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Undercooling and its importance in the formation of grain-refined microstructures is highlighted.
Grain refinement in undercooled melts is attributed to fragmentation of dendrites.
Grain Refinement through Undercooling Since the pioneering work by Walker [18] it is well known that the phenomenon of grain refinement occurs if the undercooling passes a certain critical undercooling ∆T*.
According to this model, a grain refined equiaxed microstructure is observed if ∆tbu < ∆tpl, and a coarse grained microstructure if ∆tbu > ∆tpl.
Physical Limitations of Grain Refinement through Fragmentation of Dendrites It is of great interest to produce grain refined materials because of their favourable physical properties.

As a consequence, the retained austenite grains in TRIP steels possess different carbon content or grain size, leading to a range of stability as illustrated in the literature [1].
It was experimentally determined that phosphorous refines the grain size of ferrite and bainite but no statistical data on the grain size of retained austenite is available.
Quantitative analysis of the images gives the total number, the area and the aspect ratio of each particle.
The discrepancy might be related to the shape and orientation of the austenite grains.
Therefore, neutron depolarization measurement leads to a much smaller grain size in P0.14 steel than that from the image analysis as the neutron beam transmitted through the thin part of the austenite grain.

Tensile test indicated an increasing tendency of strength by increasing the number of strands and bending tests showed that higher strength is achieved in rare bend because of inhomogeneous grain distribution after friction stir forming (FSF).
The tensile strength tends to increase with the growing number of slits as well.
Moreover, rare bend showed higher strength compared with the face bend, because the grain size became inhomogeneous after FSF.
Number of slits Number of slits Bending stress [MPa] Tensile stress [MPa] (a) (b) Fig. 8 Relation between the number of slits and strength in (a) tensile and (b) bending tests after FSF Fig. 9 Increasing the number of slits results in insufficient material flow under the shoulder Conclusions The authors have implemented FSF in various process parameters and experimentally confirmed the possibility of producing a fibre-reinforced metal.
The bending tests indicate that higher strength is achieved in rare bend because of inhomogeneous grain distribution after FSF.

The term mesh number indicates the number of linear openings per square inch on the sieves, which were used to sort the abrasive grits in the grinding wheel manufacturing process.
In an ideal case, if the wheel surface is covered with 100% of abrasive grits, the distance between the each grain center will be equal as shown in Fig 1(b).
For each and every (x, y)-coordinate, there is a chance to have several z-coordinates as several grains may pass over the same location of the workpiece surface again and again.
Kirsch, Kinematic simulation of high-performance grinding for analysis of chip parameters of single grains, CIRP Journal of Manufacturing Science and Technology 5 (2012) 164–174
Gong, Investigation of different grain shapes and dressing to predict surface roughness in grinding using kinematic simulations, Precision Engineering 37 (2013) 758–764.

The measured grain size is in the order of 5~60 nm.
Because of the high vibration frequency of the system (20KHz), the specimen surface under treatment is struck repetitively by a large number of balls within a short period of time.
The average grain size is measured to be ~7nm.
Therefore, mean grain size (µ) will be: Fig. 4 Grain size distribution histogram of 20 min treated specimen - grain size measured by TEM Fig. 3 TEM micrographs a) untreated and b) treated for 20min.
Ultra-fine grain materials.

The mass density reaches 1.86 g cm-3 at the most compact MgB2 bulk, which has an imporous microstructure, excellent grains coupling, clean grain boundaries and nano-sized grains (100~200 nm).
For the S-2 and S-3, both of them present a typical co-existence of porosity and un-porosity, but the number and area of the pores in S-3 is much smaller than them in S-2.
The grains size is about 200~500 nm.
Figure 2 S-5’ with a magnification of 50, 000 times for S-5 presents excellent grains coupling, no pores, very clean grain boundaries, nano-sized grains (100~200 nm) and more melted area than S-4.
Because the grain boundaries is thought the mainly effective flux pinning centers in MgB2, the increase of the area of grain boundaries will improve the grains boundary flux pinning and the Jc at high fields.

With a given strain, the finer the starting grain size, the finer the final TTMP grain size.
TTMP strengthening relies on grain refinement by the Hall-Petch relationship [12].
TTMP by Pressing – Warm pressing was applied as a means to impart thermomechanical energy to fine grained Thixomolded® stock to refine the grain size and to increase strength.
U.S Patent Numbers 4,694,882, 4,964,881, 5,040,589, Dow Chemical Company
Patent Number 9,017,602 B2, Method and Apparatus of Forming a Wrought Material Having a Refined Microstructure, April 28, 2015

In the last decade, the use of EBSD has been proved to be a very valuable technique in gathering information on grain structure, texture evolution and grain boundary misorientation distribution in the above-mentioned zones.
Results and discussion Texture and microstructure evolution has been analyzed by EBSD at different locations along the transversal cross-section (see areas labeled with numbers from 1 to 6 in Fig. 1).
The average grain sizes were found to be around 38 μm for the Al sheet and 33 μm for the Cu sheet.
The EBSD results acquired from the Cu side of the SZ (Fig. 2f) depict a microstructure with grains having an average grain diameter of around 10 µm and containing a significant amount of recrystallization twins.
Kokawa, Development of grain structure during friction stir welding of pure titaniumActa Mater. 57 (2009) 4519-4528

The part reason is that the number of fine aggregate is so huge that generating very close-grained structure in certain space by Monte Carlo method is impossible.
Each of them has a length which corresponds to twice the number of particles in the system.
The whole number of coarse aggregate is 490 from the table 1.
Table 1 Number and size of aggregate Coarse aggregate Size of grain Dia. of Sieve(mm) 16-20 16-9.5 9.5-4.75 4.75-2.36 Number of grain 19 363 93 15 Fine aggregate Size of grain Dia. of Sieve (mm) 4.75 4.75-2.36 2.36-1.18 1.18-0.6 0.6-0.3 Number of grain 384 2224 20856 133464 495712 The simulating progresses are as follows: first, in domain with the size of 100*100*400mm, all the spherical positions of the coarse aggregates are generated regularly, and radius are random according to ration of volume.
According to the number of fine aggregate in table 1, and after conversion, the number of fine aggregate is 654.

Moreover, it has been proved that slight changes in the hysteresis loop parameters, as a function of the number of stress cycles, have a significant influence on the obtained material data used when calculating the life.
The size of ferrite grain determined using pictorial standard models [8] amounted to 5, which corresponds to the mean diameter of grain amounting to 62.5 μm.
On the boundaries of ferrite grain in the examined steel, both single and numerous carbides of diverse size were observed, forming the so-called „continuous grid” of precipitates in some areas.
The precipitation of the M2C and MC type were revealed inside the quasi-polygonal ferrite grains, while the M7C3 precipitates were seen mostly on grain boundaries and in bainite.
Fig. 5 shows the changes of these loop parameters (Deap, Dsa), as a fuction of the number of cycles in one block of loading recorded in various periods of fatigue life.