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A statistical model of grain growth [8,9] is modified to incorporate the described alterations in grain shape.
Grain growth is assumed to evolve because of reduction in the total grain boundary area.
Both the grain boundary energy γb and the boundary mobility are taken the same for all the grains.
Microstructure evolution was traced by changes with time in the grain size distribution as well as in the mean grain diameter 〉〈D (it is calculated as the number averaged value, diameter of barrel-like grains being supposed D'), in the average diameters of spherical and columnar grains, 〉〈D s and 〉〈D c, respectively, and in the volume fraction of columnar grains, Vc.
The parabolic law Eq. 3 describes the growth and consumption of grains on condition that the grain size distribution remains self-similar.

Particularly in presence of complex geometry shapes, rare grains nucleating apart from the primary grain, become a serious problem in directional solidification, when characterized by high-angle boundaries with the primary grain, extremely brittle due the elevated amount of highly segregating elements and the absence of grain boundary strengthening elements.
In this paper, constrained dendrite growth and heterogeneous grain nucleation theories have been used to model the formation of stray grains in directional solidification of Ni-base superalloys.
It is obtained [6]: Rg=A3T∆Tcol3-∆Tn3 (1) from eq. 1 the condition for the existence of the equiaxed grain is derived: ∆Tcol3=∆Tn3 (2) It is here assumed that the critical grain radius for the successive development of a stray grain is of the order of the primary dendrite arm spacing.
For β below this limit equiaxed grains can form.
The mathematical description of the growth of constrained dendrites and of the nucleation on foreign substrates allows to evaluate the influence of a number of parameters, inherent to both the physical properties of the alloy and the way the temperature field in the liquid metal varies during the process.

In this work, the grain-boundary cavitation in polycrystalline aggregates is investigated by means of a grain-scale model.
The formulation is presented within the framework of a grain-boundary formulation, which only requires the discretization of the grain surfaces.
Grain-scale model including grain-cavitation Any comprehensive micromechanical model of creep should take into account crystal visco-plasticity and grain-boundary cavitation and sliding, which play different roles in different stages of creep evolution.
The basic grain-boundary formulation is briefly presented here.
Finally, is the number of constituent grains of the aggregate.

This is because fine-grained microstructures avoid the problem of microplasticity arising from slippage occurring among grains in the surface region.
The maximum number of cycles allowed for spectrum loading was 107.
Figure 6 shows the ferrite grain size.
Effect of ferrite grain size on fatigue properties.
Figure 12 shows the relationship between the stress amplitude and the number of cycles to failure.

This grain refinement produces significantly improved mechanical properties.
Introduction Ultra-fine-grained and nano-grained crystalline materials exhibit attractive properties such as unusually high strength combined with reasonable ductility and toughness [1-10].
Schematic explanation of the grain-switching mechanism in formation of nano-grains.
For refining the size of grains in metals down to nanometres a number of passes are required.
A total number of over 30 grain orientations were measured, for which misorientations across boundaries were determined.

Journal Title and Volume Number (to be inserted by the publisher) 5 Ultrasonic velocity.
A propagating wave is scattered at grain boundaries and absorbed by dislocations, grain boundary scattering being the dominant effect.
Journal Title and Volume Number (to be inserted by the publisher) 7 Figure 6 shows examples where the ultrasonic velocity is plotted against temperature for annealing of two different steel types.
Journal Title and Volume Number (to be inserted by the publisher) 9 Figure 8.
An alternative approach has proved to give good correlation to grain size although it cannot provide absolute values of grain size.

The water-cooled copper mold casting with slope vibration at the temperature near liquidus can obtain Al-7Si-2RE alloy with small homogeneous equiaxed grains, the average grain diameter is 48.3μm, and the average grain roundness is 1.92.
It would lead to produce different conditions of nucleation and grain growth in the solidification process.
The reason lies in two aspects: on the one hand, in the casting process, the slope can make the molten alloy roll while flowing to the mold, which plays a similar role of mechanical agitation; on the other hand, the exist of a certain temperature difference between the slope and molten alloy, makes the contact surface to meet the nucleation conditions of producing a large number of nuclei easily, and these nuclei roll into the melt alloy so as to increase the number of grains.
It can be seen from Fig.4(c) that slope water-cooled copper mold casting with a certain amount of mechanical vibration can further refine the alloy microstructure, and the grains are more rounded in the whole with its number increasing and its size reducing.
The microstructure of Al-7Si-2RE alloy from slope water-cooled copper mold casting with mechanical vibration was small and equiaxed with the average grain diameter of 48.3μm and the average grain roundness of 1.92.

These grain boundaries are often termed non-equilibrium grain boundaries [12].
(a) (b) Fig. 2: Variation of (a) grain size and (b) fraction of HAGB in the gage section of Al – 7 % Si samples as function of number of HPT turns.
As shown in Fig. 3, irrespective of the number of HPT turns, the hardness of the alloy increased with HPT processing.
Fig. 3: Variation of micro-hardness as function of radial distance from the center of the disk with number of HPT turns.
It should also be noted that instances of such transition increased with the number of HPT turns.

In addition, the solution of strengthening second phase element at the grain boundaries into the grains further reduces hardness in the overheated area.
Unlike the fine acicular ferrite and lower bainite in the as-received material, a large number of massive ferrite is formed in the incompletely recrystallized area after welding, which weakens the effect of fine grain strengthening.
This can be attributed to the complete recrystallization process, which results in significant grain growth and weakening of fine grain strengthening.
There exist a large number of dislocations in the bainite lath and the lath boundary is composed of dislocation walls.
There are a small number of fine retained austenite and M/A islands between the lath.

It can be seen that there exists a sufficient number of visible particles, which are shown in yellow in Fig. 1 (a) for one grain.
It can be also seen from Fig. 1 (a) that there are a number of particles that are located in grain interior.
Although this process greatly reduces the number of diffraction spots related to a particular grain, there are still a number of retained diffraction spots for a particular grain.
Since grain 1 is fairly large compared to Grain 2, a greater number of grains are illuminated while the X-ray pencil beam intersected Grain 1.
Black lines super posed on the strain contour maps indicates grain boundaries, and white figures indicate grain number.