Transcript UNDERSTANDING THE MICROSTRUCTURAL EVOLUTION AND

COMING FROM?
Polytechnic University of Madrid
IMDEA Materials Institute (GETAFE)
Vicente Herrera Solaz 1
Javier Segurado 1,2
Javier Llorca 1,2
1 Politechnic University of Madrid
2 Imdea Materials Institute
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
An inverse optimization strategy to determine
single crystal mechanics behavior from polycrystal
tests: application to Mg alloys
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
1. Introduction
2. Crystal Plasticity Model
3. Optimization Strategy
4. Results
5. Conclusions
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
1. Introduction
• Magnesium
Useful for the industry due  
High ANISOTROPY
Low strength and ductility
 limits its use
• Anisotropy: very different CRSS (Critical Resolved Shear Stresses) of their
slip and twinning systems besides strong initial texture
• News alloys and different manufacturing systems are
• The influence of the alloyed elements
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
• Macroscopic Properties (E, sy..) Mechanical Tests
• Microscopic Properties (grains)  Hard estimation
nº slip and twinning def systems

Micromechanical Tests

Lower scale Models (MD, DD)

Inverse analysis of mechanical tests with FE models
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
• Objetives:

Develop a CP model for HCP materials + twinning

Apply CP in a Polycrystalline homogenization Model

Implement an optimization technique  Inverse analysis  crit , sat , h0  ??
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
2. Crystal Plasticiy Model
• Multiplicative decomposition of the deformation gradient is considered
F  F e ·F p
• Composite material model: parent and twin phases
• The velocity Gradient Lp contains three terms:
p
Lp  Lslp  Ltw
 Lrep  sl
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
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With:
 N tw α  N sl i i
L  1   f    γ s sl  m isl
  1  i 1
p
sl
L
L
  f α γ tw s αtw  m αtw
 1
 N sltw i* i*

  f   γ s sl  m i*sl 
 1
 i*1

N tw
p
re sl
N tw
p
tw
α
• Three slip deformation modes (basal, prismatic and pyramidal [c+a]) and
tensile twinning (TW) have been considered .
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
• A viscoplastic model is assumed for the shear and twinning rate: depends
on the resolved shear stress 𝝉:
•
•
τ i  S : si  m i
 Shear rate
f Twinning rate
i
  0  i
g

1
m

 ·sign  i


 
1
m
 i 
f   f0    ·sign  i
g 


 

• The evolution of the CRSR for each slip and twin system follows: g i , g 
a sl
N tw


τj 
τβ 
i
j
g  qsl  sl  h0j 1  j  γ  qtw sl  h 0tw 1  tw 
j 1
 1
 τ sat 
 τ sat 
N sl
atw
γ β

τβ
α
g  qtwtw  h 0tw 1  tw
 1
 τ sat
N tw
atw

 α
 f γtw

• The crystal plasticity model has been programmed using a subroutine
(UMAT) in ABAQUS and was resolved on an implicit scheme.
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
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• The behavior of the polycristal:
 Numerical Homogenization: Calculation by FE of a boundary
problem in a RVE of the microstructure.
• Different RVEs can be used:
Voxels model with
1 element/crystal
Voxels model with
23 element/crystal
Dream 3D model with Realistic
microstructure (grain size and shapes)
≈ 200 elements/crystal
• Uniaxial tension and compression are simulated under periodic
boundary conditions
• The grain orientations are generated by Montecarlo to be statistically
representative of ODF
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
3. Optimization technique
Experimental
curves
Micromechanical
properties
(known)
Comparison
Validation
numerical model
Numerical curves
Experimental
curves
Micromechanical
properties
(????)
Inverse analysis
Numerical curves
Comparison
Micromechanical
properties fit
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
3. Optimization technique
Subjective
Time
Trial-error
Inverse analysis
Optimization algorithm
(Levenberg-Marquardt)
Objective, Automatic
Time
Experimental
curves
Micromechanical
properties
(????)
Inverse analysis
Numerical curves
Comparison
Micromechanical
properties fit
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
3. Optimization technique
IMPLEMENTATION
Inverse analysis
Optimization algorithm
(Levenberg-Marquardt)
Objective, Automatic
Time
Experimental
curves
Micromechanical
properties
(????)
Inverse analysis
Numerical curves
Comparison
Micromechanical
properties fit
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
• Experimental data: pair of n points (xi, yi) defining an experimental curve y(x)
• Numerical data: pair of n points (xi, yi*) defining a numerical curve, where:
yi*=f(xi,β)=f(β) and β
a set of m parameters on wich our
numerical model depends
 crit , sat , h0 
• Objective function: O(β):
n
O(β)   yi  f ( xi , β)  y  f (β)
i 1
• If we do small increases d
in the β parameters , the response
(modified numerical curve)
can be written as:
f (    )  f (  )  J
f ( x  dx)  f ( x) 
df
dx
dx
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
• Where J is the Jacobian Matrix, obtained here numerically
pert
f ( xi ,  ) f ( xi ,  j )  f ( xi ,  j )
J ij 

 j
 j
• The perturbance of parameters δ which results in a minimum of the
objective function is obtained with the following linear system of equations
J
T
J    diag (J T J )   δ  J T [y  f (β)]
• The new set of β parameters will be:
 *j   j   j
• The minimization process is iterative, each iteration k is based on the
KEYPOINTS
of the k-1.
The loophierarchically:
iteration endsFrom
whensimplistic
a goal is RVEs
reached
or it is ones
 results
The procedure
is applied
to realistic
impossible
to minimize the error.
→ Time saving
• The
initial set of
parameters
arbitrary

Experimental
data
used haveisto
be representative: Number of curves, load
• The
optimization
algorithm
hassolutions
been programmed in python
direction
→ To avoid
multiple
 The values obtained have to be critically assessed: Predictions of
independent load cases → Validation
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
4. Results
• Fitting done on several Mg alloys: AZ31, MN10 and MN11
• Validation
• Initial and Final textures
• Temperature influence on MN11
Param 1
AZ31
2
2
3
4
5
6
40 25 4 40 150
7
8
9
10
11
12
85 20 20 3000 1500 100
Param 1
2
3
4
5
6
7
8
9
10
11
12
MN10 17 66 68 28 40 150 85 20 20 3000 1500 100
Param 1
2
3
4
5
6
7
8
9
MN11 18 40 51 49 40 150 85 54 20
10
11
12
3000 1500 100
Param 1
AZ31
4
2
3
4
73 46 3
5
6
7
8
9
4 159 106 20 0
10
11
12
3900 2830 112
Param 1
2
3
4
5
6
7
8
9
MN10 13 72 62 25 89 136 81 28 1
10
11
12
2287 1500 100
Param 1
2
3
4
5
6
7
8
9
MN11 13 47 41 48 40 129 51 53 20
10
11
12
2398 1500 100
Param 1
AZ31
2
3
4
4 105 81 2
5
6
7
8
9
4 162 129 28 0
10
11
12
3900 2830 112
Param 1
2
3
4
5
6
7
8
9
MN10 12 79 62 19 89 127 80 28 1
10
11
12
2287 1500 100
Param 1
2
3
4
5
6
7
8
9
MN11 15 50 45 46 40 120 50 51 20
10
11
12
2328 1500 100
Param 1
AZ31
2
3
4
9 105 89 5
5
6
7
8
9
9 167 109 24 0
10
11
3900 2830
12
Param 1
87
MN10 11 78 62 19 89 127 79 28 1
2
3
4
5
6
7
8
9
10
11
12
2287 1500 100
Param 1
2
3
4
5
MN11 20 53 52 41 169
6
7
8
9
62 60 48 463
10
11
12
181
1456
97
Param 1
AZ31
2
3
4
5
6
23 88 80 35 25 179
7
8
9
10
11
94 59 20 2990 2831
12
Param 1
24
MN10 12 75 65 24 109 151 79 27 2
2
3
4
5
6
7
8
9
10
11
12
1082 1500 128
Param 1
2
3
4
5
MN11 40 50 46 42 316
6
7
8
9
66 56 77 471
10
11
12
1
693
353
Param 1
AZ31
2
2
3
4
5
6
40 25 4 40 150
7
8
9
10
11
12
85 20 20 3000 1500 100
Param 1
2
3
4
5
6
7
8
9
10
11
12
MN10 17 66 68 28 40 150 85 20 20 3000 1500 100
Param 1
2
3
4
5
6
7
8
9
MN11 18 40 51 49 40 150 85 54 20
10
11
12
3000 1500 100
Param 1
AZ31
4
2
3
4
73 46 3
5
6
7
8
9
4 159 106 20 0
10
11
12
3900 2830 112
Param 1
2
3
4
5
6
7
8
9
MN10 13 72 62 25 89 136 81 28 1
10
11
12
2287 1500 100
Param 1
2
3
4
5
6
7
8
9
MN11 13 47 41 48 40 129 51 53 20
10
11
12
2398 1500 100
Param 1
AZ31
2
3
4
4 105 81 2
5
6
7
8
9
4 162 129 28 0
10
11
12
3900 2830 112
Param 1
2
3
4
5
6
7
8
9
MN10 12 79 62 19 89 127 80 28 1
10
11
12
2287 1500 100
Param 1
2
3
4
5
6
7
8
9
MN11 15 50 45 46 40 120 50 51 20
10
11
12
2328 1500 100
Param 1
AZ31
2
3
4
9 105 89 5
5
6
7
8
9
9 167 109 24 0
10
11
3900 2830
12
Param 1
87
MN10 11 78 62 19 89 127 79 28 1
2
3
4
5
6
7
8
9
10
11
12
2287 1500 100
Param 1
2
3
4
5
MN11 20 53 52 41 169
6
7
8
9
62 60 48 463
10
11
12
181
1456
97
Param 1
AZ31
2
3
4
5
6
23 88 80 35 25 179
7
8
9
10
11
94 59 20 2990 2831
12
Param 1
24
MN10 12 75 65 24 109 151 79 27 2
2
3
4
5
6
7
8
9
10
11
12
1082 1500 128
Param 1
2
3
4
5
MN11 40 50 46 42 316
6
7
8
9
66 56 77 471
10
11
12
1
693
353
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
VALIDATION
• To accept the results obtained you need to check the predictive ability of
the model in other load cases which are not included in the iterative
process.
• The independence and representativeness of the initial curves used for
the adjustment will influence in the quality of these predictions
AZ31
Fitting
1
curves
Prediction
error= 31 MPa/pt
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
VALIDATION
• To accept the results obtained you need to check the predictive ability of
the model in other load cases which are not included in the iterative
process.
• The independence and representativeness of the initial curves used for
the adjustment will influence in the quality of these predictions
AZ31
Fitting
2
curves
Prediction
error= 25 MPa/pt
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
VALIDATION
• To accept the results obtained you need to check the predictive ability of
the model in other load cases which are not included in the iterative
process.
• The independence and representativeness of the initial curves used for
the adjustment will influence in the quality of these predictions
AZ31
Fitting
3
curves
Prediction
error= 11 MPa/pt
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
VALIDATION
• To accept the results obtained you need to check the predictive ability of
the model in other load cases which are not included in the iterative
process.
• The independence and representativeness of the initial curves used for
the adjustment will influence in the quality of these predictions
MN10
Fitting
3
curves
Prediction
error= 9.5 MPa/pt
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
VALIDATION
• To accept the results obtained you need to check the predictive ability of
the model in other load cases which are not included in the iterative
process.
• The independence and representativeness of the initial curves used for
the adjustment will influence in the quality of these predictions
MN11
Fitting
3
curves
Prediction
error= 11.3 MPa/pt
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
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INITIAL TEXTURES
Numerical
Experimental
AZ31
MN10
MN11
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
FINAL TEXTURES
Numerical
Experimental
AZ31
MN10
MN11
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
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TEMPERATURE INFLUENCE on MN11
MN11(-175 C)
700
MN11(50 C)
700
ED T exp
ED C exp
ED T model
ED C model
600
true
300
[MPa]
500
400
300
true
[MPa]
400
400
300
200
200
200
100
100
100
0.05 0.1 0.15 0.2 0.25 0.3 0.35
0
0
0.05 0.1 0.15 0.2 0.25 0.3 0.35
MN11(250 C)
700
0
0.05 0.1 0.15 0.2 0.25 0.3 0.35
ED T exp
ED C exp
ED T model
ED C model
600
500
[MPa]
500
400
true
300
400
300
200
200
100
100
0
0
MN11(300 C)
700
ED T exp
ED C exp
ED T model
ED C model
600
[MPa]
0
true
0
ED T exp
ED C exp
ED T model
ED C model
600
500
Polar effect (↑Tª)
[MPa]
true
Curves Fit
ED T exp
ED C exp
ED T model
ED C model
600
500
MN11(150 C)
700
0
0.05 0.1 0.15 0.2 0.25 0.3 0.35
0
0
0.05 0.1 0.15 0.2 0.25 0.3 0.35
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
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TEMPERATURE INFLUENCE on MN11
MN11
crit(T
100
a
• The Polar effect could be attributed to
the twinning mechanism but it doesn't
appears at high Tª then…
)
crit
[MPa]
80
• The Inclusion of the non-Schmidt
stresses on Pyramidal c+a is the only way
to explain it (by modifying Schmidt law)
60
40
 eff      
20
0
-200
basal
pyramidal c+a
prismatic
twinning
-100
0
a
T
100
200
300
• In other HCP materials (Ti), Pyramidal c+a
has this role, but never on Mg.
• At high Tª, pyramidal c+a has a great
activity due to its low CRSS
WORKSHOP
STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
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5. Conclusions
• A CPFE model has been developed for Magnesium.
• An optimization algorithm has been implemented Inverse analysis.
• Numerical results Precise fit Experimental curves
• Experimental curves input (representative)  predictive capacity
• Three Mg alloys were analyzed effect of alloyed elements and Tª on
the micromechanical parameters
• Future work:
 Optimization: Texture inclusion as objective function
 Others representations of microstructures

Inclusion of grain boundary effects Crack propagation, fatigue
crack initiation, grain boundary sliding