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Si denotes one of the Q possible grain orientations number at site i.
Sj represents the orientation number of site j which is the neighbor of site i.
The above steps are repeated until the desired evolution step number is reached.
The radius of grain is calculated as , where V is the volume of one grid and A is the number of grid points within the grain.
The orientation number Q of lattice sites for matrix phase ranges from 1 to 200.

Bulk production with high accuracy can be realized by these technologies, but costs are comparably high and the number of different materials is quite limited [2].
Thus, metal forming technologies become more and more demanding in the production of micro parts and a number of investigations are currently being done in recent years.
The physical meaning of T/D is the total number of grains across the material thickness.
Since the number of grains in cross-section area are less than 20 grains, the decreasing trend in the flow stress curve may be related to the smaller constraint among the grains .This can be explained by the increasing relation of surface grain to grain inside the materials.
That is because the grain number in the cross-sectional area is less than 20 grains.

Metallic materials used in technical applications are polycrystalline in nature and are composed of a large number of grains which are separated by grain boundaries.
The grain size hereby has an approximately diameter of 1 µm.
The number of intercepts per unit length of the test-line at the grain boundary of the cross-sections can then be used to estimate the grain size.
L is the length of the test-line, z the number of lines, nk the number of intercepts the line makes at the grain boundaries and V the magnification factor of the micrograph. 25 test-lines each with a length of 10 µm are applied for the microstructural analysis.
At rß of 30 µm the grain size in a depth of 1µm is 200 nm, whereas at rß of 150 µm the grain size is 120 nm.

SEM and FESEM analysis displayed that the microstructure in healing area is should be was mainly ferrite, and ferrite grain growth across the interface of inner crack, and there exists should be exist many polyangular grains of several hundreds nanometer in ferrite of healing area.
The microstructure in healing area mainly is ferrite, and ferrite grain growth across the interface of inner crack.
In the healing area also a certain number of voids left, number of the voids in the grain more than in grain boundaries.
Further reduce of the void size in the grain were achieved by volume diffusion.
A further reduce of the void size in the grain boundaries were achieved by volume diffusion and surface diffusion, therefore, the number of the voids in the grain boundaries is relatively small.

Titanium alloy plates have a penetration internal microstructure with a large number of ASBs.
It was indicated that the number of ASBs increased and its average width increased slightly with the increase of the velocity.
A large number of studies suggested that bifurcation phenomenon in the expansion process of ASBs was very common, and it appeared in the middle or the end of the expansion of ASBs.
It will increase the number of ASBs.
We classified grains as deformed grains, subgrains and dynamic recrystallized grains according to the value of 3°and 5.7°.

A large number of sub-crystals remained on the surface of the thick plate and was accompanied by much recrystallization.
It can be seen that coarse particles are distributed in grains and along original grain boundaries.
After etched in chromic acid, the large white bright structure was recrystallized structure, and the non-recrystallized structure contained a large number of sub-grain boundaries.
Many recrystallized grains which had no red sub-grain boundary inside were formed between the original grains.
Otherwise, due to the high degree of deformation, a large number of substructures existed in the grain, Δσgb was high and dominant.

Fine oxide particles appear to pin grain boundaries and result in inhibition of grain growth in the alloy matrix.
Table 2 Microstructural parameters of materials tested: Grain Size (D), Particle Size (d), Particle Number Density (N), Interparticle Spacing (Nd), Particle Types.
Materials Grain Size, D (μm) Particle Size, d (nm) Particle Number Density, N (m-3) Interparticle Spacing, (Nd)-0.5 (nm) Particle Types 19Cr-ODS Ferritic FCY1 3.07 9.6 5.41×1022 43.9 YCrO3, Cr2O3, Y2O3 FCY2 4.23 10.0 4.64×1022 46.4 FCY3 5.48 11.7 2.85×1022 54.8 FCY4 7.25 12.6 1.73×1022 67.7 Table 2 summarizes grain size, particle size, particle number density, and interparticle spacing for each 19Cr-ODS ferritic alloy determined by the SEM and TEM micrographs.
This is because an increase in consolidation temperature and time (1100 °C × 4 h to 1200 °C × 8 h) results in the decrease of number density and the size increase of dispersed oxide particles.
N and d refer to the particle number density and the mean particle diameter.

The grains visualized through optical microscope consist of an average grain size of 115 µm indicating homogeneous and equiaxed grains.
(b)-(d) provides the micrographs of cryorolled samples and with the increase in strain, the grains become elongated and the evolution of sub grains within the grains occur.
In addition, distinction between the grain boundaries and grain interior is difficult due to severe fragmentation of the grains.
With the reduction in grain size, the grain boundaries rose up leading to grain boundary strengthening according to Hall-Petch equation.
There is presence of dimples which is evolving in number and some micro cracks is also visible with the help of SEM due to reduced grain size by processing of cryorolling.

For this, it is necessary to solve a number of emerging problems: analyze how the effect of heat treatment on the grain size occurs; determine the value of the factor of different grain size serving as the main criterion for the dispersion of the system as a whole, and also try to find correlation between it and the ultimate strength and coercive force of the steel under study; explain the changes occurring with the grain size factor during heat treatment.
The grain size factor is calculated by the following formula: (1) where fi – share of grain with a certain score, %; fmax – share of grain occupying maximum area on the microsection, %; Zi – grain score; Zmax – score of the grain occupying maximum area on the microsection.
Methods for identifying and determining grain size’[2].
Histogram of percentage distribution of grains by score for microsections of heat-treated specimens made of 15KHSND steel by grain size Fig. 3.
Methods of Detection and Determination of Grain Size].

With biaxial deformation, grain boundary slide occurred more frequently than with uniaxial deformation, causing grain boundary separation and formation of micro-voids between the grains.
Observation of a vertically sectioned surface showed an initial grain size of 17 µm and equiaxial grains.
In addition, the Mg alloy, which as a small number of slip systems, has a high strength coefficient F which enhances the difficulty of deformation in sheet thickness.
In uniaxial deformation elongated grains are observed at έ = 2.7 ×10-1 and equiaxial grains at 2.7×10-4 s-1.
Grain rotation is hardly observed, and microvoids are formed at the grain boundaries