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Measuring and Modeling Grain, Grain Boundary and Grain Edge Average Curvature and their Application to Grain Growth P.
TA is the number of points of tangency between a sweeping test line and a curved grain boundary trace on a section per unit test area [2].
NA is the number of grains per unit of area and SV is the interface area per unit of volume of the aggregated grains that is one-half of the interface area per unit of volume for the disintegrated polycrystal.
In this case, the key measurement is PA, the number of triple points per unit of area of a planar section.
TA, the number of points of tangency between a sweeping test line and a curved grain boundary trace on a section per unit test area was measured as well as NA and PA[12].

Such a map can be easily used to analyze grain boundaries.
One of the most important results given by LSM is the fraction (length or number fraction) of a particular GB type.
Let us consider the number fraction of High Angle Grain Boundaries as an example.
It is given by the following equation: fHAGB= NHAGBNtot, (1) where NHAGB is the number of line segments corresponding to HAGB and Ntot is the number of line segments calculated for misorientation usually higher than some limit (1 or 2 degrees).
Authors gratefully acknowledge NCN for the grant number 2011/01/D/ST8/07399.

anahavandi.mahdi@gmail.com, bazmah@upm.edu.my Keywords: Lead-free solder, Bi-Ag solder alloy, Bi-Sb solder alloy, intermetallic compound (IMC), grain boundary grooving, multiple reflow number Abstract.
The Ag-rich phase like Cu-rich and grain boundary grooving, were observed in all three reflow number.
Cu-rich phase and grain boundary grooving were observed in Figs. 6.
It can be concluded that, the addition of Ag in molten Bi-Ag solder alloy, Sb in molten Bi-Sb alloy, and the number of reflow process affect the dissolution behavior of Cu grain boundaries into solder bulks which lead to changes of Cu-rich phase particle size.
Grain boundary grooving was observed in all compositions except Bi-5Sb.

It is observed that such ferrites usually form at the grain edges or grain corners.
The arrows in Fig. 2b start from the interested ferrite and point to the prior austenite, with the numbers on which show the deviation from K-S relationship between ferrite and this prior austenite.
The arrows start from the ferrite and point to the surrounding prior austenite grain, the numbers on which show the deviation from K-S relationship between ferrite and pointed austenite grain.
The arrows start from the ferrite and point to the surrounding prior austenite grains, the numbers on which show the deviation from K-S relationship between ferrite and pointed austenite.
Table 3 The statistical results about crystallographic relationship between the allotriomorphic ferrite and prior austenite grains in samples A1-A6 Sample ID Number of ferrites counted Number of prior austenite grains that have K-S relationship with the interested allotriomorphic ferrite One Two Non A1 24 18（75%） 3（13%） 3（12%） A2 27 23（85%） 3（11%） 1（4%） A3 21 13（62%） 1（5%） 7（33%） A4 26 13（50%） 2（8） 11（42%） A5 9 9（100%） 0（0%） 0（0%） A6 13 12（92%） 0（0%） 1（8%） The crystallographic relationship between allotriomorphic ferrite and prior austenite grains in Sample A1-A6 is statistically investigated and the results are shown in Table 3.

The theory developed represents individual network elements-the polyhedral cells or grains-as a set of objects called average N-hedra, where N, the topological class, equals the number of contacting neighbors in the network.
As will be shown, however, ANH's can serve as accurate "proxies" for irregular polyhedral grains, providing they have the same number of faces, N.
DeHoff [27] and others [28] showed that the average number of faces per grain in a threedimensional polycrystal is �N ≈ 13.4, corresponding to the "ideal" flat-faced grain.
Of course such an "ideal" grain does not exist, because it lacks an integer number of faces.
Using the average number of faces per grain derived by DeHoff, �N ≈ 13.397, the total free energy of an isotropic polycrystal is found to be ∆Epoly ≈ 2.63γgb � i V 2 3 i . 5.

When estimating the grain size, the processing effect for complex grain boundaries is often poor.
Fig.1 Procedure of estimation of grain size The part of image processing include enhancement of the image, extraction of grain boundary, thinning of grain boundary and reconstruction of grain boundary etc.
For area method, the calculation of grain number NA within unit area is needed to determine the grain size number G.
For intersection method, the calculation of intersected points PL within unit length is needed to determine the grain size number G.
The method to obtain grain boundary is designed, the edges of the grain are automatic found, and then the grain boundaries were modified accurately.

By the same approach, the topological relationships between number of grain faces, grain size, number of corners and edges and how these can be calculated in a real microstructure by means of a statistical approach are discussed.
The special relationship (number of faces versus grain size) in 3-D As shown in [5], the average number of faces of a grain of size i is given by the following equivalent expressions: (eq. 3) Namely, here a general expression for the number of faces of a generic grain i as a function of its geometrical parameters depends on the grain surface Ai in a non-linear way, due to the dependency of pij on i.
An approximation for the 3-D special relationship (number of faces - grain size) It is also possible to obtain a simplified equation for the number of faces of a grain in 3-D given by Eq. (3).
Once a 3-D grain size distribution φ(R) is available, by eq. (5) it is possible to calculate the number of faces corresponding to any grain size and also the average number of faces for the given grain size distribution .
One of the main results shown is the relationship between the number of grain faces versus the grain diameter (mesh size).

We try to apply to the several figures provided by JIS and compare with the grain size number and homology value.
Experimental procedure This is a table of corresponding size and grain size number.
Table 1 Grain size number The grain number par 1mm2 The average diameter of grains length of a line segment -1 4(3~6) 0.500 0.477 0 8(6~12) 0.345 0.320 1 16(12~24) 0.250 0.226 2 32(24~48) 0.177 0.160 3 64(48~96) 0.125 0.113 We have a relation of the grain size number G and the grain number par 1mm2 m.
Table 2 The relation of grain size number, the value of formula (1) and homology b1 Grain size number G -1 -0.5 0 0.5 1.0 1.5 2.0 2.5 The value of formula (1) 4 4.8 8 11.2 16 22.4 32 44.8 Homology value b1 14 24 30 41 55 75 100 The figure of grain size of -0.5 does not provide by JIS.
There are other indicators (the average diameter of the grains, the average size of crystal grains, the average segment length of the test line across one grain in the grain, the average number of supplemental grain per mm of measurement line). 4.

The results suggest that Extra-Hardening may be an effect of strengthening by SGB, which changes significantly with the number of passes and annealing, and constant LAGB strengthening.
UFG microstructure can be obtained by a number of processes.
Result and Discussion Figure 1 shows FE-SEM images for each pass number.
Strength, Δσex, and dislocation density increased with increasing number of passes and decreased with annealing.
The grain boundary fraction increased with increasing number of passes and annealing.

A simulation model based on graphic variable for implementing stored grain management was developed in order to implement simple and high efficiency management for a number of stored grain warehouses.
There are a number of methods for providing the knowledge, which include experimental investigations, and theoretical calculations as well as numerical simulation [5].
Grain Bulk System Components 1) Biotic · Grain
UNSAFE STATUS MANAGEMENT Because our graphical variable model can produce a number of snaps to as precisely as possible predict temperature and moisture variation of stored grain, the work to analyze and check various unsafe states become very easy.
Thus, the analysis of unsafe states in stored grain can be in practice changed into the analysis of relationships about the minimal distance between a numbers of curves.