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In addition, the grain size is decreased when the number of cycles is increased.
When the number of cycles increased up to eighth cycle, the dislocation density at grain interior was lower than those after fourth cycle.
In addition, the SAD pattern became more ring-like with increasing the number of ARB cycles, indicating the increment of a portion of high angle boundary.
After the first cycle (Fig. 4(a)), a large number of dislocations can be observed and grain boundaries are not clear.
With increasing the number of ARB cycles up to 6 (Fig. 4(c)), the fraction of the ultrafine grained increases and the microstructure was filled with the small and equiaxed continuous recrystallized grains containing nano scale twins (white lines in the micrograph).

Large Nb-rich particles may enhance sub-grain structure development (formation of twins and deformation bands), which in steels may lead to an increase in the number of ferrite nucleation sites and ferrite grain refinement [10, 11].
After deformation the observed grain size was almost similar in all the three alloys, although the NbC number density varied significantly.
In alloy M, due to a maximum of the NbC number density, the recrystallisation proceeded at a lower rate but the grain growth was also significantly retarded.
During holding after deformation the grain growth took place, which was accompanied by the particle coarsening and a decrease in the particle number density.
The grain growth was similar in alloys M and H, although the NbC number density after deformation was higher in alloy M than that in alloy H.

Figure 1 presents a transmission electron micrograph of ECAP processed Cu showing the evolution of the microstructure with strain (number of passes).
Equiaxed grains separated mostly by high angle grain boundaries were observed almost in all areas investigated.
These boundaries are obviously closer to an equilibrium state than after smaller numbers of ECAP passes.
Two populations of grains were found.
The average grain size of bigger grains was approximately 500 nm while that of the population of finer grains ranged between 200- 300 nm.

Two neighbouring grains are considered, cf.
Fig. 1, where the grain boundary between the two grains, GB1, is modelled as a low angle grain boundary [5].
Grain boundary modeling The first grain boundary GB1, cf.
Fig. 1, is considered as a low angle grain boundary while the second grain boundary, GB2, is assumed to be a high angle grain boundary.
Number of dislocations piling up against GB1 before breakthrough and b.

Effect of grains boundaries self-orientation caused by grains deformation can be identified on metallographic cut.
In undeformed state (Fig. 1a) the structure is isotropic, the grains have isometric dimension and grain boundaries are not oriented.
It is very difficult to describe actual shape of the grain [5], therefore an idealized grain shape is used in the model.
From the relative number (number to unit of length) of parallel test lines intersections with grain boundaries (PL)P and perpendicular lines ones (PL)O was total relative surface area (SV)TOT of grains estimated according Eq. 3.
This model of conversion of grain boundary degree orientation to grain deformation is independent on the initial grain size – strain depends only on shape of grain and it does not depend on grain dimensions.

AGG-grains are not equiaxed.
The same procedure was adopted for 30 nuclei giving a total number of 515 misorientations.
In average, over 10,000 grains were counted for calculations.
Penelle, Influence of the Goss grain environment during secondary recrystallisation of conventional grain oriented Fe–3%Si steels, Scripta Mater. 47 (2002) 725-730
Morawiec, On abnormal growth of Goss grains in grain-oriented silicon steel, Scripta Mater. 64 (2011) 466-469

These models consider inter-granular deformation, grain boundary sliding, grain boundary diffusion and grain growth.
Finite element simulation has reduced the number of trial forming experiments in these processes.
Grain boundary mechanisms include: grain boundary sliding (GBS), grain boundary diffusion (GBD) and grain growth.
(5) The superscript indicates the number of assumed boundary planes which ranges from one to 12 in this case.
Grain boundary mechanisms include: grain boundary sliding (GBS), grain boundary diffusion (GBD) and Grain growth.

Experiment and conditions Grain-arranged diamond wheel.
Many grains on the wheel surface have been traced using the microscope system and almost all the grains remained by 500th pass, total removal depth of 10mm.
Fig. 5(a) shows the grains before grinding under vertical lighting.
Fig. 9 shows a change of normal and tangential grinding forces, FBnB, FBtB, and Fig. 10 shows a variation of surface roughness, RByB, with the number of grinding passes, n.
Therefore, the less grain size and the more grain density are recommended for the stable production of mirror surfaces without grinding burn.

(ii) The number of grains Ni in the observed area (viz. region surrounding 748 grains) at the i-th deformation is counted.
(iii) We express the true grain size di at the i-th deformation by using Ni as follows, , (1) where d0 and N0 are the initial true grain size and the initial number of grain (before superplastic deformation).
Fig.6 Change in the number of grains Ni.
As shown in Fig.6, the number of grains Ni gradually decreases as superplastic deformation advances.
This corresponds to that there are two big domains in Fig.11(a), and also that the number of domains is increased but the size of each domain becomes smaller in Fig.11(b)-(d).

The assumptions numbered 1 and 2 and 3 are essentially correct.
Statistically, any grain moves along its grain boundary a mean “flight time” .
The increment of section can be calculated if we consider that in a time period a number of grains N0 moves a distance w, N0 being the number of grains of the sample.
Let W be the average volume of a grain (W»d3, d being the average grain size).
Accordingly, the number of grains displaced to the lateral faces of the sample will be given by: (2) Since the number of available sites along the lateral faces is SL/d2 (being SL the lateral faces area), the radial increment of the surface per unit time would be .