UNIFIED THEORY OF TRANSFORMATION PLASTICITY AND THE EFFECT ON QUENCHING SIMULATION

Tatsuo INOUE and Hideto WAKAMATSU Department of Mechanical Systems Engineering, Fukuyama University,

Gakuen-cho 1, Fukuyama, Japan E-mail: inoue@fume.fukuyama-u.ac.jp, Internet: http://www.fukuyama-

u.ac.jp/gakubu/mecha/staff/inoue/ index.htm fsb.hr/hdtoip

Abstract A phenomenological mechanism of transformation plasticity (TP) is discussed, in the first part of the paper, why it takes place under a stress level even lower than the characteristic yield stress of mother phase: This is princi pally based on the difference in thermal expansion coefficient in both phases. Bearing in mind that it i s also a kind of plastic strain, a unified plastic flow theory is derived by intr oducing the effect of pr ogressing new phase into the yi eld function of stre ss, temperature and plasticity related parameters. Experimental results of the TP coefficient will be shown in the following section for so me stee ls under tens ile and compressiv e st ress. As examples of the application of the theory and the data, metallo-thermo-mechanical simulation for a quenched blank gear wheel is executed to demonstrate the effect of TP. Keywords transformation plasticity, unified transformation-thermoplasticity theory, metallo-thermo- mechanics, quenching

1. Introduction Most constitutive laws for transform ation plasticity, or T P [Greenwood, Johnson, 1965], [Barbe, Quer, Taleb, Cursi, 2005], [Fischer , Reisener, Werner, Ta naka, 2000], [Leblond, 1989], whic h is known to contribute a drastic effect on t he sim ulation of some practical engineering courses of therm o-mechanical processes, hav e been treated to be independent of ordinal thermo-plasticity. Considering that the m echanisms for both plastic s trains are essentially with no diff erence f rom m etallurgical viewpo int, the con stitutive eq uation f or transformation plastic strain r ate is expected to be described in relati on with plas ticity theory [Inoue, 2007a], [Inoue, 2007b], [Inoue, 2008]. This paper motivates to propose a mechanism for TP and to f ormulate unified constitutive equation of transformation and therm oplasticity by introducing the effect of incr easing phase in m other phase. Application of the theory is made to numerical simulation of quenching pro cess of a blank gear wheel to dem onstrate the effect of TP, by use of the TP coefficients identified by experiments..

2. Phenomenological Mechanism of Transformation Plasticity As mentioned above, yielding of mother phase (or austenite in this case ) is considered to occur due to progressive new phase (such as pearlite or martensite. See Fig.1) even under small tensile stress below yiel d stress of t he mother phase. De note the vol ume fractions of ne w and mother phases nξ and mξ . (In more general case to be discussed in the next section, the volume fraction of new phase will be denoted by ξ n ). When b oth elem ents ar e supposed to be connected

parallely as

1

indicated in Fig.2 being subj ected to applied stress σ , then the follo wing relations hold [Inoue, Tanaka, 2007] when considering the phase change dilatation nβξ ;

Elastic-plastic constitutive equation;

A-P transition

Dilatation due to

phase transformation

βA-P

Temperature T

Austenite αA

Pearlite αP

Tota

l st

rain

ε

Austenite αA

Martensite αM

Phase changedilatation βA-M

Temperature T

Strain

ε

Austenite αA

Martensite αM

Phase changedilatation βA-M Austenite αA

Martensite αM

Phase changedilatation βA-M

Temperature T

Strain

ε

(a) Austenite-pearlite. (b) Austenite-martensite. Fig.1 Schematic illustration of phase transformation.

ｍ

ｎ

ｍ

ｎ

ｍ

ｎ

ｍ

ｎ

ｍ

ｎ

Thermal stress

00

m

n

σσ

><

Tensile stress in ｍ is larger than in n,

approaching to the yield stress.

Tensile stress

(a)

(b)

(c)

(d)

(e)

1 mm m

m m

d dE H

⎛ ⎞= + +⎜ ⎟⎝ ⎠

δε σ mdTα , in mother phase

1 nn n n

n n

d d dTE H

δ dε σ α β⎛ ⎞

= + + +⎜ ⎟⎝ ⎠

ξ , in new phase, (1)

with mδ and nδ representing yield condition

1, 1,,

0, 0,m ms n n

m nm ms n n

if ifif if

≥⎧ ⎧= =⎨ ⎨≥ ≥⎩ ⎩

s

s

≥σ σ σδ δ

σσ σ σ σ

, (2)

Stress equilibrium condition;

in ｍ decreases to

cease the plastic deformation.

Thermal contraction by cooling

α α>

Model

Cooled, separately

Cooled, connected Tension in ｍ

Compression in ｎ

Tensile stress applied

Further development

of ｎ phase

Fig.2 A model for transformation plasticity.

m n

2

m m n n= +σ σ ξ σ ξ with 1, or and 1m n n mξ ξ ξ ξ ξ ξ+ = ≡ ≡ −

n

, (3)

Compatibility condition;

m =ε ε . (4)

Here, σ , ε , and E α with suffices m and n denote the stress, strain, Young’ modulus and linear expansion coeffici ent for mother a nd ne w phase , β is t he expansion coeffi cient due t o phase change, a nd T and sT respectively st and for var ying t emperature a nd tra nsformation start temperature.

Figures 3(a) and (b) depict the results of temperature dependence of both stresses in mother and new phase with volume f raction of new phas e respectively f or pearlite and martensite transformation from aus tenite. Th e figures show that the st ress in the mother phase i ncreases with progressing volume fraction to reach the estimated yield stress of pearlite psσ = 80MPa and for martensit e msσ =100MPa, larger than initially applied stress 40=σ MPa with E=150GPa,

mα =2.15x10-5 K-1 nα =1.61x10-5 K-1 and sT =700℃ for pearlite transformation and nα =2.0x10-

5 K-1 and sT =420℃ for martensite reaction.

Temperature T , ℃

Volu

me

fract

ion ξ

Str

ess

σ ,

M

Pa

200 250 300 350 400 4200

200

400

600

800

1000

0.0

0.2

0.4

0.6

0.8

1.0Volume fraction

Austenite

Volume fraction

Martensite

Str

ess

σ ,

M

Pa

Volu

me

fract

ion

ξ

600 620 640 660 680 7000

100

200

300

400

500

0.0

0.2

0.4

0.6

0.8

1.0

Temperature T , ℃

Austenite

Volume fraction

Pearlite

Fig.3 Stress variation in mother and new phase when applied tensile stress. (a) Austeite-pearlite reaction. (b) Austenite-martensite reaction.

Vol

um

e fr

act

ion

ξ

Str

ess

σ ,

M

Pa

600 620 640 660 680 700-100

0

100

200

300

400

500

0.0

0.2

0.4

0.6

0.8

1.0

Temperature T , ℃

Volume fraction

Austenite

Volume fraction Pearlite

The kinetic relation of the transformations are

, k n (5) ( ) ( )1 exp np sk⎡ ⎤= − − −⎣ ⎦ξ τ τ τ 62 10 , 4.2 , 352sτ

−= × = =

and for martensite reaction

Vol

ume

fract

ion

ξ

1000 1.0

200 250 300 350 400 420-200

0

200

400

600

800

0.0

0.2

0.4

0.6

0.8

Martensite

Str

ess

σ ,

M

Pa

Temperature T , ℃

Volume fraction

Austenite

Volume fraction

(a) Austenite-pearlite reaction. (b) Austenite-martensite reaction. Fig.4 Stress variation in mother and new phase when applied compressive stress.

3

( ) 1 exp[ )( ] m sT T Tξ φ= − − , 0.0219=φ (6)

in terms of the units; stress in MPa, time in sec, and temperature in ℃ with dimensionless strain and volume f raction. It is seen f rom the f igures that the stress in th e mother phase increasi ng with pr ogressing volume ftr action of new phase reaches t he esti mated yi eld st ress of pearl ite

psσ =80MPa and martensite msσ = 100MPa. On the contrary to the previous case of tensile applied stress, Fig.4(a) and (b) show the results of progressive stress in mother and new phases when compressive stress is applied. In t he cooling process, pha se tr ansformation str ain i s gener ally i ncreased si nce m > nα α , then the s tresses in both phases tend to be positi ve in the final stage of phase chan ge. Nevertheless, com pressive stress in ne w phase is rather increased i n the first stage, which would be possible to reach t he compressive yield stress.

3. Unified Transformation-Thermo-Mechanical Plasticity

In order to for mulate a constitutive equation of a body under phase tran sformation, we assume that the material point focused is composed of kinds of phases, which include all phases with the volume f raction

NIξ ( )1, 2,3,....,I N= as is shown i n Fig.5( a) and that t he mechanical and

thermophysical property χ is represented by the mixture law [Bowen, 1976] such that

1

N

I II =

=∑χ ξ χ , with 1

1N

II =

=∑ξ , (7)

Stress state related to the yielding of the I-th phase (say, mother phase) is assumed to be affected by othe r phases ( new phase) with the volu me fraction J (ζ )1, 2,3,....,J M= as i ndicated i n Fig.5(b). Then, the plasticity of the I-th phase is controlled by the yield function in the form,

( , , , , )pI I ij Iij I JF F Tσ ε κ ζ= ， ( 1, 2,...., ; 1, 2,...., )I N J M= = . (8)

Here, ,ij Tσ and Iκ are r espectively st and for uni form stress and tem perature and plasti c hardening parameter.

Applying the consistency relation and the normality law, we have the form of plastic strain rate of the I-th phase as

1

ˆ [ ]M

p I I I IIij I I kl J

Jij kl J ij

F F F FG TT

ε σ IFζσ σ ζ=

⎛ ⎞σ

∂ ∂ ∂ ∂ ∂= Λ = + +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

∑ , (9)

with the hardening modulus G . ˆI

2 ξ

N ξ

4 ξ 1

ξ

3 ξ

•

•

(b) Some phases associating plastic deformation.

1ζ

2ζ3ζ

Mζ • •

(a) All phases Fig.5 Volume fraction of phases consisting a material point.

4

The first term in Eq. (9) is the ordinal thermo-mechanical plastic strain rate in the I-th phase, which is summ ation of mechanical and thermal parts. It is noted th at the second term in Eq .(9) ccurring in th e I-th phase i s ori ginated from the progre ssing the new J-th phase Jζ , which is found to be so-called transformation plastic strain rate

1

ˆM

tp IIij I J

J

I

J ij

FG=

⎛ ⎞ F∂ ∂= ⎜ ⎟∂ ∂⎝ ⎠

∑ε ζζ σ

. (10)

Adopting the mixture law, we finally have the global strain rate in the form

e th mij ij ij ij ij

pε ε ε ε ε= + + + . (11)

Attention is focused on the unified plastic strain rate

1 1 1

ˆ [ ]N N N

p p I I Iij I Iij I I kl I I J

I I Nkl J ij

IF F FG TT

ε ξ ε ξ σ ξ ξ ζ Fσ ζ σ= = =

⎛ ⎞ ⎛ ⎞∂ ∂ ∂= = + +⎜ ⎟ ⎜

∂⎟∂ ∂ ∂⎝ ⎠ ⎝

∑ ∑ ∑ ∂⎠ , (12)

As the sum of thermo-mechanical and transformation plastic parts. The re sult indicates that both pl astic strai n rat es relat ed t o ther mo-mechanical and phas e tra nsformation e ffect i s automatically derived from yield function in the form of Eq.(9) and that the Eq.(12) is the unified plastic strain rate. Now, consi der a speci al case when uni axial str ess σ is applied. Then we ha ve the trans formation plastic strain rate,

1 1

1N Mtp IJ J

II JI I

HH

ζ ζε ξ σσ= =

⎡ ⎤= ⎢ ⎥′ ⎣ ⎦∑ ∑ (13)

Here, haardening parameters are defined with the flow stress of the I-th phase Iσ ,

II p

I

H ∂′ =∂σε

: Strain hardening parameter of the I-th phase (14)

IIJ

J

H ζ σζ∂

=∂

: Dependence of flow stress in the I-th phase on J-th phase (15)

And the constitutive equation for TP strain in case between the mother and new phase reads

1 1tp mm m mn n nm m nn nm n

m m m n n n

H H H HH H

ζ ζ ζ ζζ ζ ζ ζε ξ σ ξ σσ σ σ σ

⎡ ⎤ ⎡⎛ ⎞ ⎛= + + +⎢ ⎥ ⎢⎜ ⎟ ⎜′ ′⎢ ⎥ ⎢⎝ ⎠ ⎝⎣ ⎦ ⎣

⎤⎞⎥⎟⎥⎠ ⎦

, (16)

with the flow stress , m nσ σ of mother and new phase. Since the effect of structure on flow stress is possible;

. (17) 0, and 0mn mm nm nnH H Hζ ζ ζ≠ = = H ζ =And putting

1 , and m n nξ ξ ξ ζ ξ= − ≡ ≡ , (18) then we have

( ) ( )1 1 3 1tp mn

m m

H KH

ζ ξε ξ σ ξ σσ

= − ≡′

− . (19)

This is the well known formula of Greenwood-Johnson for TP [Greenwood, Johnson, 1965].

5

4. Application to Strain Response for Stress-Temperature Variation

The theory developed is now applied to some cases under varying stress and temperature. Total strain in such cases of varying temperature is expressed as

( )

( ) ( )( )

0 1 d 3 1 d

s

e th m tp

T T

M NT

T

T KE T

= + + +

∂⎧ ⎫⎡ ⎤= + − + + + −⎨ ⎬⎣ ⎦ ∂⎩ ⎭∫ ∫ξ

ε ε ε ε ε

σ ξα ξ α ξ β ξ σ ξ (20)

The f irst case is to draw so -called temperat ure-elongation di agram. Figure 6 de picts the experimental diagram during pearli te tran sformation of a Cr-M o steel (SCM 420) [Inoue , Wakamatsu, 2009] , which show th e strong stress de pendence. Re sult of numer ical calculation of Eq.(20) is plotted as temperature-elongation diagram is possible to be represented in Fig.7 for paerlite reaction. Here, Eq s. (5 ) an d (6 ) are em ployed f or d iffusional an d m artensite transformation wit h the data of MATEQ [I noue, Okamura, 2006], a mat erial dat abase accumulated by our gr oup of Ma terials Database, JSMS. The coeffi cient of transfor mation plasticity ( tpK =σ ε =9x10 1/MPa-5 ) for the steel is adopted by our experiment [Ohtsuka, Inoue, 2005].

-0.5

-0.3

0.0

0.3

0.5

500 600 700 800

Temperature T ,℃

Str

ain ε

,%

40MPa

30

20

100

-5-10-15

-20-0.5

-0.3

0.0

0.3

0.5

500 600 700 800Temperature T , ℃

Str

ain

ε,

%

40MPa

30

20

10

0

-5

-10-15

-20

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

200 300 400 500 600 700 800

40302010

0-5

-10-15-20

Temperature T , ℃

0.0

-0.1

-0.2

-0.3

-0.4

-0.5

-0.6

Str

ain

ε

，％

σ= MPa

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

200 300 400 500 600 700 800

40302010

0-5

-10-15-20

Temperature T , ℃

0.0

-0.1

-0.2

-0.3

-0.4

-0.5

-0.6

Str

ain

ε

，％

σ= MPa

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

200 300 400 500 600 700 800

40302010

0-5

-10-15-20

Temperature T , ℃

0.0

-0.1

-0.2

-0.3

-0.4

-0.5

-0.6

Str

ain

ε

，％

-0.006

-0.005

-0.004

-0.003

-0.002

-0.001

0

200 300 400 500 600 700 800

40302010

0-5

-10-15-20

Temperature T , ℃

0.0

-0.1

-0.2

-0.3

-0.4

-0.5

-0.6

Str

ain

ε

，％

σ= MPa

Fig.6 Experimental temperature-elongation diagrams during pearlite transformation.

Fig.7 Simulated temperature-elongation diagram.

6

5. Effect of Tr ansformation Pl asticity on Metallo-t hermo-mechanical Simulation of Quenching Process

By e mploying t he e xperimentally d etermined data f or the trans formation plasticity coefficient in the previous section, metallo-thermo-mechanical simulation is carried out for a quenching process of a blank gear wheel. Use is made of a code COSMAP developed by Ju and the present author[ ].

The wheel is m ade of Chromium steel (SCr420) with outer and inner diameter of 34 and 24 mm, and the height of 30 mm. The work is quenched by an oil, and the data for heat transfer coefficient depending on the metal surface temperature is adopted from Japanese cooperation group.

Some results of the simulation are represented in Figs.8-11. Here, attention is focused on the effect on transformation plasticity. Figure 8 pl ots the distribution of martensite fraction after quenching process along the line A in the sm all figure. Sim ilar situation is al so seen in the fraction in entire region.

On the contrary to the structure distribution, ther e a re quite larg e difference in stress distribution. Figure 9 (a) duplicates the tangential residual stress pattern as well as the distortion in the cross section of the work when considering the TP effect, while Fig. (b) is the case without the effect. Tangential and axia l stress distribution along the line A is shown in Fig. 10, and the difference in tangential stress along the line A and B is presented in Fig. 11.

From the results of the simulation, the TP is known to affect drastically the induced stress, but not so much on the progressing structure.

0 .0

0 .2

0 .4

0 .6

0 .8

1 .0

7 .5 8 8 .5 9 9 .5Radius r , mm

Volu

me

fract

ion

of m

arte

nsite

　ξ

ｍ

W i th TPWithou t TP

MPa

(a) With TP effect (b) Without TP effect

φ24

φ34

φ15

-900

-600

-300

0

300

600

7 .5 8 8 .5 9 9 .5

Radius r , mm

Resi

dual

stre

ss σ

θ,　

MPa

Wi th TP, a l ong l i ne A

a l ong l i ne B

Wi thout TP, a l ong l i ne A

a l ong l i ne B

-600

-400

-200

0

400

600

800

0 5 10 15 20 25 30

Distance from bottom z , mm

Resid

ual

str

eθ

,σｚ ,

MP

a

Tangen t ial , with TP　　　　　 wi thou t TPAxial , wi th TP wi thou t TP

0

20

ss σ

30

20

Fig.8 Martensite fraction along line B. Fig.9 Difference in tangential stress and distortion.

20

A

CB

Fig.10 Tangential and axial residual stress Fig.11 Difference of tangential residual stress depending on TP effect along line C. along lines A and B.

7

6. Concluding Remarks A discussio n on the mechanism fr om ther mo-mechanical viewpoint is carried out, and the constitutive law is d erived from unif ied thermo-mechanical-transformation plastic ity theory. Application of the theory is made to simulate a quenched gear wheel to demonstrate the effect of TP.

Acknowledgement

The author is indeb ted to e xpress his gratitude to IMS-Japa n, NEDO a nd METI for theirr financial supports to this project.

References

Barbe, F. , Quer, R. , Taleb, L . and Cursi, E,S., FE de termination of the ef fective TRIP during diffusive transformation in a volume with randomly positioned nuclei, Proceedings of 1st Int. Conf. on Distortion Engineering, Bremen, 2005, pp.149-148

Bowen, R.M., Theory of Mixture, A.C. Eringen, ed., Continuum Physics (Academic Press, New York, Vol. 3,1976, pp. 2-129.

Fischer, F.D., Reisner, G.E., Werner; Tanaka, K., Gailletand and G.; Antr etter, G., A new vie w on transformation induced plasticity, Int. J. plasticity, Vol. 16, 2000, pp.723-740.

Greenwood, G.W. and Johnson, R.H., The deform ation of metals under sm all stresses during phase transformations, Proc. Roy. Soc., Vol. 283A, 1965, pp.403-422.

Inoue, T. and Oka mura, K., Material database for sim ulation of metallo-therm o-mechanical fields, Proceedings of 5th International Symposium on Quenching and Distortion Control, ASM, 2000, pp.753-760.

Inoue, T., U nified transformation-thermoplasticity and the application (in Japanese), J. Soc. Materials Science, Japan, Vol.56, 2007, pp.354-358.

Inoue, T ., A phenom enological mechanism of tr ansformation plastic ity and the unified constitutive equation of transformation-thermo-mechanical plasticity, Proceedings of Asia Pacific Conference on Computational Mechanics, Kyoto, 2007, CD-published.

Inoue, T., Tanaka, T., Ju, D.-Y. and Mukai, R ., Transformation plasticity and the effect on quenching process sim ulation, Key Engin eering Materials, Vol.345-346, 2007, pp.915-919.

Inoue, T., On phenomenological m echanism of transform ation plasticity and inelastic behavior of a steel subjected to varying tem perature and stress --- Application of unified transformation-thermoplasticity The ory ( in Jap anese), J. So c. Materials Science, Japan, Vol 57, 2008, pp.255-230.

Ju D. Y. and Inoue. T.; Simulation and its experimental verification by the material process CAE code CO SMAP, P roceedings of 2nd International Co nference on Dist ortion Engi neering, Bremen (2008), pp.441-450.

Leblond, J.B .; Mathematical m odelling of transform ation plasticity in st eels, II Coupling with strain hardening phenomena, Int. J. Plasticity, Vol.5, 1989, pp.1371-1391.

Magee, C.L., The Nucleation of martensite, Ch.3, ASM, 1968.

Otsuka, T., Wakasu, Y. and Inoue, T., A simple identification of transformation plastic behavior and som e data f or heat treati ng m aterials, Int . J. Mat . an d P roduct Tech., V ol.4, 2005, pp.298-311.

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