Computer simulations of auxetic foams in two dimensions

Two simple models of two-dimensional auxetic (i.e. negative Poisson's ratio) foams are studied by computer simulations. In the first one, further referred to as a Y-model, the ribs forming the cells of the foam are connected at points corresponding to sites of a disordered honeycomb lattice. In the second one, coined a Delta-model, the connections of the ribs are not point-like but spatial. For simplicity, they are represented by triangles centered at the honeycomb lattice points. Three kinds of joints are considered for each model, soft, normal and hard, respectively corresponding to materials with Young's modulus ten times smaller than, equal to and ten times larger than that of the ribs. The initial lattices are uniformly compressed, which decreases their linear dimensions by about 15%. The resulting structures are then used as reference structures with no internal stress. The Poisson's ratios of these reference structures are determined by stretching them, in either the x or the y direction. The results obtained for finite meshes and finite samples are extrapolated to infinitely fine mesh and to the thermodynamic limit, respectively. The extrapolations indicate that meshes with as few as 13 nodes across a rib and samples as small as containing 16 x 16 cells approximate the Poisson's ratios of systems of infinite size and infinite mesh resolution within the statistical accuracy of the experiments, i.e. a few per cent. The simulations show that by applying harder joints one can reach lower Poisson's ratios, i.e. foams with more auxetic properties. It also follows from the simulations performed that the Delta-model gives lower Poisson's ratios than the Y-model. Finally, the simulations using fine meshes for the samples are compared with the ones in which the ribs are approximated by Timoshenko beams. Taking into account simplifications in the latter model, the agreement is surprisingly good.