A CONTINUUM SENSITIVITY ANALYSIS FOR THE DESIGN OF HOT FORMING PROCESSES

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A CONTINUUM SENSITIVITY ANALYSIS FOR THE DESIGN
OF HOT FORMING PROCESSES
Professor Nicholas Zabaras Shankar
Ganapathysubramanian Dr. Qing Li Dr. Zhong Hu
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//www.mae.cornell.edu/zabaras/
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FEDERAL INDUSTRIAL SPONSORS/COLLABORATORS
Materials Process Design Control Laboratory
Industrial Sponsors ALCOA, ATC-Materials
Process Design Program
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OUTLINE OF THE PRESENTATION
Whats this talk about ??
Continuum framework for sensitivity analysis of
large Thermo-inelastic deformations Review of
developments for kinematic, contact,
constitutive and thermal sensitivity
sub-problems
Design example Sensitivity analysis applied to
multi-stage processes
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A VIRTUAL MATERIALS PROCESS DESIGN SIMULATOR
Selection of a virtual direct process model
Selection of the design variables (e.g. die and
preform parametrization)
Selection of the sequence of processes (stages)
and initial process parameter designs
Material Process Design Simulator
Optimization algorithms
Assessment of automatic process optimization
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ESSENTIAL FEATURES OF A DESIGN SIMULATOR OF
INDUSTRIAL PROCESSES
Application Aspects
Theoretical Aspects

• Efficiency avoid extensive direct forming
  simulations (as in surface response methods)
• Provide consistent coupling of direct
  sensitivity analyses with
• knowledge based expert systems
• microstructure evolution paths
• ideal forming techniques
• Oriented towards the design of multi-stage
  processes
• Allow for realistic polycrystalline material
  constitutive models
• Allow for hot forming design and intermediate
  thermal stages
• Interface with commercial solid modelers and
  optimization tools

• Allow consistent application of remeshing, data
  transfer adaptivity techniques to sensitivity
  analysis
• Contact/frictional conditions drive most forming
  design processes and need careful consideration
• Extend assumed strain methods to sensitivity
  analyses (preserve volume)
• Mathematically consistent and accurate
  computation of sensitivity fields
• Provide a unified approach to parameter and shape
  sensitivity / optimization

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COMPUTATIONAL DESIGN OF FORMING 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
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DESIGN OF MULTI-STAGE PROCESSES
Initial Product
Based on the designer knowledge, evaluate
practicable stage number (n) and select a process
sequence p from all feasible paths (j1 m),
such that
Node Intermediate preform
1st Stage
Arc Processing Stage

• such that
• Equipment constraint (press force, ram speed,
  maximum stroke, etc)
• Process temperature constraint
• Other process constraints

ith Stage

• Number of stages - n
• Force constraints for each stage
• Stroke allocation for each stage
• Stage temperature, etc.

Finishing Stage(nth)
Final Product
Optimal Path (pth) Feasible Paths (jth)
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THERMOMECHANICAL LARGE DEFORMATION ANALYSIS
Thermo-Mechanical Coupling System
Deformation Problem
Die
Die
Hot workpiece
Thermal Sub-problem
Thermal problem
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MACROSCOPIC CONSTITUTIVE FRAMEWORK
(1) Multiplicative decomposition framework
(2) State variable rate-dependent models
(3) Radial return-based implicit integration
algorithms
(4) Damage and thermal effects
Initial configuration Temperature ?o void
fraction fo
Thermal expansion
Deformed configuration Temperature
? void fraction f
.

• Inelastic response
• Flow rule
• ? Is the viscoplastic potential
• Internal variable evolution
• Damage evolution

Hyperelastic constitutive law
Intermediate thermal configuration
Temperature ? void fraction fo
Stress free (relaxed) configuration
Temperature ? void fraction f
Mechanical dissipation
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THE DIRECT CONTACT PROBLEM
Admissible region
Impenetrability Constraints
Current configuration
Reference configuration
Coulomb Friction Law
n
r
Inadmissible region

• Coulomb friction law assumed at the die-work
  piece interface
• Augmented Lagrangian approach to enforce
  impenetrability and frictional stick conditions

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DEFINITION OF PARAMETER SENSITIVITY
B
Fr
xn
x

• Design Parameters
• Ram speed
• Shape of die surfaces
• Material parameters
• Initial state

Fn
ILn
X
Bo
o
xnxn
xx
o
B
Two stage state variable sensitivity
contour w.r.t. parameter change
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DEFINITION OF SHAPE SENSITIVITY
Fr

X X (Y ?s )
xn
x
B
Fn
X
Bo
FR
BR
ILo
Y
ILn
XX
o
xx
xnxn
o
o
Stress sensitivity contour w.r.t preform
shape change
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SCHEMATIC OF THE CONTINUUM SENSITIVITY ALGORITHM
Equilibrium equation
Contact friction constraints
Design derivative of equilibrium equation
Sensitivity Weak Form
Regularized design derivative of contact
Frictional constraints
Material Constitutive laws
Incremental Sensitivity contact sub-problem
Design derivative of the material Constitutive
laws
Time and Space discretized weak form
Incremental Thermal sensitivity sub-problem
Incremental Sensitivity Constitutive Sub-problem
Assumed kinematics
Design derivative of Energy equation
Time Space discretized Modified weak form
Design derivative of assumed kinematics
Conservation of Energy
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SENSITIVITY KINEMATIC PROBLEM
Design sensitivity of equilibrium equation
Variational form
Calculate such that
o
o
o
Pr and F,
?
Constitutive problem
Regularized contact problem
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SENSITIVITY CONSTITUTIVE FRAMEWORK
What do we need from the sensitivity constitutive
sub-problem to solve the sensitivity kinematic
problem ?
o
o
o
Calculate Linear relationship
between T and F ,
?
o
o
c
Evolution of the state sensitivity as a linear
function of F ,
?
where V T, s, Fe
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SENSITIVITY CONSTITUTIVE FRAMEWORK
o
Evolution equation for
Sensitivity of the evolution equation
o
Evolution equation for s

• Euler backward integration scheme is applied to
  the evolution equations mentioned above
• Need to calculate the linear relationship
  between Fe and F
• and Wmech and F

o
o
,
o
o
,
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SENSITIVITY ANALYSIS OF CONTACT/FRICTION
Die
?
Parameter Sensitivity Analysis
B
r
x

• Regularization introduced
• Contact sensitivity assumption
• Friction sensitivity assumption

x x ( X, t, ß p )
y y ( ? )

o
? ?
B0
o
y y y
X
B?
o
r r
?
Die
Shape Sensitivity Analysis
y y ( ? )
r
B
X
x
B0


x x ( X, t, ß s )
X X (Y ß s )
Y
?
BR
B?
r
o
o
X X
x x
B0
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SENSITIVITY ANALYSIS OF CONTACT/FRICTION
Sensitivity of Contact Tractions

• Remarks
• Sensitivity deformation is a linear problem
• Iterations are preferably avoided within a single
  time increment
• Additional augmentations are avoided by using
  large penalties in the sensitivity contact problem

Normal traction
Stick
Slip
Sensitivity of gap and inelastic slip
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THERMAL SENSITIVITY ANALYSIS
Continuum problem
Discretize
Differentiate

Design differentiate
How about design differentiation of q?
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THERMAL SENSITIVITY ANALYSIS
Thermal sensitivity problem
Thermal sensitivity boundary conditions
Neumann b.c.
Convective b.c.
o
L F F-1
Thermal sensitivity weak form
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ALGORITHM FOR THERMO-MECHANICAL SENSITIVITY
ANALYSIS

• Hot Forming Process Design
• Sensitivity problem is linear in nature
• Coupled thermo-mechanical equation shown below
• Although the deformation sensitivity problem is
  linear, we solve it iteratively for computational
  convenience (2 iterations)

Kinematic
Thermal

• Iterative Solution Scheme
• Initialize k 1
• Mechanical phase solve the sensitivity
  deformation problem with fixed thermal
  sensitivities
• Thermal phase - solve the thermal sensitivity
  problem with fixed deformation sensitivities
• Update
• IF error ? - ? lt
  TOL then EXIT
• ELSE k k1, Repeat mechanical
  phase and thermal phase

? ? x x
o
o
k
n1
o
o
o
o
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ADDITIONAL FEATURES
Remeshing and Data transfer schemes
Assumed strain sensitivity analysis
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THERMAL SENSITIVITY ANALYSIS AXISYMMETRIC FORGING
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THERMAL SENSITIVITY ANALYSIS AXISYMMETRIC FORGING
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SENSITIVITY CONSTITUTIVE FRAMEWORK
Constitutive model used
Flow function
Hardening law or function
Where
Specific values for Al-1100 at 673 K are used in
the above formulations

• Thermal boundary conditions
• Flux
• Convective
• Contact
• Radiative

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PREFORM DESIGN Open Die Example
Objective Minimize the deviation
between the work piece and the desired final
shape. Here the final shape is desired
so as to avoid barreling.
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PREFORM DESIGN Closed Die Example
Objective Minimize the flash and the
deviation between the die and the work
piece for a Preforming shape design
More flash
Much more material with a conventional design
Unfilled cavity
The same material in a conventional design
Fully filled cavity
No flash
The same material with an optimum design
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MULTISTAGE SENSITIVITY ANALYSIS

• Multi-stage sensitivity features
• Sequential transfer of sensitivities
• Shape sensitivities
• Parameter sensitivities

Validation of multistage sensitivities
State
Stress
DDM
DDM
FDM
FDM
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MULTISTAGE DESIGN Open Die Example
Preforming Stage
Finishing Stage

• Objective
• Minimize barreling
• in final product in a
• two stage thermo-
• mechanical forging
• with given finishing die
• unknown die but given
• stroke in the preforming
• stage

?
Given Preform
H
h
h
r
R
R
o
Initial
Iteration 3
Iteration 7
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CONCLUSIONS AND FUTURE RESEARCH PLANS
APPLICATIONS OF THERMOMECHANICAL SENSITIVITY
ANALYSIS

• Novel framework for large deformation
  thermo-mechanical
• sensitivity analysis yielding accurate
  design sensitivities.
• Applications include shape, microstructure
  and force
• optimization.
• Design for porous materials, analysis of
  damage evolution and
• damage control.
• Multi-length scale computational design
  forming problems.

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