• Aucun résultat trouvé

Strain localization analysis using a large strain self-consistent approach

N/A
N/A
Protected

Academic year: 2021

Partager "Strain localization analysis using a large strain self-consistent approach"

Copied!
2
0
0

Texte intégral

(1)

Science Arts & Métiers (SAM)

is an open access repository that collects the work of Arts et Métiers Institute of

Technology researchers and makes it freely available over the web where possible.

This is an author-deposited version published in:

https://sam.ensam.eu

Handle ID: .

http://hdl.handle.net/10985/10435

To cite this version :

Gérald FRANZ, Farid ABED-MERAIM, Tarak BEN ZINEB, Xavier LEMOINE, Marcel

BERVEILLER - Strain localization analysis using a large strain self-consistent approach - 2007

Any correspondence concerning this service should be sent to the repository

(2)

• The statistically stored dislocations in the cell interior, as well as the cell boundary dislocations, are represented by a single local dislocation densityρ

• The local density of immobile dislocations stored in the wallρ(wd)associated with

the {110} plane • The polarity dislocations densityρ(wp)associated with

the {110} plane

STRAIN LOCALIZATION ANALYSIS

STRAIN LOCALIZATION ANALYSIS

USING A LARGE STRAIN SELF

USING A LARGE STRAIN SELF-

-

CONSISTENT APPROACH

CONSISTENT APPROACH

G.Franz

G.Franz

1

1

, F.

, F.Abed

Abed-

-Meraim

Meraim

1

1

, T.Ben Zineb

, T.Ben Zineb

2

2

, X.Lemoine

, X.Lemoine

3

3

, M.Berveiller

, M.Berveiller

1

1

1 : LPMM CNRS UMR 7554 ENSAM CER de Metz, 4 rue Augustin Fresnel

1 : LPMM CNRS UMR 7554 ENSAM CER de Metz, 4 rue Augustin Fresnel

57078 Metz Cedex 3

57078 Metz Cedex 3

2 : LEMTA CNRS UMR 7563 ESSTIN

2 : LEMTA CNRS UMR 7563 ESSTIN

-

-

UHP, 2 Rue Jean

UHP, 2 Rue Jean

Lamour

Lamour

54519

54519

Vandoeuvre

Vandoeuvre

-

-

Lès

Lès

-

-

Nancy

Nancy

3 : Centre Automobile Produit ARCELOR

3 : Centre Automobile Produit ARCELOR

Research

Research

, S.A. Voie Romaine BP 30320 57283

, S.A. Voie Romaine BP 30320 57283

Maizières

Maizières

-

-les

les

-

-

Metz

Metz

g

,

σ

G

,

Σ

Context

Context

of

of the

the

study

study

Plastic

Plastic

mechanisms

mechanisms

of

of

ductility

ductility

loss

loss

Structural origin: wrinkling, buckling

Material origin: localization, necking

Damage

Damage

mechanisms

mechanisms

of

of

ductility

ductility

loss

loss

Cavitie Failure

Mechanisms

Mechanisms

of

of

ductility

ductility

loss

loss

Forming

Forming

Limit

Limit

Diagram

Diagram

(FLD)

(FLD)

Plastic

Plastic

anisotropy

anisotropy

evolution

evolution

-300 -200 -100 0 100 200 300 400 -30% -20% -10% 0% 10% 20% 30% 40% 50% 60%

Strain / Amount of shear

Ca uc hy S tre ss UT UT 10% SSh SSh BS 30% BS 10% ferritic steel -300 -200 -100 0 100 200 300 400 -30% -20% -10% 0% 10% 20% 30% 40% 50% 60%

Strain / Amount of shear

Ca uc hy S tre ss UT UT 10% SSh SSh BS 30% BS 10% ferritic steel TEM (Peeters, 2002) Textural anisotropy (crystalographic network + morphology) Structural anisotropy (intragranular microstructure) Plastic anisotropy evolution

Metallurgy

Metallurgy

impact (texture, grain size, …)

impact (texture, grain size, …)

Strain

Strain

path

path

dependence

dependence

UT UT PT PT EBE EBE 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 -0,6 -0,5 -0,4 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4 0,5 e2 e1 CLF : Direct V9.2 CLF : TP10 Ferritic steel Dual Phase

• Forming limit of sheet metal = state at which a localized strain initiates during forming

• Ductility loss characterization using Forming Limit Diagram (FLD) developed first by Keeler (1963) and Goodwin (1968).

• Path-dependent representation

• Ductility loss prediction for monotonous and sequential strain paths • Optimization of microstructural properties for the sheet forming steels

Take metallurgy, mechanisms, microstructure

and textures into account

Steel behaviour during sheet forming: hardening, complex loads,

instabilities, anisotropy

Scales transitions tools, micromechanic of plasticity, localization and damage criteria,

coupling with finites elements

Aims

Aims

of

of the

the

study

study

• Three main step : • Single crystal modeling, • Scale transition, • Ductility loss criterion

Single

Single crystal

crystal

modeling

modeling

Mesoscopic

Mesoscopic

scale

scale

basic slip

basic slip

process

process

Microscopic

Microscopic

scale

scale

intragranular

intragranular

microstructure

microstructure

Assumptions

Assumptions

Elasticity

Elasticity

Plasticity

Plasticity

Elastic

Elastic

-

-

plastic tangent

plastic tangent

modulus

modulus

g g p g g p S w R d

γ

γ

& & = = g g R : ˆ σ τ =& g γ g m g n 1 X 2 X

(

d d

)

trace

( )

d C p σ σˆ= : − −

(

) (

)

[

]

(

mnkl mnkl

)

h mn h gh g pj ip pj g ip g pq ijpq jk il lj ik kj il lj ik ijkl ijkl C R k M S S R C C l δ σ σ σ δ σ δ σ σ δ σ δ − − + −     + = 2 1 2 1 g l n&= :

(

+

)

−1 = h kl ijkl g ij g gh gh R C R k M with δ • Elastic-plastic behavior • Large strains formulation • Body-Centered Cubic (BCC)

• Plastic strains only due to slip processes (<110> slip direction family and {110}, {112} slip plane families)

[Peeters, 2002]

Mughrabi’s composite model

Scale

Scale

transition

transition

Ductility

Ductility

loss

loss

criterion

criterion

Assumption

Assumption

:

:

the

the

onset

onset

of

of

localization

localization

is

is

along

along

a

a

band

band

(

(

Rice

Rice

,1976)

,1976)

Field

Field

equations

equations

Ellipticity

Ellipticity

loss

loss

υ

r

N

+

,

L

+

,

G

+ − − −

L

G

N

,

,

( )

( )

=

=

=

conditions

Boundary

:

0

G

L

N

V

grad

G

N

div

T

&

&

0

)

.

.

det(

υυυυ

L

υυυυ

=

What

What

is

is

the

the

link

link

between

between

local

local

and

and

global

global

strain

strain

?

?

kl ijkl ij

B

N

n

&

=

&

kl ijkl ij

A

G

g

=

dV

n

V

N

V ij ij

=

&

&

1

dV

g

V

G

V ij ij

=

1

mnkl ijmn eff ijkl

l

A

L

=

eff pqkl mnpq ijmn ijkl

l

B

L

A

=

−1

Fourth

Fourth

order

order

localization

localization

tensors

tensors

Volumic

Volumic

average

average

Relation

Relation

between

between

A

A

and

and

B

B

Conclusions

Conclusions

Microscopic

Microscopic

validation

validation

Longitudinal plane view TEM micrograph in a grain with initial orientation (43.3°,127.8°,-42.4°) after a reverse test of 30% simple shear with SD parallel

RD and SPN parallel to TD [Nesterova & al, 2001]

Intensity

Intensity

of dislocations

of dislocations

walls

walls

Polarity

Polarity

of dislocations

of dislocations

walls

walls

TEM

TEM

micrograph

micrograph

Macroscopic

Macroscopic

validation

validation

= + − + = 6 1 0 1 i CBB ig CB g c τ ( f)τ f τ τ ρ αµb τCB=

(

w

)

i g wd i wd ig αµb ρ absm.n τ =

( )

( )

wp i w i g wp i wp ig αµb absρ m.nsignρ τ = polarity latent hardening + isotropic hardening

Forming Limit Diagrams

Forming Limit Diagrams

Direct FLD

Direct FLD

Complex

Complex

FLD:

FLD:

Equibiaxial

Equibiaxial

Expansion

Expansion

prestrain

prestrain

(10%)

(10%)

• Reproduces correctly the intragranular microstructure during monotonic and sequential loading paths • Gives better results concerning macroscopic behavior during changing loading paths than model without intragranular modeling

Mild Steel

Mild Steel Dual Phase

Complex

Complex

FLD:

FLD:

Uniaxial

Uniaxial

Tension

Tension

prestrain

prestrain

(10%)

(10%)

Mild Steel Dual Phase

Mild Steel Dual Phase

Multiscale

Multiscale

model

model

with intraganular modeling

with intraganular modeling

Multiscale

Multiscale

model

model

without intraganular modeling

without intraganular modeling

• Reproduces correctly the shape and the level of direct FLD for mild steel and dual phase

• Reproduces the strain-path dependence of complex FLD

• The level of FLD after expansion prestrain seems to be realistic. The curve is shifted down and at the right in agreement with tendancies observed in literature

• The positive side of the FLD is overestimated. This effect can be corrected by damage introduction in the model

• FLD is shifted at the left in agreement with tendancies observed in literature but the level of the lower point of the FLD is lower

Références

Documents relatifs

Using an electromagnetic analogy, Eshelby [11] derived simple solutions of inhomo- geneous wave equations for the velocity and elastic distortion fields in terms of ‘stress

2014 The thermodynamic equilibrium between bulk strain field and interface dislocations in lattice mismatched semiconducting single heterostructures, is calculated in a

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des

Similarly, a magistrate or judge in Hong Kong could issue a local search warrant to search the Hong Kong subsidiary of the WSP (if one exists), and compel it to produce

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des

The proposed model, named dynamic latent block model (dLBM), allows therefore to discover latent partitions for rows and columns, but also for the time periods.. Thus, dLBM provides

R107: Voilà, et après, heu, à un moment donné j'ai eu envie de faire autre chose et c'est pour ça que je dis que je suis attachée au groupe parce que… C'est… avec le recul, on

The failure impact assessment rating is used in conjunction with the condition rating to provide a logical and systematic means for determining the priorities for subsequent