text: The Hall–Petch relation predicts that as the grain size decreases the yield strength increases. The Hall–Petch relation was experimentally found to be an effective model for materials with grain sizes ranging from 1 millimeter to 1 micrometer. Consequently, it was believed that if average grain size could be decreased even further to the nanometer length scale the yield strength would increase as well. However, experiments on many nanocrystalline materials demonstrated that if the grains reached a small enough size, the critical grain size which is typically around , the yield strength would either remain constant or decrease with decreasing grains size. This phenomenon has been termed the reverse or inverse Hall–Petch relation. A number of different mechanisms have been proposed for this relation. As suggested by Carlton et al., they fall into four categories: (1) dislocation-based, (2) diffusion-based, (3) grain-boundary shearing-based, (4) two-phase-based.
  There have been several works done to investigate the mechanism behind the inverse Hall–Petch relationship on numerous materials. In Han’s work, a series of molecular dynamics simulations were done to investigate the effect of grain size on the mechanical properties of nanocrystalline graphene under uniaxial tensile loading, with random shapes and random orientations of graphene rings. The simulation was run at grain sizes of nm and at room temperature. It was found that in the grain size of range 3.1 nm to 40 nm, inverse Hall–Petch relationship was observed. This is because when the grain size decreases at nm scale, there is an increase in the density of grain boundary junctions which serves as a source of crack growth or weak bonding. However, it was also observed that at grain size below 3.1 nm, a pseudo Hall–Petch relationship was observed, which results an increase in strength. This is due to a decrease in stress concentration of grain boundary junctions and also due to the stress distribution of 5-7 defects along the grain boundary where the compressive and tensile stress are produced by the pentagon and heptagon rings, etc. Chen at al. have done research on the inverse HallPetch relations of high-entropy CoNiFeAlCu alloys. In the work, polycrystalline models of FCC structured CoNiFeAlCu with grain sizes ranging from 7.2 nm to 18.8 nm were constructed to perform uniaxial compression using molecular dynamic simulations. All compression simulations were done after setting the periodic boundary conditions across the three orthogonal directions. It was found that when the grain size is below 12.1 nm the inverse Hall–Petch relation was observed. This is because as the grain size decreases partial dislocations become less prominent and so as deformation twinning. Instead, it was observed that there is a change in the grain orientation and migration of grain boundaries and thus cause the growth and shrinkage of neighboring grains. These are the mechanisms for inverse Hall–Petch relations. Sheinerman et al. also studied inverse Hall–Petch relation for nanocrystalline ceramics. It was found that the critical grain size for the transition from direct Hall–Petch to inverse Hall–Petch fundamentally depends on the activation energy of grain boundary sliding. This is because in direct Hall–Petch the dominant deformation mechanism is intragrain dislocation motion while in inverse Hall–Petch the dominant mechanism is grain boundary sliding. It was concluded that by plotting both the volume fraction of grain boundary sliding and volume fraction of intragrain dislocation motion as a function of grain size, the critical grain size could be found where the two curves cross.
  Other explanations that have been proposed to rationalize the apparent softening of metals with nanosized grains include poor sample quality and the suppression of dislocation pileups.
  The pileup of dislocations at grain boundaries is a hallmark mechanism of the Hall–Petch relationship. Once grain sizes drop below the equilibrium distance between dislocations, though, this relationship should no longer be valid. Nevertheless, it is not entirely clear what exactly the dependency of yield stress should be on grain sizes below this point.