Distribution mixtures from sampling of inhomogeneous microstructures: variance and probability bounds of the properties

Abstract The mechanical properties of inhomogeneous materials are often characterised by distribution mixtures. Alternative equations regarding the variance of a distribution mixture arising from sampling of inhomogeneous microstructural zones are derived and the concept ‘contributed variance’ of the individual distributions is introduced. On its basis a methodology is developed which can be used to identify sampling from which set of individual distributions is accountable for most of the variation of a distribution mixture. In terms of inhomogeneous microstructures this means identifying sampling from which microstructural zones is accountable for most of the scatter in mechanical properties. It is also demonstrated that an efficient way of establishing sharp probability bounds for the mechanical properties from sampling of different microstructural zones is splitting the corresponding distribution mixture into its components and subsequent integration of the obtained probability density. The equations derived and the methods developed are useful tools for structural reliability calculations.