The geometry of low-density, closed-cell, polyethylene and polystyrene foams was modelled with a Kelvin foam having uniform-thickness cell faces; finite element analysis (FEA) considered interactions between cell pressures and face deformation. Periodic boundary conditions were applied to a small representative volume element. In uniaxial, biaxial and triaxial tensile stress states, the dominant high-strain deformation mechanism was predicted to be tensile yield across nearly flat faces. In uniaxial and biaxial compression stress states, pairs of parallel plastic hinges were predicted to form across some faces, allowing them to concertina. In hydrostatic compression, face bowing was predicted. The rate of post-yield hardening changed if new deformation mechanisms became active as the foam strain increased. The effects of foam density and polymer type on the foam yield surface were investigated. Improvements were suggested for foam material models in the FEA package ABAQUS.