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By increasing the numbers of turns in HPT, the hardness values increase to a saturation level and the gradient is removed.
As the microstructure develops from the centre to the half-radius and the peripheral areas, Figs 1(c) and (d), a small number of coarser grains was detected and generally there was a very small grain structure.
Accordingly, a uniform ultrafine-grained structure was achieved with increasing displacement from the centre of the discs and with rising numbers of HPT turns up to 5.
Furthermore, it is observed that the hardness increases as a result of increasing the numbers of HPT revolutions.
Fig. 2 Vickers microhardness across diameters of discs of ZK60 processed by for various numbers of turns of HPT.

The Effect of Grain Boundary Junctions on Grain Microstructure Evolution: 3D Vertex Simulation L.A.
However, a polycrystal is formed not only by grain boundaries (GBs) and grains but also by triple lines (TLs) (line where three grains meet) and quadruple points (QPs) (point where four grains meet).
In such approach only the grain boundary is discretized whereas the grain interior is not.
The sum of the forces over all facets surrounding the vertex leads to the net force ( ).∑= × ×× ⋅= n 1i i0i1 i0i1i0 sum ss sss 2 F rr rrr r γ (2) where n is the total number of facets adjoining at P0.
In our case the grain size is represented by the term x0, which scales with the grain size.

The change in the system entropy is thus calculated as { }         ΩΩΩΩ+ΩΩ=Ω= 43421321 b a BkS 543121lnln (1) where Ω is the number of ways to permute n atoms in the grain.
It is assumed that the total number of atoms redistributed by planar deformation is ⊥Ο += nnn , where ⊥n is the number of atoms forming dislocations, and Οn is the number of atoms displaced through grain boundary sliding, and are termed "bond breaking atoms".
Equation 1 is divided in the contributions of a, which accounts for the ordering of Οn atoms over the available m positions at the interface; and b, which quantifies the number of possible permutations of ⊥n atoms in the bulk of the grain.
The energy of the system is calculated with [ ]cnaGbnEU d b 2212 2 3 + += Ο (2) where Οn is the number of bond breaking atoms, dn is the number of dislocations per grain, bE is the bonding energy, G is the shear modulus, b is the burgers vector and c is the number of dislocation direction changes per dislocation.
Referring to equations 3 and 4, there are three parameters that describe the deformation behaviour: the increase of the number of dislocation production atoms per deformed plane ⊥dn , the increase of the number of dislocations per deformed plane ddn , and the increase of the number of dislocation direction changes per dislocation and deformed plane dc.

In other words, the grain-coarsening tends to occur when: (1) the matrix grains that formed as a result of the α/γ transformation during the heating for carburizing are fine (R0 is small), or (2) the degree of grain size mix is large (Z is large), or (3) the number of pinning particles is small (either f is small or r is large).
The pre-austenite grain of the test pieces that underwent the simulative carburizing treatment was observed in a section in the rolling direction and their grain size number was determined according to ASTM.
A grain size corresponding to a ASTM grain size number of #5 or lower was defined as coarse grain, and a test piece was judged as having coarse grains if such coarse grain size number was obtained in any one field of observation in the entire section.
(c) (a) (b) (d) (e) (f) 25µm Table 2 shows the ASTM grain size number of No. 5 and 9 after simulative carburizing at high temperatures.
Ariyasu: Mitsubishi Motors Technical Review Vol. 13(2001), p. 114 Table 2 ASTM grain size number after high temperature carburizing.

We determine a smaller number of selected GB plane orientations with all the 5 needed parameters for each of them.
First the orientation matrices (with i=1,2) of the two grains at the opposite sides of a grain boundary plane is determined from their respective CBED patterns.
Although these numbers are modest from a statistical point of view, their distinct distributions uncover important differences between them.
Orientation distribution for the S3 grain boundaries.
Although the number of observations did not allow for detailed statistical analysis, some general trends can be assumed from the measurements.

If the neighboring grain is smaller, a random number rs is generated; if rs > ps no state change is made 4.
Then a random number r is generated.
Nevertheless the evolution is slower than expected, and the number of grains remaining in the microstructure soon becomes insufficient for a statistically valid determination of quantitative texture.
For avoiding the problem of statistics associated with the decrease in grain number, sequential modeling was used to enable longer simulation times to be achieved in the grain growth evolution.
First, even if the number of grains is statistically sufficient with regards to the initial texture determination, it may be not sufficient to take into account subpopulation effects and also decrease rapidly during the simulation.

Characteristics of sexual reproduction of two L. chinensis ecotypes under saline-alkali soil and sandy soil habitats on Songnen plain were studied, such as number of florets, number of grains, seed-setting percentage and the weight of thousand grains.
The results showed that number of florets, number of grains, seed-setting percentage and the weight of thousand grains of two L. chinensis ecotypes fluctuated at some range.
Variation coefficients of number of florets, number of grains and seed-setting percentage were 34.63%, 79.37% and 87.2% higher, but variation coefficient of the weight of thousand grains was lower.
The number of florets, seed setting rate and the weight of thousand grains of both two L. chinensis ecotypes fluctuated within a certain range, and the number of changes was ecological plasticity.
Meanwhile, the coefficient of variation of the number of florets, the number of grains per panicle and seed setting rate were up to 34.63%, 79.37% and 87.2%, but the coefficient of variation of the weight of thousand grains was smaller, this change was in response to plant habitat adaptation.

Using an approach of grain boundary interfacial area (Sv) calculation, which attempts to account the number of sites for potential nucleation of ferrite in thermomechanically processed austenite, an essential correlation of the correspond measured grains was developed.
Although partially recrystallized, the austenite grain boundaries still account the number of sites for potential nucleation of ferrite.
In general, because the ferrite grain size is related to the number of sites for ferrite nucleation, and because this number is related to the austenite grain boundary area per unit volume, the ferrite grain size will decrease as this boundary area increases.
In addition, presumably because the smaller grains have a large number of grain boundaries per unit volume greater than the large grains, so that nucleation can occur in many sites.
Nucleation rate is proportional to the square of the number of grains per unit surface area of ferrite grains austenite.

The ability to model the grain deformation will benefit microstructure models of the extrusion process in the following ways: 1) Calculation of the grain dimensions can provide information on the grain boundary area which can then be used to calculate the stored energy due to the grain boundaries in the deformed material and how this varies spatially, as well as number of possible sites for nucleation of recrystallization on grain boundaries. 2) Calculation of the grain thickness can be used to understand and predict phenomena such as GDRX that can occur when the grain thickness goes below the average subgrain size.
Based on their calculations for all deformation conditions, the grain area increase that occurs for spherical grains is more than that of cubic grains.
Grain shape calculation.
Assuming the number of grains between the two points remains constant during the deformation, an estimate of the final thickness of the deformed grains can be done using Equation (1) shown below: Final grain thicknessOriginal grain size=dD (1) Where D is the distance between two adjacent tracked points before deformation and d is the distance between those points after the deformation (Fig. 2).
Results and Discussion Grain thickness.

An aged Al-5Zn-1.6Mg alloy with fine η' precipitates was grain refined to ~100 nm grain size by severe plastic deformation (SPD).
Its grain size was of the order of 1mm.
The pass number was 1, 3, 6, 9, and 18 for CCDP and 1, 4, 8, 12 for ECAP, respectively.
The grain boundary misorientation monotonously increased with rising number of passes, as qualitatively demonstrated by selected area diffraction (SAD) patterns (Fig. 1e), which changed from a single crystal type to a polycrystal type as SPD progressed.
The particle size followed a logarithmic normal distribution, and their mean values decreased slightly with the number of CCDP passes, but particle spacing changed not apparently.