Mechanics of very fine-grained nanocrystalline materials with contributions from grain interior, GB zone, and grain-boundary sliding

Abstract In this paper, we formulated an atomically-equivalent continuum model to study the viscoplastic behavior of nanocrystalline materials with special reference to the low end of grain size that is typically examined by molecular dynamic (MD) simulations. Based on the morphology disclosed in MD simulations, a two-phase composite model is construed, in which three distinct inelastic deformation mechanisms disclosed from MD simulations are incorporated to build a general micromechanics-based homogenization scheme. These three mechanisms include the dislocation-related plastic flow inside the grain interior, the uncorrelated atomic motions inside the grain-boundary region (the GB zone), and the grain-boundary sliding at the interface between the grain and GB zone. The viscoplastic behavior of the grain interior is modeled by a grain-size dependent unified constitutive equation whereas the GB zone is modeled by a size-independent unified law. The GB sliding at the interface is represented by the Newtonian flow. The development of the rate-dependent, work-hardening homogenization scheme is 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 is 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 from 5.21 to 3.28 nm, and the strain rate range from 2.5 × 108 to 1.0 × 109/s, the calculated results show significant grain-size softening as well as strain-rate hardening that are in quantitative accord with MD simulations [Schiotz, J., Vegge, T., Di Tolla, F.D., Jacobsen, K.W., 1999. Atomic-scale simulations of the mechanical deformation of nanocrystalline metals. Phys. Rev. B 60, 11971–11983]. We have also applied the theory to investigate the flow stress, strain-rate sensitivity, and activation volume over the wider grain size range from 40 nm to as low as 2 nm under these high strain rate loading, and found that the flow stress initially displays a positive slope and then a negative one in the Hall–Petch plot, that the strain-rate sensitivity first increases and then decreases, and that the activation volume first decreases and then increases. This suggests that the maximum strain rate sensitivity and the lowest activation volume do not occur at the smallest grain size but, like the maximum yield strength (or hardness), they occur at a finite grain size. These calculated results also confirm the theoretical prediction of Rodriguez and Armstrong [Rodriguez, P., Armstrong, R.W., 2006. Strength and strain rate sensitivity for hcp and fcc nanopolycrystal metals. Bull. Mater. Sci. 29, 717–720] on the basis of grain boundary weakening and the report of Trelewicz and Schuh [Trelewicz, J.R., Schuh, C.A., 2007. The Hall–Petch breakdown in nanocrystalline metals: a crossover to glass-like deformation. Acta Mater. 55, 5948–5958] 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 are associated with the improved yield strength of the grain interior, but the opposite trends displayed on the negative side of the plot are associated with the characteristics of the GB zone which is close to the amorphous state.