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MICROSTRUCTURE ENGINEERING IN DESIGN Henrie B. L., Lyon M., Adams B. L.

Brigham Young University, Provo, Utah 84602

Abstract

Materials are often assumed to be homogeneous and isotropic. Microstructure Sensitive Design (MSD) offers the designer an alternative. By exploiting the microstructure and allowing heterogeneous compositions, a wider range of properties is made available to the designer. Using a spectral approach, the mathematical framework for MSD is extended to include both texture and composition, building a multidimensional design space with a basis of piecewise continuous functions and generalized spherical harmonic functions. A graphical interpretation for optimization within this design space is also presented. The Hill bounds for the stiffness and compliance tensors and Taylor yield theory are integrated into this design space. With these properties and Lekhnitskii’s anisotropic solution for the stress concentration around a hole in a flat plate, the performance of a Copper plate is computed and optimal microstructures are found. The best microstructure proves to be nearly twice as strong as the worst, when crystallographic texture alone is considered. Calculated microstructures are ranked according to their texture severity index value, which gives an estimate of the cost associated with producing a given microstructure.

I. Introduction1

In engineering, materials are often assumed to be homogeneous and isotropic; in actuality, material properties change with sample direction and location. This variation is due to the anisotropy of the individual grain orientations and their spatial distribution in the material. Charts, such as Ashby charts, have been compiled to show property limits of the generic materials database, but these charts do not take into account the microstructure of the material.1 If microstructure and composition are taken into account, more property variance is predicted.

The traditional approach for material insertion is: Material Scientist → Materials Manufacturing Specialist (Processor) → Design Engineer. Material Scientists and Processors try to develop and deliver materials with improved properties to the Design Engineers, but the combination of property improvements that the Designer requires are not always known to the Material Scientist or Processor. Within the traditional approach, there exists no common language between these groups to facilitate the development of improved materials that respond directly to the design requirements.

In an effort to facilitate communication between Design Engineers, Material Scientists, and Processing Engineers, a new methodology has been developed called Microstructure Sensitive Design (MSD). The goal of MSD is to improve material performance

Copyright © 2002 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

through the creation of a mathematical language that traverses the interfaces between the traditional disciplines participating in highly-constrained design.

Adams et al. proposed the spectral approach of microstructure design to meet the requirements of multi-objective/constrained designs2. Within this spectral representation, microstructure, design properties, and processing paths can be expressed. In this paper composition is introduced as a design variable, in addition to crystallographic texture. Also, we extend the method to design problems where complex, non-linear combinations of properties are important. The example shown is the case of the hole-in-the-plate, loaded in uniaxial tension.

II. Microstructure Sensitive Design

In this section a basic review of the methodology of

MSD is described, with extensions to include composition as a scalar variable. The primary example will be the isomorphous Copper-Nickel system. a. Local State Space

The basic premise behind MSD is that salient local properties, pi, are dependent upon a small number of local state variables, h, that are presumed known to the observer,

)(hpp ii = . (1) Local states can be crystal phase and orientation, composition, reference shear stress for dislocation slip, and others. Through homogenization relationships, the local properties and their distribution in the

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization4-6 September 2002, Atlanta, Georgia

AIAA 2002-5567

Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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microstructure are linked with estimated effective (macroscopic) properties.

In this work, single-phase polycrystals have been considered. Thus, four local state variables are employed

h = (g,λ) , (2) where g is lattice orientation (three variables), and λ is composition. Thus, local state variables are defined by the ordered set ),( λg , where Γ=∈ φφGSOg /)3( , and Λ=∈ ]1,0[λ . )3(SO is the 3-dimensional special orthogonal group of rotations of the crystal lattice, and

Gφ is the crystallographic symmetry subgroup of phase φ . It follows that Γφ represents the set of all physically distinctive orientations of the crystal lattice associated with phase φ .3 λ is a scalar parameter representing composition. Any other scalar parameter can be included in a similar manner.

The local state space, H, shall consist of all possible (ordered) sets of local state variables:

Λ×Γ=∈ φHh . (3) b. Basis Functions

Following Bunge, the generalized spherical harmonic functions, )(gT mn

l , are used to represent the

orientation dependence on Γφ .4 The crystal symmetric basis functions, )(gTl

µυφ &&& , are special linear combinations of the generalized spherical harmonic functions,

∑ ∑+

−=

+

−=

=l

lm

l

ln

mnl

nl

mll gTAAgT )()( υφµφµυφ &&&&&& . (4)

The symmetry coefficients, µφ mlA&& , carry the crystal

symmetry and the coefficients, υφ nlA& , carry the sample

symmetry. The )(gT mnl functions form a complete

system of orthonormal functions with the following property:

∫Γ

′′′+=

φ

δδδ nnmmllmn

lmn

l ldggTgT

121)()( * , (5)

where * is the complex conjugate. Piecewise constant functions are used to provide the

basis for composition dependence on Λ . )(λχ r are defined to be piecewise constant functions on subintervals, rλ , of the range of λ (enumerated by the index r).5 Thus,

∈

=otherwise

ifM rrr 0

)(/1)(

λλλλχ . (6)

These form a system of orthonormal functions

∫ ′′ =

r r

rrrr M

dλ λ

δλλχλχ)(

)()( , (7)

where )( rM λ is the measure of rλ . (Note: Exponential and other types of functions could also be used as a basis for representing scalar parameters.) c. Microstructure Hull

Transforming the local state space into its Fourier basis requires transforming H into an equivalent space, H ′ . This transformation is performed by expressing each possible local state, say hj , as a Dirac function, defined here in the conventional way:

( ) ⊂∈

=−∫ otherwise 0'' if 1

''

HHhdhhh j

Hjδ . (8)

The Dirac function is then expressed in a Fourier series

( ) )()(0

)(

1

)(

1 1

λχδφ φ

µ υ

µυφµυr

l

lM lN N

rllrj gTFhh ∑ ∑ ∑∑

∞

= = = =

Λ

=− &&&&&& . (9)

From relations (5) and (7), the Fourier coefficients of the Dirac function have the form

)()()()12( *jrjljlr gTMlF λχλ µυφµυ &&&&&& += . (10)

For the homogenization relations of interest, the microstructure is specified by the local state distribution function, )(hf . )(hf defines the volume fraction

VdV / of sample volume that lies within a range dh of local state h:

dhhfVdV )(= . (11)

Here, dh is the invariant measure.4 The local state distribution function, )(hf , is normalized, as can be seen from equation (11), according to

∫ =H

dhhf 1)( , (12)

and is expressed as a series of generalized spherical harmonics functions,6

)()()(0

)(

1

)(

1 1

λχφ φ

µ υ

µυφµυr

l

lM lN N

rllr gTFhf ∑ ∑ ∑∑

∞

= = = =

Λ

= &&&&&& . (13)

It is convenient to represent )(hf as a summation of Dirac functions, weighted by the appropriate volume fractions, jν ,

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∑=

−=N

jjj hhhf

1)()( δν . (14)

The set of volume fractions, jν , are constrained to sum to one:

∑ =j

j 1ν . (15)

The set of all possible )(hf is called the microstructure hull. It consists of all possible microstructures pertinent to the chosen homogenization relations and local state space. Formally, the microstructure hull, as described in the Fourier space, is obtained from relations (9, 10, and 14) when constrained by (15).2 Graphically, this microstructure hull is difficult to view because of the infinite-dimensional nature of the Fourier space in which it resides; however, for the cubic-orthorhombic symmetry in the elastic case, the pertinent part of the Fourier space reduces to three dimensions, and is depicted graphically in Figure 1. Cubic crystal symmetry and orthorhombic statistical (sample) symmetry will be used throughout the remainder of this paper.

Figure 1 Cubic-orthorhombic

microstructure hull

III. Property Constraints with MSD a. Elasticity, Upper and Lower-Bounds

Hill demonstrated that the Voigt and Reuss averages were really upper and lower elastic bounds.7,8,9 Following Hill, we define the effective stiffness, *

ijkmC , as

kmij ijkmC εσ *= , (16)

where < > denotes volume average quantities. The effective tensor is determined from knowledge of the

single crystal tensor and the statistical properties of the microstructure. Compliance has a similar effective tensor.

The upper and lower bounds of the effective tensor can be calculated through two classical variational principles: the principle of minimum potential energy, and the principle of complementary energy.10 These principles for the effective stiffness tensor form lower and upper bounds

kmijkmijkmijkmij

kmijkmijkmijkmij

CC

CS

εεεε

εεεε

≤

≤−

*

*1

. (17)

A similar expression for *ijkmS can be derived.

Evaluation of (17) does not directly provide bounds for all components of *

ijkmC . Direct bounds are obtained

only for *iiiiC (no sum on i) and *

ijijC (no sum on i or j, ji ≠ ). Degraded bounds can be found for “off-

diagonal” elements, but some additional work is required. This paper will focus on bounds for the “on-diagonal” cases.

The upper bound from Equation (17) can be expressed for statistically homogeneous polycrystals in terms of the local state density function as

∑ ∫∫∫ ∫= Λ

=n

GSOijkmijkm dgdgChfC

1 /)3(

),()(φ

φ λλφ

, (18)

where dg is the invariant measure in orientation space, 21)sin( ϕθϕθ ddd , and g≡21 ,, ϕθϕ are the Euler

angles defined by Bunge.4 Both ),()( λgfhf = and

),( λφ gCijkm are real-valued functions on the local state space, H.

Since ),( λφ gCijkm is the elastic stiffness tensor in the sample coordinate frame, a relationship must be found to relate ),( λφ gCijkm with the elastic stiffness tensor in the reference crystal coordinate frame,

)(λφ oijkmC . This relationship is required because the

basic elastic constants for cubic crystals, )(),(),( 441211 λλλ φφφ ooo CCC , are typically measured in

the crystal coordinate frame. In this case )(λφ oijkmC is11

kmjkij

jkimjmik

kmijijkm

CCC

C

CC

δδδλλλ

δδδδλ

δδλλ

φφφ

φ

φφ

)](2)()([

])[(

)()(

441211

44

12

ooo

o

oo

−−+

++

=

. (19)

∴

∴

∴

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Using the “passive” notation∗, as used by Bunge, the direction cosines, gij, are defined as4

ojiji ege = . (20)

Rotation of the elastic stiffness from the crystal frame to the sample frame is performed as

)(),( λλ φφ oijkmdmckbjaiabcd CgggggC = . (21)

For cubic symmetry, (21) can be written in a simpler form as

rmrkrjri

jkim

jmikkmijijkm

gggg

CCC

CCgC

)](2)()([]

)[()(),(

441211

4412

λλλδδ

δδλδδλλφφφ

φφφ

ooo

oo

−−+

++=

. (22)

The relationship rmrkrjri gggg implies a summation over r from 1 to 3:

mkji

mkjimkjirmrkrjri

gggg

gggggggggggg

3333

22221111

+

+=. (23)

b. Elasticity in Ni-Cu Alloys

The method of Hartley, for estimating the material constants, )(λφ o

ijC , for the homogenous mixture of Cu and Ni, is used in this paper.12 The composition variable, λ , is taken to be the volume fraction of Ni in Cu.

Relation (22) is divided into two parts ])[()( 4412 jkimjmikkmijijkm CCA δδδδλδδλ φφφ ++= oo (24)

and

rmrkrjriijkm ggggZ )(λβφφ = (25)

with basis functions. )(λβφ given as

∑Λ

=

=

−−=N

rrr

CCC

1

441211

)(

)(2)()()(

λχβ

λλλλβ

φ

φφφφ ooo

. (26)

Also, (24) is expressed using piecewise functions,

∑Λ

=

=N

rrrijkm ijkmkA

1)()( λχφφ , (27)

and (25) as a combination of piecewise and generalized spherical harmonic functions,

∗ The passive notation is of the form )()( gfggf =oφ , where g transforms a sample fixed coordinate system, Ka, into a crystal fixed coordinate system Kb. φg is a rotation that transforms a crystal from one orientation to a symmetrically equivalent orientation, gφ∈

φΓ .

∑ ∑ ∑= = =

=4

4,0

)(

1 1

1*1* )()()(l

lN N

rrllrijkm

r

gTijkmDZφ

υ

υφυφφφ λχβ &&&&&& , (28)

since 1)4()0( == MM φφ . (The two coefficients rβφ

and )(1* ijkmDlυφ &&& may be expressed as one coefficient,

but the )(1* ijkmDlυφ &&& coefficients do not change with

composition. From this property, the )(1* ijkmDlυφ &&&

coefficients are only calculated once for a set symmetry. The rβ

φ coefficients change for each alloy system, and require much less computational time.)

Equation (28) states that l needs to be enumerated only up to 4, due to the fact that it is the product of a constant and four direction cosines:4

∑ ∑=′ =′

′′

′′=

4

4,0

)(

1

11 )()(l

lN

llrmrkrjri gTijkmDggggφ

υ

υφυφ &&&&&& . (29)

By combining equations (13), (18), (27) and (28), the stiffness upper bound becomes

µµφφ

φ µ

φ

β

δφ

11*

0

1

1

4

4,0

)(

1 1

*

)](

)([

lrlr

ll

lN N

rrijkmijkm

FijkmD

ijkmkCC

&&&&&&+

=≤ ∑∑ ∑ ∑= = = =

Λ

. (30)

From (30), the upper bound can be calculated for all pertinent microstructures and compositions. Note that the upper-bound relationship in (30) comprises, for a fixed *

ijkmC , a hyper-plane within the Fourier space. This hyper-plane consists of all Fourier components

µφ 1lrF&&& , representing microstructures, which possess the

same upper-bound on effective stiffness. The procedure for finding the elastic property

closures for the lower bound is similar to the upper bound approach, except for an inversion13

1

11*

0

1

1

4

0

)(

1 11*

)](

)([

−

= = = =−

+

=≥∑∑ ∑∑

Λ

µµφφ

φ µ

φ

α

δφ

lrlr

ll

lN N

rr

ijkmijkm

FpqstD

pqstkSC

&&&&&&

, (31)

where rαφ is

∑Λ

=

=

−−=N

rrr

SSS

1

441211

)(

)(2)()()(

λχα

λλλλα

φφ

φφφφ ooo

. (32)

Note that the lower-bound relationship (31) forms a limiting hyper-surface within the microstructure hull for fixed *

ijkmC .

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c. Taylor Plasticity

Taylor theory assumes that all grains within the material will undergo the same strain. This strain corresponds to the macroscopic strain of the sample. In a strain-imposed problem, the application of Taylor theory within MSD is straightforward. Stress-imposed problems, though, are more common in engineering design, and applying Taylor theory in this case adds some additional challenges, forcing the model to become non-linear in the microstructure coefficients.

To preserve the assumptions of Taylor while solving a stress-imposed problem requires that the macroscopic strain be a variable dependent on the microstructure14. The Taylor factor can be found as a function of this strain ratio. By Taylor theory, yielding will occur when15

cijij Mτεσ = , (33)

where cτ is the critical resolved shear stress of the

material and M is the macroscopic Taylor factor which has been calculated using the Bishop-Hill stress corners16. If the stress is applied uniaxially in the principal direction and orthorhombic symmetry is assumed, the following form for the strain tensor is valid

−−−=

)1(0000001

qqijε . (34)

The Taylor factor then becomes a function of the contractile ratio q . The correct value of )(qM is the one for which Taylor theory predicts the imposed stress field. This value will also correspond to the minimum value of )(qM over the range 0 ≤≤ q 1.

The function )(qM is a function of orientation as well as q and can therefore be expressed in terms of a series expansion as

∑ ∑ ∑∞

=′

′

=′

′

=

′′

′′=

=

4,0

)(

1

)(

1

)(

),()(

l

lN lM

ll Fqm

qgMqMφ φ

υ η

υηφυη &&&. (35)

where ),( qgM is the local (crystallite) Taylor factor,

and the )(qmlυη ′′ coefficients are calculated using the

procedure described by Bunge14. (Note that ),( qgM is independent of composition λ .) Eq. (35) also defines hyper-planes within the microstructure hull. The Taylor factor requires a higher order of expansion than elastic

properties, but has been shown to be approximated very well with relatively few terms of the expansion14. d. Hull with properties

Section II.c. introduced the microstructure hull which represents all possible microstructures, expressed by equation (14). Points outside the microstructure hull are fictitious microstructures, having no physical meaning. The hyper-surfaces therefore only have meaning at their intersection with the microstructure hull.

A graphical depiction of (30) shows an exemplary upper-bound hyper-plane in the multidimensional Fourier space, in Figure 2. In Figure 2 the >< 1212C plane delineates all microstructures that are predicted to have 63*

1212 ≤C GPa. As the hyper-planes are translated through the microstructure hull, property extremes are observed. These extremes are used to identify the property closures introduced in section III.e. >< 1212C yields higher stiffness bounds as the plane translates positively along the 14

4F&&& axis. Equation (31) produces a hyper-surface for the

elastic lower-bound. Figure 2 shows the 11111−>< S

hyper-surface within the microstructure hull. This surface delineates all microstructures that are predicted to have effective stiffness 320*

1111 ≥C GPa. The 1

1111−>< S hyper-surface yields larger property bounds if

the hyper-surface is translated along the negative 124F&&&

axis. As these hyper-surfaces are added to the

microstructure hull, they are used to find groups of microstructures that fulfill the design objectives. As more design properties are included, it becomes more difficult to find microstructures that fulfill all design constraints. Looking at two properties, such as *

1111C

and *1212C that are illustrated here, an area of the

microstructure hull is found that satisfies both constraints, and is represented as the “acceptable area” on Figure 2.

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Figure 2 Microstructure hull with selected

11111−>< S and >< 1212C hyper-surfaces.

e. Property closures

Property closures are generated by condensing the information contained within the multi-dimensional Fourier space into representations of combined properties of lower-dimensionality (e.g., 2-D). This compaction of information is beneficial for a quick determination of combined property limits for a particular alloy system. The property closures are determined by translating two or more property hyper-surfaces, in their normal directions, through the entire microstructure hull, thereby recovering the entire set of all properties combinations.

Property closures, used in conjunction with the microstructure hull and the hyper-surface construction, can identify alloy microstructures that fulfill the selected design objectives. The design criteria dictate the minimum property values that fulfill design objectives. These criteria are used to set the effective properties, e.g. property *

1111C and *1212C . If *

1111C must be greater then 320 GPa, illustrated by A in Figure 3, and *

1212C must be less than 63 GPa, illustrated by B, then, as illustrated in Figure 3, there is a small area of microstructures that can fulfill both property objectives. This is identified as the “acceptable area” in Figure2. Therefore, it is known that for these two effective properties, sets of microstructures exist that meet the design objectives.

Figure 3 Property closure of *

1111C vs. *1212C with

design constraints added at A and B

f. Extraction of Microstructures

Once an area has been identified within the property closures, one can extract microstructures. The microstructures are extracted by first locating the bounding hyper-surfaces within the microstructure hull. Having located these boundaries, an area of µ1

lrF&&& coefficients is identified. From equation (13), these

µ1lrF&&& coefficients are used to find sets of local state

distributions, )(hf . These local state distribution functions detail the microstructures that satisfy the design objectives. Figure 4 shows one microstructure extracted from the highlighted area in Figure 3, and associated with the “acceptable area” in Figure 3.

Figure 4 Inverse pole-figures for one microstructure that satisfies both the 320*

1111 ≥C

GPa and 63*1212 ≤C GPa bounds.

g. Processing Cost Factor

Not all of the extracted microstructures are currently realizable and of those that are, several will be too expensive for their use to be justified. For example, the

acceptable area

∴

∴ ∴

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additional cost of single crystal microstructures often outweighs the benefits. As a first approximation of the cost of a particular microstructure, the texture severity index is employed17. The index value is calculated as

∫=GSO

dggfJ/)3(

2)( . (36)

The value of J ranges from 1, for a completely random distribution, to ∞, for a single crystal. J , expressed in terms of the Fourier series, becomes a function of the location within the microstructure hull, making its evaluation relatively simple. Property closures can be made to display the texture index value for different regions. This allows the designer to see not only optimal microstructures, but also those that are the easiest to produce.

IV. Application to Design a. Lekhnitskii’s solution for the hole-in-a-plate

Once the properties of a microstructure are determined, they can be used in combination to solve more complicated problems. One such problem involves the stress concentration around a small hole in a large flat plate. An anisotropic solution for this stress concentration was developed by Lekhnitskii18. This hole-in-the-plate design application will be limited to a pure composition of Cu, although it can be extended to the Ni-Cu alloy system using the methods which have been developed.

Using Lekhnitskii’s solution, the stress tangent to the hole at some angle θ relative to the direction of stress can be determined. With tension in the principal direction the tangential stress in terms of the components of the compliance tensor is

−+= θθσ θθ

2

1111

222221111 cossin)1(

SSnEpS , (37)

where p is an uniaxial stress applied some distance from the hole,

1

42222

2211221212

41111

coscossin

)24(sin−

++

++=

θθθ

θθ

S

SSSE , (38)

and

1111

2222

1111

11221212 224SS

SSSn +

+= . (39)

The stress at the midsection perpendicular to the direction of the stress simplifies to

+

++=

1111

2222

1111

11221212 2241SS

SSSpθσ . (40)

The stress concentration factor k is therefore )1( n+ . The solution listed in (40) for the tangential stress

includes both “on-diagonal” and “off-diagonal” terms of the compliance matrix. If one wishes to obtain bounds on the stress concentration factor using the Hill bounds on the elastic constants, one must include the degraded bounds of the “off-diagonal” terms, which forces the bounds too large to be physically relevant. Since (40) is a ratio of the components of the compliance tensor, the upper bounds for those components can be used as an estimate for k . This approach gives the value of k ranging anywhere from 2.2 to 4.0 for Cu.

Taylor’s yield model can be employed to insure that decreasing the stress concentration around the hole doesn’t decrease the yield strength of the material as well. The material which will carry the largest load is the one with the highest value for kM c /τ . An appropriate performance closure for such a design problem would be one showing M versus k.

b. Determination of Non-linear Performance

Closures

For the case of the hole-in-the-plate, performance is based on non-linear functions of properties. Both M and k can be computed by way of the microstructure’s Fourier coefficients, but they cannot be directly fit with the Fourier series†. The determination of a performance closure in such cases requires the use of a non-linear solver. The linearity of the elastic constants and

)(qM can be manipulated to reduce the computations involved.

For the hole-in-the-plate stress concentration design problem, a variation on the simulated-annealing algorithm19 was used where the objective was not only to find the best and worst microstructure, but also to bound the full range of performance. The algorithm was also used to calculate the minimum achievable texture severity index for each point within the closure.

The algorithm used a random microstructure with 5000 individual orientations arranged into 1250 orthorhombic clusters. These orientations were † M(q) may be fit to a Fourier series for a given value of q as indicated in Eq. (35), but for stress imposed problems, the value of q which minimizes M(q) is a discontinuous function of the microstructure.

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randomly adjusted between more than 10000 orientations, which represented a discretization of the cubic-orthorhombic fundamental zone. The highest value of the texture severity index is obtained when all 1250 orthorhombic clusters are of the same orientation. The texture severity index value for this situation is 1250. The calculations were performed on a SGI Origin 3000 Super Computer using ten processors. The calculated performance closure is shown in Figure 5 with the shade of gray designating the natural log of the texture severity index (e.g. 7.1309 is 12501309.7 ≅e ).

2

2.4

2.8

3.2

3.6

4

2 2.5 3 3.5 4

M

k

Figure 5 Combined properties closure showing k versus M , with the natural log of the texture index value designated by shades of gray.

It is evident from study of Figure 5, that, not only

do we have a broad range of microstructures near optimal performance, but also little performance is lost by requiring that the microstructure be one of low cost. c. Microstructure Solutions for the Design

Based on the design requirements, microstructures can be extracted which provide interesting solutions. Figure 6 shows the performance closure again, with lines corresponding to equivalent performance and the location of three specific microstructures that were extracted. Every point lying on a line will perform equally well and those below and to the right of the line will perform better.

The individual microstructures can be computed by the same simulated annealing program. Since only one point needs to be computed, the computation time is

anywhere between a few seconds to a few minutes. Microstructure A performs the best for the hole-in-the-plate problem and is shown in Figure 7. Microstructure B has nearly equivalent performance, but has a texture severity index two orders of magnitude lower than B. Microstructure B is shown in Figure 8. Microstructure C is the performs the worst and is shown in Figure 9.

2

2.4

2.8

3.2

3.6

4

2 2.5 3 3.5 4

M

k

Figure 6 Combined properties closure showing the texture severity index, equal performance lines, and the locations of three extracted microstructures A, B, and C.

Figure 7 Inverse pole-figures for microstructure A with a ratio kM / of 1.12 and a Texture Index of 200.

A

B

C

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Figure 8 Inverse pole-figures for microstructure B with a kM / ratio of 1.10 and a Texture Index of 2.6.

Figure 9 Inverse pole-figures for microstructure C with a kM / ratio of 0.6 and a Texture Index of 1250.

The lines indicate values for the ratio kM / of 1.10,

0.83, and 0.62. Microstructure A has a performance ratio of 1.12 and texture severity index value of 200. Microstructure B has a performance ratio of 1.10 and a texture severity index value of 2.6. Microstructure C has a performance ratio of 0.60 and a texture severity index value of 1250.

Microstructure B loses less than 2% in performance, as compared to A, but has a much weaker texture. Microstructure B could cost substantially less to produce in the extent that the texture severity factor represents the production cost. Microstructure C is a single crystal microstructure that has almost half the performance of the other two microstructures.

V. Discussion and Conclusions

Through the use of Microstructure Sensitive Design, a wider range of properties are available for consideration by the designer. MSD depends upon established homogenization relations, and the qualification of these relations is clearly an important consideration in the methodology. These homogenization relations typically describe hyper-planes or surfaces in the Fourier space used to represent microstructures. Consideration of all possible translations of these surfaces within the microstructure hull enables the recovery of properties closures that may be profitably used by designers to evaluate theoretically-available microstructures, as they apply to specified sets of design constraints or objectives. In some cases, as is illustrated by the case of the hole-in-the-plate, design considerations may lead to complex combinations of basic properties. Pertinent microstructures can also be compared on the basis of their relative cost through the use of the texture severity index, which gives a first-order indication of manufacturing complexity. The tools presented here allow the designer to consider properties that previously only the material scientist knew were possible.

Acknowledgements The support of this research by the Army Research Office is gratefully acknowledged.

References

1Ashby, M. F. (1992) Materials Selection in Mechanical Design, Pergamon Press Inc., New York.

2Adams, B. L., Henrie, A., Henrie, B., Lyon, M., Kalidindi, S. R., Garmestani, H. (2001) Microstructure-Sensitive Design of a Compliant Beam, Journal of Mechanics and Physics of Solids, 49:1639-1663.

3Adams, B. L., Olsen T. (1998) The mesostructure-properties linkage in polycrystals, Progress in Material Science, 43:1-88.

4Bunge, H. J. (1982) Texture Analysis in Materials Science, Butterworths, London.

5Keener, J. P., (2000) Principles of Applied Mathematics, 2nd Ed., Perseus, Cambridge, Mass.

6Courant, R., and Hilbert, D. (1989) Methods of Mathematical Physics, Willey-Interscience, New York.

7Hill, R. (1952) The Elastic Behavior of a Crystalline Aggregate, Proc. Phys. Soc. Lond., A 65: 349-354.

American Institute of Aeronautics and Astronautics

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8Reuss, A. (1929) Berechnung der Fliessgrenze von Mischkristallen auf Grund der Plastizitaetsbedingung fuer Einkristalle, Z. Angew. Math. Mech., 9: 49-58.

9Voigt, W. (1928) Lehrbuch der Kristallphysik, Teubner, Leipzig.

10Beran, M. J., Mason, T. A., Adams, B. L., and Olson, T. (1996) Bounding elastic constants of an orthotropic polycrystal from measurements of the microstructure, Journal of Mechanics and Physics of Solids, 44:1543-63.

11Hirth, J. P. and Lothe, J. (1968) Theory of Dislocations, McGraw-Hill, Inc.

12Hartley, C. S. (2001) Single Crystal Elastic Module of Disordered Cubic Alloys Part I: Face-Centered Cubic Systems, Submitted for publication.

13Henrie, B. L. (2002) Elasticity in Microstructure Sensitive Design through the use of Hill Bounds, Thesis, Brigham Young University, etd.byu.edu.

16Bunge, H., J., Park, N., J., Klein, H., Dahlem-Klein, E., Physical Properties of Textured Materials, Cuvillier Verlag, Goettingen, 1983.

15Taylor, G., I., Plastic Strain in Metals, J. Inst. Metals, 62, 307, 1938.

16Bishop, J., F., W., Hill, R., A Theory of the Plastic Distortion of a Polycrystalline Aggregate under Combined Stresses, Phil. Mag., 42, 414, 1951.

17Sturken, E., F., Croach, J., W., Predicting Physical Properties in Oriented Materials, Trans. Met. Soc. AIME, 227, 934-940, 1963.

18Leknitskii, S., G., Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day, San Francisco, 1963.

19Kirkpatrick, S., Gelatt, C., D., Vecchi, M., P., “Optimization by Simulated Annealing.” Science, 220, 671.