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An information-theoretic approach for obtaining property PDFs from

macro specifications of microstructural variability

Nicholas Zabaras, Veera Sundararaghavan and Sethuraman Sankaran

Materials Process Design and Control Laboratory, Sibley School of Mechanical and Aerospace Engineering

188 Frank H.T. Rhodes Hall, Cornell University, Ithaca, NY 14853-3801, USA

Abstract

Probability distribution functions (PDFs) providing a

complete representation of property variability in

polycrystalline materials are difficult to obtain.

Reconstruction of a PDF of a material property based on

limited microstructural information is an inverse problem

of practical significance since many macroscopic properties

depend strongly on geometrical variability of the micro-

constituents. We characterize the unknown probabilities of

the microstructural parameters of polycrystalline alloys

making use of average values (and lower moments) of

grain sizes, average orientation distribution functions

(ODFs) and using the concepts of maximum information

entropy (MaxEnt) and stochastic geometry.

Introduction

All materials comprise of various length scales

emphasizing the different resolution levels at which

practical components may be viewed. The physical basis of

a material’s mechanical response stems from lower scales.

A particular problem of interest addressed here is the

determination of effective behavior of polycrystalline

materials based on uncertainties induced due to random

nature of microstructures. The fundamental reason behind

this uncertainty is due to the fact that microstructual images

can be obtained only at a limited number of material points

on a sample (a material point is a distinct point on the

macro scale and has a microstructure associated with it).

The information that is obtained from this limited set is not

sufficient to deterministically characterize the

microstructures that are present in the specimen. In our

formalism, microstructures are considered as realizations of

a random field. We utilize the principle of MaxEnt to find

the distribution of microstructures that satisfy the measured

information about the microstructure. The reconstructed

microstructures are interrogated to obtain statistics of their

homogenized plastic properties. This procedure is

summarized in Fig. 1 for the case of estimating plastic

properties of two-dimensional Al microstructures using

grain size and orientation distribution functions.

Figure 1: Experimental samples of microstructure are obtained using x-

ray measurements. Reconstruction is based on two features, namely the

grain size, and ODF. Both are treated as random fields, whose PDF are

obtained using the MaxEnt scheme. Microstructure samples are generated

using voronoi tessellations and grains are randomly assigned orientations

based on a PDF of ODFs. The resulting microstructures are interrogated

using [1] and the bounds of plastic properties are obtained.

The principle of maximum entropy (MaxEnt)

Suppose that we have insufficient knowledge about an

entity. The MaxEnt approach provides a rationale to obtain

the entire probabilistic variability about the entity [2][3].

Since the problem of obtaining a distribution of

microstructures using average information about

microstructural features is ill posed, we pose an additional

requirement that the entropy of the distribution of the

microstructures is maximized. It is to be noted that the

entropy function is convex [4] and in an unconstrained

problem, it achieves the maximum when all the possible

events are equiprobable. This means that when we do not

have any information about a system, the most unbiased

prediction about the behavior of the system is by assuming

that all possible outcomes are equiprobable. The knowledge

about microstructures obtained from experimental

measurements is posed as a constraint. The distribution that

is obtained using MaxEnt is the most uniform distribution

ODF

0 0.05 0.1 0.15 0.2

30

40

50

60

70

80

Equivalent strain

Plastic property bounds

Equivalent stress (MPa)

Grain size:

lower moments

Samples of microstructures

consistent with experiments

G

i

b

b

s

s

a

m

p

l

i

n

g

0 50 100 150

0

0.05

0.1

Orientation angle (in radians)

PDF of textures obtained using

MaxEnt scheme

ODFODF

0 0.05 0.1 0.15 0.2

30

40

50

60

70

80

Equivalent strain

Plastic property bounds

Equivalent stress (MPa)

0 0.05 0.1 0.15 0.2

30

40

50

60

70

80

Equivalent strain

0 0.05 0.1 0.15 0.2

30

40

50

60

70

80

Equivalent strain

Plastic property bounds

Equivalent stress (MPa)

Grain size:

lower moments

Samples of microstructures

consistent with experiments

G

i

b

b

s

s

a

m

p

l

i

n

g

0 50 100 150

0

0.05

0.1

Orientation angle (in radians)

PDF of textures obtained using

MaxEnt scheme

0 50 100 150

0

0.05

0.1

Orientation angle (in radians)

0 50 100 150

0

0.05

0.1

Orientation angle (in radians)

PDF of textures obtained using

MaxEnt scheme

Issue 1 • 3-Dimensional Materials Science

TMS Letters • Volume 3

1

satisfying the given constraints. For details of mathematical

techniques used in entropy optimization, refer [5][6][7][8].

Figure 2: The figure shows reconstructed microstructures and the

comparison of their grain size distribution with MaxEnt grain size

distribution.

Reconstruction of 3D microstructures

The problem of reconstruction of 3D microstructures

comprises of the following sub-problems: (i) representation

of 3D microstructures (ii) reconstructing microstructures

that satisfy observable information about their features (iii)

obtaining the homogenized non-linear properties of the

microstructures such as plastic stress-strain curves. The

voronoi cell tessellation technique [9][10][11][12] is

employed for representing the microstructures. A set of

polyhedra are used to represent the 3D space comprising of

the microstructures. The polyhedra themselves are chosen

so that their grain size distribution matches the PDF of

grain sizes obtained using the MaxEnt principle. Since the

task of exactly constructing voronoi tessellations of the

given grain size distribution is intractable, a simpler version

of the problem is chosen. A Monte Carlo scheme is utilized

whereby a large database of microstructures is created

using voronoi tessellations and those microstructures which

have a correlation coefficient (R

corr

>0.8) are accepted.

Figure 2 shows some microstructures which were

constructed using the

above-mentioned scheme.

All grains are assigned

orientations based on a

distribution of ODF’s

reconstructed from

average ODF

measurements. This PDF

is obtained using the

MaxEnt scheme and

samples of reconstructed

ODFs are shown in Fig. 3

using the Rodrigues-

Frank representation.

Each of these

microstructures are

interrogated using the

homogenization

method provided in

[1]. The equivalent

stress-strain curves

are plotted for these samples. The MaxEnt technique not

only provides the extreme values of the stress-strain

response but also PDFs of the response, which is useful in

stochastic simulations.

References:

[1] Sundararaghavan V, Zabaras N. Int. J. Plasticity, in press.

[2] Jaynes ET. Phys Rev 1957;106/4:620-630.

[3] Jaynes ET. Phys Rev 1957;108/2:171-190.

[4] Cover T, Thomas J. Elements of Information Theory, John Wiley

and Sons, 1991.

[5] Fang S-C, Rajasekera JR, Tsao HSJ. Entropy Optimization and

Mathematical Programming, Kluwer Academic publishers, 1997.

[6] Agmon N et al. J Comput Phys 1979;30:250-258.

[7] Darroch JN, Ratcliff D. Ann Math Stat 1972;43/5:1470-1480.

[8] Sankaran S, Zabaras N, A maximum entropy approach for property

prediction of random microstructures, Acta Mater, in press.

[9] Aurenhammer F. ACM Computing Survey 1991;23/3:345-405.

[10] Du Q et al. SIAM review 1999;41/4:637-676.

[11] Golin MJ, Na HS. Comp Geo 2003;25:197-231.

[12] Pineda E et al. Phys Rev E 2004;70:066119-1-8.

Figure 3: The figure shows reconstructed

samples of ODF's based on the given ODF

which is shown on to

p

.

Figure 4: The figure shows bounds of the plastic properties

obtained using the reconstructed microstructural samples and

employing homogenization techniques.

Equivalent strain

Equivalent stress (MPa)

0 0.5 1 1.5 2 2.5 3 3.5

x 10

-3

0

10

20

30

40

50

60

Equivalent strain

Equivalent stress (MPa)

Bounds on plastic

properties

0 0.5 1 1.5 2 2.5 3 3.5

x 10

-3

0

10

20

30

40

50

60

Equivalent strain

Equivalent stress (MPa)

Bounds on plastic

properties

0 450 900 1350

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

correlation coefficient=0.8360

pmf

grain size

0 450 900 1350

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

grain size

pmf

correlation coefficient=0.8608

0 450 900 1350

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

correlation coefficient=0.8223

pmf

grain size

P

m)

P

m)

P

m)

P

m)

0 450 900 1350

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

correlation coefficient=0.8360

pmf

grain size

0 450 900 1350

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

grain size

pmf

correlation coefficient=0.8608

0 450 900 1350

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

correlation coefficient=0.8223

pmf

grain size

P

m)

P

m)

P

m)

P

m)

P

m)

P

m)

P

m)

P

m)

2