An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals
Résumé
Variational approaches for nonlinear elasticity show that Hill's incremental formulation for the prediction of the overall behaviour of heterogeneous materials yields estimates which are too sti and may even violate rigorous bounds. This paper aims at proposing an alternative`ane' formulation, based on a linear thermoelastic comparison medium, which could yield softer estimates. It is ®rst described for nonlinear elasticity and speci®ed by making use of Hashin±Shtrikman estimates for the linear comparison composite; the associated ane self-consistent predictions are satisfactorily compared with incremental and tangent ones for power-law creeping polycrystals. Comparison is then made with the second-order procedure (Ponte CastanÄ eda, P., 1996. Exact second-order estimates for the eective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids, 44 (6), 827±862) and some limitations of the ane method are pointed out; explicit comparisons between dierent procedures are performed for isotropic, two-phase materials. Finally, the ane formulation is extended to history-dependent behaviours; application to the self-consistent modelling of the elastoplastic behaviour of polycrystals shows that it oers an improved alternative to Hill's incremental formulation.