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Modeling, Testing and Calibration of Ductile Fracture

1

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Lab facility

Custom-made biaxial testing machine MTS universal testing machine SHPB systems in France

INSTRON 9250HV

Three-point bending of a hat assembly

Tension−compression test

HydraulicVertical: 50 kNHorizontal: 25 kN

Electro-mechanical200 kN, 10kN

Max. vel: 20m/s

INSTRON 5944

2 kN, 100 NVic 2D and Vic 3D software, two digital cameras, high speed cameras,Abaqus, LS-DYNA, Hypermesh, workstations

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Plasticity

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Conventional metal plasticity

* Yield surface

- Von Mises yield criterion:

* Flow rule

- Associated flow rule:

(Plastic potential=yield function)

* Hardening law

- Swift law:- Voce law:- Extrapolation up to a large strain- Isotropic hardening

3( , ) ' : ' ( ) 0

2p pf k σ σ σ

0( ) ( )n

p pk A

p p

fd d

ε

σ

0( ) (1 exp( ))p pk k Q

0.0 0.2 0.4 0.6 0.8 1.00

400

800

1200

1600

Tru

e s

tress [

MP

a]

Equivalent plastic strain

Swift law

Voce law

Saturation type

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Von Mises yield criterion

k

k

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* Different anisotropy between yield stress and Lankford ratio (r-value),

Anisotropy and non-associated flow rule*

*Mohr et al. “Evaluation of associated and non-associated quadratic plasticity models for advanced high strength steel sheets under multi-axial loading, International Journal ofPlasticity, Vol. 26, pp. 939-956, 2010.

( , ) ( ) 0p pf k σ Pσ σ

( ) p

gg d d

σ Gσ σ ε

σ

Non-associated flow rule

( , ) ( ) 0p pf k σ Pσ σ

p p

fd d

ε

σ

Associated flow rule

w

t

dr

d

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

Yield locus

Plastic potential

k

k

pdε

0

90

45

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Validation of non-associated flow rule

* Validation with DP590 and TRIP780

DP 590

TRIP 780

0

90

45

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Hardening curve after necking

* Fracture occurs after a significant amount of necking- Need for the stress−strain curve after necking- Non-uniform deformation in the gauge section after necking

→ hard to obtain hardening curve after necking experimentally

* ‘Inverse method’ using finite element simulation- Use of so-called ‘Notch tension test’ with R20 mm- Adjustment to the stress−strain curve in the post necking area until force

−displacement curve from simulation agrees well with the one from theexperiment

- It is unavoidable to repeat simulation

* Why ‘Notch tension test’?- Its geometry always generates necking exactly in the middle of specimen

because of the minimum gauge width there- Robustness and repeatability of tests

RW

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Procedure for inverse method

0.00 0.02 0.04 0.06 0.08 0.100

200

400

600

800

1000

1200

Tru

e s

tress [

MP

a]

Equivalent plastic strain

Experimental data

Swift law

Voce law

0( ) ( )n

Swift p pk A

0( ) (1 exp( ))Voce p pk k Q

A 1501

ԑ0 0.0002

n 0.123

k0 715

Q 400

β 37.9

1/8model

Displacement measuredwith DIC

0.0 0.2 0.4 0.6 0.8 1.00

400

800

1200

1600

Tru

e s

tress [

MP

a]

Equivalent plastic strain

Experimental data

Swift law

Voce law

Mixed Swift-Voce law

( ) ( ) (1 ) ( )p Swift p Voce pk k k

Mixed Swift−Voce law

: weighting factor

0.0 0.5 1.0 1.5 2.00

10

20

30

40

Fo

rce

[k

N]

Displacement [mm]

Experimental data

Swift law

Voce law

Mixed Swift-Voce law

Post necking region

Before necking

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Strain rate dependency*

*Roth and Mohr, “Effect of strain rate on ductile fracture initiation in advanced high strength steel sheets: Experiments and modeling”, International Journal of Plasticity, Vol. 56, pp. 19-44, 2014

0

0

0

[ , , ] [ ] [ ] [ ]

( ) ( ) (1 ) ( )

1 for

[ ]1 ln for

1 for T < T

[ ]1

p p p p T

p Swift p Voce p

p

p p

p

r

T r

m r

k T k k k T

k k k

kC

k T T T

T T

for T T T

m

r m

* Rate dependent hardening model

2

3

[ ] ( : Taylor-Quinney coefficient)

0 for

(3 2 )[ ] for

1

kp p k

p

p it

p it a p it

p it p a

a it

dT w dC

w

for a it

Input bar

Pusher

Stirrup

Specimen

Output bar 1

Output bar 2

Base plate

7,8,9: guide blocks

Experimental setup

Dependency of weighting factor on strain rate

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Plasticity model for reverse loading (MYU)

* Plasticity model for reverse loading (by Stephane Marcadet)

- Compression followed by tension or tension followed by compression- New behaviors should be incorporated into plasticity model

• Bauschinger effect

• Work hardening stagnation

• Permanent softening

- Kinematic hardening is used instead of isotropic hardening- Four parameters- Calibration of the model based on uniaxial tension followed by compression- Effect of four parameters

[ , , , ]

Bauschinger transition Stress level at stagnation Duration of stagnation Slope at post stagnation

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Novel experiment technique for plasticity

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Fracture model

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1 2 3

1 2 3 1 2 3

2 2 2

1 2 3 2

1

3

, , Load vector in Cartesian coord.

' , , Hydrostatic pressure vector

' , , , ,

2' 2

3

27det[ ]1cos

3 2

Carte

m m m

m m m

Mises

ij

Mises

OP

OO

O P s s s

O P s s s J

s

sian coord. Cylindrical coord. (Haigh-Westergaard coord.)

2cot , ( :Stress triaxiality)

3

61 , ( : Normalized Lode angle, 0 60 )

( , ) : Representing specific loading path

mp

<Principal stress space>

<Deviatoric 𝜋-plane>

Loading path parameter

* Formulation of stress triaxiality 𝜂 and Lode angle 𝜃

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History of fracture models at ICL

(1)Bao and Wierzbicki, “On fracture locus in the equivalent strain and stress triaxiality space”, International Journal of Mechanical Sciences, Vol. 46, pp. 81-98, 2004.(2)Bai and Wierzbicki, “Application of extended Mohr−Coulomb criterion to ductile fracture”, International Journal of Fracture, Vol. 161, pp. 1-20, 2010.(3)Mohr and Marcadet, “Hosford−Coulomb model for predicting the onset of ductile fracture at low stress triaxialities”, submitted for the publication.(4)Roth and Mohr, “Effect of strain rate on ductile fracture initiation in advanced high strength steel sheets: Experiments and modeling”, International Journal of Plasticity, Vol. 56, pp. 19-44, 2014.

Three-branch fracture model(1) Modified Mohr−Coulomb model (2010)(2) 3 3

2

1

2

11

3ˆ 1 sec 1

62 3

1 1cos sin

3 6 3 6

f

p

n

Ac c

c

cc

Extended Mohr−Coulomb model (2014)(3), (4)

11

1 2 2 3 3 1

1

1 3

0 0

0 0

0

1( , ) (1 ) (

2

2

for

1 ln for

a a af anp

n

p

p

p

b c f f f f f f

c f f

b

bb

0 ( , )

fp p

f

p

dD

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Calibration of fracture model

* Fracture tests5 types of tests characterizing a different combination of 𝜃 and 𝜂- NTR 20: notch tension with R20- NTR 6: notch tension with R6- CHD4: central hole with D4- Butterfly: tension & shear (plane strain tension & pure shear)- Punch: biaxial tension

NTR 20 NTR 6 CHD4

ButterflyPunch

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Identification of loading path

Experiment FE analysis Comparison

Displacement measuredusing DIC(VIC2D)

Two red points are symmetric

1/8 modelC3D8R

Extract loading path

Fracture initiation point

Optimization of fracture parameters

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Extreme bending of riser with fracture

Shear dominant fracture