Comparison of different homogenization approaches for elastic–viscoplastic materials

Homogenization of linear viscoelastic and non-linear viscoplastic composite materials is considered in this paper. First, we compare two homogenization schemes based on the Mori–Tanaka method coupled with the additive interaction (AI) law proposed by Molinari et al (1997 Mech. Mater. 26 43–62) or coupled with a concentration law based on translated fields (TF) originally proposed for the self-consistent scheme by Paquin et al (1999 Arch. Appl. Mech. 69 14–35). These methods are also evaluated against (i) full-field calculations of the literature based on the finite element method and on fast Fourier transform, (ii) available analytical exact solutions obtained in linear viscoelasticity and (iii) homogenization methods based on variational approaches. Developments of the AI model are obtained for linear and non-linear material responses while results for the TF method are shown for the linear case. Various configurations are considered: spherical inclusions, aligned fibers, hard and soft inclusions, large material contrasts between phases, volume-preserving versus dilatant anelastic flow, non-monotonic loading. The agreement between the AI and TF methods is excellent and the correlation with full field calculations is in general of quite good quality (with some exceptions for non-linear composites with a large volume fraction of very soft inclusions for which a discrepancy of about 15% was found for macroscopic stress). Description of the material behavior with internal variables can be accounted for with the AI and TF approaches and therefore complex loadings can be easily handled in contrast with most hereditary approaches.