Some interface theories and Hall-Petch relationship

In polycrystalline materials, an increase in the number of interfaces strengthens the material. For example, in deformation twins, deformation experiments have shown a correlation between the deformation-twin density and flow stress. However, this relationship is empirical, and its theoretical background is not yet well understood. In this study, we explain the relationship between interface density (particularly deformation-twin density) and flow stress using interface theories. From macroscopic and mathematical perspectives, the density of the surface dislocation is equivalent to the dislocation density with differential dimensions. Therefore, the relationship between the deformation-twin density and the flow stress obtained from deformation experiments on calcite aggregates can be derived from the correspondence between the dislocation density and the deformation-twin density. Additionally, we demonstrate the mathematical equivalence between surface dislocation theory and other interface theories (rank-1 connection, Hadamard jump condition or 0-lattice theory) as applied to deformation-twin, martensite, kink and grain boundaries. Furthermore, the Hall-Petch relationship and inverse Hall-Petch relationship can be explained from the perspective of the rank-1 connection.