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It can be said that CRSS is decided by the number of dislocation sources, existence of grain boundary and dislocation density.
While, in order to express activation of dislocations due to existence of the initial dislocation sources, mobile dislocations or grain boundaries, , and are modeled as , (3) , (4) , (5) where denotes the ideal shear strength, the initial CRSS, the minimum shear stress when dislocations are released from grain boundaries, the numerical parameter, the threshold value on the number of dislocation source and the threshold value of misorientation.
Furthermore, the number of dislocation sources , the number of mobile dislocation and the parameter on misorientation are given by , (6) , (7) , (8) where means the initial dislocation density, the representative volume, the random number between zero and 1, the norm of geometrical necessary (GN) dislocation density tensor [6] and the normal direction of slip plane.
Number of tetrahedral elements of FEM is 100694 and strain rate is 0.01s-1.
A few dislocation sources exist at the initial states and the number of dislocation sources increases due to accumulated dislocations.

Sometimes cracks stop completely for a large number of cycles sometimes cracks only decelerate, both resulting in an additional number of life time cycles.
Grain A with the initial crack is surrounded by grain B.
In fact grain A with the initial notch is surrounded by grain B.
Firstly we only consider the information we normaly have from two grains and a grain boundary: The orientation of both grains which form the grain boundary and the inclination angle d between the grain boundary and the loading axis.
Now we can calculate the Schmid-factors for all slip planes and systems of grain A and grain B.

However, depending on the diffusion time and grain size a given tracer atom can be expected to cross a number of grain boundaries in its diffusion time.
It is clear that a tracer atom need not sample a large number of grains and grain boundaries in order for its effective diffusivity in the structure to be fully developed.
In a diffusion experiment in a polycrystalline material, it can be expected that tracer atoms will traverse a number of grains in the diffusion time.
To simulate a thin-film (instantaneous) tracer source, 10 6 particles are created and released from the surface and allowed to diffuse for a time t (proportional to the number of jump attempts per particle).
These concentration profiles are built up simply by determining the number of particles that have reached a given distance from the tracer source plane after a diffusion time t [10].

In literature, there is a great number of fatigue life data of copper with conventional grain size (i.e. tens of microns, denoted here as CG copper).
The differences increase with the increasing number of cycles to failure.
This is generally true for the ultrasonically cycled specimens exhibiting very high numbers of cycles to failure.
Here the halfwidth of the hysteresis loop is plotted in dependence on the relative number of cycles to fracture.
N/Nf , where Nf is number of cycles to failure.

This method is useful, because previously it was difficult to identify a limited number of very small grains which existed in a huge number of matrix grains and grew to enormous grains.
Therefore, in case that there are a huge number of very small grains, the calculation accuracy is not necessarily adequate.
However, because the grain boundaries are approximated by not curve lines but polygonal lines, the small number of vertices on the grain boundaries makes errors.
The shrinking velocity by the proposed model is the same as that by the curvature model when the number of virtual vertices is equal to or more than three.
We consider grain boundary energy and mobility depending on grain misorientation.

With increase in number of passes, the grain size is reduced to 13 μm in the third pass (Fig. 1(f)).
After subjecting the billet to six number of passes, the elongated strips of grains are arranged into a distinct lamellar structure (Fig. 1(g)).
The trend reversed again for the higher number of passes.
In case of nine and ten pass samples the number fraction of LABs reduces further and stabilizes at 15%.
In as-cast material between two high angle (>15°) boundaries of a grain, a number of low angle (<15°) boundaries exist and point-to-point misorientation angle of these boundaries are in the range of 5-10o.

Isothermes of grain boundary tension and grain boundary adsorption in Cu-Sn system Vaganov Danil1,a and Zhevnenko Sergey 1,b 1 Moscow State Institute of Steel and Alloys (Technological University), 4, Leninsky pr., Moscow, 119049, Russia a vaganov_d_v@pochta.ru, b sergeyng@mail.ru Keywords: Grain boundary tension; Grain boundary adsorption; Free surface tension; Zero creep method Abstract.
Today, there is a lack of such data due to the limited number of the measuring methods and possibilities of apparatus.
The groove is formed at the place of grain boundary exit to the surface.
The average grain size was 115 µm.
It is possible to evaluate the number of sites in surface monolayer A A m N N V Г 3 2 max −       = (5) 0)1ln( γ γ ++−= bc zRTdc d Г γ RT c −= bc1zbc + =Г 47.0)30001ln(102.0 ++ −= c GBγ 81.1)100001ln(148.0 ++ −= c sγwhere Vm is the tin molar volume and Na is the Avogadro number.

Polycrystalline materials typically contain a very large number of grains whose surrounding grain boundaries evolve over time to reduce the overall energy of the microstructure.
Thin polycrystalline films are typically composed of a large number of small grains, and the surrounding grain boundaries largely determine the microstructure of the material.
Although applications typically involve polycrystalline specimens containing a very large number of grains, it is instructive to consider small idealized systems, such as special bicrystalline systems, which can be studied in detail [2, 4, 11].
(b) The grain boundary XIII.
(b) The grain boundary XIII.

The partially transformed microstructures were characterized by measuring the transformed ferrite volume fraction fvα, the mean ferrite grain size, dα, and the number of grains per unit area, NA.
The higher the number of active nuclei the smaller the ferrite grain size will be.
After saturation of the initial grain boundaries, the prior grain interiors stay almost free of ferrite grains.
It could be argued that the occurrence of some recovery in the austenite before transformation can lead to a reduction in the number of available nucleation sites through the decrease of the dislocation density.
There is experimental evidence showing that a part of the boundaries disappear, leading to a net decrease in the number of ferrite grains during transformation.

Comparing the life cycles (N) of large grains to small grains at 172 MPa (25 ksi) up to 241 MPa (35 ksi), a dominant behavior of larger number of cycles to failure of small grains can be seen.
Comparing the two highest stresses (Table 5), a dominance in terms of the number of cycles for the large grains life cycles is noted.
The number of cycles to crack initiation (Ni) are lower than number of cycles to propagation (Np) at high stresses.
However, the high number of grain boundaries in the small grain sample appeared to slow the nucleation rate.
The fatigue crack growth rate started slowing as a result of the effect of higher number of grain boundaries of small grains.