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Based on this framework, a number of probabilistic models have been proposed in literature.
For an appropriate modeling of the RVE’s, a statistical characterization of the microstructure (grains and particles) is required.
Thus, precise information regarding the grain size distribution, the grain shape and the grain orientation (morphologic and crystallographic) were obtained.
Based on the morphologic information in the form of best-fit-ellipses, the grain size distribution, the grain shape distribution (ratio of the semi-axes) and the orientation distribution of the grains could be determined.
The particles were inserted into the grain structure at the desired location via a spherical submodel, consisting of the material of two adjacent grains and a brittle particle with user-defined size, shape and orientation at the interface of these grains.

Journal Title and Volume Number (to be inserted by the publisher) 3 {223}<582>) has yet to be fully explained.
Recrystallization in alpha fibre oriented grains is slowed, partially, due to the low misorientation between subgrains and thus, low stored energy, within grains of these orientations.
Here, the unrecrystallized alpha fibre grains, based on the previous discussion, are taken to have a lower flow stress compared to the unrecrystallized gamma fibre grains.
The nonlinear relationship between recrystallized fraction and flow stress is thus attributable to the initial loss of the 'stronger' deformed gamma fibre grains in place of the 'softer' alpha fibre grains.
Taylor simulations have been performed using the bulk textures measured by x-ray diffraction (figure 3) in Journal Title and Volume Number (to be inserted by the publisher) 5 order to simulate the experimental r-values.

It's shown that the grain size and overall amount of carbide phase has almost no effect on ICC.
n,  - number and reduction at TMP correspondingly; D - grain size;  - dislocation density; o - density of precipitates evaluated by the light microscope measurements; e - density of dispersed precipitates evaluated by TEM; F - fraction of fragmented volume; R - fraction of recrystallized volume.
It should be noted that in the substructure of steel worked with a reduction up to 30% and n=3 the structure becomes more perfect in compare with single-pass rolling, the grain boundaries and solid solution are improved as a result of dynamic strain aging and dispersed precipitation of titanium carbide distributed comparatively uniformly throughout the volume of grains.
In contrast to data obtained in [9] for a similar steel given TMP, the grain size and overall amount of carbide phase in this case has almost no effect on ICC.
Billets differing in average grain size (by factor of two) have almost the same corrosion rate.

The average grain size was about 2.8 µm.
It was found that the average grain size was about 12.0 µm.
The fine-grained TMAZ (B) microstructure within the outer region containing 50.2 % α2 phase and 49.8 % γ phase has an average grain size about 6.1 µm (Fig. 4c).
However, a large number of voids were formed at the interface and were uniformly aligned along the bond line (Fig. 5b).
The joints exhibited cracks on the γ-TAB side and large number of voids at the interface of the joints.

It is well known that submicron sized grains/ sub grains can be produced in most Al alloys using this technique.
The total strain accumulated in a material through a series of repetitive pressings, εN is given by [9], (1) where N is the total number of passes through the die.
In this paper, specimens have been designated as 2A, etc. where 2 represents the total number of extrusion passes and A represents the processing route.
The grain size of the initial solutionized material was approximately 15 µm.
After 3 passes, the grain size was reduced to 0.46 µm in route A ECAE and 0.55 µm in route C ECAE.

To refine the microstructure, alloys are subjected to a modification by introducing grain refiners into the casting mold before pouring.
For providing uniform distribution of grain refiners over alloy, one uses special mixing devices that create turbulent flow field inside the molten metal and thereby intensify the diffusion processes promoting the uniform distribution of grain refiners.
The Power number is a function of the Reynolds number, , the Froude number, and the ratio of the diameter of the agitator to that of the crucible, .
Euler number vs Reynolds number.
The Euler number decreases exponentially with respect to the Reynolds number.

The rate of annihilation is determined as the change in the number of dislocations of the same sign N depending on time: .
Pearlite grains (about 30%) consist of cementite and ferrite interlayers.
This can be explained, taking into account that the microhardness of pearlite grains is about a third higher than that of ferrite grains.
Using a model crystal with a size of 1000 b, the dependencies in the variation of the number of dislocations and strain in a pile-up on time were analyzed.
Rybin, Grain Boundaries in Metals, Metallurgy, Moscow, 1980.

The impedance spectrum of Ho-doped sample indicates that consist of semiconducting grain and moderately insulating grain boundary regions.
Generally, the formula of doped bismuth titanate is (Bi2O2)2+(Am–1BmO3m+1)2– , where A means mono-, di-, or trivalent ions, or a mixture of them; B means quadri- or quinque-valence ions, such as Ti4+, Nb5+, Ta5+; and m means integer number > 1.
Ho-doped sample exhibits randomly oriented and plate-like grains, and the average grain area of the sample was approximately 10×10μm2, and thickness was less than 2μm in general.
In general, the capacitance C of grain and grain boundary are typically of the order of pFcm–1and nFcm–1, respectively.
The Ho-doped bismuth titanates consist of grain and grain boundary regions.

The precipitates on the grain boundaries are ellipsoidal-shaped and those in the austenite grains are needle-shaped.
Besides, precipitates start growing from grain boundaries to austenite grains and there has been a small amount of precipitates inside the austenite grains as shown in Fig. 2.
Then the precipitates connect along the grain boundaries until all the grain boundaries are filled with precipitates (Fig. 4b).
With the aging time continuing to increase, the number and size of precipitates in austenite grains increasing.
The precipitates on the grain boundaries are ellipsoidal-shaped and those in the austenite grains are needle-shaped.

Furthermore, the total number of drilled holes with brazed and electroplated trepanningtool is 130 and 7, which means a boost of 18.6 times in the life of monolayer brazed diamond trepanningtool than in that of electroplated one.
The number of drilling holes and the mean drilling time for single hole for brazed trepanningtool with optimum and random grain distribution is 157 and 130, and 19s and 30s, respectively.
Journal Title and Volume Number (to be inserted by the publisher) 75 a.
Table 5 Tangential force of grinding cup wheels [N] Grinding depth [mm] Grinding cup wheel 0.02 0.06 0.1 0.2 0.3 Multilayer sintered 13 24 38 142 Unable to machine Brazed with random grain distribution 7 13 20 72 106 Brazed with optimum grain distribution 5 9 12 34 80 Journal Title and Volume Number (to be inserted by the publisher) 77 The testing results show that the grinding force of monolayer brazed grinding cup wheel is obviously lower than that of multilayer sintered ones and indicate the former has assuredly very good sharpness.
The grinding force of brazed grinding cup wheel with optimum grain distribution is evidently lower than that of brazed ones with random grain distribution, which shows a best state for grinding cup wheel with optimum grain distribution in machining.