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For a polycrystalline structure, the grains are deformed by crystallographic sliding located in the most favorably oriented systems and which support a strong constrained stress . 1.
Volume fraction of grain.
Its microstructure (number and orientation of grains) can be determined by using Euler angles defining the orientation of each grain.
From now on this study can be generalized to other model of the mechanics of the materials and especially for models having a large number of parameters.
A note on Guttman’s lower bound for the number of common factors.

The microstructure of all investigated alloys was fully recrystallized and coarse-grained with a mean grain size of 15-60 m.
During annealing up to 120 minutes dislocation density decreases, the number of grain boundaries with banded contrast characteristic of high-angle boundaries increases, grain size increases to 0.12 m.
A banded contrast typical of equilibrium high-angle grain boundaries is revealed at many grain boundaries.
With the reduction in grain size, the SP temperature decreases.
At 700C the alloy with a grain size of 0.08 m displays the same superplastic properties as the same alloy with a grain size of 0.3 m.

Pipes with large grains and a broader grain size distribution had better corrosion resistance.
To date, a number of researches have been performed to investigate its corrosion and erosion-corrosion in seawater.
Utilizing unique grain color maps in the OIM system, the grains were colored to distinguish one grain from neighboring ones and to characterize grain size distribution.
Fig. 4 presents the grain size distributions in terms of area fraction corresponding to the grains in Fig. 3.
Pipe B have a broader grain size distribution, and the grain size of Pipe B are larger than that of Pipe A.

A very fine meshing scheme was adopted, i.e., meshing seed number used was 0.005 (i.e., total number of elements ≈ 200,000).
When the meshing seed number was changed to 0.05 from 0.005, the simulations took less time to complete and the results were only 5% on higher side of the values (to be show afterwards).
When a meshing seed number of 0.001 was adopted, a comprehensive increase in time of simulations was observed with a change in values of results less than 1% from the values shown in the article.
Deflection at the inner most point of grain (Δ)
Garrison, Grain Stress Analysis Study, AD Report No.

Consider first the movement of a grain boundary.
This is quite a small amount, only a few percent of that typical for recrystallisation or equivalent to the energy of grain boundaries in a metal with a grain size of about 200µm.
However, a number of caveats may apply.
This model was applied successfully to describe an observed phenomenon of grain rotation.
However, the number of systematic investigations are few, so many conclusions are tentative at present.

Raulea[3], grain sizes range from 0.016mm2 to 600 mm2.Results show that: yield strength and tensile strength decrease with a decreasing number of grains over the thickness when the grain size is less than the sheet thickness.
This effect can attribute to grain size strengthening or Hall-Petch effect.
If grain sizes became lager than the specimen thickness, the yield strength tend to increase with grain size and a loss of reproducibility is observed.
The grain size is larger, the ratio of surface grain  by the total grain number is larger relatively as foil thickness in constant, effect of surface layer grains on springback behavior can not be ignored.
According to “surface layer model ”,the free surface grains are less constrained and unable to store and transfer  dislocation for the blank of more surface grains, therefore, it leads to the total number of dislocations, grains and grain boundaries reducing in bending process, so the springback angle becomes smaller after unloaded.

The values of strain rate sensitivity, m was determined, as a function of grain size, and it decreased from 0.45 to 0.33 to 0.18 with increasing the grain size.
FSP creates a region called the "stir zone" or "nugget", where the microstructural refinement occurs with equiaxed ultrafine grains with high angle grain boundaries.
This figure captures the abrupt change in the microstructure from elongated grains to ultra-fine equiaxed grains.
The Vickers hardness number of the base metal is about 80, whereas at the center of the nugget the hardness increases to about 95, for the 430-90 and 430-140 condition.
Table 1 presents a comparison about the grain boundary characteristics for all friction stir processed conditions in terms of grain size, average misorientation angle and percentage of high angle grain boundaries.

The specimens processed by rolling showed only elongated grains and the ones processed by ECAE (Route A) at 2 and 4 times showed elongated grains and partially equiaxial grains and in the samples fabricated by ECAE (Route Bc) at 2 and 4 times, equiaxial grains were found by optical microscopy.
The evaluation of the time until crack initiation and the number of cracks were employed to see the corrosion morphology of the samples.
ECAEed (route A) sample consisted of mainly elongated grains and partially fine equiaxial grain.
So the equiaxial grains were formed by dynamic recrystallization.
ECAEed (route A) sample had largely elongated grains and partially recrystallized grains.

The average grain size of ZnO was 13.3 µm.
In addition, the presence of the ZnV2O4 spinel phase at the grain boundariesmay also play a role in controlling the ZnO grain growth.
As can be seen, the average ZnO grain size is 13.3 µm, it shows very normal grain growth of ZnO.
At the same time, it is clear that the grain boundaries.
However, it is also clear that the pores exist at the grains and grains interior, which are bad for the electrical properties of ZnO varistor ceramics.

Let us note that all images used in this paper bear the diamond pyramid imprints, which allow us to evaluate the microhardness numbers shown in Table 2.
Table 2 – Microhardness numbers and solid solution grain sizes.
Another type medium size particles are found on the grain boundaries around the recrystallized grains thus forming a network.
Comparing the two samples we may see that there is no correlation between the number of large and medium size particles and hardness.
The number of medium size particles is higher for sample 1 while their mean size is almost the same.