2.4.3. Modified Gibson–Ashby Model

Even though the foam cell structure can be appreciated from the SEM analysis, the direct correlation between the cell walls or edges structure and mechanical properties is better elucidated by the Gibson–Ashby model [37,38,39]. It correlates the relative density (density of the foam over the density of the cell-walls (or cell-edges) material, ρ/ρ_s) with the mechanical properties of the foam, such as the relative Young’s modulus given by the ratio between the elastic modulus of the foamed material (E) and the elastic modulus of the material constituting the cell walls (Es) by Equation (2).

where C and C′ are usually equal to 1, and the factor ϕ is the fraction of volume condensate in the cell edges. This equation is usually related to foams with closed-cell structures. If ϕ = 1, then the equation becomes

which models open-cell foams. However, a more generic equation could explain how the pores constituting the foam are made:

when C = 1, it is possible to obtain the value of the exponent n for each set of data.

Here we propose a modified Gibson–Ashby equation to model the linear elasticity of our PU foams with a hybrid open/closed cells structure. We start with Equation (2) to obtain ϕ values for each foam. In this way, we collected a dataset of the average pore of the single PU foam is constituted, in terms of material distribution between edges and wall.

Once the dataset was collected, we used it to develop the modified Gibson–Ashby model that could better fit with our outcomes.

with C″ (given by Equation (6)) is a constant, depending on the variables (ϕ and n) by a power law, by means of two arbitrary constants a and b. Since the equation is non-linear, it is solved by successive attempts, in order to obtain a general value of C″, from which each out-coming best fit does not differ from the relative modulus and relative density of the corresponding material of more than 1%. In our specific case, the two constants resulted to be a = 1.7 and b = 5.

Substituting the exponent n with the values found with Equation (4) and the factor ϕ with values found with Equation (2), the presented model fit the measured mechanical parameters best and was better at elucidating the structure of the foam.