a spectral element approach for dynamic elasto-viscoplastic problems
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Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
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
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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.
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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)
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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.
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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)
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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.
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Equivalent Stress (MPa): 0 14 28 42 57 71 85 99
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Comparison of texture from Taylor and Homogenization approach
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Plane strain compression of idealized aggregate
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
)
Equivalent Stress (MPa): 17 30 43 56 69 82 95 108
<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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Equivalent Stress (MPa): 7 14 22 30 37 45 53 60
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?
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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.
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
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 ()
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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)
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Equivalent Stress (MPa): 19 27 36 45 53 62 70 79(d)
(a)
(c)
(b)
XY
Z
Equivalent Stress (MPa): 19 27 36 45 53 62 70 79(d)
Stage 1: Plane strain compression
Unloading Stage – 2 shear(a)
(c)
(b)
XY
Z
Equivalent Stress (MPa): 19 27 36 45 53 62 70 79(d)
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CCOORRNNEELLLL U N I V E R S I T Y
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?
Materials Process Design and Control LaboratoryMaterials Process Design and Control Laboratory
CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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
Equivalent Stress (MPa): 0.00 3.57 7.14 10.71 14.29 17.86 21.43 25.00
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
Equivalent Stress (MPa): 0.00 5.71 11.43 17.14 22.86 28.57 34.29 40.00
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
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CCOORRNNEELLLL U N I V E R S I T Y
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.
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CCOORRNNEELLLL U N I V E R S I T Y
CCOORRNNEELLLL U N I V E R S I T Y
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