Foam mechanics: the linear elastic response of two-dimensional spatially periodic cellular materials

A theoretical model for the linear elastic response of two-dimensional, spatially periodic cellular materials in simple shear and pure strain is developed. Hexogonally symmetric network morphology, with three identical films intersecting at each junction, is assumed. Film thickness is permitted to vary with distance from the joint. Such hexagonal networks are transversely isotropic; the effective stress does not depend upon the principal directions of strain. The effective in-plane elastic constants are explicitly related to the film compliances for bending and stretching. Film bending is the primary deformation mechanism for low-density, two-dimensional foams, as observed by other investigators. The hexagonal symmetry that is responsible for mechanical isotropy is favored by a foam synthesis route that includes a liquid state. It is shown that a ‘square’ cellular network is neither elastically isotropic nor energetically preferred during formation.