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COMING FROM?. Polytechnic University of Madrid. Vicente Herrera Solaz 1 Javier Segurado 1,2 Javier Llorca 1,2 1 Politechnic University of Madrid 2 Imdea Materials Institute. IMDEA Materials Institute (GETAFE). - PowerPoint PPT Presentation
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COMING FROM?
IMDEA Materials Institute (GETAFE)
Polytechnic University of Madrid
Vicente Herrera Solaz 1
Javier Segurado 1,2
Javier Llorca 1,2
1 Politechnic University of Madrid2 Imdea Materials Institute
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. Introduction2. Crystal Plasticity Model3. Optimization Strategy4. Results5. 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 ?? ,, 0hsatcrit
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
2. Crystal Plasticiy Model• Multiplicative decomposition of the deformation gradient is considered
pe FFF ·
• The velocity Gradient Lp contains three terms:
pslre
ptw
psl
p LLLL
• Composite material model: parent and twin phases
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
With:
isl
isl
1
i
1
αpsl msγf1L
sltw N
i
N
αtw
αtw
1tw
αptw msγfL
twN
i*sl
i*sl
1*
i*
1
αpslre msγfL
twsltw N
i
N
• 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)
• The crystal plasticity model has been programmed using a subroutine (UMAT) in ABAQUS and was resolved on an implicit scheme.
γττ
1hγττ
1g βtwsat
β
10twj
sat
j
10j
twtw
slsl
aN
sltwj
aN
jslsl
i qhq
fττ
1hg αtwsat
β
10tw
αtw
aN
twtw γqtw
tw
• The evolution of the CRSR for each slip and twin system follows: gg i ,
• A viscoplastic model is assumed for the shear and twinning rate: depends on the resolved shear stress :
• Shear rate• Twinning rate
f
: i i iτ S s m
im
i
i
signg
·
1
0
i
mi
signg
ff
·
1
0
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
• The behavior of the polycristal: Numerical Homogenization: Calculation by FE of a boundary problem in a RVE of the microstructure.
Voxels model with 23 element/crystal
Dream 3D model with Realistic microstructure (grain size and shapes)
≈ 200 elements/crystal
Voxels model with 1 element/crystal
• Uniaxial tension and compression are simulated under periodic boundary conditions
• The grain orientations are generated by Montecarlo to be statistically representative of ODF
• Different RVEs can be used:
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
3. Optimization technique
Experimental curves
Micromechanical properties(known)
Numerical curves
Comparison
Micromechanical properties
(????)
Experimental curves
Numerical curves
Inverse analysis Comparison
Micromechanical properties fit
Validation numerical model
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
3. Optimization technique
Inverse analysisTrial-error
Optimization algorithm(Levenberg-Marquardt)
SubjectiveTime
Objective, Automatic Time
Micromechanical properties
(????)
Experimental curves
Numerical curves
Inverse analysis Comparison
Micromechanical properties fit
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
3. Optimization technique
Micromechanical properties
(????)
Experimental curves
Numerical curves
Inverse analysis Comparison
Micromechanical properties fit
Inverse analysis
Optimization algorithm(Levenberg-Marquardt)
Objective, AutomaticTime
IMPLEMENTATION
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
1
( ) ( , ) ( )n
i ii
O y f x
β β y f β
• 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
• If we do small increases d in the β parameters , the response (modified numerical curve) can be written as:
Jff )()(
• Objective function: O(β):
,, 0hsatcrit
dxdxdf
xfdxxf )()(
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
• The perturbance of parameters δ which results in a minimum of the objective function is obtained with the following linear system of equations
( ) [ ( )]T T TJ J diag J J δ J y f β
• The new set of β parameters will be:
jjj *
• The minimization process is iterative, each iteration k is based on the results of the k-1. The loop iteration ends when a goal is reached or it is impossible to minimize the error.
• The initial set of parameters is arbitrary• The optimization algorithm has been programmed in python
j
jipertji
j
iij
xfxfxfJ
),(),(
),(
• Where J is the Jacobian Matrix, obtained here numerically
KEYPOINTS The procedure is applied hierarchically: From simplistic RVEs to realistic ones
→ Time saving Experimental data used have to be representative: Number of curves, load
direction → To avoid multiple solutions 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
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12
AZ31 2 40 25 4 40 150 85 20 20 3000 1500 100 MN10 17 66 68 28 40 150 85 20 20 3000 1500 100 MN11 18 40 51 49 40 150 85 54 20 3000 1500 100
Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12
AZ31 4 73 46 3 4 159 106 20 0 3900 2830 112 MN10 13 72 62 25 89 136 81 28 1 2287 1500 100 MN11 13 47 41 48 40 129 51 53 20 2398 1500 100
Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12
AZ31 4 105 81 2 4 162 129 28 0 3900 2830 112 MN10 12 79 62 19 89 127 80 28 1 2287 1500 100 MN11 15 50 45 46 40 120 50 51 20 2328 1500 100
Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12
AZ31 9 105 89 5 9 167 109 24 0 3900 2830 87 MN10 11 78 62 19 89 127 79 28 1 2287 1500 100 MN11 20 53 52 41 169 62 60 48 463 181 1456 97
Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12
AZ31 23 88 80 35 25 179 94 59 20 2990 2831 24 MN10 12 75 65 24 109 151 79 27 2 1082 1500 128 MN11 40 50 46 42 316 66 56 77 471 1 693 353
Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12
AZ31 2 40 25 4 40 150 85 20 20 3000 1500 100 MN10 17 66 68 28 40 150 85 20 20 3000 1500 100 MN11 18 40 51 49 40 150 85 54 20 3000 1500 100
Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12
AZ31 4 73 46 3 4 159 106 20 0 3900 2830 112 MN10 13 72 62 25 89 136 81 28 1 2287 1500 100 MN11 13 47 41 48 40 129 51 53 20 2398 1500 100
Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12
AZ31 4 105 81 2 4 162 129 28 0 3900 2830 112 MN10 12 79 62 19 89 127 80 28 1 2287 1500 100 MN11 15 50 45 46 40 120 50 51 20 2328 1500 100
Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12
AZ31 9 105 89 5 9 167 109 24 0 3900 2830 87 MN10 11 78 62 19 89 127 79 28 1 2287 1500 100 MN11 20 53 52 41 169 62 60 48 463 181 1456 97
Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12 Param 1 2 3 4 5 6 7 8 9 10 11 12
AZ31 23 88 80 35 25 179 94 59 20 2990 2831 24 MN10 12 75 65 24 109 151 79 27 2 1082 1500 128 MN11 40 50 46 42 316 66 56 77 471 1 693 353
VALIDATION
PredictionAZ31
Fitting 1 curves
error= 31 MPa/pt
• 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
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
VALIDATION
PredictionAZ31
Fitting 2 curves
error= 25 MPa/pt
• 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
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
AZ31
Fitting 3 curves
Prediction
VALIDATION
error= 11 MPa/pt
• 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
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
PredictionMN10
Fitting 3 curves
VALIDATION
error= 9.5 MPa/pt
• 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
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
PredictionMN11
Fitting 3 curves
VALIDATION
error= 11.3 MPa/pt
• 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
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
AZ31 MN10 MN11
Expe
rimen
tal
Num
eric
al
INITIAL TEXTURES
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
AZ31 MN10 MN11
Expe
rimen
tal
Num
eric
al
FINAL TEXTURES
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
Curv
es F
it
Pola
r effe
ct (↑
Tª)
TEMPERATURE INFLUENCE on MN11
0
100
200
300
400
500
600
700
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
true
[MPa
]
MN11(-175 C)ED T expED C exp
ED T modelED C model
0
100
200
300
400
500
600
700
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
true
[MPa
]
MN11(150 C)ED T expED C exp
ED T modelED C model
0
100
200
300
400
500
600
700
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
true
[MPa
]
MN11(300 C)ED T expED C exp
ED T modelED C model
0
100
200
300
400
500
600
700
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
true
[MPa
]
MN11(50 C)ED T expED C exp
ED T modelED C model
0
100
200
300
400
500
600
700
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
true
[MPa
]
MN11(250 C)ED T expED C exp
ED T modelED C model
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
0
20
40
60
80
100
-200 -100 0 100 200 300
crit
[MPa
]
T a
MN11 crit (T a)
basalpyramidal c+a
prismatictwinning
• The Polar effect could be attributed to the twinning mechanism but it doesn't appears at high Tª then…
• The Inclusion of the non-Schmidt stresses on Pyramidal c+a is the only way to explain it (by modifying Schmidt law)
• 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
TEMPERATURE INFLUENCE on MN11
eff
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
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
WORKSHOP STOCHASTIC AND MULTISCALE INVERSE PROBLEMS
PARIS (2-3 October)
Thanks for your attencion