A micromechanical analysis for the linear elastic behavior of a low-density foam with open cells is presented. The foam structure is based on the geometry of a Kelvin soap froth with flat faces: 14-sided polyhedral cells contain six squares and eight hexagons. Four struts meet at every joint in the perfectly ordered, spatially periodic, open-cell structure. All of the struts and joints have identical shape. Strut-level force-displacement relations are expressed by compliances for stretching, bending, and twisting. We consider arbitrary homogeneous deformations of the foam and present analytic results for the force, moment, and displacement at each strut midpoint and the rotation at each joint. The effective stress-strain relations for the foam, which has cubic symmetry, are represented by three elastic constants, a bulk modulus, and two shear moduli, that depend on the strut compliances. When these compliances are evaluated for specific strut geometries, the shear moduli are nearly equal and therefore the elastic response is nearly isotropic. The variational results of Hashin and Shtrikman are used to calculate the effective isotropic shear modulus of a polycrystal that contain grains of Kelvin foam.

A theoretical model for the linear elastic properties of three-dimensional open-cell foams is developed. We consider a tetrahedral unit cell, which contains four identical half-struts that join at equal angles, to represent the essential microstructural features of a foam. The effective continuum stress is obtained for an individual tetrahedral element arbitrarily oriented with respect to the principal directions of strain. The effective elastic constants for a foam are determined under the assumption that all possible orientations of the unit cell are equally probable in a representative volume element. The elastic constants are expressed as functions of compliances for bending and stretching of a strut, whose cross section is permitted to vary with distance from the joint, so the effect of strut morphology on effective elastic properties can be determined. Strut bending is the primary distortional mechanism for low-density foams with tetrahedral microstructure. For uniform strut cross section, the effective Young’s modulus is proportional to the volume fraction of solid material squared, and the coefficient of proportionality depends upon the specific strut shape. A similar analysis for cellular materials with cubic microstructure indicates that strut extension is the dominant distortional mechanism and that the effective Young’s modulus is linear in volume fraction. Our results emphasize the essential role of microstructure in determining the linear elastic properties of cellular materials and provide a theoretical framework for investigating nonlinear behavior.

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