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In their model recrystallisation, recovery and precipitation are described by sub-models with dislocation density, precipitate number density and average precipitate size as the internal variables coupling them together.
Grain growth.
Such an approach, however, has a number of complications.
To determine the migration factor (i.e. the proportional factor between the migration rate and the driving force) from recrystallisation data (i.e. the volume fraction recrystallised as a function of time) it is necessary to know the size of the driving force, namely the stored energy, and the number density of recrystallizing grains, both quantities as a function of time.
This research was carried out under the project number K41.2.09354 in the framework of the Research Programme of the Materials innovation institute (M2i) (www.m2i.nl).

For higher number of passes the grain size did not change anymore.
However, the misorientation rose constantly with increasing number of passes, as demonstrated by the SAD patterns in Fig. 2 and 3.
Fig. 4: Mean grain size along the longitudinal direction of both alloys as evaluated from the grain size distributions of 700-1000 grains for each sample.
With increasing number of CCDP passes the yield stress and peak stress of both alloys grew due to strain hardening and grain size strengthening (Hall-Petch hardening) caused by severe plastic deformation.
Furthermore, both alloys showed strain softening after reaching a peak stress during compression. 250 300 350 400 450 500 0 2 4 6 8 10 12 14 16 18 Al-1.5Mn (P) Al-1.5Mn (S) CCDP passes Mean grain size d0, nm aa bb c d cc dd The higher the number of CCDP passes the stronger the strain softening (Fig. 5b-c).

The laser processing variables were pulse energy and and number of pulses.
With the increasing number of pulses was increased material loss by laser ablation mechanism.
Laser spots are affected by the specific heat and vaporization, but the crater depth is dependent on the number of pulses.
The pulse number dependent effect is clearly seen in Fig. 2.
Murakami, Recent development of low-loss grain-oriented silicon steel, J Magn.

The Vickers hardness number for the as-received Y-TZP material decreases to a very small extent after 560 thermal cycles and increases approximately 2%, after 1200 thermal cycles.
Theory of residual stress measurement and grain size evaluation.
Vickers hardness and Young’s modulus of zirconia The Vickers Hardness (HV) numbers of the Y-TZP sample were evaluated both as-received and up to 1200 thermal cycles.
Vickers hardness number for the as-received Y-TZP material decreases to a very small extent after 560 thermal cycles and increases approximately 2% after 1200 thermal cycles.
Vickers hardness number for the as-received Y-TZP material decreases to a very small extent after 560 thermal cycles and increases approximately 2%, after 1200 thermal cycles.

In this figure, a grain in the first line is represented by the number 1.1.
The grains in the second and third lines are numbered as 2.1, 2.2, 3.1, 3.2 and 3.3.
Numbering each abrasive grain, displacement of each grain is measured for the reference.
It can be seen that each displacement of abrasive grain around contacting grain is different depending on each position of grain in analysis.
On the other hand, displacements of the abrasive grains around contacted grain were same displacement for each grain.

The inhibition of grain boundary migration caused by the dispersed particles is of great practice importance in controlling the grain size of material.
Yu[10] applied the CA method to simulate the processes of normal grain growth and abnormal grain growth in steel.
The models mentioned above successfully describe the dynamic behavior of grain growth and the distribution of grain size.
Obviously, the reasons why the phenomena mentioned above take place are that under the same volume fraction of second phase particles, the smaller is the average size of second phase particles, the more is their number, and the greater is the degree of their dispersion, and the higher is the chance that they locate at the grain interfaces, edges and corners, so that grain boundary may be pinned by more second phase particles, and then the grain growth slows down.
This means that n is not a constant number and decreases with the increase of simulation time when the matrix contains second phase particles.

To have reasonable statistics and accuracy for the grain size distribution the number of grains measured was in the range of 428 to 1700.
From the sets of measured grain diameters, the two dimensional log-normal grain size distribution, f (d) [23] is arrived by using the following relationship: ……(2) where s is the std. deviation, d is the grain diameter and dg is the peak grain diameter.
Peak temperature, °C Heating rate, °C/s Number of grains Mean volumetric grain size (μm) Standard deviations 950 10 1497 6.0 0.52 950 100 1700 4.4 0.47 950 1000 1741 4.2 0.51 1150 10 670 15 0.60 1150 100 839 11 0.54 1150 1000 596 11 0.62 1350 10 763 61 0.53 1350 100 514 33 0.52 1350 1000 428 32 0.42 Fig. 3a: Representative austenite grains for 100oC/s heating rate and 1150oC austenitizing temperature.
Fig.4 represents the precipitate size distribution in terms of number and volume of precipitates in the base metal and depicts that the precipitates can be broadly divided into three families: Ti-rich (TiN), Nb-rich (NbCN) and Mo-rich (Mo-C).
The relatively sluggish grain growth at the lower times appears to be due to the presence of precipitates that pin the grain boundaries and resist grain growth.

Abnormal grain growth (AGG) proceeds in case that normal grain growth is inhibited.
Then how the overall grain structure evolves during grain growth in this situation?
This means that we can put a restriction on the maximum number of positive phase-fields coexisting on a grid, with negligible affect on the grain growth dynamics, but with significant saving of the required memory space.
As can be cleary seen in Fig. 3, the grain growth is undergoing abnormally; a number of grains are abnormally growing by sweeping away the matrix grains.
In fact the crucial one determining a grain's destiny at the initial stage of grain growth is its nearest neighbor grains.

The orientation field variables η were chosen as η1(r, t), η2(r, t), η3(r, t)...ηp(r, t). p is the possible number in the system, and it’s considered to be 512 in the simulation, in order to show more realistic grain growth evolution.
It’s shown that the grains of abnormal growth are much larger than the matrix grains.
The local low grain boundary may be the small angle grain boundaries such as boundaries of twin grains or coherent grain boundaries.
Parsing abnormal grain growth.
Phenomenology of abnormal grain growth in systems with nonuniform grain boundary mobility.

Parameters of the Fe diffusion Matrix A§ Em (eV/atom) liquid Al95Fe05 7.4·10-7 0.47 liquid Al90Fe10 4.4·10-6 0.63 liquid Al80Fe20 2.7·10-4 1.01 <100> S5 non-symmetric GB in impure Al 1.7·10-15 0.61 <100> S5 symmetric GB in impure Al 2.9·10-14 0.82 <111> S7 symmetric GB in impure Al 7.7·10-12 1.28 The grain boundary diffusion was determined as follows: , (1) where W is the atomic volume, A is the grain boundary area and d is the grain boundary width, NGB is the number of atoms in the GB region and Dxi and Dyi are displacements of atom I in the GB plane.
Fe-Fe coordination number in liquid Al-Fe alloys and in GBs.
Indeed, as can be seen from Fig. 4 the Fe-Fe coordination number reaches some limit which ranges from 3.0 to 3.6 depending on the average Fe concentration.
Rodin: "Grain Boundary Diffusion and Grain Boundary Segregation" In Proc.
In: The Nature and Behavior of Grain Boundaries, H.