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A variety of models have been proposed to interpret this relation 1-5, and a number of experimental investigations have studied dislocation behavior in the vicinity of grain boundaries 6-10.
The grain boundary is indicated by arrow-heads.
As the indenter penetrated further, a large number of dislocations were emitted on the far side from the indenter tip (left and lower side on the micrograph) of the grain boundary into the adjacent grain.
Li: Petch relation and grain boundary sources.
(a) low-angle grain boundary and (b) high-angle grain boundary.

This precipitates could avoid the grain growth during subsequent annealing treatments [6].
In this condition, the microstructure is essentially composed by equiaxed austenitic grains within stacks of thin martensite plates, annealing twins and a number of large precipitates.
These precipitates were located mainly at the triple junction and grain boundaries.
The deformed grains present some amount of stress induced martensite plates and shear bands.
Kaufman, Influence of grain austenite grain size on mechanical properties of stainless SMA.

As mentioned above, subgrains were frequently found inside some larger grains.
Micrographs of a larger grain and a subgrain boundary within the larger grain in the HPT Al-0.5Mg alloy: (a) TEM bright-field image of the larger grain; (b) HRTEM [110] image taken from the right frame in (a), showing the sub-boundary formed within the large grain and a SF appearing nearby the boundary; (c) inverse Fourier image from the frame in (b), showing 60° perfect dislocations at a sub-boundary.
A large number of excess dislocations for slip trapped at non-equilibrium GBs (Fig. 3c), as well as dislocations stored near the grain boundaries (Fig. 4), can facilitate grains to slide or rotate at room temperature, and therefore increase the ductility [27].
In the ultrafine-grained (UFG) regime, i.e. grain size in the 100−1000 nm range, the traditional dislocation mechanisms may still remain dominant in controlling plastic deformation.
In the lower nanometer regime having a grain size below 10 nm, deformation processes could be controlled by the grain boundaries, e.g. such as grain rotations and grain boundary sliding.

If it is assumed that nucleation only occurs at grain boundaries, then a given number of nuclei per length of grain boundary in a fine grained material will lead to more homogeneous recrystallisation than the same number in a coarse grained material.
Therefore, a large initial grain size provides fewer nucleation sites due to the reduction in grain boundary area per unit volume.
Note that after a reduction of 50%, the original fine-grained (A) material gave an appreciably finer grain size after recrystallisation than the original coarse-grained (C) material.
However, after a 70% reduction, the grain size after recrystallisation was similar regardless of different initial grain size.
The recrystallised grain size decreased markedly with increasing deformation and decreasing initial grain size.

The second analysis was based on the number of grains located on the active surface of the tool - calculation of abrasive grains density [8].
As expected, the number of grains was lower after machining than before, what is shown in the images of the tool active surface.
The number of abrasive grains B64 and B107 on the active surface of electroplated tools are presented in Fig. 6 and Fig. 7.
The number of abrasive grains B64 has decreased by 44% for area #1, 56% for area #2 and over 70% for area #3 - Fig. 6.
The second analysis was based on the number of grains located on the active surface of the tool before and after machining.

Recently, ultrafine grained microstructures with a grain size of less than one micrometer were obtained after annealing of severely deformed alloys [3-6].
Despite the large number of studies on continuous recrystallization operating in various metals and alloys processed by large strain deformations, some characteristics of this mechanism, e.g. texture evolution, are not known in detail.
In this case, the grain coarsening takes place homogeneously without preferential growth of any individual grains.
Number Fraction, Ni / N 0.0 0.1 0.2 0.3 0.4 0 10 20 30 40 50 60 0.0 0.1 0.2 0.3 0.4 Misorientation, θ (deg) 0 10 20 30 40 50 60 0 10 20 30 40 50 60 ε = 2.0 ε = 4.4 ε = 2.0 625o C, 30 min ε = 2.0 625 oC, 2 hours ε = 4.4 625o C, 30 min ε = 4.4 625 oC, 2 hours Fig. 6.
The development of primary recrystallization associated with grain boundary migration over long distances results in a decrease in the total number of low-angle subboundaries, while the fraction of high-angle grain boundaries increases.

Voronoin method[6,7] is a common method to construct nano-polycrystalline materials: First, a number of points whose Z coordinate are controlled to obey the linear probability density function are randomly generate as the center of the grains,and the distance between grain centers is adjusted to avoid the two grain centers being too close.Then, calling the Voronoin function of MATLAB to generate the grain boundary topology of GNG model.Finally, a C++ script from the literature[8] was developed to fill each grain with the face entered cubic (FCC) copper atom lattice.
The initial positions of crack tips in the three samples are in the grain, along the grain boundary, perpendicular to the grain boundary, respectively.
(4) Figure 2 Schematic diagram of relative crack calculation In the process of tensile deformation, as Fig. 3 shows, a large number of hexagonal close packed (HCP) structures occur in the interior of larger grains and near the crack tip, which reflects the intense dislocation activity and a large number of stacking faults in these two regions.
Heterogeneous lamella structure unites ultrafine-grain strength with coarse-grain ductility, J.
Fracture behavior of precracked nano-grained materials with grain size gradients, J.

The number and the density of surface defect were increased.
CURM then produced a large number of sub-grains.
With the strain increasing, dislocation cells increased in the number and reduced in the size.
Finally there are many sub-grains within the grain, the size of grains on the metal surface is smaller.
It needed greater resistance to transform, which played the role of grain boundary strengthening, Smaller grain size in the same volume needs the larger number of grains in the same deformation.

The strong elastic stress accumulated by shearing texture is responsible for oriental growing of a great number of grains during recrystallization.
A number of unusual properties have been reported in ultrafine grain materials produced by severe plastic deformation (SPD).
AARB effects on grain refinement.
From fig.3(a) and (b),we can see that gains were squashed and there were subgrains which have low angle grain boundaries appear.This was because that friction of inner grains and grains was intensified,and grains in the place of grain boundaries occurs slide,and crystal plane in part grains occurs glide when rolled pieces pass crossover area,meanwhile,because of the effect of AARB,2/3 large grains transform numerous subgrains which have low angle grain boundary ,i.e.2/3 large grains were broken into numerous small subgrains whose size are about 5um;from fig.
The strong elastic stress accumulated by shearing texture is responsible for orientable growing of a great number of grains during recrystallization.

This confirms that large grains are already recrystallized after ECAE. 0 5000 10000 15000 20000 30 50 70 90 110 130 Image quality Number ECAE 30s 7min30s Fig. 5 : Number of indexed points versus image quality.
Inspection of grain size distributions shows that, after 30s annealing, the number of the smallest grains (<0.5 µm) has slightly increased whereas the number of large grains (>1.5 µm) has remained constant (Fig. 6a).
Indeed, the average size of dynamically recrystallized grains (IQ>70) remains constant and equals 1.4 µm after 30s annealing. 0 100 200 300 4000.1 1 10 Grain size (diameter) (µm) Number ECAE 30s 7min30s (a) (b) Fig. 6 : a) Evolution of grain size during copper annealing at 473 K, b) Distribution of image quality of small grains (<0.5µm) after 30s annealing.
After 7min30s annealing, the number of large grains increases at the expense of the small ones.
As already suggested by the EBSD analysis and the hardness curve, this result confirms that, after 30s annealing, the copper is still recovering and the number of new oriented recrystallized grains, is not sufficient to change the texture.