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Form the model we can found that the larger the ultrasonic power and amplitude, conditions can be more easily satisfied; simlarly the higher the ultrasonic frequency, angle crossing number, grain structure defects, the longer action time to make greater fatigue, all this factors can also make the conditions more easily satisfied.
Further expand the Eq. 6 as δ ϕ ϕ ≥ ∂ ∂ + ∂ ∂ ),( )( )( ),( xtp x x x x xtp (7) and assuming the ultrasound as a plane harmonic longitudinal wave that pass through the grain spread along the x-axis positive, then the displacement fluctuation function can be written as )cos(),( kxtAxtp − = ω (8) where A is the input displacement amplitude of ultrasound, ω = 2πf is angular frequency, f is ultrasonic frequency, k = ω/c0 = 2π/λ is the angle wave number, λ is the wavelength of the ultrasound transmit in the grain, respectively.
In the factors that affecting the conditions of dendrite fragmentation, amplitude and angular wave number depend respectively on the ultrasonic power and frequency, while the variance ratio of grain cross-section M and the shape function φ(x) depend mainly on the defects of grain shape and size along the ultrasonic propagation direction.
From Eq. 12 we can found that the larger the ultrasonic power and amplitude, conditions can be more easily satisfied; simlarly the higher the ultrasonic frequency, angle crossing number, grain structure defects, the longer action time to make greater fatigue, all this factors can also make the conditions more easily satisfied.
Conclusions This article elaborated the forced movement of the particle within grains and its influence on the integrity of the grain itself, established the mathematical theory model of ultrasonic grain crushing.

Indentation was made at various positions, that is, on the grain boundary and about 210μm distant from the grain boundary.
As-aged specimen, not electropolished, showed low hardness near the grain boundary.
According as the surface was removed successively, the hardness number increased, particularly near the grain boundary.
After a removal of 100μm total thickness, the hardness became independent of the position of indentation relative to the grain boundary.
In Fig.2 stress amplitude is plotted against the number of cycles to failure for the Al-12%Zn alloy heat-treated in the same way as above.

Plasticity and fracture mechanism of refine-grain AZ31 magnesium alloy sheet was discussed.
Research has shown that the organization of the annealing processing, the second phase number is greatly reduce in the substrate and in grain boundary [4-12].
The relationship of annealing temperature and grain size as shown in Fig. 3 at a certain annealing time, along with the annealing temperature increases, annealing grain size increase gradually.
This is because the relationship between the interface migration rate B (grain boundaries migration velocity was produced by units driving force) and the grain boundaries diffusion coefficient D grain boundaries is[14]：B=D grain boundaries /KT.
Grain boundaries diffusion coefficient increase with temperature into index [D = D0exp (Q grain boundaries P/KT)] ，Because its effects exceeds 1 / T in the type, with the rise of temperature the speed of grain boundaries move increases rapidly, and the trend of the grain growth also more and more quickly.

Since the grain sizes vary with current density, the grain density defined as the average grain number per cm2, as a function of current density can be derived, as shown in Fig. 5.
Fig. 5 The average grain number per cm2 vs. current density curve.
According to Fig. 6(a), when applying ACP during liquid inoculation stage, the macrostructure is coarse columnar and equiaxed grains, the grain sizes are larger than that without ACP.
The grains are remarkably refined, and its grain sizes are almost no different from that applying ACP with density of 230 Acm-2 during the whole solidification process.
The grain is not refined compared with macrostructure without ACP.

The sample 1, of which the RE content was 0.0048 wt.%, was also ferrite, but duplex grain structure, there were ferrite with small grain size along the original grain boundaries.
(a) and (b) Ultrafine grain size ferrite at grain boundaries.
First, there were large numbers of dislocation walls, as Fig 4a shown, and this was typical specialty of the recovery and recrystallization.
(b) Clean grains with few dislocations.
(c) Second phase at grain boundaries.

The average grain sizes (d) were determined by the linear intercept method, given by d=1.56L/MN, where d is the average grain size, L is the total length of test line used, N is the number of intercepts, and M is the magnification of the photomicrograph [12].
Zn2TiO4 assembled grain boundary and inhibited the grain growth of the ZnO varistors ceramics.
Larger average grain size means less grain number per unit thickness, which resulted in lower potential gradient.
Although increasing the amount of TiO2 addition can promote the grain growth of Zn-Bi varistors ceramics, more TiO2 addition will induce the abnormal grains, which results more small grains.
Therefore, the average grain size decrease and the grain boundary layers per unit increase, the potential gradients also increase.

(a) (b) Nucleus Frequency and Orientation The recrystallization kinetics as well as the grain size distribution strongly depend on the number of grains nucleating at each particle.
The number of new grains observed in the vicinity of particles was counted, and the dependency of nucleus frequencies on particle size was determined (Fig. 3b).
Fig. 8 (a) Grain size in dependence of the nucleus frequency; (b) Grain size vs. number of nuclei per particle for different particle volume fractions.
The presence of a deformation zone was considered as well as the number of nuclei per particle.
The grain size decreased in a non-linear fashion with increasing nucleus frequency at particles and especially for low nuclei numbers as well as low particle volume fractions.

The first type is the grains whose c axes parallel to the tensile direction of sheet (grains with this orientation are called P grains here for short).
The c axis of grains of this type tends to parallel to the normal direction of sheet (grains this orientation are called N grains here for short).
N grains have a small proportion.
P grain tends to show tensile twinning and N grain tends to compressive twinning
Acknowledgements The authors express their sincere thanks to the support of the Natural Science Foundation of China with the Grant Number: 50775211 and the cooperation with Prof W.

Over 1500 grains were analyzed for each UFG sample.
Distributions of grains on size and grain boundaries (GBs) between the adjacent grains on their misorientation angle q were determined from the EBSD maps.
The samples in the initial coarse-grained state with a grain size of about 100 µm failed.
The samples in the coarse-grained state with a grain size of about 100 µm failed.
Edalati, Room-temperature superplasticity in an ultrafine-grained magnesium alloy, Scientific Reports. 7 (2017) Article number: 2662

The Dynamic Behavior of Ultra-Fine-Grained Copper Fabricated by Equal Channel Angular Pressing Y.C.
Dalla-Torre et al. [6] reported the SRS of pure copper increased from 0.007 to 0.023 along with increasing the number of passes of equal channel angular extrusion (ECAE) from 1 to 12.
This phenomenon could be explained by a heavy increase of dislocation density along with a small number of ECAP passes.
Flow stress as a function of passes after ECAP deformation Fig. 5 demonstrates the variety of flow stresses versus the number of the ECAP passes, which shows that the flow stress was growing along with the accumulation of number of ECAP passes at any strain rate, and the strength enhanced with the increasing strain rate.
They showed that strain rate sensitivity increased with the reducing of grain size, although their studies have been focused on the coarse grain sizes.