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Alternately, quasi-equiaxed ultrafine-grained structure can be generated after ultrasonic treatment only in the surface layer of Armco-iron specimens characterized by a large number of defects.
Beside that, EBSD – images indicate that in the modified surface layer of CP-Ti specimens a large number of deformation twins intersecting in different crystallographic orientations is generated.
The twin length is limited by a grain size.
It should be noted that the fraction of LAB in fine-grained specimens is less than in coarse-grained specimens.
EBSD – maps of the lateral side microstructure (a) and grain-boundary misorientation distribution (b) of fine-grained titanium subjected to UIT.

At first the grain structure is mapped onto a two-dimensional random number lattice.
Here the random numbers should assume numbers between 1 and 64.
A grain was defined as a collection of points that have the same orientation number.
In other words, two adjacent grid points having the same orientation number are considered to be a part of the same grain.
The simulation time was defined by a dimensionless number known as Monte Carlo Step (MCS), which was related to the number of re-orientation attempts.

(a) Small-grain-size (118 µm) sample (b) Large-grain-size (219 µm) sample Figure 1.
Given the calculation time, we selected two areas with relatively smaller grain sizes and a sufficient number of grain boundaries for simulation.
(a) Small-grain-size (3.02 µm) sample (b) Large-grain-size (5.45 µm) sample Figure 5.
Oxygen concentration distributions in (a) small-grain-size and (b) large-grain-size samples.
These results may be attributable to the smaller number of grain boundaries within the large-grain-size sample, which plays a role as a fast diffusion path for oxygen, compared with the small-grain-size sample.

Besides, a great number of dispersed MnS particles with the size of 20-30nm were observed in the hot rolled strips.
The average grain size is 378μm.
Thus, the coarse columnar grain was developed.
After hot rolling with 20% and 50% reduction respectively, a large number of fine MnS precipitates were observed in both strips with the average size of 20~30nm.
A great number of fine and widely distributed MnS particles in the size of 20~30nm were formed in the hot rolled strip.

To apply a numerical method with a series of ordinary differential equations, the material parameter w is used for the model parameter, and the number of nuclei can be expressed as,where is the maximum number of nuclei (new grains) at the current condition of deformation
Then the average size of the new grains can be calculated from the following equation .The average grain size of the new and old grains during recrystallization can be calculated on the basis of the new grain size and the initial grain size ,.
The model is only used for non-recrystallized grains.
(8) where is the number of curves, is the curve number, is the weight number of the curve with number , is the point number, is the number of points on the curve with the number , is the measured flow stresses for the point number in the curve number , is the calculated flow stress for the point number in the curve number , and is the maximum value of the flow stress for the curve number .
The grain size changed little at this point.

Every lattice cell represents one part of a grain and is marked with a grain identification number.
The smallest abnormal grains were measured and defined as the grain size limiting normal and abnormal grains.
The area of each grain at a given time step is directly calculated from the microstructure by counting the number of cells within a grain.
In order to define a relation for conversion of simulation time, in CAS, to the real time in seconds the common definition is stated (9), where dcell stands for the simulation lattice constant and q is the number of different orientations (grain identifications) at the beginning of the simulation process [5, 8, 13].
Evaluation of grain sizes limiting normal and abnormal grains.

Temporal Evolution of Grain Size Distributions in Two- Dimensional Pinned Cells C.
Elapsed time: 703 h Cell Number area (a.u.)
Initially, the distribution is dominated by a large number of "small" sized (quasi ordered) cells but, as time is going on, the cell scattering appears evolving to a bimodal pattern.
As shown, the best fit is obtained with two bell shaped Gaussian curves. 0 1000 2000 3000 4000 5000 6000 0 2 4 6 8 10 12 14 16 18 Cell Number Area (a.u.)
Herrera, p. 379 in Recrystallization and Grain Growth (Springer Verlag, 2001) [10] C.

A large number of fine-grained polycrystalline solids, metals, ceramics, and intermetallics exhibit superplasticity at elevated temperatures [1].
The state of a grain is classified according to its number of faces f , or coordination number.
Euler’s equation gives the law of conservation for the topological parameters of a polyhedron, the number of faces F , the number of edges E , and the number of vertices V : 2 F E V� + = (1) The average number of faces for one grain F was about 13.9 in a monodispersed structure with identical grains, and about 13.0 in a polydispersed structure with grain size distribution.
The strain, which is associated with grain switching, is represented by /F N � ��� = � , where F� is the mean cumulative number of grain switching events.
When the cumulative number of grain switching events is smaller than N�� , ��� is less than �.

deformation mechanisms, grain boundaries, grain boundary sliding Abstract.
Numbers in the curves captions stand for the number of ECA-pressings and the strain path is designated as Bc A common feature of the strain-stress behavior of SPD-manufactured metals is that the ultimate strength is reached shortly after yielding.
Typical cyclic stress-strain curves for ECAP Fe-36Ni (a) after different number of ECA-pressings (double-logarithmic coordinates) and for ECAP Cu-0.36Cr (b) after different processing routes (linear coordinates).
Figure 8 illustrates both the fine dislocation slip relief in the grain interior and coarse surface displacement along the grain boundary.
Grain boundary Grain boundary sliding Slip lines Slip lines T.A. 0 0 100 20 100 200 200 300 300 400 400 500 [nm] Grain boundary Grain boundary sliding Slip lines Slip lines Fig.8.

(a)Volume fraction of dispersoids and β’-Mg2Si across the grain, calculated using Thermo-Calc (b) SEM image showing size and number area fraction variation across a grain in LF condition.
It is also observed that the volume fraction of dispersoids is not affected as severely as those of β’-Mg2Si, hence the number of nucleation sites or number density of dispersoids would be affected more prominently compared to the volume fraction across the grains.
The grain boundaries have the highest supersaturation for Mg and Si atoms which can translate into higher number and volume fraction of nucleation sites and consequently higher number density and higher fraction of dispersoids.
PF is the highest heating rate in this study and hence the retention of concentration gradient would be highest for this case, also implying higher fraction of regions (grain centre) with lower number density of dispersoids and a greater number of such grains (Fig.2,a).
It is observed that the number density of dispersoids reduce whereas the size of dispersoids increase from grain boundary to grain centre. 2.