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The change in the system entropy is thus calculated as { }         ΩΩΩΩ+ΩΩ=Ω= 43421321 b a BkS 543121lnln (1) where Ω is the number of ways to permute n atoms in the grain.
It is assumed that the total number of atoms redistributed by planar deformation is ⊥Ο += nnn , where ⊥n is the number of atoms forming dislocations, and Οn is the number of atoms displaced through grain boundary sliding, and are termed "bond breaking atoms".
Equation 1 is divided in the contributions of a, which accounts for the ordering of Οn atoms over the available m positions at the interface; and b, which quantifies the number of possible permutations of ⊥n atoms in the bulk of the grain.
The energy of the system is calculated with [ ]cnaGbnEU d b 2212 2 3 + += Ο (2) where Οn is the number of bond breaking atoms, dn is the number of dislocations per grain, bE is the bonding energy, G is the shear modulus, b is the burgers vector and c is the number of dislocation direction changes per dislocation.
Referring to equations 3 and 4, there are three parameters that describe the deformation behaviour: the increase of the number of dislocation production atoms per deformed plane ⊥dn , the increase of the number of dislocations per deformed plane ddn , and the increase of the number of dislocation direction changes per dislocation and deformed plane dc.

Effects of grain size on fatigue crack propagation in copper film K.
The fatigue crack propagation in the film with large grains was often decelerated, so the crack propagation rate of the film with the large grain was lower than that of the film with the small grain.
When the crack propagation was decelerated, the crack opening displacement obtained from the film with large grain size was smaller than that obtained from the film with small grain size.
Especially, a grain size in the film is considered to have a greater effect on fatigue fracture properties than in the bulk materials because of a small number of crystals against to its thickness.
As a result, the fatigue crack propagation rate of the film with the large grain was lower than that of the film with the small grain. 2) When the crack propagation was decelerated, the crack opening displacement distribution obtained from the film with large grain size was smaller than that obtained from the film with small grain size. 3) The relationship between the fatigue crack propagation rate and the stress intensity factor estimated from the crack opening displacement was identical for the cracks in the film with the large grain and the small grain.

An aged Al-5Zn-1.6Mg alloy with fine η' precipitates was grain refined to ~100 nm grain size by severe plastic deformation (SPD).
Its grain size was of the order of 1mm.
The pass number was 1, 3, 6, 9, and 18 for CCDP and 1, 4, 8, 12 for ECAP, respectively.
The grain boundary misorientation monotonously increased with rising number of passes, as qualitatively demonstrated by selected area diffraction (SAD) patterns (Fig. 1e), which changed from a single crystal type to a polycrystal type as SPD progressed.
The particle size followed a logarithmic normal distribution, and their mean values decreased slightly with the number of CCDP passes, but particle spacing changed not apparently.

In other words, the grain-coarsening tends to occur when: (1) the matrix grains that formed as a result of the α/γ transformation during the heating for carburizing are fine (R0 is small), or (2) the degree of grain size mix is large (Z is large), or (3) the number of pinning particles is small (either f is small or r is large).
The pre-austenite grain of the test pieces that underwent the simulative carburizing treatment was observed in a section in the rolling direction and their grain size number was determined according to ASTM.
A grain size corresponding to a ASTM grain size number of #5 or lower was defined as coarse grain, and a test piece was judged as having coarse grains if such coarse grain size number was obtained in any one field of observation in the entire section.
(c) (a) (b) (d) (e) (f) 25µm Table 2 shows the ASTM grain size number of No. 5 and 9 after simulative carburizing at high temperatures.
Ariyasu: Mitsubishi Motors Technical Review Vol. 13(2001), p. 114 Table 2 ASTM grain size number after high temperature carburizing.

The dependences of the number of dislocations emitted by a crack on grain size (ranging from 10 to 300 nm) in Cu and 3C-SiC (the cubic phase of silicon carbide) are calculated which characterize the grain size effect on crack blunting that crucially influences ductility of these materials.
Effect of grain size on crack blunting Let us calculate the number � of dislocations emitted from a crack along one slip plane as a function of grain size d.
The number � characterizes crack blunting induced by lattice dislocation emission.
With this calculation procedure, we have calculated the number � of lattice dislocations emitted along the same plane as a function of grain size d, for the typical nanocrystalline metal (Cu) and ceramic (3C-SiC) (Fig. 2).
The maximum number � of edge dislocations (that can be emitted from the crack tip along one slip plane) as a function of grain size d in nanocrystalline Cu (a) and 3C-SiC (b). 34] that some nanocrystalline fcc metals exhibit a ductile-to-brittle transition with decreasing grain size.

An alternative method could be the use of ultra-fine grained materials in which the grain size effects may become insignificant.
A number of non-conventional methods such as rapid solidification, power metallurgy and vapour condensation can be used to produce very small grains in metals.
Without the reduction of the initial billet cross-section, the process can be repeated a number of times and the billet is rotated by different routs (A, Ba, Bc and C) [17] between consecutive passes in order to obtain uniform strain.
Grain Size and Hardness.
Sections of extruded cups heat treated to obtain a fine-grained microstructure with a grain size of about 4 µm.

On the other hand, the number of the effective cutting-edges also can be identified based on the working surface, but, this method requires the determination of the typical grain shape.
From the experiment, it is confirmed that the grain shape should be almost spherical for making the numbers of the effective cutting-edge identified from the working and ground surfaces equal.
The grinding wheel WA60J7V, which has alumina grains of average grain diameter d0= 250 µm, is used.
(3) With considering the typical grain shape, the number of the effective cutting-edges can be also determined from the measurements of the working surface profile and the grinding force
(4) By assuming the grain shape as spherical, the number of the effective cutting-edge identified from the working surface topography well corresponds to that identified from the ground surface topography.

Fig.2 Restructured topography of a diamond grain 3D Grain Contour Protection Versus Reference Point Number and the Noise-removal.
Figure.3 shows the grain contour protection versus reference point number nr and the noise-removal threshold t.
Fig.3 3D grain contour protection versus reference point number and noise-removal threshold 3D Noise Generation.
It is also found that the larger the reference point number, the more the noise number.
Moreover, the larger the noise-removal threshold, the less the noise number.

The grain sizes were measured by linear intercept method.
The numbers indicate the boundary misorientations in degrees.
CDRX Grain Size.
The transformation of low-angle subboundaries into high-angle grain boundaries results from recovery-assisted absorption of dislocations in the subboundaries, and the change in subboundary misorientation during deformation can be expressed as dq/de = (Ak/(2nGa2)) rsp [7, 16], where A is the fraction of absorbed dislocations, n is the number of dislocation sets in the boundary, and r is the recovery parameter (log r = -0.0026sp + 0.9698 [17]).
Langdon, Twenty-five years of ultrafine-grained materials: Achieving exceptional properties through grain refinement, Acta Mater. 61 (2013) 7035-7059

The fraction of nanoscale grains increases with strain.
New grains grow consuming deformed ones.
It is seen that relatively large recrystallized grains are surrounded by fine grains.
Schematic presentation for variation of structural mechanisms responsible for new grain development during annealing of cold-worked Ni-20%Cr alloy, where Nrex.nuclei – number of recrystallization nuclei, Nsubgrains – number of subgrains.
In this case the continuous grain growth leads to the formation of homogenous structure with a grain size of 60 nm.