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An information-theoretic approach for obtaining property PDFs from macro specifications of microstructural variability

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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.
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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
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Half-title pageSeries pageTitle pageCopyright pageDedicationPrefaceAcknowledgementsContentsList of figuresHalf-title pageIndex
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The task of reconstruction of microstructures from their limited description is posed as a maximum entropy (MaxEnt) problem. Microstructural descriptors are taken in the form of volume fractions, correlation functions and grain sizes. Morphological and size quantifications are used as features of microstructures and samples consistent with these features are reconstructed. The non-uniqueness of the reconstructed distribution is effectively encountered by choosing the distribution with the maximum entropy. Properties of random microstructures are characterized statistically using the MaxEnt solution. Microstructures reconstructed from correlation measures are interrogated to obtain elastic properties. For estimating plastic property statistics, grain size and orientation distribution information are incorporated. Analysis of plastic properties is performed in two steps, firstly by reconstructing microstructures with macro-specifications of grain size and secondly by attributing an orientation to each grain drawn from the MaxEnt distribution of the orientation distribution function (ODF). The MaxEnt ODF distribution is obtained by constraining the expected ODF over a sufficiently large number of microstructure samples to match with the given ODF information. Further, the effect of incorporating a larger amount of information on the variation of the effective behavior is studied. Numerical examples demonstrating the method for one- and two-dimensional microstructures are discussed.
  • S-C Fang
  • Jr Rajasekera
  • Hsj Tsao
Fang S-C, Rajasekera JR, Tsao HSJ. Entropy Optimization and Mathematical Programming, Kluwer Academic publishers, 1997.
  • E Pineda
Pineda E et al. Phys Rev E 2004;70:066119-1-8.
  • Q Du
Du Q et al. SIAM review 1999;41/4:637-676.
  • J N Darroch
  • D Ratcliff
Darroch JN, Ratcliff D. Ann Math Stat 1972;43/5:1470-1480.
  • E T Jaynes
Jaynes ET. Phys Rev 1957;106/4:620-630.