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1. Introduction
Transformation induced plasticity (TRIP) can be defined as the anomalous plastic
strain observed when metallurgical transformation occurs under an external stressmuch lower than the yield limit. From a technological point of view it plays an
essential role in many problems, in particular for the understanding of residual stres-
ses resulting from welding operations and for their prediction in practically significant
cases. From an experimental point of view, it is usually characterized in a TRIP test
(called also sometimes creep test) where a constant stress is applied during transfor-
mation under prescribed cooling conditions as presented byTaleb et al. (2001).
From a microstructural point of view, two mechanisms are usually considered to
explain TRIP,
theMagee (1966)mechanism corresponds to the formation of selected mar-
tensitic variants resulting from the applied stress,
the GreenwoodJohnson (1965) mechanism corresponds to the micro-
mechanical plastic strain arising in the parent phase from the expansion of the
product phase.
The relative importance of these two mechanisms depends on the material and the
transformation under consideration. Strictly speaking, both mechanisms are gen-
erally present in diffusional and diffusionless transformations.
We are interested here especially in welding conditions in a 16MND5 steelused for pressure vessels in which Magee mechanism seems not be significant
(Grostabussiat et al., 2001). Most existing models for TRIP based on the
GreenwoodJohnson mechanism finally result for a low applied stress, in the
following equation,
E:
tp k:z:z::
3
2S
giving the TRIP rate as a function of the volume fraction of the product phase z and
the applied deviatoric stress S. This relation introduces a material parameter k
(TRIP coefficient) and a normalized function z governing the TRIP kinetics.Different forms have been proposed for this function z see for instance the reviewofFischer et al. (1996).
During the last years several experimental, theoretical and numerical studies have
been performed on the TRIP phenomenon (Fischer et al., 2000a, b; Cherkaoui et al.,
1998; Nagayama et al., 2000, 2001a, b; Taleb et al., 2001; Grostabussiat et al., 2001;
Coret et al., 2002).
We shall focus our attention here on the model developed by Leblond (1989)and
which, although rather old, still is one of the most widely used for practical applica-
tions and which is implemented in the finite element codes SYSWELD and ASTER.
This model is obtained from a micromechanical analysis based on the determina-tion of the plastic strain induced in a spherical parent phase by the growth of a
spherical product phase core. This model is very crude:
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The assumed spherical shape, which allows an analytical solution, obviously
is not true in reality, however it may be expected that this is not essential at
the beginning of the transformation which dominates TRIP (85% of the
whole transformation plasticity takes place during the first half of thetransformation after Leblonds model),
Many supplementary assumptions are made which will be discussed in the
following.
The purpose of the present paper is to discuss these approximations, evaluate their
influence and release some of them resulting thus in a more complete model. It will
then be shown that these improvements may modify the TRIP description at the
beginning of the transformation and lead to lower TRIP value at the end of the
transformation. The obtained results will be analyzed through further experiments
and more refined finite element micromechanical analyses as those presented in
(Barbe et al., 2001a, b).
2. Leblonds model
2.1. General background
A macroscopic volume elementVcontaining two phases is considered byLeblond
et al. (1986). The starting point is the decomposition of the local (total) strain foreach phase into elastic, thermo-metallurgical and plastic strains:
"t "e "thm "p
In fact "thm and "p should be considered as respectively the isotropic and devia-toric parts of the non elastic strain"t "e, which generally includes thermal expan-sion, plastic deformation and transformation deformation. This means that
followingLeblond et al. (1986), the transformation strain has been included in "thm
for its isotropic part and"p for its deviatoric part.It follows from the assumed elastic homogeneity that the macroscopic elastic
strain is the average of the microscopic one as established by Hill (1967),Ee
"e
h iV.The same therefore holds for the isotropic and deviatoric parts of the non elastic
strain, Ethm "thm
Vand Ep "ph iV
The macroscopic elastic strain Ee will be obtained from the macroscopic elastic
stress Sby the homogeneous elastic law. The thermo-metallurgical part Ethm can be
observed experimentally (for an isotropic material) by varying the temperature
under no applied mechanical stress (free dilatometric test, S 0).
As mentioned before, the macroscopic plastic strain must include in addition to
the classical plastic term, the contribution of the deviatoric transformation strain.
The plastic strain rate therefore must account for the variation of the geometric
extent of each phase, thus leading to the following expression:
E:
p 1z ":p
1
V1
z":p
2
V2
z:
"p1! 2
F 1
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where, z is the volume fraction of the newly formed phase (2), ":p
i is the microscopic
plastic strain rate tensor in the phase i, "p1! 2 is the deviatoric part of the trans-formation strain and "p1! 2 F expresses the average value of "p1! 2 over thetransformation front F.
As discussed in Leblond et al. (1989), the last term in Eq. (1) which accounts
for the average deviatoric transformation strain disappears when there is no pre-
ferred orientation (GreenwoodJohnson mechanism). On the contrary it would be
the dominant term for the Magee mechanism. It should be noted that this term
would disappear during reaustenitization so that the resulting strain should be
recoverable upon heating. This would not be true for the other terms. This remark
provides an experimental procedure for validating the respective role of both
mechanisms.
The second term inEq. (1)will also disappear assuming only elastic behavior for
the product phase.Eq (1)finally reduces to,
E:
p 1z ":p
1
V1
2
Now the evolution of"p1 in the parent phase will result from the variations of theloading conditions (temperatureT
:and the applied macroscopic stress
:) and from
the transformation process.
Dependence ofE:
p upon T:
and :
corresponds to the usual macroscopic thermo-
plastic behavior while transformation plasticity refers to the evolution of E:
p from
the variation ofz at constant temperature and macroscopic stress.
E:
p E:
cp E:
tp;
where,
E:
tp 1z "p1
z
V1
z:
3
where "p1
z can be evaluated by changing z under constant temperature and macro-
scopic stress (T
:
:
0).Using Von Mises standard plasticity and assuming a uniform yield stress y1 , this
relation is transformed into:
E:
tp 1z 3
2
1
y1
"eq1z
s1
V1
z:
4
where, y1 is the yield stress of the parent phase (phase 1), s1is the deviatoric tensorof the microscopic stress in the phase 1,"eq1 is the Von Mises equivalent microscopicstrain in the phase 1.
It should be noted that as discussed inLeblond (1989), the assumption of uniformy1 does not allow a precise description of the local non uniform hardening. How-ever, hardening can be roughly taken into account by allowing this uniform yield
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stress y1 to depend on some macroscopic averaged hardening variable. This will befurther discussed in the application.
The last assumptions consist of neglecting the correlation betweens1and "eq1
z
and
identifying the average S1 of s1 in V1 to the overall average S(respectively H4 and
H5 inLeblond et al. 1989), finally reducingEq. (4)to
E:
tp 1z 3
2
S
y1
"eq1z
V1
z:
5
It should be noted that according to Leblonds discussion, this identification ofS1to S can only be used as a first approximation. A numerical micromechanical ana-
lysis (Fig. 7 inLeblond et al. 1989) in fact suggests that S1/Sis a decreasing function
ofz. The following second approximations,
S1 1z S or S1 1z2
S;
appear reasonable. We shall come back later when dealing with applications.
2.2. Micromechanical model
In order to evaluate the average "eq1
z
V1
which appears in(5), Leblond et al. use
the micromechanical model of a spherical product phase expanding in a sphe-
rical shell of the parent phase (austenite) without external loading (seeFig. 1).A variationz ofzcorresponds to an expansionR2 of the product phase radius:
z3R 22 R2
R 31
Two further assumptions are made:
Fig. 1. Geometry considered by Leblond et al. for the evaluation of "eq1
z
V1
.
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H2a: Elastic strains are much smaller than plastic strains in the austenitic phase,
H2b: The (compressive) forces exerted by the crust on the core have a negli-gible effect on the volume of the latter.
The transformation strain ("12) therefore appears as an imposed displacement:
u R2V
V 3"12R2
on the R2R2 boundary of the austenitic shell.V=Vis the relative difference of volume between the two phases.The micro-incompressibility condition resulting from H2a in the austenitic phase
directly leads to
u C
x2
3"12 R2R2 2R2
x2
3"12R22 R2
x2 6
Cis a constant.
The Von Mises equivalent strain is then easily computed and integrated over the
austenitic phase, finally resulting in:
"eq1z
V1
2"121z
:Ln z 7
It should be noted that this model is based on the assumption that the plasticdeformation in the austenitic phase essentially results from the internal stress arising
from the expansion of the daughter phase and is not significantly affected by the
applied external stress. This is entirely consistent with Leblonds assumption (H4 in
Leblond et al., 1989) and may be expected to remain true for small values of this
stress.
2.3. Leblonds model
Having "eq1
z V1
byEq. (7),Eq. (5)which gives the transformation plasticity rate
becomes:
E:
tp 2"12
y1:Lnz:z
::
3
2S 8
Comparing the predictions of the above model with some test results given by
Desalos (1981), Leblond et al. concluded that at the beginning of the transforma-
tion, Eq. (8) overestimates the transformation plasticity rate. To account for this
discrepancy and also to avoid the singularity at z=0, they proposed to replace (8)
by
E: tp 0 if z4
0:03
2D"12y1
:Lnz:z::
3
2S if z> 0:03
8 1 in the plastic case). The corresponding values inthe special cases discussed above (seeTable 1) are 6, 12.9 and 13 respectively at the
beginning of the ferritic, bainitic and martensitic transformations in a 16MND5
steel. This clearly shows the limitation of Leblonds assumption H2. More precisely,
for small values ofz (at the beginning of the stransformation), the parent phase is
not entirely plastified. Complete plastification will occur when reaches R1 whichcorresponds to a limit value zlofz given by,
zl
1
a ;
which is respectively equal to about 0.2 for ferritic transformation and 0.09 for bai-
nitic and martensitic transformations in a 16MND5 steel.
3.5. Plastic solution
For z > zl, the parent phase is completely plastic, the solution is now,
. If 0< r4R2,Eq. (17)remains valid with one constant 2,
. IfR2
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The complete solution therefore involves 3 constants namely2,and which aredetermined from the boundary conditions,
. No external force is applied,
Eq. (23)gives,
R1e13 25
. Continuity ofr at r R2,Eqs. (17), (23) and (25)lead to,
2 D"122y13K
LnR2
R1
26
. Continuity ofuat r R2,
fromEqs. (17), (24), (25) and (26)we have,
R 32D"12 27
A complete solution of the elasticplastic problem has thus be obtained.
4. Evaluation of the TRIP
The transformation plasticity may now be evaluated from Eq. (5). It should
however be noted that when deriving Eq. (2) from (1) the plastic contribution
resulting from the newly formed phase still disappears due to the hydrostatic stress
state in the product phase 0< r 4R2 [Eq. (17)] which induces a purely elasticdeformation.
4.1. The plastic case
According toEq. (15)and remarking that in the parent phase, >0,
"eq1 2"p
The plastic strain "pis obtained as the difference between the total strain " andthe elastic strain"e[remember that in the parent phase"
thm 0, seeEq. (16)].
" is given byEq. (12),
"u1r ;
where u1 is obtained fromEqs. (24), (25) and (27),
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u1 2y13K
rLn r
R1
D"12
R 32r2
;
so that,
"2y13K
Ln r
R1
D"12
R 32r3
:
The elastic strain results from Hookes law [Eqs. (14)] where1and1are deducedfromEqs. (23) and (25),
1 2y1 Ln
r
R1 2
3y1 ; 1
y13
so that;
"e2y13K
Ln r
R1
2
9
y1K
y16
The plastic component "ptherefore results as,
"p D"12R 32r3
2y19K
y16
28
and,
"pz
D"12R 31r3
;
so that,
"eq1z
2D"12R 31r3
and;
"eq1
z
V1
1
V1
R1
R2
2D"12R 31
r
3 4r2dr 2D"12
4R 314
3 R31 R 32
Ln r R1R2 2D"12
1z
Ln z
Finally, according toEq. (5),
E:
tp 2D"12
y1Ln z z
:3
2S 29
This relation coincides with Leblonds results. It follows from this analysis that
taking into account the elastic deformations in both phases does not change the final
result. This would not be true if elasticity had been taken into account in one phase
only as noted byTaleb (1999).For the reasons mentioned inSection 2.3., Leblond et al. have proposed to cut off
the transformation plasticity belowz 0:03 leading toEq. (9).
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4.2. The elasto plastic case
As shown inSection 3.4, when z< zlthe parent phase is not entirely plastic. Pro-
ceeding as inSection 4.1,Eqs. (20)(22)combined withEqs. (12), (14) and (15)lead to,
"p D"12R 32r3
2y19K
y16
30
which is the same asEq. (28)obtained in the plastic case. So that,
"eq1z
2D"12R 31r3
and;
"eq1
z
V1
1
V1a
13R2
R2
2D"12R 31
r3 4r2dr 2D"12
4R 31
43
R 31 R32
Ln r a13R2
R2
2D"12Ln a
1z
Finally, according toEq. (5),
E:
tp 2D"12
y1Ln zl z
:3
2S 31
So that the following new model is proposed extending properly Leblonds modelto low values ofz,
E:
tp
2D"12y1
Ln zl z:3
2S if z4 zl
2D"12y1
Ln z z:3
2S if z> zl
8>>>:
with zl y12D"12
43K
9K
It follows from this that the singularity obtained for z 0 in the original
Leblonds analysis clearly results from the unnecessary assumption H.2.
5. Application
Let us now come back to the experimental results presented byTaleb et al. (2001)
and which were already mentioned in Section 2.4. In these experiments, transfor-
mation induced plasticity was characterized by the difference between a free dilato-
metric test (no applied external stress) under prescribed cooling conditions and aTRIP test corresponding to a fixed external stress applied during the transformation
and with the same cooling conditions.
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surprising agreement with Leblonds model is observed when the threshold is chosen
equal to 0.03 (Leblond 0.03 inFig. 4).
When this threshold is taken equal to 0.01, the new model leads to lower trans-
formation plasticity value at the end of transformation which is closer to experi-mental result, however the latter seems still overestimated. The reason probably is
Fig. 4. Bainitic transformation under applied stress 24 MPa (about a quarter of the austenitic conven-
tional yield stress at the beginning of the transformation). Transformation induced plasticity (a) and
normalized transformation induced plasticity (b) versus the volume fraction of the formed bainite.
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that as discussed in Section 2.1 and by Leblond et al. (1989), S1=S is rather adecreasing function of z. The modified theoretical curve obtained by replacing S1 by
1z2 S(New model : S*(1z*z) inFig. 4) leads to a better agreement.More discrepancy between Leblonds model and the new one is expected for a
ferritic transformation in the same steel because ofzlis much higher.
6. Conclusion
A modified micromechanical model for Leblonds approach to TRIP has been
proposed, the main features of this model are:
(a) It allows to remove some assumptions made by Leblond et al. namely : elastic
behavior of the product phase, rigid plastic behavior of the parent phase,
(b) Leblonds model can be now properly extended to low z values,
(c) It leads to lower TRIP values at the end of transformation, which is in
agreement with experimental values.
This improved agreement is realized by accounting for both plastic and elastic
regions in the parent phase.
The analysis leads to the introduction of the dimensionless material parameter,
zl y1
2D"12
43K
9K
, which appears as an essential characteristic of transformation
induced plasticity.
Further experiments and more refined finite element micromechanical analyses
taking into account the recent developments on this subject, are needed and will be
performed in the future in order to confirm the proposed model.
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