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For example, consider a system of N magnetic grains of size a, uniformly distributed in a nonmagnetic matrix.
According to [1, 4], N1 is the number of particles, whose magnetic moments of both phases are parallel to the external field H; N2 is the number of particles with the magnetic moments of the first phase that are parallel to H and the magnetic moments of the second phase are antiparallel to H.
The corresponding numbers of particles in each of these states we denote as N1, N2, N3 and N4.)

Secondly, increasing the grain size reduces the number of defects, such as grain boundaries, the magnetization of which is less in the grain.
An active grain growth is observed with an increase in the SPS temperature, and a partial reduction of hematite to magnetite occurs at a temperature of 900°C and above.
In this regard, the magnetic properties of the oxide with weak ferromagnetism change with an increase in the SPS temperature, an increase in saturation magnetization is observed due to the formation of the Fe3O4 magnetic oxide phase, as well as a decrease in the coercive force due to active grain growth in the resulting ceramics.
Densification and properties of bulk nanocrystalline functional ceramics with grain size below 50nm.

The presented work is on welded specimens, made of unalloyed fine grained carbon steel, with different fatigue crack lengths.
The material used for the specimens was an unalloyed fine-grained mild construction steel (steel grade 1.0546) with a yield strength (Re) of 360 MPa.
In particular, the samples were loaded to a pre-defined number of load cycles and stopped resulting in different fatigue crack lengths at the weld toe.
The stress range Ds used for testing is plotted over the corresponding number of load cycles N for a given failure criterion (e.g. specimen fracture or a given crack length).

To improve antifriction of higher pair contacting components, the paper proposes a new method by which a large number of micro-deep holes are fabricated on the surface of the contacting component.
The basic idea is that a large number of micro-deep holes are processed regularly on the surface of components, and these micro-deep holes can store extra lubricant.
Therefore, micro concave spots can not improve the lubrication state obviously by fabricating large number of micro concave spots on the surface of metal components [6, 7].
In the process of analyzing the fundamental of self-lubricating in organisms, it is found that there is not only close grain existing on the surface of organisms but also lots of capillary leading to the inner skin.
Based on this, a large number of micro-deep holes are fabricated on the surface of metal components to simulate the capillary in the skin layer of organisms.

It was noted that an addition of Mg to the Zn matrix refines the grains but more than 3wt-% of Mg to the matrix forms clusters which deteriorate the property of the alloy in the as-cast condition.
Alloy number Alloys used in the present investigation A B C D E F Zn-Al-3Si Zn-Al-3Si-1Mg Zn-Al-3Si-2Mg Zn-Al-3Si-3Mg Zn-Al-3Si-4Mg Zn-Al-3Si-5Mg 2.
Fig.4 depicts the variation of hardness with alloy number.
Fig.5 (a) and 5 (b) represents the variation of the UTS and ductility with alloy number.
However, the value of ductility decreases and then increases with the alloy number The behavior of the alloy is very similar to the hardness versus alloy number.

A large number of white and green are skillfully used, giving people the feeling of solemness and beauty, and the layouts are properly arranged, outlining the beauty of simple lines.
There are kinds of patterns, grains, gourds, melons, eggplants and so on, created by brick and wood carving technology.

The influence of an increasing number of freeze-thaw cycles on capillary absorption of water is studied first, then chloride penetration before and after exposure to a certain number of freeze-thaw cycles was determined experimentally.
For all five types of concrete ordinary Portland cement Type 42.5, river sand with a maximum grain size of 5 mm, and crushed gravel with maximum diameter of 25 mm were used.
Strength of concrete type C decreases with increasing number of freeze-thaw cycles.
Figure 2: Relative dynamic elastic modulus as function of the number of freeze-thaw cycles.
Figure 3: Capillary absorption of concrete type A after different numbers N of freeze-thaw cycles.

However, for bending strength, when the number of elements in model is less than about 9000, there exists obvious size effect, with the increasing of the number of the elements, the size effect will become less and less until disappeared.
The value of fracture toughness decreases with the increasing of the number of the model elements.
Generally, ceramics are polycrystalline, the factors influencing the mechanical properties of ceramics are significant and the distribution of crystal grain and porosity as well as the flaws of crystal lattice in materials.
For every model, we kept the ratio of height to width constant and increased the number of model elements gradually.
The biggest uniaxial compressive model used here is 0.9×1.8mm (the number of particles is 12032), and the biggest three point bending model is 2.7×0.54mm (the number of particles is 10828), the smallest uniaxial compressive model is 0.08×0.16mm (the number of particles is 95) and the smallest three point bending model is 0.4×0.08mm (the number of particles is 237), as shown in Fig.2

The high-density ammonium polyvanadate was prepared by means of fine-grained ammonium polyvanadate (APV) produced in vanadium plant.
So far, APV is mostly used to smelting ferrovanadium alloys, which requires the density of APV above 0.6g/cm3 and the denser the better [3], but a number of low-density APV is unfavorable to smelt V-Fe alloys.
[3] Yi Peng, Qianwen Li, Qingrong Xia, Xiaojiang Wang, Mingfu Peng, Shengyou Zhu, Ronghua Liao, Publication Number CN1594106A, in Chinese

FEA modelling of the process has been used to reduce the total number of practical tests, and hence, to conserve the limited material supply.
At present the full capabilities of this approach are unknown and the number of variables is extensive making experimental investigation complex and time consuming.
To reduce the number of practical trials, FEA welding simulations were produced using sequentially coupled thermal and elastic-plastic stress analyses.
Numerous alloying additions are needed to achieve this high strength and high temperature resistance and each one contributes to the overall performance of the alloy, be it in the form of the matrix, gamma prime, grain boundaries, carbide formation, or oxide scale.
This increased the complexity of the pattern and increased the number of programmed points to over 4500.