The optimum grain size for strength-ductility combination in metals

A strength-ductility trade-off usually occurs when grains are refined to increase strength. A question arises on if there exists a grain size for the best strength-ductility combination, i.e., with the highest possible strain energy density limit and strength simultaneously. This issue is crucial for guiding the design of strong and tough structural materials. Here we reveal an optimum grain size (d_optimum) on the order of a few micrometers, at which the strain energy density limit, estimated as the product of strength and uniform elongation, reaches a maximum while maintaining reasonably high yield strength. The d_optimum is found to exist in a series of single-phase FCC, BCC and HCP materials, indicating it as a universal phenomenon. Theoretical models on the grain size-dependence of uniform elongation and ultimate strength are developed by considering dislocation accumulation in grain boundary affected region (Gbar) and grain interior based on the classical Kocks-Mecking-Estrin model. Combined with the Hall-Petch relationship, the models accurately predict the d_optimum. Importantly, the models disclose this d_optimum to be close to twice of the characteristic width of Gbar (l_Gbar), suggesting that it is exactly at or near the critical grain size with the strongest intragranular strain gradient effects.