Modeling Heterogeneous Materials via Two-Point Correlation Functions: I. Basic Principles

TL;DR

This paper develops a framework for modeling heterogeneous materials using two-point correlation functions, introducing new realizability conditions, basis functions, and an efficient algorithm for generating material microstructures.

Contribution

It presents a novel approach to model and categorize heterogeneous materials through two-point correlation functions, including new realizability conditions and an isotropy-preserving generation algorithm.

Findings

New realizable two-point correlation functions provided

Efficient Lattice-Point algorithm developed

Framework enables material microstructure modeling

Abstract

Heterogeneous materials abound in nature and man-made situations. Examples include porous media, biological materials, and composite materials. Diverse and interesting properties exhibited by these materials result from their complex microstructures, which also make it difficult to model the materials. In this first part of a series of two papers, we collect the known necessary conditions on the standard two-point correlation function S2(r) and formulate a new conjecture. In particular, we argue that given a complete two-point correlation function space, S2(r) of any statistically homogeneous material can be expressed through a map on a selected set of bases of the function space. We provide new examples of realizable two-point correlation functions and suggest a set of analytical basis functions. Moreover, we devise an efficient and isotropy- preserving construction algorithm, namely, the Lattice-Point algorithm to generate realizations of materials from their two- point correlation functions based on the Yeong-Torquato technique. Subsequent analysis can be performed on the generated images to obtain desired macroscopic properties. These developments are integrated here into a general scheme that enables one to model and categorize heterogeneous materials via two-point correlation functions.