Theoretical upper limit of dislocation density in slightly-ductile single-crystal ceramics

Upper limit of dislocation density without fracture is numerically calculated for slightly- ductile single-crystal ceramics for which the Griffith criterion for fracture and the Bailey-Hirsch type relationship between applied stress and the dislocation density are nearly valid simultaneously in order to obtain useful information to improve functional, electrical, and mechanical properties of ceramics by the introduction of appropriate dislocations. Two models of fracture as a function of dislocation density are constructed; simple model and probability model. If the diameter of pre-existing microcracks is sufficiently small, the dislocation density could be as high as the crystallographic limit (∼1018m^-2). Even if the typical diameter of pre-existing microcracks is not so small, there is some probability that the dislocation density could be as high as the crystallographic limit if the number of microcracks in the specimen is very small. Accordingly, the increase in ionic conductivity by several orders of magnitude without dendrite formation by introducing appropriate dislocations into single-crystal solid electrolytes with the dislocation density higher than about 10¹⁷m^-2theoretically predicted by the authors may be practically possible.