Materials Process Design and Control Laboratory

1
Multi-scale Computational Techniques for Design
of Polycrystalline Materials
Veera Sundararaghavan
Thesis Defense (B-Exam) Sibley School of
Mechanical and Aerospace Engineering Cornell
University
Technical Presentation Date May 18, 2007
Materials Process Design and Control Laboratory
2
ACKNOWLEDGEMENTS

• PhD COMMITTEE
• Prof. Nicholas Zabaras, M. A.E., Cornell
  University
• Prof. Subrata Mukherjee, M. A.E., Cornell
  University
• Prof. Shefford Baker, M.S. E., Cornell
  University
• Prof. Thorsten Joachims, COM S., Cornell
  University

• FUNDING SOURCES
• Air Force Office of Scientific Research
• Army Research Office
• NSF
• SPECIAL THANKS
• Materials Process Design and Control Laboratory
• Sibley School of Mechanical Aerospace
  Engineering
• Cornell Theory Center

3
PRESENTATION OUTLINE

• Motivation of microstructure sensitive design
• Texture-process-property maps
• Adaptive reduced order optimization for obtaining
  tailored material properties
• Microstructure homogenization
• Design of polycrystalline microstructures using
  sensitivity analysis
• Multi-scale design of industrial forming
  processes
• Future extensions

Materials Process Design and Control Laboratory
4
DEFORMATION PROCESS DESIGN SIMULATOR

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

Enhanced strength
Materials Process Design and Control Laboratory
5
RESEARCH 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.

Materials Process Design and Control Laboratory
6
MULTISCALE NATURE OF METALLIC STRUCTURES
Meso
Grain/crystal
Twins
Inter-grain slip
Grain boundary
Material-by-design
Micro
Nano
Atoms
precipitates
Titanium armors with high specific strength.
Materials Process Design and Control Laboratory
7
POLYCRYSTALLINE MICROSTRUCTURES

• Crystallographic orientation
• Rotation relating sample
• and crystal axis
• Properties governed by orientation during
  deformation

8
Design and control of properties in
polycrystalline materials using
texture/property/process maps
Materials Process Design and Control Laboratory
9
MOTIVATION
GRAPHICALLY REPRESENT THE SPACE OF
MICROSTRUCTURES, PROPERTIES AND PROCESSES
Process
Applications (i) Identify microstructures that
have extremal properties. (ii) Identify
processing sequences that lead to desired
microstructures and properties.
Microstructure representations
Property-process space
Process-structure space
A1000
Property-structure space
A80
a
Process paths
A100
Materials Process Design and Control Laboratory
10
FIRST ORDER STRUCTURE SPACE
Example E lt E1x1 E2x2 E3x3 (upper bound
theory) E1,2,3 are the Youngs modulus of each
phase x1,2,3 are the volume fractions of each
phase Find microstructures with Youngs Modulus
lt E
Microstructures with Youngs Modulus lt E
x1 x2 x3 1 Microstructure plane for 3
phase material
Materials Process Design and Control Laboratory
11
FIRST ORDER REPRESENTATION OF MICROSTRUCTURES
Crystal/lattice reference frame

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

Sample reference frame
crystal
RODRIGUES REPRESENTATION FCC FUNDAMENTAL REGION
n
Any property can be expressed as an expectation
value or average given by
Particular crystal orientation
Cubic crystal
Kumar and Dawson 1999, Ganapathysubramaniam and
Zabaras, IJP 2005.
12
FINITE ELEMENT INTEGRATION IN RF SPACE
Normalization

1
ri, Ai
Integration point 0.25,0.25,0.25
Materials Process Design and Control Laboratory
13
MATERIAL PLANE
Applications (i) Identify microstructures that
have extremal properties. (ii) Identify closures
of properties.
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 gt 0
• Crystallographic Symmetry, r Gr

SPACE OF ALL POSSIBLE ODFs
Aa
Materials Process Design and Control Laboratory
14
UPPER BOUND THEORY LINEARIZATION
Upper bound of a polycrystal property can be
expressed as an expectation value or average
given by
Materials Process Design and Control Laboratory
15
EXTREMAL PROPERTY POINTS

• Constraints and objectives are linear in the ODF
  problem
• Identify microstructures that maximize
  properties in a particular direction (eg.
  Clt1111gt)

LINEAR PROGRAMMING
For minima
For maxima
Extremize property
Normalization
positiveness
Number of variables 448 Number of linear
inequality constraints 448 Number of linear
equality constraints 1
and lb 0
Materials Process Design and Control Laboratory
16
EXTREMAL TEXTURES
ODF for maximum Taylor factor along TD (3.668)
ODF for maximum Taylor factor along RD (3.668)
ODF for maximum C44 (74.923 GPa)
ODF for maximum C55 (74.923 GPa)

• Taylor factor calculated through Bishop-Hill
  analysis

Materials Process Design and Control Laboratory
17
UPPER BOUND PROPERTY CLOSURES
Closure for stiffness constants (C11,C22,C66)
Closure for Taylor factor computed along RD, 45o
to RD and TD
Materials Process Design and Control Laboratory
18
TEXTURE EVOLUTION
Represent the ODF as
Viscoplastic rate dependent model, no hardening
(Acharjee and Zabaras, 2003) Taylor hypothesis
Reduced model for the evolution of the ODF
X-axis compression
Initial conditions
Materials Process Design and Control Laboratory
19
MODEL REDUCTION POD
Method of snapshots
Suppose we had an ensemble of data (from
experiments or simulations) for the ODF
Eigenvalue problem
where
Is it possible to identify a basis
such that it can represent the ODF as

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

POD technique Proper Orthogonal Decomposition
Materials Process Design and Control Laboratory
20
PROCESS PLANE
f3
a1f1a2f2a3f3
a3
a1
a2
f1
f2

• Normalization
• Lower bound
• Crystallographic Symmetry

Equation of a plane

• Basis already includes symmetries.

Materials Process Design and Control Laboratory
21
TEXTURE PLANES FOR SOME PROCESSES
Plane strain compression
Tension/compression
Y-Z shear
X-Y shear
Materials Process Design and Control Laboratory
22
PROCESS PATH REPRESENTATION
0.17 initial strain
initial textures
Compression path
Tension path
Final textures
90 accurate reconstruction width
0.07 initial strain
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)
Materials Process Design and Control Laboratory
23
PROPERTY CLOSURE OF A PROCESS PLANE
Process plane
Process-property plane
C
Taylor factor along TD
R
Taylor factor along RD
Materials Process Design and Control Laboratory
24
PROCESS SELECTION

• 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.

Materials Process Design and Control Laboratory
25
Classification And Adaptive Reduced Order
Optimization For Deformation Process Sequence
Selection
Materials Process Design and Control Laboratory
26
ODF CLASSIFICATION
2. Compute input ODF features
Input Texture
1. Input desired ODF
fiber path integral
3. Feature based classification
A machine learning approach
4. Identify processing paths from the ODFs in
this class
z-axis lt110gt fiber (BB)
Sundararaghavan and Zabaras (Cornell University),
Acta Materialia 2005
Materials Process Design and Control Laboratory
27
K-MEANS CLUSTERING
Find the cluster centers C1,C2,,Ck such that
the sum of the 2-norm distance squared between
each feature xi , i 1,..,n and its nearest
cluster center Ch is minimized.
Each class is affiliated with multiple processes
Cost function
Feature Space
DATABASE OF ODFs
Clusters
Identify clusters
Materials Process Design and Control Laboratory
28
IDENTIFICATION OF PROCESSING PATHS

• Automatic class-discovery without class labels.
• Hierarchical Classification model
• Association of classes with processes, to
  facilitate data-mining
• Can be used to identify multiple process routes
  for obtaining a desired ODF

ODF 2,12,32,97
One ODF, several process paths
Data-mining for Process information with ODF
Classification
Materials Process Design and Control Laboratory
29
DATABASE STRUCTURE
DATABASE
Process sequence-2 New process parameters ODF
history Reduced basis
Process sequence-1 Process parameters ODF
history Reduced basis
New dataset added
Desired texture/property
Classifier
Adaptive basis selection
Process
Reduced basis
Optimization
Probable Process sequences Initial parameters
Stage - 1
Stage - 2
Optimum parameters
Materials Process Design and Control Laboratory
30
DESIGN FOR DESIRED ODF A MULTI STAGE PROBLEM
Desired ODF
Optimal- Reduced order control
20x faster than full optimization. Gradients are
obtained from reduced order sensitivity analysis.
Stage 1 Plane strain compression (a1 0.9472)
Stage 2 Compression (a2 -0.2847)
Initial guess, a1 0.65, a2 -0.1
Materials Process Design and Control Laboratory
31
DESIGN FOR DESIRED MAGNETIC PROPERTY
Crystal lt100gt direction. Easy direction of
magnetization zero power loss
h
External magnetization direction
Stage 1 Shear 1 (a1 0.9745)
Stage 2 Tension (a2 0.4821)
Materials Process Design and Control Laboratory
32
MICROSTRUCTURE DESIGN USING MULTI-SCALE
HOMOGENIZATION
Materials Process Design and Control Laboratory
33
MOTIVATION
g orientation of crystal

• Microstructure response depends on
• Crystal orientations,
• Higher order correlations of orientations,
• Grain size distribution

f(g)
One point statistic Texture
two point statistics
f(g,gr)
Final response
Homogenization
Process? Strain rate?
Property
Desired response
Time
Materials Process Design and Control Laboratory
34
HOMOGENIZATION OF DEFORMATION GRADIENT
Macro-deformation is an average over the
microscopic deformations (Hill, Proc. Roy. Soc.
London A, 1972)
Macro
Micro
Decompose deformation gradient in the
microstructure as a sum of macro deformation
gradient and a micro-fluctuation field
implies
Mapping
Use BC
0 on the boundary
Note 0 on the microstructure volume is the
Taylor assumption
Materials Process Design and Control Laboratory
35
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)
Macro
Micro
Apply boundary condition
Homogenized stresses
Must be valid for arbitrary variations of dF
Sundararaghavan and Zabaras, International
Journal of Plasticity 2006.
Materials Process Design and Control Laboratory
36
MACRO-SCALE TANGENT MODULI
Rearrange the global stiffness matrix of the
converged microstructure problem to relate
increment in forces on boundary nodes to
increment in boundary node displacements.
Macro
Micro
Linearization of homogenized PK-I stress
Apply boundary condition
Tangent moduli at macro-scale
Note For Taylor assumption, the macro-tangent
modulus is just the volume average of micro-scale
tangent moduli
Materials Process Design and Control Laboratory
37
MICROSTRUCTURE DEFORMATION
Thermal effects linking assumption
Bn1
Fn1
Equate macro and micro temperatures
X
B0
F F (X, t)
F deformation gradient

• Hexahedral meshing using CuBIT.

Materials Process Design and Control Laboratory
38
FCC SINGLE CRYSTAL RESPONSE TO IMPOSED DEFORMATION

• Stress
• Evolution of slip system resistances
• Shearing rate

Athermal resistance (e.g. strong precipitates)
Thermal resistance (e.g. Peierls stress, forest
dislocations)
If resolved shear stress lt athermal resistance
, otherwise
Balasubramaniam and Anand, Int J Plasticity, 1998
Materials Process Design and Control Laboratory
39
SINGLE CRYSTAL CONSTITUTIVE ANALYSIS
Reference configuration
Crystallographic slip and re-orientation of
crystals are assumed to be the primary mechanisms
of plastic deformation
Fn1
na
Fn
Bn
Bn1
ma
_
B0

na
ma
Fr
_

ma
na
_
e
Fn
e
p
Bn
Fn
Ftrial
Deformed configuration
Evolution of plastic deformation gradient
na
e
Intermediate configuration
ma
Fn1
p
Fn1
_
Fc
Bn1
na
The elastic deformation gradient is given by
ma
Intermediate configuration
Evolution of various material configurations for
a single crystal as needed in the integration of
the constitutive problem.
(Anand and Balasubramaniam, 1998)
40
MICROSTRUCTURE RESPONSE VALIDATION
Experimental results from Anand and Kothari (1996)
Materials Process Design and Control Laboratory
41
IMPLEMENTATION OF MULTI-SCALE HOMOGENIZATION
APPROACH
Forming process Update macro displacements
Macro-deformation gradient
Homogenized (macro) stress, ct
Boundary value problem for microstructure Solve
for deformation field
Micro-scale stress, ct
Micro-scale deformation gradient
Integration of constitutive equations Dislocation
plasticity
Materials Process Design and Control Laboratory
42
OPTIMIZATION OF MICROSTRUCTURE RESPONSE
Select optimal process parameters to achieve a
desired property response.
Initial microstructure
Need to evaluate gradients of objective function
(deviation from desired property) with respect to
process variables.
Die shape
Forging rate
Initial preform shape
Sundararaghavan and Zabaras, IJP 2006.
Materials Process Design and Control Laboratory
43
OPTIMIZATION FRAMEWORK
Gradient methods
Continuum equations

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

Design differentiate
Discretize

• Direct differentiation technique
• (Chenot et al., Grandhi et al.)
• Discretization sensitive
• Sensitivity of boundary condition
• Coupling of different phenomena

CSM -gt Fast Multi-scale optimization Requires 1
Non-linear and n Linear multi-scale problems to
compute gradients

• Continuum sensitivity method
• (Zabaras et al.)
• Design differentiate continuum
• equations
• Complex physical system
• Linear systems

44
DESIGN DIFFERENTIATION

• Directional derivative

(Badrinarayan and Zabaras, 1996)
45
MICROSTRUCTURE SENSITIVITY ANALYSIS
Calculate such that
Sensitivity of equilibrium equation
Microstructure
o
o
o
Pr and F,
?
Constitutive problem
Kinematic problem
Sensitivity of single crystal response

• 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.

Materials Process Design and Control Laboratory
46
MATERIAL POINT SENSITIVITY ANALYSIS
Sensitivity of (macro) properties
Perturbed macro deformation gradient
Solve for sensitivity of microstructure
deformation field
Perturbed Mesoscale stress
Perturbed meso deformation gradient
Integration of sensitivity constitutive equations
Materials Process Design and Control Laboratory
47
SENSITIVITY OF THE CRYSTAL CONSTITUTIVE PROBLEM
Sensitivity of (macro) properties
Perturbed macro deformation gradient
Solve for sensitivity of microstructure
deformation field

• Sensitivity hardening law

Perturbed Mesoscale stress
Perturbed meso deformation gradient

• Sensitivity constitutive law for stress

Integration of sensitivity constitutive equations

• Derive sensitivity of PK-I stress

Materials Process Design and Control Laboratory
48
DESIGN VARIABLES FOR A MATERIAL POINT PROBLEM

• 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 (W)

Design variables
Materials Process Design and Control Laboratory
49
PROCESS DESIGN FOR STRESS RESPONSE AT A MATERIAL
POINT
c
b
Sundararaghavan and Zabaras, IJP 2006.
Materials Process Design and Control Laboratory
50
Microstructure-sensitive design of industrial
forming processes
Materials Process Design and Control Laboratory
51
COMPUTATIONAL DESIGN OF DEFORMATION PROCESSES
BROAD DESIGN OBJECTIVES Given raw material,
obtain final product with desired microstructure
and shape with minimal material utilization and
costs
OBJECTIVES
VARIABLES
CONSTRAINTS
Press force
Material usage
Preform shape
Press speed
Properties
Die shape
Product shape
Microstructure
Forging rate
Cost
Materials Process Design and Control Laboratory
52
MULTISCALE MODEL OF DEFORMATION USING ODF
REPRESENTATION
Largedef formulation for macro scale Update macro
displacements
Macro
Macro-deformation gradient
Homogenized (macro) stress
ODF evolution update Polycrystal averaging for
macro-quantities
Meso
Macro-deformation gradient
Microscale stress
Integration of single crystal model Dislocation
plasticity
Micro
Parallel solver PetSc (Argonne Labs) KSP-Solve
Materials Process Design and Control Laboratory
53
MULTI-LENGTH SCALE SENSITIVITY ANALYSIS
The velocity gradient depends on a macro design
parameter
A micro-field depends on a macro design
parameter (and) the velocity gradient as
Sensitivity of the velocity gradient driven by
perturbation to the macro design parameter
Sensitivity of this micro-field driven by the
velocity gradient
54
COMPUTING SENSITIVITY OF PROPERTIES USING ODF
REPRESENTATION
time t
time t 0
Reorientation map
r
s
ODF at time t
Initial ODF
Polycrystal conservation (Lagrangian)
Macro-Property
Sensitivity of ODF to perturbations in
macro-variables
Sensitivity of macro-properties perturbations in
macro-variables
Sundararaghavan and Zabaras, Int J Plasticity, in
preparation.
Materials Process Design and Control Laboratory
55
MACRO-MICRO SENSITIVITY ANALYSIS
Design sensitivity of equilibrium equation
Calculate such that
Variational form -
o
o
o
Pr and F,
?
Constitutive problem
Kinematic problem
Regularized contact problem
Material point sensitivity analysis
Materials Process Design and Control Laboratory
56
EXTRUSION DESIGN PROBLEM
Microstructure evolution is modeled using an
orientation distribution function
Objective Design the extrusion die for a fixed
reduction such that the deviation in the Youngs
Modulus at the exit cross section is
minimized Material FCC Cu
Minimize Youngs Modulus variation across
cross-section
Die design for improved properties
Materials Process Design and Control Laboratory
57
DESIGN PARAMETERIZATION OF THE PROCESS VARIABLE
r(a)
Objective Minimize Youngs Modulus variation in
the final product by controlling die shape
variations
Identify optimal Ci that results in a desired
microstructure-sensitive property
Materials Process Design and Control Laboratory
58
CONTROL OF YOUNGS MODULUS ITERATION 1
First iteration Objective function Minimize
variation in Youngs Modulus
Materials Process Design and Control Laboratory
59
CONTROL OF YOUNGS MODULUS ITERATION 2
Intermediate iteration
Materials Process Design and Control Laboratory
60
CONTROL OF YOUNGS MODULUS ITERATION 4
Optimal solution
Materials Process Design and Control Laboratory
61
MULTISCALE EXTRUSION VARIATION IN OBJECTIVE
FUNCTION
Small die shape changes leads to better
properties
Objective function variance (Youngs Modulus)
Materials Process Design and Control Laboratory
62
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
Multi-objective optimization

• Increase Volumetric yield
• Decrease property variation

Fill cavity
Materials Process Design and Control Laboratory
63
UPDATED LAGRANGIAN SHAPE SENSITIVITY FORMULATION
Sensitivity to initial preform shape
Materials Process Design and Control Laboratory
64
MULTI-SCALE DESIGN FOR OPTIMUM STRENGTH
ITERATION 1
Large Underfill
Yield strength (MPa)
variation in yield strength
Materials Process Design and Control Laboratory
65
MULTI-SCALE DESIGN FOR OPTIMUM STRENGTH
ITERATION 2
Smaller under-fill
Yield strength (MPa)
variation in yield strength
Materials Process Design and Control Laboratory
66
MULTI-SCALE DESIGN FOR OPTIMUM STRENGTH
ITERATION 7
Underfill
Optimal fill
Yield strength (MPa)
Optimal yield strength
Materials Process Design and Control Laboratory
67
COMPARISON OF FINAL PRODUCTS AT DIFFERENT
ITERATIONS
Cost function f(underfill,variance of yield
strength)
Iteration number
Materials Process Design and Control Laboratory
68
CONCLUSIONS
Microstructure evolution

• 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.
• Ability to mathematically represent the space of
  microstructures using lower order features for
  optimization of properties.

Multi-scale optimization
Materials Process Design and Control Laboratory
69
FUTURE DIRECTIONS
Materials Process Design and Control Laboratory
70

• Simulations such as forming of a cross-link
  (right) requires remeshing and data-transfer
  operations.
• Remeshing has been incorporated into the
  multi-scale code using NetGEN library. This would
  allow multi-scale simulation of complex forging
  processes

1/8 symmetry used for modelling
71
DESIGN OF TITANIUM ALLOY MICROSTRUCTURES
Ti turbine blades
Account for additional physical effects,
twinning, slip-twin interaction, twin saturation
Multiplicative decomposition accounting for
twinning
Materials Process Design and Control Laboratory
72
HIGHER ORDER STRUCTURE-PROCESS-PROPERTY MAPS
Higher order features High dimensionality How
to identify property iso-contours in
microstructural space?
Non-linear property surfaces Multiple solutions?
Lankford R parameter 1.0253 surface (along RD)
on the tension basis
Active contouring techniques
Materials Process Design and Control Laboratory
73
MODELING GRAIN BOUNDARY SLIDING AND SEPARATION
Grain boundary modeling Molecular dynamics
simulation, cohesive zone models using Finite
elements
MD simulation of Cu bicrystal (45º
misorientation) under tension, EAM potential
(above) Cohesive Model Veera and Zabaras (2007)
74

• Current models explain but do not predict
  recrystallization textures, making the process of
  DESIGNING properties extremely challenging.
• Need to model the effect on grain size on
  homogenized properties. The model must take into
  account the strengthening attributed to grain
  size refinement.

• KEY QUESTIONS FOR MODELLING
• How many nuclei, what nucleation rates, which
  sites, what orientations?
• Growth rate of grains?
• Which models to use? (MC, phase field)
• Changes to the constitutive model to include
  grain size effect (eg. Acharya and Beaudoin, 2000)

Thermal processing
Stored energies
Deformation processing
ODF
75
MODELING MATERIAL UNCERTAINTIES
Microstructure samples (Voronoi models) from the
PDF
Property variability
Micro-scale variability
SOLUTIONS Robust design to avoid catastrophic
failures Spectral, collocation approaches for
modelling microstructural variabilities such as
uncertainty in ODF