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In the 'ALAMEL' model [5], grains do not deform homogeneously.
Grains are divided into domains, and those domains are assumed to deform homogeneously.
Linear stacking of N number of homogeneously deforming domains is assumed to repeat periodically.
Fig. 2: (a) Macroscopic stress-strain response and (b) evolution of the average number of active slip systems, ⟨Sact⟩, of an idealized two-dimensional polycrystal when subjected to tension.N domains, so that MN = ng.
Fig. 2(b) shows the evolution of average number of active slip-systems for various stack size.

After 25% reduction the grains elongate in the direction of rolling.
The majority of grains contain twins (Fig.1b) which grow through the whole grain interior and interacting with each other refine the microstructure.
With strain the number of the twinned grains increases and the initial grain boundaries become less distinct.
The number of such fragments increases with strain (Fig. 1d), so that after 96% rolling the microstructure of the sheet is mainly homogeneous with the mean size of the fragments of about 0.15 µm (Fig. 1e).
At further strain the size of grains decreases gradually.

And the number of {111} twinning is far less than the other two types of twinning .
In order to describe the twinning number of different deformed structures quantitatively, the ratio of grain area with twinning and microscopic orientation image area is calculated and defined as twinning fraction.
The twinning grains in Fig. 4(a) and untwinning grains in Fig. 4(b) and (c) are marked by black lines.
Additionally, a number of parallel twinnings are existed in grains during the deformation process which means that more grains participate in the dynamic deformation through twinning behavior.
The number of the three types of twinning is 66, 11 and 39 respectively.

For site-saturated reactions and random distribution of the grains, the microstructural path is described by: ( ) 3 2 1 1 ln1             − −= V V V V VBS (2a) 3/1 3 4 3       π = VN B (2b) where SV is the interface area between the recrystallized grains and the matrix and NV is the number of grains per unit of volume.
The interfacial area density per unit of volume separating recrystallized grains and the recovered matrix (Sv) was measured by optical microscopy superimposing a grid of straight lines on a planar section and counting the number of intercepts between the straight lines and the interfaces [7].
In consequence, the substructure developed within grains and the corresponding amount of stored energy varies significantly from one grain to another.
The nucleation of these grains is far from to be regarded random.
Therefore, a relatively small number of grains were measured that unavoidably led to the high scatter observed.

Since 1999 there has been a steady growth in conferences and topical workshops on nanotechnology and in parallel, comparable growth in the number of symposia on ultrafine-grained (UFG) metals.
This number compares with a total of 33 patent actions found in 2002 [5].
Ultrafine Grained Materials.
Ultrafine Grained Materials II.
Utrafine grained materials III.

And according to the optical micrographs, the mean grain size of the TM-addition alloy is much smaller than the base alloy, which is attributed to the grain refinement of TM.
The fine precipitates are formed in Co-, Ni-, Cr- and Fe-addition alloys with a larger number density distribution than the base.
The coarse precipitates are formed in Mn-, Y- and Gd-addition alloys with a smaller number density distribution than the base alloy.
Conclusions The grain size decreases for TM-addition alloys in this work, which is attributed to the grain refinement effect of TMs.
The decrease of the grains further increases the as-quenched hardness of TM-addition alloys.

The reference numbers used are as follow: forsterite (00-034-0189), periclase (00-043-1022) and enstatite (00-011-0273).
L is the average interception length, C is the total length of the test line, M is the magnification of SEM and N is the number of intercepts [10].
Smaller grain size was known to produce better mechanical properties for ceramic as more grain boundaries in unit of volume will serve as barrier for crack propagation.
Acknowledgements This study was supported under UMRG grant number RP024B-13AET, UMRG grant number RP011B-13AET, PPP grant number PG129-2014A and Esciencefund grant number SF010-2014.
Mendelson, Average grain size in polycrystalline ceramics, J.

133 Variant selection in Zr alloys: how many variants generated from one beta grain?
The minimum number of variants needed to reach this minimum is shown to be 6, and in this case, the variants have very specific volume fractions.
We will assume the β grain is spherical, and behaves according to isotropic elasticity.
Volume fractions of variants allowing to reach the minimum energy with the minimum variant number (6).
Concluding remarks It is reasonable to assume that a minimum number of variants will nucleate, to reduce the surface energy of the system.

As the concentration of Fe2+ ions increases the number of oxygen vacancies increases, and this increases the sintering rate, lowering the maximum temperature of densification [13].
As can be seen in Fig. 3, the SFO1OOSiC sample (SrFe12O19), appears to have a liquid phase on the grain and also across the grain, as can be observed in other samples, but further studies are needed for confirmation.
In the BFO90 sample there was not any change in grain size, while in the sample SFO100 the dopant made a decrease in size of the grains.
Table 3: Grain size variation of the samples with and without additives, sintered at 1000 °C/2h.
In the BFO100 and BFO15 samples, there was a significant growth in the grains, while in the sample BFO90 the grains remained practically the same size and the sample SFO100 had a decrease in grain size.

When the balls are resonated, the sample surface is impacted by a large number of flying balls over a short period of time.
Key to realize the surface self-nanocrystallization of a sample is the introduction of a large number of defects and interface into surface layer of the sample.
In other words, a grain refinement process need to nanoscale surface while coarse-grained structure matrix is constant.
It is known that nanocrystalline materials possess ultrafine grains with a large number of grain boundaries that may act as fast atomic diffusion channels.
A large number of grain boundaries with various kinds of nonequilibrium defects also constitute a high excess stored energy that may further facilitate their chemical reactivity.