Stereological Estimation of Anisotropic Microstructural Features

Abstract

Microstructural banding is explored in 2D and 3D using serial sectioned micrographs. The banding is quantified using two parameters called the Band Continuity Index Cb and the Perpendicular Continuity Index Cp. The indexes determined from the 2D micrographs is shown to be sufficient for estimating the distribution of the bands in 3D. A stereological model employing oriented circular cylinders is established to represent the bands in steel microstructures. This model assumes that cylinders are distributed randomly inside of a box with their symmetry axes oriented in the same direction. The box is then cut parallel to the symmetry axes and rectangles are observed on the cut plane. The inverse relationship between the rectangles and the cylinders is established, and the distribution functions and expectations various quantities of interest, such as the cylinder radius, height, surface area and volume, as well as the covariance between the radius and height, are estimated directly from the rectangle observations. The asymptotic behavior of the empirical, isotonic and kernel smoothed estimators is determined and used to obtain confidence intervals for the expectations of the quantities of interest when the model is applied to the steel microstructures. The limitations of the model are explored and the model is shown to be reasonably robust against deviations from its rather stringent requirements. This makes the model useful in a practical setting, and the results obtained for the steel microstructures are reasonable. This model is not limited in application to banded microstructures, but can also be used for any microstructural feature that can be thought of as being plate-like, rod-like or needle-like. Features such as inclusions, voids and dendrites often take on these shapes.