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Somewhat different assumptions are accepted for non-equiaxial grains.
Prolate grains were inclined with respect to the extrusion direction. 2.
Grains were relatively thin plates parallel to the rolling plane.
ASTM grain size number is defined as G = 3.322 log - 2.954, which gives 16.6 for the Al sample and by extrapolating the ASTM Tables ending at G = 14.3 we find a good agreement concerning the above estimated values of NV and NL.
[4] ASTM E-112: Standard Methods for Determining Average Grain Size.

The effects of centroidal process on the distributions of grain size and number of grain corners, facet and edge are analysed.
In microforming process, as the specimen size approaches the dimension of grain size, there are a small number of grains locating in the specimen as shown in Fig. 1 [3].
The number of grain with the average volume increase significantly as shown in Fig. 5b, which indicates the centroidal process make the grain size more even.
As the influence of CS on the geometrical features, Fig. 6 shows the distributions of the number of corners, facets and edges per grain Nc, Nf and Ne.
The analysis indicates that the distributions of grain size and number of geometrical features per grain including grain corners, grain facets and edges appear towards a value of the steady-state polycrystalline structure.

The real-space distributions of texture and grain/crystallite size of HPTed steels of four torsion numbers were quantitatively visualized at once.
Five samples of the same rotation number were stacked for statistics averaged general evaluation.
The numbers of the rotation were 1/4, 1/2, 1 and 10 in the same direction.
Fig. 3: Torsion number dependence of texture and grain/crystallite size over the whole area of HPTed ultralow-carbon steels visualized by the neutron Bragg imaging combined with the RITS code.
The iteration number of FBP-EM was 30 times (k = 30).

It has a 'trap' effect on the hard large grains that can prevent defect effectively on the surface of the workpiece which is caused by large grains.
But the practical grain size is not uniform, when the larger grains from abrasive or the debris which come off from workpiece into the machining zone, the load is borne by small number of large grains if using the hard tool plate (cast iron, copper, tin etc.), which will lead to the workpiece's cutting depth increased and then form scratching or pit damages (Fig. 1b).
When the large grain enters into the machining zone, grains surrounding the large grain may generate position displacement to form 'trap' space (Fig. 3), and the large grain and abrasive will be at same high.
Therefore it can avoid surface defect of workpiece caused by large grain.
The equipment used to test roughness is Mahr Perthometer S2 (vertical resolution: 0.8 nm, sampling numbers: 11,200).

The results show that it is feasible to fabricate nanocrystalline copper by explosively dynamic plastic deformation of coarse-grained copper and the grain size of the NC copper can be controlled less than 100 nanometer; higher strain at high strain rate is beneficial to the grain refining; the distribution of the grain size is not uniform along the loading direction; dynamic yield strength of the NC copper enhences with the decreasing of the average grain size and increasing of the strain rate.
Analysis of the grain sizes The mean grain size of the NC copper samples were determined by XRD of D/Max2500VL/PC.
According to the Scherrer formula, the grain size can be obtained[15].
According to the Scherrer equation, the mean grain size of sample 1 was obtained the number was 31.4nm, by the same way, the mean grain size of all the other samples can be obtained.
So Ls-dyna3d finite element programm is used in this paper to research the effect of strain on the grain sizes and distribution rule of the grain sizes.

The metal is composed of a plurality of grains.
For the finer the grain, the more plastic deformation may be dispersed in the grains.
Assuming that the area and force are the same, the force apportioned to each grain size is less and the fatigue crack growth rate is decreased if the grain size is small; on the contrary, the force apportioned to each grain size is great and the fatigue crack growth rate is increased if the grain size is large.
In the same area, each grain is allocated more force if the grain size is larger, and the number of grain is greater.
On the contrary, each grain is allocated less force.

It may be noted that the formation of a large number of low angle boundaries is attributed to the multidirectional deformation processing which promotes rapid formation of intersecting sub-boundaries compared to uniaxial compressive deformation [7,8].
Although some sub grains and ferrite grains elongated in the rolling direction have been retained, a large number of equiaxed ultrafine ferrite grains in the submicron range surrounded by high angle grain boundaries were observed.
However, among the grain boundaries with misorientation angles of 1.50 or higher, high angle grain boundaries of 150 or more and low angle grain boundaries with relatively large misorientation of each account for approximately 35% of total grain boundary length.
(I). work hardened grains; (II). mixed grains consisting of work hardened grains and dynamically recrystallized (DRX) grains and (III).
DRX grains.

The microstructures of the solidified ceramics presented a number of fine TiB2 platelets embedded in TiC grains, Cr-Ti alloy or between TiC grains and Cr-Ti alloy.
FESEM images and EDS results showed that a large number of TiB2 platelets were embedded in the irregular TiC grains, Cr-Ti metallic phases (showed by the white area in Fig. 2) or between TiC grains and Cr-Ti metallic phases, as shown in Fig. 2.
However, mole concentration of Ti and B actually decreases because of the addition of liquid metallic Cr in liquid TiC-TiB2, making a number of TiB2 platelets grow more hardly.
As discussed above, the increased mass fraction of Cr binder not only makes fine-grained even ultrafine-grained microstructure achieved in the ceramic, but also brings about the increased plastic phases of Cr-Ti alloy.
FESEM images and EDS analyses showed that current solidified TiC-TiB2 composites were composed of TiB2 platelets, irregular TiC grains, Cr-Ti metallic phases and a few of isolated Al2O3 inclusions, and a number of fine TiB2 platelets were embedded in TiC grains, Cr-Ti alloy or between TiC grains and Cr-Ti alloy.

The phase field variable varies smoothly across the interface from fi = 1 in the ith grain to fi = 0 in other grains.
The time evolution equations for the phase-field and carbon concentration variable, C, are given by the following equations: (1) (2) Here, n is the number of phase fields at an arbitrary point.
According to a study of the ferrite nucleation during the austenite-to-ferrite transformation during continuous cooling reported by Umemoto et al. [7], the number of ferrite grains nucleated on a unit area of the austenite grain boundary in a unit of time is calculated by the following equation: (6) where Dg is the diffusion coefficient of a carbon atom in the austenite phase.
In Fig. 2(a), the translucent gray region indicates the austenite grain boundary and the colored regions represent the ferrite grains.
Figure 3 Size distribution of ferrite grains.

In this case grain 2 Title of Publication (to be inserted by the publisher) coarsening by grain growth may dominate over grain refinement by dynamic recrystallization.
Fo39 Fo40 Fo41 Fo42 Fo43 Fo44 HIP4 HIP5 Deformation (D) or HIPped-only (H) D D D D D H H H Initial sample length 16 16 16 5.3 16 16 5 5 Initial sample diameter 8 8 8 8 8 8 11 11 HIP-duration [hrs] 3 24 9 14 15 17 9 40 Deformation-duration [hrs] 17.8 18.8 17.9 30.2 35.4 - - - 4.80 9.30 1.13 5.23 10.80 Strain rate [10-6 s -1 ] 5.07 5.01 5.12 5.85 1.26 - - - 17.3 31.3 9.9 41.1 83.0 Flow stress average [MPa] 33.1 32.0 24.8 32.3 31.6 - - - Final: Axial shortening strain [%] 28.0 27.9 27.1 45.6 28.4 - - - Mean grain size [µm] 1.4 1.2 1.1 1.5 1.3 0.8 0.6 0.7 Standard error 0.06 0.05 0.04 0.08 0.05 0.02 0.01 0.01 Average Feret90/0 ratio 1.1 1.1 1.1 1.1 1.0 1.1 1.1 1.1 Number of grains analysed 203 280 347 180 274 772 1207 1005 Journal Title and Volume Number (to be inserted by the publisher) 3 0 10 20 30 40 50 60 70 80 0 10 20 30 strain [%] stress [MPa] Fo39 0 10 20 30 40 50 60 70 80 0 5 10 15 20 25
Quantitative grain size analysis.
The low n-value, the weak LPO and the lack of any change in average feret90/0 ratio of the grains are observations that point to a GSS, possibly grain boundary sliding (GBS) dominated, Journal Title and Volume Number (to be inserted by the publisher) 5 Fig. 4 Average grain size-time plot of HIPedonly and deformed samples; error-bars show standard error of the average value.
The model of Ashby and Verall [12] involves grain switching during GBS, which increases the probability of high-sided grains coming into contact with low-sided grains, therefore stimulating grain growth.