In this paper, new 3-D plastic potentials for a porous solid with a von Mises matrix are obtained. First, a strain rate based potential is derived, the noteworthy result being its centro-symmetry. Moreover, it is revealed that the couplings between invariants are very specific, the most important influence of the third invariant being for axisymmetric states. It is demonstrated that the exact stress-based potential of the porous material should have the same key properties. Furthermore, it is deduced a new analytic 3-D stress-based potential that satisfies these properties. Compared to the existing criteria for porous solids with a von Mises matrix, this model is the only one that captures the specific couplings between all stress invariants and is exact for axisymmetric states.