Second-order estimates of the self-consistent type for viscoplastic poly crystals

The 'second-order' homogenization procedure of Ponte Castaneda is used to propose new estimates of the self-consistent type for the effective behaviour of viscoplastic polycrystals. This is accomplished by means of appropriately generated estimates of the self-consistent type for the relevant 'linear thermoelastic comparison composite', in the homogenization procedure. The resulting nonlinear self-consistent estimates are the only estimates of their type to be exact to second order in the heterogeneity contrast, which, for polycrystals, is determined by the grain anisotropy. In addition, they satisfy the recent bounds of Kohn & Little for two-dimensional power-law polycrystals, which are known to be significantly sharper than the Taylor bound at large grain anisotropy. These two features combined, suggest that the new self-consistent estimates, obtained from the second-order procedure, may be the most accurate to date. Direct comparison with other self-consistent estimates, including the classical incremental and secant estimates, for the special case of power-law creep, appear to corroborate this observation.