Analysis of internal structures of composite materials by second-order property of mosaic patterns

Properties of composite materials depend on their internal structures which in most cases are highly random. Therefore, high-order information is often required for quantitative characterization of morphological features in spatial distributions of the constituent phases. The present study develops a general expression for a second-order moment, that is, autocorrelation function Rxυ, of multiphase heterogeneous composites. The obtained Rxυ depends on three types of parameters: volume fractions of the constituent phases, values of the phase function, and π (the probability that two points with distance υ are in the same basic cell). π depends on the constitution of the basic cell, and thus the specific characteristics of the internal structure of the material. To determine completely the autocorrelation function, probability π is derived based on a morphological model called mosaic patterns. Derivation of π for two types of mosaic patterns indicates that π [and thus Rx(υ)] has different formulations for different internal structures, although all of them depend on the same set of parameters: coarseness of microstructures, λ, volume fraction of constituent phases, Φ, and values of the chosen phase function. For a two-phase composite, with increasing Φ1 (volume fraction of phase 1), the value of Rxυ increases; while, with increasing λ, the structure of the composite becomes more and more fine grained, and thus the correlation between two points with a fixed distance υ becomes weaker. As an example of applications of the present result, the autocorrelation function is applied to predict the fluctuation of local volume fractions of two-phase composite materials.