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The number of grains, n, in the analyzed area is determined so that the resultant mean grain-size should correspond approximately to the size measured in experiment.
Schematic illustration of grain-structure and stress state in i-th grain.
Stress State in Modeled Grain.
The grain size d(i) of the i-th grain will be used as the slip-band length of the grain in the crack initiation analysis.
Employing three series of uniform random numbers, three distinct modeled microstructures composed of differently-shaped grains, which have distinct directions of slip-lines and slip-planes, are analyzed for one testing condition.

They found that the finer grain size and more uniform microstructure were obtained along with significant improvement in both strength and ductility with increasing the number of RU passes, AZ31 alloy exhibits an average grain size of 1.3 um and yield strength of 294 MPa and tensile strength of 354 MPa after 5 RU passes at 250℃.
Xia et al. [21] evaluated the effect of MDF process on microstructure, texture and tensile properties of Mg-Gd-Y-Nd-Zr magnesium alloy, the relations between number of MDF passes and grain refinement as well as that between number of MDF passes and mechanical properties were investigated.
In Fig. 2(a), it can be seen that eutectic compounds are distributed at grain boundaries, and the average grain size is about 50 μm.
Some fine lamellar LPSO structures pass through the eutectic phase in the grain boundary, and grow to the grain interior.
LPSO structures with different forms are found in the alloy: the block-shaped ones at grain boundaries and lamellar ones in the inner grains.

As illustrated in Fig. 2, the microstructure of AF, QF, GB, and DP exhibited irregular-shaped grains, and the EBSD analysis was applied to estimate corresponding effective grain size.
The effective grain was mainly composed of high-angle boundary.
There are general two methods to define effective grain size of microstructure.
The other one is equivalent diameter, which can be estimated by equivalent the irregular-shaped grain to circle, and the diameter of circle is regarded as effective grain size [11, 12].
The dimples and tearing rides involved in the fracture surface were typical features of ductile fracture, implying that large number of energy was consumed during fracture.

The neighbouring cells of type Liq from its nearest environment (N, S, W and E) are captured by the growing grain, thus creating type Int-2 with all the attributes of a grain of this type (i.e. number, orientation angle of crystal lattice, and phase type in the case of multiphase growth).
The neighbouring cells of type Liq present in the nearest vicinity are absorbed by the growing grain and acquire type Int-2 with all the attributes of a grain of this type.
The neighbouring cells of type Int-3, belonging to this grain, change their type into Int-2.
The attributes of none of the 8 closest cells will change, if the cells have already been captured by other grains.
In these equations the subscripts n and n+1 denote the iteration number, and the superscripts denote the indices of the neighbouring cells (see: Fig. 1).

The concentration stress is related to the number of dislocations and the length of slip bands in dislocation accumulation group, so the fracture stress and crystal grain size have the following relation: 2/1 0 − += dkgg σσ (1) σ0 is intergranular fracture force, and kg is a constant that is related to the binding energy among the grain boundaries, d is the diameter of grain.
According to the formula, the smaller grain size, the greater fracture stress along the grain and the greater the intensity [4].
The smaller the grain size, the more the grain boundary and the greater the energy consumed by plastic deformation, which is the main portion of the crack propagation resistance.
In the samples, there was occasionally a small number of glass phase among the high melting point crystals (Figure 6(c)), and in most cases, the grain boundaries were very clean (Figure 6(d)), which indicated that there was little glass phase in the samples, and the grains were combined with each other most directly, which would improved the condition of the grains boundary, furthermore increased strength and toughness of materials [4].
The distribution probability of nano- aperture was the largest, and the volume of micron- aperture was the maximum, but all pores were less than 4µm in diameter; (4) Grains of the samples were fine, mostly below 2µm; there were little glass phase among grains, which lead to enhance of direct combination among grains; the samples had micro pore structure.

However, elongated grains appear in the Lu-α-sialon ceramic, coexisting with small equiaxed grains.
Yb-stabilized α-sialon has fine equiaxed grains with a size of 1.0 µm or so.
Large numbers of nuclei formed, which leads to premature grain impingement during grain growth, therefore, the entirely equiaxed grains appeared in the Yb-α-sialon ceramic.
It means that enough room is provided for the growth of elongated grains.
Lu-α-sialon ceramic with elongated grains presents a high toughness, but for Yb-α-sialon ceramic, the equiaxed grain morphology is responsible for the relatively lower toughness.

The scanning electron microscopy (SEM) experimental results indicated that: under the force, non-crystalline materials distorted, crystal faces were torn along grain boundaries and crystal grains were disrupted.
So crystal size became smaller, more grain boundaries and more voids were formed, which supplied more free space for the movement of crystal grains and chain segments.
Introduction Ultra-high molecular weight polyethylene (UHMWPE) is a linear polymer with number average molecular weight in the range from one to ten million.
So crystal size became smaller, more grain boundaries and more voids were formed, which supplied more free space for the movement of chain segments and crystal grains.
So crystal size became smaller, more grain boundaries and more voids were formed, which supplied more free space for the movement of crystal grains and chain segments.

In addition to coarsening of M23C6 carbides and an equiaxed fine grains in FGHAZ, intermetallic Fe2(Mo,W)Laves phase precipitated on grain boundaries during creep is probably the significant factor caused the type failure.
There have been a number of studies that have investigated the fracture characteristics and cause of type crack in welded joints of P91 and P122[3,4].
The microstructure near the fracture location in the specimen fractured at 100MPa after 1546.5h, as shown in Fig.4, this area exhibits a finer and an equiaxed grains of average size 10μm, whereas initial martensite lath and relative coarse grain with prior austenite grains of size about 125μm microstructure are observed in base metal area of weldment.
A large number of creep cavites on grain boundaries could be determined in the FGHAZ, while hardly any cavities could be observed in the other position of weldment.
●Except for coarser M23C6 carbides and an equiaxed fine grains, intermetallic Fe2(Mo,W) Laves phase precipitated on grain boundaries in FGHAZ during creep is probably the significant factors caused the type failure.

The decrease of coercivity and remanence with the number of impacts per turn is clear.
The size determine the FM or SPM behavior of the grains, and only the number of grains above the SPM limit contribute to the hysteresis and coercivity [9,10].
If remanence and coercivity are taken as directly related to the number of FM Fe grains, the samples with high ipt should have very few or none of such FM grains or clusters.
A direct observation of the number and size of the clusters can be performed by AFM.
Samples b) and c) have much smaller iron grains.

It is suggested that the AlN precipitates restrict the grain grow in one direction, promoting the appearance of an elongated grain in the rolling direction, which is fundamental to obtain a steel with deep drawing quality (DDQ)[3-7] .
It will be characterized by the aspect ratio, defined as the ratio between the grain length and grain width.
At these conditions, the recrystallized ferrite grain shows a "pancake" type of morphology, characterized for elongated grains oriented in the longitudinal direction.
It is observed that a 40°C/h and 100°C/h, a pancake type of structure is obtained, at 200°C/h, the material exhibits a mix microstructure, comprised by a combination of elongated and equiaxial grains, accompanied by a decrease in grain size.
In consequence, the nucleation of new ferrite grains is favored, and as result, the number density of small recrystallized grains increases with a tendency to be equiaxial [3,10,11] .