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Concerning grain growth, the statistical model is based on the assumption of [10]: · Super-position of average grain curvatures in individual grain boundaries; · Homogeneous surroundings of the grains.
The integration of all the above assumptions in the model leads to the following final form of the grain growth rate equation: dRidt= Mj1Rj-1Rinj4πRj2jnj4πRj2 (1) Where Ri [cm] and Rj [cm] are the radius of grain belonging to class i and j and nj and nj are the total numbers of grains in class i and j.
The effect of reduction rate has been exploited maintaining the temperature constant at 1100°C and varying the cold reduction rate jointly to the dislocation density (Δρ) and the initial numbers of deformation nuclei (N) measured by means of X ray diffraction analysis.
The amount of strain affects also the recrystallization rate because the strain modifies the amount of stored energy and the number of effective nuclei.
Four temperatures, ranging from 700°C to 1100°C, have been simulated for AISI 304 steel grade while, at the same time, cold reduction rate (90%), dislocation density Δρ and the number of nuclei N were maintained constant.

Therefore, those simulations were limited to small grains in order to keep the number of grains adequate for statistically analysis.
Number of faces vs. grain size.
One important topological feature of the microstructure is the correlation between the number of faces s per grain and the relative grain size x (cf.
Figure 1: a - Number of faces vs. relative grain size for all grains together with quadratic leastsquares fit; b - Number of faces vs. relative grain size divided into size classes for three different time steps.
Figure 1a shows the number of faces of the individual grains as a function of the relative grain size for the 500 th MCS.

Its grain output reached 111.41 billion kg in 2011; commodity grain output amounted to 90 billion kg and became the first major grain-producing province beyond Henan.
Meanwhile, Heilongjiang also ranks first in total grain output, commodity grain output and grain self-sufficiency rate. 30 billion kg of 90 billion kg commodity grain is contributed by the Heilongjiang reclamation.
The rate of its commodity grain is as high as 90% and the grain farmers make great contributions to guarantee national grain security in Heilongjiang reclamation.
It includes: the number of labor force, whether have cadre relatives or not, the number of underage children, the number of boys, the number of girls, the highest education in family and the number of non-healthy persons. (1) The number of labor force has a positive impact on farmers’ income.
It includes: the number of land blocks, sown area, the proportion of plain land, deeply plowing the land and land Huancha. (1) The number of land blocks has a negative impact on farmers’ income.

In this paper, hydroxyapatite was used as a grain growth inhibitor additive to get nano-titania ceramics with different grain size, and the effect of grain size on the bioactivity was studied in vitro.
In this paper, hydroxyapatite was used as a grain growth inhibitor additive to get nano-titania ceramics with different grain size, and the effect of grain size on the bioactivity was studied in vitro.
Ltd, China) were used as grain growth inhibitors.
The result of MTT assay showed that the number of viable cells was not statistically significantly different between the HT and the control (HA) (p<0.05), but the number of cells on the MT was much lower than the number on the control (p<0.05).
Both HA additive and the grain size of HT might be responsible for the bioactivity of HT.

The size of grains at the top and bottom of the rolled plate converged to 0.65µm, while that of grains at the center of the plate increased with the number of ARB cycles.
The size of grains at locations of NT and NB ranges from 0.7µm to 0.8µm and remains more or less unchanged with the ARB cycle number.
However, the grain size at the NC position varied a lot with the cycle number.
The grain size increased continuously from 0.56µm to 1µm as the cycle number increased from 3 to 6.
However, the size of grains at the center of the processed plate increased with the cycle number.

For interface controlled abnormal grain growth there may be a number of processes associated with atomic attachment at the interface that can be the controlling mechanism.
In the past decade, a number of researchers have invoked the idea of a non-linear relationship between driving force and grain boundary velocity to explain abnormal grain growth during nucleation limited interface controlled grain growth.[8-10,19-20] A schematic of this is shown in Fig. 4.
The nucleation limited interface controlled abnormal grain growth mechanism has been invoked to describe abnormal grain growth in single phase and pseudo-single phase systems.[16,21] Systems where only a small number of the grain boundaries contain a nanoscale intergranular film may be considered to be pseudo-single phase.
If the grain with the defect can not grow any faster than the normal grains then this grain can not grow abnormally.
Arrhenius behavior has been observed in alumina in a number of studies, where the driving force varied with temperature.[4,26,29-31] It then remains to explain how abnormal grain growth occurs by a diffusion controlled mechanism in pseudo-single phase alumina.

The evolved substructures with nodes of the Fe precipitates gradually changed to new grains surrounded by low- and high-angle boundaries with increasing number of the repeated processes.
Introduction Numerous numbers of thermo-mechanical processes (TMPs) have been applied for grain refinement of bulky metallic materials.
Because RX normally involves grain coarsening due to grain boundary migration, lower limit of the minimum grain size seems to exist.
It is evident that fine grains were gradually evolved with increasing number of cycles.
Such slight grain coarsening took pace more significantly where fine grains were evolved in groups.

In 2-dimensional case, the Betti numbers are consisting of two numbers.
The other is b1= H1(X), which is the number of holes.
Because the Betti numbers are invariant, the shape of the grain has nothing to do with the Betti numbers.
The number of b1 in the substrate area and HAZ area also has not changed except for the fine grains, but WMZ is decreasing.
The number of grains of more than 6µm2 seems to be the same of WMZ and HAZ.

During grinding a large number of micrometer or sub-micrometer grains remove material from the surface of a workpiece, acting as cutting tools.
They were able to determine the origins of lattice distortion of the substrate and the dependence of material removal rate on the ratio of sp2 to sp3 phases by conducting analyses of RDF and coordination number.
At a later moment, all grains are involved in the process, with the two back grains removing less material than the two front ones and forming chip in the area between the two rows of abrasive grains.
It can be seen that the number of atoms in a deformed lattice constitute a small part of the total number of atoms, as expected, because depth of cut is less than the 1/10 of total workpiece height and thus, deformation occurs at a few atomic layers beneath the surface.
The number of atoms belonging either to the plastic deformation region or the cutting chip range is larger with larger depth of cut.

Note that the initial number of grains are 50 in all cases, and the results for Ks > 3.0 were mostly the same as that shown in Fig. 3 (e), because the grain separation, i.e. increase in the number of grains, never occurs in this model.
(a) Grain distribution (b) Variation of grain number Fig. 4.
Effect of the initial number of grains and the size of calculation domatin on the steady state grain size.
identification numbers, large-number grains depicted in yellow-red color show decaying tendency, and vanish at last, as shown in Fig. 5 (a).
On the other hand, when the small-number grains are weighed, large-number grains survive, as shown in Fig. 5 (c).