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If the support stiffness of a single abrasive grain kg can be obtained, and the number of abrasive grains in contact approximated, a reasonable estimate of the contact stiffness can be obtained by the product of these values.
In this initial grinding situation, since the contact area between grinding wheel and workpiece is obtained from the geometric contact length lg and wheel width b, the number of abrasive grains in contact can be approximated by the product of the contact area and the number of abrasive grains per unit on wheel surface.
To calculate the number of abrasive grains in contact, a knowledge of the abrasive grain density per unit area on the wheel surface is needed.
The number of abrasive grains on the wheel surface can therefore be estimated by multiplying the number of cutting points by 1/2.
Consequently, in the grinding situation as shown in Figure 3(b), an initial contact stiffness of the grinding wheel K'con is obtained from the product of the number of abrasive grains in contact and the support stiffness of a single abrasive grain kg as follow

The effect of ultrasonic attenuation and phase velocity dispersion due to grain scattering is included in the predictions.
An inhomogeneous material is composed of numerous discrete grains, each having a regular atomic structure.
The elastic properties of grains are anisotropic.
For pulse echo set up, the voltage received at the transducer is given by Kim et al. [10] and is expressed as (1) where is the surface of the flaw or defect. k is the wave number and is the area of the transducer.
Panetta, Ultrasonic attenuation as influenced by elongated grains, Review of Progress in QNDE, 21(2003), 109-116

The grain is coarse and cellular dendrite brochus is large.
The secondary dendrite arm spacing is 15~20 µm, average grain size (diameter) is 40~50µm, and the maximum grain diameter is 200µm.
By applying compound energy field, most of precipitations transform into Al( Fe, Mn) Si, whose number is larger than the number of (Fe, Mn) Al6.
The participations of plates under the compound energy field are relatively uniform, less in number and the participate phase Al( Fe, Mn) Si are dominated.
Potency of high-intensity ultrasonic treatment for grain refinement of magnesium alloys[J].

When substituting an actual value of undercooling, one can determine the actual grain density after solidification.
The ratio of the density of nucleation sites (those a microscopic examination of the bulk t which a nucleus has already formed and growth has begun) and the total number of sites per unit volume λ is dependent on the maximum undercooling and is a non-decreasing function.
As a consequence the relative grain density NV(∆T)/λ, i.e. the ratio of grain density to the total density of substrates can be described by a continuous non-decreasing function of the undercooling with the following characteristics: ( )    ∞→∆ =∆ = λ ∆ T T TNV for1 0 for0 . (1) Let ( ) ( )( ) ( )TTNTn V ∆d d λ∆=∆ be the first derivative of the function NV(∆T)/λ with respect to d(∆T), then λ·n·d(∆T) determines the change of grain density in the interval from ∆T to ∆T + d(∆T).
Grain density versus undercooling according to the lognormal and Oldfield models.
Vol. 20A (1989), pp. 311-322 [iii] Fraś E. at al.: Theoretical Model for Heterogeneous Nucleation of Grains During Solidification.

In as cast samples, main alloying elements Si and Cu were located between aluminum grains along grain boundaries.
At higher heat treatment temperatures, precipitates were observed to have grown in size due to Ostwald ripening and their number per unit area was reduced (Figure 5).
Precipitate size is larger at this temperature but their number is reduced.
The grains became larger and spherical.
Precipitate size is larger at this temperature but their number is reduced.

The gamma to alpha phase transformation caused by the diffusion of Si and Al determines the grain size and morphology resulting in columnar grain growth.
The grain growth is apparently not affected by the diffusion process.
As silicon and aluminum diffuse from the coating the substrate is enriched in Si and Al and the grains transform into α phase.
These grains grow from the surface as a consequence of the diffusion flow resulting in a columnar grain morphology.
According to Szpunar [10], this decrease of magnetically favorable textures, such as (001)||RD, during the grain growth is caused by the different mobility of grains with these textures if compared with other textures, such as (111)||ND, which can grown much faster and consume the lower mobility grains.

By EBSD analysis, it was found that crystallographic variant selection was observed not only across those prior α/α grain boundaries, but also within the α grain interior.
It can be found that many slip bands in a phase grains and bulk b phase accumulated in triple grain junctions.
The dark grey grains refer the a grains and the light grey grains illustrate the different Euler angles of b phase.
From Fig. 1, it is sure that the matrix is the a phase, and lots of random b phase accumulated along grain boundaries and within prior a grains.
And it is sure that the orientation of b nucleating from a grain boundaries and within a grains is best described by the K-S orientation relationship.

Rotation of grain crystal lattice is the basic mechanism of texture formation and of anisotropic behavior of metals during plastic deformation.
Also the intensity of grain-matrix interaction plays an important role in the prediction of the above quantities.
Besides the lattice rotation, also the intensity of grain-matrix interaction has a strong influence on the predicted results, therefore it was also studied.
The increment of displacement gradient of a grain resulting from a slip with δg shear is: (2) The plastic strain increment of a grain is the symmetric part of : (3) The rigid body rotation (plastic rotation) increment of a grain is the asymmetric part of : (4) Two definitions of crystal lattice rotation Classical definition (CL) Let us consider that total sample rotation, as well as a resulting rotation of each grain, is zero.
This work was financed by the Polish National Centre for Science (NCN) basing on the decision number: DEC-2011/01/B/ST8/07394.

In every sample 3 measurements were done and the mean was rounded to the nearest whole number. 1.
Mean grain’s cross sectional area varied from 1112 to 2262 µm2.
The presence of steel tubes did not affect significantly the grain size.
Fine grained microstructure - sample 4 (a), coarse grained microstructure - sample 6 (b) The mean volume fraction of eutectic areas varied from 0,44 to 2,70%.
Voids’ shape and distribution were typical for shrinkage porosity (longitudinal pores situated along grain boundaries and triangular voids at the junction of 3 grains).

Effect of a Si Additive on an Al Grain Boundary: a First-principles Investigation Ying Zhang1, Guang-Hong Lu1,a, Han Zhang 2 Tianmin Wang1, Shenghua Deng1 and Xuelan Hu1 1 School of Science, Beijing University of Aeronautics and Astronautics, Beijing 100083, China 2Department of Physics, Peking University, Beijing 100871, China a lgh@buaa.edu.cn Keywords: Al grain boundary; Si; electronic structure; first-principles Abstract.
The electronic structure of an Al grain boundary (GB) with Si as an additive has been investigated by a first-principles method.
Impurity can largely change the mechanical properties of the Al alloys by its segregation in grain boundary (GB).
Italic numbers indicate the Al-Si atomic distances.
S A B Q x y (a) Fig. 1 Supercell of an Al 9(221)/[110] Σ tilt grain boundary.