Numerical determination of diffusional transformation plasticity from computations of random microstructures Fabrice B ARBE , Romain Q UEY, Lakhdar T ALEB Laboratoire de M ´ ecanique de Rouen Institut National des Sciences Appliqu ´ ees de Rouen ICTAM 2004, Warsow, August 2004 . – p.1
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Numerical determination ofdiffusional transformation plasticity
from computations of randommicrostructures
Fabrice BARBE, Romain QUEY, Lakhdar TALEBLaboratoire de Mecanique de Rouen
Institut National des Sciences Appliquees de Rouen
ICTAM 2004, Warsow, August 2004
. – p.1
Motivations: to reproduce experimental observations
Experimental procedure for the Pre-Hardening Testfor a bainitic transformation
� �� �� � �� ��
�����
Strain decomposition:
� ��� � �� � � � �� � ��� � � ��
� � : total strain (measured by dilatometry)
� � : elastic strain� �� � (imposed)
� � � �: thermo-metallurgical strain
(density variation imposed)� ��� : transformation plasticity (TRIP):
inelastic strain accomodating internal stresses
� �� : classical plastic strain from pre-hardening
. – p.2
Motivations: to reproduce experimental observations
Experimental observation with the Pre-Hardening Test
for a bainitic transformation
TRIP is influenced bythe plasticity ofthe austenitic phase
� Evidence ofthe interaction betweenclassical plasticity
and TRIP
. – p.3
Motivations: to reproduce experimental observations
Comparison with predictions of Leblond modelwith kinematic hardening
� current modelingsfail to reproduce
the effects of thisinteraction
� Objective:to determine a configurationof FE modeling forthe analysis of the
interactionplasticity-TRIP
. – p.4
Main assumption for the material properties
Modeling of aMacro-Volume (MV)extracted fromthe specimen:
the new phase
��
�
is a homogeneouselastoplastic materialembedded in ahomogeneouselastoplasticmatrix ( � phase)
diffusive coolingtransformationfrom domainγ
homogeneousmaterial
� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �
� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �
� � � � � �
� � � � � �
� � � � � �
� � � � � �
� � � � � �
� � � � �
� � � � �
� � � � �
� � � � �
� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �� � � � �
crystallographic orientationscristals with different
2−D ANALOGY
germs
2−D ANALOGY
Macrovolume (MV)
Macrovolume (MV)Specimenneglecting of the grain anisotropy
and the material heterogeneities(inclusions, etc.)
extraction of a MV
. – p.5
(1) Spherical growth and regular pattern (Ganghoffer et al, 1993)
uniform normal displacement
+ normal force leading to the
on each cube face:
top & bottom faces:
externally applied stressoverall
FE modeling:germ (in the corner)and 5 successives growthlayers
x
y
z
x
y
z
x
y
z
x
y
z
2−D ANALOGY2−D ANALOGY
product phase germs
initially regularpattern of
Macrovolume (MV) Macrovolume (MV)
(a)
Spherical growth
previouslysame BC as
. – p.6
(1) Spherical growth and regular pattern: results
Effect of the mesh size on the mean TRIP as function of �:
0
0.0005
0.001
0.0015
0.002
0 0.2 0.4 0.6 0.8 1
TR
IP in
the
dire
ctio
n of
load
volumic fraction of product phase
spherical, size 14spherical, size 16spherical, size 20spherical, size 24
5 random nucleates, size 145 random nucleates, size 165 random nucleates, size 24
experimental, after Petit-Grostabussiat et al, IJP (2004)
� the larger the mesh size, the smaller the TRIP. – p.7
(2) Spherical growth and random distribution: (Ganghoffer et al, 1993)
xy
z
xy
z
xy
z
xy
z
2−D ANALOGY
Macrovolume (MV)
Random growth
set of randomlypositionned germs
2−D ANALOGY
Macrovolume (MV)
. – p.8
(2) Spherical growth and random distribution: results
Effect of the position of the nucleates:100 realisations of microstructures differing by the positions of the nucleates.
mesh size
� � � � � � � � � �
nucleates for each realisation.
0
0.0005
0.001
0.0015
0.002
0 0.2 0.4 0.6 0.8 1
TR
IP
volumic fraction of the product phase
large dispersion � need to perform ensemble averaging over a largenumber of random microstructures
. – p.9
(2) Spherical growth and random distribution: results
Effect of the mesh size on the mean TRIP:
0
0.0005
0.001
0.0015
0.002
0 0.2 0.4 0.6 0.8 1
TR
IP in
the
dire
ctio
n of
load
volumic fraction of product phase
spherical, size 14spherical, size 16spherical, size 20spherical, size 24
5 random nucleates, size 145 random nucleates, size 165 random nucleates, size 24
experimental, after Petit-Grostabussiat et al, IJP (2004)
� the bigger the mesh, the larger the TRIP
� effect opposite to the one in modeling (1)
. – p.10
(3) Improved modeling: (i) new morphology of the product phase
� 3 principal morphologies available in pre-processing:
x
y
z
x
y
z
x
y
z
Cubic
Spheric
Polyhedric (Voronoï)
. – p.11
(3) Improved modeling: (ii) growth from outside the microstructure
Example of the growth in a sub-domaincontaining 2 nucleates:
x
y
z
x
y
z
x
y
z
x
y
z
. – p.12
(3) Improved modeling: (iii) ensemble averaging in a Macro-Volume
εtp
nminε
tp
nmax
max(proportion p )
maxn
sub−domainscontaining n germs
εtp
nmax
k=nmin
z
initial distributionsaverage over several
weighted average
germs in thesub−domains:
n +1 n −1
min(proportion p )
minn
sub−domainscontaining n germs
z
initial distributionsaverage over several
number of
min max
z
average curve,representative of the MV
εtp
)Σ (p .k k
on the sub−domainsCalculations
several cases with differentgerm positions
x
y
z
x
y
z
x
y
z
x
y
z
2−D ANALOGY
Macrovolume (MV)
set of randomlypositionned germs
2−D ANALOGY
Macrovolume (MV)
based studySub−domains
�� : proportion of sub-domainscontaining
�
nucleates
�� �� �
� � � � �
where
is the meannucleate densityper sub-domain
Assumption:ergodic medium
. – p.13
Improved modeling: results about mean TRIP
Effect of the mean nucleates densitysize of the MV:
�� � � �� � � �� �
- size of the sub-domain:
� � � � � � � �
0
0.00025
0.0005
0.00075
0.001
0.00125
0.0015
0.00175
0.002
0.00225
0 0.2 0.4 0.6 0.8 1
TR
IP
global volumic fraction
density=5, cube
density=4, cube
density=3, cube
density=2, cube
density=1, cube
density=3, sphere
density=2, sphere
density=1, sphere
experiment (Petit-Grostabussiat et al, IJP 2004)
� the higher the mean nucleates density, the smaller the TRIP
� no significant difference between spherical and cubic shape
. – p.14
Improved modeling: distribution and dispersion of TRIP in a Macro-Volume
Effect of the mean nucleates density and growth speedsize of the MV:
�� � � �� � � �� �
- size of the sub-domain:
� � � � � � � �
0
200
400
600
800
1000
1200
1400
-0.0005 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004
TRIP
sphere, density=3sphere, density=2sphere, density=1
0.2
0.25
0.3
0.35
0.4
0.45
0.5
1 1.5 2 2.5 3 3.5 4
rela
tive
disp
ersi
on
mean density (1st and 2nd curves) / growth speed (3rd curve)
cubes, density effectspheres, density effect
cubes, growth speed effect
� the higher the mean nucleates density, the smaller the dispersion
� the higher the growth speed, the higher the dispersion
Distribution Relative dispersion( = square_root(variance) / mean_value )
. – p.15
Conclusion and perspectives
a new numerical model based on random FEcomputations has been proposed
significant sensitivity to mesh size, boundary conditionsand nucleates density
decreasing dispersion with increasing nucleates density
(ergodicity hypothesis may be valid)
�� �
perspective: parallel computations on larger meshes
�� �
perspective: study of the interaction betweenclassical plasticity and TRIP (pre-deformation tests)
General perspective: perform new experiments andanalyse the results at the light of the corresponding FEcomputations.
. – p.16
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