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Fig. 1 shows the general algorithm for Markov Chain - Monte Carlo based model of intergranular crack propagation where N is the number of Monte Carlo steps, n is the number of different GBCDs, M is the maximum crack length and Lavg is the average crack length.
[Rh] = [V(A1), V(A2), V(A3), … … … …, V(An)] [Ri1] = [V(B1), V(B2), V(B3), … … … …, V(Bm)] [Ri2] = [V(C1), V(C2), V(C3), … … … …, V(Cm)] n = number of type 1 TJ m = number of type 2 TJ TJ type 1: T11(i) = 1 - V(Ai) T12(i) = V(Ai) i = 1, 2, 3, … … …, n TJ type 2: T22(j) = 1 - {V(Bj)+V(Cj)-V(Bj)*V(Cj)} T23(j) = V(Bj) T24(j) = V(Cj) j = 1, 2, 3, … … … …, m C1 1 3 4 A1 B1 2 Table 1: Transition Matrix for Fig. 2 Table 2: Grain Boundary Ranking for HM The initial crack location can be expressed in matrix form of the same column size of Tr: p0 = [1 0 0 0]
The susceptibilties (hypothetical) of grain boundaries are given in Table 2.
Contrary to the equiaxed hexagonal grain structure, the voronoi microstructure (see, e.g.
It will enable one to incorporate different grain shapes, grain size distributions etc. in the model microstructure and, determine their effects on the crack propagation behaviour.

For the latter case, even if the number fraction of larger grains in the nanostructure is low, their volume fraction can be sufficiently high to contribute to dislocation-based plasticity in the material [2].
The material was processed using the so-called route Bc, as detailed in [18], but for number of passes as high as 16.
It is clear from Fig. 2a that the strength of the material increases gradually with the number of passes from 1 to 8.
(a) 10 µm 5 µm (b) 5 µm 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 5500 6000 0 10 20 30 40 50 60 70 80 90 100 100 Above 1000 nm 500-1000 nm 100-500 nm Number of grains (%) Grain diameter (nm) milled with Y milled without Y atomized Figure 4: Graph showing the number of grains and their average size within the +100/-500 nm, +500/-1000 nm and +1000 nm grain size fractions for different types of initial powder (atomized FeAl powder, milled FeAl powder, milled FeAl powder with Y2O3 addition).
Fig. 4 gives a plot showing the number of grains contained within the +100/-500 nm, +500/-1000 nm and +1000 nm grain size fractions for the case of the processing of different types of FeAl powders.

The constitutive relations of grain boundry.
Periodic assumption is used to ensure the grain geometry is continuum and material parameters of grain interiors are uniform in the grains of each boundary of the opposite edges pair.
In the present study, the intragranular material properties are allowed to vary from grain to grain but the individual grains are taken to be isotropic with different Young moduli, and parameters of cohesive interfacial model are the same for all grain boundaries in the sample.
From the cyclic stress-strain curves result we can find that cyclic strain hardening is more obvious with the increase of tension-compression cycle numbers.
Meanwhile the plastic dissipation energy is smaller with the with the increase of cycle numbers, in which intergranular cracks are initiation and extended.

Impedance spectrum was analysed to obtain the behaviour of grain core and grain boundary responses by a fitting procedure using brick-layer model.
Further calculation of grain size by means of ImageJ software revealed that the grain size of the modified BCZY was around 88 nm.
Grain arc could not exist at such temperature due to the incapability of impedance spectroscopy to measure grain response according to Barison S. et al. [7].
Consequently, the total effect on the conductivity is smaller than that of microcrystalline materials, although the number of grain boundaries per unit length increases in smaller grain size condition and eventually provide higher conductivity value for nano-sized sample.
The Electrical Characterization of Grain Boundaries in Ultra-Fine Grained Y-TZP, Mater.

Type-I AEs with higher frequency components were detected during the pit growth and supposed to be produced by falling-off of surface grains due to intergranular attack, while a number of Type-II AEs (approximately 12,500 counts) with low frequency components were detected during SCC propagation and supposed to be produced by cracking of the chromium oxy-hydroxides.
We concluded that AEs were produced by falling-off of grains.
No AE data in the three gray bands means the exceeded memory capacity of the personal computer, since a number of AE signal was produced during this period.
The first AE was detected at 320 ks and the number of AEs increased gradually.
Cracking of the oxide produces a number of secondary AE.

The porosity decreased and the grain size increased with increasing the number of coatings.
The thickness of the PZT-6(6: number of coatings) films was about 60~65µm.
The grain size and the densification of the thick films increased with increasing the number of sol coating.
It can be understood in terms of the effect of the increment of grain size and the decreasing porosity, as shown in Fig. 2.
These properties can be understood in terms of the effect of the increasing densification and the ferroelectric grain growth with increasing the number of 0 2 4 6 30 40 50 60 70 Atomic percent [%] Number of coatings Zr Ti (a) PZT-0 (b) PZT-2 (c) PZT-4 (d) PZT-6 (e) PZT-0 (f) PZT-2 (g) PZT-4 (h) PZT-6 0 2 4 6 250 300 350 400 450 500 550 Relative dielectric constant Number of coatings PZT(30/70) precursor solution coatings[7].

The sequence of phase transformations of zirconium-doped ultrafine-grained alloy Al-Mg-Li in heating is revealed.
Temperature dependence of the lattice parameter for ultrafine-grained alloy at in-situ heating in the diffractometer chamber (1).
After annealing at 380 °C and subsequent quenching metastable S-phase particles of type I are observed to dissolve with increasing number and volume fraction of more stable S-phase particles of type II that are precipitated at high-angle boundaries of general-type grains (Fig. 2b).
With increasing temperature the coagulation of grain-boundary S-phase particles leads to recrystallization.
Microstructure of the ultrafine-grained alloy after 20-minute annealing at the temperatures 300°С (a) and 380°С (b).

The chord length l of a grain is the length of the intersection of the grain with a test line.
In traditional stereology, the number of intersections Nis of a (random) test line of length L with grain boundaries is counted.
The number density NL = Nis/L of intersections is the inverse of the mean chord length ¯l = 1/NL.
From the number Ncr of such occurrences and the total line length L = N∆x the intersection density NL = Ncr/N∆x and the average chord length ¯l = N∆x/Ncr can be inferred.
The accumulated frequency distribution Pθ(θcr) = 1− Ncr N (1) is intimately linked to the number Ncr of incidences of point pairs with angles above the critical angle.

As the number of ARB cycles increased to 3, the cell structure was well defined and the dislocation density increased (Fig. 2b).
However, it exhibited an ultrafine grained structure in which the grains hardly contain the dislocations in the interior and were surrounded by sharply defined and straight boundaries.
The SAD pattern clearly indicates that the misorientation between the grains is very large.
However, the grain boundaries were sharply defined.
The mean grain size of ultrafine grains formed by LDR-ARB was smaller than that by HDR-ARB.

Since it included large number of high-misorientation grain boundaries, the foil was miserably brittle with no sign of plastic deformation in the tensile test [15].
However, after the grain growth at 1273K/0.5h, most of grains had {110} textures (green-colored) with few high-misorientation grain boundaries.
Other types of grains exist but are minor.
Note that among these minor grains only the grains with the cold-rolled texture are surrounded by the special, high-mobility grain boundary in the 40˚{TTP}730 <111> rotated recrystallization texture.
Then, in the next stage, i.e. in the grain growth, only the grains with the cold-rolled texture have the advantage of growing faster over the other grains, yielding the texture returning to the original, cold-rolled texture.