The sections' fractal dimension of grain boundary

The fractal dimensional increment of the experimentally dynamic recrystallized grain boundary is proportional to logarithm of Zener-Hollomon parameter. The fractal dimensional increment is defined as the fractal dimension of the grain shape minus the Euclidean dimension of certain transection. To draw the geometrical image of the fractal dimensional increment, the basic rule of the sections' fractal dimension is introduced. The geometrical implication of the fractal dimensional increment is concluded as the fractal dimension of the crossing point distribution on the grain boundary transected by the circumscribing circle or ellipse with the equivalent-area of the grain, and a power law relationship between the Zener-Hollomon parameter and the number of crossing points is found. Therefore, summarizing power laws among the Zener-Hollomon parameter, the differential stress and the number of the crossing points on the grain boundary, the number of crossing points could respond to the differential stress.