Communication Modeling Anisotropic Multiphase Heterogeneous Materials via Directional Correlation Functions: Simulations and Experimental Verification SUDHANSHU S. SINGH, JASON J. WILLIAMS, YANG JIAO, and NIKHILESH CHAWLA The effective properties of heterogeneous materials critically depend on their complex microstructure. In this article, by successfully reconstructing the three- dimensional microstructure of a multiphase alloy with orientated anisotropic inclusions from two-dimensional slices, we show that such materials can be modeled by the two-point correlation functions of the inclusion phases along the three orthogonal characteristic directions of the inclusions. The reconstructions are compared to the actual microstructure obtained from high-resolution X-ray tom- ography experiments by quantifying certain directional cluster statistics associated with the microstructures. DOI: 10.1007/s11661-012-1451-7 Ó The Minerals, Metals & Materials Society and ASM International 2012 Heterogeneous materials, such as composites materi- als or metallic alloys, are ubiquitous. In composites, the effective properties are determined by their complex microstructures (e.g., the volume fraction, morphology, and spatial distribution of different phases). [1,2] Alumi- num alloys and steel almost always have second phase inclusions and particles with impurities that are present in the microstructure. A quantitative understanding of the structure–property relationship in such materials has begun to emerge over the past few decades, mainly because of the development of advanced experimental and computational materials microstructure character- ization techniques. [3] In particular, X-ray tomography techniques have been widely used to obtain high-resolution three-dimen- sional (3D) microstructure for a wide range of hetero- geneous materials. X-ray tomography is an excellent technique that eliminates destructive cross-sectioning, and allows for superior resolution and image quality with minimal sample preparation. [4,5] 3D visualization and quantification of heterogeneous microstructures by X-ray tomography have been successfully performed in Sn-rich alloys, [6] powder metallurgy steels, [7] metal matrix composites, [8–12] and lightweight alloys. [13–17] In addition to visualization, such microstructural datasets can be incorporated into finite element models to predict the onset of local damage mechanisms and the macro- scopic deformation behavior. [18–22] One of the most time-consuming parts of the 3D X-ray tomography process is segmentation of gray-scale images and 3D reconstruction of the segmented image dataset. Such a 3D dataset is a prerequisite for quan- tification of salient microstructural features. [11,23] A simpler and more efficient method for developing realistic 3D microstructures, which are statistically and visually descriptive of the actual microstructure of the material, is required. Recently, it has been suggested that the complex microstructures of a wide class of heterogeneous mate- rials can be modeled by certain statistical morphologic descriptors associated with the materials, i.e., lower- order spatial correlation functions of the material phases. [24,25] Efficient microstructure’s reconstruction procedures incorporating such correlation functions have been developed, which enable one to ascertain the amount of structural information contained in these statistical descriptors. [26–32] Specifically, the standard two-point correlation functions S 2 (defined below) have been employed to successfully characterize statistically homogeneous and isotropic heterogeneous materi- als, [24,28] although in general, S 2 is not sufficient to uniquely determine a microstructure. [33,34] In this letter, we report on a methodology to model anisotropic multiphase heterogeneous materials via directional two-point correlation functions. In particu- lar, we show that the microstructure of an aluminum (Al) alloy with orientated anisotropic Fe-rich and Si-rich inclusions in an Al alloy matrix (due to rolling of the alloy) can be successfully reconstructed from the S 2 associated with the inclusion phases. More importantly, the information is computed from two-dimensional (2D) slices of the material perpendicular to a set of orthog- onal directions. Figure 1 shows the 3D microstructure of the alloy obtained from the X-ray tomography experiment. It can be clearly seen that both the Fe-rich and Si-rich inclusions are elongated along the longitu- dinal direction, leading to an overall anisotropic micro- structure. Also shown are three 2D slices perpendicular to the three characteristic directions associated with the anisotropic inclusions, i.e., the longitudinal (L), trans- verse (T), and short transverse (S) directions. Details of the X-ray tomography process, conducted at the Advanced Photon Source, Argonne National Labora- tory, are reported elsewhere. [35] In general, the microstructure of the Al alloy with anisotropic inclusions in the Al matrix can be uniquely determined by specifying the indicator functions asso- ciated with all of the individual phases (the inclusions and the matrix): [1] SUDHANSHU S. SINGH, Graduate Research Assistant, JASON J. WILLIAMS, Research Scientist, and NIKHILESH CHAWLA, Fulton Professor, are with the Materials Science and Engineering, Arizona State University, Tempe, AZ 85287-6206. Contact e-mail: Nikhilesh.Chawla@asu.edu YANG JIAO, Postdoctoral Fellow, for- merly with the Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton, NJ 08540, is now Assistant Professor with Materials Science and Engineering, Arizona State University. Manuscript submitted: July 9, 2012. METALLURGICAL AND MATERIALS TRANSACTIONS A