Materials Process Design and Control Laboratory DESIGN OF MICROSTRUCTURE-SENSITIVE PROPERTIES IN ELASTO-VISCOPLASTIC POLYCRYSTALS USING MULTISCALE HOMOGENIZATION

Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory

CCOORRNNEELLLL U N I V E R S I T Y


DESIGN OF MICROSTRUCTURE-SENSITIVE PROPERTIES IN ELASTO-VISCOPLASTIC

POLYCRYSTALS USING MULTISCALE HOMOGENIZATION TECHNIQUES

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, S. Sankaran and Nicholas Zabaras

peoplepeople




RESEARCH SPONSORS

U.S. Air Force Partners

Materials Process Design Branch, AFRL

Computational Mathematics Program, AFOSR

NATIONAL SCIENCE FOUNDATION (NSF)

Design and Integration Engineering Program

CORNELL THEORY CENTER

U.S. Army research office (ARO)

Mechanical Behavior of Materials Program




MULTISCALE MODELING

grain/crystal

Inter-granular slip

Twins

atoms

Me

so

-sc

ale

Mic

ro-s

cale

Nano

Continuum scale

Metallic materials are composed of a variety of features at different length scales




MULTISCALE MODELING

grain/crystal

Inter-granular slip

Twins

atoms

Me

so

-sc

ale

Mic

ro-s

cale

Nano

Continuum scale

Homogenization

Me

ch

anic

s of slip

MD

Material property evolution is dictated by different physical phenomena at each scale.




CONTROL PROPERTY EVOLUTION THROUGH PROCESS DESIGN

f(g)

f(g,g’|r)

One point statistic: Texture

two point statistics

g: orientation of crystalMicrostructures are complex and the response depends on

•crystal orientations,

•higher order correlations of orientations,

•grain boundary and defect sensitive properties.

Control these features through careful design of deformation processes

Process?

Strain rate?

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): 7 14 22 30 37 45 53 60

Pro

pert

yTime

Desired response

Final response




MULTI-LENGTH SCALE CONTROL

Meso-scale representationForging

Properties

Evolving microstructure

Process: Cold working

Control process parameters

Identification of stagesIdentification of stages

Number of stagesNumber of stages

Preform shapePreform shape

VARIABLESVARIABLES

Intermediate step

Forging ratesForging rates

Design properties

OBJECTIVESOBJECTIVESMaterial usageMaterial usage

MicrostructureMicrostructure

Desired shapeDesired shape

Desired propertiesDesired properties




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

PHYSICAL APPROACH TO PLASTICITYPHYSICAL APPROACH TO PLASTICITY




CONVENTIONAL MULTISCALING SCHEMES

APPROXIMATION 1:

All grains will take the same deformation – TAYLOR

Relaxed constraints model: takes grain shapes into account for relaxing certain stress components

APPROXIMATION 2:

All grains have the same stresses – SACHS ASSUMPTION

APPROXIMATION 3:

Assume each grain is surrounded by an equivalent medium: Identify an interaction law between a grain and its surroundings – Self consistent scheme

Satisfies compatibility, Equilibrium across GBs

fails

Strong kinematic constraint: gives

stiff response

(upper bound)

Gives softest response

(lower bound)

How does macro loading affect the microstructure

Failure to predict evolution of texture within grains

Failure to predict GB misorientation development

Taylor assumption

Sachs assumption




Homogenization scheme

(a) (b)

How does macro loading affect the microstructure

1. Microstructure is a representation of a material point at a smaller scale

2. Deformation at a macro-scale point can be represented by the motion of the exterior boundary of the microstructure. (Hill, R., 1972)




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 xMacro

Meso

x = FXx = FX

y = FY + w

N

n

Macro-deformation can be defined by the deformation at the boundaries of the microstructure (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).

Sundararaghavan and Zabaras, IJP 2006.




Virtual work considerations

How to calculate homogenized stresses?

Hill Mandel condition: The variation of the internal work performed by homogenized stresses on arbitrary virtual displacements of the

microstructure is required to be equal to the work performed by external loads on the microstructure.

Apply BC

Homogenized stresses

Must be valid for arbitrary variations of F




Equilibrium state of the microstructure

An equilibrium state of the micro-structure is assumed

This assumes stress field variation is quasi-static and inertia forces are instead included in the equations of motion of the homogenized continuum.

Is assumed and used to calculate the averaged Cauchy stress

Thermal effects linking assumption

Equate macro and micro temperatures

Macro dissipation = average micro dissipation

Assumed (Nemat-Nasser, 1999)




Single crystal constitutive laws

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

Incorporates thermal effects on shearing rates and slip

system hardening(Ashby; Kocks; Anand)

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

• Constitutive law for stress

• Evolution of slip system resistances

• Shearing rate

• Coupled system of equations for slip system resistances and stresses at each time step is solved using Newton-Raphson algorithm with quadratic line search




Constitutive integration scheme

Athermal resistance (e.g. strong precipitates)

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

If resolved shear stress does not exceed the athermal resistance

, otherwise




Consistent tangent moduli at meso-scale

Implicit solution schemeT = Eetrial

• Definition of stresses

• Variation in Cauchy stress

The consistent tangent moduli required for non-linear solution of the microstructure equilibrium problem is calculated using a implicit solution scheme by direct differentiation of crystal constitutive equations.




Implementation

Boundary value problem for microstructure

Solve for deformation field

Integration of constitutive equations

Continuum slip theory

Consistent tangent formulation (meso)

Macro-deformation information

Homogenized (macro) properties

Mesoscale stress, consistent tangent

meso deformation gradient

(a) (b)

Macro

Meso

Micro




Pure shear of an idealized aggregate

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

1.2

X Y

Z

ODF5.05004.39293.73573.07862.42141.76431.10710.4500

0 0.5 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2


X Y

Z

4.428793.844583.260372.676162.091941.507730.9235190.339306

X Y

Z

8.750777.584576.418365.252154.085952.919741.753540.58733

Equivalent strain

Eq

uiva

len

tstr

ess

(MP

a)

0 0.1 0.2 0.3

10

20

30

40

50

60

continuum Taylor

Discrete Taylor

FEM homogenization




Comparison of texture from Taylor and Homogenization approach




Plane strain compression of idealized aggregate




Three-dimensional shear of an idealized aggregate

-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

10

20

30

40

50

60

70

80

0.00 0.05 0.10 0.15 0.20 0.25 0.30

2D microstructure (400 grains)

3D microstructure (512 grains)

Experimental results

Equivalent strain

Equ

ival

ent s

tress

(M

Pa

)


<111> <110>

<111> <110>

(a)

(b)

(c) (d)

Experiment (Carreker and Hibbard, 1957)

Homogenization with Taylor-calibrated parameters from (Balasubramanian and Anand 2002)

Initial texture

Final texture




Homogenization of real microstructures

X

Y

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

Equivalent Strain: 0.04 0.08 0.12 0.16 0.2 0.24 0.28

(a)

(c)

(b)

XY

Z

Equivalent Stress (MPa): 19 27 36 45 53 62 70 79(d)

X Y

Z

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

XY

Z

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

XY

Z0

10

20

30

40

50

60

0.000 0.010 0.020 0.030 0.040 0.050 0.060

Equivalent plastic strainE

quiv

alen

t str

ess

(MP

a)

Simple shear

Plane strain compression

(a) (b)

3D microstructure from Monte Carlo Potts simulation

24 x 24 x 24 Pixel based grid

1000 mins on 60 X64 Intel processors with a clock speed of 3.6 GHz using PetSc KSP solvers on the Cornell theory center’s supercomputing facility




Design of microstructure-sensitive properties

Process?

Strain rate?

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


Pro

pert

y

Time

Desired response

Final response

Design Problems:

Microstructure selection: How do we find the best features (e.g. grain sizes, texture) of the material microstructure for a given application?

Process sequence selection: How do we identify the sequences of processes to reach the final product so that properties are optimized?

Process parameter selection: What are the process parameters (e.g. forging rates) required to obtain a desired property response?




PROCESS DESIGN FOR STRESS RESPONSE AT A MATERIAL POINT

dv

• Sensitivity of a homogenized property

z: D

evia

tion

from

de

sire

d p

rope

rty

x: Strain rate of stage 1 y: Strain rate of stage 2

starting point

Given an initial microstructure

Problem 1) Selection of optimal strain rates to

achieve a desired property response during

processing?

Problem 2) A more relevant problem: What should be the straining

rates during processing so that a desired response can be obtained after

processing?

Steepest descent: Need to evaluate gradients of

objective function (deviation from desired property) with respect to

strain rates.




CONTINUUM SENSITIVITY METHOD FOR MICROSTRUCTURE DESIGNCONTINUUM SENSITIVITY METHOD FOR MICROSTRUCTURE DESIGN

1. Discretize infinite dimensional design space into a finite dimensional space

2. Differentiate the continuum governing equations with respect to the design variables

3. Discretize the equations using finite elements

4. Solve and compute the gradients

5. Gradient optimization

Bo

X

Bn+1

B’n+1

Linking and homogenization

Sensitvity linking and perturbed homogenization

Sundararaghavan and Zabaras, IJP 2006.

COMPUTE GRADIENTS

Microstructure homogenization

Perturbed homogenization

• Definition of homogenized velocity gradient

• Decomposition of homogenized velocity gradient into basic 2D modes – Plane Strain Compression, Shear and Rotation

•Design objective – to minimize mean square error from discretized desired property ()




Design variables and objectives

Design variables

• perturbed homogenized deformation gradient

•Sensitivity linking assumption:

The sensitivity of the averaged deformation gradient at a material point is taken to be the same as the sensitivity of the deformation gradient on the boundary of the underlying microstructure, in the reference frame.

• Sensitivity equilibrium equation (Total Lagrangian)




Multi-scale sensitivity analysis

Solve for sensitivity of microstructure deformation

field

Integration of sensitivity constitutive equations

Sensitivity of (macro) properties

Perturbed Mesoscale stress, consistent tangent

Perturbed meso deformation

gradient

perturbed macro deformation gradient




Sensitivity equations for the crystal constitutive problem

• Sensitivity hardening law

• Sensitivity flow rule

• Sensitivity constitutive law for stress

• From this derive sensitivity of PK 1 stress

Integration of sensitivity constitutive equations

Sensitivity of (macro) properties

Perturbed Mesoscale stress, consistent tangent

Perturbed meso deformation

gradient

perturbed macro deformation gradient

Solve for sensitivity of microstructure deformation

field




Initial microstructures for the examples

• Contains 151 and 162 grains, respectively, generated using a standard Voronoi construction

• Meshed with around 4000 quadrilateral elements. Mesh conforms to grain boundaries.

• An initial random ODF is assigned to the microstructures as shown in the pole figures

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4

<111> <110>

<110><111>

0 2 4 6 8 100

10

20

30

40

50

60

1 2 3 4 5 6 70

50

100

150

200

250

300




Problem 1: Design of process modes for a desired response

Desired response

Final microstructure of the design solution

Iterations

Cos

t fu

nct

ion

Time (sec)

Equ

iva

lent

str

ess

(M

Pa

)

(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

)

Change in Neo-Eulerian angle (deg)

9.81

7.05

4.28

1.52

-1.24

-4.00

-6.76

Misorientation map




Design of multi-stage processes

Modeling unloading• Unloading process is modeled as a non-linear (finite deformation) elasto-static

boundary value problem.

• Assumptions during unloading:No evolution of state variable during unloadingUnloading is fast enough to prevent crystal reorientation during

unloadingThe bottom edge of the microstructure is held fixed in the normal

direction during unloading

(a)

(c)

(b)

XY

Z


(a)

(c)

(b)

XY

Z


Stage 1: Plane strain compression

Unloading Stage – 2 shear(a)

(c)

(b)

XY

Z





Design for response in the second stage after unloading

0 0.002 0.004 0.006 0.008

5

10

15

20

25

30

Equivalent strain

Eq

uiv

ale

nt

stre

ss (

MP

a)

Iterations of the design problem

Stage 1: Shear

Stage 2: Compression

What should be the strain rate used in the first stage be for getting desired microstructure-response in the second stage after unloading?




Design for response in the second stage after unloading

a

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 6.43 12.86 19.29 25.71 32.14 38.57 45.00

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


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


1 2 3 4 50

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

Iterations

Cos

t fun

ctio

n

b c

d e f

0 0.5 1 1.5 2

x 10-3

5

10

15

20

25

Equivalent plastic strain

Equi

vale

nt s

tress

(MPa

)

InitialIntermediateFinal

0.1 0.15 0.2 0.25 0.3 0.35 0.4

23.5

24

24.5

25

25.5

26

26.5

Second stage time (sec)

Equi

vale

nt s

tress

(MPa

)

Initial responseIntermediate responseFinal responseDesired response

At the end of stage 1 After unloading During stage 2




Conclusions and Future work

Design extensions

• Address process sequence selection and initial feature selection to obtain a desired response after loading (- Statistical learning problems)

• Model thermal processing stages, designing thermal stages

• Inclusion of grain boundary accommodation and failure effects

Conclusions

• A multi-scale homogenization approach was derived and employed for modeling elasto-viscoplastic behavior and texture evolution in a polycrystal subject to finite strains.

• The model was validated with ODF-Taylor, aggregate-Taylor and experimental results with respect to the equivalent stress–strain curves and texture development.

• A continuum sensitivity analysis of homogenization was developed to identify process parameters that lead to desired property evolution.




INFORMATIONINFORMATION

RELEVANT PUBLICATIONSRELEVANT PUBLICATIONS

S. Ganapathysubramanian and N. Zabaras, "Modeling the thermoelastic-viscoplastic response of polycrystals using a continuum representation over the orientation space", International Journal of Plasticity, Vol. 21/1 pp. 119-144, 2005

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

188 Frank H. T. Rhodes HallCornell University

Ithaca, NY 14853-3801Email: [email protected]

URL: http://mpdc.mae.cornell.edu/

V. Sundararaghavan and N. Zabaras, "Design of microstructure-sensitive properties in elasto-viscoplastic polycrystals using multi-scale homogenization", International Journal of Plasticity, in press

Prof. Nicholas Zabaras

CONTACT INFORMATIONCONTACT INFORMATION

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Materials Process Design and Control Laboratory DESIGN OF MICROSTRUCTURE-SENSITIVE PROPERTIES IN ELASTO-VISCOPLASTIC POLYCRYSTALS USING MULTISCALE HOMOGENIZATION