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(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.

Tensile strength under different heat input Number Heat Input /kJ·cm-1 Tensile Strength /MPa Average/MPa 1 7.5 915 946 930 4 9.6 864 884 874 5 11.5 835 805 820 6 12.0 785 815 800 7 18.0 713 707 710 Table 3.
The coarse grain zone in the heat affected zone is the zone between the two broken lines in the figure, and the lower part of the figure is the fine grain zone in the heat affected zone.
It can be seen from Fig. 4(b) that the grain size of the coarse-grained zone in the heat-affected zone is very coarse, and the austenite grain boundary is very clear.
The grain size of the coarse-grained zone can be determined by the cut line method to be about 31 μm.
It can be seen from Fig. 4(c) that the fine-grained grains in the heat-affected zone are fine, mostly equiaxed, and the microstructure is polygonal ferrite and quasi-polygonal ferrite.

The stress of 400 MPa at the crack end in the columnar grain region was about two-fold larger than that of 180 MPa in the equiaxed grain region.
The plane consisted of a columnar and an equiaxed grain region.
The maximum size of the columnar grains was 5 mm.
The collimator size and oscillation range were optimized to obtain a continuous Debye-Scherrerring from a sufficient number of grains.
Debye-Scherrer ring from a sufficient number of grains in an area of 9 mm2.

Taking this into account, in this article we have developed methods for modeling grain geometry, grain boundary surface and geometry of grain contacts of doped BaTiO3-ceramics.
The small-grained microstructure, with the grains tending to form clusters, was observed.
Denote the grain by G.
Choose wi such that h(T, T*) < e, where e is a given positive number and h is the Hausdorff distance.
Conclusion In this article we have developed methods for modeling grain geometry, grain boundary surface and geometry of grain contacts of doped BaTiO3.

Compared to the uniform deformation field of specimen, many more coarse original grains remain and new recrystallized grains form at the region in the vicinity of grain boundary in the large deformation field as shown in Fig. 4.
DRX is fewer in number and finer with the strain increase.
Cracks are more in number and larger at 20 s-1 compared to the strain rate 2 s-1.
So the cracks are more in number and higher in length in specimens of 20 s-1.
Cracks are more in number and larger at 20 s-1 compared to the strain rate 2 s-1.

a Stuart.Hampshire@ul.ie, b Michael.Pomeroy@ul.ie Keywords: Oxynitride glass, Silicon nitride, SiAlON glass, viscosity, grain boundary, creep Abstract.
As Y:Al ratio increases, fracture toughness also increases which is indicative of easier debonding at the grain interfaces [8].
A number of studies [1, 5-8, 11-14] have shown that oxynitride glasses have higher glass transition temperatures, elastic moduli, viscosities and microhardness values than the equivalent silicate glasses due to extra cross-linking within the glass network as a result of substitution of oxygen by nitrogen.
Overall, these effects can be assumed to be related to changes in the density of the glass network and the numbers of non-bridging oxygens as Al changes from a network ion in 4-fold co-ordination (AlO4) to a modifying role in 6-fold co-ordination (AlO6).
Those cations with ionic radius smaller than Y exhibit higher viscosities and should provide grain boundary glasses with higher creep resistance.

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.

These excellent properties are resulted from the grain size reduction and the existence of large numbers of grain boundaries in the microstructure [1].
The grains are large and tend to agglomerate.
For CoFe nanocrystalline coating prepared at 90 minutes deposition, the small grain size is due to the existence of higher number of interfaces such as grain boundaries in the coating microstructure.
The higher proportion of boundary atoms in the grain boundaries compared to those inside the grains has cause the grain size reduction in CoFe coating.
The grain size reduction has resulted in the high volume fraction of the boundary atoms inside the grain boundaries.

The mean grain-lamellar colony size 512 mm was transformed to fully lamellar structure containing g and a2 phases having the mean grain size 229 mm.
Grain size was calculated from the measured average grain area.
Only at grain boundaries (see Fig. 3b) discontinuous coarsening of new gamma grain with thick lamellae morphology can be observed.
Eutectoid transformation affects grain size only negligibly (discontinuous coarsening).
The principal effect on the grain size has recrystallization in the alpha phase region. 3.

According to Callister [10] the particle growth is a result of the movement of grain boundaries, which is conducted through two processes: grain boundary diffusion and migration of grain boundaries.
Both processes promote the densification, but the grain boundary migration that occurs at a higher temperature promotes faster grain growth.
Measurements of average grain size and interfacial area per unit of volume were carried out (Figure 5) [13], counting the number of intersections between the grain boundary and straight lines with known length, which were designed on the image with the program ImageJ of free access.
The number of intersections per image was more than 400, aiming to achieve better measurement accuracy.
(Numbers denote the logarithm of frequency).