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The obtained shear strain depends on the geometry of the sample and number of revolutions applied and is calculated according Eq. 1, where r, t and n are the radius, the thickness of the sample and the number of revolutions.
In the present study, the number of revolutions was chosen in such a way that a certain amount of plastic strain is reached at a radius of 3mm.
Nuclei A, C, D and E grow each to a single grain.
In Ultrafine Grained Materials III
In Ultrafine grained materials III, page 629, 2004

On its basis the SAR number values for all investigated materials were calculated.
The SAR number values for the alumina matrix and composites.
The wear proceeded not only by the alumina grain crushing, but also by the weakening of grains boundaries and the removal of whole grains as a result.
In effect whole grains or grains agglomerates could be removed.
This counteracts rapid grain boundary erosion.

The values of grain size and strain obtained from this plot, presented in table 1, indicates that grain size increased from ~13 nm to ~36 nm as the deposition temperature increased from 27 0C to 45 0C.
The SEM images indicate that with increase in deposition temperature the number of crystallites increases leading to formation of more homogeneous film.
The optical spectroscopy is fast and simple pointer to crystal size, since band gap-size correlations have been made for a number of colloids and films [27].
The average grain size is found to increase with increase in deposition temperature.
The SEM images shows that deposited films are homogeneous and free form any pinholes or cracks with increase in number of crystallites with increase in deposition temperature.

The grain size was determined by the average grain equivalent diameter d2.
In all the samples, the average grain size determined on transverse sections was about 70 nm and the nano-grains in the various regions of the rods were similar in the shape.
Samples cut for TEM studies and microhardness measurements Results and Discussion The homogeneity of the bulk nanocrystalline titanium was examined in two HE-produced rods that differed by their diameters, applied strain and the number of the extrusion passes.
The grain size distribution in the Ø7 mm rod: a) top, b) end a) b) Fig.6.
The grain size distribution in the Ø10 mm rod: a) top-centre, b) top-surface Figures 5 and 6 show examples of the grain size distribution determined in various regions of the Ø7 and Ø10 rods.

Results and discussion The metallographic microstructure of as-extruded ZA 31 magnesium alloy consists of α-Mg and the small quantity of MgZn distributed along grain boundary, with average grain size of about 10.3μm.
As the Figure shows, a large number of cavities take shape along grain boundary, and obviously, there are more cavities in sample tensile structure at temperature of 623 K than around sample fracture at temperature of 608K.
When superplastic deformation is conducted by dint of grain boundary sliding mechanism, micro cavities will shape between gains and then expand into crack, which will thus lead to the fracture of deformation sample along grain boundary [4, 5].
Some literatures point out that a certain quantity of cavities which exist independently in scattered state and small size will effectively adjust grain boundary sliding in plastic deformation process and further the grain sliding process [6].
When grain boundary encounters the difficulty in moving in triangular grain boundary, cavity can be used for relaxation and enhancement of plasticity.

In the end of the 20th century, LU [3] proposed the concept of metal surface auto-nanocrystallization, namely, by using external mechanical load to make the metal surface occur a strong plastic deformation and introduce a large number of non-equilibrium defects and interface to refine the conventional coarse grain to nanocrystals.
Observed from the morphology, we can find that the surface grains have been refined into the equiaxed nanometer grains with the size of 10nm-30nm.And it is difficult to identify their apparent grain boundaries.
This makes the material surface get the continuous multi-directional high stress and strain rate, which results in a large number of defects.
Due to the expansion of the accumulated compressive stress and tensile stress, the twins move leads to more crystal surface slip, which eventually forms a large number of dislocations.
After HEBMT, the surface grain can be refined to 10nm

More recently Hutchinson et al. [15] have shown that care should be taken when analysing the number of 40º<111> boundaries because even in a completely randomly textured material, a choice of a particular rotation axis for example <111> results in a very strong weighting for certain angular ranges.
Stereographic projections of grain interface segment normals for the grain shown in Fig. 1.
This facet exists and extends in size as the grain is growing.
The total displacement of facet #1 (see Figs. 3 and 4) along the facet normal as a function of annealing time (here expressed as scan number).
The numbers on this graph are the scan numbers.

In this model, it is assumed that a huge number of defects are located at the grain boundary and these defects catch free carriers in the polycrystalline grains.
They were re-crystallized by ELA (XeCl, wave number: 308 nm, energy density: 460 mJ/cm2 x 5shots) at room temperature (RT) in a vacuum chamber.
The determination of the grain boundary in the TEM observation is not easy [13], but we decided that the grain boundary is the boundary between grains with different orientations.
(A) (B) insufficient to orient the poly-Si grains, so the stress might not be induced in the poly-Si grains.
In this model, a huge number of defects are located at the grain boundary, and these defects catch free carriers in the polycrystalline grains.

The grain size of primary α-Al grains was measured using ASTM E1382 linear intercept method.
The shape factor of primary α-Al grains and the secondary Al grains were also determined.
Number, size and distribution of pores were thus determined using the software package Volume Graphics Studio Max 3.3.
The number density is only 0.024 mm-3.
The number density is 159 mm-3.

We should calculate the differences between the two sides of Eq. (1) in Eq. set(1) and that of Eq.(2~11) in Eq. set(1), and then let (a¢=2, 3, ¼ , 11), we can obtain (2) According to the EET theory we can know that the total number of covalent electron ånc of all atoms in a structure unit should be equal to the sum of the number of covalent electron of all covalent bonds in the same structure unit, i.e. ånc=n1+n2+¼+n11= n1åIa¢ra¢.
In ref.[9], the PSF of sN was defined as the number of atom state groups which satisfying the bond length difference DDa<0.005 nm, it is considered that the larger the number of probable atom state groups sN in a phase, the stabler the phase.
In ref.[10], the PSF F was defined as F=åna[(fu+fv)/2]Ia, where, na is the number of covalent electron pairs on the covalent bond a, Ia is the equivalent bond number of bond a, fu and fv are forming-bond abilities of the two forming-atoms u and v, their values were shown in ref.[7].
From the VESs of Al8Fe4Ce and Al4Ce in Table 1 and 2, We can see the numbers of covalent electron pairs n1 in their strongest bond are =0.9611 and ＝0.6189 separately, they are bigger than that of the matrix α-Al (=0.2086), so they can hinder the movement of dislocation and the slipping of the grain boundary strongly.
That with one another seizes Ce atoms with the center of Al atoms and restricts the growth of grains result in the quantity of crystal nucleus increased and the difficulty of the grains growth.