The aim of this paper is to present new approximate macroscopic models
for porous viscoplastic materials, based on partial but exact results
applicable to such media. Available results are first supplemented by
providing a new inequality (which, in addition to its intrinsic inter
est, allows one to rederive in a simpler way some previous bounds of P
onte-Castaneda and Talbot and Willis), and by exhibiting the exact for
m of the overall potential of a typical porous viscoplastic volume ele
ment, namely a hollow cylinder loaded in generalized plane strain. App
roximate expressions for the macroscopic viscoplastic potentials of ma
terials containing cavities of cylindrical or spherical shape are then
proposed, based on these and other results; these expressions satisfy
, in particular, the three following natural requirements: (i) reprodu
ce the exact solution of a hollow cylinder or sphere loaded in hydrost
atic tension or compression; (ii) be a quadratic form of the overall s
tress tensor in the extreme case of a Newtonian (linear) behavior; and
(iii) yield the currently accepted Gurson criterion in the other extr
eme case of an ideal-plastic behavior.