Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 1, pp. 141-153, Warsaw 2017
DOI: 10.15632/jtam-pl.55.1.141

A SEMI-THEORETICAL MODEL FOR TRANSFORMATION PLASTICITY

OCCURRED DURING BAINITIC TRANSFORMATION

Mahmoud Yaakoubi, Fakhreddine Dammak

Laboratory of Mechanics, Modeling and Production (LA2MP), ENIS, Sfax, Tunisia

e-mail: mahmoud5yakoubi@gmail.com; fakhreddine.dammak@enis.rn.tn

This studyaddresses the taskofpredicting the transformationplasticity inducedduringpha-
se transformationof the 16MND5carbon steel fromaustenite tobainite under lowexternally
applied stress using a semi-theoreticalmodel based on the Greenwood-Johnsonmechanism.
Both models proposed by Leblond et al. (1989) and Taleb and Sidoroff (2003) sufficiently
describe the evolution of the TRansformation Induced Plasticity (TRIP) during continuous
cooling of the austenitic phase. Nevertheless, TRIP values predicted by these models unde-
restimatemeasured data through the first half of the transformation and overestimate them
through the second half. So, we propose in this paper amethod to improveTaleb’smodel in
order to remove discrepancies between theoretical and experimental results throughout the
whole transformation and obtain a better description of experimental data.

Keywords: bainitic transformation, transformation induced plasticity (TRIP), traction
loading

1. Introduction

In some thermo-mechanical manufacturing processes, especially heat treatment and welding,
phase transformations can occur and can affect significantly themechanical behavior and struc-
tural properties of quenched or welded steel parts. Indeed, when metallurgical transformations
occur under small external stress lower than the yield stress of the weaker phase (austenite), a
supplementary plastic strain is observed (Leblond et al., 1986, 1989; Fischer et al., 1998; Taleb
et al., 2001; Taleb and Sidoroff, 2003; Meftah et al., 2007; Hoang et al., 2008; Moumni et al.,
2011). This plastic strain increment is called TRansformation Induced Plasticity (TRIP) and
has a significant effect on the distribution of the residual stresses, distortions and mechanical
properties (Leblond et al., 1986, 1989; Taleb et al., 2004; Dan et al., 2008; Tahimi et al., 2012;
Deng andMurakawa, 2013; Song et al., 2014). In the literature, there are twomechanisms pro-
posed to explain the origin of the TRIP: Magee’s mechanism which is proposed for displacive
transformations and Greenwood-Johnson’s mechanism which is well suitable for the diffusio-
nal transformations (Meftah et al., 2007; Hoang et al., 2008). According to Magee, TRIP is
due to privileged orientation of martensitic plates during transformation in presence of exter-
nal stress.While Greenwood and Johnson supposed that accommodation between differences in
compactness and dilation coefficients of the parent and the product phase leads to apparition
of local dislocations in the vicinity of the interface between phases. When deviatoric stress is
applied, dislocations are oriented in the direction of the applied stresswhich induces transforma-
tion plasticity at themacroscopic scale. However, it was revealed through experimental analysis
that Magee’s mechanism was not dominant for low applied stresses; hence, it was admitted
that Magee’s mechanism might not be considered for carbon steels; however it is normally de-
emed for shape memory alloys (Moumni et al., 2011). In addition, it was illustrated that only
Greenwood-Johnson’s mechanismwas considered whenmodeling TRIP for both diffusional and
shear transformations (Taleb et al., 2001).



142 M. Yaakoubi, F. Dammak

It is well known thatmuchworks have been done in the last thirty years for bettermodeling
of the evolution of TRIP during phase transformations of steel alloys under different types of
loadings. So this additional strain increment originating fromphase transformation is accounted
in the development of constitutive behavior of a multiphasic material in order to perfectly
simulate thematerial response in continuummechanical computation.Theapproachesdescribing
the evolution of theTRIPduring phase transformations can be classified into phenomenological
models (Mohr and Jacquemin, 2008), micromechanics-basedmodels (Leblond et al., 1989; Taleb
and Sidoroff, 2003; Sun et al., 2009) and discrete dislocation-transformation model (Shi et al.,
2010).

In this paper, we focus only on themicromechanics-based model proposed by Leblond et al.
(1989) and improved later by Taleb and Sidoroff (2003). This theoretical model was established
by considering Greenwood-Johnson’s mechanism where an elementary volume of austenite ha-
ving spherical shape inwhich a spherical core ofα-phasewas growing.Thehomogeneity of strain
and stress fields in the transforming elementary volumewas assumed. So,micromechanical ana-
lysis permits establishing the theoretical model describing TRIP during phase transformation.

The purpose of this work is to present a summarization of the hypothesis and theoretical
development carried out by the authors to obtain their TRIPmodels, appraise their simulation
results, and improve someassumptions resulting then inabettermodelwhichpredicts effectually
the evolution of theTRIPduring phase transformation. Simulations obtained by the newmodel
will be comparedwith experimental results provided in the literature in order to investigate the
efficiency of our modeling.

2. Basic framework

We are interested in the theoretical model developed by Leblond et al. (1989) which is one of
the most widely used for practical applications and which is implemented in the finite element
codes such as SYSWELD and ASTER. This model was improved later by Taleb and Sidoroff
(2003).

Fig. 1. Geometry considered by Leblond to illustrate phase transformation of austenitic nuclei

Leblond’smodel is obtained fromamicromechanical analysis of stress and strain fieldswhich
evolve in an austenitic spherical nuclei occurring during continuous cooling. The growth of a
spherical product phase core is carried out in the center of austenitic spherical nuclei as shown in
Fig. 1.Rγ andRα are radii of parent andproductphase, respectively.One shouldnote thatRα is
nil before the beginning of transformation and it grows progressively during transformation until
it reaches Rγ. δRα is the radius increasing of the spherical phase α during a time increment δt.
Because of the positive volume change induced by the transformation, points located originally
at Rα+ δRα come to a new location Rα+ δRα + δu. As revealed by Leblond, the macroscopic
plastic strain rate generated during phase transformation under external loading depends only



A semi-theoretical model for transformation plasticity... 143

on the shape variation of each phase. Indeed, the author assumed through its hypothesis 1 that
the effect of local anisotropy due to a small difference between elastic parameters of each phase
is negligible with respect to the stresses and deformations due to volume differences between
phases α and γ. So, the general expression for the plastic strain rate is given by the following
equation

Ėp =(1−z)〈ε̇pγ〉Vγ +z〈ε̇pα〉Vα + ż〈∆εpγ→α〉F (2.1)

where z is the volume fraction of the product phase, ε̇pγ and ε̇
p
α are the microscopic plastic

strain rate tensors in phases γ and α, respectively, ∆εpγ→α is the deviatoric component of the
transformation strain tensor and 〈∆εpγ→α〉F expresses the average value of ∆εpγ→α along the
transformation front F.
Theauthorassumedthat theaverage ofdeviatoric transformation strain tensor on the frontF

is negligible since there is no favorite orientation. Subsequently, the last term in equation (2.1)
is omitted. The second hypothesis proposed by the author is that for small or moderately high
applied stresses, the austenitic phase is entirely plastic, but the α-phase remains elastic or its
plastic strain rate remains alwaysmuch smaller than that of the γ-phase. Afterward, the second
term in the right-hand side of equation (2.1) disappears, and this later is reduced to

Ėp =(1−z)〈ε̇pγ〉Vγ (2.2)

Given that the plastic strain in the parent phase is the sum of the classical plastic term due to
variation of the loading conditions and transformation induced plastic term corresponding to
the evolution of new phase fraction z, then

Ėp = Ėcp+ Ėtp (2.3)

Ėcp is the classical plastic termand the transformation induced plastic term iswritten as follows

Ėtp =(1−z)
〈δεpγ
δz

〉

Vγ
ż (2.4)

The third hypothesis used by the author is that material obeys the Von Mises criterion and
possesses an ideal-plastic flow. By assuming a uniform austenitic yield stress σyγ, equation (2.4)
can be transformed into

Ėtp =
3(1−z)
2σ
y
γ

〈δεeqγ
δz

sγ
〉

Vγ
ż (2.5)

where εeqγ is the von Mises equivalent microscopic plastic strain in the parent phase (phase γ),
σyγ and sγ are respectively the yield stress and the deviatoric tensor of themicroscopic stress in
this phase.
At this stage, Leblond assumed through hypothesis 4 and 5 that correlation between δεeqγ /δz

and sγ can be neglected and the average Sγ of sγ within the volume Vγ is equal to the overall
average S of s in the whole nuclei. Then

Ėtp =
3(1−z)
2σ
y
γ

〈δεeqγ
δz

〉

Vγ
Sż (2.6)

Using a spherical coordinate systemand considering apurely radial displacement, the solution of
themechanical problem is performed using the dynamic equilibrium equation in the continuous
mediums. Finally, it is found that

δεeqγ
δz
=
2∆εαγR

3
γ

r3
(2.7)



144 M. Yaakoubi, F. Dammak

So

Ėtp =−
3∆εαγ
σ
y
γ
ln(z)żS (2.8)

where ∆εαγ is the volume change that corresponds to phase transformation.
Because equation (2.8) includes a singularity at the beginning of the transformation (z =0),

the author proposed to cut off the TRIP below z =0.03 leading then to the following model

Ėtp =











0 if z ¬ 0.03

−
3

2
k ln(z)żS if z > 0.03

(2.9)

Fig. 2. Geometry considered by Taleb and Sidoroff (2003): γ-phase is composed by an outer elastic
layer Le around an inner plastic layer Lp

Taleb and Sidoroff (2003) developed their model by following the micromechanical scheme
assumed by Leblond to formulate its transformation plasticity kinetic model. They extended
the Leblond model by keeping all hypotheses except hypothesis 2. Afterwards, the behavior of
the austenitic phase has been considered elastoplastic with ideal plasticity. Indeed, according to
Taleb and Sidoroff (2003), the product phase remains elastic while the parent phase consists of
an outer elastic layer Le around an inner plastic layer Lp with an elastic-plastic boundary at
r = ξ where Rα ¬ ξ ¬ Rγ as shown in Fig. 2. The boundary between these layers increases
progressively during transformation until ξ becomes equal to Rγ. At this instant, the remainder
of the parent phase turns into completely plastic. By executing the solution of the mechanical
problem using the dynamic equilibrium equation in the continuous medium, Taleb and Sidoroff
(2003) found that

ξ = 3

√

2∆εγα
σ
y
γ

9Kµ

4µ+3K
Rα (2.10)

where K and µ are respectively the bulk and shear elastic moduli.
Finally, Taleb’s model assuming elastoplastic parent phase and extending Leblond’s one to

low values of z; is the following

Ėtp =



















−2∆εγα
σ
y
1

ln(zℓ)ż
3

2
S if z ¬ zℓ

−
2∆εγα
σ
y
1

ln(z)ż3
2
S if z > zℓ

(2.11)

with

zℓ =
σyγ
2∆εαγ

4µ+3K

9Kµ



A semi-theoretical model for transformation plasticity... 145

Experimental tests onbainitic transformation of the16MND5 steel under small applied stres-
ses were performed and their results were given by Taleb and Sidoroff (2003). These tests allow
comparison between simulation and experiment results. Indeed, theoretical and experimental
curves are presented in Fig. 3 providing the evolution of the TRIP against the product phase
fraction. So, a coincidence betweenTaleb’s andLeblond’smodels is observedwhen the threshold
of Leblond’s model is equal to 0.03. Theoretical prediction given by Taleb’s model that agrees
with Leblond’s forecast (Leblond 0.03) illustrates slow transformation plasticity kinetic during
the first half of the transformationwith respect to the experimental resultwhile the latter seems
overestimated at the end of the transformation. Therefore, the aim of the following Section is to
revise some assumptions made by the authors; that lead to formulate enhanced transformation
plasticity kinetics ensuring then a better congruence with the experimental curve through all
the transformation.

Fig. 3. Transformation plasticity evolution during bainitic transformation in the 16MND5 steel under
applied stress (24MPa) versus volume fraction of the formed bainite

3. Numerical procedure

Our newmodel will be developed basing on themicromechanical analysis presented above after
reviewing some assumptionsmade by the authors. Indeed, some hypothesis will bemore discus-
sed and improved leading thus to amore refinedmodel that better agrees with the experimental
results.
Hypothesis 4 suggested by the author which assumes that

〈δεeqγ
δz

sγ
〉

Vγ
=
〈δεeqγ
δz

〉

Vγ
〈sγ〉Vγ

is mathematically inaccurate because the integral of the product of two functions is different to
the product of their integrals. So, we suppose that the previous equation can be calibrated by
introducing a function m(z) as follows

〈δεeqγ
δz

sγ
〉

Vγ
= m(z)

〈δεeqγ
δz

〉

Vγ
〈sγ〉Vγ (3.1)

In order to have an idea about the evolution of the function m(z), let us consider Fig. 4a
which is available in Leblond et al. (1989) that points out the simulation of 〈(δεeqγ /δz)sγ〉Vγ and
〈δεeqγ /δz〉Vγ〈sγ〉Vγ versus z. Basing on this result, we remark that disagreement between the two



146 M. Yaakoubi, F. Dammak

curves is moderately small; thus we admit that the function m(z) is not varying much in the
interval [0,1]; so we can substitute equation (3.1) by

〈δεeqγ
δz

sγ
〉

Vγ
≈〈m(z)〉[0,1]

〈δεeqγ
δz

〉

Vγ
〈sγ〉Vγ (3.2)

Then hypothesis 4 is replaced by hypothesis 4’, thinking that

〈δεeqγ
δz

sγ
〉

Vγ
= m
〈δεeqγ
δz

〉

Vγ
〈sγ〉Vγ (3.3)

with m is the average of the function m(z) in the interval [0,1].

Fig. 4. Verification of: (a) hypothesis 4 and (b) hypothesis 5

Hypothesis 5 assuming that the average stress deviator in the parent phase is almost equal
to the overall average stress deviator (Sγ = S with Sγ = 〈sγ〉Vγ and S = 〈s〉V ) is not verified.
Indeed, Fig. 4b which is taken from Leblond et al. (1989) illustrates the simulation of Sγ and S
versus the product phase fraction z in the uniaxial case, and an important discrepancy between
them has been shown. More precisely, the curves in this figure prove that Sγ/S is a decreasing
function versus z. Afterwards, we propose that the function Sγ/S has a style of 1−zn defining
then hypothesis 5’. The new relationship between Sγ and S is the following

Sγ =(1−zn)S (3.4)

with n is a constant.
Replacing hypotheses 4 and 5 by hypotheses 4’ and 5’ respectively, one can obtain

〈δεeqγ
δz

sγ
〉

Vγ
= m(1−zn)

〈δεeqγ
δz

〉

Vγ
S = χ(z)

〈δεeqγ
δz

〉

Vγ
S (3.5)

with

χ(z)= m(1−zn) (3.6)

The functionχdepends on twoparametersm andn and should accomplishmore coincidence
between quantities 〈(δεeqγ /δz)sγ〉Vγ and 〈δεeqγ /δz〉VγS. Afterwards, the newmodel of transforma-
tion plasticity evolution is defined as follows

Ėtp(z)= χ(z)ψ(z)ż (3.7)

with

ψ(z) =



















−2∆εαγ
σ
y
γ
ln(zℓ)

3

2
S if z ¬ zℓ

−
2∆εαγ
σ
y
γ
ln(z)
3

2
S if z > zℓ

(3.8)



A semi-theoretical model for transformation plasticity... 147

with

zℓ =
σyγ
2∆εαγ

4µ+3K

9Kµ

According to the newmodel, Eq. (3.7), the transformation plasticity increment that corresponds
to a product phase increment produced during a time increment is the following

∆Etp(zj)= χ(zj)ψ(zj)(∆z)j = χ(zj)∆TPT(zj) (3.9)

where zj is obtained until j-th time increment by accounting from the beginning of bainitic

transformation zj =
∑i=j
i=1(∆z)i, (∆z)j is the increment of the product phase formed during the

j-th time increment. ∆TPT(zj) represents the transformation plasticity increment generated
during the j-th time increment according toTaleb’smodel. Indeed, the function χ(zj) is defined
as a quotient obtained by dividing ∆TPExp(zj) by ∆TPT(zj) for a non nil value of zj

χ(zj)=
∆TPExp(zj)

∆TPT(zj)
zj 6=0 (3.10)

where ∆TPExp(zj) designates the experimental value of the plasticity transformation incre-
ment developed during the j-th time increment. ∆TPExp(zj) and ∆TPT(zj) are determined
graphically from curves illustrated in Fig. 3. χ(1) is directly equal to zero from equation (3.6).
Subsequently, the curve characterizing the evolution of function χ against z is illustrated in
Fig. 5. One can remark then that the function χ which depends on m and n parameters is

Fig. 5. Evolution of the function χ versus z

decreasing versus z. The parameters m and n can be determined by evaluating the derivative of
the function χ for two different values z1 and z2 as follows

χ′(z1)=−mnzn−11
χ′(z2)=−mnzn−12

}

⇒







n =
ln
χ′(z1)
χ′(z2)

ln z1
z2

+1 m ≈ 1
2

( χ(z1)

1−zn1
+

χ(z2)

1−zn2

)

(3.11)

The estimation of the function χ′ at a given value z is accomplished by applying the following
formulation

χ′(z)=
χ(z+~)−χ(z −~)

2~
(3.12)

with ~ being a parameter of too small value (it is chosen equal to 0.02 in our case). χ(z + ~)
and χ(z − ~) are determined graphically from Fig. 5. It is found that χ′(0.2) = −1.98 and
χ′(0.7)=−0.96. So

n =0.405 m =1.93 (3.13)



148 M. Yaakoubi, F. Dammak

Finally, the function χ is determined

χ(z)= 1.93(1−z0.405) (3.14)

According to equations (3.7) and (3.8), thenewmodel forTRIPkinetics,which is formulated
to get an improved agreement with the experimental result, is defined by

Ėtp =



















−
2∆εαγ
σ
y
γ

χ(z)ln(zℓ)ż
3

2
S if z ¬ zℓ

−2∆εαγ
σ
y
γ

χ(z)ln(z)ż
3

2
S if z > zℓ

(3.15)

with

χ(z)= 1.93(1−z0.405) zℓ =
σyγ
2∆εαγ

4µ+3K

9Kµ

Now, the new model will be investigated through comparison between numerical simulations
andmeasured TRIP generated during bainitic transformation of 16MND5 steel specimens.

4. Experimental validation

In this Section, free dilatometry and TRIP tests carried out by Coret et al. (2002) are deemed.
Specimens were 16MND5 steel tubular cylinders having inner and outer diameters equal to
22.4mmand 23.4mm, respectively. The feeble thickness of the specimen enables obtaining low a
radial thermal gradient and, subsequently, homogenous stress and strains fields. Specimenswere
austenitized by induction current at 900◦C for 30s and then cooled by injecting argon inside.
The heating and cooling rate were 10◦C/s and −3◦C/s, respectively. We consider in this paper
three experimental results of dilatometric tests provided by Coret et al. (2002). The first was
the free dilatometric test while the second and the thirdwere theTRIPdilatometric tests under
uniaxial traction loading equal to 30MPaand60MPa, respectively. Traction loadingwas applied
during the cooling stage when temperature reached 600◦C (somewhat before the beginning of
bainitic transformation at 560◦C) and released at the end of the test. In this study, dilatation
curves obtained by these tests were adjusted in such away that there was no difference between
them before reaching temperature 600◦C. In addition, only difference due to elastic strains was
considered for temperature between 600◦C and 560◦C.This procedure takes away experimental
uncertainty and allows getting reliable results. Dilatometric curves are plotted together in Fig. 6
for the temperature range 700◦C-390◦C which includes bainitic transformation during cooling.
Thedifference betweenTRIPcurves and free dilatometric curve is due to elastic strain caused by
the external loading and essentially to TRIP generated by phase transformation under external
stress.
The total strain occurred during a TRIP test is supposed to be the sum of thermo-

-metallurgical strain, elastic strain due to external loading and plasticity transformation strain
generated through phase transformation (Taleb et al., 2001; Coret et al., 2002; Dutta et al.,
2013). Then

εpt(T)= εtot(T)−εthm(T)−εe(T) (4.1)

with εtot(T) being the total strain issued fromTRIP curve. εthm(T) corresponds to strain obta-
ined from the free dilatometric curve and εe(T) is the elastic strain. It is given by

εe(T)=
σ

E(T)
(4.2)

where σ is the external applied stress and E(T) is the thermal dependent Youngmodulus.



A semi-theoretical model for transformation plasticity... 149

Fig. 6. Dilatometric curves obtained during bainitic transformation under different tension loadings

The evolution of TRIP against the temperature is estimated for two TRIP tests assuming
that there is noTRIP evolution before starting of the bainitic transformation. For the 16MND5
steel, when the austenite transforms under cooling rate equal to −3◦C/s, the obtained phase
proportions are respectively 87% of bainite and 13% of martensite (Moumni et al., 2011). Sub-
sequently, wewill not take into account of theTRIP occurred below 390◦Cwhen estimating the
evolution of TRIP against temperature because we consider in this study only TRIP occurred
through bainitic transformation. Experimental results for the evolution of TRIP according to
temperature during cooling are illustrated in Table 1. Now, these experimental results will be
used to evaluate numerical simulations performed in the following Section.

Table 1.TRIP obtained from dilatometric tests

Temperature [◦C] 560 540 520 500 480 460 440 420 390

εtp [%], σ =30MPa 0 0.052 0.118 0.152 0.169 0.182 0.188 0.192 0.194

εtp [%], σ =60MPa 0 0.099 0.212 0.290 0.335 0.359 0.378 0.390 0.402

5. Simulations and discussions

The simulation of the quenching process is performed through calculation of temperature evo-
lution in the specimen during treatment. It is coupled with calculation of metallurgical phases
distributions followed by the solution of the mechanical problem by the finite element method.
The ABAQUS software linked to many subroutines is used to simulate heat treatment phase
transformation histories and strain fields asmentioned byYaakoubi et al. (2013b). The thermal
cycle recorded by Coret et al. (2002) is used as the boundary condition to carry out the simu-
lation (Fig. 7). Phase transformation kinetics is modeled by using the JMAK formalism (Barbe
et al., 2008; Yaakoubi et al., 2013a). Thermo-physical properties of the material are available
in Moumni et al. (2011). The analysis is realized two times for each loading case by using in
the first time Taleb’s model to predict TRIP evolution and using the new model in the second
time. It is found that the maximum value of bainite proportion obtained by simulation is equal
to 0.883, which is very close to value (0.87) obtained byMoumni et al. (2011).

The comparison between numerical and measured TRIP (εtp = f(T)) appears in Fig. 8. It
is evident that the transformation plasticity predicted by the newmodel is considerably better
thanTaleb’s predictions for the loading case of 30MPa.However, for the loading case of 60MPa,
newpredictions are not adequate because they showoverestimations through the first half of the
transformation and underestimations through the second half. We observe that as the tension



150 M. Yaakoubi, F. Dammak

Fig. 7. Thermal cycle used as the boundary condition to carry out the simulation

Fig. 8. Transformation plasticity evolution versus temperature during transformation plasticity tests
under (a) 30MPa, (b) 60MPa. Comparison between experimental results and predictions of Taleb’s and

the newmodel

load becomes larger, the discrepancies of the new model become increasingly significant. This
fact is explained by that the function χ which appears in the new model is identified for the
loading case of 24MPa. So, we think that this newmodel can be refined bymaking parameters
m and n (those define the function χ) dependent on the applied stress σ. Indeed, analysis of the
function χ shows that the increasing of the parameter m increases the predictions of TRIP at
the beginning of transformation, and that the increasing of the parameter n increases them at
the end of the transformation and vice versa. The fitting of numerical simulations conducted to
define parameters m and n versus σ is as follows

m =
10√
σ+3

n =
σ

60
(5.1)

Subsequently, the function χ becomes

χ(z,σ) =
10√
σ+3

(1−z)
σ

60 (5.2)

Then, the final newmodel that describes the evolution of TRIP during bainitic transformation
of the 16MND5 steel under low applied stress is

Ėtp =



















−2∆εαγ
σ
y
γ

χ(z,σ)ln(zℓ)ż
3

2
S if z ¬ zℓ

−
2∆εαγ
σ
y
γ

χ(z,σ)ln(z)ż
3

2
S if z > zℓ

(5.3)



A semi-theoretical model for transformation plasticity... 151

with

χ(z,σ) =
10√
σ+3

(1−z)
σ

60 zℓ =
σyγ
2∆εαγ

4µ+3K

9Kµ

The results of the new refined model and experimental results are illustrated by Fig. 9.
It has been found that new predictions achieved by the refined model, Eq. (4.5), are more
convenient with experiments than those obtained from the preliminary version of thismodel for
all considerate cases of the applied stress. Indeed, the refined new model is capable to capture
not only the fast transformation plasticity observed experimentally at the beginning of the
transformation but also the deceleration of this plasticity rate during the second half of the
transformation; a profit which cannot be accomplished by the previous models.

Fig. 9. Transformation plasticity evolution versus temperature during transformation plasticity tests
under (a) 30MPa, (b) 60MPa. Comparison between experimental results and predictions of Taleb’s and

new refinedmodels

6. Conclusion

In this study, a new (semi-theoretical) model to predict TRIP induced during bainitic trans-
formation under an external tension loading is developed by upgrading the existing models in
the literature. A way to improve these models would be the revising some simplifying assump-
tions suggested by authors during analysis of themicromechanical approach. Subsequently, new
substitute assumptions are reasonably suggested leading then to multiplying Taleb’s formula
by an appropriate function that is numerically established. The new semi-theoretical model for
predicting TRIP produced during bainitic transformation under low tension stress is finally
established.
The investigation of the accuracy of the newmodel is performed in the light of comparison

between numerical simulations and experimental results provided in the literature. It has been
found that predictions obtained by the refined new model are significantly better than Taleb’s
forecasts. Furthermore, this new model leads not only to elevate TRIP values at the beginning
of the transformation but also to lower them during the second half of the transformation; a
result that is experimentally perceived and cannot be described by priormodels. Further studies
are needed in order to extend this newmodel for the case of high applied stress and other kinds
of transformations.

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Manuscript received October 9, 2015; accepted for print June 15, 2016