A uniform rule between hardness and structure hierarchy of ceramics in both direct and inverse Hall–Petch regimes

The MicroStructure Hierarchy Descriptor ((Figure presented.) SHD) has built a bridge in linking the structure hierarchy and properties of ceramic materials. Here, a new variable (Figure presented.), which stands for the average of scales present in a ceramic microstructure, based on the (Figure presented.) SHD, is constructed and calibrated. It is found that the (Figure presented.) is not only related to the size scales, but also the morphological features in microstructures, due to an inheritance from the (Figure presented.) SHD. The (Figure presented.) is positively linear correlated with the average grain size d, but the relationship changes to be nonlinear when the morphology of grains is considered. In the Hall–Petch regime, the Vickers hardness H_V of both, ZrO₂–SiC composite ceramics and a class of high entropy metallic borides, is found to increase with the decrease of (Figure presented.) index. For the case study of ZrO₂ ceramic, a uniform rule is found to be valid in both the direct and inverse Hall–Petch regimes that the (Figure presented.) varies in an opposite trend as the H_V varies with d.