Interpreting the softening of nanomaterials through gradient plasticity

Experimental and simulation studies have shown that decreasing the grain size below a critical value results in softening rather than hardening in both the yield stress and flow stress of nanomaterials. In this work, a gradient plasticity framework is presented that can capture this softening behavior by treating grain boundaries as a separate phase with a finite thickness. The theoretical expression obtained for the yield stress as a function of the grain size can capture numerous experimental data that exhibit this "normal" to "abnormal" Hall-Petch transition, and an analytical equation is obtained that can predict the grain size at which this transition occurs. Furthermore, analytical expressions are obtained for the flow stress in nanomaterials, and they are in precise agreement with atomistic simulations on nanocrystalline Cu, which predict that below a critical grain size the flow stress decreases proportional to it.