Space-Filling Polyhedra as Mechanical Models for Solidified Dry Foams

Abstract

The simulation of the mechanical behavior of idealized cellular structures is important as it gives insight into the principal deformation mechanisms that govern the mechanical behavior of real cellular structures, such as polymer foams or metallic foams, making accessible at least qualitative information about properties that are difficult to determine otherwise, for example the effective strength under hydrostatic loading. For capturing the mechanics of a closed-cell foam material in a meaningful way, three-dimensional models are employed in the context of the Finite Element method. The modeling approach presented in this paper employs generic, periodic unit cell models with the extension of providing physically sound microstructures by basing these models on surfaces of minimal energy calculated with the program Surface Evolver. This program is able to minimize the energy of a surface subject to given constraints, for example a prescribed initial geometry and, where required, periodicity. Accordingly, space-filling polyhedra with cells being separated by walls of minimal total area can be predicted, such as Lord Kelvin's tetrakaidecahedra or the Weaire-Phelan partition, the latter being energetically more favorable. These physics-based configurations are good representations of solidified (dry) foams. The results obtained by Finite Element stress and deformation analyses comprise the full tensor of elasticity and its dependence on the effective density; the non-planar faces resulting from the surface energy minimization are shown to influence the behavior for very low effective densities. By studying the behavior up to the onset of yielding on the effective, macro-mechanical level including the effects of multi-axial loading conditions, valuable information for the homogenized representation of closed-cell foams is obtained.