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
• 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 ijB
N
n
&
=
&
kl ijkl ijA
G
g
=
dV
n
V
N
V ij ij=
∫
&
&
1
dV
g
V
G
V ij ij=
∫
1
mnkl ijmn eff ijkll
A
L
=
eff pqkl mnpq ijmn ijkll
B
L
A
=
−1Fourth
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 hardeningForming 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