An atomically-equivalent continuum model was formulated here in order to study the viscoplastic behavior of nanocrystalline materials with special reference to the low end of grain sizes that was typically examined using molecular dynamics simulations. Based upon the morphology disclosed by molecular dynamics simulations, a two-phase composite model was constructed, in which three distinct inelastic deformation mechanisms deduced from molecular dynamics simulations were incorporated so as to build a general micromechanics-based homogenization scheme. These three mechanisms included dislocation-related plastic flow inside the grain interior, uncorrelated atomic motions inside the grain-boundary region (the grain-boundary zone), and the grain-boundary sliding at the interface between the grain and grain-boundary zone. The viscoplastic behavior of the grain interior was modeled by a grain-size dependent unified constitutive equation whereas the grain-boundary zone was modeled by a size-independent unified law. The grain-boundary sliding at the interface was represented by the Newtonian flow. The development of the rate-dependent, work-hardening homogenization scheme was based on a unified approach starting from elasticity to viscoelasticity through the correspondence principle, and then from viscoelasticity to viscoplasticity through replacement of the Maxwell viscosity of the constituent phases by their respective secant viscosity. The developed theory was then applied to examine the grain size- and strain rate-dependent behavior of nanocrystalline Cu over a wide range of grain size. Within the grain-size range of 5.21 to 3.28nm, and for strain-rates ranging from 2.5 x 108 to 1.0 x 109/s, the calculated results revealed significant grain-size softening as well as strain-rate hardening that were in quantitative accord with molecular dynamics simulations (Schiotz et al., 1999). The theory was also used to investigate the flow stress, strain-rate sensitivity and activation volume over the wider grain size range from 40nm to as low as 2nm under these high strain rate loading, and it was found that the flow stress initially exhibited a positive slope and then a negative one in the Hall–Petch plot, that the strain-rate sensitivity first increased and then decreased and that the activation volume first decreased and then increased. This suggested that the maximum strain rate sensitivity and the lowest activation volume did not occur at the smallest grain size but, like the maximum yield strength (or hardness), they occurred at a finite grain size. These calculated results also confirmed the theoretical prediction of Rodriguez and Armstrong (2006) on the basis of grain boundary weakening and the report of Trelewicz and Schuh (2007) on the basis of hardness tests. In general the higher yield strength, higher strain rate sensitivity, and lower activation volume on the positive side of the Hall–Petch plot were associated with the improved yield strength of the grain interior, but the opposite trends obtained on the negative side of the plot were associated with the characteristics of the grain-boundary zone which was close to the amorphous state.

Mechanics of Very Fine-Grained Nanocrystalline Materials with Contributions from Grain Interior, GB Zone, and Grain-Boundary Sliding. P.Barai, G.J.Weng: International Journal of Plasticity, 2009, 25[12], 2410-34