell edge is defined by points which are equidistant from three (or more) cell centers. If the edges are retained but the cell walls are absent, an open-cell Voronoi tessellation results. In a parallel study of open-cell foams we have shown that the Young's modulus of the open-cell foam is
E / E_s = 0.93 ( φ / φ_s )^2.04 for φ / φ_s < 0.5, for in good agreement with scaling arguments.

To test the intermediate cases we consider tessellations with 20 %, 40 %, 70 % and 85 % of the faces removed at random. The underlying edges of the open-cell tessellation are left intact. Examples of the microstructure are shown in Figs. 4 and 5 for the cases where 70 % and 100 % of the cell walls have been deleted. The results are plotted in Fig. 6. If 20 % of the faces are deleted, we find E / E_s = 0.64 ( φ / φ_s ) ^1.4 and when 40 % of the faces are deleted, the result is E / E_s = 0.76( φ / φ_s )^1.7 . For 70 % and above, the Young's modulus follows the open-cell result.

Figure 4: The Voronoi tessellation model with 70 % of the faces deleted. The reduced density is ρ / ρ _s ≈ 0.09.

Figure 5: The open-cell Voronoi tessellation model with reduced density ρ / ρ _s =0.05. (All the faces have been deleted.)

Figure 6: The effect of deleting faces from the closed-cell Voronoi tessellation. The symbols correspond to deletions of 20 % ($\circ$), 40 % ($\Box$), 70 % ($\triangle$) and 85 % ($\diamond$). The extreme cases of no-deletions (---) and 100 % deletions (-- --) are also shown. The dotted lines correspond to the empirical fits described in the text.

It would be theoretically useful to partition the results for partial deletion into a contribution from edge-bending and plate stretching, similar to Eq. (4). For each model, we can directly measure the respective solid fractions p_e and p_f of the edges and faces, with p_e + p_f = ρ / ρ _s. In the absence of cell walls, the edges have a modulus of E / E_s ≈ p ²_e , and the cell walls should contribute a term which depends linearly on p_f. Thus, if the contributions can be linearly combined, a plot of F (p_f = (E / E _s − p²_e ) / p_f vs. p_f should yield a constant value. This is seen not to be the case in Fig. 7. Indeed F (p_f ) is seen to increase nearly linearly with p_f indicating that the additional contribution of the mass in the cell walls to the Young's modulus approximately follows a quadratic law. We conclude that it is not possible to describe the Young's modulus of the partially open cell model in terms of a contribution of 'edge-bending' and 'plate-stretching'. Clearly both mechanisms are active in deformation, but they combine non-linearly. Our evidence suggests that the data can be best represented by a power law with a non-integer exponent 1 < n < 2.

Figure 7: If the modulus of the partially-open tessellation model could be linearly resolved into a contribution from the cell faces and edges (with fractions p_f and p_e) then we would expect E / E_s − p²_e) / p_f to be independent of p_f. This is seen not to be the case. The line has a slope of one. The symbols correspond to deletions of 0 % ( $\bigtriangledown$), 20 % ($\circ$), 40 % ($\Box$), 70 % ($\triangle$) and 85 % ($\diamond$).