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With the increase of the number of deformation passes, refinement effect becomes weakened gradually, the grain size tends to stabilize and the organization is more uniform.
Currently, this technology has been successfully used to refine grain of pure metals of aluminum, copper, but the study on the grain refinement of alloy is rarely reported.
Initial grain size of the material: 90.
The grain size is refined from the initial 90 to 31 or so, but the grain size of the billet is inhomogeneous.
The grain size becomes more uniform in the follow-up passes.

To apply a numerical method with a series of ordinary differential equations, the material parameter w is used for the model parameter, and the number of nuclei can be expressed as,where is the maximum number of nuclei (new grains) at the current condition of deformation
Then the average size of the new grains can be calculated from the following equation .The average grain size of the new and old grains during recrystallization can be calculated on the basis of the new grain size and the initial grain size ,.
The model is only used for non-recrystallized grains.
(8) where is the number of curves, is the curve number, is the weight number of the curve with number , is the point number, is the number of points on the curve with the number , is the measured flow stresses for the point number in the curve number , is the calculated flow stress for the point number in the curve number , and is the maximum value of the flow stress for the curve number .
The grain size changed little at this point.

It is established that the grain boundaries in coarse-grain copper have significantly lower relative energy in contrast to the grain boundaries of ECAP-treated copper.
Additionally, the curves obtained using the proposed model are shown, highlighting the individual groups of grain boundaries that represent the overall distribution, numbered 1 to 5 The application of the proposed model, shown in Fig. 1, made it possible to approximate the experimental distributions with sufficient accuracy.
It was found that grain boundaries in coarse-grained copper have significantly lower relative energy as compared to grain boundaries in ECAP treated copper.
Acknowledgments The research was carried out within the State Assignment (theme “Function”, State registration number AAAA-A19-119012990095-0).
Rabkin, Relative grain boundary energies in ultrafine grain Ni obtained by high pressure torsion, Scr.

The crack lies at the interface of two adjacent NC grains with the crack tip intersecting at the grain boundary of the coarse grain.
Let us calculate N--the number of dislocations emitted from a crack tip and retarded at the opposite grain boundaries of the coarse grain.
Fig.5 The maximum number N of edge dislocations that can be emitted from the crack tip along one slip plane as a function of coarse grain size D in NC copper.
It is because that the number of emitted dislocations N increases with increasing the coarse grain size D which can prove more shielding effect on crack.
The dependence of both the maximum number of dislocations, emitted from a crack, and the critical stress intensity factor on grain size d of the NC matrix (ranging from 20 to 100 nm) as well as to the coarse grain size D (ranging from 1 to 10 μm) for Cu were calculated.

It can be caused by the fact that a standard grit number is defined in terms of grain sizes corresponding to five sieves [3].
The grain distribution standard allows a wide variation of the number of grains of various size fractions in the grain distribution of the same grit number.
So, the actual grain size distribution of close grit numbers can be almost identical.
As it is established by our experiments, the number of such cutting edges on the working surface of an abrasive tool depends on the material type.
The values of the BP and NP parameters depend on the grit number (grain size), the sieve-shaking procedure of abrasive grains and research depth. 2.

A Brief History of the Grain Refinement of Cast Light Alloys D.H.
Today we understand the main factors that lead to good refinement for all alloys systems: growth restricting solute [5,6] and potent particles of optimum particle number density and size distribution [7].
Prior to the 1990s a number of theories were developed [1] to explain grain refinement observations (Table 1).
Reports of good grain refinement.
Prediction capability needs further refinement to quantitatively take into account the simultaneous increase in solute Zr and Zr particle number density.

So far we have established that the rate of grain growth follows the usual L ~ t1/2 scaling law when the grain boundary energy is independent of the misorientation angle between neighboring grains.
In the standard phase field model the number of different grain types decide the dimensionality of the orientation field vector.
Where as here, irrespective of the number of grain types, we time evolve a 4 dimensional vector only.
Once grains are formed further evolution is driven by the interfacial energy between grains i.e., grains grow in size via coarsening (Fig.1) and average grain size follows the usual t1/2 law (Fig.2).
During coarsening, a quaternion at the interface between two grains is forced by noise to lift off from the minima of one grain and fall into the minima of a neighboring grain; thus the neighboring grain gets bigger at the cost of another grain.

It is observed that grains with more sides than a critical number (equal to six in two dimensions) grow while those below the critical number shrink.
(1) In two dimensions, the rate of change of grain area (A) with time can be related to the number of grain edges (n) (Eq. 2).
Grains having edges greater than six grow and grains having edges lesser than six shrink [12,13]
In the present model, a two-dimensional square lattice with size either 256x256, or 512x512 is populated with a specific number of grains (1000 and 2000 grains respectively) by using a Voronoi tessellation routine, which produces an initial microstructure with near-random distribution of boundary inclinations.
(5) Only those neighbors contribute to the Hamiltonian which are of unlike spin number [1].

However, the relationship between the intensity of the Bauschinger effect and the grain size that varied with the number of passes of ECAP was not shown clearly.
Distributions of misorientation angle after various numbers of ECAP passes.
Subsequently, the yield and flow stresses slightly increase with the number of ECAP passes.
Fig. 6(a) shows the changes in the proof stresses and with the number of ECAP passes for the TC and CT tests.
Results after different numbers of ECAP passes: (a) proof stress and ; (b) average grain size and .

We postulated a softening model involving grain rotation that results in diffusion-accommodated grain-boundary sliding.
Shan et al. [10] also confirmed the presence of visible grain orientation and deteceted that grains grew into an elongated equiaxed shape as a result of grain orientation.
Here we consider a two-dimensional hexagon ranged structure element that contains a large number of grains.
Then denotes as the number fraction of grains having the soft orientation aligned with the shear direction, where is the number of grains that have the soft orientation aligned with the shearing direction.
Homogenized representation of grain rotation.