On Application of a Model of a Crack Having a Dislocation-Free Zone and a Slip Band to the Strength of Polycrystalline Aggregates

The grain-size dependence of the yield stress of polycrystalline metals in the range of normal grain sizes is known as the Hall-Petch relation. However, recent experiments by Armstrong et al. show that failure of this relation occurs in the range of ultrafine grain sizes. In the present paper, in order to elucidate the above phenomena over both ranges of grain sizes, we propose the dislocation pile-up model, Which consists of a crack, a dislocation-free zone and a slip band blocked at the grain boundary. Analyzing the above model by the method of the continuously distributed theory of dislocations, we obtain an analytical expression which gives the relationship between the macroscopic applied stress and the grain size D. It can be shown that the behaviors of the macroscopic yield stress vs. D-1/2 explain the experimental results well over both ranges of grain sizes. The applicability of this model to the grain-size dependence of the fracture stress of engineering ceramics is discussed when the friction stress of the dislocation is reduced to zero.