Iterated linear comparison bounds for viscoplastic porous materials with “ellipsoidal” microstructures

Analytical estimates are obtained for the effective constitutive response of porous viscoplastic materials consisting of aligned ellipsoidal voids that are distributed randomly with “ellipsoidal” symmetry in the matrix material. These estimates are obtained by means of a novel iterative homogenization strategy recently proposed by Ponte Castañeda (2012), and can be shown to be bounds for certain classes of multi-scale microstructures. By design, the resulting constitutive model agrees exactly with the earlier “variational linear comparison” model at the first iteration (N  =1), and provides estimates that get progressively more accurate as the number of iterations increases (N→∞N→∞), especially for high-triaxiality loading conditions, and low porosity and strain-rate sensitivity. However, in practice, a small number of iterations (N≈10N≈10) is sufficient to get very accurate results. It is important to emphasize that, unlike other models that have been proposed in the literature, the new model requires no fitting parameters, solely depending on the properties of the matrix phase and microstructural information, such as the porosity, the average void shape and orientation, as well as the generally different shape and orientation of their distribution. Results are given for the yield and gauge surfaces of ideally plastic and power-law viscoplastic porous materials for the special cases of aligned spheroidal and ellipsoidal voids, and the results are compared with available numerical results and with the results of other models. Compared to available numerical results, the new estimates are found to be quite accurate, while they also provide more flexibility than competing models in terms of the characterization of the microstructure. In particular, it was found that the effect of different shapes for the average pore shape and distribution on the yield surfaces of the porous materials can be significant at high triaxialities, even for very small porosities. In addition, consistent estimates are obtained for the average strain rate in the voids, and explicit results have been computed for initially spherical voids, which exhibit a strong effect of the triaxiality, especially at low porosities and strain-rate sensitivities.