Constitutive equations are developed for the elastic behavior of polymer foams under three-dimensional deformations with finite strains. These relations provide an extension of the Danielsson–Parks–Boyce model for a compressible matrix material in a representative volume element, whose volumetric deformation is governed by the specific work of external forces. The mechanical response of foams is described by the governing equations with four adjustable parameters. Two of them are material constants independent of porosity ϕ. The other two coefficients are functions of ϕ, for which simple equations are suggested. The model is applied to the analysis of observations on an extensive number of foams. Good agreement is demonstrated between results of simulation and experimental data in uniaxial tensile tests, compressive tests, and tension-compression tests. Predictions of the model for mechanically induced evolution of the microstructure of foams are in accord with available observations.