Distribution of properties from sampling inhomogeneous materials by line transects

Abstract

A theory related to the statistics of the structure and properties of inhomogeneous materials has been proposed, dealing with the distribution of properties from sampling of inhomogeneous structures by line transects. It is demonstrated that the empirical cumulative distribution functions of the intercepts from the microstructural constituents and their variances are important fingerprints of the inhomogeneous structures. Simulated sampling of the images of the microstructure by line transects has been proposed as a powerful method for characterization the distribution of properties of inhomogeneous structures. One of the advantages of the method based on the expected values of the order statistics of the intercepts is that the probability bounds of the property are determined without prior knowledge of the functional relationship between the property and the intercepts from the structural constituents. The concept intercept variance has an important application in determining the minimum transect length that stabilizes the variation of the intercept at a low value. Additionally, the concept ‘intercept variance’ can be used for topological optimization of the microstructure regarding the risk of intercepting a large amount of the weaker constituent. Equations regarding the variance of the intercepted fraction characterizing transect lengths with a specified distribution have also been derived.