Abstract
Methodologies to quantify the microstructural homogeneity, or uniformity, have been developed based on the proposed statistical homogeneity theory. Two kinds of homogeneities are considered, for the size and orientation distributions, respectively. In the case of size distribution, the homogeneity is quantified using two parameters, H 0.1 and H 0.2, which are defined as the probabilities falling into the ranges of μ ± 0.1μ and μ ± 0.2μ, respectively, where μ is the mean size. Whereas in the case of orientation distribution, three parameters are used to quantify the homogeneity: H R, the mean resultant length that is a simple measure of the angular data concentration, and H 0.1 and H 0.2, which are the probabilities in particular angular ranges of the circular or spherical data. These homogeneity quantities are formularized using the common statistical models, and typical examples are demonstrated.
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Abbreviations
- a, b, c:
-
Grain axes
- COV or σ/μ:
-
Coefficient of variation
- COV(d mean):
-
Coefficient of variation of the mean near-neighbor distance
- \( \left( {\overline{C} ,\,\overline{S} } \right) \) :
-
Coordinates of mean resultant vector
- D, D0.1 and D0.2:
-
Dispersion quantities
- f :
-
Probability density function
- G v :
-
Grain homogeneity parameter
- H, H0.1 and H0.2:
-
Homogeneity quantities
- H R :
-
Directional homogeneity quantity
- HPq :
-
Dimensionless homogeneity parameter
- K :
-
Curvature
- L :
-
Length
- N :
-
Total number, or measurement number
- p :
-
Fitting parameter
- \( \overline{R} \) :
-
Mean resultant length
- r :
-
Correlation coefficient
- s :
-
Sample standard deviation
- U size :
-
Size uniformity
- U sp :
-
Spatial uniformity
- \( \overline{V} \) :
-
Mean volume
- \( \overline{x} \) :
-
Sample mean
- (x, y):
-
Mass center
- \( (x_{\text{G}} ,y_{\text{G}} ) \) :
-
Mass gravity center
- \( (x_{\text{S}} ,y_{\text{S}} ) \) :
-
Microstructural center
- α and β:
-
Fitting parameters
- ϕ:
-
Longitude
- κ:
-
Fitting parameter
- μ:
-
Population mean
- θ:
-
Angle, or colatitude
- ρ:
-
Population mean resultant length
- σ:
-
Population standard deviation
- \( \sigma_{\text{ga}} \) :
-
Standard deviation of the grain areas
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Appendices
Appendix 1: Size distribution
Normal and lognormal distribution
The normal and lognormal distributions were studied previously [23]. For a normal distribution,
For a lognormal distribution,
Gamma distribution
The density distribution of a gamma distribution is defined by
Its mean and variance are
Here, the gamma function Γ(α) in Eq. 21 is expressed as \( \Upgamma (\alpha )= \int_{ \, 0}^{ \, \infty } {t^{\alpha - 1} {\text{e}}^{ - t}\,{\text{d}}t} . \) The probability distribution function for the gamma distribution is F(x) = 0 for x ≤ 0 and \( F(x)=\gamma ({\frac{x}{\beta }};\alpha ) \) for x > 0. Here, γ is the incomplete gamma function, \( \gamma (x;\alpha )= {\frac{1}{\Upgamma (\alpha )}}\int_{ \, 0}^{ \, x} {t^{\alpha - 1} {\text{e}}^{ - t}\,{\text{d}}t} , \) whose value can be found in statistical references or online programs.
From Eq. 22, \( \mu /\sigma = \sqrt \alpha , \) thus \( \alpha = (\mu /\sigma )^{2} . \) Hence, the homogeneity
From these two equations, H 0.1 and H 0.2 are functions of a single variable μ/σ. For convenience, they are regressed as
Note that the gamma distribution yields a special case of Erlang distribution when α = k is an integer, or the chi-square distribution when α = ν/2 and β = 2 [24]. As H 0.1 and H 0.2 are only related to the ratio μ/σ but not related to specific α or β, the homogeneity equations, as expressed in Eqs. 23a, 23b and 24a, 24b, remain valid for both Erlang and chi-square distributions.
Weibull distribution
The density distribution of a Weibull distribution is defined by:
Its mean and variance are
Thus, we have
From Eq. 27, the relationship between α and (μ/σ) can be established, and their relationship is expressed as:
The probability distribution function for the Weibull distribution is F(x) = 0 for x ≤ 0 and \( F(x) = 1 - {\text {e}}^{{ - (x/\beta )^{\alpha } }} \) for x > 0. Therefore, the homogeneity degree of the Weibull distribution is:
Here, \( \mu /\beta = \, \Upgamma \left( {{\frac{\alpha + 1}{\alpha }}} \right) \) according to Eq. 26. For a given μ/σ, the value α can be evaluated from Eq. 28, and thus H 0.1 and H 0.2 can be calculated. For convenience, these equations are regressed as:
Note that the Weibull distribution yields a special case, Rayleigh distribution, by taking α = 2 and \( \beta = \sqrt 2 \eta \) [24]. As H 0.1 and H 0.2 are not related to specific α or β values in these H equations, they can also be used for the homogeneity quantification of the Rayleigh distribution.
Beta distribution
The beta distribution is defined as:
where α and β are two independent fitting parameters, and B(α,β)=Γ(α)Γ(β)/Γ(α+β). Its mean \( \mu = \alpha / (\alpha + \beta ) \) and variance \( \sigma^{2} = \alpha \beta / [ (\alpha + \beta )^{ 2} (\alpha + \beta + 1 ) ] , \) and thus \( (\mu /\sigma )^{2} = \alpha (\alpha + \beta + 1 ) /\beta . \) For a given pair of α and β, particular H values can be computed numerically by integrating \( f(x) \) in Eq. 31 over the range of μ ± 0.1μ or μ ± 0.2μ. Figure 8 shows some of the computation results of the beta distribution for 0 < α, β ≤ 10, as plotted as H versus \( (\mu /\sigma )= \sqrt {\alpha (\alpha + \beta + 1 ) /\beta } . \) The normal distribution is included for comparison. Different to the previous distributions, the homogeneity parameters H 0.1 and H 0.2 here are functions of the ratio μ/σ as well as α (or β), i.e., for a given μ/σ, H 0.1, and H 0.2 are no longer monotonic but may possess different values depending on α or β. Only when α = β, H 0.1 and H 0.2 are monotonic functions of μ/σ, which as expressed as:
As shown in Fig. 8, the H values with α = β are highlighted, which are getting close to the normal distribution at higher ratio μ/σ.
Moreover, the homogeneity H functions of the uniform, exponential, Laplace, Pareto, or any other distributions [24] can be formularized from their specific integration forms, while these models are not commonly used for the size distribution modeling.
Appendix 2: Orientation distribution
Circular data
Wrapped normal distribution
The density distribution of the wrapped normal distribution is described as
Its mean is μ, and mean resultant length is ρ. By numerical computation, the relationship of H R, H 0.1, and H 0.2 with ρ are obtained, as shown in Fig. 9, which are expressed as follows:
Wrapped Cauchy distribution
The density distribution of the wrapped Cauchy distribution, with mean direction μ and mean resultant length ρ, is described as:
The relationships of H R, H 0.1 and H 0.2 with ρ are shown in Fig. 10, which are expressed as follows:
von Mises distribution
The widely used von Mises distribution, with mean direction μ, is described by a density function of
where I 0 denotes the modified Bessel function of the first kind, with order 0 (p = 0). In general with the order p, it is defined by
Its mean resultant length is \( \rho = I_{1} (\kappa )/I_{0} (\kappa ). \) According to these relationships, H R, H 0.1, and H 0.2 are numerically computed, as shown in Fig. 11. In the range of κ ≤ 20 that covers most practical values, they are expressed as:
Cardioid distribution
The cardioid distribution, with a mean direction at μ, has the density function as follows:
Its mean resultant length is ρ, thus
According to Eq. 40, the density function at ρ = 0.5, 0, −0.5 are shown in Fig. 12a. When ρ = 0.5, the maximum f is at θ = 0, so μ = 0. Therefore, H 0.1 and H 0.2 include areas on the right side of θ = 0 and left side of θ = 2π, as shown in the shadowed areas in Fig. 12a. While when ρ = −0.5, H 0.1 and H 0.2 are obtained from the areas with μ = π. Along with H R, the calculated results of H 0.1 and H 0.2 are shown in Fig. 12b, which are computed as:
Triangle distribution
The triangle distribution, with mean direction at μ, has the density function as follows [33]:
From this function, it is computed that
For other distribution models, H R, H 0.1, and H 0.2 then can be computed in similar ways. Even when the ρ of a model is not available, for example, the nonsymmetrical models given by Fu and Lauke [8], the H 0.1 and H 0.2 can be calculated numerically.
Spherical data
Fisher distribution
The density function of Fisher distribution [35], with mean direction along the z axis, is expressed as:
where κ is the concentration parameter, which controls the curved shape, as plotted for some values in Fig. 13a. Its
Accordingly, the H values are calculated for different κ (Fig. 13b), and regressed as:
Watson distribution
The density function of Watson distribution [35], with mean direction along the z-axis, is expressed as
where \( C_{\text{W}} = 1/ [4\pi \int_{ \, 0}^{ \, 1} {{\text{exp(}}\kappa u^{2} ) {\text{d}}u} ] \). When κ ≥ 0, the distribution f is plotted in Fig. 14a, with H 0.1 and H 0.2 areas indicated. As the another half at the supplementary angle (π − θ) in not included, the maximum H value is 0.5. Its ρ is not available. However, the H quantities are calculated numerically as shown in Fig. 14b for 0 ≤ κ ≤ 40, which are expressed as:
However when κ < 0, as plotted in Fig. 14a, the f maximum value is at 0.5π, i.e., at the equator, which is the girdle case. It is more reasonable to define the homogeneity, denoted as H′, around 0.5π, as shown in Fig. 14a. Within −40 ≤ κ < 0, the H′ is expressed as:
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Luo, Z.P. Statistical quantification of the microstructural homogeneity of size and orientation distributions. J Mater Sci 45, 3228–3241 (2010). https://doi.org/10.1007/s10853-010-4330-x
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DOI: https://doi.org/10.1007/s10853-010-4330-x