On the control of microstructural degrees of freedom in deformation processes

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On the control of microstructural degrees of
freedom in deformation processes
V. Sundararaghavan and Nicholas Zabaras
Materials Process Design and Control
Laboratory Sibley School of Mechanical and
Aerospace Engineering188 Frank H. T. Rhodes
Hall Cornell University Ithaca, NY
14853-3801 Email zabaras_at_cornell.edu URL
http//mpdc.mae.cornell.edu/
Materials Process Design and Control Laboratory
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MULTISCALE MODELING
Design properties
Control process parameters
Improve performance
Materials Process Design and Control Laboratory
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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
COMPUTATIONAL PROCESS DESIGN Design the forming
and thermal process sequence Selection of stages
(broad classification) Selection of dies and
preforms in each stage Selection of mechanical
and thermal process parameters in each
stage Selection of the initial material state
(microstructure)
OBJECTIVES
VARIABLES
CONSTRAINTS
Material usage
Identification of stages
Press force
Plastic work
Number of stages
Press speed
Uniform deformation
Preform shape
Processing temperature
Die shape
Microstructure
Geometry restrictions
Mechanical parameters
Desired shape
Product quality
Thermal parameters
Cost
Residual stresses
Materials Process Design and Control Laboratory
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ANALYSIS TO DESIGN
Process design
Conventional design objective MAXIMIZE
VOLUMETRIC YIELD
Accurate process modeling Methodology for
design Control problems as optimization
problems Accurate computation of gradients
Thermal effects
Numerical methods
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DEFORMATION PROCESS DESIGN SIMULATOR
Research Objectives To develop a mathematically
and computationally rigorous gradient-based
optimization methodology for virtual materials
process design that is based on quantified
product quality and accounts for process targets
and constraints.

• Current capabilities
• Development of a general purpose continuum
  sensitivity method for the design of multi-stage
  industrial deformation processes
• Deformation process design for porous materials
• Design of 3D realistic preforms and dies
• Extension to polycrystal plasticity based
  constitutive models with evolution of
  crystallographic
• texture

Initial guess Optimal preform




Materials Process Design and Control Laboratory
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DEFORMATION PROCESS DESIGN
DETERMINISTIC PROCESS DESIGN BASED ON CONTINUUM
SENSITIVITY METHOD (CSM)
Preform design for a steering link
Reference problem large flash
First iteration underfill
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DEFORMATION PROCESS DESIGN - CONTD
Preform design for a steering link
Final iteration flash minimized and complete
fill
Objective function
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Process Optimization using phenomenological
material models
Preform optimization of a Ti-64 automotive cross
shaft
Flash
Initial Iteration
Underfill
Intermediate Iteration
Maximize volumetric yield
Final
Final Iteration
No Flash
Materials Process Design and Control Laboratory
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INCORPORATING THE MESO-SCALE

• Process Material modeling
• Phenomenological
• approach
• Polycrystal
• plasticity
• Design methodology
• Parameter selection
• Gradient based
• Continuum sensitivity
• analysis

Numerical methods

Microstructure models

Friction, contact effects
Thermal effects
Plasticity models
Design
Damage modeling
Phenomenological approach
Polycrystal analysis
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PHYSICAL APPROACH TO PLASTICITY

• 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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REPRESENTING POLYCRYSTALS
Crystal/lattice reference frame

• CRYSTAL SYMMETRIES?

Same axis of rotation gt planes Each symmetry
reduces the space by a pair of planes
Sample reference frame
crystal
RODRIGUES REPRESENTATION FCC FUNDAMENTAL REGION
n

• ORIENTATION SPACE
• Euler angles symmetries
• Neo Eulerian representation

Particular crystal orientation
Rodrigues parametrization
Cubic crystal
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ORIENTATION DISTRIBUTION FUNCTION (ODF)
Why continuum approach for ODF?
EVOLUTION EQUATION FOR THE ODF (Eulerian)
Conservation principle Texture can be described,
quantified compared Based on the Taylor
hypothesis Eulerian Lagrangian forms
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 in each
• crystal of the
• polycrystal
• is the macroscopic
• deformation.
• Compute the
• reorientation velocity
• from the spin

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PROCESS MODELING DESIGN A POLYCRYSTAL PLASTICIT
Y BASED APPROACH
Micro problem driven by the velocity gradient L
Macro problem driven by the macro-design variable
ß
Bn1
L L (X, t ß)
Polycrystal plasticity
L velocity gradient

• Kumar Dawson, CMAME 1997
• Zabaras,
• IJP 2004, Acta. Mat 2003

• Design variables (ß) are macro
• design variables
• Die shapes
• Preform shapes
• Processing conditions
• Etc.

Design objectives are micro-scale averaged
material/process properties
Materials Process Design and Control Laboratory
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REORIENTATION TEXTURING
Polycrystal average of orientation dependent
property
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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.
Rate-independent model (Anand and Kothari, 1996)
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Implementation of the direct problem
Largedef formulation for macro scale Update macro
displacements
Macro
Macro-deformation gradient
Homogenized (macro) stress, Consistent tangent
ODF evolution update Polycrystal averaging for
macro-quantities
Meso
microscale stress, consistent tangent
Macro-deformation gradient
Integration of single crystal slip
laws Consistent tangent formulation (meso)
Parallel solver PetSc (Argonne Labs) KSPSolve
Materials Process Design and Control Laboratory
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CONTINUUM SENSITIVITY METHOD - BROAD OUTLINE

• Discretize infinite dimensional design space into
  a finite dimensional space
• Differentiate the continuum governing equations
  with respect to the design variables
• Discretize the equations using finite elements
• Solve and compute the gradients
• Gradient optimization

CSM -gt Fast Multi-scale optimization Requires 1
Non-linear and n Linear multi-scale problems
Materials Process Design and Control Laboratory
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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
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SENSITIVITY KINEMATIC PROBLEM
Design sensitivity of equilibrium equation
Calculate such that
Variational form -
o
o
o
Pr and F,
?
Constitutive problem
Regularized contact problem
Sensitivity of ODF evolution
Materials Process Design and Control Laboratory
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COMPUTING SENSITIVITY OF PROPERTIES
time t 0
time t
Reorientation map
s
r
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
Materials Process Design and Control Laboratory
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Verification of polycrystal plasticity model
X-axis Compression Taylor model FCC Copper
Fundamental region at time t 135 sec
Strain rate 1e-3 /s, at time t 135 sec
Comparison with results from Anand and Kothari
(1996)
Materials Process Design and Control Laboratory
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Multi-scale forging ODF property computation
Computation of yield strength Material point
sub-problem -ODF at each integration point is
subject to a tension test -Deformation under
uniaxial tension at a strain rate of 0.02/sec,
for a time of 0.1 sec -Equivalent stress measured
corresponds to 0.2 yield strength
B
ODF at point B
Yield strength contours
Materials Process Design and Control Laboratory
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DESIGN FOR MICROSTRUCTURE SENSITIVE PROPERTIES
Yield stress
Design Problem
ltMgt polycrystal Taylor factor (computed using
Bishop-Hill analysis)
Normalized objective function
Normalized yield stress
Desired value a 0.02,0.86,0,0,0T , Initial
guess a 0,0.2,0,0,0T Converged solution a
0.019,0.86,-0.018,0.006,-0.012T, t 0.1
s Reduced order solution using basis IV
contains data from plane strain compression,
uniaxial tension and mixed deformation tests to
ensure accurate representation of mixed
deformation components in the desired and
intermediate solutions.
Materials Process Design and Control Laboratory
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DESIGN FOR MICROSTRUCTURE SENSITIVE PROPERTIES -
R value - planar variation
(from Hills anisotropic yield criterion)
Design Problem ? F,G,H,NT
R value
Normalized objective function
Desired value a 1.2,0,0,0,0T, Initial
guess a 0.5,0,0,0,0T Converged solution a
1.19,0.05,0.001,0,0T, Reduced order solution
(basis II) at t 0.1 s
Materials Process Design and Control Laboratory
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DESIGN FOR SPECIFIC MATERIAL RESPONSE
Design for the strain rate such that a desired
material response is achieved
Material 99.98 pure f.c.c Al
Materials Process Design and Control Laboratory
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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
Multi-objective optimization

• Increase Volumetric yield
• Decrease property variation

Materials Process Design and Control Laboratory
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Multi-scale design OFHC Copper closed die
forging Iteration 1
Large Underfill
Yield strength (MPa)
variation in yield strength
Materials Process Design and Control Laboratory
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Multi-scale design OFHC Copper closed die
forging Iteration 2
Smaller underfill
Yield strength (MPa)
variation in yield strength
Materials Process Design and Control Laboratory
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Multi-scale design OFHC Copper closed die
forging Iteration 7
Underfill
Optimal fill
Yield strength (MPa)
Optimal yield strength
Materials Process Design and Control Laboratory
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Comparison of final products at different
iterations
Cost function f(underfill,variance of yield
strength)
Iteration number
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Extrusion design problem
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
Young's Modulus Distribution Material point
sub-problem -
In sample coordinates
In crystal coordinates
Minimize Youngs Modulus variation across
cross-section
ODF
Die design for improved properties
Polycrystal average in sample coodinates
Youngs modulus (along sample x-axis)
Materials Process Design and Control Laboratory
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Multiscale Extrusion Control of Youngs Modulus
Iteration 1
First iteration Objective function Minimize
variation in Youngs Modulus
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Multiscale Extrusion Control of Youngs Modulus
Iteration 2
Intermediate iteration
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Multiscale Extrusion Control of Youngs Modulus
Iteration 5
Optimal solution
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Multiscale Extrusion Variation in Objective
function
Small die shape changes leads to better
properties
Die Shape
Objective function variance (Youngs Modulus)
Youngs Modulus (GPa)
Iteration number
Materials Process Design and Control Laboratory
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ONGOING WORK AT MPDC LAB
Higher order homogenization techniques to account
for stereology and control of polycrystal
properties (right) (Sundararaghavan and Zabaras
IJP, in press)
Develop multi-scale tools for materials design
Stochastic variational multiscale techniques to
account for uncertainties in properties
(left) (Badri VA and Zabaras JCP, in press)
Materials Process Design and Control Laboratory
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ONGOING WORK AT MPDC LAB
Stochastic design
Computation of microstructure induced
uncertainties (MaxEnt approach)
Property statistics
Property variability due to uncertain initial
void fraction
Homogenization
Statistical variability in material property
(uniaxial stress-strain curve)
Reliability based design
Microstructure samples (Voronoi models) from the
PDF
Actual limit state surface
(Present approach) Full order reliability method
Unsafe state Z(g)lt0
Grain size lower order statistics (average grain
sizes, shape data) from manufacturer
Design point
SORM Approximation
Safe state Z(g)gt0
MaxEnt and Gibbs sampler
Materials Process Design and Control Laboratory
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LOOKING TO THE FUTURE

• STATISTICAL LEARNING TECHNIQUES TO CHOOSE INITIAL
  DESIGN
• EFFECTS OF STEREOLOGY and PHASE DISTRIBUTION
• REDUCED REPRESENTATIONS FOR ACCELERATED DESIGN.
• MODELING MICROSTRUCTURE INDUCED UNCERTAINTIES

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INFORMATION
RELEVANT 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
CONTACT INFORMATION
Prof. Nicholas Zabaras
Materials Process Design and Control
Laboratory Sibley School of Mechanical and
Aerospace Engineering188 Frank H. T. Rhodes
Hall Cornell University Ithaca, NY
14853-3801Email zabaras_at_cornell.edu URL
http//mpdc.mae.cornell.edu/
Materials Process Design and Control Laboratory