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A multi-scale design approach for tailoring the
macro-scale properties of polycrystalline
materials
Babak Kouchmeshky Admission to Candidacy Exam
Presentation Date April 22,2008
Materials Process Design and Control Laboratory
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Outline

• Modeling HCP polycrystals deforming by slip and
  twinning
• A design approach for tailoring the processing
  parameters that lead to desired macro-scale
  properties
• A microstructure-sensitive design approach for
  controlling properties of HCP materials
• Future work

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I. Constitutive model
Theory
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DEFORMATION TWINNING

• Twinning produces a rotation of crystal lattice.
• Important deformation mode for HCP materials
• Dominant at room temperature. Twinning is lesser
  at high temperatures.

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SLIP AND TWIN PLANES
Orthogonal system used for modeling
Basal slip system
Pyramidal slip
Prismatic slip
Pyramidal twin
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TWINS MODELED AS PSEUDO-SLIP
Dawson and Myagchilov (1999)
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CONSTITUTIVE MODEL FOR SLIP AND TWINNING
For slip and twin systems
Velocity gradient
Isotropic flow rule to account for grain boundary
accommodation
Hardening law
(Anand,IJP 2003)
no hardening
Condition for slip and twinning
Consistency condition
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INCREMENTAL KINEMATICS
Deformed configuration
Reference configuration
Crystallographic slip, twinning and
re-orientation of crystals are assumed to be the
primary mechanisms of plastic deformation
Fn1
na
Fn
Bn
Bn1
ma
B0

na
Fr

_
ma
_
na
e
Fn
p
Bn
Fn
na
e
Intermediate configuration
ma
Fn1
p
Fn1
_
Fc
Bn1
na
ma
Intermediate configuration
Evolution of various material configurations for
a single crystal as needed in the integration of
the constitutive problem.
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Constitutive theory
Polycrystal plasticity
Initial configuration
Deformed configuration
s0
s
n
n0
s0
n0
Stress free (relaxed) configuration
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CONSTITUTIVE MODEL FOR SLIP AND TWINNING
Solve for shearing rates on slip and twin systems
Solve for the reorientation velocity and the rate
of change in volume fraction of twins
Rate of change of volume fraction
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Arresting of twinning systems
Three stages of strain hardening obtained using
current model.
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Orientation distributions
ORIENTATION DISTRIBUTION FUNCTION A(r,t)

• 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

ODF EVOLUTION EQUATION EULERIAN DESCRIPTION
reorientation velocity
Any macroscale property lt ? gt can be expressed
as an expectation value if the corresponding
single crystal property ? (r ,t) is known.
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Different methodologies in representing the
texture

• 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
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Angle-axis and Rodrigues representation
ANGLE AXIS REPRESENTATION

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

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

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LATTICE REORIENTATION DUE TO TWINNING
Twin plane normal

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

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.

• In quaternion representation, Tq 0,n1,n2,n3
• Convert crystal axis h to the quaternion
  representation hq
• Perform quaternion product Q Tq hq
• Project Q to the fundamental region (QF) based on
  crystal symmetries
• Convert QF to Rodrigues representation

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ORIENTATION DISTRIBUTION FUNCTION (ODF)
Why continuum approach for ODF?
EVOLUTION EQUATION FOR THE ODF
Conservation principle Texture can be
described, quantified compared
J Jacobian determinant of the reorientation of
the crystals r orientation of the crystal. A
is the ODF, a scalar field
Constitutive sub-problem Taylor
hypothesis deformation in each crystal of the
polycrystal is the macroscopic deformation.
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ODF EVOLUTION EQUATION WITH TWINNING
Calculate crystal reorientation
Initial random ODF 1.2158 at all nodal points
HCP Fundamental region
Total Lagrangian formulation
Source term due to twinning
Volume fraction lost
Volume fraction gained from other orientations
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VOLUME FRACTION OF TWINS
Orientation space
r
rk
Volume fraction lost due to transfer of
orientation from r to rk
Dawson and Myagchilov (1999)
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Numerical results
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I. Tension mode

• modeling tension on an initially textured
  Magnesium alloy AZ31B rod

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Initial texture
initial texture used in the simulation
initial texture in the experiment by anand and
staroselsky , 2003
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Material properties
Elastic constants
Slip and twining systems
Slip resistances
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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
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II. Compression mode

• Modeling compression on an initially textured
  Magnesium alloy AZ31B rod

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Material properties
Elastic constants
Slip and twining systems
Slip resistances
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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
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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
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III. Shear mode
In this problem texture evolution and
stress-strain curve is examined for Titanium.
A shear mode is assumed.
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Material properties
Elastic constants
Slip and twining systems
Slip resistances
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Result (cont.)
Experiment
This work
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)
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Result (cont.)
Experiment is done by Wu et al. (2007)
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IV. Plain strain compression mode
In this problem texture evolution is examined for
Titanium .
A plane strain compression mode is assumed.
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Material properties
Elastic constants
Slip and twining systems
Slip resistances
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Result
Results obtained by Myagchilov and Dawson, Simul.
Mater. Sci. Eng. 7 (1999)975-1004
Evolved texture at effective strain of 0.5
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CONCLUSION(part 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
• Twinning is accounted through pseudo shear and
  reorientation of crystals
• Twin saturation is phenomenologically accounted
  for.
• ODF conservation equation modified to include the
  source and sink terms due to twinning.
• The constitutive model is tested for Titanium and
  Magnesium alloy AZ31B.

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Enhancing properties of polycrystals through a
sensitivity problem that spans macro and micro
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.
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Definition 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
The hardness and ductility are presented by Bulk
modulus (B), Shear modulus and B/G.
Convex hull of B,G,B/G
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Problem 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

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Optimizing the processes to get the optimum
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.

A1
A2
L2
T
2T
L2 and L1 are the design parameters
A1
L1
T
0
Process 1 is supposed to start from a random
texture
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The sensitivity problem
The sensitivity problem with respect to design
parameters (L1,L2)
The sensitivity problem with respect to the field
A1
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Sensitivity of the reorientation velocity
Constitutive sensitivity problem
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Definition of the adjoint problem
Lagrange identity for obtaining the adjoint
operator
Gradient of the objective functional
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The texture that provides the maximum hardness
and ductility
The optimum texture from material space
B 124.8 GPa, G58.87 GPa B/G2.12
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The reduced order model
second process (plain strain compression , with
initial texture selected from the convex hull of
all textures available from process 1.)
first process (simple tension, with random
initial texture)
Convex hull of textures for process 1
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3- Find the optimum texture from process plane
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.
Prioritizing the objectives
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3- Find the optimum texture from process space
Optimum texture from Process space
Optimum texture from material 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
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Verify
Verify sensitivity problem
Relative error 0.3
From sensitivity analysis,1st process t5 sec
From Finite difference, 1st process t5 sec
Verify supplementary problem
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Verify
To verify
define
t(sec)
Relative error
0.50
0.014
10
0.2084
0.20733
0.020
0.61
0.1982
0.1969
8
0.58
0.2429
0.029
6
0.2415
0.65
0.036
0.2600
0.2617
5
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Objective function for process 1
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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).

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Forging
Multi-scale polycrystal plasticity

• Goals
• Minimal material wastage due to flash
• Filling up the die cavity

Material Ti

• Why multi scale?
• The evolution of the material properties at the
  macro scale has a strong correlation with the
  underlying microstructure.

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Multi-length scale design environment
Selection of the design variables like preform
parameterization
Coupled micro-macro direct model
Coupled micro-macro sensitivity model
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Implementation of the direct problem
Macro
formulation for macro scale Update macro
displacements
Macro-deformation gradient
Meso
Texture evolution update Polycrystal averaging
for macro-quantities
Macro-deformation gradient
microscale stress
Micro
Integration of single crystal slip and twinning
laws
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THE DIRECT CONTACT PROBLEM
Impenetrability Constraints
Augmented Lagrangian approach to enforce
impenetrability
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Continuous representation of texture
Polycrystal average of orientation dependent
property
REORIENTATION TEXTURING
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A sample direct problem
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SCHEMATIC OF THE CONTINUUM SENSITIVITY METHOD
Equilibrium equation
Material constitutive laws
Sensitivity weak form
Design derivative of equilibrium equation
Design derivative of the material constitutive
laws
Incremental sensitivity constitutive sub-problem
Time and space discretized weak form
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
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Representing the preform shape
Curved surface parametrization Cross section
can at most be an ellipse Model semi-major and
semi-minor axes as 6 degree bezier curves
Design vector
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DEFINITION OF PARAMETER SENSITIVITY
Fr

X X (Y ?s )
xn
x
B
Fn
X
Bo
FR
BR
ILo
Y
ILn
XX
o
xx
xnxn
o
o
state variable sensitivity contour w.r.t.
parameter change
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SENSITIVITY KINEMATIC PROBLEM
Continuum problem
Differentiate
Discretize
Design sensitivity of equilibrium equation
Calculate such that
Variational form -
Constitutive problem
Kinematic problem
Regularized contact problem
Sensitivity of ODF evolution
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Method of snapshots
Suppose we had a collection of data (from
experiments or simulations) for the ODF
Solve the optimization problem
Is it possible to identify a basis
where
such that it is optimal for the data represented
as
Eigenvalue problem
where
POD technique Proper Orthogonal Decomposition
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Reduced model for the evolution of the ODF
The reduced basis for the ODF ensemble has been
evaluated, say
where
Using this basis, ODF represented as follows
Initial conditions
This representation of the ODF leads to a
reduced-model in the form of an ODE.
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EVOLUTION EQUATION FOR THE SENSITIVITY OF THE ODF
Sensitivity of reorientation velocity
Gradient of the sensitivity of the velocity
Assumption extended Taylor hypothesis for the
continuum sensitivity analysis i.e. we make no
distinction between the sensitivity of the
crystal velocity gradient and the sensitivity of
the macroscopic velocity gradient.
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Reduced model for the evolution of the ODF
The reduced basis for the ODF ensemble has been
evaluated, say
where
Using this basis, ODF represented as follows
Initial conditions
This representation of the ODF leads to a
reduced-model in the form of an ODE.
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Validation of the reduced-order model
Sensitivity analysis
The same set of regions are used for the
sensitivity problem
Full model (Continuum sensitivity method)
FDM solution
Reduced model
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Design
Objective function
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Design
Last iteration of optimization
First iteration of optimization
Preform shape for the last iteration
Preform shape for the first iteration
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Distribution of texture for some points on macro
scale
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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

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Future direction
Macro problem driven by the macro-design variable
ß
Multi-scale Computation
Micro problem driven by the velocity gradient L
Bn1
B0
Fn1
L L (X, t ß)
Polycrystal plasticity
L velocity gradient
Data mining techniques
Reduced Order Modes
Database
Design variables (ß) are macro design variables
Processing sequence/parameters
Design objectives are micro-scale averaged
material/process properties
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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.
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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.

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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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