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The nonmagnetic phase in the grain boundary can reduce the actual DC-bias superposition in the grains when the sample was subject to DC-bias superposition.
For sample 2 doping with 0.75wt% Bi2O3 which has a dual microstructure with some very large grains surrounded by many small grains.
For NiZn ferrite, when the grain size D≤2.8um [7], the domain size is equal to D, many grains present mono-domain states.
On the left side, a microscopic description has been chosen, where different internal fields in the grains (H2) and grain boundary (H1) are considered.
(3) where N is the number of wire turns, I is the current, is the equivalent length of the cores.

XRD, FESEM and EDS results showed that the solidified TiC–TiB2 were composed of a number of TiB2 primary platelets, irregular TiC secondary grains, and a few of isolated Al2O3 inclusions and Cr-based alloy.
Ferber [2] has pointed out that the micro-cracks in the material can initiate rapid crack propagation if the size of TiB2 grains was larger than 15 µm, thereby deteriorating sharply mechanical properties of the materials.
Recently, a rapid and economical processing, i.e. combustion synthesis in high-gravity field has been taken to prepare high-hardness bulk solidified TiC-TiB2 composites, and high performance of TiC-TiB2 composites are achieved due to the presence of fine-grained microstructure [3-4].
FESEM images and EDS results showed that a large number of randomly-orientated, fine TiB2 platelets were uniformly embedded in or distributed around irregular TiC grains, and discontinuous band of Cr metallic phases were located between TiC and TiB2 crystals, whereas a few of α-Al2O3 inclusions were also observed in the form of the isolated particles, as shown in Fig. 2.
XRD, FESEM and EDS results showed that the solidified TiC–TiB2 was composed of a number of TiB2 primary platelets, irregular TiC secondary grains, and a few of isolated Al2O3 inclusions and Cr-based alloy.

Results and Discussion The main microstructural factors closely examined here are the grain size, solid volume fraction, tungsten-tungsten interfacial fraction (contiguity) and number of solid-solid contacts per tungsten grain (Connectivity).
The mean grain size slightly increases with the effective sintering time due to ostwald ripening.
The contiguity (the fraction of internal surface area of the phase shared with grains of the same phase in two phase alloys) and connectivity (the number of solid- solid contacts per grain) are the mean values for all grains.
It can be seen that the grain size, contiguity and connectivity is decreasing with wt% oxide impurity content.
Higher contents of oxide impurity reduced the tungsten grain size at the same sintering temperature and time.

(1) Where 0 0 90 is the cone-apex angle of a grain KId is the dynamic fracture toughness of the ceramics material (GPa); H is the Vickers diamond hardness of the ceramics material (MPa); =1.8544 is the geometric factor of a grain and 4 0 10)6.1~0.1(   is coefficient.
y x Diamond wheel x,y two-domensional ultrasonic vibration ,fA ,fB Worktable wv n a m p l i t u d e A w i d t h d e p t h A b r a s i v e g r a i n s u r f a c e d e p t h 工 件 s u r f a c e W i t h u l t r a s o n i c s w o r k p i e c e W i t h o u t u l t r a s o n i c s h u m p c o n  u l t  Fig.1 Illustration of the experimental set-up Fig.2 A grain motion model of with and without ultrasonic According to the grinding mechanics, the ratio of the brittle material removal volume unit time and the number of effective abrasive particles in the working area is the average volume of cutting lay in WTDUVG , so the average cutting thickness of a grain in WTDUVG can be expression: ds Pw sgg bNv bav lba  
(2) where, ga  is the average cutting thickness of a grain, gb  is the average cutting width of a grain, ls is the grinding arc length vw is the velocity of the workpiece (m/s), ap is the grinding depth (mm)；b is the grinding width of wheel (mm), vs is the the spindle speed (m/s) and Nd the number of effective abrasive particles in the working area (mm-2).
Because of the number of effective abrasive particles in the working area increases with the larger of worktable speed, the maximum cutting thickness increased, As Fig.4(c) shows that brittle regime grinding of ceramics can be realized, when the maximum cutting thickness of a grain more than critical depth of cut.
The equations for the maximum cutting thickness of a grain of vibration grinding deduced and verified.

The grains on the top surface of Fe3Al are refined to nanocrystallines and the grain size of nanocrystallines is about 35nm.
It is obviously that the grains were refined to nanometer range and the grain boundaries of nanocrystallines were blurry.
The grain boundaries of these ultrafine grains were wide and high density dislocations piled up among the grain boundaries area.
The grain boundaries with high density dislocations subdivided the ultrafine grains.
A large number of dislocations are induced by high strain.

It was found that the grain growth was restrained with the addition of Ba2+.
It is widely accepted that the transport properties in these polycrystalline ceramic varistors are mainly controlled by the electrically active grain boundaries between the semiconducting ZnO grains [4].
The l˚Cal defects on the grain boundary regions are ionized on both sides of the grain which gives rise to the depletion region.
The average grain size (l) is determined by the lineal intercept method, given by l=1.56L/MN, where L is the total line length on the micrograph, M is the magnification of the photomicrograph, 1.56 is the proportionality constant, and N is the number of the grain boundaries by lines[7].
This is mainly attributed to the increase in the number of grain boundaries caused by the decrease in the ZnO grain size.

However, the hexagonal lattice of magnesium and magnesium alloys is the reason behind the small number of slip systems, which make the cold deformation difficult or even impossible.
It consists of a large number of lath shaped grains with a length of 3 μm and a thickness of 0.4 μm, as well as very fine equiaxed grains with a diameter of 0.5 μm.
The smaller grains of deformed microstructure favor more uniform distribution of the grain size.
Hence, the number of ECAP passes had an influence on refining microstructure in the semi-solid state.
The grain growth in alloys at temperatures of the semi-solid range is caused by a number of phenomena including coalescence, coagulation as well as Ostwald ripening [2, 3].

The number of “effective” crystallization contacts is determined from the correlation of approached particles sizes and a number of particles of different size in a dispersive system.
The optimal structure providing maximum of contacts is formed under the condition of one grain of small size being placed between grains with greater sizes.
Small sized grains fill in porous space like liquids since they easily pass through big-sized.
The optimal granularity ensures the most compact packing in a unit space with maximum of contacts of varying-sized grains(see Fig. 8 a).
The equations for the coordination number in disordered systems.

Grain size.
Total number of cell state variables.
depends on the total number of cells in each array, and .
The distribution of grains size was analysed in terms of the ferret diameter, and Weibull analysis was performed to obtain Weibull grain size parameters, in order to incorporate the distribution of grains size in CAFE model using random number generators.
A misorientation angle of grains was incorporated in the brittle cells of CAFE model using random number generators, with following Weibull parameters: shape parameter = 1. 1931, scale parameter = 39.832.

The grain is defined as the group of dendrites, which grew in the same direction.
Following this definition, it is found that the crack propagated along the grain boundary, also the segregation line was observed on the grain boundary.
Thus, the number of sharp peaks (�v) and the length of hot tear or segregation line as a function of copper content have been analyzed.
On the other hand, in the case of high solute content, such as Al-10 mass%Cu in this study, the number of crack formation is high but the length of propagation may be short.
Thus, hot hear indicates open crack. 3) In the case of Al-10 mass%Cu alloy, the number of crack formation is high and the crack may be healed by residual liquid.