materials process design and control laboratory veera sundararaghavan and nicholas zabaras sibley...

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


Veera Sundararaghavan and

Nicholas ZabarasSibley School of Mechanical and Aerospace Engineering

Cornell University

Supported by AFOSR, ARO

Design of materials with enhanced properties: A multi-length scale

computational approach.

Technical Presentation




PRESENTATION OUTLINE

• Motivation of microstructure sensitive Motivation of microstructure sensitive designdesign

• Microstructure homogenizationMicrostructure homogenization

• Multi-scale deformation process Multi-scale deformation process simulationsimulation

• Multi-scale sensitivity analysisMulti-scale sensitivity analysis

• Design resultsDesign results

• Development of a multi-scale continuum sensitivity method for multi-scale deformation problems• Design processes and control properties using multi-scale modeling



Materials Process Design and Control Laboratory

DEFORMATION PROCESS DESIGN SIMULATORDEFORMATION PROCESS DESIGN SIMULATOR

Enhanced strength

RESEARCH OBJECTIVESRESEARCH OBJECTIVES




Info from NASA ALLSTAR network, 2005

-Materials design is a slower process than engineering design.

-Replace empirical approaches to design with physically sound multi-scale approaches.

-Integrate materials design into engineering design.




MULTISCALE NATURE OF METALLIC STRUCTURES

Grain/crystal

Inter-grain slip

Grain boundary

Twins

precipitatesAtoms

Meso

Micro NanoMaterial-by-design

Titanium armors with high specific

strength.




COMPUTATIONAL DESIGN OF DEFORMATION PROCESSESCOMPUTATIONAL DESIGN OF DEFORMATION PROCESSES

Press forcePress forcePress speedPress speed

Product shapeProduct shapeCostCost

CONSTRAINTSCONSTRAINTSOBJECTIVESOBJECTIVES

Material usageMaterial usagePropertiesProperties

MicrostructureMicrostructure

Preform shapePreform shapeDie shape Die shape

VARIABLESVARIABLES

BROAD DESIGN OBJECTIVESGiven raw material, obtain final product with desired

microstructure and shape with minimal material utilization and costs

Forging rateForging rate




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 during deformation

POLYCRYSTALLINE MICROSTRUCTURESPOLYCRYSTALLINE MICROSTRUCTURES

• Stress

• Evolution of slip system resistances

• Shearing rate




FCC SINGLE CRYSTAL RESPONSE TO IMPOSED DEFORMATION

Athermal resistance (e.g. strong precipitates)

Thermal resistance (e.g. Peierls stress, forest dislocations)

If resolved shear stress < athermal resistance

, otherwise

Balasubramaniam and Anand, Int J Plasticity, 1998




Crystallographic slip 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.

Evolution of plastic deformation gradient

The elastic deformation gradient is given by

Rate-independent model(Anand and Kothari, 1996)

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

Ftrial

p

p

e

ee

Fr

Fc

Intermediateconfiguration

Deformedconfiguration

Intermediateconfiguration

Reference configuration

0

1Sapp FF

0SI tFF etrial

e

SINGLE CRYSTAL CONSTITUTIVE ANALYSISSINGLE CRYSTAL CONSTITUTIVE ANALYSIS




HOMOGENIZATION OF DEFORMATION GRADIENT

Use BC: = 0 on the boundary

Note = 0 on the volume is the Taylor assumption, which is the upper bound

X x

Macro

Meso

x = FXx = FX

y = FY + w

N

n

Macro-deformation is an average over the microscopic deformations (Hill, Proc. Roy. Soc. London A, 1972)

Decompose deformation gradient in the microstructure as a sum of macro deformation gradient and a micro-fluctuation field

Mapping implies that

(Miehe, CMAME 1999).

Macro

Micro




VIRTUAL WORK CONSIDERATIONS

Hill Mandel condition: The variation of the internal work performed by macroscopic stresses on arbitrary virtual displacements of the microstructure is required to be equal to the work performed by external loads on the microstructure. (Hill, J Mech Phys Solids, 1963)

Apply boundary condition

Homogenized stresses

Must be valid for arbitrary variations of F

Sundararaghavan and Zabaras, International Journal of Plasticity 2006.

Macro

Micro




MICROSTRUCTURE DEFORMATION

Thermal effects linking assumption

Equate macro and micro temperatures

An equilibrium state of the microstructure is assumed Updated Lagrangian formulation

XY

Z

Equivalent Stress (MPa): 20 30 40 50 60 70 80

Bn+1

x = x(X, t) F = F (X, t)

F = deformation gradient

Fn+1

B0X

Hexahedral meshing using CuBIT.




IMPLEMENTATION

Forming process

Update macro displacements

Boundary value problem for microstructure

Solve for deformation field

Integration of constitutive equations

Dislocation plasticity

Macro-deformation gradient

Homogenized (macro) stress

Micro-scale stress

Micro-scale deformation gradient

(a) (b)

Macro

Micro




MICROSTRUCTURE RESPONSE VALIDATION

0 0.1 0.2 0.3 0.4 0.5 0.60

0.1

0.2

0.3

0.4

0.5

0.6

Equivalent Stress (MPa): 0.00 5.71 11.43 17.14 22.86 28.57 34.29 40.00

y FY w

yY

0 0.05 0.1 0.15 0.230

40

50

60

70

80

Equivalent strain

Eq

stre

ss (

MP

a)

Experimental

Computed

Experimental results from Anand and Kothari (1996)




OPTIMIZATION OF MICROSTRUCTURE RESPONSEz:

Dev

iatio

n fr

om

desi

red

pro

pert

y

x: Process variable 1 y: Process variable 2

Initial guess

Select optimal process parameters to achieve a desired property response.

Need to evaluate gradients of objective function (deviation from desired property) with respect to

process variables.

Sundararaghavan and Zabaras, IJP 2006.

Scale linking, homogenization

Die shape

Initial preform shape

Forging rate

Initial microstructure




OPTIMIZATION FRAMEWORKOPTIMIZATION FRAMEWORK

Gradient methods

Finite differences (Kobayashi et al.) Multiple direct (modeling) steps Expensive, insensitive to small perturbations

Direct differentiation technique (Chenot et al., Grandhi et al.)

Discretization sensitive Sensitivity of boundary condition Coupling of different phenomena

Continuum sensitivity method(Zabaras et al.)

Design differentiate continuum equations Complex physical system Linear systems

Continuum equations

Design differentiate

Discretize

CSM -> Fast Multi-scale optimization

Requires 1 Non-linear and n Linear multi-scale problems to compute gradients




DESIGN DIFFERENTIATIONDESIGN DIFFERENTIATION

(Badrinarayan and Zabaras, 1996)

Directional derivative




MATERIAL POINT SENSITIVITY ANALYSISMATERIAL POINT SENSITIVITY ANALYSIS

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

FFrr and and xxoo

Pr and F,o

o o

Constitutive problem

Kinematic problem Sensitivity of single crystal response

Sensitivity of equilibrium equation

L = L (X, t; β)

I + Ls

x + x = x(X, t; β+Δ β)

o

F + Fo

L + L = L (X, t; β+Δ β)

o

F

x = x(X, t; β)

L = velocity gradient

X

•Sensitivity linking assumption:

The sensitivity of the deformation gradient at macro-scale is the same as the average of the sensitivities of deformation gradients in the microstructure.




MATERIAL POINT SENSITIVITY ANALYSISMATERIAL POINT SENSITIVITY ANALYSIS

Solve for sensitivity of microstructure deformation

field

Integration of sensitivity constitutive equations

Sensitivity of (macro) properties

Perturbed Mesoscale

stress

Perturbed meso deformation

gradient

Perturbed macro deformation gradient

SENSITIVITY DEFORMATION PROBLEM

Derive a weak form for the shape sensitivity of the equilibrium equation

Primary unknown of the weak form

x – sensitivity of the deformed configuration




SENSITIVITY OF THE CRYSTAL CONSTITUTIVE PROBLEM

• Sensitivity hardening law

• Sensitivity constitutive law for stress

• Derive sensitivity of PK-I stress

Integration of sensitivity constitutive equations

Sensitivity of (macro) properties

Perturbed Mesoscale

stress

Perturbed meso deformation

gradient

Perturbed macro deformation gradient

Solve for sensitivity of microstructure deformation

field




PROCESS DESIGN FOR STRESS RESPONSE AT A MATERIAL POINT

b c

0 2 4 6 8 100

10

20

30

40

50

60

Iterations

Cos

t fu

nct

ion

Time (sec)

Equ

iva

lent

str

ess

(M

Pa

)

(a) (b)

(c) (d)

0 2 4 6 8 100

10

20

30

40

50

60

Initial responseIntermediateFinal responseDesired response

Time (sec)

Equ

iva

lent

str

ess

(M

Pa

)

1 2 3 4 5 6 70

50

100

150

200

250

300

Change in Neo-Eulerian angle (deg)

9.81

7.05

4.28

1.52

-1.24

-4.00

-6.76

Sundararaghavan and Zabaras, IJP 2006.




Cubic crystal

FIRST ORDER REPRESENTATION OF MICROSTRUCTURESFIRST ORDER REPRESENTATION OF MICROSTRUCTURES

RODRIGUES’ REPRESENTATIONRODRIGUES’ REPRESENTATIONFCC FUNDAMENTAL REGIONFCC FUNDAMENTAL REGION

Crystal/lattice

reference frame

e2^

Sample reference

frame

e1^ e’1

^

e’2^

crystalcrystal

e’3^

e3^

n

Particular crystal

orientation

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

Any property can be expressed as an expectation value or average given by

Kumar and Dawson 1999, Ganapathysubramaniam and Zabaras, IJP 2005.




EVOLUTION OF CRYSTAL VOLUME FRACTIONS WITH DEFORMATION

Conservation principle

Solve for evolution of the ODF with deformation

Based on the Taylor hypothesis

EVOLUTION EQUATION FOR THE ODF (Eulerian)

v – re-orientation velocity: how fast are the crystals reorienting

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

Constitutive sub-problem

Taylor hypothesis: deformation gradient (F) in each crystal of the polycrystal is same

as the macroscopic deformation gradient.

Compute the reorientation velocity from the elastic deformation gradient

macro microF FLinking assumption

Sundararaghavan and Zabaras, Acta Materialia, 2005




Equivalent strain

Equiv

alen

tStres

s(M

Pa)

0 0.1 0.2 0.30

50

100

150

200

250

300

350

400

300K

195K

140K

20K

Material: 99.987% pure polycrystalline f.c.c Aluminum

Process: Simple shear motion

-0.4

-0.2

0

0.2

0.4

Z

-0.4

-0.2

0

0.2

0.4

-0.25

0

0.25

TL ODF30.025.721.417.112.9

8.64.30.0

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

Z

-0.4

-0.2

0

0.2

0.4

-0.4

-0.2

0

0.2

0.4

X Y

Z

-0.4

-0.2

0

0.2

0.4

Z

-0.4

-0.2

0

0.2

0.4

X

-0.4

-0.2

0

0.2

0.4

Y

VERIFICATION OF ODF EVOLUTION MODELVERIFICATION OF ODF EVOLUTION MODEL

Experimental analysis addressed in Carreker and Hibbard, 1957

Benchmark problem in Balasubramanian and Anand 2002.




MULTISCALE MODEL OF DEFORMATION USING ODF REPRESENTATION

Largedef formulation for macro scale

Update macro displacements

ODF evolution update

Polycrystal averaging for macro-quantities

Integration of single crystal model

Dislocation plasticity

Macro-deformation gradient

Homogenized (macro) stress

Microscale stressMacro-deformation gradient

(a) (b)

Macro

Micro

Meso

Parallel solver: PetSc (Argonne Labs) KSP-Solve

Meso

Micro

Macro




0 ~Ω + Ω = Ω (r, t; L+ΔL)r – orientation parameter

Ω = Ω (r, t; L)~I + (Ls)n+1

Fn+1 + Fn+1

o

x + x = x(X, t; β+Δ β)

o

Bn+1

L + L = L (X, t; β+Δ β)

o

Fn+1

x = x(X, t; β)

Bn+1

L = L (X, t; β)L = velocity gradientB0

Ls = design velocity gradient

ODF: 1234567

ODF: 1234567

The velocity gradient – depends on a macro design parameter

Sensitivity of the velocity gradient – driven by perturbation to the macro

design parameter

A micro-field – depends on a macro design parameter (and) the velocity

gradient as

Sensitivity of this micro-field driven by the velocity gradient

Sensitivity thermal

sub-problem

Sensitivity constitutive sub-problemsub-problem

Sensitivity kinematic

sub-problem

Sensitivity contact & friction

sub-problemsub-problem

MULTI-LENGTH SCALE SENSITIVITY ANALYSISMULTI-LENGTH SCALE SENSITIVITY ANALYSIS




MACRO-MICRO SENSITIVITY ANALYSISMACRO-MICRO SENSITIVITY ANALYSIS

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

Pr and F,o

o o

Constitutive problem

Regularized contact problem

Kinematic problem Material point sensitivity analysis




EXTRUSION DESIGN PROBLEM

Objective: Design the extrusion die for a fixed reduction such that the deviation in the Young’s Modulus at the exit cross section is minimized

Material:FCC Cu

Microstructure evolution is modeled using an orientation distribution function

Minimize Youngs Modulus variation across cross-section

Die design for

improved properties




DESIGN PARAMETERIZATION OF THE PROCESS VARIABLE

Objective: Minimize Young’s Modulus variation in the final product by controlling die shape variations

Identify optimal Ci that results in a desired microstructure-sensitive property

r()




CONTROL OF YOUNGS MODULUS: ITERATION 1

121

114

115

116

117

118

119

120

122

Youngs Modulus

(GPa)

First iteration

Objective function:

Minimize variation in Youngs Modulus





121

114

115

116

117

118

119

120

122

Youngs Modulus

(GPa)

Intermediate iteration





121

114

115

116

117

118

119

120

122

Youngs Modulus

(GPa)

Optimal solution

1 2 3 40.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1




MULTISCALE EXTRUSION –VARIATION IN OBJECTIVE FUNCTION

Obj

ectiv

e fu

nctio

n: v

aria

nce

(You

ng’s

Mod

ulus

)

Iteration number

Die Shape

Youngs Modulus

(GPa)

121

114

115

116

117

118

119

120

122

Uniform Youngs modulus

Small die shape

changes leads to better

properties




DESIGN PROBLEM

Objective: Design the initial preform such that the die cavity is fully filled and the yield strength is uniform over the external surface (shown in Figure below).

Material:FCC Cu

Uniform yield strength desired on this surface

Fill cavity

Multi-objective

optimization

• Increase Volumetric yield

•Decrease property variation




UPDATED LAGRANGIAN SHAPE SENSITIVITY FORMULATION

Sensitivity to initial preform shape




MULTI-SCALE DESIGN FOR OPTIMUM STRENGTH: ITERATION 1

70

80

90

100

110

120

130Large

Underfill

variation in yield strength

Yie

ld s

tren

gth

(M

Pa)




70

80

90

100

110

120

130

Yie

ld s

tren

gth

(M

Pa)

Smaller under-fill

variation in yield strength





70

80

90

100

110

120

130Underfill

Yie

ld s

tren

gth

(M

Pa)

Optimal yield strength

Optimal fill


1 2 3 4 5 6 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1




COMPARISON OF FINAL PRODUCTS AT DIFFERENT ITERATIONS

Uniform yield strength

Cos

t fun

ctio

n:

(und

erfil

l,var

ianc

e of

yie

ld s

tren

gth) Initial preform design

After forging

Iteration number

XY

Z

Equivalent Stress (MPa): 20 30 40 50 60 70 80




CONCLUSIONSCONCLUSIONS

• First-ever effort to optimize macro-scale properties of materials using multi-scale design of deformation processes.

• Ability to relate process variables to microstructure evolution and directly control microstructure-dependent properties.

Microstructure evolution

Multi-scale optimization

materials process design and control laboratory veera sundararaghavan and nicholas zabaras sibley...

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