Materials with a cellular structure are increasingly used in engineering. Proper design requires an understanding of the response of the materials under complex stress states. The behavior of cellular solids under multiaxial loads has, up until now, hardly been studied at all:​ no comprehensive attempt to establish multiaxial failure criteria exists. In this study, we attempt to do this.

The mechanical behavior of cellular materials is primarily governed by bending and axial deformations of their cell walls. Bending deformation dominates when the material is loaded uniaxially;​ but it can be entirely suppressed under a uniform (hydrostatic) stress state, in which case cell wall stretching controls the behavior.

We first model the elastic buckling, plastic yield, and brittle collapse of two-dimensional hexagonal honeycomb-like structures under in-plane stresses to develop equations describing their failure surfaces. In the absense of shear stresses, the failure envelopes for plastic or brittle materials are extremely elongated along the axis of equal biaxial stress. For the tensile fracture of brittle honeycombs a maximum principal stress criterion (based on linear elastic fracture mechanics) is employed, resulting in a box-like cutoff in the tension-tension quadrant. The elastic buckling envelope which, too, is almost box-like, acts as a cutoff in the compressive quadrant.

We next examine three-dimensional cellular materials. The failure criteria obtained for isotropic plastic and brittle materials are functions of both the mean stress and the deviatoric stress. While they require a limited number of parameters to be defined, they are by no means the result of curve fitting procedures. The failure surfaces follow the results obtained for honeycombs:​ when plotted in the principal stress space, they are extremely elongated along the hydrostatic axis. They are truncated by an almost box-like elastic buckling surface in the compressive octant;​ for brittle materials, the failure surface is also truncated by a box-like cutoff in the tensile octant corresponding to a maximum principal stress criterion. Failure criteria have also been developed for materials with hollow, porous cell walls such as some foamed ceramics. By appropriately distorting the failure surfaces criteria for axisymmetric and orthotropic foams are also developed.

The failure criteria are compared with the results of an extensive experimental program that included uniaxial, biaxial, and triaxial (axisymmetric) testing of elastomeric, elastic-plastic, and elastic-brittle cellular solids. A novel experimental technique was developed for testing cylindrical specimens of materials with a continuous porosity under radial tensile stresses and axial compressive or tensile stresses;​ this technique is, remarkably, capable of producing hydrostatic tension in open-cell foams. The overall agreement between the tests and the models is satisfactory.

A constitutive model for isotropic elastic-perfectly plastic foams is developed based on the plastic yield failure criterion and the associated flow rule. The model can be implemented using a finite element technique. It is useful for describing the post-yield behavior of cellular materials, of particular relevance in designing protective padding and packaging for absorbing the energy of impacts. It can also be used to estimate the post-yield behavior of structural sandwich panels with foam cores.