This Is AuburnElectronic Theses and Dissertations

Numerical Modeling of Random 2D and 3D Structural Foams Using Voronoi Diagrams: A Study of Cell Regularity and Compression Response




Sotomayor, Oscar

Type of Degree



Mechanical Engineering


Representing the complex morphology of cellular solids in general and metal foams in particular for understanding key factors governing their mechanical behavior is typically achieved using periodic unit cells. Due to the intricacies of foam microstructures, however, such idealizations have many limitations. In this context, the Voronoi tessellation technique in different Euclidean spaces is employed in the present thesis to better represent structural foams. A procedure to generate dry and wet Voronoi structures through stochastic generation of nuclei in a control space based on Poisson probability distribution in 2D and 3D is attempted. Moreover, a process of controlling the degree of irregularity of Voronoi 2D and 3D foams through the application of a Simple Sequential Inhibition (SSI) process is demonstrated. A suitable and intuitively simple index for quantifying the regularity (δ) of a specific foam structure both in 2D and 3D has been identified. Further, the extremely low probability of generating highly regular foam configurations based on pseudo-random generation of nuclei has led to the development of a perturbation process to complement the SSI process. The possibility of creating real Voronoi foams using Additive Manufacturing methods has also been demonstrated. A finite element modeling procedure incorporating solid modeling methods is developed to analyze the influence of the cell regularity and density on the compression response of Voronoi honeycombs (2D) of low relative densities (ρ ̅) in the range of 3% to 9%. An isotropic response is seen in all the configurations with only a limited directional dependency in case of regular honeycombs of higher densities. An inverse relationship between regularity and elastic modulus is also identified. A fully irregular (δ = 0) configuration is 66% stiffer than a completely regular counterpart (δ = 1). In contrast, the plastic-collapse strength of a regular foam is {28%, 31%, 35% and 50%} higher than a fully irregular Voronoi honeycomb of relative density ρ ̅ = {9%, 7%, 5%, 3%}. Simulations also show that elastic modulus and plastic-collapse strength of Voronoi honeycombs scale with ρ ̅3 and ρ ̅2, respectively. The influence of regularity has also been studied in the context of syntactic foam-filled Voronoi honeycombs as a 2D representation of an Interpenetrating Phase Composite (IPC). Voronoi wet honeycombs with δ = {0, 0.5, 0.7, 0.8 and 1} and ρ ̅ = 9% are used for simulations. Irregular configurations show an isotropic response with an inverse monotonic relation between mechanical properties and cell regularity in the elastic and plastic regimes. In the elastic range, the stress-strain response is bounded by those of regular geometries compressed in two orthogonal directions. The analysis presented for Voronoi honeycombs is also extended to the 3D space to simulate open cell Voronoi foams. The elastic modulus of foams with different regularities and relative densities in the range of 3% to 9% are compared with the analytical and empirical models available in the literature. Direct relationships between elastic modulus, plastic-collapse strength and plateau strength with the degree of regularity are observed. A regular foam based on a 3D array of regular tetrakaidecahedron cells is {41.3%, 43.7%, 46.5% and 49.5%} stiffer than a fully irregular Voronoi foam for ρ ̅ = {9%, 7%, 5%, 3%}, respectively. Simulations show that elastic modulus of Voronoi foams can be described in terms of relative density (ρ ̅) using a second order polynomial whereas the plastic-collapse strength scales with ρ ̅1.5. It is shown that the deleterious effect of sudden collapse of regular 3D foams at yield can be mitigated by introducing a small perturbation to the regularity, say, δ = 0.95.