Statistical modelling the scatter of properties from sampling of inhomogeneous materials

A theory dealing with the distribution of properties from notch-sampling of inhomogeneous microstructures is proposed. Various relationships and efficient estimates have been derived regarding the variance of distribution mixtures resulting from notch-sampling of inhomogeneous microstructures. It is demonstrated that the variance of a distribution mixture can be minimised if the individual distributions (components), whose means lie far from the common mean of the distribution mixture are sampled with small probability. It has also been demonstrated that the empirical cumulative distribution of the intercepts from the microstructural constituents is a very important `fingerprintʹ of the inhomogeneous microstructures, which contains valuable information regarding the risk of poor properties. In cases where the scatter in the investigated property depends predominantly on the intercept from the second structural component, very efficient probability bounds of the property are proposed based on the mean order statistics of the intercepts determined by a Monte Carlo simulation. The latter consists of random placement of the notch on microstructural sections and building the distribution of the intercepts. It has also been demonstrated that maximising the log-likelihood function by methods which use only function values (no derivatives) is an efficient way of splitting distribution mixtures consisting of two normal components.