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However, the effects of grain size in the very small grain range on the mechanical properties, especially fracture mechanism, are not fully clarified.
In nano-crystalline models, the larger grain size model is called analysis Model 2- (L), and the smaller grain size model is called analysis Model 2-(S).
The crystal orientation of the grain, which contains the initial crack-tip position, is the same as that of Model 1-(b), and all grains are oriented in the [100] direction to the thickness direction.
We choose the same crack orientation and the same grain shape around the crack in both grain-size models in order to exclude the influences from other than grain sizes (see Fig. 6 (i) (a) and Fig. 6 (ii) (a)).
Therefore, the crack opening and extension are affected by the grain size.

The film deposited from SiO2(CuOxSnOy) sol revealed grains with different sizes and grain boundaries.
The grain size of the film is found to be in the range of 150 to 500 nm.
А large number of well defined regular shaped grains with good crystalline nature are observed as shown in fig. 1 (b).
It is also observed that small grains agglomerate to form larger grains.
In any case, by adding the SnCl4 into the sol the grain shape changes from flower-like to regular shaped.

(011) (110) (011) [111] M Shear direction 0.2 μm (b) diffractive pattern (a) BF Image Figure 3 TEM micrographs of deformed shear band (112)Fe3C (200) M (b) diffractive pattern (a) BF Image Figure 4 TEM micrographs of transformed shear band Figure 4 (a) is a BF image, showing that there are a large number of very fine equiaxed grains, about 100-200nm in diameter, with a low density of dislocations.
Meyers and Pak [8] proposed that such equiaxed grains in the center of shear band could be the result of dynamic recrystallization.
In other words, the recrystallization machnism via grain boundary migration based on atomic diffusion should not be the case for the formation of the fine equiaxed grains in the adiabatic shear banding here.
The center of the transformed band is composed of very fine equiaxed grains with a low dislocation density.
However, the formation mechanism of the fine equiaxed grains in the adiabatic shear banding is still unclear and a further study is necessary.

At the same time, the inner and outer surfaces of the sample are grayish black, and there are a large number of oxide scales.
In addition, the microstructure grains are coarse and the cracks are different to some extent.
The grain size at the crack is relatively large, with an average grain size of 3.84, and plastic deformation occurs obviously.
The grain is significantly elongated to the crack tip.
The other locations are equiaxed grains with an average grain size of 3.31 at the inner wall far from the crack, 2.97 at the center and 3.05 at the outer wall.

A fine-grained matrix with coarse particles of the lead phase and eutectoid, distributed mainly along the grain borders, is obtained after solidification.
The structure after crystallization represents coarse grains of a-phase, lead and eutectoid impurities, which are uniformly distributed inside the matrix grain.
a) lead impurities matrix grains b) lead impurities matrix grains Figure 2.
When the temperature of the casting rises, the grains of lead and eutectoid are refined, and the matrix grains become coarse.
Brinell hardness number is the highest for the bronze, casted in a chill mould at 900°С, and the lowest for the bronze, casted in aluminum-chromium-phosphate moulds at 1220°С.

The RETiNbO6 compounds RE atomic number 57-63 are reported [6-9] as orthorhombic with space group Pnma, Z=4, and aeschnite structure.
The compounds with RE atomic number 64-71 show a euxenite structure, having space group Pcan.
So, grain sizes, densification and porosity are the main effect factors.
With the density increased there were less porous and grain sizes is the main factor when the sample is fully dense [14].
There are many pores on the LaTiNbO6 grain surface as well as abnormal growth and it deteriorated the microwave dielectric properties of samples.

The increase of Bi content in the coatings causes grain refinement effect.
A fine grain size can improve the yield strength and hardness of the plated coating by increasing the chance of stopping the dislocation movement at the grain boundaries.
The synthesized coatings show globular surface morphology, and the increase of Bi content in the coatings causes reduction of the globule size and the number of pores.
Stability of nanocrystalline metals: The role of grain-boundary chemistry and structure[J].
On the analysis of grain size in bulk nanocrystalline materials via x-ray diffraction[J].

Number of the X-ray diffraction data is 3850.
These grain boundary phases can impede growth of TiB2, and help obtain materials with fine grain structure.
Fi : the Drive force of grain boundary.
The grain boundary phase can reduce the grain boundary diffusion coefficient remarkably [9].
Acknowledgements The authors are grateful to The National High-Tech Research and Develop Project (863) for this work under Grant number: 2001AA333020 Reference [1] J.Casting, P.

The mean grain size is reduced to 10.2±0.6 μm.
After ECAPed 4 passes, the grain size was significantly reduced to 5.3±0.4 μm, and small equiaxial grains can be found in black region of β phase.
It is shown in Fig.3a that, the resulting grain size after ageing at 150oC for 30h is slightly increased compared to that of ECAP-4P in Fig.2c, but the grain sizes are more uniform.
A large number of twins can be found in grains.
The number of twinned grains in deformed specimen after 300oC ageing is greater than that after 150oC ageing, implying the contribution of continuous precipitates to induce twinning.

Fig.4 micrographs evaluate that the microstructure became finer and the number of pores (black dots on the microstructures) were reduced after aluminum modification.
Additionally, the coarse elongated α-Al grains were significantly refined into fine equiaxed grains with eutectically distributed lamellar Si grains.
Grain boundary (a.)
and Al grain (b.).
Kung, Grain Refining in A356 Alloys, J.