Variational estimates of the self-consistent type for the effective behaviour of some model nonlinear polycrystals

A new class of self‐consistent estimates for nonlinear polycrystals is obtained by making use of the classical self‐consistent estimates for the ‘linear comparison polycrystal’ in the nonlinear variational procedure of deBotton and Ponte Castañeda (1995). A distinctive feature of the new self‐consistent estimates is that they satisfy the recent bound of Kohn and Little (1997) for two‐dimensional ideally plastic polycrystals, which is known to be significantly more restrictive than the Taylor bound at large grain anisotropy. In addition, the new self‐consistent estimates are found to satisfy the Taylor and Hashin–Shtrikman bounds, which are sharper than the Kohn–Little bound at low grain anisotropy. This result suggests that the new self‐consistent estimates may also be more accurate than other self‐consistent estimates, like those arising from the incremental model of Hill (1965), for three‐dimensional nonlinear polycrystals.