Materials Process Design and Control Laboratory Design and control of properties in polycrystalline...

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


Design and control of properties in polycrystalline materials using texture-

property-process maps

Materials Process Design and Control LaboratorySibley School of Mechanical and Aerospace Engineering

188 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801

Email: [email protected]: http://mpdc.mae.cornell.edu/

V. Sundararaghavan and Prof. Nicholas Zabaras

Supported by AFOSR, ARO




MOTIVATION

GRAPHICALLY REPRESENT THE SPACE OF MICROSTRUCTURES, PROPERTIES AND PROCESSES

Applications:

(i) Identify microstructures that have extremal properties.

(ii) Identify processing sequences that lead to desired microstructures and properties.

3.05 3.06 3.07 3.08 3.09 3.1 3.11 3.123.03

3.04

3.05

3.06

3.07

3.08

3.09

3.1

3.11

3.12

Taylor factor along RDT

aylo

r fa

ctor

alo

ng T

D

b

2. Rolling

3. Rolling followed by drawing

1. Drawing

R

C

ad

fe

Process

Property-process space

-1

-0.5

0

0.5

-1-0.5

00.5

1-1.8

-1.7

-1.6

-1.5

-1.4

-1.3

Microstructure representations

Process-structure space

Process pathsA100

A1000

A80

Property-structure space




Example:

E <= E1x1 + E2x2+ E3x3 (upper bound theory)

E1,2,3 are the Young’s modulus of each phase

x1,2,3 are the volume fractions of each phase

Find microstructures with Young’s Modulus <= E

x1 + x2+ x3 = 1:

Microstructure plane for 3 phase materialx1

x2

x3

E <= E1x1 + E2x2+ E3x3

Property plane

Property Iso-line

Microstructures with Youngs Modulus <= E

FIRST ORDER STRUCTURE SPACE

x1

x2

x3




FIRST ORDER REPRESENTATION OF MICROSTRUCTURESFIRST ORDER REPRESENTATION OF MICROSTRUCTURES

Crystal/lattice

reference frame

e2^

Sample reference

frame

e1^ e’1

^

e’2^

crystalcrystal

e’3^

e3^

n

r = n tan(/2)

Cubic crystal

RODRIGUES’ REPRESENTATIONRODRIGUES’ REPRESENTATIONFCC FUNDAMENTAL REGIONFCC FUNDAMENTAL REGION

Particular crystal

orientationr

ODF representation for design

• Spectral (Adams, Kalidindi, Garmestani)

• FE discretization of RF space (Dawson)

High dimensional space, difficult to visualize process-structure-property maps




FINITE ELEMENT INTEGRATION IN RF SPACE

Integration point [0.25,0.25,0.25]

=

ri, Ai

Normalization:

= 1




MATERIAL PLANE

Applications:

(i) Identify microstructures that have extremal properties.

(ii) Identify closures of properties.

SPACE OF ALL POSSIBLE ODFs

Mathematical representation of all possible ODFs using FE degrees of freedom.

Three constraints define the space of first order microstructural feature (ODF):

• Normalization, qTA = 1

• Lower bound, A >= 0

• Crystallographic Symmetry, r’ = Gr

A

A100

A1000

ODF at

A80

~103 dimensions




UPPER BOUND THEORY: LINEARIZATION

Upper bound of a polycrystal property can be expressed as an expectation value or average given by

A100

A1000

ODF at

A80

~103 dimensions




LINEAR PROGRAMMING

Geometrically, the linear constraints define a convex polyhedron, which is called the feasible region.

-- all local optima are automatically global optima. ---optimal solution can only occur at a boundary point of the feasible region.

http://en.wikipedia.org/wiki/Convex

http://en.wikipedia.org/wiki/Polyhedron




EXTREMAL PROPERTY POINTS

Constraints and objectives are linear in the ODF problem

• Identify microstructures that maximize properties in a particular direction (eg. C<1111>)

and lb = 0

Normalization

Extremize property

positiveness

LINEAR PROGRAMMING

Number of variables: 448 Number of linear inequality constraints: 448Number of linear equality constraints: 1

For minima

For maxima




EXTREMAL TEXTURES

•Taylor factor calculated through Bishop-Hill analysis

500

400

300

200

100

0

500

400

300

200

100

0

(b)

(c)

(a)

0

3000

2000

1000

(d)

X Y

Z

0

3000

2000

1000

ODF for maximum Taylor factor along RD (=3.668)

ODF for maximum Taylor factor along TD (=3.668)

ODF for maximum C44 (=74.923 GPa)

ODF for maximum C55 (=74.923 GPa)




UPPER BOUND PROPERTY CLOSURES

168

188178

198208

218228

237

3343

5363

7383

9397

168

188

178

208

218

237

198

228

M a

lon

g T

D

M along RD

M at 45 degrees to RD

2.4

2.8

3.2

3.6

3.5

3.0

2.52.5

3.0

3.5

(b)

C11

(G

Pa

)

C 66 (GPa)

C22 (GPa)

(a)

Closure for stiffness constants (C11,C22,C66)

Closure for Taylor factor computed along RD, 45o to RD and TD




OPTIMIZATION ON MATERIAL PLANE

Given Initial ODF find the closest ODF (A) that satisfies the desired property (d)

Minimize scalar: r0

Such that:Positiveness

Uniform norm constraint

NormalizationA100

A1000

ODF at

A80

~103 dimensions

Initial ODF

ro

Desired property




BOUND PROBLEM

Number of variables: 449Number of linear inequality constraints: 1344Number of linear equality constraints: 2Solution: r0 = 7.8569 (No other solution can be confined within A +/- r0)

Input ODF(A) C<1111> = 209.0696 GPa

Optimized ODFC<1111> = 231.0 GPa (this is close to

extreme value of 236.82 GPa)




TEXTURE EVOLUTION

Represent the ODF as

Reduced model for the evolution of the ODF

Initial conditions

Viscoplastic rate dependent model, no hardening (Acharjee and Zabaras, 2003)

Taylor hypothesis

X-axis compression




MODEL REDUCTION

Suppose we had an ensemble of data (from experiments or simulations) for the ODF:

such that it can represent the ODF as:

Is it possible to identify a basis

POD technique – Proper Orthogonal Decomposition

Method of snapshots

Eigenvalue problem

where

Other features• Generated basis can be used in interpolatory as well as extrapolatory modes• First few basis vectors enough to represent the ensemble data




PROCESS PLANE

-1

-0.5

0

0.5

-1-0.5

00.5

1-1.8

-1.7

-1.6

-1.5

-1.4

-1.3

X Y

Z

6.54

4.77

3.00

1.23

-0.54

-2.31

-4.07

-5.84

2.10

1.23

0.36

-0.51

-1.38

-2.25

-3.13

-4.00

-1.29

-1.41

-1.53

-1.65

-1.77

-1.90

-2.02

-2.14

a1+a2a3

a1

a2

a3

4.84.54.23.93.73.43.12.92.62.32.11.81.5

Equation of a plane

• Basis already includes symmetries.

•Normalization

•Lower bound

•Crystallographic Symmetry




TEXTURE PLANES FOR SOME PROCESSES

Plane strain compressionTension/compression

X-Y shear Y-Z shear




PROCESS PLANES FOR YIELD STRENGTH ALONG RD

Plane strain compressionTension/compression

Y-Z plane shearX-Y plane shear




PROCESS PATH REPRESENTATION

-0.2 0 0.2 0.4

-0.2

0

0.2

0.4

0.6

-1.7

-1.65

-1.6

-1.55

-1.5

-1.45

-0.1 0 0.1 0.2-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

-1.75

-1.7

-1.65

-1.6

-1.55

-1.5

B

B’

A

A’ C’

Cb

b’ a’

a c

c’m

m’

n

n’

a1

a2

a3

a1

a2

a3

A

A’

R

R’

B

B’

m

m’

n

n’

Process plane for x-axis tension (ensemble obtained by processing an initial random texture to 0.1 strain)

Process plane for y-axis rolling followed by x-axis tension (initial random texture processed to 0.2 strain)

90% accurate reconstruction width

Compression path

Ten

sion

pat

h

initial textures

Final textures

0.17 initial strain

0.07 initial strain




BOUNDING IN A PROCESS BASISFind the closest distance of a desired ODF in the

material plane to an ODF in the process plane

Desired property

Deviation of the optimal ODF from the basis

ODF space

process basis

Microstructure with the desired property

Closest solution

Process microstructures

Minimize bound on the deviation

Process plane equation

Normalization




PROCESS PLANE SOLUTION FOR DESIRED STIFFNESS

(c)

5.95.55.14.74.34.03.63.22.82.42.01.61.3

0

50

100

150

200

250

Val

ues

(GP

a)

Desired values

Solution (Rolling-Tension)

C11 C22 C66 100*qTA

Normalization constraint

(a) (b)(d)

6.05.65.24.84.44.03.63.22.82.31.91.51.1

-0.2 -0.1 0 0.1 0.2

-0.4

-0.2

0

0.2

0.4

0.6

-1.75

-1.7

-1.65

-1.6

-1.55

-1.5

A

A’

12

a1

a2

a3

- Desired stiffness properties {c11 = 210.85 GPa, c22 = 210.42 GPa, c66 = 66.31 GPa}.

- Process plane: x-axis tension preceded by y-axis rolling of a random texture to 0.1 strain.

Optimal ODF

Final ODF in the process plane Exact solution

in the material plane

Optimal process path in the second stage




3.03800

3.04300

3.04800

3.05300

3.05800

0 20 40 60 80Angle from rolling direction

Ta

ylo

r F

act

or

Desired propertyBest solution (Rolling)Best solution (Tension)Best solution (Shear)

(a)

(b)

(c)

3.93.73.53.23.02.82.52.32.01.81.61.31.1

3.33.23.13.02.92.82.72.62.52.32.22.12.0

PROCESS PLANE SOLUTION FOR DESIRED STRENGTH

Desired Taylor Factors

• Design for obtaining a desired Taylor factor distribution.

• Three different process (x-axis rolling, tension and x-y shear) are tested.

• Provides the ability to select best processing pathsOptimal ODF in Tension process

Optimal ODF in rolling process




PROPERTY CLOSURE OF A PROCESS PLANE

Where,

With normalization constraint

Maximize (or minimize) properties

And positiveness constraint

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

3.055

3.06

3.065

3.07

3.075

3.08

3.085

3.09

3.095

3.055 3.06 3.065 3.07 3.075 3.08 3.085 3.09 3.095

3.05

3.06

3.07

3.08

3.09

3.1

3.11

Taylor factor along RD

Tay

lor

fact

or a

long

TD

R

C

Process plane

Process-property plane




CROSS-PLOTS

3.055 3.06 3.065 3.07 3.075 3.08 3.085 3.09 3.095

3.05

3.06

3.07

3.08

3.09

3.1

3.11

(a)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

3.055

3.06

3.065

3.07

3.075

3.08

3.085

3.09

3.095

(b)a1

a2

M along RD

(c)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

3.05

3.06

3.07

3.08

3.09

3.1

3.11

a1

a2

M along TD

4.84.54.23.93.73.43.12.92.62.32.11.81.5

(d)


Tay

lor

fact

or a

long

TD

R

C

Identify optimal textures from

the intersections of

property curves on the

structure-property space




PROCESS SELECTION

3.05 3.06 3.07 3.08 3.09 3.1 3.11 3.123.03

3.04

3.05

3.06

3.07

3.08

3.09

3.1

3.11

3.12


Tay

lor

fact

or a

long

TD

b

4. z-axis rolling following y-axis rolling and x-axis tension

2. y-axis rolling

3. y-axis rolling followed by x-axis tension

1. x-axis tension

R

C

ad

fe

• Property space of Taylor factors for x- and y- direction loading corresponding to various process planes.

• Desired property is C and initial property is R

• Multiple processes can be identified by superposing the property closures of different process planes on the property space.

• R-C and R-a-b-C are two possible routes.




NON-LINEAR PROPERTY SURFACES

Lankford R parameter = 1.0253 surface (along RD) on the tension basis

Youngs Modulus = 145.3GPa (along RD) surface on the tension basis

Two solutions

Solution on the reduced material plane

Non-linearly related to ODF

Linearization?




FUTURE CHALLENGES

Development of basis functions for complex strain

paths

Construction of property iso-surfaces in higher order

feature spaces using statistical learning

techniques

Identification of error limits in reduced order models as part of design procedure




CONCLUSIONS

Linear analysis of texture–property relationships using process-based representations of Rodrigues space

Acta Materialia, Volume 55, Issue 5, March 2007, Pages 1573-1587Veera Sundararaghavan and Nicholas Zabaras

• The concept of a ‘material plane’ in Rodrigues space was employed to identify optimal or extremal ODFs

•A new concept of a ‘process plane’ was established that represents the space of reduced-order coefficients for a given process.

• The process plane is capable of extrapolating several different processing paths.

• Linear programming methods were constructed to solve problems involving identification of ODFs on the process plane that are as close as possible to desired ODFs on the ‘material plane’.

• Graphical solution to the process sequence selection problem was enabled through the identification of process paths on property closures of process planes.

Materials Process Design and Control Laboratory Design and control of properties in polycrystalline...

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