Comparison of foam models from regular and irregular arrays of Gibson-Ashby open-cells

The paper studies effective elastic properties of foam or cellular materials modeled by a set of Gibson-Ashby open cells with regular or irregular structures. Currently, there are many papers that present results of studying cellular materials using theoretical, numerical and experimental methods. However, these papers consider either regular lattices, or a single cell, or representative volume models based not on the Gibson-Ashby models. In this paper, in addition to the regular lattice, irregular structures were numerically studied. A mathematical formulation of the homogenization problem based on the energy equivalence of a foam-like material and on a homogeneous comparison medium is described. Formulations of six boundary value problems are presented. The solutions of these problems allow us to determine a complete set of effective stiffness modules for foams with different types of physical and geometric anisotropies. All stages of the numerical study were implemented in the ANSYS finite element package. Two algorithms for forming solid-state and finite-element models of irregular Gibson-Ashby lattices with small and large porosity are described in detail. As an example, numerical calculations are carried out for polycarbonate foams. The values of the effective elastic modules for regular and irregular lattices and for the Gibson-Ashby analytical model are compared. The results of numerical experiments showed that the Gibson-Ashby model describes the behavior of highly porous materials quite well (for porosity more than 75%), but this model gives a less satisfactory prediction in case of lower porosity. It is noted that for a large number of cells, regular and irregular lattices statistically give similar results for effective modules. However, for individual structures of irregular lattices, especially with strongly differing cells in individual directions, the effective moduli can have significantly different values, and the effective homogeneous medium can have pronounced anisotropic properties. These effects are due to geometric anisotropy and stress concentration in long connecting beams and at the joints of beams of various sizes in highly irregular Gibson-Ashby lattices. Examples of such lattices are given. We analyze the scatter of value for relative modules, which characterizes the anisotropy of such structures.