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Cyclic behavior of a type 304 stainless steel in biaxial stress states at elevated temperatures
International Journal of Plasticity, Vol. 4, pp. 77-89, 1988
0749-6419/88 $3.00 + .00 Copyright© 1988PergamonJournalsLtd.
Printedin the U.S.A.
C Y C L I C B E H A V I O R O F A T Y P E 304 S T A I N L E S S S T E E L I N BIAXIAL STRESS STATES AT ELEVATED TEMPERATURES
MASATERU OHNAMI, MASAO SAKANE a n d SEHCHI NISHINO Ritsumeikan University (Communicated by J.L. Chaboche, GIS Ruptune/t Chaud)
Abstract-The results of companion, incremental/decremental, and stepup fatigue experiments on austenitic stainless steel tubes (type 304) are presented. The experiments include proportional and nonproportional loading conditions at ambient as well as elevated temperatures. Empirical relations are developed between van Mises effective stress and strain, and these relations are shown to describe the cyclic behavior during proportional companion as well as incremental/ decremental tests. In case of nonproportionai incremental/decremental experiments, the material behavior is not accurately modeled either by using the van Mises effective stress and strain, or by relating Tresca's maximum shear stress and strain.
I. INTRODUCTION
The study of the constitutive relation is one of the important research fields of plasticity as well as the yield condition and the hardening rule. Recent progress in numerical analyses such as FEM and BDM especially requires the constitutive relation that accurately expresses the actual material stress-strain relation. Numerous studies have been carried out on the cyclic stress-strain relation in uniaxial stress state (BaAT ~ LAIRD [1979]; POL~, KLESNIL& H~LESlC [1982]; VOYL~DrtS[1984]) and many constitutive equations have been proposed. In order to summarize the characteristics of constitutive equations, the Subcommittee on Inelastic Analysis and Life Prediction of High Temperature Materials, the Society of Materials Science, Japan, recently made a benchmark project on inelastic deformation of 21/4Cr-lMo steel in uniaxial creep-fatigue condition at 873 K. They studied 17 constitutive equations with six loading patterns and identified the constitutive equations relating to the life prediction in creep-fatigue condition [1985a, 1985b, 1987]. In spite of such an extensive study of constitutive equation, studies in biaxial cyclic condition are not many (BROWN~ Mm~R [1979]; DE Los RIOS & BROWN [1981]; ELLIS, ROmNSON ~ PU6H [1983]). The constitutive equation should be verified in multiaxial stress states. In biaxial stress states, since combination of mechanical stress with thermal stress sometimes yields nonproportional loading, the study of the constitutive relation in nonproportional condition is important as well as in proportional condition. Experimental studies on the nonproportional plasticity have been published (LAunA R, SmEnorroM [1978]; KANAZAWA,MtLLERS~BaOWN [1979]; IO~a'Z ~ LU [1984]) but they have only treated plasticity at room temperature; high temperature studies on nonproportional plasticity are scarcely made. The objective of this article is to examine the cyclic behavior in biaxial stress states during proportional and nonproportional loading conditions at elevated temperatures. As the cyclic stress-strain relation was expected to be affected by the test method, test results in two main methods were compared; one is the companion method using several 77
78
M, ()HNAMI el ¢l],
specimens to draw a cyclic stress-strain curve, and the other is the incremenlai, decrcmental method using the strain history that varies strain amplitude at each cycling where only one specimen completes a cyclic stress-strain curve. The companion lest ,aa~ ca~ ried out in proportional condition whereas the incremental/decremental test wa~ carried out in proportional and nonproportional conditions.
II. EXPERIMENTAL PROCEDURE
Test specimen used in this study was a thin walled cylinder of a solution heat treated Type 304 austenitic stainless steel, of which inner diameter, outer diameter, and gage length were 9 mm, 11 ram, and 20 mm, respectively. Test apparatus was a computer assisted electrohydraulic servo machine that can apply an axial load in combination with a torsional load (O~INAUl & HAMADA [1982]; HAMADA, SAKANE & OHNAMI [1984]; NISHINO, HAMADA, SAKANE, OHNAMI, MATUMURA & TOKIZANE [1986]) in any strain history. Axial and shear strains were measured by an originally designed extensometer composed of two thin tubes, measuring the axial displacement and the relative rotation angle between the tubes by two LVDT's (OHNAMI, SAKANE & HAMADA [1985]). A heater was inserted into the hollow cylindrical specimen to obtain elevated temperature and a sheath subheater was used to obtain uniform temperature distribution along the gage length. In order to examine the effect of test method on cyclic stress-strain relation, three strain histories were employed (as shown in Fig. 1). Strain history (a) in the figure has a constant strain amplitude while strain history (b), with variable strain amplitude in each cycle, has 25 cycles in one block of strain sequence. As to strain history (c), after straining 10 cycles with constant strain amplitude, the amplitude is increased to the next step that has the larger strain amplitude, and then this strain sequence is continued until the specimen fails. The strain histories (a), (b), and (c) are called the companion test, the incremental/decremental (I/D) test, and the stepup test, respectively. All strain histories in Fig. 1 have a Mises's equivalent strain rate of 10-3/s. Table 1 lists the test program carried out in this study. Experiments were mostly per.formed in strain controlled condition, with some performed in stress-controlled conditions in order to examine how the loading mode affects the cyclic behavior. Also, the majority of experiments were performed at 823 K, while only the I / D and the companion tests were carried out at 923 K as well as 823 K in order to examine the effect of temperature. Nonproportional loading was only applied in the I / D test at 823 K. All experiments were made in air. Figure 2 shows the loading sequence of nonproportional tests, where principal strain ratio, ~b, is defined as e3/el- el and e3 are the maximum and the minimum principal
Table 1. Test program Temperature
823 K
923 K
Strain-controlled test
I/D test" Companion test Step up test
I/D test Companion test
Stress-controlled test
Companion test
"Both proportional and nonproportional tests were carried OUt.
Biaxial stress states at elevated temperatures
-t-
79
+
lu5 ~= IC)3 Is
#_. = 10-3/s
c
.c_
llll,
o c
time
0
._> o
"5
~o
(I) u~
"viIIi
time
(b)
(a) it.j
~, +1
~ [ ~:,o-3/s
oL,, AAAA"AJA AAAAAAAAAA AAA
; '"""l"'"'"'Ivvvvvvvvvvvvvv -
~ I !/Ocycles! AV_meosuredcycle (c)
Fig. 1. Strain histories for proportional and nonproportional tests. (a) Companion test. (b) Incremental/ decremental (I/D) test. (c) Stepup test.
strains. Loading paths (~) and (~) correspond to the compression-tension and the reversed torsion tests, respectively, and the loading paths (~) and (~) are the test having the principal strain ratio of 4) = - 2 / 3 and ~b = - 5 / 6 , assuming that the Poisson's ratio is 0.5 in the low cycle fatigue case. Nonproportional test of Id01 = 1/2, for example, has the strain history, (~)-(~)-(~)-(~), and in that test the principal strain direction alternates 45 degrees in each cycle. In the I / D test, the principal strain direction alternates after each subcycle. The strain history of the other nonproportional tests is also listed in Fig. 2. Usually the nonproportional loading experiment employs the sinusoidal waves with a phase difference between combined compression-tension and reversed torsion loadings. In that test, however, the Mise's equivalent strain does not have a sinusoidal form but rather has a kind of trapezoidal shape, so the test result using this strain history includes the nonproportional loading, as well as hold-time effects. In this sense, the out-of-phase experiment is not suitable for examining the cyclic constitutive relation.
80
M. OHNAMIet al,
.............
-
Non-propor't lona t Id ¢1 = ~ / 6
=1/3 =1/2
Of]
loadJn9 ~
@------.~ @ ..... _ > @
--->®
@---> @ ...._>
--->@ .... > @
@ .... ~ (9 @ . . . . .> @
Fig. 2. Loading sequence of nonproportional tests.
In order to obtain an effect of only nonproportional loading on cyclic constitutive relation, the test method shown in Fig. 2 was used in this study. 111. E X P E R I M E N T A L RESULTS A N D D I S C U S S I O N
III. 1 Cyclic stress-strain behavior during proportional tests Figures 3(a), (b), and (c) show the cyclic behavior during the I / D , the companion, and the stepup test, respectively, (Mises' equivalent stress-strain). Here, the following equations were used in calculating the Mises' equivalent stress and strain. 0 = sign(S)~'tS j S = sign(a)o 2 + 3 s i g n ( r ) r 2
(1) ,
[ - 7
= slgn(N),J ]NJ N = sign(e)e 2 + (1/3)sign(3,)3 ,2. In these equations, a and r are, respectively, the normal stress and shear stress calculated from the axial load and the torque, while e and y are, respectively, the normal strain and the shear strain. Conventionally, the Mises' equivalent stress and strain have no negative sign, but both the negative and positive signs are employed here to emphasize the cyclic loading. The tensile loading is taken as positive for a and e, and one predetermined twisting direction is taken as the positive for r and % In the I / D test, the Mises' equivalent stress and strain correlate well with the cyclic stress-strain relation in biaxial stress state. In the companion test, Mises' criterion appears to be also effective, but the scatter of the data is larger in comparison with the I / D test. In the companion test, the maximum difference in the stress amplitude of 21°70
Biaxial stress states at elevated temperatures
~_ I1~
SUS304,
823K,
SUS304
8 2 3 K , in Air ' e .~" Proportionol cm ~"~ loading 1 ~,'0
, in ,Air _
I ~,D,., I
1
4001 Propgrtionol loading
81
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200
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-200 i
2
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o
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-I.5 -I.O -Q5 Mises' type
-It.O -C)5 O 0.5 ID 1.5 Mises' type equivalent strain E,%
I
L
0 Q5 1.0 1.5 equivalent strain E , %
(b)
(a)
~_ 50O
SUS304, i
~. = io-3/s
40(
Proportional
823K~ i in Air i
f
Io,
Ib"
t~
.-"
20£ E -~ O
0
:-=ooi /
]---c=.=oo test
07
- 4 0 0 ~
o STep up rest
o Mises'
d5
type equivalent
E, (c)
strain
%
Fig. 3. Cyclic stress-strain curves in strain controlled proportional test at 823 K plotted on Mises' equivalent stress/strain basis. (at I/D test. (b) Companion test. (c) Stepup test.
occurs at 0.5% strain amplitude in the results of Fig. 3(b). The difference of 21% in stress amplitude is comparatively large, so that we have to treat the cyclic stress-strain relation carefully in the companion test if the precise analysis is required. Figure 3(c) compares the cyclic stress-strain relation between the I/D test (solid line) and the companion test (dotted line) together with the data in the stepup test. There exists a slight difference in the cyclic stress-strain relation between I/D and companion tests, but the data in the stepup test are very close to the results of the companion test. If the cyclic stress-strain relation is expressed as eqn (2), the material constants in the equation have the values listed in Table 2.
82
M. OHNAMI e t
al.
Table 2. Material constams in constitutive equation (2)
Mode
t£ (× 10 ~ MPa)
K' (MPa)
,~
~:, IMPa?
C o m p a n i o n test
Proportional loading
I. 25
480
0.265
? i?
lncremental/decremental test (I/D test)
Proportional loading Nonproportional loading [d4~l = 1/6 = 1/3
1.25
4t4
0.107
;~
1.30 1.40 1.50
440 480 495
0.102 0.099 0.092
3;'~ 409 427
Test method
= I/2
e=
+
~7
(2)
where ~ is Mises' equivalent total strain, o a Mises' equivalent stress, E ' the tangential modulus representing the gradient of the elastic part in o - ~ diagram, K ' the cyclic stress strength, and n' the cyclic stress exponent. Note that E ' has the similar value as Young's modulus but is not identical to Young's modulus. In the table, the material constants in nonproportional loading are also listed but they will be discussed in the next section. K ' and n' in the companion test have larger values compared with those in the I / D test, especially n' value in the companion test is twice as large as that in the I / D test. However, 0.2% p r o o f stress, oo.z, in the companion test is smaller than that in the I / D test. Although there exists a difference in material constants given in Table 1, the absolute values of the stress amplitude in the two testing methods do not differ much with each other as shown in Fig. 3(c). The companion test simulates the operating condition of practical components but needs m a n y specimens. So, if we employ the I / D test, we can complete a cyclic stress-strain curve with only one specimen. Since the cyclic stress amplitude in the I / D test does not differ much from that in the companion test, we can use the cyclic stress-strain relation in the 1/D test instead of the companion test. The stepup test is also useful instead of the companion test. Besides the Mises' equivalent stress and strain, the arrangement of the data using the Tresca's criterion is also made. Figure 4 shows an example. The data in the I / D test appears to be correlated with Tresca's criterion, but in the companion test the scatter o f the data becomes large. In both the test methods, the Tresca's criterion enlarges the scatter of the data in comparison with the Mises' criterion, so we can conclude that the Mises' criterion is a better parameter to correlate cyclic stress-strain relation in biaxial stress state. In this regard, KANBIL, MILLER & BROWN [1982] reported that the Tresca's criterion correlates well the biaxial cyclic stress-strain relation for a type 316 stainless steel in the companion test at 823 K. However, the experimental data in this study contradict their conclusion. The cause of the different trend of the data is not clear now but m a y be attributed partly to the test condition or material. In order to discuss the difference in shapes of the cyclic stress-strain curve between the I / D and the companion tests, Fig. 5 shows the hysteresis loops obtained in experiments, together with the schematic representation of them in the two tests. Bilinear representation is employed as the schematic figure, since in the bilinear modeling it is easy to understand the essential difference in the cyclic stress-strain relation due to the test method. In the I / D test, since each hysteresis loop has nearly equal n' value, and since the yield stress o f each hysteresis loop agrees with the stress amplitude of the cyclic
Biaxial stress states at elevatedtemperatures
3001_8U8,5044823K, inr~qAir o?
! I/D test Proportional L loading 200F g IO-S/S
83
300 SUS,304? 8213K,in Air
f
Companion test j ~ Proportiono I I
• -~~,.~
: 200~"loading ,~ )-~ IOOi~='d3/s Ii/ nO
=
~oo~ o~
o
.
.
.
.
.
.
i
.
~, -IO0~-
E
I
I I
-2oo
-~nnL -I.5 /
1 L I I -I.0 -0.5 0 0.5 1.0 1.5 Maximum shear strain Trnax/2,%
(a)
o
L
(,° o
,.o
,is
Maximum shear strain r m a x / 2 , %
(b)
Fig. 4. Cyclic stress-strain curves in strain controlled proportional test at 823 K on maximum shear stress/strain basis. (a) I/D test. (b) Companion test.
stress-strain relation, the cyclic stress-strain relation expressed by dotted lines occurs. The cyclic stress-strain relation in the I/D test agrees with the hysteresis loop in the test of the maximum strain amplitude as shown in Fig. l(b). In the companion test, however, the stress strain relation of each hysteresis loop, after yield, differs from each other and the stress amplitude increases with increasing plastic strain range. Then, the cyclic stress-strain relations as shown by dotted lines occur. From this, the cyclic stress-strain relation in the companion test has a larger strain hardening exponent, n', in comparison with the I/D test. The difference in the shape of the cyclic stress-strain curve due to the test method has resulted, of course, from the difference in the strain history. In the companion test, the material is loaded by the constant strain amplitude, whereas in the I/D experiment the specimen is subjected to various strain amplitudes during each strain block. The material in the I/D test especially experiences the maximum strain amplitude at each block and affects the shape of the cyclic stress-strain relation (Cagaocrm ROUSSELmR[1983b]. This effect is well known as the strain memory effect and is confirmed in the present study. The discussion above uses the stress-strain relation in order to clarify the essential point of difference in the shape of the hysteresis loop. The real material exhibits curved cyclic stress-strain relation, but the essential part is covered by bilinear modeling. Figure 6 shows the test results in the proportional companion test at 923 K together with that at 823 K expressed by solid line. Cyclic stress-strain curve has smaller stress amplitude at 923 K in comparison with that at 823 K. At 923 K, as well as at 823 K, the stress-strain relation can be expressed by the Mises' criterion with somewhat larger scatter in the data. The results with the larger value of n' exhibits smaller stress amplitude. 1II.2 Cyclic Stress-Strain Behavior During Nonproportional Loading Figure 7 shows th e cyclic stress-strain relation in the nonproportional I/D test. Nonproportional loading increases the stress amplitude, and the stress amplitude becomes
84
M.
i/D
OHNAMI
el al.
test
i:~:o m P a nion
test
I
i'
irS
35%
05 %
(a) I/D
test
(b)
. }
Companion test ~
f
i
,
t
/////j////,/ --Hysteresis ---
Cyclic (c)
loop constitutive relation
--
Hysteresis
---
Cyclic
loop
constitutive relation (d)
Fig. 5. Actual hysteresis loops and schematic figure. (a) Actual hysteresis loops in I/D test. (b) Actual hysteresis loops in companion test. (c) Schematic figure in I/D test. (d) Schematic figure in companion test.
larger as [d$] increases. In the nonproportional experiment, cyclic stress-strain behavior cannot be appropriately expressed by using the Mises' type equivalent stress-strain. The cause of the difference in the cyclic stress amplitude due to various nonproportional loading will be discussed in the next section in relation to the microstructures of the material. Table 2 lists the material constants in the proportional and the nonproportional I/D tests. Values of n' and K ' in the proportional companion test are larger than those in the proportional I / D test, whereas in the nonproportional test, an increase in ]dq51 increases the value of K ' and decreases that of n'. The tangential modulus in the elastic part increases as ]d4~[ increases. In order to correlate cyclic stress-strain curves in the nonproportional test, a simplified cyclic stress-strain relation is considered. The relation is of a modified power law type and is expressed by eqn (3),
Biaxial stress states at elevated temperatures
SUS304,
~-
--
Ib" 4 0 0
923K,
85
in Air
[ I Companion test Proportional I oadin~ /iA~ / ~ = I0"°/s i
I
200
c
~
o
o"
~ -200 M
/l
.~ - 4 0 0
-I.5 -I.0 -0.5
0 0.5 ID Mises' type equivalent strain £,%
1.5
Fig. 6. Cyclic stress-strain curves in strain controlled proportional test at 923 K.
o 5OO S U S S 0 4
Q_
400
I/D
82
test
Non- proportionol loading 200
c
O
0
~
-200
~ -400
I lol~i~l,l
~_sool.sL 40-o.s
o o,s
Mises' type equivalent ~, %
LO 1.5 strain
Fig. 7. Cyclic stress-strain curves in strain controlled non-proportional I/D test at 823 K.
Aa = IA - ~~pm.x p 1) m[K'A~p} n',
(3)
The equation only expresses the relation between the plastic strain range and stress range. If the total strain range-stress range relation is required, an elastic part should be added. In the equation, A ~ and A~p are the stress range and the conventional plastic strain range of Mises' type. The increase in the stress amplitude due to the nonproportional loading is taken into account by the term in parentheses, where za~p,~ is the equivalent strain of Mises' type but calculated from the maximum range in each
86
M. OHNAMIel al.
strain component for the case of nonproportional loading. For example, in the case ol Idq~l = 1/2, A~:prna x specified a s A~prna x = [A• 2 q- (1/3)(A'~p) 2] 1/2, assuming that the axial plastic strain range is A~p and shearing plastic strain range A,,fp. In that case, ~3~, is calculated by A~p = (A(~2)1/2 or Agp = [l/3(Ayv)2 ] 1,2. The exponent 07 in eqn (3) is the material constant and has the value of 0.5 in the case of this study. The solid lines in Fig. 7 show the cyclic stress-strain relation calculated from eqn (3), with the addition of the elastic part to the equation. Most of the experimental data are very close ',o the predicted lines o f eqn (3), so that we can conclude that eqn (3) is useful to express the cyclic stress-strain relation under nonproportional loading condition as shown in Fig. 2. It should be pointed out that the eqns {2) and (3) discussed so far have restriclions on usage. The equations presented here express the relation between the strain amplitude/range and stress amplitude/range so that the equations are not applicable io the generalized proportional/nonproportional loading conditions. However, the equations of course are valuable in calculating the stress amplitude from the strain amplitude in proportional/nonproportional conditions for most commonly used strain histories [ h e equations, which have the incremental form and are applicable to wider loading conditions, should be used. In order to understand the cause of the increase in stress amplitude due to the nonproportional loading, the authors examined the microstructures of the material strained in the compression-tension test (P), reversed torsion test (T), and nonproportional companion test ( A P T ) of [&b[ = 1/2 at A~ = 1.0% (NISmNO et al. [1986]). Dislocation structure in the compression-tension experiment is mostly ladder, partly maze, and somewhat cellular, whereas in the reversed torsion test it is mostly maze, partly ladder, and slightly cellular. A P T loading yields only cellular structure. These ladder and maze walls under compression-tension or reversed torsion loading are on {111 } planes, which are primary slip planes. In the A P T test, on the other hand, the maximum principal shear strain direction change by 45 degrees in each cycle, so that larger interaction of the slip system occurs. In the A P T test, the larger interaction results in cellular s~ructures, because it overcomes the rearrangement of the dislocations with the assistance of the thermal process. The resistance to the dislocation glide is, of course, larger in the cellular structure so that the larger cyclic hardening results (Ntsmyo et al. [1986])~ 1II.3 Cyclic Stress Amplitude and Cyclic Yield Stress Loci In addition to the cyclic stress-strain behavior, it is also important to describe the hardening behavior in the proportional and the nonproportional loadings in order to assist the general understanding of the cyclic plasticity in those loadings. In the following, we discuss the hardening behavior of the prestrained material in proportional and nonproportional loadings referring to the previous result of the authors. Figure 8(a) and (b) (NIsI-IINO et al. [1986]) show the equivalent stress amplitude and cyclic yield stress ellipses in P and A P T modes, respectively, with those of the virgin material at the Mises' equivalent strain range of 1.0°7o. The equivalent stress amplitude is obtained from the tensile and compressive peak stresses, and the cyclic yield stress is 0.4% offset stress from the peak strain as shown in the figure. The figure shows that the virgin material exhibits an almost isotropic stress response because the data of the material are on the Mises' ellipse. However, the cyclic stress response is not isotropic for the prestrained material in the P mode. The shape of that material is being elongated along the shear stress axis. The data points of the stress amplitude are almost on the ellipse expressed by the equation, a 2 + 2r 2 = 0 2, 0 2 + r 2 ----:
Biaxial stress states at elevated temperatures
SUS504,
823K,
AE
| o400+
~
Peak
~J~t~------__a
stress
~
~
|
~
~
in
Air
0 Stress amplitude of prestrained material m P I o a d ~ • Stress amplitude of prestroined material in APTIoading
= % ..... V =O.I t-'17
2
87
z
2 oo ~-----._~
-"-,n
z-z
~
i
I
M -z P T z -. ÷ _ ~ = 0.
-I1
Stress virgin
•
amplitude of the material
I (a)
SUS 3 0 4 ,
823 K ,
1E , oJ ¢. : , /o J/ =0.1Hz
~ o | ~ &.200~ ~ ~ |
0.4%
~p"
offset
in Air
OCyclic yield stress of prestrained material in P loading OCyclic yield stress of prestrained
i
material in APT loading
~ 37-2: 0"veer
2o _
\
1, ~
6o'
ol
\'X~ o
~
~
'
1
~
,
.
.
/
~
,'oo/r' ..//
/
/
O
Axial stress , MPa
(7
~D.4'% /
2
c yield curve of the virgin material
z" _z
(3- + 7- = Oyp 2 0 0
(b) Fig. 8. Equivalent stress amplitude and cyclic yield stress ellipses in P and APT mode at 823 K (NISHINO et al., [1986]). (a) Equivalent stress amplitude ellipses. (b) Equivalent yield stress ellipses.
e2p. As with the virgin material, the cyclic response for the prestrained material in the the A P T mode is isotropic, but the ellipse expands from the virgin material. The equivalent stress amplitude in A P T mode is 1.4 times larger than that in the P mode. This tendency of the data was also confirmed at a strain range of 0.7°70.
88
M. OHNAMIel a/.
IV. CONCLUSIONS Mises' type e q u i v a l e n t stress a n d strain are effective in describing the biaxJal cyclic stress-strain behavior during p r o p o r t i o n a l c o m p a n i o n a n d p r o p o r t i o n a l I / D tcst:,. They are effective in both strain controlled a n d stress controlled conditions at 823 K and strain controlled condition at 923 K. However, the Mises' type equivalent stress and strai~t were n o t effective for the biaxial cyclic data in the n o n p r o p o r t i o n a l I / D test. Tresca'~ Iype e q u i v a l e n t stress a n d strain was less effective in c o m p a r i s o n with the Mises' type defin i t i o n s o f e q u i v a l e n t stress a n d strain. The shape o f the cyclic stress curve in the I / D test differs f r o m that in the c o m p a n ion test. W h e n the cyclic stress-strain relation is expressed by power law type e q u a t i o n , however, the c o m p a n i o n test has a larger strain hardening exponent t h a n that in the I / D test. The data in the stepup test are very close to the cyclic stress a n d strain data in the c o m p a n i o n test. C h a n g i n g the direction o f the principal strain axis increases the cyclic stress amplitude. I n order to take a c c o u n t o f the increase in the stress a m p l i t u d e due to the n o n p r o p o r t i o n a l loading, we p r o p o s e d the simplified e q u a t i o n . The e q u a t i o n can predict results that correlate well with the cyclic stress-strain relation in both p r o p o r t i o n a l as well as n o n p r o p o r t i o n a l loadings. A c k n o w l e d g e m e n t - T h i s work was supported by Grant-in-Aid for ScientificResearch B 60460084, the Ministry
of Education, Japan. The authors wish to express gratitude to Mr. Y. Hanato, Mr. Y. Hamano, and Mr. M. Fujiwara, the senior students of Ritsumeikan University, for carrying out the experiment.
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1979 1979 1979 1981 1982 1982
1982 1983 1983 1984 1984 1984 1985
LAMBA,H.S. and SIDEBOTTOM,O.M., "Cyclic Plasticity for Nonproportional Paths: Part 1 and Part 2-Cyclic Hardening, Erasure of Memory, and SubsequentStrain Hardening Experiments," J. Engng. Mater. Tech., 100, 96. BHAT,S.P. and LAIRO,C., "High Temperature Cyclic Deformation of Nickel," Fatigue Engng. Mater. Struct., 1, 59. BRowav,M.W., and MILLER,K.J., "Biaxial Cyclic Deformation Behaviour of Steels," Fatigue Engng. Mater. Struct., 1, 93. KANAZAWA, K., M~LER,K.J., and BROWN,M.W., "Cyclicdeformation of a 1°70Cr-Mo-V Steel under Out of Phase Loads," Fatigue of Engng. Mater. and Struct., 2, 217. DE Los Rms, E.R. and BROWN,M.W., "Cyclic Strain Hardening of 316 Stainless Steel at Elevated Temperatures," Fatigue Engng. Mater. Struct., 4, 377. POLAK,J., KLESNIL,M., and HELESIC,J., "The Hysteresis Loop 2. An Analysis of the Loop Shape," Fatigue Engng. Mater. Struct., 5, 33. OHNAMI,M. and HAMADA,N., "Biaxial Low-Cycle Fatigue of SUS304 Stainless Steel at Elevated Temperature," Proceedings of the 25th Japan Congress on Materials Research, Japan Society of Materials Science, Kyoto, 93. KANDIL,F.A., MILLER,K.J., and BROWN,M.W., "Creep and Ageing Interactions in Biaxial Fatigue of Type 316 Stainless Steel," ASTM STP, 853,651. ELLIS,J.R., ROmNSON,D.N., and PUGH,C.E., "Time Dependencein BiaxialYield of Type 316 Stainless Steel at Room Temperature," J. Engng. Mater. Tech., 105, 250. CHAm~HE,J.L. and ROUSSELmR,G.R., "On the Plastic and ViscoplasticConstitutiveEquations, Part I and Part It," J. Pressure Vessel Tech., 105, 153. VOYIADJIS,G.Z., "Experimental Determination of the Material Parameters of Elasto-Plastic WorkHardening Metal Alloys," Mater. Science Engng,, 62, 99. KREMPL,E. and Lu, H., "The Hardening and Rate-DependentBehavior of Fully Annealed AISI Type 304 Stainless Steel under Biaxial In-Phase and Out-of Phase Strain Cycling at Room Temperature," J. Engng. Mater. Techn., 106, 376. HAMADA,N., SAKANE,M., and OrlANAma,M., "Creep-Fatigue Studies under a Biaxial Stress State at Elevated Temperature," Fatigue Engng. Mater. Struct., 7, 85. The subcommittee on Inelastic Analysis and Life Prediction of High Temperature Materials, Japan Society of Materials Science, Kyoto. "Inelastic Behaviors of 21/4Cr-lMo Steel under Plasticity-creep
Biaxial stress states at elevated temperatures
1985 1985 1986 1987
89
Interaction Condition-Results of Joint Work," Trans. 8th Int. Conf. Structural Mechanics in Reactor Technology, Brussels Vol. 1, 49. "Inelastic Behaviors of 21/4Cr-1Mo Steel under Plasticity-creep Interaction Condition-An Interim Report of the Bench Mark Project by Subcommittee on Inelastic Analysis and Life Prediction of High Temperature Materials," JSMS Nucl. Eng. Design, 90, 287. O m ~ , M., S~:A~r~, M., and HAMADA,N., "The Effect of Changing Principal Stress Axes on LowCycle Fatigue Life in Various Strain Wave Shapes at Elevated Temperatures," ASTM STP, 856, 622. NismNo, S., HAMAO^,N., SAKANE,M., Om~AMLM., MATU~LrRA,N., and TOKIZASE,M., "Microstructural Study of Cyclic Strain Hardening Behaviour in Biaxial Stress States at Elevated Temperature," Fatigue Fract. Engng. Mater. Struct., 9, 65. "Report of the Bench Mark Project on Inelastic Deformation and Life Prediction of 21/4Cr-lMo Steel at 600 C," Proc. 2nd Int. Conf. Constitutive Laws and Short Course for Engineering Materials, Tucson; The Subcommittee on Inelastic Analysis and Life Prediction of High Temperature Materials, Japan Society of Materials Science, to appear.
Department of Mechanical Engineering Faculty of Science and Engineering Ritsumeikan University 56-1 Tojiin Kita-machi Kita-ku Kyoto, 603, Japan
(Received 21 July 1986; in final revised form 18 March 1987)
0749-6419/88 $3.00 + .00 Copyright© 1988PergamonJournalsLtd.
Printedin the U.S.A.
C Y C L I C B E H A V I O R O F A T Y P E 304 S T A I N L E S S S T E E L I N BIAXIAL STRESS STATES AT ELEVATED TEMPERATURES
MASATERU OHNAMI, MASAO SAKANE a n d SEHCHI NISHINO Ritsumeikan University (Communicated by J.L. Chaboche, GIS Ruptune/t Chaud)
Abstract-The results of companion, incremental/decremental, and stepup fatigue experiments on austenitic stainless steel tubes (type 304) are presented. The experiments include proportional and nonproportional loading conditions at ambient as well as elevated temperatures. Empirical relations are developed between van Mises effective stress and strain, and these relations are shown to describe the cyclic behavior during proportional companion as well as incremental/ decremental tests. In case of nonproportionai incremental/decremental experiments, the material behavior is not accurately modeled either by using the van Mises effective stress and strain, or by relating Tresca's maximum shear stress and strain.
I. INTRODUCTION
The study of the constitutive relation is one of the important research fields of plasticity as well as the yield condition and the hardening rule. Recent progress in numerical analyses such as FEM and BDM especially requires the constitutive relation that accurately expresses the actual material stress-strain relation. Numerous studies have been carried out on the cyclic stress-strain relation in uniaxial stress state (BaAT ~ LAIRD [1979]; POL~, KLESNIL& H~LESlC [1982]; VOYL~DrtS[1984]) and many constitutive equations have been proposed. In order to summarize the characteristics of constitutive equations, the Subcommittee on Inelastic Analysis and Life Prediction of High Temperature Materials, the Society of Materials Science, Japan, recently made a benchmark project on inelastic deformation of 21/4Cr-lMo steel in uniaxial creep-fatigue condition at 873 K. They studied 17 constitutive equations with six loading patterns and identified the constitutive equations relating to the life prediction in creep-fatigue condition [1985a, 1985b, 1987]. In spite of such an extensive study of constitutive equation, studies in biaxial cyclic condition are not many (BROWN~ Mm~R [1979]; DE Los RIOS & BROWN [1981]; ELLIS, ROmNSON ~ PU6H [1983]). The constitutive equation should be verified in multiaxial stress states. In biaxial stress states, since combination of mechanical stress with thermal stress sometimes yields nonproportional loading, the study of the constitutive relation in nonproportional condition is important as well as in proportional condition. Experimental studies on the nonproportional plasticity have been published (LAunA R, SmEnorroM [1978]; KANAZAWA,MtLLERS~BaOWN [1979]; IO~a'Z ~ LU [1984]) but they have only treated plasticity at room temperature; high temperature studies on nonproportional plasticity are scarcely made. The objective of this article is to examine the cyclic behavior in biaxial stress states during proportional and nonproportional loading conditions at elevated temperatures. As the cyclic stress-strain relation was expected to be affected by the test method, test results in two main methods were compared; one is the companion method using several 77
78
M, ()HNAMI el ¢l],
specimens to draw a cyclic stress-strain curve, and the other is the incremenlai, decrcmental method using the strain history that varies strain amplitude at each cycling where only one specimen completes a cyclic stress-strain curve. The companion lest ,aa~ ca~ ried out in proportional condition whereas the incremental/decremental test wa~ carried out in proportional and nonproportional conditions.
II. EXPERIMENTAL PROCEDURE
Test specimen used in this study was a thin walled cylinder of a solution heat treated Type 304 austenitic stainless steel, of which inner diameter, outer diameter, and gage length were 9 mm, 11 ram, and 20 mm, respectively. Test apparatus was a computer assisted electrohydraulic servo machine that can apply an axial load in combination with a torsional load (O~INAUl & HAMADA [1982]; HAMADA, SAKANE & OHNAMI [1984]; NISHINO, HAMADA, SAKANE, OHNAMI, MATUMURA & TOKIZANE [1986]) in any strain history. Axial and shear strains were measured by an originally designed extensometer composed of two thin tubes, measuring the axial displacement and the relative rotation angle between the tubes by two LVDT's (OHNAMI, SAKANE & HAMADA [1985]). A heater was inserted into the hollow cylindrical specimen to obtain elevated temperature and a sheath subheater was used to obtain uniform temperature distribution along the gage length. In order to examine the effect of test method on cyclic stress-strain relation, three strain histories were employed (as shown in Fig. 1). Strain history (a) in the figure has a constant strain amplitude while strain history (b), with variable strain amplitude in each cycle, has 25 cycles in one block of strain sequence. As to strain history (c), after straining 10 cycles with constant strain amplitude, the amplitude is increased to the next step that has the larger strain amplitude, and then this strain sequence is continued until the specimen fails. The strain histories (a), (b), and (c) are called the companion test, the incremental/decremental (I/D) test, and the stepup test, respectively. All strain histories in Fig. 1 have a Mises's equivalent strain rate of 10-3/s. Table 1 lists the test program carried out in this study. Experiments were mostly per.formed in strain controlled condition, with some performed in stress-controlled conditions in order to examine how the loading mode affects the cyclic behavior. Also, the majority of experiments were performed at 823 K, while only the I / D and the companion tests were carried out at 923 K as well as 823 K in order to examine the effect of temperature. Nonproportional loading was only applied in the I / D test at 823 K. All experiments were made in air. Figure 2 shows the loading sequence of nonproportional tests, where principal strain ratio, ~b, is defined as e3/el- el and e3 are the maximum and the minimum principal
Table 1. Test program Temperature
823 K
923 K
Strain-controlled test
I/D test" Companion test Step up test
I/D test Companion test
Stress-controlled test
Companion test
"Both proportional and nonproportional tests were carried OUt.
Biaxial stress states at elevated temperatures
-t-
79
+
lu5 ~= IC)3 Is
#_. = 10-3/s
c
.c_
llll,
o c
time
0
._> o
"5
~o
(I) u~
"viIIi
time
(b)
(a) it.j
~, +1
~ [ ~:,o-3/s
oL,, AAAA"AJA AAAAAAAAAA AAA
; '"""l"'"'"'Ivvvvvvvvvvvvvv -
~ I !/Ocycles! AV_meosuredcycle (c)
Fig. 1. Strain histories for proportional and nonproportional tests. (a) Companion test. (b) Incremental/ decremental (I/D) test. (c) Stepup test.
strains. Loading paths (~) and (~) correspond to the compression-tension and the reversed torsion tests, respectively, and the loading paths (~) and (~) are the test having the principal strain ratio of 4) = - 2 / 3 and ~b = - 5 / 6 , assuming that the Poisson's ratio is 0.5 in the low cycle fatigue case. Nonproportional test of Id01 = 1/2, for example, has the strain history, (~)-(~)-(~)-(~), and in that test the principal strain direction alternates 45 degrees in each cycle. In the I / D test, the principal strain direction alternates after each subcycle. The strain history of the other nonproportional tests is also listed in Fig. 2. Usually the nonproportional loading experiment employs the sinusoidal waves with a phase difference between combined compression-tension and reversed torsion loadings. In that test, however, the Mise's equivalent strain does not have a sinusoidal form but rather has a kind of trapezoidal shape, so the test result using this strain history includes the nonproportional loading, as well as hold-time effects. In this sense, the out-of-phase experiment is not suitable for examining the cyclic constitutive relation.
80
M. OHNAMIet al,
.............
-
Non-propor't lona t Id ¢1 = ~ / 6
=1/3 =1/2
Of]
loadJn9 ~
@------.~ @ ..... _ > @
--->®
@---> @ ...._>
--->@ .... > @
@ .... ~ (9 @ . . . . .> @
Fig. 2. Loading sequence of nonproportional tests.
In order to obtain an effect of only nonproportional loading on cyclic constitutive relation, the test method shown in Fig. 2 was used in this study. 111. E X P E R I M E N T A L RESULTS A N D D I S C U S S I O N
III. 1 Cyclic stress-strain behavior during proportional tests Figures 3(a), (b), and (c) show the cyclic behavior during the I / D , the companion, and the stepup test, respectively, (Mises' equivalent stress-strain). Here, the following equations were used in calculating the Mises' equivalent stress and strain. 0 = sign(S)~'tS j S = sign(a)o 2 + 3 s i g n ( r ) r 2
(1) ,
[ - 7
= slgn(N),J ]NJ N = sign(e)e 2 + (1/3)sign(3,)3 ,2. In these equations, a and r are, respectively, the normal stress and shear stress calculated from the axial load and the torque, while e and y are, respectively, the normal strain and the shear strain. Conventionally, the Mises' equivalent stress and strain have no negative sign, but both the negative and positive signs are employed here to emphasize the cyclic loading. The tensile loading is taken as positive for a and e, and one predetermined twisting direction is taken as the positive for r and % In the I / D test, the Mises' equivalent stress and strain correlate well with the cyclic stress-strain relation in biaxial stress state. In the companion test, Mises' criterion appears to be also effective, but the scatter of the data is larger in comparison with the I / D test. In the companion test, the maximum difference in the stress amplitude of 21°70
Biaxial stress states at elevated temperatures
~_ I1~
SUS304,
823K,
SUS304
8 2 3 K , in Air ' e .~" Proportionol cm ~"~ loading 1 ~,'0
, in ,Air _
I ~,D,., I
1
4001 Propgrtionol loading
81
Comlxmion'test
4O0
it:)"
~'0
= I O-S/s
200
ID
g-
/
~ .A.¢ -4°° I -200
-200 i
2
5
I
.2 -4OO
-I.5
o
-~
i/
i
I
I
L
-I.5 -I.O -Q5 Mises' type
-It.O -C)5 O 0.5 ID 1.5 Mises' type equivalent strain E,%
I
L
0 Q5 1.0 1.5 equivalent strain E , %
(b)
(a)
~_ 50O
SUS304, i
~. = io-3/s
40(
Proportional
823K~ i in Air i
f
Io,
Ib"
t~
.-"
20£ E -~ O
0
:-=ooi /
]---c=.=oo test
07
- 4 0 0 ~
o STep up rest
o Mises'
d5
type equivalent
E, (c)
strain
%
Fig. 3. Cyclic stress-strain curves in strain controlled proportional test at 823 K plotted on Mises' equivalent stress/strain basis. (at I/D test. (b) Companion test. (c) Stepup test.
occurs at 0.5% strain amplitude in the results of Fig. 3(b). The difference of 21% in stress amplitude is comparatively large, so that we have to treat the cyclic stress-strain relation carefully in the companion test if the precise analysis is required. Figure 3(c) compares the cyclic stress-strain relation between the I/D test (solid line) and the companion test (dotted line) together with the data in the stepup test. There exists a slight difference in the cyclic stress-strain relation between I/D and companion tests, but the data in the stepup test are very close to the results of the companion test. If the cyclic stress-strain relation is expressed as eqn (2), the material constants in the equation have the values listed in Table 2.
82
M. OHNAMI e t
al.
Table 2. Material constams in constitutive equation (2)
Mode
t£ (× 10 ~ MPa)
K' (MPa)
,~
~:, IMPa?
C o m p a n i o n test
Proportional loading
I. 25
480
0.265
? i?
lncremental/decremental test (I/D test)
Proportional loading Nonproportional loading [d4~l = 1/6 = 1/3
1.25
4t4
0.107
;~
1.30 1.40 1.50
440 480 495
0.102 0.099 0.092
3;'~ 409 427
Test method
= I/2
e=
+
~7
(2)
where ~ is Mises' equivalent total strain, o a Mises' equivalent stress, E ' the tangential modulus representing the gradient of the elastic part in o - ~ diagram, K ' the cyclic stress strength, and n' the cyclic stress exponent. Note that E ' has the similar value as Young's modulus but is not identical to Young's modulus. In the table, the material constants in nonproportional loading are also listed but they will be discussed in the next section. K ' and n' in the companion test have larger values compared with those in the I / D test, especially n' value in the companion test is twice as large as that in the I / D test. However, 0.2% p r o o f stress, oo.z, in the companion test is smaller than that in the I / D test. Although there exists a difference in material constants given in Table 1, the absolute values of the stress amplitude in the two testing methods do not differ much with each other as shown in Fig. 3(c). The companion test simulates the operating condition of practical components but needs m a n y specimens. So, if we employ the I / D test, we can complete a cyclic stress-strain curve with only one specimen. Since the cyclic stress amplitude in the I / D test does not differ much from that in the companion test, we can use the cyclic stress-strain relation in the 1/D test instead of the companion test. The stepup test is also useful instead of the companion test. Besides the Mises' equivalent stress and strain, the arrangement of the data using the Tresca's criterion is also made. Figure 4 shows an example. The data in the I / D test appears to be correlated with Tresca's criterion, but in the companion test the scatter o f the data becomes large. In both the test methods, the Tresca's criterion enlarges the scatter of the data in comparison with the Mises' criterion, so we can conclude that the Mises' criterion is a better parameter to correlate cyclic stress-strain relation in biaxial stress state. In this regard, KANBIL, MILLER & BROWN [1982] reported that the Tresca's criterion correlates well the biaxial cyclic stress-strain relation for a type 316 stainless steel in the companion test at 823 K. However, the experimental data in this study contradict their conclusion. The cause of the different trend of the data is not clear now but m a y be attributed partly to the test condition or material. In order to discuss the difference in shapes of the cyclic stress-strain curve between the I / D and the companion tests, Fig. 5 shows the hysteresis loops obtained in experiments, together with the schematic representation of them in the two tests. Bilinear representation is employed as the schematic figure, since in the bilinear modeling it is easy to understand the essential difference in the cyclic stress-strain relation due to the test method. In the I / D test, since each hysteresis loop has nearly equal n' value, and since the yield stress o f each hysteresis loop agrees with the stress amplitude of the cyclic
Biaxial stress states at elevatedtemperatures
3001_8U8,5044823K, inr~qAir o?
! I/D test Proportional L loading 200F g IO-S/S
83
300 SUS,304? 8213K,in Air
f
Companion test j ~ Proportiono I I
• -~~,.~
: 200~"loading ,~ )-~ IOOi~='d3/s Ii/ nO
=
~oo~ o~
o
.
.
.
.
.
.
i
.
~, -IO0~-
E
I
I I
-2oo
-~nnL -I.5 /
1 L I I -I.0 -0.5 0 0.5 1.0 1.5 Maximum shear strain Trnax/2,%
(a)
o
L
(,° o
,.o
,is
Maximum shear strain r m a x / 2 , %
(b)
Fig. 4. Cyclic stress-strain curves in strain controlled proportional test at 823 K on maximum shear stress/strain basis. (a) I/D test. (b) Companion test.
stress-strain relation, the cyclic stress-strain relation expressed by dotted lines occurs. The cyclic stress-strain relation in the I/D test agrees with the hysteresis loop in the test of the maximum strain amplitude as shown in Fig. l(b). In the companion test, however, the stress strain relation of each hysteresis loop, after yield, differs from each other and the stress amplitude increases with increasing plastic strain range. Then, the cyclic stress-strain relations as shown by dotted lines occur. From this, the cyclic stress-strain relation in the companion test has a larger strain hardening exponent, n', in comparison with the I/D test. The difference in the shape of the cyclic stress-strain curve due to the test method has resulted, of course, from the difference in the strain history. In the companion test, the material is loaded by the constant strain amplitude, whereas in the I/D experiment the specimen is subjected to various strain amplitudes during each strain block. The material in the I/D test especially experiences the maximum strain amplitude at each block and affects the shape of the cyclic stress-strain relation (Cagaocrm ROUSSELmR[1983b]. This effect is well known as the strain memory effect and is confirmed in the present study. The discussion above uses the stress-strain relation in order to clarify the essential point of difference in the shape of the hysteresis loop. The real material exhibits curved cyclic stress-strain relation, but the essential part is covered by bilinear modeling. Figure 6 shows the test results in the proportional companion test at 923 K together with that at 823 K expressed by solid line. Cyclic stress-strain curve has smaller stress amplitude at 923 K in comparison with that at 823 K. At 923 K, as well as at 823 K, the stress-strain relation can be expressed by the Mises' criterion with somewhat larger scatter in the data. The results with the larger value of n' exhibits smaller stress amplitude. 1II.2 Cyclic Stress-Strain Behavior During Nonproportional Loading Figure 7 shows th e cyclic stress-strain relation in the nonproportional I/D test. Nonproportional loading increases the stress amplitude, and the stress amplitude becomes
84
M.
i/D
OHNAMI
el al.
test
i:~:o m P a nion
test
I
i'
irS
35%
05 %
(a) I/D
test
(b)
. }
Companion test ~
f
i
,
t
/////j////,/ --Hysteresis ---
Cyclic (c)
loop constitutive relation
--
Hysteresis
---
Cyclic
loop
constitutive relation (d)
Fig. 5. Actual hysteresis loops and schematic figure. (a) Actual hysteresis loops in I/D test. (b) Actual hysteresis loops in companion test. (c) Schematic figure in I/D test. (d) Schematic figure in companion test.
larger as [d$] increases. In the nonproportional experiment, cyclic stress-strain behavior cannot be appropriately expressed by using the Mises' type equivalent stress-strain. The cause of the difference in the cyclic stress amplitude due to various nonproportional loading will be discussed in the next section in relation to the microstructures of the material. Table 2 lists the material constants in the proportional and the nonproportional I/D tests. Values of n' and K ' in the proportional companion test are larger than those in the proportional I / D test, whereas in the nonproportional test, an increase in ]dq51 increases the value of K ' and decreases that of n'. The tangential modulus in the elastic part increases as ]d4~[ increases. In order to correlate cyclic stress-strain curves in the nonproportional test, a simplified cyclic stress-strain relation is considered. The relation is of a modified power law type and is expressed by eqn (3),
Biaxial stress states at elevated temperatures
SUS304,
~-
--
Ib" 4 0 0
923K,
85
in Air
[ I Companion test Proportional I oadin~ /iA~ / ~ = I0"°/s i
I
200
c
~
o
o"
~ -200 M
/l
.~ - 4 0 0
-I.5 -I.0 -0.5
0 0.5 ID Mises' type equivalent strain £,%
1.5
Fig. 6. Cyclic stress-strain curves in strain controlled proportional test at 923 K.
o 5OO S U S S 0 4
Q_
400
I/D
82
test
Non- proportionol loading 200
c
O
0
~
-200
~ -400
I lol~i~l,l
~_sool.sL 40-o.s
o o,s
Mises' type equivalent ~, %
LO 1.5 strain
Fig. 7. Cyclic stress-strain curves in strain controlled non-proportional I/D test at 823 K.
Aa = IA - ~~pm.x p 1) m[K'A~p} n',
(3)
The equation only expresses the relation between the plastic strain range and stress range. If the total strain range-stress range relation is required, an elastic part should be added. In the equation, A ~ and A~p are the stress range and the conventional plastic strain range of Mises' type. The increase in the stress amplitude due to the nonproportional loading is taken into account by the term in parentheses, where za~p,~ is the equivalent strain of Mises' type but calculated from the maximum range in each
86
M. OHNAMIel al.
strain component for the case of nonproportional loading. For example, in the case ol Idq~l = 1/2, A~:prna x specified a s A~prna x = [A• 2 q- (1/3)(A'~p) 2] 1/2, assuming that the axial plastic strain range is A~p and shearing plastic strain range A,,fp. In that case, ~3~, is calculated by A~p = (A(~2)1/2 or Agp = [l/3(Ayv)2 ] 1,2. The exponent 07 in eqn (3) is the material constant and has the value of 0.5 in the case of this study. The solid lines in Fig. 7 show the cyclic stress-strain relation calculated from eqn (3), with the addition of the elastic part to the equation. Most of the experimental data are very close ',o the predicted lines o f eqn (3), so that we can conclude that eqn (3) is useful to express the cyclic stress-strain relation under nonproportional loading condition as shown in Fig. 2. It should be pointed out that the eqns {2) and (3) discussed so far have restriclions on usage. The equations presented here express the relation between the strain amplitude/range and stress amplitude/range so that the equations are not applicable io the generalized proportional/nonproportional loading conditions. However, the equations of course are valuable in calculating the stress amplitude from the strain amplitude in proportional/nonproportional conditions for most commonly used strain histories [ h e equations, which have the incremental form and are applicable to wider loading conditions, should be used. In order to understand the cause of the increase in stress amplitude due to the nonproportional loading, the authors examined the microstructures of the material strained in the compression-tension test (P), reversed torsion test (T), and nonproportional companion test ( A P T ) of [&b[ = 1/2 at A~ = 1.0% (NISmNO et al. [1986]). Dislocation structure in the compression-tension experiment is mostly ladder, partly maze, and somewhat cellular, whereas in the reversed torsion test it is mostly maze, partly ladder, and slightly cellular. A P T loading yields only cellular structure. These ladder and maze walls under compression-tension or reversed torsion loading are on {111 } planes, which are primary slip planes. In the A P T test, on the other hand, the maximum principal shear strain direction change by 45 degrees in each cycle, so that larger interaction of the slip system occurs. In the A P T test, the larger interaction results in cellular s~ructures, because it overcomes the rearrangement of the dislocations with the assistance of the thermal process. The resistance to the dislocation glide is, of course, larger in the cellular structure so that the larger cyclic hardening results (Ntsmyo et al. [1986])~ 1II.3 Cyclic Stress Amplitude and Cyclic Yield Stress Loci In addition to the cyclic stress-strain behavior, it is also important to describe the hardening behavior in the proportional and the nonproportional loadings in order to assist the general understanding of the cyclic plasticity in those loadings. In the following, we discuss the hardening behavior of the prestrained material in proportional and nonproportional loadings referring to the previous result of the authors. Figure 8(a) and (b) (NIsI-IINO et al. [1986]) show the equivalent stress amplitude and cyclic yield stress ellipses in P and A P T modes, respectively, with those of the virgin material at the Mises' equivalent strain range of 1.0°7o. The equivalent stress amplitude is obtained from the tensile and compressive peak stresses, and the cyclic yield stress is 0.4% offset stress from the peak strain as shown in the figure. The figure shows that the virgin material exhibits an almost isotropic stress response because the data of the material are on the Mises' ellipse. However, the cyclic stress response is not isotropic for the prestrained material in the P mode. The shape of that material is being elongated along the shear stress axis. The data points of the stress amplitude are almost on the ellipse expressed by the equation, a 2 + 2r 2 = 0 2, 0 2 + r 2 ----:
Biaxial stress states at elevated temperatures
SUS504,
823K,
AE
| o400+
~
Peak
~J~t~------__a
stress
~
~
|
~
~
in
Air
0 Stress amplitude of prestrained material m P I o a d ~ • Stress amplitude of prestroined material in APTIoading
= % ..... V =O.I t-'17
2
87
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2 oo ~-----._~
-"-,n
z-z
~
i
I
M -z P T z -. ÷ _ ~ = 0.
-I1
Stress virgin
•
amplitude of the material
I (a)
SUS 3 0 4 ,
823 K ,
1E , oJ ¢. : , /o J/ =0.1Hz
~ o | ~ &.200~ ~ ~ |
0.4%
~p"
offset
in Air
OCyclic yield stress of prestrained material in P loading OCyclic yield stress of prestrained
i
material in APT loading
~ 37-2: 0"veer
2o _
\
1, ~
6o'
ol
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~
'
1
~
,
.
.
/
~
,'oo/r' ..//
/
/
O
Axial stress , MPa
(7
~D.4'% /
2
c yield curve of the virgin material
z" _z
(3- + 7- = Oyp 2 0 0
(b) Fig. 8. Equivalent stress amplitude and cyclic yield stress ellipses in P and APT mode at 823 K (NISHINO et al., [1986]). (a) Equivalent stress amplitude ellipses. (b) Equivalent yield stress ellipses.
e2p. As with the virgin material, the cyclic response for the prestrained material in the the A P T mode is isotropic, but the ellipse expands from the virgin material. The equivalent stress amplitude in A P T mode is 1.4 times larger than that in the P mode. This tendency of the data was also confirmed at a strain range of 0.7°70.
88
M. OHNAMIel a/.
IV. CONCLUSIONS Mises' type e q u i v a l e n t stress a n d strain are effective in describing the biaxJal cyclic stress-strain behavior during p r o p o r t i o n a l c o m p a n i o n a n d p r o p o r t i o n a l I / D tcst:,. They are effective in both strain controlled a n d stress controlled conditions at 823 K and strain controlled condition at 923 K. However, the Mises' type equivalent stress and strai~t were n o t effective for the biaxial cyclic data in the n o n p r o p o r t i o n a l I / D test. Tresca'~ Iype e q u i v a l e n t stress a n d strain was less effective in c o m p a r i s o n with the Mises' type defin i t i o n s o f e q u i v a l e n t stress a n d strain. The shape o f the cyclic stress curve in the I / D test differs f r o m that in the c o m p a n ion test. W h e n the cyclic stress-strain relation is expressed by power law type e q u a t i o n , however, the c o m p a n i o n test has a larger strain hardening exponent t h a n that in the I / D test. The data in the stepup test are very close to the cyclic stress a n d strain data in the c o m p a n i o n test. C h a n g i n g the direction o f the principal strain axis increases the cyclic stress amplitude. I n order to take a c c o u n t o f the increase in the stress a m p l i t u d e due to the n o n p r o p o r t i o n a l loading, we p r o p o s e d the simplified e q u a t i o n . The e q u a t i o n can predict results that correlate well with the cyclic stress-strain relation in both p r o p o r t i o n a l as well as n o n p r o p o r t i o n a l loadings. A c k n o w l e d g e m e n t - T h i s work was supported by Grant-in-Aid for ScientificResearch B 60460084, the Ministry
of Education, Japan. The authors wish to express gratitude to Mr. Y. Hanato, Mr. Y. Hamano, and Mr. M. Fujiwara, the senior students of Ritsumeikan University, for carrying out the experiment.
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Department of Mechanical Engineering Faculty of Science and Engineering Ritsumeikan University 56-1 Tojiin Kita-machi Kita-ku Kyoto, 603, Japan
(Received 21 July 1986; in final revised form 18 March 1987)