Analysis of two- and three-dimensional hyperelastic model foams under complex loading conditions

Subject of the present study is the numerical analysis of hyperelastic two- and three-dimensional model foams at large strains. The macroscopic stress–strain relationships are determined by means of a strain energy based homogenization procedure from the behavior of the cellular structure at the mesoscopic level. The proposed homogenization procedure is based on the assumption that a representative volume element with the cellular microstructure and a volume element containing the homogeneous effective medium are macroscopically equivalent if both volume elements hold the same amount of strain energy. As a first and simplifying approach spatially periodic 2-D and 3-D lattices are adopted for representing open-cell foams. The 2-D approximation is the commonly used honeycomb microstructure, whereas its 3-D counterpart is a regular lattice with tetrakaidecahedral cells. Subsequently, the effective mechanical response of these models is compared under uniaxial and multi-axial loading cases. On the macroscale, it is observed that the 2-D model foam covers most of the basic features of the three-dimensional cellular structure. Also on the mesoscale the same principal deformation mechanisms like cell wall bending and stretching are observed. However, the effect of different modeling dimensions of a solid foam should be taken into account if quantitative predictions are required.