Journal of Materials Science

, Volume 45, Issue 12, pp 3228–3241

# Statistical quantification of the microstructural homogeneity of size and orientation distributions

Article

## 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.

### List of symbols

a, b, c

Grain axes

COV or σ/μ

Coefficient of variation

COV(dmean)

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

Gv

Grain homogeneity parameter

H, H0.1 and H0.2

Homogeneity quantities

HR

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

Usize

Size uniformity

Usp

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

## Notes

### Acknowledgement

The author thanks three reviewers for their in-depth critical comments and constructive suggestions to improve this article.

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