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The microstructure is discretized in pseudo-grains and each pseudo-grain contains only the matrix material and one family of inclusions.
Each pseudo-grain is homogenized with Mori-Tanaka formulation.
In the next step all pseudo-grains are homogenized to a single material using Voigt scheme as can be seen in Fig.1.
In case of orthotropic materials the total number of virtual experiments would be six (1 tensile and 1 shear for each direction) as presented below.
The size of the RVE, the number of the generated inclusions and the proximity between them, are some of the parameters that are not taken into account in the mean field homogenization approach, but in the numerical analysis those parameters affect greatly the homogenization results.

The number of models and programs is designed to calculate ν from the δ known.
The data for calculating the average grain size and distribution histograms of grain size were obtained by resulting dark-field image processing.
For each condition about 400-450 grains were counted.
The numbers near the curves denote the thermal diagrams of the three regimes.
Herzer, Grain structure and magnetism of nanocrystalline ferromagnets, IEEE Trans.

(3) Sand transport: Sand erosion detaches sand grains into a perforating cavity or wellbore.
Some of the grains are transported to the surface while others settle into the perforation tunnel or into the well hole.
The criterion for sanding is: CBHP≥ 3σ1-σ2-U2-A-poA2-A (1) The model predicts the rate of sand production by utilizing the non-dimensionalized concepts of Loading Factor, LF (near-wellbore formation stress normalized by strength), Reynolds number (Re) and water production factor.
In 2010, Isehunwa and Olanrewaju [7] proposed a new model for sand production, considering the effects of flow rate, fluid viscosity and density, grain size, and cavity height.
The radius of sand production cavity is: Ra=Qf μ49Rs2pHρfg (11) The volume of sand produced can be expressed as: Vsp=pRa2H (12) where, Qf is the fluid flow rate, Rs is the grain radius, H is the cavity height, and g is the gravitational acceleration.

A number of studies are known where the influence of the material structure onto fatigue cracks origination [1-5] studied while a large amount of data on fractographic inspection of locomotive axels fractured during operation are available and descussed.
Electron- microscopy studies revealed the presence of structurally free ferrite, as well as pearlite colonies located in different parts of ferrite grains (fig. 1, a, b).
It is found by the analysis of SEM-micrographs that the ferrite grains occupy about 30 % of the tested area in the specimen under investigation.
This indicates activation of shears inside pearlite grains [9, 10].
Increasing the number of such regions ensures accumulation of differences between fracture rate at macro- and microlevel of under crack growth in the OsL steel.

(2) Table 1 The rock mechanics parameters of the simulated area Rock properties Uniaxial compressive strength（MPa） Uniaxial tensile strength（MPa） Bulk density（kg/m3） Elastic modulus（GPa） Poisson's ratio loess — — 1800 0.07 0.24 Close-grained sandstone 42.6 1.49 2570 8 0.22 Close-grained mudstone 30 1.17 1640 13 0.28 coal seam 125 0.03 1400 3.3 0.33 Close-grained sandstone and mudstone 27 1.22 2590 10 0.27 According to rock mechanics parameters, specific material ratio are shown in table 2.
Table 2 The results of similar material ratio Material ratio Rock seam Sand (%) Calcium carbonate (%) Gypsum (%) Mica (%) sawdust (%) loess 86 2 2 — 10 Close-grained sandstone 84 5 8 3 — Close-grained mudstone 84 6 6 4 — coal seam 85 3 3 — 10 Close-grained sandstone/ mudstone 86 5 5 4 — Mining And Observation Model mining.
According to the curves of experiment, some laws can be found:(1) Forming a flat subsidence basin, multiple strip mining must be need, and when the number of strip mining is to the limited, the maximum subsidence value of ground is constant or only a little increasing;(2) Forming The flat subsidence basin is because the coal pillar was pushed into base plate, the rock pillar, the rock strata which support the load were compressed, and the rock strata above strip mining were bent.

It is known that during operation of a coil material in the creep mode, one of the main roles is played by the strength of the grain boundaries of the steel structure.
All phases formed at the grain boundaries have a significant effect on the strength of steel during its operation under voltage at high temperatures [9, 10].
Consequently, the response to the directional electric signal effect deep into the metal will depend on the number of vacancies, voids and structural transformations that can give information about the unsoundness of the structure [6].
The increase in relative impact viscosity (KCUo ⁄KCUi - the ratio of impact viscosity in the condition of supply to its value after a different period of operation) (Fig. 2) is explained by numerous phase separation with different content of alloying elements at the grain boundaries and in the grain body.
This type of carbide depending on the nature and morphology at the determination on the boundaries of the grain in case of cellular forms, this adversely affects all the mechanical properties of the coil metal and conversely if it has a globular shape [23].

All of the heat content introduced in the calcined system after formation of the desired manganite compound, leads to sintering processes followed by intensive densification and grain growth, inducing subsequent decrease in the specific-surface area of samples.
Average grain size, calculated from the measured specific-surface areas by using theoretical density derived from the respective unit cell volume and stoichiometry, together with the approximation of spherical grain shapes, thus, from the values of ~ 300, 500 and 900 nm for x = 0.5, x = 0.24 and x = 0.16 samples, respectively, jumps to ~ 100 µm for the LaMnO3 (x = 0) sample (Fig. 2b).
The Curie temperature of rare-earth manganite compounds is known to vary with a number of synthesis parameters, such as calcination temperature and the corresponding average grain size [7], sintering temperature [8], oxygen nonstoichiometry (parameter δ) [9], La/Mn ratio [10], overall stoichiometry [11], porosity [12], pressure [13], additional silica content [14], etc.
In general, small inclination of M = f(T) curve might be a sign of either the wide particle size distribution or weak inter-domain interactions, whereas in our case, the smaller the grains (as calculated from the specific-surface area measurement) of the sample, the smaller the slope of the M = f(T) curve in the range of Curie phase transition (Fig. 3a).
It was also found out that specific-surface area linearly decreases with strontium content (due to decrease in the manganite formation temperature with strontium content), as well as that Curie temperature follows a trend of linear increase between x = 0.16 (Tc = 45 oC) and x = 0.33 (Tc = 100 oC) stoichiometries, after which it remains constant at 100 oC with further increase in strontium content, showing also the trend of increasing the phase transition sharpness with the increase in average particle size of the manganite compound, being the consequence of weaker inter-domain interactions within the systems comprising smaller grains.

Fig.4 Schematic diagram of crystalline grain (CG) and non-crystalline grain boundary (NCGB) regions in polycrystalline LMO.
The polycrystalline LMO samples that we have produced consist of crystalline grain (CG) and non-crystalline grain boundary (NCGB) regions.
The NCGB regions have a large number of dangling bonds per unit volume.
The polycrystalline LMO samples consist of crystalline grain (CG) and non-crystalline grain boundary (NCGB) regions.

Therefore, in open circuit conditions (Fig.1 / electron transfer cannot occur at the electrode level) according to the zero current condition (Je + Σ ji Zi = 0), one obtains: −−OJ = RTODOC − τe R , with R = z O ∂ ∂µ , (7) where τe is the electronic transference number.
The results have been obtained with yttria (9 mol%)doped zirconia polycristals (YSZ) which contains 1.6 wt % SiO2 leading to lenticular glassy precipitates at grain boundary triple points [10].
Fig.3 shows that the grain boundary conductivity (σgb) increases when the cooling rate at the end of sintering increases.
Figure 3- Influence of the cooling rate at the end of sintering on the grain boundary conductivity of YSZ samples (Figs.1,5), coated with the same electrode material, between reversible electrodes subjected to different PO2 ( PO2 I and PO2 II ).
We have also shown the influence of the cooling rate, at the end of sintering, on cation redistribution processes near the interfaces and on the grain boundary transport processes.

In UAG, grinding forces and temperature, and consequently the wear of the abrasive grains are relatively low.
During the conventional grinding process an increase in the grinding forces leads to an increase in cutting temperature and wear of the abrasive grains.
Hence the abrasive grains will become dull, in turn the grinding forces and temperature will increase and the surface roughness will reduce.
This can be due to macro fracture, grains pullout and wear of the bond at progressed wear stages, as a consequence of high grinding forces and temperatures.
Rabiey, M.: Ultrasonic assisted dry grinding of 42CrMo4, International Journal of Advanced Manufacturing Technology, Volume 42, Numbers 9-10, 2009, p. 883-91