Linear elastic behavior of a low-density Kelvin foam with open cells

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 pat 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.