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In author’s previous work [1], damage evolution Eq.1 on rock under uniaxial compression has been established based on the assumption that the compressive strength of grains in rock keeps to the Weibull’s distribution [2]: ])(exp[1 0 m D ����= (1) where damage variable D reflects damage extent of rock specimen loaded;¦is the apparent strain of rock specimen;¦0 is scale parameter, which is related to the mean strain value of all grains in the rock specimen; The parameter m is a shape parameter of Weibull’s density function that defines the degree of material homogeneity, and is referred to as the homogeneity index.
In RFPA 2D , the model is discretized into large number of small elements to take into account the local variations of the material heterogeneity.
The reason for this deviation is that the complicate interaction among the grains of the rock is unable to be reflected in the theoretical model using the strain as damage variable.
In theoretical model, it was assumed that the grains in rock followed the linear deformation, i.e., the minor strain of individual grains kept to the superposing principles.
In addition, it is noted from the experimental curves in Fig. 2 that the number 1 specimen is more brittle than the number 2 specimen, which agrees well with the results calculated of homogeneity index m in Table 2. 4.3 Comparison between numerical and experimental results � � � � � � � � � � � � � � � � � � � � � � � � � �
 � � � � � � � � � � � � � � � � � � � � � � � � � � Normalized Strain Normalized Stress Experimental Curve Numerical Curve Fig.3 Comparison of experimental curves and numerically simulated curves for specimens Fig. 4 Failure patterns of model specimens with different homogeneity index m Based on the experimental results on rock specimens in uniaxial compression and numerical tests by RFPA, the dimensionless stress-strain curves were depicted in Fig. 3.

The microstructure is ferrite with equiaxe grains.
The ferrite grain size is included in 10 to 40 mm.
In the stage I (Fig. 3b), the trace of the grains in which Slips Marks assimilated at Persistant Slips Bands (PSB) have appeared as previously published [1, 10] are visible.
For this test where the temperature was monitored, the temperature profile along the specimen width was extracted for the 168 latest pictures captured by the camera (Fig. 5a), that is to say from 3.2086.107 cycles to 3.2153.107 cycles (number of the cycles at the failure).
Assuming that for R =-1, ΔKeff ≈ Kmax, a plot of ΔKeff at the transition between the initiation and transition propagation stage according to the fatigue stress amplitude (Fig. 8) and number of cycles (Fig. 9) shows that this crack propagation threshold is constant and equals to 3.99 MPa√m.

Longer holding time however gradually diffused Cr2N at the ferrite grains at the substrates.
Numbers of successful techniques have been used in increasing the wear resistance and surface hardness of DSS.
The diffusion of Cr2N seem to happen mostly at the α grain and gradually increased according to longer holding treatment time.
This occurrence happen may be due to higher chromium content at the α grain than γ grain.

Compared to counterparts of Bi0.9La0.1FeO3 (BLFO) film, the grain refinement of all films is obvious.
BLFSO thin films show a uniform grain structure with a mean grain size of 80–100 nm.
In the Sc-doped BLFO thin films, grains are refining and packed more densely and the surface are smooth.
Moreover, compared to BLFO, the grain boundary tends to be ambiguous gradually with the increase of Sc doping level.
It has been reported that remanent polarization of perovskite-type ferroelectric material are affected by various factors, such as the displacement of polar ions, domain pinning by defects, and orientation [18-20], In this study, enhanced values of 2Pr seem to be related to Sc-substitution of Fe-sites by proper cations in BLFO thin films to effectively reduce the number of oxygen vacancies.

The grain refinement effect increases and then decreases with increasing pressure.
And these defects can provide a number of new nucleation sites, resulting in increasing neucleation efficiency[9].
The reason for this is that the atoms on the grain boundaries are disordered and many vacancies exist, resulting in the lower density and the higher energy than that in the grain.
And the grain with a higher potential would be protected as a cathode.
It is well known that the finer the grain of the alloy is, the larger the proportion of grain boundary area is.

The LSMO properties are modified by sintering temperature, which ruled the grain size and grain boundary properties [3, 4].
The SEM images showed LSMO samples have different grain size where the grain size increases with increasing sintering temperature.
This implies a decreasing number of grain boundaries by increasing sintering temperature, in agreement with Ref. [3].
Table 1 summarized the properties of grain size, µHC and MSP of LSMO samples.
We remark that MR values are not ruled by the physical properties i.e. grain size as reported in Refs. [3].

To avoid grain coarsening of the microstructure of sintered specimen, SPS was conducted below the γ prime temperature, which is about 1140 °C (1413 K) [29].
Experimental record of a number of SPS-parameters for sintering Ni-alloys powders at 1273 and 1323 K and a hold-time 15 min: a) undoped (as-received) and b) Pt-doped.
Fig. 2 shows experimental record of a number of SPS parameters such as the densification of Ni-alloys powders as function of time.
During this period, grain boundaries were deformed and reorganized, especially due to the plastic deformation of the β phase rich in aluminum.
A short hold-time also helps to avoid grain ripening due to diffusion controlled by phase transformations.

Huadong Fu [8] et al. reported that the grain orientation and precipitation were found to be crucial parameters for obtaining significant superelasticity.
When the number of variables increases, it becomes more complex and time-consuming [12,13].
It reduces the number of experiments as well as quantifies the influence of process variables on the output.
This is due to the occurrence of recrystallization at 1150℃.New grain formation takes place,leading to grain refinement, which enhances the strength of the material and thereby developing greater resistance to deformation.
At high temperature (1200℃) and low strain rate (0.1s-1), recrystallized grain growth occurs, which decreases the strength of the material.

In order to produce the desired surface texture and remove the physisorption layer, a part of the investigated samples was mechanically processed with a coated abrasive tool (grain P320) for 30 seconds.
The number of necessary measurements in the main tests was defined on the basis of the scatter analysis and assumed materiality level [12].
“i”, y - arithmetic mean, n- total number of measurements.
Table 4 lists roughness profiles of the surface of the samples made of 316L steel before and after mechanical treatment with a coated abrasive tool (grain P320).
Notably, the level of the polar component of the SFE after mechanical treatment with a coated abrasive tool (grain P320) more than doubled in comparison to samples prior to mechanical treatment. 3.

Total number of experimental runs is 9.
Nugget region has grain structure which is randomly oriented.
The grain size is finer than base metal at nugget zone.
The grain size normally increases with higher rotational speed due to high heat input.
Observed higher strength of weld can be attributed to refined grain structure.