Materials Process Design and Control Laboratory Babak Kouchmeshky Admission to Candidacy Exam...

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


Babak Kouchmeshky

Admission to Candidacy Exam Presentation

Date April 22,2008

A multi-scale design approach for tailoring the macro-scale properties of

polycrystalline materials

OutlineOutline• Modeling HCP polycrystals deforming by slip and Modeling HCP polycrystals deforming by slip and

twinningtwinning

• A design approach for tailoring the processing A design approach for tailoring the processing parameters that lead to desired macro-scale parameters that lead to desired macro-scale propertiesproperties

• A microstructure-sensitive design approach for A microstructure-sensitive design approach for controlling properties of HCP materialscontrolling properties of HCP materials

• Future workFuture work




I. Constitutive modelI. Constitutive model




TheoryTheory

DEFORMATION TWINNINGDEFORMATION TWINNING

Twinning produces a rotation of crystal lattice.

Important deformation mode for HCP materials

Dominant at room temperature. Twinning is lesser at high temperatures.




SLIP AND TWIN PLANESSLIP AND TWIN PLANES

Basal slip system Prismatic slip Pyramidal slip Pyramidal twin

a2 - axis

c - axis

a1 - axis

a3 - axis

e1

e2

e3

Orthogonal system used for modeling




TWINS MODELED AS PSEUDO-SLIPTWINS MODELED AS PSEUDO-SLIP




Dawson and Myagchilov (1999)

1

twb

twn

1

Sx x

( )tw twSb x n

CONSTITUTIVE MODEL FOR SLIP AND TWINNINGCONSTITUTIVE MODEL FOR SLIP AND TWINNING

Velocity gradient

Hardening law

Isotropic flow rule to account for grain boundary accommodation

For slip and twin systems

no hardening

Consistency condition

(Anand,IJP 2003)




Condition for slip and twinning

0i is

0 if

0 if th

th

s

s

Crystallographic slip, twinning and re-orientation

of crystals are assumed to be the primary mechanisms of

plastic deformation

Evolution of various material configurations for a single crystal as needed in the integration of the

constitutive problem.

B0

m

n

n

m

m

n

n̂

m

n^ m

__

Bn

Bn Bn+1

Bn+

1

_

_

Fn

Fn

Fn

Fn+1

Fn+1

Fn+1

p

p

e

Ftrial

e

e

Fr

Fc

Intermediateconfiguration

Deformedconfiguration

Intermediateconfiguration

Reference configuration

INCREMENTAL KINEMATICS





Constitutive theoryConstitutive theory

D = Macroscopic stretch = Schmid tensor = Lattice spin W = Macroscopic spin = Lattice spin vector



Polycrystal plasticityInitial configuration

Bo BF*Fp

F

Deformed configuration

Stress free (relaxed) configuration

n0

s0

n0

s0

ns

(2) Ability to capture material properties in terms of the crystal properties

(1) State evolves for each crystal

CONSTITUTIVE MODEL FOR SLIP AND TWINNINGCONSTITUTIVE MODEL FOR SLIP AND TWINNING

Solve for the reorientation velocity and the rate of change

in volume fraction of twins

Solve for shearing rates on slip and twin systems

Rate of change of volume fraction







Arresting of twinning systemsArresting of twinning systems

Three stages of strain hardening obtained using current model.

is a uniform distribution between 0.3 and 1.0.

is supposed to be 0.2.

fP

satf


Orientation distributionsOrientation distributions

Any macroscale property < χ > can be expressed as an expectation value if the corresponding single crystal property χ (r ,t) is known.

• Determines the volume fraction of crystals within a region R' of the fundamental region R• Probability of finding a crystal orientation within a region R' of the fundamental region• Characterizes texture evolution

ORIENTATION DISTRIBUTION FUNCTION – A(r,t)

– reorientation velocity

ODF EVOLUTION EQUATION – EULERIAN DESCRIPTION

v



Crystal/lattice

reference frame

e1^

e2^

Sample reference

frame

e’1^

e’2^

crystalcrystal

e’3^

e3^Crystallographic orientation

Rotation relating sample and crystal axis Properties governed by orientation

Discrete aggregate of crystals (Anand et al.)

Comparing & quantifying textures

Continuum representation Orientation distribution function (ODF) Handling crystal symmetries Evolution equation for ODF

Different methodologies in representing the textureDifferent methodologies in representing the texture







Hexagonal crystal (HCP)

Angle-axis and Rodrigues representationAngle-axis and Rodrigues representation

RODRIGUES REPRESENTATION

Neo-Eulerian representation of orientation Rotations about a fixed axis trace straight

lines in parameter space Set of orientations equidistant from two

rotations is always a plane Helps reduce symmetries to between a pair

of planes – fundamental region

( ) tan2

f ( )r nf

Rodrigues vectorr

ANGLE AXIS REPRESENTATION

Any orientation can be uniquely represented by a rotation about an axis n by an angle Φ

Φ

n

( ) cos( )( ) sin( )R n n n I n n I n 1 1

cos ( 1)2

traceR 1

sinn vectR

LATTICE REORIENTATION DUE TO TWINNINGLATTICE REORIENTATION DUE TO TWINNING

Twin plane normal

1( 2 )x n n x xI T

angle

Crystal Axis = h 180o

Twin mapping is a reflection

• Can be represented as a rotation of crystal axis about twin normal through 180o

1( )T I T

In quaternion representation, Tq = [0,n1,n2,n3]

1) Convert crystal axis h to the quaternion representation hq

2) Perform quaternion product Q = Tq hq

3) Project Q to the fundamental region (QF) based on crystal symmetries

4) Convert QF to Rodrigues representation




We take advantage of Quaternions in here. They prove useful for coordinate transformations. The quaternion method is the natural choice when the coordinate systems keep moving.

ORIENTATION DISTRIBUTION FUNCTION (ODF)ORIENTATION DISTRIBUTION FUNCTION (ODF)

Conservation Conservation principleprinciple

Texture can be Texture can be described, described, quantified & quantified & comparedcompared

Why continuum approach for

ODF?

EVOLUTION EQUATION FOR THE ODF EVOLUTION EQUATION FOR THE ODF

J – Jacobian determinant of the reorientation of the crystals

r – orientation of the crystal.A – is the ODF, a scalar field;

Constitutive sub-problemConstitutive sub-problem

Taylor hypothesis:Taylor hypothesis:deformation in each deformation in each crystal of the polycrystal crystal of the polycrystal is the macroscopic is the macroscopic deformation.deformation.




( , ) ( , ) ( ,0)J r t A r t d A r d

ODF EVOLUTION EQUATION WITH TWINNINGODF EVOLUTION EQUATION WITH TWINNING

Calculate crystal reorientation

Initial random ODF = 1.2158 at all nodal points

HCP Fundamental region

Source term due to twinning

Volume fraction lost

Volume fraction gained from other orientations

Total Lagrangian formulation




VOLUME FRACTION OF TWINSVOLUME FRACTION OF TWINS

Volume fraction lost due to transfer of orientation from r to rk

r

rk

Orientation space




Dawson and Myagchilov (1999)

Numerical resultsNumerical results




I. Tension modeI. Tension mode




• modeling tension on an initially textured modeling tension on an initially textured Magnesium alloy AZ31B rodMagnesium alloy AZ31B rod

Initial texture Initial texture




initial texture in the experiment by anand and

staroselsky , 2003

initial texture used in the simulation

Material properties Material properties




Elastic constants:

11 12 13 33

55

C = 58 GPa; C = 25 GPa; C = 20.8 GPa; C = 61.2 GPa;

C = 16.6 GPa.

basal prismatic

pyramidal twin

s = 0.55 MPa; s = 105 MPa;

s = 105 MPa; s = 180 MPa.

Slip resistances:

Slip and twining systems:

Final texture Final texture




Texture of the Mg rod at the tensile strain of 15% in the experiment by anand and staroselsky, 2003

Texture of the Mg rod at the tensile

strain of 15%

Comparison between stress-strain curve from experiment, this work and numerical simulation by anand and staroselsky , 2003

II. Compression modeII. Compression mode




• Modeling compression on an initially Modeling compression on an initially textured Magnesium alloy AZ31B rodtextured Magnesium alloy AZ31B rod





Elastic constants:

11 12 13 33

55

C = 58 GPa; C = 25 GPa; C = 20.8 GPa; C = 61.2 GPa;

C = 16.6 GPa.

basal prismatic

pyramidal twin

s = 0.55 MPa; s = 105 MPa;

s = 105 MPa; s = 180 MPa.

Slip resistances:


Final texture Final texture




Texture of the Mg rod at the tensile strain of 18% in the experiment by Anand and

Staroselsky, 2003

Texture of the Mg rod at the tensile strain of 18%

Stress-strain curve Stress-strain curve




Comparison between stress-strain curve from

experiment, this work and numerical simulation by

anand and staroselsky,2003

3 different stages in the normalized strain hardening

response

III. Shear modeIII. Shear mode




A shear mode is assumed.

In this problem texture evolution and stress-strain curve is examined for Titanium.





Elastic constants:

11 12 13 33

55

C = 256.6 GPa; C = 0.36 GPa; C = 69.7 GPa; C = 325.13 GPa;

C = 46.71 GPa.

basal prismatic

pyramidal twin

s = 150.0 MPa; s = 30.0 MPa;

s = 120.0 MPa; s = 125 MPa.

Slip resistances:


Result (cont.)Result (cont.)




Experimentally measured texture of Ti at effective strain 1 for the shear mode

Numerically predicted texture of Ti at effective strain 1 for the shear mode X. Wu et al, Acta Materialia 55 (2007)

Experiment This work

Result (cont.)Result (cont.)




Experiment is done by Wu et al. (2007)

IV. Plain strain compression IV. Plain strain compression modemode




A plane strain compression mode is assumed.

( , )( ( , ) ( , )) ( ,0)J r t A r t r t d A r d

In this problem texture evolution is examined for Titanium .





Elastic constants:

11 12 13 33

55

C = 256.6 GPa; C = 0.36 GPa; C = 69.7 GPa; C = 325.13 GPa;

C = 46.71 GPa.

basal prismatic

pyramidal twin

s = 8.0 MPa; s = 1.0 MPa;

s = 10.0 MPa; s = 1.25 MPa.

Slip resistances:


ResultResult




Results obtained by Myagchilov and Dawson, Simul. Mater. Sci. Eng. 7 (1999)975-1004

6543210

Evolved texture at effective strain of 0.5




CONCLUSION(part 1)CONCLUSION(part 1)

1) Continuous representation of texture

• Eliminates the need for splitting existing elements to account for new orientations caused by twinning

• Provides a natural tool for calculating sensitivities needed for design problem

2) Twinning is accounted through pseudo shear and reorientation of crystals

3) Twin saturation is phenomenologically accounted for.

4) ODF conservation equation modified to include the source and sink terms due to twinning.

5) The constitutive model is tested for Titanium and Magnesium alloy AZ31B.


Enhancing properties of polycrystals through Enhancing properties of polycrystals through a sensitivity problem that spans macro and a sensitivity problem that spans macro and micro scalesmicro scales



The orientation of crystals in a poly crystal sample has a direct influence on the properties of the specimen in the macro scale.

Crystals reorient during the deformation. So macro scale processing parameters like velocity gradient affect the crystal reorientation.


Definition of the problemDefinition of the problem



The aim of this problem is to design the processing parameters in a sequence of two processes such that a micro structure with desired properties is obtained

Desired qualities:

High hardness and ductility

Convex hull of B,G,B/G

The hardness and ductility are presented by Bulk modulus (B), Shear modulus and B/G.


Problem statementProblem statement



Sub problems:

1- Find the texture that provides the maximum hardness and ductility

2- Find the reduced order model for the processes

3- Find the optimum texture from process plane

4- Define the design problem as two coupled optimization problems where each represent a process.

5- Find the convex hull of textures obtainable from process 1

6- Define a supplementary problem for reducing the computational efforts needed for the inverse problem

7- Solve for the constrained optimization

X Y

Z


Optimizing the processes to get the Optimizing the processes to get the optimum textureoptimum texture

• Step1: Find the design parameter L2 and initial texture A1 such that at the end of process 2 the desired texture is obtained. There will be a constraint on A1 based on the textures obtainable from texture 1.

• Step2: find the design parameter L1 in process 1 that leads to final texture A1.



T 2T

A1 A2L2

L2 and L1 are the design parameters

Process 1 is supposed to start from a random texture

T

A1L1

0

ˆ( , ) ˆ( , ) ( , ) 0A s t

A s t v s tt

0

ˆ( ,0) ( )A s A s


The sensitivity problemThe sensitivity problem



( , )( , ) ( , ) ( , ) ( , ) 0

oo oA s t

A s t v s t A s t v s tt

with the initial condition : ( ,0) 0o

A s

The sensitivity problem with respect to The sensitivity problem with respect to design parameters (L1,L2)design parameters (L1,L2)

The sensitivity problem with respect to the The sensitivity problem with respect to the field A1field A1

1with the initial condition : ( ,0) 0 ( )o

A s A s

( , )( , ) ( , ) 0

ooA s t

A s t v s tt


Sensitivity of the reorientation velocitySensitivity of the reorientation velocity



1 1 1 1 1 1

1 1 11 1 1

2

2

o o

n nn n n n n

o o o

n n nn n n

tr r r r

tr r r

I I Ωo

o rv

t

1 e eo o oeT e eT eT

n nt

Ω R R R R R Ro o

vect Ω

1 1 1o o o

e e e e e e eT e e esym sym

R F F R R U F F F R

Constitutive sensitivity problemConstitutive sensitivity problem


Definition of the adjoint problem Definition of the adjoint problem



1 1 12 2 2ˆ ˆ ˆ2 ( ( ) ( )) ( , )) = 2 ( ,0) ( ,0))A des A f AJ A s A s A s t d s A s d

2

( , ) ˆ( , ) ( , ) ( ( ) ( )) ( )

des f

s ts t v s t A s A s t t

t

with initial condition

( , ) 0fs t

2 22 2 ( ) 2 ( )( ( , ) , ( , )) ( ( ,0), ( ,0))o o

des f l lA s t A A s t s A s

12 2( , ) ( , )o

AA s t D A s tLagrange identity for obtaining the adjoint operator

*2 2( , ) ( , ) 0

o o

L A L A

Gradient of the objective functional

where is the solution of the following problem




The texture that provides the maximum hardness and ductility

where

2

1 1

1 2 3

21 1

1

(1 )

, ,

1

(1 )

1

ji ji i i

i i i i ii

i i

N N

i i i

f X Jr r

BX B X G X G

y A

Q I

P Jr r

b

1

2

3

such that

Py=b

Qy 0

T

T

T

f y

Max f y

f y

Convex hull of B,G and B/G values obtainable for a single crystal

X Y

Z

B= 124.8 GPa, G=58.87 GPa B/G=2.12

The optimum texture from material space


The reduced order model



first process (simple tension, with random initial texture)

second process (plain strain compression , with initial texture selected from the convex hull of all textures available from process 1.)

Convex hull of textures for process 1( , ) 0, A(s)d =1A s t

1

( , ) ( ) ( )M

j jj

A s t c t s


3- Find the optimum texture from process plane



11

21

31

1

1

such that

=b

0

bT

m mm

bT

m mm

bT

m mm

bT

m mm

b

m mm

f c

Max f c

f c

p c

Qc

where

2

1 1

1 2 3

1

21 1

1

(1 )

, ,

1

(1 )

1

ji ji i i

i i i i ii

b

m mm

N N

i i i

f X Jr r

BX B X G X G

A c

Q I

P Jr r

b

Prioritizing the objectives

The emphasize is given on the ductility(B/G). Other parameters are treated by inequality constraints which forces them to be greater than two third of the maximum values obtainable.


3- Find the optimum texture from process space



The optimum texture obtained from the process plane of the sequence of a tension process followed by a plain strain compression process

Process space contains all the plausible textures obtainable from a sequence of processes

X Y

Z

Optimum texture from material space

Optimum texture from Process space


Verify



From sensitivity analysis,1st process t=5 sec

From Finite difference, 1st process t=5 sec

1 2 2 1 2 2 2 1 2( , , , ) ( , , , ) ( , ,0) ( , , ,0)o o

desA A L s t A A A L s t d s L A A L s d

1 2 2 1 2( , , , ) ( , , , ) 0.20733o

desA A L s t A A A L s t d

12 2 1 2( , ,0) ( , , ,0) 0.2084As L D A A L s d

111 2 ( ,0,...,0) 2 11 12 1( , , , ) ( , , , ,..., )o

A nA A L s t D A L t A A A

Verify sensitivity problem

Relative error: 0.3%

Verify supplementary problem


Verify



1 12 2 1 2 1 2 2 1 2( , ,0) ( , , ,0) ( , , , ) ( , , , )A des As L D A A L s d A A L s t A D A A L s t d

To verify

define1 12 2 1 2 1 2 2 1 2( , ,0) ( , , ,0) ( , , , ) ( , , , )A des AB s L D A A L s d A A L s t A D A A L s t d

12 2 1 2Relative Error = / ( , ,0) ( , , ,0)AB s L D A A L s d

t(sec) Relative error%2L

0.01410

8

6

5

0.020

0.029

0.036

0.50

0.61

0.58

0.65

1 2

1 2

0.2084 0.20733

0.1982 0.1969

0.2429 0.2415

0.2617 0.2600




-1-0.5

00.5

1

-1-0.5

00.5

1

Z

-0.200.2

32.72.521.81.510.70.50

-1-0.5

00.5

1

-1-0.5

00.5

1

Z

-0.200.2

Objective function for process 1

Objective function for process 2




Conclusion (part 2)

• The problem of obtaining metallic alloys with optimum hardness and ductility through cold processing is addressed.

• The problem of finding optimum macro-scale properties was converted to that of polycrystalline texture through linear homogenization methodology.

• The optimum texture was projected from the material space to process space which contains all textures obtainable from the sequence of two parameters.

• Process parameters for a sequence of two deformation modes are optimized through two coupled optimization problems

• A functional optimization methodology is used for addressing the infinite dimensional optimization problem defined.

• Solution of an adjoint problem is used for calculating the gradient of the objective function with respect to a field parameter (initial texture of the second process).




Forging Forging

• Goals : Goals :

1.1. Minimal material Minimal material wastage due to flashwastage due to flash

2.2. Filling up the die cavityFilling up the die cavity

Material : Ti

• Why multi scale?Why multi scale?

– The evolution of the material properties at the macro The evolution of the material properties at the macro scale has a strong correlation with the underlying scale has a strong correlation with the underlying microstructure.microstructure.

Multi-scale polycrystal plasticity

Multi-length scale design environment

Coupled micro-macro direct model

Selection of the designSelection of the design variables like variables like

preform parameterizationpreform parameterizationCoupled micro-macro Coupled micro-macro

sensitivity modelsensitivity model

Reduced modeling of

microstructure/texture evolution for meaningful and

effective control.

Design based on:

Polycrystal plasticity, evolution of texture, multi-length scale analysis

Deformation problem:

-Updated Lagrangian framework

-Connection to the micro scale through Taylor hypothesis

- Microstructure represented as orientation distribution function (ODF) in Rodrigues space.







Implementation of the direct problem

Meso

Macro

formulation for macro scale

Update macro displacements

Texture evolution update

Polycrystal averaging for macro-quantities

Integration of single crystal slip and twinning laws

Macro-deformation gradient

microscale stressMacro-deformation gradient

Micro

( , ) ( , ) ( ,0)J r t A r t d A r d




THE DIRECT CONTACT PROBLEM

r

n

Inadmissible region

Referenceconfiguration

Currentconfiguration

Admissible region

ImpenetrabilityImpenetrability ConstraintsConstraints

Augmented Lagrangian Augmented Lagrangian approach to enforce approach to enforce impenetrabilityimpenetrability

Polycrystal average of orientation

dependent property

Continuous representation of texture




REORIENTATION & TEXTURINGREORIENTATION & TEXTURING


A sample direct problem



Equivalent stress (MPa)

390380375370365360350345340330

350340330325320310300


610600590580570560550530520510490


Equivalent stress for the direct problem in different time steps




SCHEMATIC OF THE CONTINUUM SENSITIVITY METHOD

Advantage : Fast Multi-scale optimization

Requires 1 Non-linear and n Linear multi-scale problems for

each step of the optimization algorithm.

n: number of design parameters

Equilibrium equation

Design derivative of equilibrium

equation

Material constitutive

lawsDesign derivative

of the material constitutive laws

Incremental sensitivityconstitutive sub-problem

Time and space discretized weak form

Sensitivity weak form




Curved surface parametrization – Cross section can at most be an ellipse

Model semi-major and semi-minor axes as 6 degree bezier curves

6

1

51

3 33

2 55

( ) ( )

(1.0 ) (1.0 5.0 )

20.0(1.0 )

6.0(1.0 )

i ii

R

4 2

2

2 44

66

( ) 0

15.0(1.0 )

15.0(1.0 )

R

/z H

Design vector

1 2 3 4 5 6{ , , , , , }T β

H

Representing the preform shape




DEFINITION OF PARAMETER SENSITIVITY

state variable sensitivity contourstate variable sensitivity contourw.r.t. parameter changew.r.t. parameter change

X = X (Y; s )

oFR + FR

Y

X

X+Xo

xn+xn

o

xn

oFn + Fn

FR

Fn

BR

Bo

I+Lo

x+xoo

Fr + Fr

x B

xn + xn = x (Y , tn ; s + s)o ~

Qn + Qn = Q (Y, tn ; s + s)o ~

x = x (xn, t ; s)^

B n

xn = x (X, tn ; s )~

Qn = Q (X, tn ; s )~

I+Ln

Main FeaturesMain Features

• Mathematically rigorous Mathematically rigorous definition of sensitivity fields definition of sensitivity fields

• Gateaux differentials (directional Gateaux differentials (directional derivatives) referred to fixed derivatives) referred to fixed YY in the configuration in the configuration BB RR

o

X + X= X (Y; s + s)o ~

~ Fr

x + x = x (x+xn , t ; s + s)

^o o




SENSITIVITY KINEMATIC PROBLEM

Continuum problem Differentiate Discretize

Design sensitivity of equilibrium equation

Calculate such that x = x (xr, t, β, ∆β )oo

Variational form -

FFrr and and xxoo o

λ and x o

Constitutive problem

Regularized contact problem

Kinematic problemSensitivity of ODF evolution

Pr and F,o

o o

Enormous degrees of freedom& number of PDEs to be solved

limits the scope of design

Computational issues

Microstructure-model reductionwithout significant loss of

accuracy

Possible Solution




COMPLEXITY OF MULTILENGTH SCALE DESIGN




INTRODUCE MICROSTRUCTURE-MODEL REDUCTION

Suppose we had a collection of data (from experiments or simulations) for the ODF:

such that it is optimal for the data represented as

Is it possible to identify a basis

POD technique – Proper Orthogonal

Decomposition

Solve the optimization problem

Method of snapshots

where

Eigenvalue problem

where




REDUCED ORDER MODEL FOR THE ODF

( , )( , ) ( , ) ( , ) 0

A s tA s t v s t s t

t

The reduced basis for the ODF ensemble has been evaluated, say

Using this basis, ODF represented as follows

This representation of the ODF leads to a reduced-model in the form of an ODE.

Reduced model for the evolution of the ODF

where

Initial conditions

( ( , ) ( , ) )ij j i jR v s t s t d




POLYCRYSTAL SENSITIVITY ANALYSIS

EVOLUTION EQUATION FOR THE SENSITIVITY OF THE ODF

Assumption: extended Taylor hypothesis for the continuum sensitivity Assumption: extended Taylor hypothesis for the continuum sensitivity analysis – analysis – i.e. we make no distinction between the sensitivity of the crystal velocity i.e. we make no distinction between the sensitivity of the crystal velocity gradient and gradient and the sensitivity of the macroscopic velocity gradient.the sensitivity of the macroscopic velocity gradient.

Sensitivity of reorientation velocity

Gradient of the sensitivity of the velocity

( , )( , ) ( , ) ( , ) ( , ) ( , ) 0

A s tA s t v s t A s t v s t s t

t

1 1( ) ( )

2 2v r r r r r I r

,

ii i kj kjj

vv v

r




REDUCED ORDER MODEL FOR THE ODF SESNSITIVITY

The reduced basis for the ODF ensemble has been evaluated, say

Using this basis, ODF represented as follows

This representation of the ODF leads to a reduced-model in the form of an ODE.

Reduced model for the evolution of the ODF

where

Initial conditions

( ( , ) ( , ) )ij j i jG v s t s t d

( , )( , ) ( , ) ( , ) ( , ) ( , ) 0

A s tA s t v s t A s t v s t s t

t

o

b G b H

( ( , ) ( , ))i iH A s t v s t d

(0) ( ,0) ) 0i ib A s d




IMPLEMENTATION OF REDUCED MODEL

-Regions suspected to come into contact (2 sets, before andafter coming into contact)

-Other part of the macro scale (1 set)

Time varying boundary conditions during the deformation

3 different set of Reduced order modes for the microstructure are constructed for different regions in the macro scale.

large number of reduced order modes




IMPLEMENTATION OF REDUCED MODEL




Validation of the reduced-order model: Sensitivity analysisValidation of the reduced-order model: Sensitivity analysis

The same set of regions are used for the sensitivity problem

FDM solution Full model

(Continuum sensitivity method)

Reduced model

ODF: 2 2.625 3.25 3.875 4.5




DesignDesign

Objective function




DesignDesign

First iteration of optimization Last iteration of optimization

Preform shape for the first iteration

Preform shape for the last iteration




Distribution of texture for some points on macro Distribution of texture for some points on macro scalescale




Conclusion (part 3)

A multi scale design methodology is applied for the problem of forging Ti alloy

- The goal has been to fill the die cavity and minimize the wastage of material

- Continuum sensitivity approach using extended Taylor hypothesis is used .

- Reduced order modeling is used to address the large amount of computational effort needed




Future direction

Micro problem driven by the velocity gradient L

Macro problem driven by the macro-design

variable βBn+1

Ω = Ω (r, t; L)~Polycrystal

plasticityx = x(X, t; β)

L = L (X, t; β)

L = velocity gradient

Fn+1

B0

Reduced Order Modes

Data mining techniques

Multi-scale Computation

Design variables (β) are macrodesign variables Processing sequence/parameters

Design objectives are micro-scale

averaged material/processproperties

Database




Future direction

In addition to preform shape , parameters like forging velocity and initial texture of the workpiece should be considered

Obtaining optimized distribution of macroscale properties like Young modulus , Yield strength, etc.

E

(with respect to design parameter )1




Plan for future work

•Extend the design methodology for sequence of processes to multi-scale polycrystal plasticity

•Extend the graphically based selection of processes to mathematically rigorous method using nonlinear model reduction.




Publication and presentations since Aug. 2006

•Journal:

•B. Kouchmeshky and N. Zabaras, "Modeling the response of HCP polycrystals deforming by slip and twinning using a finite element representation of the orientation space", Int. J. Plasticity, submitted.

•B. Kouchmeshky and N. Zabaras, "A designing approach with reduced computational effort for tailoring the processing parameters that lead to desired macro-scale properties in HCP polycrystals", in preparation.

•Conference:

•B. Kouchmeshky and N. Zabaras, "A microstructure-sensitive design approach for controlling properties of HCP materials", presented at the TMS Annual Meeting, New Orleans, Louisiana, March 9-13, 2008

•B. Kouchmeshky and N. Zabaras, "A simple non-hardening rate- independent constitutive model for HCP polycrystals deforming by slip and twinning", presented at the TMS Annual Meeting, New Orleans, Louisiana, March 9-13, 2008

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