Micromechanics model for three-dimensional open-cell foams using a tetrakaidecahedral unit cell and Castigliano's second theorem

A micromechanics model for three-dimensional open-cell foams is developed using an energy method based on Castigliano's second theorem. The analysis is performed on a tetrakaidecahedral unit cell, which is centered at one lattice point of a body-centered cubic lattice and is subjected to compression on its two opposite square faces. The 36 struts of the unit cell are treated as uniform slender beams undergoing linearly elastic deformations, and the 24 vertices as rigid joints. All three deformation mechanisms of the cell struts (i.e., stretching, shearing and bending) possible under the specified loading are incorporated, and four different strut cross section shapes (i.e., circle, square, equilateral triangle and Plateau border) are treated in a unified manner in the present model, unlike in earlier models. Two closed-form formulas for determining the effective Young's modulus and Poisson's ratio of open-cell foams are provided. These two formulas are derived by using the composite homogenization theory and contain more parameters than those included in existing models. The new formulas explicitly show that the foam elastic properties depend on the relative foam density, the shape and size of the strut cross section, and the Young's modulus and Poisson's ratio of the strut material. By applying the newly derived model directly, a parametric study is conducted for carbon foams, whose modeling motivated the present study. The predicted values of the effective Young's modulus and Poisson's ratio compare favorably with those based on existing models and experimental data.