Second-order estimates for the effective behaviour of viscoplastic polycrystalline materials

This paper is concerned with the application of the "second-order" nonlinear homogenisation procedure (Ponte Castañeda, J. Mech. Phys. Solids 44 (6) (1996) 827) to generate estimates of the self-consistent type for the effective behaviour of fcc and hcp viscoplastic polycrystals. The method has the distinctive property that it leads to estimates that are exact to second-order in the heterogeneity contrast, and which are expected to be more accurate, particularly when compared to rigorous bounds, than those resulting from earlier homogenisation schemes such as the Hill "incremental" method or its "total" formulation (Hutchinson) for pure power-law viscous materials. Special attention is paid to large grain anisotropy leading to correspondingly large heterogeneity contrast, and to highly nonlinear behaviour. Comparisons are also carried out with estimates derived from other more recent homogenisation schemes such as the "tangent" and "affine" methods. The results, illustrated for zirconium- and ice-type polycrystals, show that the second-order procedure offers the potential for significantly improved results, at least relative to the Hill incremental formulation.