em

llw

Uni

am

ed fne 3

The complete set of 2-point correlations for a composite material system with a large number of local states (e.g. polycrystalline met-

At its core, the eld of materials science and engineering

It is not usually practical or even necessary, to quantifycompletely the microstructure of a large sample in its fullspatial extent. The assumption of statistical homogeneity

domains extracted randomly from the sample. A rigorous

point selected randomly in the microstructure. Expandingon this basic concept, the 2-point correlations, f h; h0 j r,capture the probability density associated with nding anordered pair of specic local states at the head and tail ofa randomly placed vector r into the microstructure. In avery similar manner, n-point statistics can be extracted

* Corresponding author.E-mail address: skalidin@coe.drexel.edu (S.R. Kalidindi).

Available online at www.sciencedirect.com

Acta Materialia 56 (2008) 528552is concerned with understanding and modeling the rela-tionships between a materials internal structure, itsmacro-scale properties and its processing history. Funda-mental to establishing these relationships is the quantita-tive representation of the materials internal structure,which includes not only an identication of the constituentlocal states, but also their spatial placement. As interest istypically in the micro-scale features, the materials internalstructure is referred to as the microstructure in this paper.However, the approach and the results presented here canbe applied at any length scale.

framework dening the statistics of the microstructure isavailable in the literature in the form of n-point correla-tions or n-point statistics [15]. These correlations providea hierarchy of statistical measures of the microstructure.The simplest of these are the 1-point correlations, f(h),which essentially reect the volume fractions of the variousdistinct constituents (denoted by h and also referred to aslocal states; the complete space of local states is referredto as the local state space H). These are termed 1-point sta-tistics because they reect the probability density associ-ated with nding a specic local state of interest at aals) forms a vast and unwieldy data set containing a large amount of redundant information. The interrelations in these correlations havebeen well characterized for composite material systems with two local states, but only a small number have been delineated for the com-posite systems with many local states. This paper presents an analysis of interrelations between the complete set of 2-point correlationsfor composite material systems through their spectral representations via discrete Fourier transforms. These interrelations are used todelineate a compact and convex space that bounds the set of all physically realizable 2-point correlations called the 2-point correlationshull. The representation of any given microstructure in this hull, and the techniques to produce a representative volume element are alsoexplored in this paper. 2008 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: 2-Point correlation; Microstructure; Discrete Fourier transform; Pair correlation function; Statistics

1. Introduction permits one to approximate the relevant statistics of inter-est in the sample as an average from an ensemble of sub-Delineation of the spacin a composite

S.R. Niezgoda a, D.T. FuaDepartment of Materials Science and Engineering, Drexel

bMechanical Engineering Department, Brigh

Received 4 March 2008; received in revisAvailable onli

Abstract1359-6454/$34.00 2008 Acta Materialia Inc. Published by Elsevier Ltd. Alldoi:10.1016/j.actamat.2008.07.005of 2-point correlationsaterial system

ood b, S.R. Kalidindi a,*

versity, 32nd Chesnut Street, Philadelphia, PA 19104, USA

Young University, Provo, UT 84602, USA

orm 30 June 2008; accepted 1 July 2008August 2008

www.elsevier.com/locate/actamat

92rights reserved.

from the microstructure through the placement of a com-plete set of n-vertex polyhedra.

This paper focuses on the 2-point correlation functions

structures that are theoretically predicted to exhibit adesigner-specied combination of macro-scale propertiesor performance characteristics. A critical element ofMSD is the delineation of the complete space of physicallyrealizable microstructures, which for several of the prob-lems being currently studied amounts to the delineation

5286 S.R. Niezgoda et al. / Acta Materialia 56 (2008) 52855292for composite material systems. Two-point correlationshave been the focus of several investigations in current lit-erature [610]. They capture the rst-order information onthe morphology of the microstructure. There is a tremen-dous leap in the amount of information contained in the2-point correlations, when compared with the 1-point sta-tistics. For most real microstructures, the 2-point correla-tions form a very large and unwieldy set of data. Forexample, in single-phase polycrystalline samples, the localstate is dened by the lattice orientation g, which is oftendened by a set of three angles called the BungeEulerangles [11]. Thus, for this relatively simple class of materi-als, the 2-point correlations need to be dened in a spaceconsisting of nine independent variables. The continuousorientation and spatial spaces spanned by these variablesare often discretized by binning. For example, a coarse dis-cretization of the orientation space into 512 bins produces262,144 2-point distributions, each of which is a function ofthe three-dimensional vector r. Fortunately, this completeset of 2-point correlations has a large number of interde-pendencies, only some of which have been outlined byprior studies in this eld [1214]. This paper presents a rig-orous treatment of the interdependencies among thecomplete set of 2-point correlations, leading to the estab-lishment of a compact high-dimensional space whereineach point represents a complete set of 2-point statistics.This is accomplished largely through spectral representa-tion of these correlations, and exploiting several of theknown properties of discrete Fourier transforms (DFT).Although, every physically realizable microstructure1 willhave a representation in this high-dimensional space, it isnot true that every point in this space will correspond toa physically realizable microstructure. The set of pointsin this space that correspond to all physically realizable2-point correlations is referred to in this paper as the 2-point correlations hull. In other words, a point inside the2-point correlation hull will correspond to one or morephysically realizable microstructures, whereas points out-side this hull will not correspond to any physically realiz-able microstructure.

In recent work [1518], a novel mathematical frameworkcalled Microstructure Sensitive Design (MSD) was formu-lated for establishing invertible structurepropertypro-cessing linkages in materials. In a parallel eort, it wasfound to be quite ecient to quantify the microstructureusing 2-point correlations [9,10,15]. Second-order struc-tureproperty linkages for elastic response of multi-phaseand polycrystalline microstructures were successfully estab-lished. MSD aims to identify the complete set of micro-

1 Physically realizable microstructures include all those that can beimagined (or digitally created). It is recognized that a very large fraction of

these have not yet been realized by currently employed materialsprocessing routes.of the space of physically realizable correlations or the 2-point correlation hull mentioned earlier. This paper, usingthe established interdependencies among the 2-point corre-lations in a given microstructure, demonstrates the delinea-tion of the 2-point correlation hull for a highly simpliedclass of one-dimensional microstructures. To the best ofthe authors knowledge, this is the rst report of the suc-cessful delineation of the 2-point correlation hull. Althoughthe example selected for this demonstration deals with agrossly simplied microstructure, the underlying mathe-matical framework presented is quite general and easilyextendable to a much broader class of material systemsand microstructures.

The concepts presented in this paper also have signi-cant implications for the notion of a representative volumeelement (RVE). In characterizing the microstructure of amaterial system, it is necessary to collect microstructureinformation from dierent samples produced with nomi-nally the same processing history, and from dierent loca-tions in each sample. The benet of obtaining multiplemicrostructure data sets is that the averaged statistics fromthis ensemble of micrographs is expected to represent moreaccurately the overall microstructure statistics for thematerial system. It should be recognized, however, that itis not necessarily possible to produce a single instantiationof the RVE that exhibits the ensemble averaged 2-pointcorrelations. With a simple example case study, it is dem-onstrated here that it is often impossible to identify a singleRVE that reects precisely the ensemble averaged 2-pointcorrelations. Viable alternatives for constructing RVEsfrom an ensemble of microstructures are discussed in thispaper.

2. Microstructure function, 2-point correlations and DFT

The application of DFT techniques to the rapid calcula-tion of 2-point statistics has been discussed previously[19,20] and will be reviewed briey here. Consider a micro-structure data set extracted from a material sample. Let thespatial position in the microstructure be identied by thevector x, and the local state of interest by h. One can denethe microstructure function m(x,h) as a distribution on thelocal state space for each spatial position as2 [9].

2 In some prior work,M was used to denote the microstructure function.In this paper, the authors choose to follow the conventions used in thesignal processing literature and use m for the microstructure function inreal space and the M for the Fourier transform of the microstructurefunction. In addition, the Einstein summation convention for repeated

indices is not used. For clarity, all summations will be explicitly indicatedwith R notation.

atemx;hdh hVV

x

;

ZHmx;hdh 1;

ZHdh 1; 1

where V denotes the volume of material probed in theimmediate neighborhood of x, and hV denotes the compo-nent of V that is associated with the local state h to withinan invariant measure dh.

The microstructure function and the local state spaceare assumed to be continuous in the denitions presentedabove. Microstructure data, however, are often collectedin a digital form on a discrete grid of spatial locations. Adiscrete description of m(x,h) is therefore a natural reec-tion of the available information. Let the local state spaceY be binned into N discrete local states labeled n = 1,2, . . .,N. Let the spatial domain of the microstructure be binnedinto a uniform grid of S cells, whose nodes are enumeratedby the ordered triplet [s1,s2,s3] where each component isnumbered si = 0,1,2, . . ., Si1. For convenience, thisordered triplet is represented by the vector s, the vectorfrom the origin to [S11,S21,S31] as S1, and the zerovector as 0. This discrete microstructure function may bedenoted in a condensed form as nms. The following proper-ties can be readily established from Eq. (1)

XNn1

nms 1; 0 6 nms;XS1s0

nms nVS 2

where nV is the volume fraction of the local state n in themicrostructure.

Next, 2-point correlations f h; h0jr are dened as

f h; h0jr 1volX

ZXmx; hmx r; h0dx 3

where X represents the spatial domain of the volume ele-ment of the microstructure being investigated, and volXdenotes its volume. In this denition, it is implicitly assumedthat the volume element being investigated is repeated peri-odically to generate the microstructure of interest. This spe-cic denition is adopted here because of the authorsintention to use DFT representations. In the discretizedrepresentation, the 2-point correlations are expressed as

npft 1SXS1s0

nmspmst; 4

where the complete space of vectors r, introduced in Eq.(3), is conveniently binned using the same uniform grid thatwas used for binning the spatial domain X .

The denitions of the 2-point correlations presented inEqs. (3) and (4) allow their ecient computation usingestablished fast Fourier transform (FFT) algorithms. TheDFT of the microstructure function is expressed as

nMk Inms XS1s0

nmse2pisk=S jnMkjei nhk : 5

The term j nMk j will be referred to as the amplitude of then np

S.R. Niezgoda et al. / Acta MFourier transform and hk as the phase. The DFT of ft iscomputed asnpF k Inpft 1SnMk

pMk

1SjnMkjjpMkjei nhkei phk 6

where nMk denotes the complex conjugate ofnMk. Taking

n = p in Eq. (6), one arrives at a special set of correlationstermed the autocorrelations. Note that the autocorrelationis a real-valued even function, and therefore its DFT arealso real. However, n 6 p leads to cross-correlations, andtheir DFT are, in general, complex.

In a previous work [8], it was found convenient to classifymicrostructures as eigen and non-eigen microstructures. Ifnms 2 f0; 1g (i.e. they are only allowed to take values of either0or 1) for all n, s, thatmicrostructure is referred to as an eigenmicrostructure, otherwise it is non-eigen.

The complete set of 2-point correlations for any givenmicrostructure constitutes a very large data set. If a micro-structure of interest comprises N distinct local states, thecomplete set of its 2-point statistics contains N2 correla-tions for each value of the three-dimensional vector r.The full set of correlations is conveniently visualized asan N N array

npft

11ft 12ft . . . . . . 1Nft

21ft . .. ..

.

..

. . .. ..

.

..

. . .. ..

.

N1ft . . . . . . . . . NNft

26666666664

37777777775: 7

It has long been known that this set exhibits many inter-dependencies. Frisch and Stillinger [21] showed that onlyone of the four correlations is independent for a two-phasematerial (i.e. if 11ft is known, 12ft, 21ft and 22ft can be calcu-lated). More recently, Gokhale et al. [13] showed that, atmost, 1

2NN 1 correlations are independent for a mate-

rial system consisting of N distinct local states. In thispaper, exploiting the known properties of DFT, it is dem-onstrated that the number of independent correlations isonly N1. By careful consideration of the degrees of free-dom of the microstructure and statistics, one can intuitivelyarrive at the correct result. It can be seen from Eq. (2) thatthe degree of freedom in the microstructure function isS(N1). As the 2-point statistics are a derived quantity,the number of degrees of freedom must be less than orequal to that of the microstructure function. In a previouswork [19], it was shown that any original eigen-microstruc-ture can be exactly reconstructed (to within a linear trans-lation and/or inversion) from its statistics, and hence holdsan equivalent information content. In other words the sta-tistics and microstructure must possess the same number,S(N1), of degrees of freedom, and therefore only (N1)correlations can be independent. This constitutes a dra-matic reduction in the size of the data set needed to specify

rialia 56 (2008) 52855292 5287uniquely the complete set of 2-point correlations for anygiven microstructure.

ateThe denition of the discrete 2-point correlations in Eq.(4) requires that npft pnft. In the Fourier space, thisrequirement translates to

npF k 1S jnMkj jpMkjeinhkeiphk pnF k: 8

Furthermore, the product of two of the DFT of the 2-pointcorrelations exhibits the following property:

npF kpqF k 1S2

jpMkj jpMkj jnMkjeinhk jqMkjeiqhk

ppF knqF k: 9Using Eq. (8), Eq. (9) can be recast into a more conve-

nient form as

nqF k pnF k

pqF kppF k

: 10

Eq. (10) is a key result of this paper. This equationimplies that, if pnF k (and by extension pnft) are known forany one choice of p and all n or vice versa (i.e. either a col-umn or a row in the array in Eq. (7) is known), all the othercorrelations can be calculated.

In addition to the redundancies described abovebetween the various 2-point correlations, a number ofother constraints or bounds on the values of npF k exist.The following relationships are easily seen from the deni-tions in Eqs. (2)(6)

npF 0 SnV pV ; 0 6 npF 0 6 S; jnpF kj 6 S2 11These bounds allow one to dene an additional redun-

dancy; Eq. (2) (together with Eqs. (4)(6)) requires that

XNn1

npF k S pV ; if k 00; if k 6 0

12

Since pV is already specied by ppF 0 (Eq. (11)), it followsthat there are only (N1) independent correlations in thecomplete set shown in Eq. (7). Although the interdepen-dencies shown in Eqs. (10) and (12) are formulated betweenthe DFT of the 2-point correlations, the same relationshipscan be expressed in terms of their continuous Fouriertransforms or classical Fourier series. However, as the planis to exploit the signicant advantages associated with FFTcalculation of DFT, all further discussion in this paper willbe restricted to DFT.

Substituting Eq. (11) into Eq. (12) allows Eq. (12) to berecast completely in terms of the DFT as

XNn1

npF k SppF 0

p; if k 0

0; if k 6 0

(13

For eigen microstructures (nms 2 f0; 1g), one additionalredundancy can be established by summing over all fre-quencies in Fourier space asXk

npF k dnpS

p nnF 0

p14

5288 S.R. Niezgoda et al. / Acta Mwhere dnp denotes the Kronecker delta.As a nal note, the reader is reminded that, as all npft arereal valued, there is an implicit symmetry in their DFT thatmay be expressed asnpF k npF Sk 15for all k 6 0. In other words, about half the transforms aresimply complex conjugates of the other half.

3. Hull of 2-point correlations

Having established the interdependencies in a given setof 2-point correlations, attention is now turned to identify-ing the complete set of theoretically feasible sets of 2-pointcorrelations for a given material system. This problem hasbeen tackled in various guises in the past, but never com-pletely solved. For example, using the WienerKhinchintheorem, it has been demonstrated [22,23] that a semi-posi-tive denite 2-point correlation in one-dimensional spacerelates to a real random process. The reader is referred tothe work of Torquato [12,14] and Gokhale [13] for otherimportant results obtained to date using such approaches.

Jiao et al. [24] have recently proposed a novel approachto exploring a certain sub-space of the 2-point correlationspace described here. The space explored by these authorsis called the S2 space and captures only the auto-correla-tions of eigen microstructures. In particular, for compositeswith more than two distinct local states, the S2 space will bemissing the important phase information that is only pres-ent in the cross-correlations (i.e. the nhk described earlier).

Previous work [18] demonstrated the advantages of visu-alizing in Fourier spaces the set of all theoretically feasible1-point statistics in a given material system. In these con-structs, the 1-point statistics of any microstructure are visu-alized as a point in the Fourier space whose coordinates areidentied by the Fourier coecients of the selected 1-pointstatistics. More specically, it was demonstrated that thespace corresponding to the set of all theoretically feasible1-point statistics in a given material system is compactand convex. These constructs have been broadly referredto as microstructure hulls in previous work, and have beenfound to be of tremendous value in designing componentsfor optimized performance characteristics [16]. It is partic-ularly emphasized here that, in material systems with acontinuous local state space, the hulls for 1-point statisticsare most easily established in the Fourier space. This isbecause the values of the distribution (probability densityfunction) used to dene the 1-point statistics could beunbounded for many microstructures. For example, indescribing the 1-point statistics in polycrystalline micro-structures, the values taken by the orientation distributionfunction (ODF) have to be positive, but are otherwiseunbounded. In fact, the ODF for a single crystal micro-structure would be a Dirac-delta function. However, theFourier coecients corresponding to any theoretically fea-sible texture occupy a bounded region, making it very con-

rialia 56 (2008) 52855292venient to visualize the complete set of all theoreticallypossible 1-point statistics in a given material system. In

rithms [26]. All the concepts described above will be clari-ed through a very simple case study in the next section.

4. Case study: one-dimensional two-phase microstructure

As a simple case study that is mainly intended for clar-ication of the various concepts presented earlier, a set ofhypothetical one-dimensional two-phase microstructuresextracted from a larger two-dimensional sample shown inFig. 1 is investigated. These represent multiple instantia-tions of eigen microstructures extracted from a hypotheti-cal sample. The two distinct phases present in thesemicrostructures are shown as black and white, respectively.Furthermore, the spatial domain in these microstructureswas divided into only 10 bins, as shown in Fig. 1. As thereare only two phases, only one 2-point correlation is neededto represent the microstructure. Let 11F k denote the DFTselected for representation in a 2-point hull. As S has beenselected as 10 in this simple case study, the relevant DFT

aterialia 56 (2008) 52855292 5289the present work, because of the use of discrete sampling ofthe microstructure in both the spatial domain and the localstate space, hulls for 2-point correlations could be theoret-ically delineated either in the vector space of pnft or in spec-tral space of pnF k. In fact, the discrete sampling employedin dening pnft is tantamount to the use of a primitive Fou-rier basis called the indicator functions or characteristicfunctions [25], and therefore both descriptions can be inter-preted as Fourier representations. However, as shown inthe previous section, the use of DFT allows many of theredundancies that exist in the set of pnF k to be identiedmuch more easily than in pnft. Therefore, this work focusesexclusively on the delineation of the hulls of 2-point corre-lations in the DFT space.

The concept of eigen microstructures described earlier,where all nms values are either 0 or 1, is central to the con-struction of the microstructure hulls. Let pn~F k denote theDFT of the 2-point statistics of an eigen microstructureof a material system of interest. Let ~H denote the set ofall theoretically feasible pn~F k. Then the convex regiondened by the elements of ~H represents the hull of 2-pointcorrelations sought here. Eqs. (10)(15) explicitly denethe hull in a 2N 3dS 1=2e -dimensional space(where dxe denotes the ceiling function which returnsthe smallest integer P x). The 2N 3 term comes fromthe fact that (N1) correlations are necessary to describea N local state material system (Eq. (10)) and the observa-tion that the Fourier terms in cross correlations are com-plex valued and real valued in autocorrelations. ThedS 1=2e term comes from Eq. (15) and the symmetryof each Fourier transform. It is anticipated that the eigenmicrostructures occupy an extremely small portion of thepoints in the hull. The points in the hull that are not occu-pied by eigen microstructures can be interpreted in two dif-ferent ways. One option is to interpret them as belonging tonon-eigen microstructures. However, investigations haverevealed that it may not be possible to assign a (single) spe-cic non-eigen microstructure, of spatial extent S, to everypoint in the hull. It is important to note that the inability togenerate a single instantiation is the byproduct of the dis-creteness in space. It is expected that, in the continuouslimit, a single non-eigen instantiation could be found forall points in the hull. The second option is to interpret thesepoints as the ensemble averaged 2-point statistics over mul-tiple instantiations of the microstructure drawn either froma single sample or from multiple samples with nominallythe same processing history. The authors prefer the secondinterpretation for the following two main reasons: (i) den-ing a representative microstructure in terms of the ensem-ble average of statistics from multiple instantiations(rather than from a single instantiation) assures one thatthe microstructure statistics for the sample are capturedas accurately as possible; (ii) ensemble averaging alsoassures one that the hull is convex and compact, which inturn makes it possible to search for microstructures with

S.R. Niezgoda et al. / Acta Moptimal combinations of properties using the availablecomputationally ecient quadratic programming algo-space here is six-dimensional (taking advantage of Eq.(15) and the fact that the autocorrelation coecients 11F kare all real). This is the main reason for selecting a highlysimplied idealized microstructure for this case study. Asone can see, the dimensionality of the space of the 2-pointhulls described in this work grows approximately propor-tionally with the product SN. However, it is only possibleto view projections of this hull in, at most, a selectedthree-dimensional subspace. Although the mathematicalframework presented above can be employed on micro-structures with much larger values of both S and N, thevisualization of their 2-point hulls in three-dimensionalsubspaces is not particularly insightful. Nevertheless, thereader might be interested to know that the mathematicalframework presented here has been used successfully onmicrostructures with S values of about two million and Nvalues of about ve hundred [19].

Fig. 1. A set of 10 one-dimensional two-phase samples extracted from a

very large hypothetical sample. It is expected that the large sample can berepresented as an ensemble average of these smaller samples.

Fig. 2 shows the projections of the computed 2-pointhull for the selected one-dimensional material system inthe 11F 0, 11F 1; 11F 2 subspace and the 11F 0, 11F 4; 11F 5 sub-space. The starred points inside the hull in Fig. 2 denotethe set of all possible eigen microstructures in this materialsystem. Although one might expect a total of 210 eigenmicrostructures (based on S = 10 and N = 2), the numberof eigen microstructures shown in Fig. 2 is substantiallyless. Two factors can help explain this discrepancy. (i) Avery large number of eigen microstructures share the exactsame representation in the 2-point hull shown in Fig. 2. Infact, in the DFT representation 2-point correlations areinvariant under a translation and/or an inversion of themicrostructure. It is easy to see from the denitions inEqs. (3) and (4) that the microstructure data sets nms andnmsa would produce identical 2-point correlations, wherea denotes a translation of the microstructure data set by an structure can be interpreted as representing the ensemble

Fig. 3. The projection of the ensemble of microstructures from Fig. 1 intothe 2-point correlation hull. The average statistics for the ensemble aredenoted by the black circle.

5290 S.R. Niezgoda et al. / Acta Materialia 56 (2008) 52855292integer number of grid points on the spatial domain of themicrostructure. (ii) The projection of the six-dimensionalhull into a three-dimensional subspace causes some of thedistinct points in the six-dimensional space to occupy thesame location in the smaller three-dimensional space.

The ensemble of eigen microstructures shown in Fig. 1have been identied as black squares in Fig. 3. These area subset of the starred points shown in Fig. 2. Note thateigen microstructures 1 and 3 are related to each other bya translation and therefore occupy the same location inFig. 3. The ensemble averaged 2-point statistics, denotedh11F ki, for the entire set of microstructures shown in Fig.1 is shown as a lled circle in Fig. 3. The ensemble averagedstatistics in this case did not correspond to any of the pos-sible eigen microstructures for the selected material system.Given the relatively small number of points in the hulloccupied by the eigen microstructures, it should beexpected that there is only an exceedingly low chance thatthe ensemble averaged 2-point statistics would correspondto an eigen microstructure. As discussed earlier, all thepoints in the hull that do not belong to an eigen micro-Fig. 2. Projections of the computed 2-point correlation hull for the selected onthe set of all possible eigen microstructures in this material system.averaged 2-point statistics of a set of microstructuresextracted from one or more samples with nominally thesame processing history.

One method of visualizing the points in the 2-point hullis through reconstruction techniques. Recent advances inreconstruction techniques using phase retrieval algorithmshave resulted in rapid restoration of microstructures froma given set of 2-point statistics. The procedures used forthe reconstructions have been described in other papersfrom the authors group [19,20] and have been highly suc-cessful in non-eigen reconstructions and exact reconstruc-tions of eigen microstructures (up to an arbitrarytranslation or inversion). However, eorts to reconstructa non-eigen microstructure corresponding to the ensembleaveraged statistics in Fig. 3 have resulted in only limitedsuccess. In general, reconstructions from ensemble aver-aged statistics result in multiple solutions where the statis-tics of the reconstructed microstructures are close to theensemble averaged statistics, but the normalized error isstill outside the bounds of what would be considered asuccessful reconstruction [19,24]. This leads one toe-dimensional material system. The starred points inside the hull represent

believe that there are points in the 2-point hull that wouldnot correspond to any single non-eigen microstructure(dened at the adopted spatial resolution). An example ofthe reconstructed non-eigen microstructure that comes clos-est to capturing the ensemble averaged statistics is shown inFig. 4. This inability to capture the ensemble averaged sta-tistics is a direct result of the limited size of the spatial dis-cretization used in this example. Given the extremelylimited spatial discretization (only 10 spatial points in thesample) it would be impossible to build a microstructurefor each point in the convex hull. In principle, if one is will-ing to rene the spatial grid indenitely, it will become pos-sible to nd a non-eigen microstructure with the statistics ofthe ensemble structures to a pre-dened approximation.Any non-eigen microstructure will eventually become an

eigen microstructure (to a given approximation) as the spa-tial grid on which it is discretized is made ner. Alterna-tively, the non-eigen microstructures can be thought of asan eigen microstructure on a coarse spatial grid.

Instead of the above approach, the authors prefer torepresent the ensemble averaged 2-point statistics for theset of microstructures shown in Fig. 3 using a weightedset of RVE. For this purpose, a Euclidian distance betweenany two points in the 2-point correlation hull, normalizedby the size of the 2-point hull (i.e. the largest distance ofthe hull vertices from the origin) is dened and used toidentify a set of eigen microstructures that are close tothe ensemble averaged statistics of the given set of micro-structures. Fig. 5 describes examples of how a weightedset of eigen microstructures can approach the ensembleaveraged 2-point statistics. As expected, the more micro-structures that can be used in the RVE set, the closer theensemble averaged statistics can be approximated.

5. Summary and conclusions

n-Point statistics are an important method of describingmaterial structure in a statistical manner. This paper hasdescribed an ecient manner for representing the completehull of 2-point statistics an essential step towards explor-

Fig. 4. Best possible reconstruction of a non-eigen microstructurecorresponding to the ensemble averaged statistics shown in Fig. 3. Thestatistics of the reconstructed microstructure dier from the ensembleaveraged statistics by 1%.

S.R. Niezgoda et al. / Acta Materialia 56 (2008) 52855292 5291Fig. 5. RVE for a large hypothetical sample represented as the weighted fractions of ve eigen-microstructures.

ing the spaces of n-point statistics for material analysis anddesign. The framework presented above made use of spec-tral representation through DFT techniques to examineand enumerate many of the interdependencies in the com-plete set of 2-point microstructure correlation functions forcomposite material systems with multiple local states. Theinterdependencies presented demonstrate that the numberof independent correlations necessary to dene an N localstate eigen-microstructure completely is N1.

These interdependencies in Fourier space have beenused to delineate the bounded space of all physically mean-ingful 2-point correlations, termed the 2-point correlationshull. Visualizations of the hull have been presented for asimplied one-dimensional two-phase microstructure.

It was found that it is often impossible to capture exactlythe ensemble averaged statistics in a single representativeeigen or non-eigen microstructure. Instead, it was foundpossible to approach the ensemble averaged statistics usinga small weighted set of RVE. The utility of the hull in pro-ducing such an RVE set for a bulk sample was demon-

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Acknowledgments

The authors acknowledge nancial support for thiswork from the Oce of Naval Research, Award No.N000140510504 (Program Manager: Dr Julie Christodou-lou). SRN has been supported by the National ScienceFoundation Drexel/UPenn IGERT Program in NanoscaleScience & Engineering NSF Award No. DGE-0221664.2004;20:1561.[17] Wu X, Proust G, Knezevic M, Kalidindi SR. Acta Mater

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Delineation of the space of 2-point correlations in a composite material systemIntroductionMicrostructure function, 2-point correlations and DFTHull of 2-point correlationsCase study: one-dimensional two-phase microstructureSummary and conclusionsAcknowledgmentsReferences