Lonsdaleite Model of Open-Cell Elastic Foams: Theory and Calibration

Abstract

We formulate a unit-cell model of open-cell elastic foams. In this model, a foam consists of four-bar tetrahedra arranged in the hexagonal diamond structure known as Lonsdaleite. The parameters of the model are the Young’s modulus of the bars and a few geometric parameters, the values of which may be roughly estimated for any given foam. We use the model to simulate a set of experiments in which elastic polyether polyurethane foams in a broad range of densities were tested under five loading conditions, namely tension along the rise direction; compression along the rise direction; compression along a transverse direction; simple shear combined with compression along the rise direction; and hydrostatic pressure combined with compression along the rise direction. With a suitable choice of values of the parameters of the model, the stress–stretch curves that we compute using the model compare favorably with the stress–stretch curves that were measured in the experiments. In some of the experiments a stress plateau in the stress–stretch curve was accompanied by heterogeneous stretch fields, even though the attendant stress fields were homogeneous. For these experiments we show that the model can be used to predict the occurrence of a second-order phase transition, so that the plateau stress can be interpreted as a Maxwell stress and the attendant heterogeneous stretch fields as two-phase fields, consistent with the experimental evidence. In other experiments the stress–stretch curve evinced a sudden and pronounced loss of stiffness, but no genuine stress plateau, and the attendant stretch fields remained homogeneous. For these experiments we show that the model can be used to predict the occurrence of a bifurcation of equilibrium in which the stress keeps rising as the deformation continues to increase in the post-buckling stage, so that the stretch fields remain homogeneous throughout, consistent with the experimental evidence. In general, to appraise the goodness of our model we put emphasis on the relation between the stress–stretch curve measured in an experiment and the nature of the attendant stretch fields. We submit that this emphasis should remain a guiding methodological trait in the appraisal of constitutive models of open-cell elastic foams.