On the overall properties of nonlinearly viscous composites

The overall mechanical properties of a power-law viscous material, as weakened by an isotropic distribution of voids, or reinforced by rigid particles, are investigated; the overall stress potential being estimated from a variational principle recently developed by Talbot & Willis (IMA J. appl. Math. 35, 39 (1985)). ‘Self-consistent’ estimates are generated by two alternative conditions which coincide for linear behaviour; the general superiority of one is demonstrated. A rigorous upper bound for the stress potential of the voided material is established, which reduces to the classical Hashin–Shtrikman bound in the particular case of linear constitutive behaviour. It always lies below the elementary ‘Voigt’ bound, which is obtained from the classical principle of minimum energy. Dual results, based on the strain-rate potential, are also obtained and discussed. Comparisons are made between previous estimates of overall properties and those obtained in this work.