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.

Textural equilibrium melt geometries around tetrakaidecahedral grains

“…For low dihedral angles, this occurs close to φ = 0.11, but for larger dihedral angles there is an overlap where both 'c' and 's' topologies exist. The wetting of square faces at around φ = 0.11 is also seen in calculations of wet Kelvin foams [19][20][21][22][23][24]. Wet Kelvin foams are essentially a limiting case of the problem considered here: the case where the dihedral angle approaches 0 • .…”

Comparison of elastic properties of open‐cell metallic biomaterials with different unit cell types

“…Several studies have been carried out on selecting the appropriate unit cell types to manufacture biomimetic scaffolds aimed for bone tissue engineering. 14,15 Cube, 1,16,17 rhombic dodecahedron, 2,16,18,19 tetrakaidecahedrons, [20][21][22] Weaire-Phelan, 23,24 and diamond 12,16 are the most popular unit cell types which have been investigated mechanically before. Other micro-lattice structures such as body-centred cubic structure (BCC), 25 body-centered cubic with vertical pillars (BCC-Z), 25 rhombicuboctahedron, 26 truncated cube, 27 facet-centered cubic with vertical pillars (FCC-Z), 28,29 and truncated cuboctahedron 30 have also been investigated by different researchers.…”

Aluminium Alloy Foam Modelling and Prediction of Elastic Properties Using X-ray Microcomputed Tomography

“…The same tendency is observed for the open-cell foams (Figure 19b), but with greater accuracy. The Representative Volume Element models perform very well when compared to the theoretical and numerical models as well as the experimental results discussed in Figure 15a [53,69]. Again, the µCT model created here shows a value of relative Young's modulus just slightly above the predicted ones, consequently, demonstrating that it might be unnecessary to use such complex techniques.…”

“…An alternative to the proposed procedures based on µCT scans is using Representative Volume Element (RVE) models which try to recreate the different foams with a simplified structure which can be periodically repeated to construct foams of different sizes [53,61,69]. For this reason, they are far less detailed, but simpler and require less processing power, a feature that is highly desirable in case the results are accurate.…”