Lakhdar Taleb; François Sidoroff -- A Micromechanical Modeling of the Greenwood–Johnson Mechanism...

8/12/2019 Lakhdar Taleb; Franois Sidoroff -- A Micromechanical Modeling of the GreenwoodJohnson Mechanism in Transfor

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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:

1822 L. Taleb, F. Sidoroff / International Journal of Plasticity 19 (2003) 18211842


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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

L. Taleb, F. Sidoroff / International Journal of Plasticity 19 (2003) 18211842 1823


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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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