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Braginsky et al. [7] extended this model to simulate sintering in a complex powder compact consisting of a large number of particles.
Lattice sites having the identical Q number are considered as a grain, and a grain boundary segment is defined to lie between sites of different Q number.
NA and NB are the sites number of phase A and B.
NC is the smaller sites number of that of phase A or B. δQiQj is Kronecker delta function, Qi and Qj denote the grain orientation state of the neighboring sites i and j
It can also be seen when the initial size of matrix particles is increased, the number of nano- particles inside matrix grains decreases, even those nano-particles within the matrix grains are mostly located at the place near grain boundaries.

As a result, the elongation of alloy is decreased with pass number increasing.
In Fig. 2(b), a number of new fine grains appear along the initial grain boundaries.
The sub-grain boundaries consist of low-angle grain boundaries (LAGBs).
This is why elongation is decreased with pass number increasing.
The elongation is decreased with pass number increasing.

We determine the number of grains (n1), completely inside the circle (square, rectangle) and the number of grains (n = 2), cut by the circumference.
Fig. 3 Effect of T.T.C cycle number on the size of the austenite grain of 42 CD4 steel: a)- water cooling, b)- air cooling Fig. 3 illustrate the influence of the number of T.T.C cycles on the grain index G for a slow heating of the order of 4 °C.min-1 and for two different cooling.
As number of cycles increases, an increase in the index of the grain G was observed.
The possibility of subjecting a steel grain refinement by a T.T.C is subject to three factors: heating rate, cooling rate and number of cycles (n).
All the results obtained represent the variation of the grain size as a function of the number of cycles (TTC) for two types of cooling and different cooling rates.

TEM image of the two kinds of precipitates in as-rolled tested steel A great number of precipitates, see Fig.1, which appear to be nearly rectangular or spherical, in tested as-rolled microalloyed steel were revealed under TEM.
As seen from the Fig.4(a), the finer austenite grains distribute inhomogeneously as new austenite grains contact with each other and the grain boundary is bent.
Under the temperature of 1100℃, the average grain size of austenite is less than 50μm; from 1100℃ to 1270℃, the grain size grows up gradually; at the temperature above 1270℃, the grain size increases rapidly.
However, under the same condition, effect of holding time on the grain size is less obvious than heating temperatures on the grain size of austenite.
Consequently, the grain size increases tremendously.

The texturing development in these superconductors decreases in an efficient way the number of high-angle grain boundaries, increasing the values of the critical current densities.
The texturing development in these superconductors decreases in an efficient way the number of high-angle grain boundaries, increasing the values of the critical current densities [1-3].
In the study was calculated the grain size distribution.
Table 1: Results of grain size in the samples.
The micrographs show grains in the form of small grains flattened (flake-shaped) plates.

When the recrystallization reaches certain degree, the new-deformed core number and the swallowed number of the original grains are almost the same, and the average diameter is with little change.
The distribution of grain diameter and the number of side.
(a) (b) Fig 3 Distribution of grain diameter and the number of side （a）distribution of grain diameter（b）distribution of the number of side Fig 3 shows the distribution of grain diameter and the number of side in the simulation when .
In the initial time, the number of side is distributed around 4~9, and the rate of the grain whose distribution around 5~7 is up to 80% or more.
Many new grains are formed around some original deformed grains, which increases the number of side, even up to 15 sides in certain moment.

With reference to the characteristics of polycrystalline materials, according to contemporary terminology UFG materials may be defined as polycrystals having very small grains with average grain sizes less than ~1 mm [1].
Mechanical properties of steel AISI 321 and structure parameters after HTMP with different numbers of passes n and different rolling reduction, e.
The volume density of the precipitates increases and their size decreases up to 0,1 mm as the number of passes increases.
Langdon, Twenty-five years of ultrafine-grained materials: Achieving Exceptional properties through grain refinement.
Micro-Mechanical Responses of Ultrafine-Grained Materials Processed through High-Pressure Torsion.

The average grain size (dAve) is related to recrystallized grains and sub-grains, and their relationship is as follows: dAve= drex*φrex+ dsub*(1-φrex)
Low deformation temperature will increase the number of recrystallization, thereby increasing the brittle fracture characteristics along the recrystallized grain boundaries.
When the crack propagates in the deformed grains containing sub-grains, due to the low strength of the sub-grain boundary, the crack preferentially propagates to the sub-grain boundary.
Therefore, when the grain size decreases, the grain cross point increases with the decrease of grain size, the number of cross points through the crack propagation increases, and the probability of crack propagation increases.
The mathematical relationship between deformation temperature and sub-grain size, recrystallized grain size and recrystallized grain fraction was established.

The fraction of the spot overlap depends on the number of grains, the sample-to-detector distance, the detector resolution, the crystal structure of the sample and the mosaic spread of the grains.
The overlap fraction for the TTF-CA samples was studied as function of the grain number, the sample-to-detector distance, and the data resolution.
Fig.1 shows the overlap fraction as a function of the sample-to-detector distance for 100 grains(a), and the data resolution and the grain number at the sample-to-detector distance of 50mm(b).
It is evident that the overlap fraction increases with the increasing number of the grains.
Crystallographic accuracy of structure solution for individual grains from the TTF-CA polycrystalline sample Grain R(int) R(sigma) GooF R1 1 0.0269 0.0268 1.130 0.0234 2 0.0229 0.0212 1.101 0.0203 3 0.0232 0.0230 1.090 0.0193 4 0.0253 0.0261 1.025 0.0195 Conclusion The overlap fraction for the TTF-CA samples was studied as function of the grain number, the sample-to-detector distance and the data resolution.

Development of Micro-alloying Ultra-fine Grain High Strength Steel Produced by EAF-TSCR J.X.
In order to control the number of inclusions in molten steel and form, the processes of deep desulfurization, deep deoxidation and calcium treatment must be carried out subsequently.
Fig.5 The precipitation analysis and EDAX analysis of the sample of finished steel SEM observation was performed on the specimens of finished steel, a large number of round micro-alloy precipitates were found in the specimens with intra-granular dispersion, as shown in Figure 5.
Ultra-fine grains steel the theory and technical progress.
Ultra-fine grain steel and its production technology lecture courses[M].