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Effect of nonproportional loading on creep-fatigue properties of 304 stainless steel at low strain ranges near the elastic region
Nuclear Engineering and Design 139 (1993) 299-309 North-Holland
299
Effect of nonproportional loading on creep-fatigue properties of 304 stainless steel at low strain ranges near the elastic region Tai Asayama, K a z u m i A o t o a n d Y u u s a k u W a d a Materials Development Section, Oarai-Engineering Center, Power Reactor and Nuclear Fuel Development Corporation, 4002 Narita, Oarai-machi, Ibaraki 311-13, Japan
Received 3l August 1992
The effect of nonproportional strain path on fatigue/creep-fatigue properties was investigated with 304 stainless steel at 550°C under strain controlled biaxial conditions. The fatigue/creep-fatigue life reduction due to nonproportional strain path occurred even at the lowest strain range investigated, that is, 0.2% for fatigue loading and 0.3% for creep-fatigue loading. The Mises-type path-dependent equivalent strain range was employed in order to evaluate the fatigue/creep-fatigue strength under nonproportional loading conditions. Stress relaxation behavior under nonproportional loading was examined. It was shown that stress relaxes proportionally toward the origin of stress plane even under nonproportional loading. Fatigue damage and creep damage were calculated based on the linear damage summation rule. Life prediction was shown to be possible within an accuracy of a factor of about 2 for nonproportional loading along with other waveforms including pure axial loading, pure torsional loading and combined proportional loading.
I. Introduction Damage assessment under multiaxial loading conditions is one of the most important concerns on the improvement of creep-fatigue damage evaluation methods in the high temperature structural design codes. The material behavior under multiaxial loading conditions is complicated compared with that under uniaxial loading. But the damage assessment procedure is required to be a simple as possible and have a sufficient experimental background. From this viewpoint, the authors have been developing a multiaxial creep-fatigue damage assessment method based on the concept of uniaxialization of strain histories of structural component which are subjected to multiaxial loading [1-4]. This approach enables us to evaluate multiaxial creep-fatigue on the basis of the uniaxial material properties which the authors have extensively accumulated for a number of structural materials for fast breeder reactors. It is widely known that considerable fatigue life reduction and the additional strain hardening occur
under nonproportional combined loading [5,6]. But most of the studies reported so far have dealt with fatigue properties at relatively higher strain ranges. Few studies have been reported concerning the fatigue properties at low strain ranges near the elastic region for creep-fatigue properties under nonproportional loading, which have practical importance. The objectives of this paper are to provide observations regarding the fatigue/creep-fatigue strength of SUS304 stainless steel under biaxial loading at relatively low strain ranges near the elastic region and to propose a creep-fatigue damage assessment method. The minimum strain range employed for nonproportional loading was 0.2% for fatigue tests and 0.3% for creep-fatigue tests. The fatigue life reduction and the additional strain hardening due to the nonproportionality of the strain path imposed was examined and an equivalent strain range was employed for the uniaxialization of strain histories. Based on this equivalent strain range, fatigue damage and creep damage were evaluated and the effects of the nonproportional strain path on fatigue damage and creep damage were dis-
0 0 2 9 - 5 4 9 3 / 9 3 / $ 0 6 . 0 0 © 1993 - Elsevier Science Publishers B.V. All rights reserved
T. Asayarna et al. / Effect of nonproportional loading
300
cussed. Life prediction was conducted by the linear damage summation rule utilizing the uniaxial material properties the authors had extensively accumulated.
2. Experimental procedure
2.1. Material and test specimen The material used in the present study was SUS304 stainless steel from a hot rolled plate (1000 x 1000 × 40 ram) which was solution annealed (1100°C ×48 rain). The chemical composition and the mechanical properties at room temperature are shown in table 1. Hollow cylinder specimens of 30 mm parallel gauge length with 22 and 15 mm outside and inside diameters respectively were machined after the heat treatment. The configuration of the specimen is shown in fig. 1.
mum life reduction is observed when a strain hold period is imposed at a tensile strain peak. For pure torsional loading, a strain hold period of 1 hour was imposed at an extreme value of the torsional strain. For combined proportional loading and nonproportional loading, both axial strain and torsional strain were held simultaneously. The loading conditions are summarized in table 2. The strain rate in the loading/unloading portion was a nominal value of 0.1%/s which was defined by Aeeq/r, where Aeeq and v are ASME type equivalent strain range (see 3.1 in this paper) and cycle time respectively. Fatigue/ creep-fatigue life Nf was defined as a number of cycles when a 25 percent fall in axial load or torsional load was observed.
3. Results of experiment
3.1. Stress-strain behavior under biaxial conditions 2.2. Test facility and test conditions
A hydro-servo axial-torsional fatigue test machine was utilized. The fatigue and creep-fatigue tests were conducted under strain controlled conditions at 550°C in air. The axial strain and torsional strain were controlled and measured independently. A triangular waveform and sinusoidal waveform were employed. Loading conditions included uniaxial loading, pure torsional loading, combined proportional loading and nonproportional loading. For combined loadings, a biaxiality factor defined by A = A y / A e was 1.7. For nonproportional loading, a phase angle of 90 degrees was introduced. For all of these loading conditions, fatigue tests (no strain hold period is imposed) and creep-fatigue tests (a strain hold period is imposed) were conducted. For creep-fatigue tests, a strain hold period of 1 hour was imposed at a tensile peak of axial strain except pure torsional loading, considering the fact that under uniaxial loading conditions the maxi-
ASME type equivalent strain range Ae~q and Mises equivalent stress range Atreq are widely used in evaluation of stress-strain response under multiaxial conditions. Those are defined by the following equations. AEeq = (Ae 2 + A y e / 3 ) '/z,
(1)
3 AT2)1/2,
(2)
atreq = (a~r z +
where de, a y , Atr, zl~- are axial strain, torsional strain, axial stress and torsional stress, respectively. Figure 2 shows the Mises equivalent stress range obtained in the present study as a function of ASME type equivalent strain range. The additional strain hardening due to the nonproportionality of the strain path [7,8] is obviously observed. Mises equivalent stress range under nonproportional loading increased as much as about 30% above a strain range of 0.5% compared with uniaxial fatigue loading. But the amount of increase
Table 1 Chemical composition (wt%) and mechanical properties of the tested material 1100°C x 48 min WQ
C
Si
Mn
P
S
Ni
Cr
Mo
Ti
AI
N
Nb + Ta
0.050
0.60
0.87
0.026
0.002
8.94
18.59
0.11
< 0.002
0.013
0.019
< 0.003
Yield strength (MPa)
Tensile strength (MPa)
Elongation (%)
213.6
623.3
71.9
T. Asayama et al. / Effect of nonproportional loading
301 C0.5
+i
3.2S (,q
70.0
90.0
-,-
260.0
Fig. 1. Test speomen.
path dependent strain range AEpth [1] was employed instead of the ASME type equivalent strain range. /l~-pth is defined as follows.
was less at lower strain ranges and it was about 15% at a strain range of 0.2% near elastic region. Almost the same amount of increase in Mises equivalent stress range was observed for nonproportional creep-fatigue loading, too. For proportional loadings, the pure torsional fatigue/creep-fatigue loading showed a somewhat smaller stress range compared with uniaxial loading and combined proportional loadings. In order to estimate the additional strain hardening under nonproportional loadings, the Mises type strain
A%, h = / d e e q ,
(3)
where deeq is an increment of Mises type equivalent strain. Integration is conducted between the two points in a strain cycle where the axial strain takes extreme values. In eq. (3), AEpth is calculated based on the total strain instead of plastic strain which is used in the original Mises equivalent strain range. The relationship between the Mises type path dependent equivalent strain r a n g e Aept h and Mises equivalent stress range A(req is shown in fig. 3. It is observed that the estima-
Table 2 List of waveforms
800
7/,,3 Uniaxial loadi~
Triangular ~avefor~ Strain hold period = O, 1 hour
e
~
700
t
600
k=O, ~=0
• v
Torsional loading Trlaugul ar ~ f ot-~ Strain hold period = O, 1 hour
o~ k=~, ¢=0 ,
proportional
4
loadi~
500 400
k=l.7, ¢=0
"5 o-
ta
Si=idal
~fom
Strain hold period = 0 hour
k.A_/ a =1.7, ¢=90*
hold period
= I hour
UniaxialFatigue \0.1%/'s~.
/// ~,
• 304 Steel,
k
/
0
/
550°C
¢ (°l
th(min) 0
60 i
I
0
--!
W
1,7
•x
1.7
o'.6 Strain
/
/
,~/ i 200
~-
Nouproportinnal Loading
~// ~
v, /
=
Trla~ul ar wave f orm
Strain hold period = O, i hour
"~ i
7/~3
C~in~d
v
I'.0
[ 60 I
[
90
1'.5
0
°° 0 6O
2'.o
ASME Type Equivalent Strain Range ,3.~eq%
k=l.7, ¢=90°
Fig. 2. Stress-strain response represented by ASME type strain range.
T. Asayama et al. / Effect of nonproportional loading
302
A~eq =1.0% k =1.7 =90' t h =60min N f = 164cycle
800 z~ 700
o
v
g b
600
v ,.or.v / "
~o 500 /
x
-I00 304 Steel,
c
_~ 300 200
. . " [] \Uniaxial Fatigue z~ 0~'1% "sec'
0
I
/
X I--
r- 400 zE I
/ I'"
Z& /V// |
400
ore
/
/v
i-_
X
0
550°C
I I _lJO0 -400 I~00 -200
¢,{°) ! t h (rain) 0 60
/
©
O °o
I
I
1.7 1.7
90
I
1.0
1.5
(7
MPa
7°°o/
0 60
lO0
015
40 2Go -100
O
;~00
u_400
2t.0
Path DependentEquivalentStrainRange ~ pth % Fig. 3. Stress-strain response represented by the Mises type path dependent strain range.
tion of additional hardening was improved by Aepth, although the scatter is not negligible.
3.2. Stress relaxation behauior under biaxial conditions It is very important to examine the behavior of stress relaxation under nonproportional strain path to check if the creep damage evaluation method for uniaxial loading can also be applied to the nonproportional strain waveforms. Figure 4 shows the stress path of nonproportional loading. Stress moves counterclockwise during the loading/unloading portion and relaxes proportionally toward the origin of the stress plane during the strain hold period, just like the proportional combined loading. The fact that the stress relaxation occurs proportionally toward the origin of the stress plane even under nonproportional loading indicates the possibility that creep damage evaluation under nonproportional loading can be performed based on the same method as the uniaxial loading. The details will be examined later. It is reported that under certain multiaxial conditions, the amount of stress relaxation is reduced compared with that under uniaxial conditions [7,8]. Figure 5 shows the amount of stress relaxation (Mises equivalent stress) under various loading conditions as a func-
Fig. 4. Proportionality of stress relaxation under various loading conditions. tion of initial stress. It is observed that the amount of stress relaxation is reduced under biaxial conditions, particularly under nonproportional loading.
3.3. Fatigue ~creep-fatigue life under biaxial conditions The results of biaxial fatigue/creep-fatigue tests are shown in fig. 6 in t e r m s o f a relationship between A S M E type equivalent strain range Aeeq and the n u m b e r of cycles to failure Nf (see 2.2 for the definition o f Nf).
Fatigue life For proportional loadings, there was a slight difference in fatigue life. The fatigue life was in the order of pure 120
304 Steel,
550"C
~ lO0 i io
[_
k ~ ~l°:Iith(min~
80 °
g- 6o
I•
q
-7i~"
00/
/ /
/
•,~
~- 40 'S
=
e/ /m
/
•
/
20 0
/
I O0
200
Initial Stress of Relaxation
300 o" max
400
MPa
Fig. 5. A m o u n t of stress relaxation under nonproportional
loading.
T. Asayama et aL / Effect of nonproportional loading g
1o I
304 Steel, 550"(2
. . . .
l0 0
/ v ~UUniaxLalFatigue, 0.1% see
lO-1 102
103
104
Number of Cycles to Failure
105
n
106
107
Cycles
Fig. 6. Result of biaxial fatigue/creep-fatigue tests.
torsional loading, combined proportional loading and uniaxial loading. For nonproportional loadings, a magnificent fatigue life reduction was observed. Life reduction was still observed at the lowest strain range investigated, that is, 0.2%, where hysteresis loops of this specimen showed that the deformation was almost elastic. The amount of life reduction showed no particular dependence on the strain range and the fatigue life of nonproportional combined loading was reduced to 1 / 4 - 1 / 2 0 of that of uniaxial loading. It was revealed that the effect of the nonproportional strain path on the fatigue strength of this material still existed at a strain range near the elastic region.
Creep-fatigue life Creep-fatigue life was reduced compared with fatigue life under the same ASME type strain range for all types of loading, including nonproportional loading. The amount of life reduction was almost the same irrespective of the strain waveform. It was made clear that creep-fatigue life reduction occurs under nonproportional loading just as under proportional loadings. 3.4. SEM observations For uniaxial loading, a clear relationship has been observed between loading condition and fracture mode with 304 stainless steel [10] and this observation forms the background of the creep-fatigue evaluation method. That is, fracture was transgranular for fatigue loading and intergranular for creep-fatigue loading. In the present study, a scanning electron microscope (SEM) observation was made to check if the above observation obtained for uniaxial loading also hold true for biaxial loading conditions including nonproportional loading.
303
Figure 7 shows the result of the SEM observation. A Fracture surface of non-proportional fatigue loading (fig. 7d) is similar to that of torsional fatigue loading (fig. 7b) which showed a transgranular fracture mode with typical rubbing marks. Striation marks which were observed in the case of uniaxial loading (fig. 7a) and combined proportional loading (fig. 7c) were not observed in the case of nonproportional fatigue loading. For nonproportional creep-fatigue loading (fig. 7d), the fracture surface is similar to those of uniaxial creep-fatigue loading (fig. 7a) and combined proportional creep-fatigue loading (fig. 7c), showing a typical intergranular fracture mode. Therefore it was made clear that the fracture mode was transgranular for fatigue loading and intergranular for creep-fatigue loading under nonproportional loading conditions, which is the same as the case of uniaxial loading. And as far as fracture surface observations concerns, the effect of the nonproportionality of the strain path is evident only for fatigue loading and is not observed for creep-fatigue loading. This is because the strain hold period was imposed at a tensile peak of axial strain when the torsional strain is zero.
4. Evaluation of the test results
4.1. Creep-fatigue evaluation method for uniaxial condition The authors have developed a creep-fatigue life prediction method for uniaxial loading conditions. The method [11-14] is summarized here. The following relationships are used in the method. All those relationships were obtained based on uniaxial test results with various heats of 304 steel. (a) fatigue life versus strain range, (b) strain range versus stress range at mid-life, (c) creep strain versus time, (d) stress versus creep rupture time. The method is as follows: (1) Fatigue damage Df is calculated as a fatigue life fraction consumed in the cyclic loading as expressed by the following equation:
Df = N / N f I a,,
(4)
where Nf is the fatigue life that corresponds to the applied strain range Ae in the relationship between fatigue life versus strain range. N is the number of strain cycles imposed.
304
T. Asayama et al. / Effect qf nonproportional loading
(2) Creep damage is calculated as the lime fraction consumed during the strain hold periods as expressed by the following equation: th
D~= Nf dt/tr(~r), o
(5)
where t~(o-) is the uniaxial creep rupture time under stress o-. t h is the strain hold time per cycle. The integration is conducted for one cycle. The stress relaxation behavior needed to calculate eq. (5) is determined by the following procedure. (3) The initial stress at a stress relaxation is obtained as a half of a stress range that corresponds to the applied strain range on the relationship between strain range versus stress range at mid-life.
(4) Stress relaxation curve is estimated based on the relationship between time versus uniaxial creep strain, and the classical strain hardening law. (5) A creep-rupture time for a given stress is obtained by the relationship between stress versus rupture time. (6) Creep-fatigue life is calculated based on the Campbell criterion.
4.2. Fatigue damage under biaxial condition The authors have proposed that the Mises type strain path dependent equivalent strain range depth bc used for estimation of the additional strain hardening that occurs under non-proportional loading. Based on
(a) Fatigue t h=0 hour Aa eq = 1 . 0 %
Nf=1339
l
u
l
50/.zm Creep-Fatigue th----1 hour Ae eq = 1 . 5 %
N f--304cycles
I _ _ I
50,u m
Fig. 7. SEM observation; (a) uniaxial loading, (b) torsional loading, (c) combined proportional loading, and (d) nonproportional loading.
T. Asayama et al. / Effect of nonproportional loading this fatigue damage under biaxial condition is assumed to be expressed as follows:
Df = N/Nt-la%th.
(6)
For proportional loading, the ASME type equivalent strain range and Mises type path dependent equivalent strain range give the same value. For nonproportional loading, the Mises type path dependent equivalent strain range gives a larger value compared with the ASME type equivalent strain range and the difference between the two is maximum at a phase angle of 90 degrees. In this case, AEpt h equal ~r/2 times AEeq.
4.3. Creep damage under biaxial condition The initial stress of the stress relaxation was calculated to be half of the stress range that corresponds to
305
the Mises type path dependent equivalent strain range obtained by eq. (3). It is to be noted that by eq. (3) the initial stress for nonproportional loading is estimated to be lower than the actual value. It is because the estimation of the additional strain hardening by eq. (3) is not enough as mentioned in section 3.1. In the present study, for simplicity, no further discussion was made on this point. Based on the proportionality of stress relaxation under nonproportional loading (see fig. 4), the biaxial stress relaxation curve was assumed to be estimated by the relationship between time versus creep strain, where the creep strain is calculated using the Mises equivalent stress defined by eq. (2). The amount of stress relaxation depends on the type of loading as shown in fig. 5. But in the present study, for simplicity, the difference of the amount of
(b) Fatigue t h=0 hour A e e q = l .0% N f =2691 cycles
I
I
50~um Creep-Fatigue t h = l hour Aeeq = 1.0% N f = 1328cycles
I
I
50/~m Fig. 7 (continued).
7".Asayama et al. / Effect of nonproportional loading
306
relaxation was not taken into account. The relationship between the initial stress and the amount of relaxation estimated by the relationship between creep strain and time obtained under uniaxial condition is shown in fig. 5 by a broken line. This relationship was used for the evaluation of stress relaxation behavior of both the proportional loading and the nonproportional loading. It was also assumed that stress versus time curve in stress relaxation under biaxial conditions was identical to that under uniaxial condition. The relationship between stress and time to failure, where stress is represented in terms of the Mises equivalent stress, was used to evaluate the creep damage under biaxial conditions. Creep damage under
biaxial conditions was assumed to be expressed by the following equation.
D~
=
N jdt /tr(cr~q).
(7)
4.4. Result of life prediction The fatigue damage and the creep damage calculated by the above procedure are plotted on the Campbell diagram in fig. 8. It is seen that damage assessment was fairly reasonable except for pure torsional loading which was estimated somewhat overconservatively. The result of life prediction is shown in fig. 9. For most of the data, creep-fatigue life could be pre-
{c) Fatigue t h = 0 hour
A~ eq = 1 . 0 % N f =2274cycle-~
k
I
50t~ m Creep-Fatigue t h = l hour A~ eq = 1 . 0 % N f=727cycles
50/z m Fig. 7 (continued).
T. Asayama et al. / Effect of nonproportional loading dicted within an accuracy of factor of 2. Nonproportional fatigue loading at the lowest strain range investigated and pure torsional creep-fatigue loading were estimated conservatively. 4.5. Estimation of creep damage at low strain ranges near elastic region From the above observations, it was shown that fatigue/creep-fatigue strength under proportional loading and nonproportional loading was reasonably evaluated by the method proposed in the present study at least for strain ranges above 0.2% for fatigue strength and 0.3% for creep-fatigue strength, although the nonproportional fatigue at 0.2% was estimated conservatively. In this section we will discuss the creep-fatigue life evaluation at strain ranges below 0.3%.
307
As stated above, by the method proposed in the present study, the creep damage is calculated based on three key properties obtained under uniaxial conditions, that is, the initial stress of stress relaxation, the amount of stress relaxation and the creep strength. In the present study, a simplification was made in the evaluation of the initial stress and stress versus time curve in stress relaxation under biaxial conditions. That is, the initial stress was assumed to be obtained as half of the stress range that corresponds to the Mises type path dependent equivalent strain range and the stress versus time curve was assumed to be obtained based on the uniaxial creep strain versus time curve by using Mises equivalent stress. Therefore, effects of the simplification at lower strain ranges are to be considered. In the low strain ranges near the elastic region, the difference of the initial stress between experiment and
(d)
Fatigue t h---0 hour A~eq = 1 . 0 % N f =312cycles
50/1 rn
Creep-Fatigue th--1 hour A~eq =1.0% N f=164cycles
[
I
50/z m
Fig. 7 (continued).
T. Asayama et aL / Effect of nonproportional loading
308
damage evaluation method proposed in the prcsent study is also valid at lower strain ranges below 0.3%.
304 Steel, 550°C I0 0
~
•
5. Conclusion
121
bO
Campbell
r~ 10-1
~ k ~ # (°)
l,:ili-:
°
;,t 10 -2
1
th(min)
0
17 i 9~-~ l
J
I
i
lO -l
lO 0
Of
Fatigue Damage
Fig. 8. Creep-fatigue damage assessment based on the method proposed in the present study.
estimation would be small because the amount of the additional strain hardening due to nonproportional strain path is very small at low strain ranges as indicated in fig. 3. The amount of stress relaxation is smaller at lower strain ranges compared with that at higher strain ranges so that the dependence of the amount of stress relaxation on waveform would be smallcr as inferred from fig. 5. To sum up, the effect of the simplifications made in the present study is considered to be smaller at lower strain ranges and it is implied that the creep-fatigue
10 6 u~
(3
SUS304Steel, 550°C
Factor of 2 I f z / Z / / ~
105
/ ~/ / // t t/v
104
//
J "IO
J'/
/t "'//"//
A series of axial-torsional biaxial creep-fatigue tests were performed with 304 stainless steel at 550°C to check the effect of the strain path imposed on the fatigue and creep-fatigue strength. The following observations were made: (1) Fatigue strength under nonproportional loading conditions reduced to 1 / 4 - 1 / 2 0 of that under proportional loading. This tendency was still observed at a low strain range of 0.2% which is very close to the elastic region. (2) Creep-fatigue u n d e r n o n p r o p o r t i o n a l loading showed life reduction compared with fatigue under nonproportional loading. The amount of life reduction due to strain hold period was almost the same as in the case of proportional loading. (3) Proportional stress relaxation was obscrvcd even under nonproportional loading. (4) Transgranular fracture was observed for nonproportional fatigue loading and intergranular fracture was observed for nonproportional creep-fatigue loading. It was the same observation as that made for proportional loading. (5) Fatigue damage was defined based on the Mises type path dependent equivalent strain range and creep damage was defined based on the Mises equivalent stress. Fatigue damage and creep damage were evaluated based on the uniaxial material properties extensively obtained by PNC. (6) The result of damage evaluation was fairly reasonable and life prediction was possible for most of the data within an accuracy of factor of 2.
,,g/~: ~;,g'/
10 3
References
/1 / /
/
I°
o
I~.~
t ~-t ' o UE1
(3
n
102
103 Observed Life
104
105 NfOBS
Fig. 9. Result oflifeprediction.
106 Cycles
[1] T. Sakon et al, Proc. of 9th SMiRT, Vol. L (1987) 267. [2] T. Asayama, K. Aoto and Y. Wada, Trans. of the Japan Society of Mechanical Engineers 56 (1989) 532. [3] T. Asayama, K. Aoto and Y. Wada, Proc. of the 27th Symposium of Strength of Materials at High Temperatures (1989) 112. [4] T. Asayama, K. Aoto and Y. Wada, The 6th Int. Conf. on Mechanical Behavior of Materials 2 (1991) 311. [5] E. Krempl and H. Lu, Trans. of ASME. J. of. Eng. Mater. Technol. 106 (1984) 376. [6] A. Benallal and D. Marquis, Trans. of 9th SMiRT, Vol. L (1987) 385.
T. Asayama et al. / Effect of nonproportional loading [7] L.K. Severud and B.V. Winkel, Proc. of 9th SMiRT, Vol. L (1987) 123. [8] L.K. Severud, PVP 163 (1989) 1. [9] Y. Wada et al., PVP 123 (1987) 37. [10] M. Morishita et al., Nuclear Engineering and Design 83 (1984) 367.
309
[11] K. Aoto et al., PVP 123 (1987) 43. [12] Y. Wada et al., Proc. Int. Conf. on Computational Mechanics 1 (1986) 4. [13] A. Yoshitake et al., Proc. Int. Conf. on Creep (1986) 441. [14] H. Kawasaki, H. Kagawa, K. Aoto, Y. Wada and I. Nihei, PVP 172 (1989) 29.
299
Effect of nonproportional loading on creep-fatigue properties of 304 stainless steel at low strain ranges near the elastic region Tai Asayama, K a z u m i A o t o a n d Y u u s a k u W a d a Materials Development Section, Oarai-Engineering Center, Power Reactor and Nuclear Fuel Development Corporation, 4002 Narita, Oarai-machi, Ibaraki 311-13, Japan
Received 3l August 1992
The effect of nonproportional strain path on fatigue/creep-fatigue properties was investigated with 304 stainless steel at 550°C under strain controlled biaxial conditions. The fatigue/creep-fatigue life reduction due to nonproportional strain path occurred even at the lowest strain range investigated, that is, 0.2% for fatigue loading and 0.3% for creep-fatigue loading. The Mises-type path-dependent equivalent strain range was employed in order to evaluate the fatigue/creep-fatigue strength under nonproportional loading conditions. Stress relaxation behavior under nonproportional loading was examined. It was shown that stress relaxes proportionally toward the origin of stress plane even under nonproportional loading. Fatigue damage and creep damage were calculated based on the linear damage summation rule. Life prediction was shown to be possible within an accuracy of a factor of about 2 for nonproportional loading along with other waveforms including pure axial loading, pure torsional loading and combined proportional loading.
I. Introduction Damage assessment under multiaxial loading conditions is one of the most important concerns on the improvement of creep-fatigue damage evaluation methods in the high temperature structural design codes. The material behavior under multiaxial loading conditions is complicated compared with that under uniaxial loading. But the damage assessment procedure is required to be a simple as possible and have a sufficient experimental background. From this viewpoint, the authors have been developing a multiaxial creep-fatigue damage assessment method based on the concept of uniaxialization of strain histories of structural component which are subjected to multiaxial loading [1-4]. This approach enables us to evaluate multiaxial creep-fatigue on the basis of the uniaxial material properties which the authors have extensively accumulated for a number of structural materials for fast breeder reactors. It is widely known that considerable fatigue life reduction and the additional strain hardening occur
under nonproportional combined loading [5,6]. But most of the studies reported so far have dealt with fatigue properties at relatively higher strain ranges. Few studies have been reported concerning the fatigue properties at low strain ranges near the elastic region for creep-fatigue properties under nonproportional loading, which have practical importance. The objectives of this paper are to provide observations regarding the fatigue/creep-fatigue strength of SUS304 stainless steel under biaxial loading at relatively low strain ranges near the elastic region and to propose a creep-fatigue damage assessment method. The minimum strain range employed for nonproportional loading was 0.2% for fatigue tests and 0.3% for creep-fatigue tests. The fatigue life reduction and the additional strain hardening due to the nonproportionality of the strain path imposed was examined and an equivalent strain range was employed for the uniaxialization of strain histories. Based on this equivalent strain range, fatigue damage and creep damage were evaluated and the effects of the nonproportional strain path on fatigue damage and creep damage were dis-
0 0 2 9 - 5 4 9 3 / 9 3 / $ 0 6 . 0 0 © 1993 - Elsevier Science Publishers B.V. All rights reserved
T. Asayarna et al. / Effect of nonproportional loading
300
cussed. Life prediction was conducted by the linear damage summation rule utilizing the uniaxial material properties the authors had extensively accumulated.
2. Experimental procedure
2.1. Material and test specimen The material used in the present study was SUS304 stainless steel from a hot rolled plate (1000 x 1000 × 40 ram) which was solution annealed (1100°C ×48 rain). The chemical composition and the mechanical properties at room temperature are shown in table 1. Hollow cylinder specimens of 30 mm parallel gauge length with 22 and 15 mm outside and inside diameters respectively were machined after the heat treatment. The configuration of the specimen is shown in fig. 1.
mum life reduction is observed when a strain hold period is imposed at a tensile strain peak. For pure torsional loading, a strain hold period of 1 hour was imposed at an extreme value of the torsional strain. For combined proportional loading and nonproportional loading, both axial strain and torsional strain were held simultaneously. The loading conditions are summarized in table 2. The strain rate in the loading/unloading portion was a nominal value of 0.1%/s which was defined by Aeeq/r, where Aeeq and v are ASME type equivalent strain range (see 3.1 in this paper) and cycle time respectively. Fatigue/ creep-fatigue life Nf was defined as a number of cycles when a 25 percent fall in axial load or torsional load was observed.
3. Results of experiment
3.1. Stress-strain behavior under biaxial conditions 2.2. Test facility and test conditions
A hydro-servo axial-torsional fatigue test machine was utilized. The fatigue and creep-fatigue tests were conducted under strain controlled conditions at 550°C in air. The axial strain and torsional strain were controlled and measured independently. A triangular waveform and sinusoidal waveform were employed. Loading conditions included uniaxial loading, pure torsional loading, combined proportional loading and nonproportional loading. For combined loadings, a biaxiality factor defined by A = A y / A e was 1.7. For nonproportional loading, a phase angle of 90 degrees was introduced. For all of these loading conditions, fatigue tests (no strain hold period is imposed) and creep-fatigue tests (a strain hold period is imposed) were conducted. For creep-fatigue tests, a strain hold period of 1 hour was imposed at a tensile peak of axial strain except pure torsional loading, considering the fact that under uniaxial loading conditions the maxi-
ASME type equivalent strain range Ae~q and Mises equivalent stress range Atreq are widely used in evaluation of stress-strain response under multiaxial conditions. Those are defined by the following equations. AEeq = (Ae 2 + A y e / 3 ) '/z,
(1)
3 AT2)1/2,
(2)
atreq = (a~r z +
where de, a y , Atr, zl~- are axial strain, torsional strain, axial stress and torsional stress, respectively. Figure 2 shows the Mises equivalent stress range obtained in the present study as a function of ASME type equivalent strain range. The additional strain hardening due to the nonproportionality of the strain path [7,8] is obviously observed. Mises equivalent stress range under nonproportional loading increased as much as about 30% above a strain range of 0.5% compared with uniaxial fatigue loading. But the amount of increase
Table 1 Chemical composition (wt%) and mechanical properties of the tested material 1100°C x 48 min WQ
C
Si
Mn
P
S
Ni
Cr
Mo
Ti
AI
N
Nb + Ta
0.050
0.60
0.87
0.026
0.002
8.94
18.59
0.11
< 0.002
0.013
0.019
< 0.003
Yield strength (MPa)
Tensile strength (MPa)
Elongation (%)
213.6
623.3
71.9
T. Asayama et al. / Effect of nonproportional loading
301 C0.5
+i
3.2S (,q
70.0
90.0
-,-
260.0
Fig. 1. Test speomen.
path dependent strain range AEpth [1] was employed instead of the ASME type equivalent strain range. /l~-pth is defined as follows.
was less at lower strain ranges and it was about 15% at a strain range of 0.2% near elastic region. Almost the same amount of increase in Mises equivalent stress range was observed for nonproportional creep-fatigue loading, too. For proportional loadings, the pure torsional fatigue/creep-fatigue loading showed a somewhat smaller stress range compared with uniaxial loading and combined proportional loadings. In order to estimate the additional strain hardening under nonproportional loadings, the Mises type strain
A%, h = / d e e q ,
(3)
where deeq is an increment of Mises type equivalent strain. Integration is conducted between the two points in a strain cycle where the axial strain takes extreme values. In eq. (3), AEpth is calculated based on the total strain instead of plastic strain which is used in the original Mises equivalent strain range. The relationship between the Mises type path dependent equivalent strain r a n g e Aept h and Mises equivalent stress range A(req is shown in fig. 3. It is observed that the estima-
Table 2 List of waveforms
800
7/,,3 Uniaxial loadi~
Triangular ~avefor~ Strain hold period = O, 1 hour
e
~
700
t
600
k=O, ~=0
• v
Torsional loading Trlaugul ar ~ f ot-~ Strain hold period = O, 1 hour
o~ k=~, ¢=0 ,
proportional
4
loadi~
500 400
k=l.7, ¢=0
"5 o-
ta
Si=idal
~fom
Strain hold period = 0 hour
k.A_/ a =1.7, ¢=90*
hold period
= I hour
UniaxialFatigue \0.1%/'s~.
/// ~,
• 304 Steel,
k
/
0
/
550°C
¢ (°l
th(min) 0
60 i
I
0
--!
W
1,7
•x
1.7
o'.6 Strain
/
/
,~/ i 200
~-
Nouproportinnal Loading
~// ~
v, /
=
Trla~ul ar wave f orm
Strain hold period = O, i hour
"~ i
7/~3
C~in~d
v
I'.0
[ 60 I
[
90
1'.5
0
°° 0 6O
2'.o
ASME Type Equivalent Strain Range ,3.~eq%
k=l.7, ¢=90°
Fig. 2. Stress-strain response represented by ASME type strain range.
T. Asayama et al. / Effect of nonproportional loading
302
A~eq =1.0% k =1.7 =90' t h =60min N f = 164cycle
800 z~ 700
o
v
g b
600
v ,.or.v / "
~o 500 /
x
-I00 304 Steel,
c
_~ 300 200
. . " [] \Uniaxial Fatigue z~ 0~'1% "sec'
0
I
/
X I--
r- 400 zE I
/ I'"
Z& /V// |
400
ore
/
/v
i-_
X
0
550°C
I I _lJO0 -400 I~00 -200
¢,{°) ! t h (rain) 0 60
/
©
O °o
I
I
1.7 1.7
90
I
1.0
1.5
(7
MPa
7°°o/
0 60
lO0
015
40 2Go -100
O
;~00
u_400
2t.0
Path DependentEquivalentStrainRange ~ pth % Fig. 3. Stress-strain response represented by the Mises type path dependent strain range.
tion of additional hardening was improved by Aepth, although the scatter is not negligible.
3.2. Stress relaxation behauior under biaxial conditions It is very important to examine the behavior of stress relaxation under nonproportional strain path to check if the creep damage evaluation method for uniaxial loading can also be applied to the nonproportional strain waveforms. Figure 4 shows the stress path of nonproportional loading. Stress moves counterclockwise during the loading/unloading portion and relaxes proportionally toward the origin of the stress plane during the strain hold period, just like the proportional combined loading. The fact that the stress relaxation occurs proportionally toward the origin of the stress plane even under nonproportional loading indicates the possibility that creep damage evaluation under nonproportional loading can be performed based on the same method as the uniaxial loading. The details will be examined later. It is reported that under certain multiaxial conditions, the amount of stress relaxation is reduced compared with that under uniaxial conditions [7,8]. Figure 5 shows the amount of stress relaxation (Mises equivalent stress) under various loading conditions as a func-
Fig. 4. Proportionality of stress relaxation under various loading conditions. tion of initial stress. It is observed that the amount of stress relaxation is reduced under biaxial conditions, particularly under nonproportional loading.
3.3. Fatigue ~creep-fatigue life under biaxial conditions The results of biaxial fatigue/creep-fatigue tests are shown in fig. 6 in t e r m s o f a relationship between A S M E type equivalent strain range Aeeq and the n u m b e r of cycles to failure Nf (see 2.2 for the definition o f Nf).
Fatigue life For proportional loadings, there was a slight difference in fatigue life. The fatigue life was in the order of pure 120
304 Steel,
550"C
~ lO0 i io
[_
k ~ ~l°:Iith(min~
80 °
g- 6o
I•
q
-7i~"
00/
/ /
/
•,~
~- 40 'S
=
e/ /m
/
•
/
20 0
/
I O0
200
Initial Stress of Relaxation
300 o" max
400
MPa
Fig. 5. A m o u n t of stress relaxation under nonproportional
loading.
T. Asayama et aL / Effect of nonproportional loading g
1o I
304 Steel, 550"(2
. . . .
l0 0
/ v ~UUniaxLalFatigue, 0.1% see
lO-1 102
103
104
Number of Cycles to Failure
105
n
106
107
Cycles
Fig. 6. Result of biaxial fatigue/creep-fatigue tests.
torsional loading, combined proportional loading and uniaxial loading. For nonproportional loadings, a magnificent fatigue life reduction was observed. Life reduction was still observed at the lowest strain range investigated, that is, 0.2%, where hysteresis loops of this specimen showed that the deformation was almost elastic. The amount of life reduction showed no particular dependence on the strain range and the fatigue life of nonproportional combined loading was reduced to 1 / 4 - 1 / 2 0 of that of uniaxial loading. It was revealed that the effect of the nonproportional strain path on the fatigue strength of this material still existed at a strain range near the elastic region.
Creep-fatigue life Creep-fatigue life was reduced compared with fatigue life under the same ASME type strain range for all types of loading, including nonproportional loading. The amount of life reduction was almost the same irrespective of the strain waveform. It was made clear that creep-fatigue life reduction occurs under nonproportional loading just as under proportional loadings. 3.4. SEM observations For uniaxial loading, a clear relationship has been observed between loading condition and fracture mode with 304 stainless steel [10] and this observation forms the background of the creep-fatigue evaluation method. That is, fracture was transgranular for fatigue loading and intergranular for creep-fatigue loading. In the present study, a scanning electron microscope (SEM) observation was made to check if the above observation obtained for uniaxial loading also hold true for biaxial loading conditions including nonproportional loading.
303
Figure 7 shows the result of the SEM observation. A Fracture surface of non-proportional fatigue loading (fig. 7d) is similar to that of torsional fatigue loading (fig. 7b) which showed a transgranular fracture mode with typical rubbing marks. Striation marks which were observed in the case of uniaxial loading (fig. 7a) and combined proportional loading (fig. 7c) were not observed in the case of nonproportional fatigue loading. For nonproportional creep-fatigue loading (fig. 7d), the fracture surface is similar to those of uniaxial creep-fatigue loading (fig. 7a) and combined proportional creep-fatigue loading (fig. 7c), showing a typical intergranular fracture mode. Therefore it was made clear that the fracture mode was transgranular for fatigue loading and intergranular for creep-fatigue loading under nonproportional loading conditions, which is the same as the case of uniaxial loading. And as far as fracture surface observations concerns, the effect of the nonproportionality of the strain path is evident only for fatigue loading and is not observed for creep-fatigue loading. This is because the strain hold period was imposed at a tensile peak of axial strain when the torsional strain is zero.
4. Evaluation of the test results
4.1. Creep-fatigue evaluation method for uniaxial condition The authors have developed a creep-fatigue life prediction method for uniaxial loading conditions. The method [11-14] is summarized here. The following relationships are used in the method. All those relationships were obtained based on uniaxial test results with various heats of 304 steel. (a) fatigue life versus strain range, (b) strain range versus stress range at mid-life, (c) creep strain versus time, (d) stress versus creep rupture time. The method is as follows: (1) Fatigue damage Df is calculated as a fatigue life fraction consumed in the cyclic loading as expressed by the following equation:
Df = N / N f I a,,
(4)
where Nf is the fatigue life that corresponds to the applied strain range Ae in the relationship between fatigue life versus strain range. N is the number of strain cycles imposed.
304
T. Asayama et al. / Effect qf nonproportional loading
(2) Creep damage is calculated as the lime fraction consumed during the strain hold periods as expressed by the following equation: th
D~= Nf dt/tr(~r), o
(5)
where t~(o-) is the uniaxial creep rupture time under stress o-. t h is the strain hold time per cycle. The integration is conducted for one cycle. The stress relaxation behavior needed to calculate eq. (5) is determined by the following procedure. (3) The initial stress at a stress relaxation is obtained as a half of a stress range that corresponds to the applied strain range on the relationship between strain range versus stress range at mid-life.
(4) Stress relaxation curve is estimated based on the relationship between time versus uniaxial creep strain, and the classical strain hardening law. (5) A creep-rupture time for a given stress is obtained by the relationship between stress versus rupture time. (6) Creep-fatigue life is calculated based on the Campbell criterion.
4.2. Fatigue damage under biaxial condition The authors have proposed that the Mises type strain path dependent equivalent strain range depth bc used for estimation of the additional strain hardening that occurs under non-proportional loading. Based on
(a) Fatigue t h=0 hour Aa eq = 1 . 0 %
Nf=1339
l
u
l
50/.zm Creep-Fatigue th----1 hour Ae eq = 1 . 5 %
N f--304cycles
I _ _ I
50,u m
Fig. 7. SEM observation; (a) uniaxial loading, (b) torsional loading, (c) combined proportional loading, and (d) nonproportional loading.
T. Asayama et al. / Effect of nonproportional loading this fatigue damage under biaxial condition is assumed to be expressed as follows:
Df = N/Nt-la%th.
(6)
For proportional loading, the ASME type equivalent strain range and Mises type path dependent equivalent strain range give the same value. For nonproportional loading, the Mises type path dependent equivalent strain range gives a larger value compared with the ASME type equivalent strain range and the difference between the two is maximum at a phase angle of 90 degrees. In this case, AEpt h equal ~r/2 times AEeq.
4.3. Creep damage under biaxial condition The initial stress of the stress relaxation was calculated to be half of the stress range that corresponds to
305
the Mises type path dependent equivalent strain range obtained by eq. (3). It is to be noted that by eq. (3) the initial stress for nonproportional loading is estimated to be lower than the actual value. It is because the estimation of the additional strain hardening by eq. (3) is not enough as mentioned in section 3.1. In the present study, for simplicity, no further discussion was made on this point. Based on the proportionality of stress relaxation under nonproportional loading (see fig. 4), the biaxial stress relaxation curve was assumed to be estimated by the relationship between time versus creep strain, where the creep strain is calculated using the Mises equivalent stress defined by eq. (2). The amount of stress relaxation depends on the type of loading as shown in fig. 5. But in the present study, for simplicity, the difference of the amount of
(b) Fatigue t h=0 hour A e e q = l .0% N f =2691 cycles
I
I
50~um Creep-Fatigue t h = l hour Aeeq = 1.0% N f = 1328cycles
I
I
50/~m Fig. 7 (continued).
7".Asayama et al. / Effect of nonproportional loading
306
relaxation was not taken into account. The relationship between the initial stress and the amount of relaxation estimated by the relationship between creep strain and time obtained under uniaxial condition is shown in fig. 5 by a broken line. This relationship was used for the evaluation of stress relaxation behavior of both the proportional loading and the nonproportional loading. It was also assumed that stress versus time curve in stress relaxation under biaxial conditions was identical to that under uniaxial condition. The relationship between stress and time to failure, where stress is represented in terms of the Mises equivalent stress, was used to evaluate the creep damage under biaxial conditions. Creep damage under
biaxial conditions was assumed to be expressed by the following equation.
D~
=
N jdt /tr(cr~q).
(7)
4.4. Result of life prediction The fatigue damage and the creep damage calculated by the above procedure are plotted on the Campbell diagram in fig. 8. It is seen that damage assessment was fairly reasonable except for pure torsional loading which was estimated somewhat overconservatively. The result of life prediction is shown in fig. 9. For most of the data, creep-fatigue life could be pre-
{c) Fatigue t h = 0 hour
A~ eq = 1 . 0 % N f =2274cycle-~
k
I
50t~ m Creep-Fatigue t h = l hour A~ eq = 1 . 0 % N f=727cycles
50/z m Fig. 7 (continued).
T. Asayama et al. / Effect of nonproportional loading dicted within an accuracy of factor of 2. Nonproportional fatigue loading at the lowest strain range investigated and pure torsional creep-fatigue loading were estimated conservatively. 4.5. Estimation of creep damage at low strain ranges near elastic region From the above observations, it was shown that fatigue/creep-fatigue strength under proportional loading and nonproportional loading was reasonably evaluated by the method proposed in the present study at least for strain ranges above 0.2% for fatigue strength and 0.3% for creep-fatigue strength, although the nonproportional fatigue at 0.2% was estimated conservatively. In this section we will discuss the creep-fatigue life evaluation at strain ranges below 0.3%.
307
As stated above, by the method proposed in the present study, the creep damage is calculated based on three key properties obtained under uniaxial conditions, that is, the initial stress of stress relaxation, the amount of stress relaxation and the creep strength. In the present study, a simplification was made in the evaluation of the initial stress and stress versus time curve in stress relaxation under biaxial conditions. That is, the initial stress was assumed to be obtained as half of the stress range that corresponds to the Mises type path dependent equivalent strain range and the stress versus time curve was assumed to be obtained based on the uniaxial creep strain versus time curve by using Mises equivalent stress. Therefore, effects of the simplification at lower strain ranges are to be considered. In the low strain ranges near the elastic region, the difference of the initial stress between experiment and
(d)
Fatigue t h---0 hour A~eq = 1 . 0 % N f =312cycles
50/1 rn
Creep-Fatigue th--1 hour A~eq =1.0% N f=164cycles
[
I
50/z m
Fig. 7 (continued).
T. Asayama et aL / Effect of nonproportional loading
308
damage evaluation method proposed in the prcsent study is also valid at lower strain ranges below 0.3%.
304 Steel, 550°C I0 0
~
•
5. Conclusion
121
bO
Campbell
r~ 10-1
~ k ~ # (°)
l,:ili-:
°
;,t 10 -2
1
th(min)
0
17 i 9~-~ l
J
I
i
lO -l
lO 0
Of
Fatigue Damage
Fig. 8. Creep-fatigue damage assessment based on the method proposed in the present study.
estimation would be small because the amount of the additional strain hardening due to nonproportional strain path is very small at low strain ranges as indicated in fig. 3. The amount of stress relaxation is smaller at lower strain ranges compared with that at higher strain ranges so that the dependence of the amount of stress relaxation on waveform would be smallcr as inferred from fig. 5. To sum up, the effect of the simplifications made in the present study is considered to be smaller at lower strain ranges and it is implied that the creep-fatigue
10 6 u~
(3
SUS304Steel, 550°C
Factor of 2 I f z / Z / / ~
105
/ ~/ / // t t/v
104
//
J "IO
J'/
/t "'//"//
A series of axial-torsional biaxial creep-fatigue tests were performed with 304 stainless steel at 550°C to check the effect of the strain path imposed on the fatigue and creep-fatigue strength. The following observations were made: (1) Fatigue strength under nonproportional loading conditions reduced to 1 / 4 - 1 / 2 0 of that under proportional loading. This tendency was still observed at a low strain range of 0.2% which is very close to the elastic region. (2) Creep-fatigue u n d e r n o n p r o p o r t i o n a l loading showed life reduction compared with fatigue under nonproportional loading. The amount of life reduction due to strain hold period was almost the same as in the case of proportional loading. (3) Proportional stress relaxation was obscrvcd even under nonproportional loading. (4) Transgranular fracture was observed for nonproportional fatigue loading and intergranular fracture was observed for nonproportional creep-fatigue loading. It was the same observation as that made for proportional loading. (5) Fatigue damage was defined based on the Mises type path dependent equivalent strain range and creep damage was defined based on the Mises equivalent stress. Fatigue damage and creep damage were evaluated based on the uniaxial material properties extensively obtained by PNC. (6) The result of damage evaluation was fairly reasonable and life prediction was possible for most of the data within an accuracy of factor of 2.
,,g/~: ~;,g'/
10 3
References
/1 / /
/
I°
o
I~.~
t ~-t ' o UE1
(3
n
102
103 Observed Life
104
105 NfOBS
Fig. 9. Result oflifeprediction.
106 Cycles
[1] T. Sakon et al, Proc. of 9th SMiRT, Vol. L (1987) 267. [2] T. Asayama, K. Aoto and Y. Wada, Trans. of the Japan Society of Mechanical Engineers 56 (1989) 532. [3] T. Asayama, K. Aoto and Y. Wada, Proc. of the 27th Symposium of Strength of Materials at High Temperatures (1989) 112. [4] T. Asayama, K. Aoto and Y. Wada, The 6th Int. Conf. on Mechanical Behavior of Materials 2 (1991) 311. [5] E. Krempl and H. Lu, Trans. of ASME. J. of. Eng. Mater. Technol. 106 (1984) 376. [6] A. Benallal and D. Marquis, Trans. of 9th SMiRT, Vol. L (1987) 385.
T. Asayama et al. / Effect of nonproportional loading [7] L.K. Severud and B.V. Winkel, Proc. of 9th SMiRT, Vol. L (1987) 123. [8] L.K. Severud, PVP 163 (1989) 1. [9] Y. Wada et al., PVP 123 (1987) 37. [10] M. Morishita et al., Nuclear Engineering and Design 83 (1984) 367.
309
[11] K. Aoto et al., PVP 123 (1987) 43. [12] Y. Wada et al., Proc. Int. Conf. on Computational Mechanics 1 (1986) 4. [13] A. Yoshitake et al., Proc. Int. Conf. on Creep (1986) 441. [14] H. Kawasaki, H. Kagawa, K. Aoto, Y. Wada and I. Nihei, PVP 172 (1989) 29.