Stress-induced transformation in a carburized steel—Experiments and analysis

Acta metall, mater. Vol. 40, No. 9, pp. 2257-2268, 1992 Printed in Great Britain. All rights reserved

0956-7151/92 $5.00 + 0.00 Copyright ~: 1992 Pergamon Press Ltd

STRESS-INDUCED T R A N S F O R M A T I O N IN A CARBURIZED STEEL--EXPERIMENTS A N D ANALYSIS R. W. N E U t and H U S E Y I N S E H I T O G L U ~ : Department of Mechanical and Industrial Engineering, University of Illinois, 1206 West Green Street, Urbana, IL 61801, U.S.A. (Received 24June 1991; in revised form 17January 1992) Akstract--Stress-induced transformation of austenite to martensite was studied experimentally and numerically to gain an understanding of the increase in transformation strain with increasing tensile stress and decreasing temperature. In addition, the anisotropy of the transformation strains was determined both experimentally and numerically. The material chosen for the study was a carburized 4320 steel with 35% retained austenite content. Monotonic and cyclic experiments were conducted in the range from - 6 0 to 150°C. At 22°C, the volumetric transformation strain reached 0.006 at fracture in the uniaxial tensile test and 0.004 in the torsion test. Numerical calculations of the volumetric transformation strain and the anisotropy of the transformation strains were obtained with a modified Eshelby model where planes favorably oriented gradually transformed as stress was applied. The analysis predicted the experimentally observed transformation strains under both uniaxial and torsional loadings. The model also predicted the anisotropy of the transformation strains that was observed in experiments. R@sum~-Une 6tude, exp6rimentale et num~rique, de la transformation aust~nite --~ martensite induite par contrainte a &6 men6e pour comprendre l'accroissement de la d&ormation de transformation lorsque la contrainte de traction augmente et lorsque la temperature d6croit. On a d~termin6 en outre, exp6rimentalement et par des calculs num6riques, l'anisotropie des d6formations de transformation. Un acier 4320 au carbone ~ 35% d'aust6nite r6siduelle a 6t6 choisi comme mat6riau d'6tude. Des essais de d6formations monotone et cyclique ont 6t~ r6alis~s entre - 6 0 et + 150°C. A 22°C, la dbformation de transformation en volume atteint 0,006 ~ la rupture en traction uniaxiale et 0,004 ~i la rupture en torsion. Les calculs num6riques de la d6formation de transformation en volume et de l'anisotropie des d&ormations de transformation ont 6t~ obtenus ~ipartir d'un module d'Eshelby modifi~ o~ les plans favorablement orient,s se transforment au fur et fi mesure que la contrainte est appliqu6e. Cette analyse pr~dit les d&ormations de transformation observ6es exp~rimentalement aussi bien sous charge uniaxiale que sous charge de torsion. Ce module pr6dit 6galement l'anisotropie des d6formations de trasformation que l'on observe exp6rimentalement.

Zusammenfassung--Die spannungsinduzierte Umwandlung des Austenit in Martensit wird experimentell und numerisch studiert, um zu einem besseren Verst~indnis des Anstiegs der Umwandlungsverzerrung mit zunehmender Zugspannung und abnehmender Temperatur zu gelangen. AuBerdem wird die Anisotropie der Umwandlungsverzerrungen experimentell und numerisch bestimmt. Hierzu wird Kohlenstoffstahl 4320 mit 35% Restaustenit verwendet. Monotone und zyklische Verformung wird im Temperaturbereich zwischen - 6 0 und 150°C durchgefiihrt. Bei 22°C erreicht die volumetrische Umwandlungsverzerrung beim Bruch im einachsigen Zugversuch 0,006, in Torsion 0,004. Numerische Berechnungen der volumetrischen Umwandlungsverzerrung und der Anisotropie der Umwandlungsverzerrungen ergeben sich aus einem modifizierten Eshelby-Modell, in dem gfinstig orientierte Ebenen sich langsam umwandeln, wenn die Spannung angelegt wird. Die Analyse sagt die experimentell beobachteten Umwandlungsverzerrungen sowohl bei einachsiger wie auch Torsionsbelastung voraus. Das Modell sagt auch die Anisotropie der im Experiment beobachteten Umwandlungsverzerrungen voraus.

BACKGROUND U n d e r a n imposed stress, retained austenite transforms to martensite a n d there is a n associated

tPresent address: Systran Corp., Wright Laboratory Materials Directorate, WL/MLLN, Wright-Patterson AFB, OH 45433-6533, U.S.A. :[:Present address: Program Director of Mechanics and Materials, National Science Foundation, Room 1108, 1800 G St NW, Washington, DC 20550, U.S.A.

increase in volume o f a b o u t 4 % [1]. T r a n s f o r m a t i o n of austenite to martensite is a solid state structural change which is displacive, diffusionless, a n d domin a t e d by the strain energy arising from shear-like displacements [2]. R i c h m a n a n d L a n d g r a f [3] conducted uniaxial fatigue tests on a carburized 4027 steel at r o o m temperature. They f o u n d t h a t a b o u t 5 0 % o f the initial retained austenite c o n t e n t transformed d u r i n g the test. In a n o t h e r study, a q u e n c h e d a n d t e m p e r e d 0 . 1 7 % C - 2 ! % N i steel was tested u n d e r

2257

2258

NEU and HUSEYIN SEHITOGLU: STRESS-INDUCED TRANSFORMATION IN STEEL

very large strain amplitudes (AG > 0.020). It was found [4] that most of the austenite with an initial level of 17% transformed when the tempering temperature during the heat treatment was below 200°C. When the tempering temperature was between 200 and 400°C, some retained austenite remained (9%) after cycling. As the tempering temperature increased, the austenite became more stable. This stress-induced (or stress-assisted) transformation occurs at martensitic nucleation sites present in the parent phase (austenite) [5]. When new nucleations sites are created by plastic deformation, which causes additional transformation, the term straininduced transformation is used [5-7]. For stressinduced transformation, the yield strength in tension sharply increases with increasing temperature; whereas for strain-induced transformation, the yield strength decreases with increasing temperature [2, 5]. The initial yield behavior indicates whether stressinduced or strain-induced transformation occurs. However, with further deformation the distinction between stress-induced and strain-induced transformation becomes less apparent. For transformation to proceed, it has been found that the stress must be tensile or shear [8]. A compressive stress tends to suppress transformation. Supplementing the chemical driving force contribution, the applied stress contributes a mechanical driving force that either increases or decreases the total thermodynamic driving force [2]. Since a volume increase is associated with the transformation, tensile stress provides additional space to accommodate this volume expansion, lowering the activation energy required for the transformation. In complicated stress states such as those observed under contact loading conditions, high shear stresses superimposed with compressive hydrostatic stresses could result in phase transformations [9-11]. Certain planes may still undergo tensile stresses under these conditions facilitating the transformation. There have been investigations to define the kinetics of stress-induced transformation [12-17]. Olson and Cohen [12] studied the stress-assisted isothermal martensitic transformation in metastable austenitic steels, also named TRIP (transformation-induced plasticity) steels. They found that the fraction of transformed martensite is linearly related to the plastic strain at small plastic strains under uniaxial monotonic loading. Other investigators [8, 13-17] suggest that the rate of martensitic transformation is a function of temperature, Ms temperature, hydrostatic and equivalent stress (or normal and shear stress), carbon content, and the fraction of remaining austenite. As discussed earlier, these models suggest a tensile stress or large shear stress is necessary for stress-induced transformation. These kinetic models generally give the fraction of the austenite that has transformed, but they do not provide any information on the anisotropy of the strains o.r transformation under repeated (cumulative) straining

conditions. Furthermore, determination of volumetric strains with micromechanics models at different stress, temperature levels, and stress states has not been systematically considered. Another question is whether the transformation strains are anisotropic. This is important both from the fundamental and engineering standpoint, as these strains could contribute to the overall dimension changes and residual stresses in components such as carburized cones in bearings. In their study of residual stresses under the raceway, Hahn et al. [18] showed that transformation of austenite to martensite would have to be anisotropic to account for these stresses. Because transformation is determined by the activation energy, the volume accommodation would proceed in the direction of least resistance, which is the maximum tensile direction. Stress-induced transformation is strongly influenced by the temperature. A 20% Ni and 0.5% C steel was examined in an early study of stress-induced transformation [8], which had M s = - 37°C, where Ms refers to the temperature at which martensitic transformation begins when cooling from the fully austenitized structure. Therefore, the steel was still austenite at room temperature. When the steel was cooled to - 3 7 ° C , transformation began to occur. Tensile tests conducted at temperatures above Ms showed that the amount of stress-induced transformation linearly dropped off as the temperature was increased above Ms temperature [8]. At 0°C, very little transformation occurred. At room temperature, no stress-induced transformation occurred. So there was a very narrow range where stress-induced transformation occurred. This ranged from Ms to about 40°C above M s. From the chemical composition of carburized 4320 steel typically used in bearings, Ms was about 70°C [19]. At room temperature, only a fraction of the steel was austenite. The retained austenite could transform with applied stress near the final quenching temperature (22°C), but based on Ref. [8], no stress-induced transformation would be expected more than 40°C above the quenching temperature. The motivation behind this work has been to develop an understanding of transformation of carburized 4320 steel used in railroad bearings. Specifically, the purpose of this study was to: (1) perform tensile and compressive monotonic experiments at temperatures from - 6 0 to 150°C and measure transformation-induced strains under uniaxial and multiaxial stress states, and (2) develop a model that is capable of predicting the magnitude and anisotropy of the stress-induced transformation strains. EQUIPMENT AND MATERIAL

Smooth fatigue specimens with 0.5 in. (12.7 mm) gage length and 0.200 in. (5.08 mm) diameter were used. To obtain a through-carburized gage section

NEU and HUSEYIN SEHITOGLU:

STRESS-INDUCED TRANSFORMATION IN STEEL

Table 1. Heat treatments Carburized 4320 steel Material (Aret ~ 35%) Carburizing temp. (°F) Time (h) Hardening temp. CF) Time (h) Quenching medium Quenching temp. (°F) Tempering temp. (~'F) Time (h)

X-ray diffraction technique This technique for determining retained austenite content by X-ray diffraction is described in Ref. [19]. The surface of the specimen gage section was shot with X-rays through a 0.6 mm hole. Four independent volume percent retained austenite values were determined on each specimen utilizing different diffraction peaks. The average of the four readings is reported. As determined from the X-ray diffraction measurements, the average retained austenite content (Aret) in the gage section of all specimens was 35%. The variation among specimens ranged from 27.0% to

1800 18~ 1575 1 oil 70 350 1.5

aDouble carburization treatment (two 9 h treatments).

with uniform structure, a special specimen preparation was required. The gage section of each specimen was first rough machined to 0.230 in. (5.84 mm) diameter. The carburization in an endothermic atmosphere with CO= gas and hardening treatment were performed on these specimens (Table 1). Then the specimens were machined to final dimensions utilizing a slow grind process. The carbon content in the gage section was 1.10+0.05%. The Vickers hardness of the steel was 763. The uniaxial experiments were conducted on a uniaxial 20 kip (89 kN) ATS Long Term Fatigue Machine utilizing a SoMat Model 1100 Test Control and Data Acquisition System. Two MTS high temperature extensometers with a strain resolution of 4.0 × 10 -5 were used to measure both the axial (q) and diametral (q) strains. These extensometers were attached to the specimen at the locations shown in Fig. 1(a). During all experiments, the load, temperature, axial strain, diametral strain, and time data were digitally stored for later analysis. Induction was used to heat the specimens. For cooling specimens below 22°C, a copper coil, containing a controlled flow of liquid nitrogen, surrounded the specimen. For both heating and cooling, the temperature was controlled within + I°C. The same specimen design was used to study phase transformations under torsional loading. The specimen was twisted under rotation control on an MTS 20 kip - 50 kip.in. (89 kN - 5.65 kN. m) tension-torsion machine. For these experiments, strain gage rosette readings were recorded and stored with a computer. The strain measurements 6A, EB, EC are indicated in Fig. l(b). The maximum shear strain is EB-- EC. The axial (q) and diametral (Ed) strain components are given by

Ea= E. + Ec - E A

42.8%.

Direct measurement of volumetric strain For the uniaxial tests, the direct volumetric strain

(AV/V) measurement utilized axial and diametral extensometers on the specimen. The volumetric strain is determined by summing the strains AV -V- = E. + 2q.

(a)

13a

Cb)

il

I ~ J 0.5 inch ( 12.7 m m )

~ 13o "//2

~l-Ed (2)

(3)

Higher order terms were neglected, because they were smaller than the resolution of the extensometers.

(1)

and Ed ~---~A"

2259

11'

Ec

MEASURING TRANSFORMATION Two methods were used to detect transformation of retained austenite: (1) X-ray diffraction techniques and (2) direct volumetric strain measurements.

Fig. 1. (a) Fatigue specimen with location of axial strain (E,) and diametral strain (Ed)measurements for uniaxial loading. (b) Location of strain measurements eA, %, EC from strain rosette attached to specimen for torsional loading.

2260

NEU and HUSEYIN SEHITOGLU:

STRESS-INDUCED TRANSFORMATION IN STEEL

Since the axial and diametral strains were measured independently, the anisotropy of the volumetric strains could also be determined. The elastic and thermal strains constitute recoverable volumetric strain, AVr°C/V

Corburized 4320 Steel (Initial Aret = 35%) M o n o t o n i c Tests T=22oc

ession

2000

AV r~: o-01 - -

-

V

E

(1 -

2v) + 3g(T

-

To)

(4)

where 0-01 is the applied stress in uniaxial loading, E is the elastic modulus (E = 200,000 MPa), v is Poisson's ratio (v = 0.3), ~ is the coefficient of thermal expansion (~ = 1.6 x 10 _5 1/°C), T is the temperature, and To is the reference temperature. The first term on the right-hand side of equation (4) is the elastic volumetric strain and the second term is the thermal volumetric strain. The volumetric transformation strain, Avtr/v, is defined by A V tr - -

V

-

A V

A V rec

A V pl

V

V

V

AV tr AV -

V

= EA + EB +

Ec.

1500

1 OOO

Tension

5OO

0

r

I

I

i

P

i

i

i

i

I

.010

i

I

i

i

I

.02O

I

i

i

t

.0~0

I .040

Axial Strain

Fig. 2. Stress-strain curves for monotonic tests of carburized 4320 steel (h m ~ 35%) conducted at 22°C.

(5)

where AVpl/v is the plastic volumetric strain. The plastic volumetric strain included the effects of any increase in volume from vacancy generation or void and crack growth during plasticity. As will be explained later, the volumetric transformation strain was generally much larger than any increase from plasticity alone. Consequently, A VP~/V ~ O. Under torsional loading, the rosette strain gages measured the strain components EA, EB, CC [see Fig. l(b)], which are related to the axial and diametral strain by equations (1) and (2). If transformation occurs, Ea is no longer equal to Ec in magnitude. The volumetric strain is given by equation (3), and for a torsional loading at constant temperature, A V / V = = O. Then, the direct measure of volumetric transformation strain is given by

V

v

(6)

no significant increase was measured up to an axial strain of 3%. The asymmetry of the yield strength of the carburized 4320 steel in tension and compression (as seen in Fig. 2) was the direct result of transformation. If no or very little transformation occurred, as in compression, the yield strength depended on slip processes. However, in tension a phase transformation occurred, resulting in an increase of volume before the stress was high enough to cause slip. Because there was an increase in axial strain resulting from the transformation, the stress-strain response became non-linear at a stress level lower than that found when no transformation occurred. Consequently, transformation can be considered an alternate deformation mechanism. Comparing tensile tests conducted at 22 and 150°C (Fig. 4), the elastic modulus at 150°C was slightly lower, but no transformation took place at 150°C. Consequently, the apparent yield strength of the steel at 150°C was much higher than at 22°C. The

STRESS-INDUCED TRANSFORMATION EXPERIMENTS

To examine stress-induced transformation, both monotonic and cyclic tests were conducted at constant temperatures. The cyclic tests indicated how reversed plasticity affected the rate of transformation. The stress-strain curves for both monotonic tension and compression tests at 22°C are shown in Fig. 2. The yield stress was much lower in tension than in compression. The decrease in the yield stress in tension was attributed to transformation of retained austenite to martensite. The amount of volumetric transformation strain recorded during the monotonic tests at 22°C is shown in Fig. 3. The volumetric transformation strain increased in tension with axial strain up to the fracture strain. The volumetric transformation strain at fracture was 0.006. In compression, a small increase in the volumetric transformation strain (less than 0.001) was observed, but

.008

Carburized 4320 Steel (Initial Aret = 35%) M o n o t o n i c Tests T = 22 ° C

.007

.£ 0

.006

(~)

,005

n s i o n

u

~.~ .004 0

"~ .oo~ r"

{3

I--- .002 0 ~-~ .001 0

E

o

O >

-.oo~ --.O02

, 0

,

,

,

I

.010

i

i

i

,

I

,

,

.O2O

i

l

I

.030

p

i

,

,

I

.04O

Axial Strain

Fig. 3. Stress-induced volumetric transformation strain for monotonic tests of carburized 4320 steel (Am ~ 35%) at 22°C in tension and compression.

NEU and HUSEYIN SEHITOGLU: 2000

STRESS-INDUCED TRANSFORMATION IN STEEL 1600-

Carbufized 4320 Steel Initial A~e~= 35% Tensile "rests

2261

Carburlzed 4320 Steel 22° C

Are t = 3 5 %

14OO2 1200 2

1500

1ooo2 T= 1~ , , i ' x

S

:E

600. ~o 2~ o 0.005

0,~0

o 0000

0.010 Axial Strain

i 0005

0010 Axial Strain

0015

0020

Fig. 4. Tensile tests of carburized 4320 steel (Aret ~ 35%) conducted at 22 and 150°C.

0.020

0.012 _c co

c o

yield strengths of carburized 4320 steel in tension a n d c o m p r e s s i o n are given in Table 2. A t a t e m p e r a t u r e w h e n t r a n s f o r m a t i o n occurred (22°C), the tensile s t r e n g t h was m u c h less t h a n the compressive strength (Fig. 2). W h e n t r a n s f o r m a t i o n did n o t occur (150°C), the yield strengths in tension a n d c o m p r e s s i o n were comparable. Experiments were also c o n d u c t e d below the q u e n c h i n g t e m p e r a t u r e (22°C): at - 6 0 , - 4 0 a n d 0°C. The volumetric t r a n s f o r m a t i o n strain d u r i n g the tensile loading for these tests are s h o w n in Fig, 5. These tests showed t h a t a small increase in t e m p e r a t u r e (10-20°C) would result in a significant decrease in the volumetric t r a n s f o r m a t i o n strain. T h e volumetric t r a n s f o r m a t i o n strain was as high as 0.01 at - 6 0 ° C at specimen fracture. D u r i n g cooling to - 6 0 ° C at zero stress, a t h e r m a l t r a n s f o r m a t i o n initiated at - 4 5 ° C , a n d therefore at - 6 0 ° C Avtr/v=o.o045 before the stress was applied. U p o n d e f o r m a t i o n at - 6 0 ° C , further transf o r m a t i o n occurred with increasing stress. As these experiments illustrated, stress-induced t r a n s f o r m a t i o n was sensitive to the testing temperature. In Fig. 6, the stress-induced volumetric t r a n s f o r m a t i o n strain at E. = 0.005 as a function of t e m p e r a t u r e is shown. Tests c o n d u c t e d at 35 a n d 50°C were also included in this figure. The volumetric t r a n s f o r m a t i o n strain decreased nearly linearly with

0.015

~_o

-60° C 0.010

~ / "

0.008

-40° C 0° C

0.004 0.002

.~

0.000 ,

0.000

,

H

I

,

0.005

,

,

,

I

. . . .

0.010

I

H

0.015

'

'

I

0.020

Axial Strain

Fig. 5. Stress-strain behavior and volumetric transformation strain during monotonic tests conducted on carburized 4320 steel (Aret ~ 35%) at four temperatures: - 6 0 , - 4 0 , 0 and 22°C.

testing temperature. The plot suggests t h a t very little stress-induced t r a n s f o r m a t i o n would be expected a b o v e 60°C. The influence of stress state on t r a n s f o r m a t i o n was e x a m i n e d with torsion experiments. The m a x i m u m shear s t r e s s - m a x i m u m shear strain curve for carburized 4320 steel at 22°C is s h o w n in Fig. 7. The specimen fractured in a brittle m a n n e r at a m a x i m u m shear strain of 0.056. It is w o r t h noting t h a t the m a x i m u m resolved tensile stress in this experiment was 2000 M P a which far exceeded the tensile fracture stress o f 1300 M P a u n d e r uniaxial loading. The corres p o n d i n g volumetric t r a n s f o r m a t i o n strain for this torsion test is s h o w n in Fig. 8. The volumetric

Table 2. Yield strength and ultimate tensile strength for carburized 4320 steel 0.2% 0.02% 0 . 0 1 % 0.001% offset offset offset offset yield yield yield yield strength strength strength strength ksi ksi ksi ksi Heat Ant T e m p . Loading treatment (%) (°C) T or Cb (MPa) (MPa) (MPa) (MPa) 87.6 75.6 66.2 63.0 Carburized 32.6 22 T (604) (522) (456) (435) 149.1 96.1 82.9 63.0 Carburized 35.5 150 T (1028) (663) (571) (434) Carburized 42.8 22 C 190.3 106.1 99.9 78.0 (1312) (732) (689) (538) a 93.0 81.9 56.7 Carburized 35.4 150 C (641) (565) (391) "Test did not give this information. bT = Tension; C = Compression.

Ultimate tensile strength ksi (MPa) 189.1 (1304) 195.5 (1348) --~ --~

2262

NEU and HUSEYIN SEHITOGLU:

STRESS-INDUCED TRANSFORMATION IN STEEL 0.005

0.010 -

-

Carburized 4320 S t e e l Are t = 35%

.~ P

Carburized 4320 S t e e l Are ~= 35% T=22°C

0,004.

0.008

,S ,_

._c 2

a = 0.005 0.006

0003.

Torsion Test

g

o

0.002. ~-

o

0.OO4

o 0.001 t) ~ >

0.002

o

0.000

....

I ....

- 100

I ....

-50

0

I ....

50

I .... 100

I~ . . . . 150

E o >

0.000

I

-QO01

200

Temperature (°C)

Fig. 6. Volumetric transformation strain as a function of temperature for carburized 4320 steel (Ant.~ 35%) in the temperature range from - 6 0 to 150°C. transformation strain at failure was nearly 0.004. The transformation strain at fracture in the torsion experiment was lower than observed in the uniaxial test. Under a torsional loading, the shear stress (and consequently the principal tensile stress) decreased from a maximum value at the surface to zero at the center of the cross section. Therefore, only a fraction of the volume of the specimen experienced the maximum tensile stress. The volumetric transformation strain for a number of cyclic tests is compared to the monotonic tension test in Fig. 9. The maximum peak stress has been normalized by the tests at which transformation commenced in each test (average was 485 MPa). The solid line in Fig. 9 represents the monotonic loading. Under this loading, all stress levels represented an instantaneous peak stress, establishing that transformation was approximately proportional to the maximum stress level. The data symbols in Fig. 9 represent the volumetric transformation strain at the maximum peak stress of cycles 1, 2, 3, 4 and 8 in

.

2500

Carburized Are t = 35%

2000

~

-0.002 500 1000 1500 Maximum Shear Stress (MPa)

Fig. 8. Volumetric transformation strain for torsion test of carburized 4320 steel (Aret ~ 35%) conducted at 22°C.

the cyclic loadings. Under cyclic conditions, when there was reversed plasticity (R, = - 1 ) , the amount of volumetric transformation strain initially increased with each cycle. After cycle 8, no detectable transformation occurred with further cycling and the cyclic response was stable. As the axial strain range (AE,) increased, the cyclic plastic strain was greater, resulting in an increase in the volumetric transformation strain with cycling. However, in the cyclic R, = 0 test or small strain range (Ae, = 0.008) test, where there was little reversed plasticity, the deviation from the monotonic curve was not observed. A small change in volume from the introduction of vacancies in the material from dislocation plasticity was expected [20], but this plastic volumetric strain was much smaller than changes from stress-induced transformation. At large strains, such as those experienced in the compression monotonic test, there could be a discernable volume change. It could be as much as 0.02% [20], but this is still much smaller than the

4320 Steel

0.010 -

T = 22 ° C

1500

Carburized 4320 I n i t i a l A,et = 3 5 %

._C 0.0O9,

p

~,

•

'~ 00o5 : o 0004 -

1000

Steel

T = 22 ° C

0.008

2~ 0°06-.:

£ E

2000

Z~-~ = 0010, Rt = 0

• Z~a = 0 008, Re=-1 D &r~== O010, RE=-1 0 Z ~ = 0.020, Rc =-I

0-8 0 0

0

.E ooo3 o 8-

00022 o

500-

oooi. 0000 1.0 . . . .

0

'

0.00

' ' ' I ' ' ~ ' I ' ' ' ' I ' ' ' ' I '

0.01

002 0.03 0.04 Maximum Sheor Strain

' ' ' I ....

0,05

1!5 . . . . 2!0 . . . . 2!5 . . . . N o r m a l i z e d M a x i m u m P e a k Stress

i

0,06

Fig. 7. Maximum shear stress-maximum shear strain curve for carburized 4320 steel (Ant ~ 35%) conducted at 22°C.

3!0

9 . Volumetric transformation strain for both monotonic and cyclic loadings as a function of the maximum peak stress normalized by the stress when transformation commenced ( 4 8 5 M P a ) . Fig.

NEU and HUSEYIN SEHITOGLU:

STRESS-INDUCED TRANSFORMATION IN STEEL

average volumetric transformation strains measured in the tests. At the end of each experiment conducted at 2Z:C, the average amount of retained austenite, as measured by X-ray diffraction, was typically 28%. No clear evidence of any cyclic strain range effect on the amount of transformed austenite could be deduced from X-ray diffraction measurements. However, the effect of strain range was clearly recognized when the volumetric transformation strain was measured directly. X-ray diffraction measurements indicated that the total amount of retained austenite did not significantly change during the monotonic tests, even in the tensile loading test. The differences between the initial and final readings were within the accuracy of the X-ray measurements. However, with the more precise volumetric strain measuring technique utilized in present investigation, a significant volumetric change during tensile loading was clearly observed. This discrepancy may be explained by the fact that the X-ray technique measures austenite at a point on the surface, whereas the volumetric strain technique averages the transformation throughout the volume of the specimen. At the end of the - 4 0 ° C tests, the average retained austenite content had decreased to 21.8%, which verified that additional transformation occurred during the sub-zero tests. EFFECT OF STRESS AND TEMPERATURE

ON TRANSFORMATION The stress at which initial transformation occurs is dependent on the test temperature and the quenching and tempering temperatures, all of which are factors affecting the stability of the austenite. Even steels quenched to room temperature tend to become somewhat more stable in a short period of time (order of I h) after quenching [21]. Tempering further stabilizes the austenite. Consequently, athermal transformation does not occur until after sufficient undercooling. Carburized 4320 steel had to be cooled to - 4 5 ° C before a burst of athermal transformation occurred. When the austenite has been stabilized with time or by tempering, the stress must be raised to some threshold value before stress-induced transformation commences. This threshold was about 485 MPa when stressed at 22°C. This threshold value decreased with decreasing temperature. Since transformation rapidly decreased as the testing temperature increased (Fig. 6), stress-induced transformation, rather than strain-induced transformation, was responsible for initial yielding. An explanation of stress-induced transformation requires the examination of the transformation thermodynamics [2, 12]. A schematic of the free energy vs temperature is shown in Fig. 10. The free energy of austenite (A) and martensite (M) are shown as solid lines. A decrease in free energy is the driving force for the transformation: therefore, transformation to martensite becomes favorable when the free energy of

Athermal Transformation

[

I t

I

i

t

--.. _

!I A G QT a T ~

....

Stress-Induced Transformation

i

[ J

.....

2263

q.. I ""

"'"

h


Increasing Transformation

1

. .... J

~AIto, Transt°.rn .....

,,'5, ,,=, ,,'H,

Quenching Temperature

TEMPERATURE

Fig. 10. Schematic of the free energy as a function of temperature showing the austenite (A) and martensite (M) free energy curves.

austenite is greater than that of maretensite. Since there is a free energy barrier that must be overcome when the transformation occurs, the steel must generally be undercooled below the intersection of the martensite and austenite free energy curves before transformation commences. At the quenching temperature (22°C), there is a free energy difference between austenite and martensite, AG QT(see Fig. 10). At this temperature a fraction of the austenite still has not transformed. To get further transformation, an increase in AG is necessary. AG can be expressed as the sum of the change in chemical free energy (difference between the austenite and martensite free energy curves), AG oh, and the change in the mechanical free energy, AG mech, which is associated with the strain energy generated by a tensile stress AG = AGch + AG mech.

(7)

Athermal transformation occurs if the additional free energy contribution comes only from the AG ~h component. Stress-induced transformation occurs if the additional free energy comes from the application of a tensile stress, AG m°ch.Since AG ch decreases as the temperature increases, AG meshmust increase (indicating that the tensile stress must be greater) as the temperature increases in order to obtain the same amount of transformation. The amount of the stress-induced volumetric transformation strain rapidly decreased with temperature and no transformation was measured at any temperature higher than 60°C in the carburized 4320 steel (Fig. 6). Results of other experiments on a fully austenitic 20% Ni and 0.5% C steel also support this narrow temperature range for stress-induced transformation [81. MODELING OF ANISOTROPIC STRESS-INDUCED TRANSFORMATION

In both monotonic and cyclic tests, the stressinduced volumetric transformation strain was not

2264

NEU and HUSEYIN SEHITOGLU: STRESS-INDUCED TRANSFORMATION IN STEEL

purely isotropic. There tended to be preferential expansion in the direction of the largest tensile stress. A micromechanical model of stress-induced transformation is used to further examine the effect of the loading and orientation of martensite plates on the anisotropy of the transformation strains. Briefly, the model utilizes Eshelby's inclusion method [22] to determine the transformation strains in a threedimensional body containing a distribution of transformed martensite plates. The decrease in free energy is the criterion to determine which orientations of plates will transform. A transformation strain is imposed within the plates that have transformed and the average transformation strain in the body is determined. The advantage of this model is that any loading condition can be studied. The model predicts that transformation strain is greater in the direction of the maximum tensile stress, agreeing with experiments.

Theory The size of the martensite plate is of the order of the austenite grain size, since the individual plates do not cross grain boundaries. The habit plane, which has the characteristic such that it does not change dimensions when transformation occurs [23], is parallel to the long axes of the martensite plate. Within the transformed martensite plate, there are a series of alternating twins at some angle to the habit plane. Because the habit plane remains unchanged, the resulting transformation strain in the plate is perpendicular to the habit plane. This assumption has been used in other investigations of martensitic transformations [24-26]. A single transformed martensite plate in a matrix of untransformed body is shown in Fig. 11. The transformed martensite plate is represented by an ellipsoidal inclusion with al =a~>>a 3. The transformation strain within the transformed martensite inclusion, f~, is e*3., which is the strain parallel to the x~ direction. The x'-coordinates belong to the

Oil °

Gij°

¢¢¢¢¢¢

¢¢¢¢¢¢

Ontransformed

Transformed

Fig. 12. Schematics of untransformed and transformed

inclusions.

coordinate system associated with the transformed martensite inclusion, and the x-coordinates are associated with the direction of the applied stress and resolved transformation strain. All orientations of inclusions in three-dimensional space are included when tk (see Fig. 11) is rotated about the x~-axis from 0 to n and 0 (not shown in Fig. 11) is rotated about the x3-axis from 0 to 2x. The orientations which contain a transformation strain are determined by the model. Transformation is driven by the decrease in free energy, AG, when transformation takes place. Transformation occurs when AG is less than some critical free energy, AGcrit, which has a negative value. For example, consider a body under tensile loading with an inclusion, ~ (Fig. 12). The elastic constants throughout the body are the same, because the elastic properties of austenite and martensite are the same. In the untransformed body (Fig. 12), no transformation occurs, so no transformation strains are imposed inside fL The free energy of the untransformed body, Gum..... which arises from only the applied stress, is equal to the elastic strain energy, W0 [27]

(~33

¢

L

×;

Gu,uans = W0= ~l f v ai:ui,jdV ° ° (~23

•

(~22

where a ° is the applied stress and ui°a is the elastic strain. The volume of the body including the matrix and inclusions is V. If transformation occurs, a transformation strain, e*, is imposed within f~. In the analysis e* = 1,3,lj3, e ~'3'

~

(8)

2a 1

Fig. 11. Schematic of a transformed inclusion, fL

(9)

where l~ is the coordinate transformation matrix and the transformation strain, (..3,3, , * is equal to 0.040, the assumed volumetric transformation strain. The free energy after transformation, Gvrans, is GT. . . .

~--

Wo+ w * + AW

(10)

NEU and HUSEYIN SEHITOGLU:

STRESS-INDUCED TRANSFORMATION 1N STEEL

where W0 is the elastic strain energy due to the applied stress, W* is the additional strain energy due to the stress caused by the transformation strain, and A W is an interaction energy between the applied stress and the transformation strain. W0 was given by equation (8). The other two energy terms are [27] W* = - -

';o

(11)

t" jn

(12)

*dV 2 a°E!i

--

o 6*dV. O'qE

The integrals in equations (11) and (12) are evaluated inside the transformed martensite inclusion domain, f~. The stress, a~j, in equation (11) is a function of the geometry of the inclusion, the elastic properties, and the magnitude of E*, and is given in Ref. [27]. Transformation begins to occur when AG = GT. . . .

- - G u m . . . . < AGcrit.

(13)

Note that AGcrit is negative, because some undercooling is needed. Gunt.... and GT.... are also functions of temperature, but the free energy from temperature cancels when the temperature is held constant. Substituting equations (8) and (10) into equation (13) AG = W* + AW.

(14)

Thus, the orientation of the inclusions that have transformed depends on the applied loading, a °, the dimensions of the inclusion, a~ and a 3, the elastic constants, and the transformation strain, E~.y. The transformation strain interacts with the applied strain to lower the overall strain in the vicinity of the transformed martensite. When the strain is reduced, the elastic strain energy and consequently the free energy is lowered, which is the driving force for the transformation. The value of AGcrit defines the magnitude of the reduction in free energy required to initiate transformation. An orientation of an inclusion transforms when its AG is less than AG¢r,t. The number of transformed martensite plates is dependent on the magnitude of AG - AGcrit at each orientation. The relative potency of this thermodynamic driving force is represented by the constant P~G. The total thermodynamic driving force is determined by the integration of driving force, AG -AGcrit , for all orientations, which is then normalized by PA~ to determine the actual fraction of austenite that transforms. The value of AGo~itis determined by the value of AG at q~ = 0 ° when transformation just commences under a uniaxial tensile loading. Other loading conditions at the same temperature are examined assuming the s a m e AGcrit. At different temperatures, AGcrit will be adjusted by the change in AG ch with AT. The conAM 40,'9

M

stant, PaG, is also determined from the uniaxial tensile loading by examining the rate that the volumetric transformation strain increases with stress. Note that no information about the anisotropy of the transformation strains is required to determine the constants.

Computationof transformationstrains The average transformation strain, (E~), is determined as follows:

and AW

2265

(1) The parameters for the problem are entered (Table 3). (2) An increment of stress is imposed. An incremental solution is necessary because the fraction of retained austenite remaining is dependent on the amount that transforms during previous increments. (3) The volume fraction of transformed inclusions, c, is determined by integrating over all possible orientations [28] ('2n ('n

1

fRAjo J0 o ;sin, d d0. PA6

(15)

C -~---

In this equation, fRA is the current fraction of retained austenite, p ~ is a constant representing the net thermodynamic driving force necessary to transform all retained austenite, and ~uA6,the force distribution function, is given by

I AGi-AG,,

l

71aa=(-1)llGi-AGcrit

ifAG~t AAGcrit G , _ and otherwise.

(16)

The value of c, which represents the rate of transformation, ranges from 0 (no transformation) to f ~ (transformation of all retained austenite). (4)The average transformation strain for the current stress increment, (E~)k, is

C(Eq)k+

(1

C)(~)7

(17)

where (ES)~ is the average transformation strain from all orientations of transformed inclusions and (E,:i) T m k is the average strain in the matrix resulting from the transformations. The increment in Table 3. Parameters used in stress-inducedtransformation model for carburized4320 steel Dimensionsof martensite a~ 10,um ellipsoid a3 0.5 ,um Elastic constants u 76,920MPa v 0.3 Applied stress a° Varies Transformationstrain in martensiteplate ~~j * 0.040 Critical free energy change

Potency of thermodynamic driving force Initial fraction of retained austenite

,~Gcrit

P6G JRA

- 2 . 5 × l0 9j

2.0 × l0 0.35

9j

2266

NEU and HUSEYIN SEHITOGLU: STRESS-INDUCED TRANSFORMATION IN STEEL

transformation strain in the inclusions and matrix is determined from

0.010 -

0.008

f;f: =-c.

~

(% - e~)~t~-~ sinq~ d4~ dO

fo .i= jo ~t ~I

sin~ dO dO

o Axial (Cyclic Test) [] Diametral (Cyclic Test) 0.C06,

(~j + %) ~r

Axial (Model)

(18) 0.004,

sin~bd~b dO

~ =

Carburized 4320 Steel Aret = 35% T = 22° C

0.002.

(19)

f;" f[ ~'r~ sin4~d~ dO where the transformation distribution function, ~Pr, has the value of 1 if transformation occurs and 0 if it does not occur, as determined by equation (13). In equations (18) and (19), e0 is the volume average disturbance strain in the matrix domain because there is more than one transformed inclusion, and % is the constraint strain resulting from the stress-free transformation strain, q *i ' Since c already accounts for the fraction of austenite that transforms, the computation of the average transformation strains should not change this fraction. This is achieved by dividing the integrated transformation strain by the fraction of orientations that have transformed. When all orientations of inclusions transform, ~0 is given by [29]

0.000

Axial (Monotonic Test) J

,,"

Y ,.0%¢

Diametral (M°n°t°n'c Z

' ' '1 ' ' ' I '' 200 400

~

i

J

/

," ! ~ Diametral ,,-'" 1/ ~ (Model) . . . . . . ' ~ . °.. -~2_ ~ -

' I ' ' ' I' ' '1' ''1' '' I 600 800 1000 1200 1400 Axial Stress (MPa)

Fig. 13. Transformation strains under tensile loading at 22°C for both experiments and model.

strain component, both the elastic and plastic strains were subtracted from the total strain. The plastic strain was estimated from the plastic strain of a tensile test conducted at 150°C where no transformation occurred. Both the model and experiment agree that the transformation is not isotropic under a tensile loading, and that the transformation strain in the loading (axial) direction is greater than the diametral direction. As the stress increases, other less favorably e'U = - - C (eij - - E ~ ) . (20) orientations transform, which contribute more to the diametral strain, slightly decreasing the anisoSince all orientations do not necessarily transform, tropy as shown by the model prediction. The greater integration over all orientations is performed as in anisotropy shown in the experiment at higher stresses equation (18). indicates that the synergistic effect of plastic deforEshelby [22] has shown a linear relation between e U mation accommodating the transformation also and e* influences the anisotropy. In addition, plastic deformation creates new potential nucleation sites for * (21) eij - - S i j k l E kl martensitic transformation [23]. Under cyclic conwhere S~ikt is the Eshelby tensor. This tensor is a ditions when reversed plasticity occurs, it was shown function of the dimensions of the martensite ellipsoid that the volumetric transformation strain increases and the Poisson's ratio of the matrix material. The (Fig. 9). It is now shown in Fig. 13 that the increase equations for this tensor are known [27] for the in volumetric strain tends to be an increase in the geometry of interest (i.e. al = a2 ~ a3). diametral strain, causing %/E a t r tr to approach nearly (5)The average transformation strain, (e~), is unity by cycle 8. In contrast, ed/E, tr tr ranges from 0.05 incremented by = + k (22) conditions are given in Table 4. Under a tensile and the fraction of retained austenite decreases by c hydrostatic loading (Case I in Table 4), the transfRA = fRA -- c. (23) formation is isotropic and all retained austenite transforms. The volumetric strain (0.014) is equal to (6) Stress is incremented and the calculation is the fully transformed value [fRA*£yy(0.35 *0.04)]. repeated starting at step 3. In simple shear (Case II), a greater transformation strain is in the maximum principal stress direction. Results of the model For a shear stress of 1000 MPa, AVtr/V is predicted The transformation strains from the model and to be 0.003. Under fully reversed loads in shear, experiments under uniaxial loadings are compared the transformation strains <•ITI> and (22T2> reverse. in Fig. 13. The experimental data shown as a solid Therefore, the favorable orientations for transline in this figure is from a tensile test conducted formation change with cycling. Under uniaxial loadat 22°C. To obtain the individual transformation ing, the favorable orientations remain unchanged

NEU and HUSEYIN SEHITOGLU: STRESS-INDUCED TRANSFORMATION IN STEEL Table 4. Results of simulations for stress-inducedtransformation model at 22"C Case I (hydrostatic tension) a°,(MPa)

5i0

0 500

=

0 ] 0

0 I0.0~467

T

CONCLUSIONS

/~ Vtr - - = 0.0140 V Case II (shear)

,ooo o il 0 - 1000

a ° (MPa) =

0

0

i0o ,,o 0.000136 o

(E ij )tot =

0 0.000284

0 A V tr = 0.0030 V

during cycling. Consequently, more orientations of inclusions become favorably oriented at some point within each cycle under cyclic shear loads. In addition, the resolved tensile stress in the torsion experiment reached 2000MPa, which was much higher than the 1300 MPa reached in the uniaxial test. The cases shown in Table 4 indicate that one of the principal stresses must be positive for transformation to occur. However, severe deformation with large shear strains will increase potential nucleation sites for transformation, thus increasing the probability for transformation. The volumetric transformation strain in the torsion experiment and the model are compared in Fig. 14. The volumetric transformation strain corresponding to a maximum shear stress of 2000 MPa is near 0.004 in the experiments. For the prediction from the model, the gradient of the shear stress from the outer radius to the center of

0,008

._

Carburized Are t = 35% T=22°C

0.006

4320 Steel

Inc., Petersburg, Va and the Association of American Railroads (AAR), Technical Center, Chicago, Ill. The cooperation of Mr Robert Lawrence and Mr Arun Dhir of Brenco, Inc. and Dr Dan Stone, Dr Keith Hawthorne and Mr Michael Fec of AAR is appreciated. Discussions with Dr Gerald Moyar were insightful and highly valuable. Thanks goes to Dr Peter Kurath and Mr Craig Payne for conducting the torsion tests.

Torsion Test

e

0004

t3 E

0.002 .

.

1. Stress-induced transformation of retained austenite to martensite was observed by measuring volumetric transformation strain in carburized 4320 steel with 35% retained austenite content. Stress-induced transformation occurred under tensile loading at temperatures below 60°C. 2. The a m o u n t of volumetric transformation strain increased with decreasing temperature. At fracture of a monotonic tensile loading, the volumetric transformation strain was 0.6% at 22°C and 1.0% at - 60oc. 3. Under cyclic loads, when the stress was increased above the stress level of the previous cycle, some additional transformation occurred during the increase in stress. Under steady state cycling when the stress reached the same peak level on each cycle, no further transformation was detected. The volumetric transformation strain based on maximum stress level reached was greater for the cycling when Ae, ~> 0.010. 4. The model of stress-induced transformation based on Eshelby's inclusion method indicated that stress-induced transformation would occur under shear loading which was confirmed with the experiments in torsion. 5. Stress-induced transformation strains were anisotropic when the applied stress was not isotropic. The largest transformation strain was in the largest positive stress direction. Under uniaxial loading, the transformation strain anisotropy ratio (Ed/E a t r tr) for stress-induced transformation ranged from 0.05 to 0.25. The prediction of the anisotropy effect under uniaxial loading was in general agreement with the experiments. The a m o u n t of anisotropy decreased with cycling and was attributed to reversed plasticity. During cycling, [d/E tr tr a increased to 0.85. Acknowledgements--This work was supported by Brenco,

S E

the cross section was taken into account by integrating the volumetric transformation strain over the cross section. The model predicts the experimental trends accurately.

500

0 ] o 0.00467 0 0.00467

(• ij)tot :

2267

~

x

p

e

r

m

e

n

t

REFERENCES >

0000

-0002

'

'

'

'

I ' 500

'

Maximum

Fig.

'

'

I ' ] 000

'

'

'

I f ] 500

'

'

'

I 2000

S h e o r Stress ( M P o )

14. Volumetric transformation strain under torsional loading at 22°C for both experiment and model.

1. C. S. Roberts, J. Metals Trans. A.LM.E. 203 (1953). 2. M. Cohen and C. M. Wayman, Metallurgical Treatises (edited by J. K. Tien and J. F. Elliott), pp. 445~468 T.M.S.-A.I.M.E. (1981). 3. R. H. Richman and R. W. Landgraf, Metall. Trans. 6A, 955 (1975). 4. Yunoshin Imai and Shin-ichiro Kumagai, Sci. Rep. Res. Inst. T6hoku Unh~. 24A, 1 (1972).

2268

NEU and HUSEYIN SEHITOGLU:

STRESS-INDUCED TRANSFORMATION IN STEEL

5. G. B. Olson and M. Cohen, J. less-common Metals 28, 107 (1972). 6. M. A. Zaccone and G. Krauss, Int. Conf. on Processing and Performance (edited by George Krauss), pp. 285-290. ASM International, Metals Park, Ohio (1989). 7. M. A. Zaccone, J. B. Kelley and G. Krauss, Heat Treatment '87, Proc. Int. Conf. included in Materials '87, pp. 93-101. The Institute of Metals, London (1988). 8. S. A. Kulin, M. Cohen and B. L. Averbach, J. Metals Trans. A.LM.E. 661 (1952). 9. V. Bhargava, G. T. Hahn and C. A. Rubin, A.S.M.E. J. appl. Mech. 52, 67 (1985). 10. J. E. Merwin and K. L. Johnson, Proc. Inst. Mech. Engrs 177, 676 (1963). 11. A. D. Hearle and K. L. Johnson, J. appl. Mech. 54, 1 (1987). 12. G. B. Olson and M. Cohen, Metall. Trans. 13A, 1907 (1982). 13. S. Denis, A. Simon and G. Beck, Heat Treatment "83, Shangnai, Proc. of 3rd lnt. Congress on Heat Treatment of Materials, pp. 5.68-5.79 (1983). 14. T. Inoue, T. Yamaguchi and Z. Wang, Mater. Sci. Tech. 1, 872 (1985). 15. J. C. Lambropoulos, Adv. Struct. Ceram. Materials Research Society Symp. Proc., Vol. 78, pp. 35-41 (1987). 16. S. Suresh and J. R. Brockenbrough, Acta metall. 36, 1455 (1988). 17. I-Wei Chen and P. E. Reyes-Morel, Advanced Structural Ceramics, MaterialsResearch Society Symp. Proc., Vol. 78, pp. 75-88 (1987). 18. G. T. Hahn, V. Bhargava, C. A. Rubin, Q. Chen and K. Kim, A.S.M.E.J. Tribology 109, 618 (1987). 19. C. F. Jatczak, J. A. Larson and S. W. Shin, Retained Austenite and lts Measurements by X-ray Diffraction, SP-453. SAE, Warrendale, Pa (1980). 20. W. A. Spitzig, R. J. Sober and O. Richmond, Acta metall. 23, 885 (1975). 21. S. G. Glover and T. B. Smith, The Mechanism o f Phase Transformations in Metals, pp. 265-276. The Institute of Metals, London (1956). 22. J. D. Eshelby, Proc. R. Soc. London A 241, 376 (1957). 23. C. M. Wayman, Advances in Materials Research (edited by Herbert Herman), Vol. 3, pp. 147-304. Interscience (1968). 24. M. Shibata and Kanji Ono, Acta metall. 23, 587 (1975). 25. M. Shibata and Kanji Ono, Acta metall. 25, 35 (1977). 26. K. E. Easterling and A. R. Tholen, Acta metall. 24, 333 (1976). 27. Toshio Mura, Micromechanics o f Defects in Solids. Martinus Nijhoff, The Hague (1982). 28. Suresh G. Advani and C. L. Tucker III, J. Rheology 31, 751 (1987). 29. T. Mori and K. Tanaka, Acta metall. 21, 571 (1973).

APPENDIX Nomenclature at , a2 ' a3 Ar~t c E eo ~i fr~ GTrans Gu,,~.... AG

Radii of martensite ellipsoid Initial volume fraction of retained austenite Volume fraction of transformed inclusions Young's modulus Constraint strain Disturbance strain Current fraction of retained austenite Free energy with transformation in inclusion Free energy with no transformation in inclusion Free energy difference between austenite and martensite AGeh Chemical free energy AGcrit Critical free energy AGm~h Mechanical free energy AGQT Free energy difference at quenching temperature IU Coordinate transformation matrix Ms Martensite start temperature R, Strain R-ratio (minimum strain/maximum strain) Sqkt Eshelby tensor T Temperature To Reference temperature u.0d Elastic strain V Volume AV/V Volumetric strain AVPl/V Plastic volumetric strain A v r ~ / v Recoverable volumetric strain AVtr/v Volumetric transformation strain W* Elastic strain energy from transformation strain W0 Elastic strain energy AW Interaction energy o~ Coefficient of thermal expansion Ea Axial strain Diametral strain Ed EA, EB, EC Strains measured with rosette arrangement . Transformation strain E~j Transformation strain in inclusion E~3' r E[r/E~ Transformation strain anisotropy ratio AEa Axial strain range Average transformation strain (E~) Increment of average transformation strain (E~)k Average transformation strain in inclusion (E~)~ (E~)~ Average transformation strain in matrix Shear modulus /a v Poisson's ratio 0, q~ Angles Potency of thermodynamic driving force Pac aq Stress a° Applied stress Inclusion domain Driving force distribution function ~A~ ~t Transformation distribution function