An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals

Variational approaches for nonlinear elasticity show that Hill's incrementa l formulation for the prediction of the overall behaviour of heterogeneous materials yields estimates which are too stiff and may even violate rigorou s bounds. This paper aims at proposing an alternative 'affine' formulation, based on a linear thermoelastic comparison medium, which could yield softe r estimates. It is first described for nonlinear elasticity and specified b y making use of Hashin-Shtrikman estimates for the linear comparison compos ite; the associated affine self-consistent predictions are satisfactorily c ompared with incremental and tangent ones for power-law creeping polycrysta ls, Comparison is then made with the second-order procedure (Ponte Castaned a, P., 1996. Exact second-order estimates for the effective mechanical prop erties of nonlinear composite materials. J. Mech. Phys. Solids, 44 (6), 827 -862) and some limitations of the affine method are pointed out; explicit c omparisons between different procedures are performed for isotropic, two-ph ase materials. Finally, the affine formulation is extended to history-depen dent behaviours; application to the self-consistent modelling of the elasto plastic behaviour of polycrystals shows that it offers an improved alternat ive to Hill's incremental formulation. (C) 2000 Elsevier Science Ltd. All r ights reserved.