Application of the overstress concept to inelastic behavior and evaluation of creep-fatigue damage for modified 9Cr-1Mo steel

Int. J. Pres. Ves. & Piping 44 (1990) 99-115

Application of the Overstress Concept to Inelastic Behavior and Evaluation of Creep-Fatigue Damage for Modified 9Cr-lMo Steel Kosei Tagucliifl Etsuro K a n n o , a S a t o r u Ozaki a & Testuro Uno b aNuclear Engineering Laboratory, bApplied Metallurgy and Chemistry Department, Toshiba Corporation, 8 Shinsugita-cho, Isogo-ku, Yokohama 235, Japan (Received 31 May 1989; accepted 27 February 1990)

A BSTRA CT The study reported in this paper was carried out to develop a method for both simulating the inelastic behavior and predicting the life of modified 9Cr- l M o steel at elevated temperature. A unified constitutive equation and a damagerate equation, based on the overstress concept, have been proposed. These equations are based on the same internal-state variables, which coincide with back stress and overstress determined experimentally. It is therefore possible, t5), coupling these equations, to simulate inelastic behavior and damage development simultaneously. This method was applied to the results of creep and creep-fatigue tests on modified 9 C r - l M o steel at 550°C in air. The simulated inelastic behavior and the predicted lives agreed well with the experimental results.

NOMENCLATURE DC

Dd Dr Di E

Time-dependent damage Time-dependent damage parameter (MPa s) Time-independent damage

Time-independent damage parameter (MPa s) Young's modulus (MPa) N u m b e r o f cycles to failure N, Time (s) t 99 Int. J. Pres. Ves. & Piping 0308-0161/90/$03"50 © 1990 Elsevier Science Publishers Ltd, England. Printed in Great Britain

100

o~2

gi gi gq o"

rT rC

Kosei Taguchi, Etsuro Kanno, Satoru Ozaki, Testuro Uno

Back stress (MPa) Back stress component (MPa) Back stress component (MPa) Inelastic strain Accumulated inelastic strain Total strain Stress (MPa) Overstress (MPa) Tension-going time (s) Compression-going time (s) Time-independent damage during one cycle Time-dependent damage during one cycle INTRODUCTION

A reasonably accurate constitutive equation and a life-prediction method are required in structural design at elevated temperature. Asada et al. have proposed an equation for creep-fatigue damage, based on the overstress concept, for type 304 stainless steel 1'2 and 2.25Cr-lMo steel 3'4 in a high vacuum. They experimentally determined back-stress and overstress values by analysis of the stress-strain behavior under the unloading process after a load reversal. For modified 9Cr-1Mo steel at 550°C in air, the authors have reported the applicability of this kind of damage equation to creep-fatigue life prediction 5 and have proposed a unified constitutive equation based on the overstress concept. 6 In the present paper, the authors summarize the creep-fatigue test procedures and test results, the procedures for determining back stress and overstress, the unified-constitutive equation, and the damage equation based on the overstress concept. Furthermore, a damage-rate equation and a coupling method between the unified-constitutive and the damage-rate equations are proposed and applied to the results of creep-fatigue tests for modified 9 C r - l M o steel at 550°C in air. TEST P R O C E D U R E S A N D TEST RESULTS

Test procedures The test material is a normalized and tempered modified 9Cr-1Mo forging steel, 250 m m thick. The chemical composition and heat treatment and the mechanical properties of the material are given in Tables 1 and 2,

Application o] overstress concept to inelastic behavior TABLE 1 Chemical C o m p o s i t i o n and Heat Treatment C Si Mn P S Ni

0-08 0"24 0-49 0-008 0.001 0"20

Cr Mo V Nb A N

8.54 0"96 0-19 0"06 0"010 0'036 ( W t % )

Normalized for 4 h at 1 0 5 0 C and tempered for 5h at 750 C.

TABLE 2 Mechanical Properties

Temperature

0"2% Proqf stress (MPa)

Tensile strength (MPa)

Elongation (% )

RT 55OC

490 321

627 351

23 32

TABLE 3 Variable-Strain-Rate Test Program

Step

~!, (s l)

~:t (at start)

At (s)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

10 s l0 3 10 - s -10 3 0 (hold) - 10- 3 - 10 s 10 3 10 6 10 4 0 (hold) 10 -4 --10 3 10 - 3 --10 -3

0"004 0.006 0"008 0"0 0"0 -0.004 - 0"009 0"006 0.008 0"010 0"010 0"012 0"0 0"005 0'0

400 2 200 8 3 600 4 500 15 2 000 20 3 600 20 12 5 5

101

Kosei Taguchi, Etsuro Kanno, Satoru Ozaki, Testuro Uno

102

ld; Modified 9 C r - ]Mo steel 550°C kU

/ F a t i g u e curve for symmetric " ~ cycling of ~t=lO-3s -1

<3 c"

L

10-2

/o ,o-' o I Is ,o-" o I l ~ ~o-~o-~ o I

-o

[

TM

[

~-~,o~o

L 10- 3

,

j ,

,,,,,I 103

,

102

,

....... 10 4

Number of cycles to failure Fig. I.

Nf

Relation between total strain range and number of cycles to failure for tests without strain hold.

ld t Modified 9 C r - l M o

steel

550% curve for symmetric

<3

¢

o•Fatigue g at ~t =10"3s'1

10-2

c i_. •

[0-a

0

Io"=

3600/0

0/600

io'= 0

0/36OO 3O0/30O

I0-a

600/600

i0-3

10- 3 10 2

I

I

I

i

i

i I

Number of cycles to failure Fig. 2.

i

i

, ii

10 4

10 3

Nf

Relation between total strain range and number of cycles to failure for strain-hold tests.

Application o f overstress concept to inelastic behavior

103

respectively. Solid-bar specimens were prepared with an 8-mm diameter. The test facility is an electro-hydraulic closed-loop system with a digital computer included. Push-pull strain-controlled creep-fatigue tests were carried out at 550°C in air. Creep tests were also completed under the same conditions. The creepfatigue tests include typical strain-time waveforms, such as symmetric and unsymmetric (fast-slow, slow-fast) continuous cycling, one with tensile and/or compressive hold times, and a test that is composed of a variety of strain-rate or strain-hold stages listed in Table 3.

Creep-fatigue test results The test results are summarized in Figs 1 and 2. Figure 1 shows the relation between the total strain range and the number of cycles to failure for the tests without a strain hold. Figure 2 shows the relation for the strain-hold cycling tests. The life reduction in fast-slow and compressive-strain-hold cycling is observed to the same degree as that in slow-fast and tensile-strain-hold cycling. However the life of the strain cycling with the same hold time at both tensile and compressive peaks is approximately equal to that for symmetric continuous cycling. These observation results were clarified from the relation between the number of cycles to failure and the difference between the tension-going time and the compression-going time at total strain ranges of 0-01 and 0.02, as shown in Fig. 3. The tension-going and the compression-going times are defined to include tensile and compressive strain-hold times, respectively. The authors therefore consider that the reduction in creep-fatigue life is caused by the time-dependent damage, which is induced by the difference 2~ 5xlOZ/ . I"

Modified 9Cr-IMo steel

I

.,.o ~

Ir~ . . . . }03~ -

55o-c

-- . . . . . . "

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103

. . . . . . . .

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IrT - T e l , s e e

Fig. 3.

Relation between number of cycles to failure and difference between tension-going time and compression-going time. Key as Figs 1 and 2.

104

Kosei Taguchi, Etsuro Kanno, Satoru Ozaki, Testuro Uno

between the tension-going time and the compression-going time for modified 9 C r - l M o steel at 550°C in air. 5 BACK-STRESS A N D OVERSTRESS D E T E R M I N A T I O N PROCEDURE By analyzing the stress-strain response during unloading, back stress and overstress were determined experimentally. Illustrations of stress and backstress behavior are shown in Figs 4(a) and (b). Unloading-stress-inelasticstrain curves on the first cycle and the half-life cycle for symmetric continuous cyclings of gt = 10-3 s-1 are shown in Figs 5(a) and (b). As shown in Fig. 4, a slight inelastic strain develops in the direction of prior loading, just after a load reversal at peak P. As unloading develops, the inelastic strain becomes almost constant (region Q-R) and then develops in the direction of present unloading. The back stress at peak P is defined as the stress value at point R, after which an inelastic strain develops in the unloading direction. The overstress at peak P is defined as the applied stress minus the back stress at the peak P. This procedure is almost the same as that proposed by Asada et al. ~ 4 and is based on the same concept of a 'stressdip' test. The test material shows typical behavior in cyclic-strain softening. By investigating the back-stress and overstress properties, it was shown that the overstress is almost constant. This feature is independent of both strain range and accumulated inelastic strain. 6 In Fig. 5, broken lines give the relation between back stress and inelastic strain. The monotonic-stress-inelastic-strain curve for gt = 10 7 s l, as shown in Fig. 5, is observed to be approximately parallel to the back-stress-inelasticstrain curve. The authors defined the corresponding overstress at the inelastic-strain rate of 10-Vs -~ as the stress difference for the stress inelastic-strain curve from the back-stress-inelastic-strain curve. 5'6 This O"

O"

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~:

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(b) Fig. 5.

Stress-strain behavior under unloading and determined back-stress-strain curve: (a) first cycle; (b) half-life cycle.

103

Modified 9Cr-IMo s/eel

550"C i

102

b P

o

101

~oo

i

10-7

I

10-6

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10-5

i

10-4

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10-3

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106

Kosei Taguchi, Etsuro Kanno, Satoru Ozaki, Testuro Uno

300

Modified 9Cr- IMo steel

i

'o

550~ L~Et=0.01

o,,,,,,,~

m

.~7

2OO . ~ 3_.z... . . . . .

"6

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I 10 -6

I 10 -5

I 10 -4

I 10 -3

10- 2

UnlodingStrain Rate ~JL , sec-1 Fig. 7.

Effect o f unloading-strain rate on evaluated overstress.

procedure means that the back stress is independent of the inelastic-strain rate and corresponds to the quasistatic stress, which is obtained by loading at an extremely slow strain rate. 7 The relation between overstress and inelastic-strain rate, determined by this procedure, is shown as open circles and a solid line in Fig. 6. 5.6 To verify this determination procedure, the values of the overstress determined by the stress-strain behavior during unloading with continuous cycling at a low strain rate and slow-fast and fast-slow cycling, are also plotted in Fig. 6. Because these values agree well with the previous values, which are shown as open circles in Fig. 6, the determination procedure is found to be applicable. However, there is a little difference between the overstress values, shown as open and closed circles at the 10-7 s- 1 inelastic strain rate. It was found that the evaluated overstress values increased as the unloading-strain rate decreased according to the data in Fig. 7. The difference between the two evaluated overstress values is therefore considered to be caused by the recovery of the back stress during the unloading. Determining the relation for the overstress-inelastic-strain rate, as shown in Fig. 6, will be required for the unified-constitutive and damage equations discussed later.

UNIFIED-CONSTITUTIVE MODEL A unified-constitutive equation, based on the overstress concept, has been proposed, 6 in which the internal-state variables coincide with the back stress and the overstress, determined experimentally. On the basis of the properties of the back stress and overstress, it is assumed that the overstress is a

Application of overstress concept to inelastic behavior

107

G

f" 0 ~ - /

--"a-ff

a, ~I El

~

12

Fig. 8. Illustration of constitutive model.

function of the inelastic-strain rate and that the cyclic strain-softening is caused by the softening of the back stress. Furthermore, two back stresses, a l and a 2, are assumed. The term a2 becomes constant rapidly, when an inelastic strain develops. Its constant value gradually becomes smaller, as the accumulated inelastic strain increases. Strain-hardening is simulated by the evolution of ~1. The thermal-recovery term is added to the evolution equation of ~2. The evolution equations for the back stresses are based on Chaboche's model s and are modified to correspond directly with the experimentally observed properties for the back stress. The unified-constitutive-equation model is illustrated in Fig. 8. It is expressed in the following forms under uniaxial stress conditions: d = E(~ t - ~) d, = F(la - ~1) sgn (o - a)

(3)

= (~1 "{- ~2 ~1 = C l ( a l e i -

(1) (2)

~l~i)

~2 -----C2{a2f(~l)/~i -- ~2~1} -- h2lct2la- 1~2

f(gi) = l + (1 - 1) exp ( -- 7gt) F(la - al) = A ( I a - al/K) m + B(la - :tl/K) n

(4) (5) (6) (7)

where o-, a(al, a2), a - a, e,, e~, g~, and E denote stress, back stress, overstress, total strain, inelastic strain, accumulated inelastic strain, and Young's modulus, respectively, and (') represents the partial derivative with respect to time t. Furthermore, F(la - ~1) signifies a function of the absolute value of the

108

Kosei Taguchi, Etsuro Kanno, Satoru Ozaki, Testuro Uno

400

4 0 0 ~Fost-slow cycling

I Fast-slow

300 ..~t'o.ot,~e~o~i ! 5500C /,~~

b

o~

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',

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9.

//

.[

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0.5 Et

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i

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i !

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-1

200 IOO

ID -500

30O -AEt = 0.01

Nf,,'2cycle

~°'),]rSimulatio~

-0.5

xlO - z

0 Strain

0.5 Et

xl(

Examples o f simulated results for fast-slow cycling tests.

overstress, which is represented in the form o f e q n (7), and a~, a 2, C 1, C2, l, 7, h 2, fl, A , B, m, n, and K are material constants.

The determination procedures for the material constants are as follows. First of all, the thermal-recovery term, in eqn (5), is assumed to be negligible. The constants in the F-function (eqn (7)) are determined by the relation between the overstress and the inelastic strain rate, shown in Fig. 6. The constants a~, C~, a 2, and C 2 a r e determined by the results of TABLE 4 Material Constants aI a2 a3 (/4 a5 ao A B C1 C2 E ]t 2 K l m n p q [~ ;'

88"69 (MPa) 121.2 (MPa) 2'111 x 10 4 ( M P a ) r 6'193 X 10 Iv ( M P a . s ) q 3.458x10 4(MPa) r 1.491x 10 8((MPa) °.s) 2 6 8 3 x 10 5 ( s - 1 ) 1.011 x 10 -3 (s i) 322"5 6716"0 1-85 x 105 (MPa) 3.858 x 10 2v (MPa)l-/J 173-2 (MPa) 0"244 9 3"902 24-91 1-214 2"484 11"81 0.9133

Application of overstress concept to inelastic behavior

109

1.4

1.2

~

Creep, 550"C (r =196MPo

(~:~plure)

,'I

o.e

0.2

~ ~ / "

0.%J

'

- - Experiment :.': } Simulotion

Fig. 10. Comparison between simulated and experimental results of creep strain and time~ ,ooo ,~bo 2ooo T i m e , hour dependent damage.

400= Tell.a Comp.llmln h o ~ . . - -- ~..~-" tX~,-O.02 ~ .?.~.~ .1¼cycle 500 0 n Nf/2 cycle =~ 200 Tn-600/C~~ 55o'c

"

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1 O0

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,

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~. 16 , Tens~ Comp. o 14 strain hold

j

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; 12 .zxet.o.o2,~rlO, s~I i0 .Th'6OO/6OOs 8 Nf/2cycle "o 6 550*(3 C~

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" ' - - --"

~

tz,oo

Time 1', s e c (b) Fig. I !. Comparison between simulated and experimental results of strain-cycling test, with both tensile and compressive strain hold: (a) stress-strain curve at first and half-life cycles; (b) stress and damage behavior during one cycle at half-life cycle.

110

Kosei Taguchi, Etsuro Kanno, Satoru Ozaki, Testuro Uno

O

Nf/2 cycle, , 5 ~ : ' C

4

~3 2 I

__=_.___=,__~.

f--~

8-2 Fig. 12. Comparison between simulated and experimental results of variable-strain-rate test (stress and damage behavior during one cycle at half-life cycle).

b

--

~penrnent

Z~=_} Simu~tion

-4

Time

~ sec

continuous cycling at gt = 10- 3 S 1. Based on the tensile-peak data for the first cycle at several strain ranges, the relation between back stress and inelastic strain can be assumed to be expressed by the following equation: = kl + k2{1 - e x p ( - k 3 8 i )

}

(8)

The constants k~, k2, and k 3 can be determined so as to correspond, respectively, to a2, a~, and C 1. The constant C2 is determined by the tensilepeak data for the first cycle at the minimum strain range, in order to simulate the back-stress behavior in the smaller-strain region. F o r the cyclic-softening behavior of continuous cycling at g, = 10-3 s-1, the constants l and ? in eqn (6) can be determined by regression analysis. Finally, the constants h2 and fl for the thermal recovery can be determined by the results of creep tests. Because the back stress can be considered to be steady (~ = 0) in the steady-creep region, the following equation is derived from eqns (3)-(6), and the constants h 2 and fl can be determined. Cl(a 1 - -

~l)gi

-~-

C 2 { a z f ( g i ) - ~2}g, = h2(~2) t~

(9)

Consequently, all the material constants are determined without any iterative alteration, as listed in Table 4. Some examples of the comparison between the simulated and the experimental results are shown in Figs 9-12. This constitutive model verified the possibility of using it to simulate the inelastic behavior under the creep-plasticity interaction. 6 T H E D A M A G E M O D E L A N D ITS A P P L I C A T I O N Basic damage model A creep-fatigue damage model, connected with the overstress, proposed by Asada et al. 1 - 4 was applied to the life prediction of the creep-fatigue tests for modified 9 C r - l M o steel at 550°C in air. 5

Application ~[" overstress concept to inelastic' behavior

111

The damage model is based on the following time-independent damage parameter D i and time-dependent damage parameter D d. I

(o--~)d~ i

( 1O)

Dd----- t" (a-- ~)dt dcycle

(11)

Oi = f

J~ycle

where the integration is made over one cycle. The D d definition was modified, because the creep-fatigue test results suggested that the timedependent damage during the tension- and the compression-goings developed with opposite signs, and the absolute value for the accumulated damage was time-dependent damage during one cycle. Based on the overstress, as shown in Fig. 6, Di and D d are calculated for one cycle at the half-life cycle. Furthermore, the following empirical equations are obtained from the correlation between D~ and ~b~( - 1/N r) for symmetric continuous cycling at gt = 10-3 s-x and from that between D o and ~b~( - 1 / N f - ~p~) for slow-fast cycling, in which a severe life reduction is observed: dp~ = a3D~ q5d = a4D~

(12) (13)

where ~b~ and ~bd denote time-independent and time-dependent damage during one cycle, respectively, and N r denotes the number of cycles to failure. The material constants determined (a3, a4, p, and q) are listed in Table 4. Finally, this damage model results in the following life-prediction equation: Nf(Oi + q~d)= 1

(14)

The predicted lives by eqns (10)-(14) were shown to agree well with the experimental results of creep-fatigue tests for modified 9 C r - l M o steel at 550°C in air. 5

Damage-rate model development The time-independent and the time-dependent damage parameters for the basic model previously described are defined as time integrals during one cycle and are correlated with each instance of damage during one cycle in the form of power-law equations. Difficulties thus arise when this model is applied to the result of a test in which the stress-strain loop is unclosed, such as a creep test. A damage-rate model was therefore considered to be required.

112

Kosei Taguchi, Etsuro Kanno, Satoru Ozaki, Testuro Uno

104 Modified9Cr-lMo

/

steel

//

/'/ (} , ~

" 103

O.

///// //

///////

"""

//"

2

////// ~7//

///

A/v

/ / / I0102

,A/

///~S,

550%

Symbolsas Figs.l,2

,'" , . . . . .

,J 3 ~o

.......

Io4

Nf obs. , cycles Fig. 13.

Comparison between observed and predicted lives by overstress-damage equations, based on experiments.

F r o m eqns (10) and (12), the time derivative of time-independent damage during one cycle can be expressed as follows: dt - a3 ~

(a

-- ~)gi

dt

(15)

Here, let us consider a case of symmetric continuous cycling. Assuming that the inelastic-strain rate and the overstress are constant during one cycle, the equation is extended as follows: dt

-

a3'[a

=

a3,pla

~fPI~IP~ -

--

dt

~lPlgilPtp- 1

-

(16)

where time t is reset at the start of every cycle. The following time-independent damage-rate equations can be given by substituting t for lei/g~l with the same dimensions: dDf

. -

asia

-

-

(zlp]~il p

e. pei

: asl~ - ~lPl~,l ~- 11~il

(17)

Furthermore, the time-dependent damage-rate equation can be expressed in the following same form as eqn (17), as expected from the analogy between eqns (10) and (1 1). dD¢ dt

_

a61a

_

~[ql~ilq- 1 sgn(a - ~)

(18)

Application o f overstress concept to inelastic behavior

113

Finally, a failure criterion is given as follows: Df q-IDol = Nr(~bi + q~d)= 1

(19)

In order to verify this damage-rate model, it was applied to the creepfatigue test results for modified 9Cr-1Mo steel at 550°C in air. The constants p and q have the same values as for the basic damage model. However, the constants as and a 6 were determined in the same manner as the determination of a 3 and a4. The material constants determined are listed in Table 4. The lives predicted by eqns (17)-(19) are compared with the experimental ones, as shown in Fig. 13. It was proved that the proposed damage-rate model can predict creep-fatigue lives with good accuracy.

LIFE P R E D I C T I O N BY T H E C O U P L I N G M E T H O D Since the proposed constitutive and damage-rate equations are based on the same overstress determined experimentally, it is possible, by coupling these equations, to simulate inelastic behavior and damage development simultaneously. The behavior of stress, back stress, overstress, strain, and damage is simulated by the unified-constitutive equation formulated in eqns (i)-(7) and by the damage-rate equation formulated in eqns (17) and (18). The life prediction can be obtained by the simulation, which is continued until the accumulated damage reaches the limit expressed in eqn (19). The proposed coupling method was applied to the results of creep and creep-fatigue tests for modified 9Cr-1 Mo steel at 550°C in air. Figure 10 shows the simulated and experimental creep strain. The creepstrain behavior is described fairly well by the unified-constitutive equation, except for the tertiary-creep-strain behavior, which is not taken into account in this model. The simulated time-dependent damage develops rapidly as the creep strain increases. The predicted and experimental creep-rupture times are compared in Fig. 14. The predicted rupture times agree well with the experimental results. Figure 11 shows an example of the simulated and experimental results of the strain-cycling test, with both a tensile and a compressive strain hold. This figure indicates the stress-strain curve at the first cycle and the half-life cycle, and the stress and damage behavior during a period at the half-life cycle. The simulated time-independent and time-dependent damage developments are non-linear during one cycle. Their accumulated damage values are almost constant at the end of every cycle, with a little difference caused by the cyclic strain-softening. Figure 12 shows a comparison between simulated and experimental

114

Kosei TaguchL Etsuro Kanno, Satoru Ozaki, Testuro Uno

lo4

.

Modified9Cr-IMosteel 550°C

//" / /

///~'Xf

////

.

~' ~ T°3

"Z 11 I I

~-

o

,,'o

"o O_

~//////

/

"D

I ~

j/

i

I

,,l~ v

i

,,'"

/" /

iiiii

"1 0 2

t

i

i

,

,

, , ,I

I0 3 Observed

10 4

volues N f , cycles or Tr , hours

Fig. 14. Comparison between observed and predicted lives by coupling method between constitutive and damage-rate equations, based on overstress concept. A , Test as Table 3, 0 , creep; other symbols as Figs 1 and 2.

results of a test that is composed of a variety of strain-rate or strain-hold stages listed in Table 3. It is also proved that the inelastic behavior and creepfatigue life can be predicted by the coupling method, as shown in Figs 12 and 14. Consequently, the lives predicted by the coupling method agree fairly well with the experimental results of all the creep-fatigue tests, as shown in Fig. 14.

C O N C L U D I N G REMARKS For modified 9 C r - l M o steel, creep-fatigue tests were conducted at 550°C in air. The test results suggested that the reduction in creep-fatigue life was caused by the time-dependent damage, which was induced by the difference in the tension-going time from the compression-going time. The back stress and the overstress were phenomenologically determined by investigating the unloading curve for the stress-strain response. A unified-constitutive equation, based on the overstress concept, was proposed and proved to be able to simulate inelastic behavior for creepplasticity interaction. Based on the creep-fatigue test results, the damage equation connected with the overstress was improved, and a damage-rate equation was proposed. The proposed constitutive and damage-rate equations are based on the

Application of overstress concept to inelastic behavior

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same internal-state variables, which coincide with back stress and overstress determined experimentally. By coupling these equations, it is possible to simulate inelastic behavior and damage development simultaneously. The proposed method was applied to the results of creep and creepfatigue tests and proved to be capable of predicting the lives with a good degree of accuracy.

REFERENCES 1. Morisita, M., Taguchi, K., Satake, M., Ishikawa, A. & Asada, Y., Creep-fatigue behavior of 304 stainless steel in vacuum. In Proc. ICPVT5, Part II. ASME, New York, 1984, p. 1109. 2. Morisita, M., Taguchi, K., Asayama, T., lshikawa, A. & Asada, Y., Application of overstress concept for creep-fatigue evaluation. A S T M STP 942. American Society for Testing and Materials, Philadelphia, PA, 1988, p. 487. 3. Asayama, T., Cheng, S., Tachibana, Y., Ishikawa, A. and Asada, Y., Creepfatigue behavior of2¼Cr-I Mo steel in air and vacuum. In Low-Cycle Fatigue and Elasto-Plastic Behavior of Materials, ed. K. T. Rie. Elsevier Applied Science Publishers, London, 1987, p. 265. 4. Asayama, T., Cheng, S., Asada, Y., Mitsuhashi, S. & Tachibana, Y., Creepfatigue behavior of 2¼ 1 Mo steel at 550°C in air and vacuum. In Proc. 9th SMiRT, Vol. L, 1987, p. L6/1. 5. Taguchi, K., Kanno, E., Ozaki, S. & Uno, T., Creep-fatigue life evaluation based on the overstress concept for modified 9Cr-I Mo steel. J. Soc. Mater. Sci. Japan, 38 (1989) 1316. 6. Taguchi, K. & Uno, T., Application of unified constitutive equation to inelastic deformation behavior of modified 9Cr-lMo steel at 550°C. Trans. Japan Soc. Mech. Engrs A, 55 (1989) 1560. 7. Krempl, E., Viscoplasticity based on total strain. J. E M T (A SME ) 101 (1979) 380. 8. Chaboche, J. L. & Rousselier, G., On the plastic and viscoplastic constitutive equations: Part I--Rules developed with internal variable concept. Part II-Application of internal variable concepts to the 316 stainless steel, J. P V T ( A S M E ) 105 (1983) 153.