A composite model for the grain-size dependence of yield stress of nanograined materials

Based on the observation that, in a nanograined material, a significant portion of atoms resides in the grain-boundary region and grain-boundary activity plays a key role for its plastic behavior, a micromechanics-based composite model is developed to calculate the transition of yield stress as the grain size decreases from the coarse grain to the nanograin regime. The development makes use of a generalized self-consistent scheme in conjunction with the secant moduli of the constituent phases and a field-fluctuation approach. The constituent grains are modeled by inclusions with a grain-size-dependent plastic property and, in order to reflect the atomic sliding inside the grain boundaries observed in molecular dynamic simulations, the grain-boundary phase is modeled as a soft, ductile material with a pressure-dependent property. Applications of the developed model to a high-density copper showed the distinctive—some experimentally observed—features: (1) the yield stress initially increases following the Hall-Petch equation, but as the grain size reduces to the nanorange, it will depart and decrease; (2) when the grain size drops to a critical value (called the critical equicohesive grain size), the slope turns negative, (3) there is a tension-compression asymmetry (or strength-differential effect) in the yield stress, and (4) parametric calculations for materials whose grains deform only elastically indicate that the Hall-Petch plot will exhibit a continuously decreasing negative slope over the entire range of grain size. Further application of the theory to palladium in the nanorange shows a continuous decrease of the yield strength with decreasing grain size. It can be generally concluded that the range following the Hall-Petch equation is dominated by the deformation of the grains, and the range with a negative slope is controlled by the plasticity of the grain boundaries. During the transitional stage, both grains and grain boundaries deform competitively.