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As a consequence of the transformation, the vector u becomes a new vector v given by [11]: v = P∆u + (u − ∆u) (4) When considering the formation of large number of bainite plates in many austenite grains, u traverses a polycrystalline sample of austenite so this equation must be generalised as follows [11]: v = n∑ k=1 24∑ j=1 Pkj∆ukj + (u − n∑ k=1 24∑ j=1 ∆ukj) (5) where j = 1 . . . 24 represents the 24 crystallographic variants possible in each austenite grain, and k = 1 . . . n represents the n austenite grains traversed by the vector u.
In this scenario of a large number of bainite plates, the intercepts ∆ukj can be approximated by fkj u where fkj is the fraction of sample transformed by variant j in austenite grain k.
An equiaxed grain in the form of a Kelvin tetrakaidecahedron will have fourteen faces, so that the number of bicrystal orientations that must be described per grain becomes 1 2 ×14×3N = 21N.
A typical grain size is about 10 µm so a cubic centimetre of material will contain N = 1012 grains so that its full descriptionrequires about 1015 parameters!
Reconstructive transformations have so much more freedom to develop that the number of parameters that must be included in any theory is daunting.

Coupled with the measured power value P and the energy partition value ε got in the previous section, for a circular grain contact of radius r, the heat generated per unit area of contact per unit time is: 2 ca rBlC P)-ε(1 q' π = . (3) In this equation, Ca is the number effective on the wheel surface, for kinematic considerations and assuming that the active cutting points per unit area Ca is uniformly distributed on the wheel surface.
For the circular grain contact of radius r, it must be calculated considering with the maximum grain depth of cut hmax.
Fig. 4 Illustration of maximum grain depth of cut The Effect of Grain Contact Radius on Tip Temperature.
When the grain depth of cut is below the threshold value, the material removal process is mostly ductile.
Due to the different grain protrusion of diamond grits on the wheel surface, the contact between the diamond grain and workpiece are various from each other and only a part of the grain is engaged in creating the ductile plowing.

The diamond grain size was chosen with D3 (resin bonded and metal bonded) and D7 (resin bonded).
Furthermore, reducing the grain size the number of grains increases by the square of the grain size reduction [3].
Having very small grain sizes of D3 or D7, the break-out of single grains from the top layer of the grinding wheel already influences this perfect edge.
This can be explained by the grain distribution on a grinding wheel.
In contrast to that, during grinding of a graphite line, the most protruding grains will be captured because these grains are heavily involved in the material removal.

Recently, in [1] we developed a single-crystal yield criterion that involves the correct number of anisotropy coefficients such as to satisfy the intrinsic symmetries of the constituent crystals and the condition of yielding insensitivity to hydrostatic pressure.
In the FE calculations, the polycrystal behavior is obtained by considering 250 grains per element.
It is considered that the total strain-rate of each grain belonging to a given element is equal to the overall strain-rate .
The stress of the polycrystal at the end of the increment is given by: (4) where is the stress tensor of grain i, and is the transformation matrix for passage from the crystal axes of grain i to the loading frame axes, while is the weight of the grain i.
This cubic single-crystal yield criterion is defined for any stress state and involves the correct number of anisotropy coefficients required to satisfy the intrinsic symmetries of the cubic lattice and the condition of yielding insensitivity to hydrostatic pressure.

The maximum number of cycles to failure was 5.4x10 7 cycles, and 3 specimens of 21 tested failed at numbers of cycles above 10 7.
In the enclosed ferrite grain, strong slip activity produced a rough surface.
When slip bands encounter the grain boundary, cracks are formed at the grain boundary.
When slip in grains is present, fatigue cracks may initiate at the intersection of slip lines and grain boundaries.
Close to the fatigue limit, fatigue cracks are initiated when slip in ferrite grains encounter a grain boundary.

The results show that the microstructure become into strip-shaped morphology, more granular particles appear in the grain boundaries or inside grains, and the grains are more refined with the increase of Zr additions.
Addition of Zr element refines grains, which not only improves plastic deformation ability, but also increases the grain boundaries to improve damping capacities [7].
When the Zr content is 1.5wt%, thin strip-shaped structure appears, and punctate particles appear simultaneously in both grain boundaries and grains.
Distribution of grain sizes of Mg-0.6%Zr is obviously unhomogeneous, while grains of Mg-2.5%Zr refine to uniform fine strip-shaped microstructure.
This is due to that with the increase of temperature, slip planes of magnesium matrix increase, causing number of movable dislocations increases greatly.

The grain size, the lattice defect densities as well as the hardness of the pure and composite materials were determined.
The pre-compacted discs were finally consolidated by HPT at RT or 373 K with an applied pressure and number of revolution of 2.5 GPa and 10, respectively.
The grain structure was studied using a JEOL-200EX transmission electron microscope (TEM) operating at 200 kV.
The grain size in the composite sample is about half of the value obtained for pure Cu.
Langdon, Principles of equal-channel angular pressing as a processing tool for grain refinement, Prog.

Introduction A large number of studies have shown that some suitable additions of rare earth and transition elements to Al alloys could improve their properties [1-4], including mechanical properties and recrystallization temperature.
Meanwhile, these precipitations could also hinder the movement of dislocations and grain boundaries / sub-boundaries effectively.
And in Fig. 3(c) and (d), there still were few elongated grains existing even at 525 °C and the precipitates started to coarsen.
There were obvious elongated grains and subgrains existing when isochronal aged to 450 °C.
In Fig. 5(a) and (b), there are larger and fewer precipitates after isochronal aging to 500 °C, and elongated grains coexisted.

On this premise a number of alloys from the 5xxx series were investigated to assess their ability to form superplastically.
However, a fine grain size is a prerequisite to ensure that grain boundary sliding can occur.
There are a number of ways to assess the superplastic performance.
What was less well known was the combined effect of the solute content and the number of dispersoids.
Therefore the logical extension beyond AA5083 was to increase the magnesium content while maintaining a sufficient number of dispersoids to prevent grain growth.

The dielectric constant decreases with the increase of atomic number.
The grain of the product is inhomogeneous and the grain size is about several micrometers.
Fig.3 also shows that the interface between the grains is clean and no other phases are found between the interfaces.
Fig. 4 also shows that the electric constant decrease with the atomic number.
There is the inverse relationship between dielectric constant and atomic number.