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By doing this, it evades the fact that, for example, a metallic material contains a large number of defects such as dislocations that can be organized into heterogeneous microstructures and partitioned into phases and grains with various orientations.
It also ignores the fact that the plastic deformation is produced on some crystallographic planes in small numbers: the slip systems.
In the case of tensile test, the value of the strain calculated with a reference length must include a minimum number of adjacent grains.
This effective distribution and the restriction of plastic slip upon a limited number of crystallographic planes partition the material in different volumes more or less distorted.
This modeling, with a sufficient number of compartments, is potentially able to give back complex experimental evidence related to mechanical properties of materials.

Despite the many positive properties of vermiculite, it does not find widespread use in building materials and products for two reasons - weak adhesion to smooth surfaces of grains of vermiculite, basically, to cement by binding and low mechanical strength - the fragility of grains of vermiculite owing to weak communication between separate it records. [13–16].
They make up about 75% of the mass of the Earth’s crust and a third of the total number of known minerals.
Extrusion of the grain of vermiculite at temperatures of 950, 1000 and 1050 °C, respectively At the bottom of the photo the time is indicated.
This type of swelling may be explained by the fact that the vermiculite grains have an unequal degree of vermiculization in individual areas, and, accordingly, a different capacity for swelling.
For the preparation of ceramic charge was used expanded vermiculite - Tatar and Chinese - with a grain size of 0.63 mm or less.

This feature makes the program effective for a large number of steel grades with a wide range of chemical composition variations.
Fig. 4 shows predicted variation in austenite grain size and retained strain over the strip width of Grade #1 steel with time of rolling.
Grain size, μm Retained strain Thickness, mm Thickness, mm Time, s Time, s a) b) Fig. 4 Predicted variation in austenite grain size (a) and retained strain (b) distributions over the strip width in Grade #1 steel with rolling time.
This model was calibrated using a mechanical properties data base for a number of practically important steels hot rolled on mill 2000 under different rolling regimes.
The model STAN 2000 has greatly contributed to the development of new technologies for production of pipe steels, high strength steels for mechanical engineering and shipbuilding, as well as a number of structural steels.

The samples used in this study are hereafter referred to as N20, N100, N2000 and Nf using number of cycles.
It is seen that elastic strain demonstrates the same trend as the stress amplitude with increasing of numbers of cycles (Fig. 1a and 2a).
A limited number of perfect dislocations were found in the alloy deformed to 20 cycles.
With further cyclic straining, all grains are involved in deformation; the slip bands and deformation induced ε-martensite are visible in most of these grains (Fig. 3e and 3f).
The martensite formed mainly in single direction was found in the grains with the unidirectional slip and martensite intersected in multiple directions was found in the grains with high density of intersected slip bands.

The grain size is level 9.5, and the mechanical properties of them are similar.
Table. 3 Surface roughness for three coarsening plates um plate number Ra Ry Rv Rp Sm Pc PPI 1＃ 0.98 7.78 3.84 4.58 205 31 4 2＃ 1.08 7.25 4.23 3.61 122 39 7 3＃ 0.9 7.47 4.81 2.79 74 59 9 Note: Pc- peak number / cm, Sm-average spacing of contour microscopic roughness.
Multiple crystal grains are generally surrounded by the other grains.
Consequently, its deformation needs the mutual coordinate of the adjacent grain.
Otherwise, it is difficult to deform, and to keep continuity between grains, resulting in gap and material fracture[2].

The interaction energy between cation and anion can be approximate calculated according to Eq. 2: E = [Z1Z2/n(r1+r2)]e2 (2) where Z1 and Z2 are the electrovalences of the cation and oxygen ions, r1 and r2 are the radii of the cation and oxygen ions, e is the electron charge and n is the coordination number of the cation.
MgO coating layer on the CaO grains.
As can be seen, CaO grains agglomerated in mixing and sintering processes after adding MgO powders.
MgO powders formed coating layers on the surface of CaO grains, which could effectively reduce opportunities for CaO grains to contact with water vapor outside, so that the hydration resistance was enhanced.
Also, MgO layers coating on CaO grains were more compact and the particle feeling of MgO coating layers was gradually reduced, thus the hydration resistance of CaO reduced effectively.

A refinement of the grain and an increase in the microhardness of the surface layer similar for the investigated materials were revealed.
At a depth of 40 ... 60 microns, there was a significant refinement of the grain, while the materials were hardened to the same depth.
Further, the fracture rate noticeably decreases, since linear dislocations have left the grain, and the presence of screw dislocations inhibits the increase in hardness.
Obviously, there is a certain number of dislocations per unit volume of a metal grain, which determines the value of the indicator.
Acknowledgments This research was funded by the Russian Science Foundation, grant number No. 21-19-00660, https://rscf.ru/project/21-19-00660/.

Meanwhile, the non-equilibrium solidification conditions result in microsegregation and formation of second phases mainly on the grain boundaries.
It can be seen that without UST, the structure is full of coarse-cell grains non-uniform distributed.
The number of fine-cell grains and spherical grains in the ingot center are greatly enhanced after applying UST, and the higher the ultrasonic power is, the better the refining effect becomes.
Eskin et al reported that the fine-cell grains were solute-rich while the coarse-cell grains were solute-poor with the partition coefficient K＜1 [11].
The number of the fine-cell grains largely increase in the central portion facilitates the solutes enrichment in this region.

The refinement of grain in these materials is accomplished through various Severe Plastic Deformation techniques, which apply significant shear strain, leading to the enhancement of the grain structure.
SPD is a high-pressure metalworking technique that improves material characteristics by generating ultrafine grains [1,2].
This contributes to the highest level of grain refinement and structural uniformity in this region.
Instead, other factors such as the number of passes and the material's properties may have a more significant influence on the billet's stress distribution and subsequent mechanical improvements.
Acknowledgement This research is funded by Ho Chi Minh City University of Technology (HCMUT) - VNU-HCM under grand number SVKSTN-2024-CK-42.

Recrystallization (RX) is modeled in two steps, namely nucleation and grain growth.
Nucleation is assumed to be controlled by the stored energy of a grain.
(1) where pl ε& is the plastic strain rate of the grain, 0γ& is a reference strain rate, sτ is a reference stress for the slip system s, S is the number of slip systems, sm is the geometric Schmid tensor, σ' is the deviatoric stress tensor in the grain and n is the stress exponent.
It is assumed that the recrystallized grains inherit their orientation from the parent grains [5].
Acknowledgement This work is partly supported by the Deutsche Forschungsgemeinschaft (Contract numbers Ri 329/25-1 and Ri 329/27-1).