Strain localization analysis using a large strain self-consistent approach

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Strain localization analysis using a large strain self-consistent approach

Gérald Franz, Farid Abed-Meraim, Tarak Ben Zineb, Xavier Lemoine, Marcel Berveiller

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. Shear 07, International Symposium

on Shear Behavior and Mechanisms in Materials Plasticity, Sep 2007, Nancy, France. 2007. �hal-

01232410�

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This is an author-deposited version published in: http://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

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• 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.Franz¹¹, F., F.AbedAbed--MeraimMeraim¹¹, T.Ben Zineb, T.Ben Zineb²², X.Lemoine, X.Lemoine³³, M.Berveiller, M.Berveiller¹¹

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 Fresnel57078 Metz Cedex 357078 Metz Cedex 3 2 : LEMTA CNRS UMR 7563 ESSTIN

2 : LEMTA CNRS UMR 7563 ESSTIN --UHP, 2 Rue Jean UHP, 2 Rue Jean LamourLamour54519 Vandoeuvre54519 Vandoeuvre--LèsLès--Nancy Nancy 3 : Centre Automobile Produit ARCELOR

3 : Centre Automobile Produit ARCELOR ResearchResearch, S.A. Voie Romaine BP 30320 57283 , S.A. Voie Romaine BP 30320 57283 MaizièresMaizières-- les-les-MetzMetz

g σ, G Σ,

Context

Context of of the the study study

Plastic

Plastic mechanismsmechanismsof of ductilityductilitylossloss

Structural origin:

wrinkling, buckling

Material origin:

localization, necking

Damage

Damage mechanismsmechanismsof of ductilityductilitylossloss

Cavitie Failure

Mechanisms

Mechanismsof ductilityof ductilitylossloss FormingFormingLimitLimitDiagramDiagram(FLD)(FLD) Plastic Plastic anisotropyanisotropyevolutionevolution

-300 -200 -100 0 100 200 300 400

-30% -20% -10% 0% 10% 20% 30% 40% 50% 60%

Strain / Amount of shear

Cauchy Stress

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

Cauchy Stress

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

Metallurgyimpact (texture, grain size, …)impact (texture, grain size, …) StrainStrainpathpathdependencedependence

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

Mesoscopicscalescale––basic slip basic slip processprocess MicroscopicMicroscopicscalescale––intragranularintragranularmicrostructuremicrostructure Assumptions

Assumptions

Elasticity Elasticity Plasticity

Plasticity Elastic-Elastic-plastic tangent plastic tangent modulusmodulus

g g p

g g p

S w

R d

γ γ

&

&

=

g =

g σˆ:R

τ =&

γg

mg

ng

X1

X2

(^{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

= ijkl kl^h g ij g gh

gh kRC R

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: : thetheonsetonsetof of localization

localizationisisalongalonga a bandband (

(RiceRice,1976),1976)

Field Field equationsequations

Ellipticity Ellipticitylossloss υr N⁺,L⁺,G⁺

−

−

−L G N , ,

( )( )











=

=

=

conditions Boundary

: 0

G L N

V grad G

N div ^T

&

&

0 ) . . det(υυυυ^Lυυυυ = What

Whatisisthethelinklinkbetweenbetweenlocal local andandglobal global strain

strain??

kl ijkl

ij B N

n&= &

kl ijkl

ij A G

g=

dV V n

N^&ij⁼¹∫V^&ij

dV V g G

V ij

ij=1∫

mnkl ijmn eff

ijkl l A

L =

eff mnpq pqkl ijkl lijmnB L A =⁻¹ Fourth

Fourthorderorderlocalizationlocalizationtensorstensors Volumic

Volumic average average

Relation

Relation betweenbetweenA andA andBB

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

Intensityof dislocations of dislocations walls walls

Polarity

Polarityof dislocations of dislocations walls walls TEM micrographTEM micrograph

Macroscopic

Macroscopic validation validation

∑

=

+

− +

= ⁶

1

0 1

i CBB ig CB g

c τ ( f)τ f τ

τ αµbρ τ^CB=

(g i^w)

wd i wd

ig αµb ρ absm.n

τ =

( ) ( )i^wp 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

ComplexFLD: EquibiaxialFLD: EquibiaxialExpansion 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

ComplexFLD: FLD: Uniaxial Uniaxial Tension Tension prestrain prestrain (10%)(10%)

Mild Steel Dual Phase

Mild Steel Dual Phase

Multiscale

Multiscale model model with intraganular modelingwith intraganular modeling Multiscale Multiscale model model without intraganular modelingwithout 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