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

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1. Introduction2. Crystal Plasticity Model3. Optimization Strategy4. Results5. Conclusions

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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

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• 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

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• Objetives:

Develop a CP model for HCP materials + twinning

Apply CP in a Polycrystalline homogenization Model

Implement an optimization technique Inverse analysis ?? ,, 0hsatcrit

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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

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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 .

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• 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

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• 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:

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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

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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

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3. Optimization technique

Micromechanical properties

(????)

Experimental curves

Numerical curves

Inverse analysis Comparison

Micromechanical properties fit

Inverse analysis

Optimization algorithm(Levenberg-Marquardt)

Objective, AutomaticTime

IMPLEMENTATION

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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 )()(

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• 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

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4. Results• Fitting done on several Mg alloys: AZ31, MN10 and MN11

• Validation• Initial and Final textures

• Temperature influence on MN11

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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

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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

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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

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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

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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

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AZ31 MN10 MN11

Expe

rimen

tal

Num

eric

al

INITIAL TEXTURES

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AZ31 MN10 MN11

Expe

rimen

tal

Num

eric

al

FINAL TEXTURES

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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

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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

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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

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Thanks for your attencion