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(4) where Γ is the Gibbs-Thomson coefficient, R is the dendrite tip radius, Pei is the Peclet number for element i, mi is the liquidus slope for element i, C0,i is the initial concentration for element i, ξc(Pei) is function of Peclet number, ki is the solute partition coefficient for element i, Iv(Pei) is the Ivantsov solution, and Ghkl is the average thermal gradient near the tip.
The potential weld defects of centerline grain boundary formation, stray grain formation and solidification cracking occur.
The centerline grain boundary formation is almost eliminated along the weld centerline.
The typical weld defects of centerline grain boundary, asymmetrical stray grain formation and subsequent solidification cracking are potentially prevented with high welding speed.
Stray grain formation in single crystal Ni-based superalloy welds, Journal of Applied Physics.

With hybrid heating the average grain sizes for die pressed samples were in the range 70 - 80 nm whilst for the more homogeneous slip cast samples a final average grain size of just 64 nm was achieved for a body with a final density of >99%.
However, whilst commercial nanopowders offering these properties have now been produced successfully, sometimes in relatively large quantities, a number of challenges still need to be surmounted if engineering parts are to be manufactured.
With radiant heating, there was a steady increase in grain size up to around 90 - 95% of full density when there was a sharper increase in grain size.
As for the radiantly heated samples, the rapid increase in grain growth began at densities of 90- 95%; achieving full density resulted in average grain sizes ≥160 nm.
Fig. 7: Grain size of Y2O3 as a result of two-stage sintering (after Chen et al [11]).

For both temperatures, the chromia grains morphology is similar and appears as polygonal.
For longer oxidation times, the mean grain size does not significantly evolve.
Thus, by using Gauss/Lorentz simulations (Lab Spec software), the wave number ν1 (A1g) is deduced. ν1 = 551.55 cm-1 at room temperature for the stress free value.
Then, the stress determination in the constrained oxide is obtained from the variation of the wave number with respect to this reference at each working temperature.
The variation ∆ν of ν1 (A1g) (∆ν = νC - νL, where νC and νL are, respectively, the wave numbers for constrained and unstressed oxides at each working temperature) is used to deduce the corresponding stress.

This suggests that the SMAT induced grain refinement remains stable after thermomechanical treatment.
The average mean grain size can be evaluated between 100 nm and 250 nm.
More details about grain refinement and strength increase after SMAT can be obtained in following references [1, 4, 5, 9].
This high strength level can be explained by the increased volume fraction of nano- and ultrafine grains (≈ 5%) directly related to the number of treated sheets in the stack, and by the work hardening induced during the process.
The latter depends on the number of passes imposed during the rolling process and on the reduction ratio.

The main features include: grain elastic properties, level of anisotropy and grain orientation, their morphology, grain size distribution within the aggregate, stiffness and toughness mismatch [2].
Grain boundary interfaces as well as cracks within the grains are modelled using cohesive laws which allow for nucleation and propagation of cracks under mixed mode conditions.
For each grain H=1,..., Ng, being Ng the number of grains, two integral equations can be considered: CijHx'ujHx'=SHUijHx',x tjHxdSH-SHTijHx',x ujHxdSH-ScrHTijHx',xc δujHxcrdScrH . (1) tjHxcr= niH(xcr)SHDkijHxcr,x tkHxdSH-SHSkijHxcr,x ukHxdSH-ScrHSkijHxcr,xcr δukHxcrdScrH. (2) Eq. 1 and Eq. 2 are respectively displacement and traction boundary integral equation used together in order to take into account the simultaneous presence of grain boundary and grain’s interior elements (where required). ujH and tjH are displacement and traction values while δukH represent displacement jumps, at intergranular or transgranular elements.
Fig. 1: Microstructures used for simulating effect of grain size on microcracking; from left to right: 30 grains (ASTM G=11.7), 50 grains (ASTM G=12.5), 100 grains (ASTM G=13.5).
The general trend suggests a more brittle and transgranular behaviour for coarse grains with a tendence to more intergranular failure for finer grains.

Nucleation rate and number of precipitates in V and Nb-microalloyed steels Sebastián F.
The number of precipitates was calculated by integration of the nucleation rate expression.
The interaction of these two phenomena has been widely studied by a number of researchers, who have considered different variables such as the steel composition, strain, strain rate, austenite grain size and deformation temperature, among others.
On the other hand, the integration of equation (1) would give the number of precipitates per unit of volume (N).
Precipitate number vs. time.

A new method of action on melts is proposed which is based on pulse-magnetic processing them with the aim to form a homogeneous fine-grained structure and improved technological properties of cast metal.
Microstructure of the alloy Аl-6%Si Analysis of the microstructure of the alloy Al-6%Si has shown that the grains are ground with the eutectic at the grain boundaries also becomes smaller.
The grain size and the quantitative evaluation of the structure are presented in Table 1.
The melts, subjected to PMP, immediately after introduction of silicon were processed also by the axial scheme (Fig.4a) at discharge energy of 1.5 kJ and with the number of pulses 1÷3 at temperature of 700°С.
The total processing time with the number of pulses n=3 was no more than 2 min.

The impedance spectrum of Dy-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 quinquevalence ions, such as Ti4+, Nb5+, Ta5+; and m means integer number > 1.
Dy-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 Dy-doped bismuth titanates consist of grain and grain boundary regions.

A larger number of electrodes and wiring materials mainly made of Ag, Cu and Al thin films are found in the control components of TFT-LCD.
In addition, the average grain size needs to be about 100μm, and the grain size and distribution are expected to be as uniform as possible [8-10].
When being annealed at 1373K, the grains of as-forged Mo target obviously began to grow, while the grain structures of Mo target prepared by other two modes are basically recrystallized grains and began to grow at 1423K.
Meanwhile, it can generate the impact force along the direction of forging which could effectively break the large grains into small grains, particularly break the large grains of as-sintered Mo slabs.
After being annealed at 1323K, grain structures of the Mo target with 80% total deformation were made of recrystal grains and grains began to grow from 1373K.

Table 1 Characteristics of abrasive used in experiment From Waterproof coating sand paper Nylon fabric pad Number 150# 320# 600# 800# 1200# 1500# 120# 240# W14 W0.3 Grain size (µm) 100 40 36 28 16 10 110 61 14 0.3 Results and Discussion Fig.2 shows the average of surface roughness (Ra) values before and after machining and extremum (min. and max.) of Ra after machining in different abrasives machining.
Sketch of experimental setup Fig.2 Roughness of machined surface the relation between the decrease in Ra of ground surface and abrasive grain size in fixed and loose condition, from which can be seen that drop of Ra decreases rapidly with the increase in grain size in abrasives machining, specially and interestedly, an obviously exponential relationship is found between drop of Ra and grain size in sand paper used .
The number of active abrasives and their size contribute to the active grains in unit area, so the increase in number of active abrasives or decrease in size of active abrasives or both above, may lead to a higher density of active 0 20 40 60 80 100 120 50 60 70 80 90 100 110 120 Decrease in Ra (nm) Grain size (µm) Loose Fixed Fig.3 Decrease in Ra vs. grain size 600# 800# 1200# 1500# 150# 320# Inimate (a) Fixed abrasives magnification coefficient X40 240# W14 W0.3 120# Inimate (b) Loose abrasive magnification coefficient X40 Fig.4 Morphology of machined surface abrasive grains, directly improve the machining force, resulting to a rapid decrease in Ra finally.
Deeply increase in the number of active grains in loose abrasive machining is gotten easier than that in fixed abrasive machining, which can be used to attribute to difference in drop of Ra in nearly size between fixed and loose abrasives used.
Higher number of active abrasives or smaller size of active abrasives or both them on unit area, lead to rapid decrease in Ra and improvement of defect on surface ground, finally achieving defect less transient surface after certain machining practice.