Fatigue and creep-fatigue damage of

Fatigue and Creep-Fatigue Damage of Austenitic Stainless Steels under Multiaxial Loading

J. WEISS and A. PINEAU

The objectives of the present study are to observe and model physical damage induced by cyclic muitiaxial (tension-torsion) loading of 316L stainless steel both at room temperature and at elevated temperature (600 ~ Four types of experiments were carried out on thin tubular specimens: (a) continuous pure fatigue (PF) tests; (b) PF sequential tests with different sequences of push-pull and torsional loading; (c) creep-fatigue (CF) tests with superimposed hold time at maximum tensile strain; and (d) sequential tests involving sequences of PF and CF loadings. Optical microscopy and scanning electron microscopy (SEM) were used to study quantitatively the damage, in particular, to determine the orientation of cracks and to measure the kinetics of crack nucleation and crack growth. It is shown that in pure fatigue at 600 ~ the classical crack initiation stage I is bypassed due to a strong interaction between cyclic plasticity, oxidation, and cracking. Intense slip bands act as diffusional short circuits, leading to the formation of external (Fe203) and internal ((FeCr)304) oxide scales. The orientation of the microcracks during initiation and propagation stages, which is strongly affected by oxidation effects, explains qualitatively the significant deviations observed in the sequential tests from the Miner linear damage cumulative rule. It is also shown that creep-fatigue damage, which involves intergranular damage, is a complex process rather than a simple superposition of fatigue and creep damage. A stochastic model based on a Monte-Carlo simulation is developed. This model, which accounts very well for the situations in which crack initiation and crack propagation are coplanar, includes damage equations based on quantitative metallographical observations. Damage is modeled as the continuous nucleation of a population of growing cracks which eventually coalesce to lead to final fracture. It is shown that this simulation is able to reproduce with a good accuracy the fatigue lives measured under multiaxial continuous and sequential tests.

I. INTRODUCTION

MOST structural components are subjected in service to complex loading conditions in regions of stress con- centrations. This occurs, in particular, in high-temperature components which are submitted to superimposed mechanical and thermal stresses, leading to fatigue and creep multiaxial damage. For a long time, most of the studies dealing with fatigue damage at high temperature have been limited to uniaxial (mainly tensile) conditions. Brown and Miller tz,z~ were among the first investigators to em- phasize the importance of the directional aspect of fatigue damage under multiaxial loading. At low temperature, without significant oxidation effect, fatigue cracks initiate on shear planes from persistent slip bands, whereas they propagate usually under tensile mode I, except for loadings approaching pure shear, i.e., for large values of the ratio A' = A y / A e between the shear strain range, A T, and the tensile strain range, Ae, and for large values of Ay. Crack propagation occurs under shear modes II and III in this case. t31 At high temperature (650 ~ the same behavior was observed by Sakane et al. f41 on 304 stainless steel for the propagation stage but not for the initiation stage which occurred under mode I.

This directional aspect of fatigue damage plays a key role in the explanation of the results of sequential low- cycle fatigue (LCF) tests in which different sequences of

J. WEISS, formerly with Ecole des Mines, is Visiting Scientist, Ice Research Laboratory, Thayer School of Engineering, Dartmouth College, Hanover. A. PINEAU, Professor, is with the Centre des Matdriaux, Ecole des Mines, 91003 EVRY, Cedex, France

Manuscript submitted January 11, 1993.

push-pull and torsional loading are applied. At room temperature, it is usually observed that the cumulative damage D = NffNF, + N2/NF2 is larger than one for the sequence tension -~ torsion and lower than one for the reverse.[5.6,7] The detrimental effect of sequential fatigue corresponding to the latter situation can easily be explained, at least qualitatively, by the fact that under these conditions, the initiation plane and the propagation plane have the same orientation. At elevated temperature, similar effects have not yet been reported to the knowledge of the authors.

Fatigue lives in push-pull conditions are strongly reduced at elevated temperature, as a rule, tS1 and particularly for 316 stainless steels (e.g., Reference 9). The comparison of test results carried out under air or vacuum has shown that this reduction is largely due to environmental effects, especially oxidation, 191 but very few studies have been devoted to the micromechanisms of oxidation damage. Without cyclic loading, a very thin layer of Cr203 oxide grows on the free surface of austenitic stainless steels. ~ml Several questions can be raised when the material is deformed. What is the role played by this CrzO3 film with respect to fatigue damage? Is there any change in the nature and in the growth kinetics of oxide film during cyclic loading?

At elevated temperature besides this strong environmental effect, it is also well known that the wave form signal can also have a large influence on fatigue life. I81 Bulk intergranular damage can take place, particularly in austenitic stainless steels, when a tensile hold time is superimposed to the fatigue cycle. Levaillant et al . llll showed that the interaction between fatigue damage

METALLURG[CAL TRANSACTIONS A VOLUME 24A, OCTOBER 1993--2247

(microcracks initiated from the free surfaces) and bulk intergranular damage leads to an acceleration of the crack growth rate and to a reduction of the fatigue life. This conclusion was drawn from uniaxial tests. Very few studies have been devoted to the creep and creep-fatigue (CF) interactions under multiaxial loading.

The aims of the present work were the following:

(a) To study the influence of multiaxial loading on the fatigue life and fatigue damage. For pure fatigue loading, which produces only transgranular damage, the orientation and the distribution of cracks were measured. For creep-fatigue conditions, intergranular damage was also studied. These measurements were made in order to develop a quantitative model for damage accumulation. (b) To investigate the effects of sequential multiaxial loadings in relation to the orientation of microcracks. Special attention was paid to the influence of temperature on both types of sequence, push-pull and torsion, in order to test the effect of damage directionality. This part of our work has already been partly presented elsewhere, I j21 but more details are given in the present study. (c) To identify the nature of the oxides formed at the free surface of fatigue specimens and to assess the growth kinetics of these oxide layers and their role on damage process. (d) To contribute to the understanding of the influence of alloy composition on intergranular carbide precipitation and intergranular cavitation. (e) To develop a damage model based on the interaction of a population of microcracks.

II. E X P E R I M E N T A L P R O C E D U R E S

A. Materials

Three austenitic 316L stainless steels were examined. Their compositions and tensile properties at 600 ~ are given in Tables I and II, respectively. Most of the results reported in the present study were obtained on SQ steel, which is a 316L stainless steel with a low carbon content and a controlled amount of nitrogen. An annealing treatment was applied on this heat at 1100 ~ for 90 minutes, followed by water quenching. The resulting grain size

was about 50 /zm, although SQ steel contained a few larger grains ( - 2 0 0 / z m ) .

The results obtained with this heat were compared to those corresponding to the following.

(a) A very similar 316L stainless steel, SP steel, investigated in a previous study, tlu with a boron content slightly larger than SQ material. A double annealing treatment at 1070 ~ was applied to this material, followed by water quenching. This produced a grain size similar to that of SQ alloy. (b) An extra-low-carbon stainless steel, named 316MN stainless steel, with a mean grain size of about 90/xm. This material was thoroughly investigated elsewhere. 1~3J

All these materials were tested in the as-received conditions for which the grain boundaries are free from carbide precipitation.

B. Mechanical Tests

Fatigue and CF tests were carried out using a servo- hydraulic tension-torsion machine under strain-control conditions.

Tubular specimens, similar to those used in a previous study, TM with an outer diameter of 19.5 mm and a wall thickness of 1.5 mm, mechanically polished with 3/~m of diamond paste, were tested. A radiation furnace was used for the tests carried out at high temperature (600 ~ The longitudinal temperature gradient along the gage length of the specimens was less than 2 ~ Data ac- quisition was performed on X-Y recorders and with a microcomputer.

Mechanical tests can be classified as follows.

(a) Multiaxial continuous LCF tests, carried out at room and high temperatures (600 ~ with a sinusoidal fully reversed waveform at a frequency of 0.05 Hz, including push-pull tests, torsion tests, and in-phase tension-torsion tests. (b) Low-cycle-fatigue sequential tests, at 25 0(7 and 600 ~ including two independent successive phases with different types of loading. In these tests, a variable percentage of the fatigue life is applied with a given loading mode during the first phase, followed by another type of loading applied up to failure. Both types of sequences

Table I. Chemical Composition (Weight Percent)

Material C S P Si Mn Ni Cr Mo N AI B

SQ 0.028 0.001 0.028 0.30 1.88 12.46 17.31 2.44 0.077 - - 0.001 SP 0.021 0.007 0.033 0.41 1.74 12.27 17.18 2.40 0.080 0.040 0.0035

316MN 0.011 0.0057 0.028 0.47 0.85 1 1.07 16.64 2.14 0.068 0.003 - -

Table II. Tensile Properties at 600 ~

0.2 Pct E /x Yield Stress UTS Elongation Reduction of Area

Steel (10 3 MPa) (10 3 MPa) (MPa) (MPa) (Pct) (Pct)

SQ 144 55 146 407 41 67 SP 144 55 159 404 41 67

316MN 146 56 - - 390 - - 62

2 2 4 8 - - V O L U M E 24A, OCTOBER 1993 METALLURGICAL TRANSACTIONS A

were investigated, i .e. , tension ~ torsion tests and torsion ~ tension tests. (c) Multiaxial continuous CF tests carried out at 600 ~ by imposing a hold time at maximum tensile strain. Two types of tests were carried out: push-pull tests, with a tensile hold time, and torsion tests. The applied hold times varied between 30 and 60 minutes. (d) Fatigue/creep-fatigue sequential tests, in which a first phase of PF with a given loading mode, followed by a creep-fatigue (CF) phase is applied up to failure, or the opposite (CF then PF).

A von-Mises equivalent total strain range of AEteq/2 = ~0.80 pct was chosen for all of the tests carried out on SQ and 316MN steels. The results of a number of uniaxial (tensile and torsion) tests carried out on SP steel at different total strain ranges, and described elsewhere,U ~.~4j were also used. Most of the experiments were carried out under air environment, except for a few push-pull tests on SP steel which were performed under vacuum (5- 10 -6 T o r r ) .

C. Microstructural Observations

Optical microscopy observations were performed on longitudinal sections of the specimens in order to measure the density of secondary cracks, the mean thickness of oxide scales, and the percentage of cracked grain boundaries. The directional aspect of damage was investigated by scanning electron microscopy (SEM) observations both on longitudinal sections and on the free surface, as indicated previously, lie1 These statistical re- suits concerning the distribution of the orientation of microcracks were represented by orientation roses, where the ratio of the number of microcracks measured in various directions is drawn. Scanning electron microscopy was also used to examine the fracture surfaces and to measure fatigue striation spacings. The chemical composition and crystallographic nature of oxide layers were determined by microanalysis and X-ray diffraction.

I I I . R E S U L T S

A. Mechanical Behavior

The results of continuous PF tests, carried out on SQ and SP steels both at room temperature and at 600 ~ are shown on Figures l(a) and (b), respectively. It is worth noting that PF lives do not depend on small vari- ations in alloy composition, since SP and SQ steels lead to the same fatigue life. It is observed that whatever the temperature, the fatigue life under fully reversed torsion is much larger than under push-pull conditions, whereas the endurance underin-phase tension-torsion loading (with A' = Ayp/Aep = V3) is nearly the same as under push- pull conditions. These results confirm those which were published earlier. 1~51

Figure 1 (a) also includes the results of two additional tests which were carried out in order to study the effect of oxidation. In the first experiment, the specimen was preoxidized for 4 hours at 600 ~ without any applied stress before being tested under fully reversed torsion at room temperature. This test shows that the formation of a very thin film of Cr203 oxide during high-temperature

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tors, (this study) {SOl

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tens, (vacuum:600~ {SP~ , , , , , , , , , , i , , , , , r T

1 (300 10000 N F

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_e ~a tens (this study)lSQI o ' ~

--> torsion ISal O" LU �9

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0.1 , , , , , , , , , ~ J , , J J 100 1 000 10 000

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(b)

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( a ) r o o m t e m p e r a t u r e a n d ( b ) 6 0 0 ~

exposure has no effect on the fatigue life. In the second test, the material was tested at high temperature under vacuum under push-pull conditions. It is observed that the fatigue life is the same as that determined at room temperature under air environment. The reduction of fatigue life observed at high temperature is therefore due to a strong environmental effect which takes place si- multaneously with cyclic deformation.

The results of PF sequential tests, carried out on SQ steel both at 25 ~ and 600 ~ are shown in Figures 2(a) and (b), respectively. In these figures, the damage Dl = NI/NF, represents the fraction of fatigue life applied under loading conditions corresponding to the first phase, while D2 = N2/Np 2 is the relative remaining fatigue life under loading conditions corresponding to the second one. Large deviations from the Miner linear cumulative damage rule (D~ + D2 = 1) are observed. Moreover, an anti- symmetric behavior is shown, since at a given temperature, an opposite effect is observed between push-pull

torsion sequence and torsion ~ push-pull sequence. At room temperature, the remaining push-pull fatigue life is significantly reduced by initial torsion loading, whereas prior cycling under push-pull conditions has much less effect on the remaining torsional life. These results are

M E T A L L U R G I C A L T R A N S A C T I O N S A V O L U M E 2 4 A , O C T O B E R 1 9 9 3 - - 2 2 4 9

'i1 0.8 �84

192 0.6

0.4

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Tens ~Tor

Tor~Tens

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0 0.2 0.4 0.6 0.8 1 1.2 D 1

(a)

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[ ~ Tor---~Tens

~ ~ D

~ 0.2- �9

0 , , �9 , , "~ 0 0.2 0.4 0,6 0,8 1 1.2

Da

(b)

Fig. 2 - -Cumu la t i ve fatigue damage for tension ---* torsion and torsion ~ tension sequential Ioadings at (a) room temperature and (b) 600 ~

similar to those reported by other investigators (e.g., References 5 through 7). An opposite situation is observed for the same sequences between experiments car- ded out at room and high temperatures. In particular, at 600 ~ the remaining torsional life is strongly reduced by initial push-pull fatigue. For a value of prior damage D~ = N t e n s / N F t e n s close to 0.5, the remaining torsional life is almost negligible and final failure occurs almost in- stantaneously when the specimen is cyclically deformed under torsion.

Figure 3 shows the results of CF tests carried out at 600 ~ For the same total strain range (Aete q = 1.60 pct), the numbers of cycles to failure in CF are much lower than those obtained in PF, for push-pull conditions as well as for torsion. The effect is much more

Type Hold Steel of lime

loading (mn)

0 SQ Push-Pull 33

0 SQ Torsion 33

60 0

SP Push-Pull 30

0

316MN Push-Pull 32

Pure-Fatigue and Creep-Fatigue lives

_ _ ] �9 .~-

~J , , , , , , , , , , i , , J J , ,

100 1000 10000

N F

Fig. 3 - Comparison of fatigue lives between continuous fatigue tests and CF tests carried out at 600 ~ with the same total strain amplitude (Ae,eq/2 = +-0.80 pct).

pronounced in SQ steel than in SP steel and in the extra- low-carbon material.

In order to investigate the interactions between intergranular creep damage and transgranular fatigue damage, a number of sequential PF /CF tests were carried out at 600 ~ on SQ steel. The results are given on Figures 4(a) and 4(b). Figure 4(a) shows the effect of an initial CF push-pull loading on the remaining PF push- pull life. In these tests, a cumulative damage close to the Miner linear damage rule is observed. The results of two different types of tension --~ torsion tests are reported on Figure 4(b). For the sequence tension (CF) torsion (PF), a behavior similar to that corresponding to tension (PF) ~ torsion (PF) is observed (D > 1), while an opposite situation occurs for the tension (PF) ~ torsion (CF) sequence.

B. Metallographical Observationx and Danlase Meusurements

1. Pure fatigue Pure fatigue damage corresponds to the formation of

a population of transgranular microcracks nucleated on the free surface of the specimens which propagate and coalesce as a main crack.

a. Initiation mode Figure 5 shows the orientation roses of microcracks

initiated under various conditions. These measurements were made on microcracks observed both on the free surface of the specimen and on longitudinal sections. These microcracks had a depth smaller than 25/~m, i.e., about one-half the grain size of the material. It is observed that at room temperature, microcracks initiate on maximum shear planes (Figures 5(a) and (b)) which cor- respond to stage I crack initiation, whereas at 600 ~ under air environment, microcracks nucleate in opening mode (Figures 5(c) and (d)) both under push-pull and torsional loading. Therefore, the initiation mode changes with temperature but not with loading conditions.

b. Distribution and density of microcracks It was shown elsewhere [~6J that the microcracks were

randomly distributed on specimen surfaces, following a Poisson distribution, which can be written as

2250--VOLUME 24A, OCTOBER 1993 METALLURGICAL TRANSACTIONS A

1.2-

T=600~ I 1

~, 0.8- .~. t - O

~ 0.6-

n 0.4-

0.:~-

0 i i i i 0 0.2 0.4 0.6 0.8 1

D (Tension:CP)

(a)

'ii , -

1.2

Tens(PF)-~Tor(CF)

Tens(CF)--,Tor(PF) 0.8-

D 2 0.6-

0.4-

0.2-

0 i i i ~ i 0 0.2 0.4 0.6 0.8 1 1.2

D1

(b)

Fig . 4 - - C u m u l a t i v e f a t igue d a m a g e for C F tests o n S Q steel tes ted at Ae ,~J2 = -+0.80 pct: (a) push -pu l l (CF) --~ p u s h - p u l l (PF) s e q u e n c e a n d (b) push -pu l l (CF) ---> to r s ion (PF) a n d p u s h - p u l l (PF) ---> to r s ion (CF) sequences .

P(X = h) = e - h - [1] h!

where A = X" S, X being the surface density of microcracks and S the area of a surface, h is the mean number of microcracks located on an area S, and P(X = h) is the probability that the area S contains h cracks.

The surface density of microcracks X was determined from the measurements of the linear density 77 of secondary cracks on longitudinal sections of the specimens which were converted into X by using simple stereo- logical relations. More details are given elsewhere. 1~2,~6~ The following expressions were derived from these measurements:

For push-pull loading at 600 ~

X ( N ) = Xo (Aep/2) 4" N [2]

(btm -2) (pct) (cycles)

where X0 = 8 • 10 -7. For torsional loading at room temperature, it was found

that the surface density was not dependent on the shear strain range Ayp/2, at least in a relatively small range of Ayff2 values which were explored (0.8 pct < Ayp/2 < 1.4 pct). The corresponding surface density was expressed as

X(N) = Xo" N [3] (/zm -2) (cycles)

where X0 = 8 • 10 -9.

c. Propagation mode Macrocrack propagation corresponding to the growth

of coalesced microcracks such that the crack depth is much larger than the grain size was observed to be dependent on loading mode but not on temperature. Our observations confirmed that for proportional loadings, the preferential orientation of cracks during the propagation stage depended only on the strain ratio A' = Ayp/ Aep (see also Reference 4). Cracks propagate under mode I for push-pull (A' = 0) or in-phase tension-torsion (with A' -- ~v/'3) conditions, while they grow along maximum shear planes for large values of h ' and Ayff2, particularly for pure torsion tests (h' = +~) .

Observations on fracture surfaces and simultaneous statistical measurements of crack depths and crack lengths showed that the crack front adopts a semielliptical shape with an aspect ratio q = small axis length/large axis length equal to - 1 for push-pull loading and to - 0 . 5 for torsion loading.

d. Crack growth laws Two different methods described in more detail else-

where 112A6] were used to measure crack growth rates. The first one, which applies only to mode I propagation, i.e., essentially push-pull tests, is based on the measurements of striation spacings on fracture surfaces. The striation spacing which is assumed to be equal to the crack growth rate, da/dN, is related to the depth of the main crack, a, and to the plastic strain amplitude. Push-pull tests carried out at different plastic strain ranges on SP and SQ steels led to the following expression:

da/dN = A(Aep/2)~a ~ [41

where a = 1.5 and/3 = 1.45 while A = 6.8 • 10 -4 at 25 ~ and A = 3.1 • 10 -3 at 600 ~ with (da/dN) in ~m/cyc le , Aep/2 in percent and a in micrometers.

The second method, essentially applied to torsion tests since no striations were observed in the case of pure shear mode propagation, is based on measurements of crack depth histograms on longitudinal sections (for more details, see Reference 12). The crack growth rate under pure torsion was expressed as:

da/dN = C(ATp/2)'~'a ~' [5]

where a ' = 3.7, /3' = 2.6, and C = 7.9 • 10 -7 .

METALLURGICAL TRANSACTIONS A VOLUME 24A, OCTOBER 1993- -2251

specimen axis specimen axis

longitudinal section free surface longitudinal section free surface

T : 2 5 o C TENSION T = 2 5 o C TORSION

(a) (b)

specimen axis specimen axis

longitudinal section free surface longitudinal section free surface

T = 6 0 0 o C TENSION T = 6 0 0 o [2 TORSION

(c) (d)

Fig. 5--Orientat ion of microcracks determined on specimen free surface and on longitudinal sections for (a) push-pull tests at 25 ~ (Ae,/2 = -----0.80 pct); (b) torsion tests at 25 ~ (Ay,/2= _ 1.38 pct); (c) push-pull tests at 600 ~ (Ae,/2 = +-0.80 pct); and (d) torsion tests at 600 ~ (Ay,/2 = +- 1.38 pct).

2. Sequential tests It was shown that prior torsional loading at room tem-

perature produces a large detrimental effect on the subsequent push-pull fatigue life (Figure 2(a)). This situation can qualitatively be explained by the fact that shear microcracks nucleated during the torsion phase and perpendicular to the specimen axis can easily propagate in mode I during push-pull loading (Figure 6(a)). Simi- larly, at high temperature, the microcracks initiated in opening mode during push-pull fatigue are located along preferential propagation and coalescence planes in shear

mode during torsional loading (Figure 6(b)). These observations show that a cumulative damage D lower than one is observed when initiation planes during the first phase and preferential propagation planes during the second one are coplanar. If this geometrical condition is not satisfied, the microcracks nucleated during the first part of the sequence have to bifurcate to propagate during the second phase of the sequence or a new population of microcracks has to be nucleated. A cumulative damage larger than one is observed in this case. The difference of initiation modes between room and high temperatures


Fig. 6 - - S E M observations of the free surface of specimens tested under PF sequential conditions. (a) Torsion ~ tension sequence at room temperature, Shear microcracks initiated during the torsional loading propagate under mode I during the tensile loading. (b) Ten- sion ~ torsion sequence at 600 ~ Mode I microcracks nucleated during tensile loading propagate and coalesce in shear mode during torsion fatigue.

but not between different types of loading, and the reverse in the case of propagation modes, explains, at least qualitatively, the double antisymmetry observed on cumulative diagrams of sequential fatigue tests (Figure 2).

3. Creep-fatigue experiments

a. lntergranular damage Creep-fatigue damage of austenitic stainless steels at

600 ~ is associated with the formation of the following two types of defects.

(a) A population of transgranular and intergranular microcracks initiated on the free surface of the specimen (Figure 7). The amount of intergranular cracking can be

Fig. 7 - - S Q steel: optical observation of a longitudinal section of a CF tension/compression specimen (Ae,/2 = -+0.80 pct; hold time = 33 min). The specimen axis is vertical. Intergranular cracks connected with the surface and intergranular bulk damage are observed.

determined by the ratio I,. measured on longitudinal sections, where L represents the number of intergranular microcracks over the total number of microcracks. The observation of push-pull specimens as well as torsion specimens showed that the initiation and the propagation of intergranular cracks occurred predominantly under mode I. (b) An intergranular bulk damage, i.e., intergranular decohesions appearing generally as sharp wedge cracks (Figure 7). Round cavities were also observed by SEM at larger magnification. The orientation rose reported on Figure 8 shows that these decohesions are perpendicular to the maximum principal stress o"1, positive during the hold time.

Levaillant et al. 1'~'~41 introduced the parameter DI defined as follows to measure the intergranular bulk damage:

D I = LF/L r [6]

where Lr is the cumulative length of cracked grain boundaries per unit area of section while Lr is the total length of grain boundaries on the same area. More details concerning the practical determination of D~ are given elsewhere, tm For CF conditions, the damage per cycle can be defined as:

Dc = D , / N [7]

It was shown earlier that the three investigated materials exhibited widely different sensitivities to the effect of hold time (Figure 3). It was therefore tempting

METALLURGICAL TRANSACTIONS A VOLUME 24A, OCTOBER 1993--2253

specimen axis

longitudinal section{ 1 )

l spt~m*n ~ s

perpendicular section(2

T : 6 0 0 * C TORSION (Creep-Fatigue)

Fig. 8 - - S Q steel: orientation of intergranular decohesions observed in CF torsion specimens tested at 600 ~ (Ay,/2 = _+ 1.38 pct; hold time = 60 min). Observations on two longitudinal sections, as indicated schematically.

to compare the damage introduced in these materials tested under the same conditions.

For the same loading conditions (push-pull tests; Ae,/2 = ---0.80 pct; hold time = 30 minutes), the Ic parameter is equal to 0.5 for SP and 316MN steels whereas Ic is equal to 1 for SQ steel since in this material all cracks are intergranular. The value of Dc is also much larger for SQ steel (Dc = 2.3 10 -4 cycle -~) than for SP steel (Dc = 1.5 10 -4 cycle-a). In 316 MN steel submitted to a hold time of 30 minutes, no intergranular bulk damage could be observed. For multiaxial CF experiments carried out on SQ alloy with hold times larger than 30 minutes, the fracture surface is predominantly intergranular (Figure 9) while the crack path exhibits a mixed

Fig. 9 - - S Q steel: SEM observations o f the fracture surface of CF push-pull specimens tested at 600 ~ (Ae,/2 = ---0.80 pct; hold time = 33 min).

transgranular/intergranular character in SP steel [141 and in 316 MN material. These differences between the behavior in CF of low-carbon (316L) and extra-low-carbon (316 MN) steels are in good agreement with the results published by Nakazawa et al. 071

b. Grain boundary precipitation Scanning electron microscopy observations carried out

on sections of CF specimens of SQ steel showed the precipitation of M 2 3 C 6 carbides along the grain boundaries which appeared either as isolated particles or as massive precipitates in the form of parallel platelets, crystallo- graphically oriented with one of the neighboring grains I18] (Figures 10(a) and (b)). This carbide precipitation was not observed in SP and 316MN steels tested under the same conditions (Figure 10(c)).

Intergranular carbide precipitation in austenitic stainless steels has largely been investigated in the literature (e.g., Weiss and Stickler[~9]). These authors showed that in a 316 stainless steel with a composition similar to SQ material, in the annealed condition, carbide precipitation is initiated after approximately 100 hours at 600 ~ In our study, a number of CF tests lasted less than 100 hours. This indicates that carbide precipitation is accelerated by cyclic deformation. Stress-enhanced grain boundary seg- regation and diffusion of chromium or carbon might be invoked to explain this acceleration in grain boundary precipitation. 120.211

The extra-low-carbon content of 316 MN alloy shifts the beginning of carbide precipitation to longer times. This observation is in agreement with those reported by Nakazawa e ta / . IlT] on the same 316 MN steel. The difference observed between SP and SQ alloys is probably due to two factors:

(a) a lower carbon content in SP steel ( - 2 0 0 ppm) than in SQ steel ( - 2 8 0 ppm), and (b) a larger boron content in SP steel (--35 ppm) than in SQ steel ( - 1 0 ppm), since it is well known that boron has a strong influence on creep ductility of austenitic stainless steels (e.g., Reference 22) by modifying the grain boundary energy t23] and by reducing intergranular carbide precipitation which, in turn, produces a reduction in intergranular cavitationJ TM

C. Oxidation Behavior

The following two types of oxides, different from Cr203, were identified in the three investigated materials when they were cyclically deformed.

(a) An external layer of Fe203 oxide (Hematite) (Figures 11 and 12). (b) An internal layer of (Fe,Cr)304 oxides, particularly rich in chromium near the interface between the external and the internal layers (Figures 11 and 12(b)). It was shown by X-ray analysis that the crystallography of this oxide is similar to that of chromite F e C r 2 0 4 . It is also worth noting that nickel does not seem to be oxidized.

The thickness of the external layer is very irregular, particularly on PF specimens. It can be locally as large as 50 /xm. The mean thickness of the internal layer vs testing duration was measured by image analysis (Figure 13). This figure shows that the kinetics of oxidation are different from the classical parabolic law, since


Fig. I O - - S E M observations of grain boundaries of specimens tested at 600 ~ under push-pull conditions (Ae,/2 = • pct; laold time = 33 rain). (a) SQ steel: isolated M2sC6 carbides; (b) SQ steel: cellular grain boundary carbide precipitation; and (c) 316 MN steel.

the thickness of the oxide layer increases linearly with time, within a first approximation.

Internal and external oxidation often takes place close to microcracks, as shown in Figure 11, but not exclu- sively. Oxides are often observed independently of the formation of microcracks. It seems that oxide formation occurs before crack initiation and not the reverse (Figure 12(a)). The role of cyclic plasticity on oxide formation is clearly illustrated on Figure 12(b) where preferential growth of internal (Fe,Cr)304 oxide is observed to take place along intense slip bands acting as diffusional short circuits for iron, chromium, and oxygen.

IV. DISCUSSION

The discussion is divided into two parts. First, the oxidation behavior of 316 stainless steel and the role of

oxidation on the fatigue life are discussed, since it was shown that environmental effects play a predominant role on many aspects of the fatigue damage. Then, a model based on the results of quantitative metallographical observations is proposed to simulate the fatigue life not only under continuous cycling but also under sequential loading.

A. Oxidation

It was observed that the interaction between oxidation and cyclic deformation produces three main effects: (a) a change in the nature and in the growth kinetics of oxides; (b) a modification of the initiation mode for microcracks when the test temperature is increased from room temperature to 600 ~ and


501~m

Fig. 11 - - S Q steel: optical observation of a longitudinal section of a torsional CF specimen (~y,/2 = -+1.38 pct; hold time = 60 min) illustrating both internal and external oxidation.

(c) a strong reduction of the fatigue lives at elevated temperature when the materials are tested under air environment.

During heating of the specimens before cycling, a Cr203-0xide scale thinner than 0 .5 /zm grows on the free surfaces of the specimens. The existence of this thin layer seems to have no effect on the fatigue life, since the endurance of a preoxidized specimen is similar to that of a polished specimen (Figure 1 (a)). Moreover, detailed observations similar to those presented earlier and shown in Figure 14(a) demonstrate that crack initiation occurs under shear mode in preoxidized torsion specimens tested at room temperature, similarly to the situation observed on polished specimens (Figures 14(a) and 5(b)).

The modification in the initiation mode between room temperature and 600 ~ under air is clearly due to a syn- ergistic effect between oxidation and cyclic deformation, as shown by further examinations of the microcrack orientations observed on the tensile specimen tested at 600 ~ under vacuum (Figure 14(b)). These results confirm that in the absence of oxidation effect, crack initiation takes place according to the classical stage I mechanism, even at elevated temperature.

It was observed that cyclic deformation produced drastic modifications in the nature and in the growth kinetics of oxide films. Other authors have also reported nonpara- bolic oxidation kinetics under cyclic deformation (e.g., the study by Skelton and Bucklow on Cr-Mo-V steelst251). In their study, devoted to a low alloyed steel, the de- viation from parabolic law was attributed to periodic

spalling of oxide film which results in the exposure of fresh metal surfaces to oxidation. Moreover, the oxides were striated. It seems that in the present study, the mechanism is different since the growth of two relatively thick oxide scales is directly related to the existence of intense slip bands acting as diffusion short circuits for metallic cations and oxygen. Furthermore, these oxide layers were not striated. The thickness of these oxide layers is such that it can be assumed that they are brittle and that they break easily under tensile stresses, as indicated by Schfitze. L261 These observations suggest that the interactions between cyclic deformation, oxidation, and cracking can be described by the following sketch.

(a) Formation of a protective, very thin Cr203-0xide film during prior heating. (b) Nucleation of intense slip bands during cyclic loading. These bands act as efficient diffusion short circuits for reactive species, while the Cr203 scale loses its protective properties. (c) Successive formation of an external layer of Fe203 oxide and an internal layer of (Fe,Cr)304. (d) Nucleation of microcracks into these brittle scales under the effect of tensile stresses. (e) Subsequent propagation of these cracks into the material while these cracks become, in turn, preferential oxidation sites.

It should also be added that the situation is somewhat different when the materials are tested under CF conditions, since the grain boundaries then act as diffusion short circuits and are preferentially oxidized instead of the intense slip bands. Preferential oxidation of grain boundaries can be due to carbide precipitation, either directly, the carbides being easily oxidized, or indirectly, because of the change in alloy composition, particularly chromium, at the vicinity of the grain boundaries (e.g., Reference 21).

Finally, to conclude this part of the discussion, it is also worth mentioning that oxidation does not necessarily always have a detrimental effect on fatigue life. It can even have a beneficial effect under certain circumstances, particularly when oxide formation contributes largely to crack closure. This might be the situation observed in Figure 4(b) for the tension (PF) ~ torsion (CF) sequence where a cumulative damage larger than one was observed, which is an unexpected result. It is suggested that the tensile microcracks initiated during the first part of the sequence are healed by oxide formation during the hold times applied during the second part of the loading.

B. Modeling Pure Fatigue Damage

It was clearly shown that a classical linear cumulative damage rule is unable to account for the results of sequential fatigue (Figure 2). More sophisticated rules or damage functions have to be developed. Some of them might be derived from the application of the continuum damage mechanics, introduced by Kachanov. t27j In this theory, the damage is described by a parameter which is considered to be a state variable. However, recent studies showed the limitations of this approach in the case of complex loadings, t281 This is one of the reasons why


Fig. 1 2 - - S Q steel: preferential oxidation along intense slip bands. (a) Free surface of a push-pull pure fatigue specimen (Ae,/2 = -+0.80 pct). (b) Longitudinal section of a torsion CF specimen (A3,,/2 = --+ 1.38 pct; hold time = 60 min).

the authors have preferred to develop an approach based on the physical description of fatigue damage, i . e . , the nucleation, the growth, and the final coalescence of microcracks. A short account of this model was given previously. [291

One of the interests of the authors' approach lies in the fact that it is directly based on the relations derived from metallographical observations, describing both the nucleation and the growth of a microcrack population (Eqs. [.11 through [51). The only stage of fatigue damage which is quantitatively unknown is that corresponding to crack coalescence. An influence zone was associated with each crack tip to describe this phenomenon (Figure 15). It was assumed that coalescence between two neighboring cracks occurred as soon as their associated influence zones touched and overlapped. The shape of these crack

tip plastic zones is indicated in Figure 15. Their sizes were calculated from the Dugdale-Barenblatt model, 13~ extended to large scale yielding conditions as follows:

in opening mode, R1 = wl a(cr/~r,) 2 [8a]

in shear mode, Rtr = 0)2 a('r/~'R) ~ [8bl

where o-R mad rR (= crR/X/3) are the ultimate tensile (shear) stresses, while 0)~ and w2 are constants determined by fitting experimental results from continuous fatigue tests carried out at one strain amplitude ( A e t e q / 2 = ---0.80 pct) to those derived from the simulation.

This model is based on a 2-D Monte-Carlo type simulation. The specimen surface is represented by a grain matrix of 1250 or 16,000 grains with a mean size of

METALLURGICAL TRANSACTIONS A VOLUME 24A, OCTOBER L993--2257

10

E

._o

E

2-

1 1"

§

I "

lb. /

160 260 0

o 36o 460 56o 600 testing duration (hr)

Fig. 1 3 - - S Q steel: mean thickness of internal oxide layer as a func- tion of time.

50/xm, randomly generated by a Poisson process at the beginning of each simulation�9 Initiation sites of microcracks located at the center of these grains are also chosen by a Poisson process. The size of these microcracks is taken as equal to 50/xm. The density of microcracks increases linearly each cycle according to Eqs. [2] and [3], while the length of each crack increases according to the growth laws which were determined in Eqs. [4] and [5]. The coalescence criterion is tested at each cycle for each couple of cracks. In this model, the microcrack orientation is imposed: the cracks are either perpendicular to the tensile axis of push-pull specimens or aligned along one of the shear directions of torsion specimens. This geometrical condition limits the application of this model to situations where no crack bifurcation was observed either from the initiation to propagation stage for continuous tests (i.e., torsion at 25 ~ and push-pull at 600 ~ or from the first sequence to the second one for sequential tests (i.e., torsion ~ tension at 25 ~ and tension ~ torsion at 600 ~ The values of the only unknown parameters of this model, i.e., 0)~ and w2, were determined as indicated previously and found to be equal to Wl = 0.42 and 0) 2 = 5.90 ~-- 14 0)1. The fact that w2 is much larger than 0)! indicates that crack coalescence occurs much more easily under shear mode than under opening mode (Figure 6(b)).

Illustrations showing the simulated evolution of crack population in a sequential test are reported in Figure 16 where the effect of crack coalescence is clearly observed. More details showing simulated crack population are given elsewhere, l~6'29] Here, it is sufficient to sum- marize the salient features derived from this model.

(a) This model is able to reproduce the fatigue lives observed on SP and SQ steels tested under continuous tension or torsion loading as well as the statistical scatter (Figure 17). (b) The conclusions related more specifically to the statistical aspects of this model were discussed in detail elsewhere. [z9] In particular, it was shown that for continuous fatigue, the simulated size effect was described by a Weibull expression and the scatter was more im- portant for torsion than for push-pull loading, as shown in Figure 17 at low plastic strain amplitude�9

specimen axis

longitudinal section free surface

T = 2 5 a C TORSION(Preoxidation)

(a)

specimen axis

longitudinal section free surface

T = 6 0 0 ~ TENSION(Vacuum)

(b)

Fig. 14--Orientation of microcracks: (a) SQ steel. Preoxidized specimen tested at room temperature under fully reversed torsion (Ay,/2 = -+1.38 pet). (b) SP steel. Push-pull test at 600 ~ under vacuum (Ae,/2 = -+0.80 pet).

(c) The histograms of secondary crack lengths observed on longitudinal sections of specimens were in good agreement with those obtained from the simulation. (d) The deviations from the nonlinear damage rule with a cumulative damage lower than one were well repro- duced for both sequences (tension --~ torsion at 600 ~ and torsion ~ tension at 25 ~ as observed in Figure 18. This model therefore provides a quantitative explanation of the detrimental effect of sequential tests in which crack initiation and crack growth planes are parallel.


f f

f f

: )

'< >'

1"

,,(

T

>

Fig. 15--Shapes of the influence zones for modeling crack coalescence.

- _ _ _ - - - _ _ _ - _ - _ - _

( a ) - - _ - - - - - - - - - - - -

_ - - _ _ - - - - _

- - - - - - - - - - - - - - . . . . . - ~ 2 - ~ - _ - _ _ - _ ~ _ - _ ~ - ~ _ _ - - _ _

- - _ _ _ - - - ~ - _ _ - - _ _ _ _ - - - _

- ~ _ _ _ _ _ _ -

~ - _ - : - _ ~ - _ _ _ _ _ _ -_ - _ . . . .

_ _ _ ~ ~ - - ~ - ~ - _ _ - - - _ - ~ - _ : - = -

= - _ - : - _ - - : - - _ - _ _

_ - - _ ~ - - - _ = : ( _ - - _ - - - _ : _ _ ~

- - - - _ _ - _ _ - ~ = - -

- - - _ - - : 500 pm

( b ) - : ~ ~ -__ - - -7=_- - :~ - __ _ -- -- _ - _ _ ____ _ _

- _ - - _ _ - _ _ _ _ - - ~ - - _

- L

- - - _ : _ _ - - _ _ . . . . . - _ _ : - _ _ - = _

_ _ ~ _ _- - - - ~ - _ _ _ - - / _ _ _ - _ = _ _-_ - - . - = - _

_ -_ ~ - - - - - _ - _ - _ - - - . - - _ -

_ - _ _--_ - __ __ - _ _ - - _ l

- = - - : : _ - : 5001am

Fig. 16--Monte Carlo type simulation of a sequential push-pull (Ae,/2 = --+0.80 pct) ---> torsion (AT,/2 = +1.38 pct) test. The specimen axis is vertical: (a) cracks after 150 tension/compression cycles and (b) cracks after eight torsion cycles following 150 tensile cycles.

2.5"

1.5-

N " ~ 1 �84

0 . 5

Ex+pe r imen ts

M o d e l : s c a t t e r b a n d

~ . ~ = o . 9

§

P f = 0 . 1

1 .2-

1 0 0 0 10OOO 100OOO

N F

(a)

I ~ : p r o ~ o f f a l u r o

I I

1- E x p e r i m e n t s

0 . 8 - M o d e l : s c a t t e r b a n d

o4 ~ . ~ , / P [ =0 .9

ao .6 -

0 . 4 - F'I 0 = .

0 . 2 -

0 t , , , , = , , , , , , , , , 1 0 0 1 0 0 0 1 0 0 0 0

N F

(b)

Fig. 17--Comparison between experimental and simulated continuous fatigue lives: (a) torsion, 25 ~ (b) tension/compression, 600 ~ Lines corresponding to probabilities to failure (PI) equal to 0.10 and 0.90 are drawn.

This s tochast ic m ode l was appl ied on ly to t rans- g ranu la r fat igue. Fur ther work is necessa ry to deve lop a s imi la r approach to a ccoun t for the effect o f inter- g ranu la r d a m a g e on fat igue life.

V . C O N C L U S I O N S

1. For the same equ iva l en t von Mises s t ra in range , P F l ives u n d e r tors ional load ing are larger than those de- t e rmined u n d e r push-pu l l cond i t i ons at bo th room and e leva ted (600 ~ tempera tures .

2. The fa t igue l ives measu red at 600 ~ u n d e r con t in - uous cyc l ing do not s t rongly d e p e n d on s l ight chemical c o m p o s i t i o n s in 316L type s ta in less steels. O n the o ther hand , the CFs l ives m a y largely he affected even by smal l compos i t i ona l mod i f i ca t ions , such as the con ten t in ca rbon and boron . Th i s re la t ive sus- ceptibility to the effect o f hold t ime at m a x i m u m strain is par t ly re la ted to in te rg ranu la r carb ide precipi ta t ion .

3. N o n l i n e a r c u m u l a t i v e d a m a g e is obse rved for mul t i - axial sequent ia l tests. A de t r imenta l effect o f pr ior

M E T A L L U R G I C A L T R A N S A C T I O N S A V O L U M E 2 4 A , O C T O B E R 1 9 9 3 - - 2 2 5 9

1.2 �84

0.8-

D 2 0.6-

0.4-

0.2-

0 0 1.2

e!periment " ~ simulation

Torsion >Tension § ~ T=25"C

D ~ , t ,

0.2 0.4 0.6 0.8 1 D1

(a)

1.2

exoperiment

0.8] ~ simulation

D 2 0.6- ~ Tens~Tor tz ~ T=600*C

0.4- § ~ .

0.2- ~

0 , , e~ , , 0 0.2 0.4 0.6 0.8 1 1.2

D1 (b)

Fig. 1 8 - - C o m p a r i s o n between experimental and simulated lives for sequential fatigue tests (Ae,oq/2 = +0 .80 pct): (a) torsion --* tension sequence at 25 ~ and (b) tension --* torsion sequence at 600 ~

loading on the subsequent life is observed when initiation planes associated with the first phase of the sequence are parallel to the propagation planes corresponding to the second phase. This is observed at room temperature for the torsion ~ tension sequence and at elevated temperature for the tension ~ torsion sequence.

4. At 600 ~ oxide formation at the free surface of specimens is considerably modified by cyclic deformation. An external layer of hematite (Fe203) is formed instead of Cr203 oxide. An internal layer of ( F e C r ) 3 0 4

oxide is observed to grow preferentially along intense slip bands acting as diffusional short circuits. The interaction between cyclic deformation and oxidation is also revealed by the fact that the internal oxide layer growth kinetics does not follow a conventional parabolic law but rather a linear law.

5. Observations of the orientations of the microcracks initiated from the free surface of the specimens tested under continuous fatigue show that stage I crack initiation is bypassed by stress-enhanced oxidation. At elevated temperature, microcracks initiate into the brittle oxides under opening mode and then propagate into the metal. The modifications in the orientation of crack initiation planes produced at elevated temperature by oxidation effect explain qualitatively the results of sequential fatigue tests.

6. Sequential CF tests show that as a rule, prior intergranular damage introduced during the first part of the sequence has no effect on the residual fatigue life. Moreover, in a number of circumstances, oxidation does not necessarily have a detrimental role, as observed in sequential tests in which crack healing by oxide formation can take place during the second part of the sequence. This phenomenon accounts for the fact that in these conditions, the cumulative damage may be larger than one.

7. A 2-D stochastic model based on a Monte-Carlo type simulation is shown to reproduce with good accuracy the results of fatigue tests in which no bifurcation takes place between crack initiation and crack growth. In this model, crack initiation and crack growth laws determined from quantitative metallography are introduced, while crack coalescence is simulated with overlapping influence zones attached to the tips of each microcrack. Comparison between experimental results and simulations indicates that crack coalescence occurs much more easily under shear mode than under opening mode. Moreover, this micro- structurally based simulation accounts very well for the results of not only continuous fatigue tests but also sequential tests in which crack initiation and crack propagation are coplanar.

N O M E N C L A T U R E

A, C, k, constants p , OL, f l ,

0) t ~ r

X0 a

D

Ol Dc

L.

LF

Lr

N q

R~, R.

crack depth (/zm) damage parameter (D = N/NF: number of applied cycles for a given loading mode/ number of cycles to failure for the same loading) intergranular damage (D/ = LF/Lr) intergranular damage per cycle (D, = D1/N) (cycle -I ) number of intergranular cracks/total number of cracks (for creep-fatigue specimens) cumulative length of cracked grain boundaries per unit area of section (/zm -~) total length of grain boundaries per unit area (/ .r - l )

number of cycles aspect ratio of microcracks (q = small axis length/large axis length) size of the influence zone for coalescence in modes I and II, respectively (/zm)


S

Y E

~7 A

o"

o'R ,./-

re

X

surface area shear strain (pct) tensile strain (pct) linear density of microcracks (/xm -~) mean number of microcracks located on an area S(A = X" S) strain ratio (A' = Ayp/Aep) tensile stress (MPa) tensile strength to failure (MPa) shear stress (MPa) shear strength to failure (re = ~rR/k/3) (MPa) surface density of microcracks (/xm -2)

Subscripts

eq equivalent (von Mises) F failure CF creep fatigue PF pure fatigue p plastic t total

A C K N O W L E D G M E N T S

Financial support from the Service Central de la Sfiret6 des Installations Nucl6aires (SCSIN) is acknowledged.

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1. M.W. Brown and K.J. Miller: Proc. Inst. Mech. Eng., 1973, vol. 187, pp. 745-55.

2. M.W. Brown and K.J. Miller: Fatigue Eng. Mater. Struct., 1979, vol. 1, pp. 231-46.

3. V. Doquet and A. Pineau: in Fatigue under Biaxial and Multiaxial Loading, ESIS 10, K. Kussmaul, D. McDiarmid, and D. Socie, eds., Mechanical Engineering Publications, London, 1991, pp. 81-101.

4. M. Sakane, M. Ohnami, and M. Sawada: J. Eng. Mater. Technol., 1987, vol. 199, pp. 236-43.

5. K.J. Miller: Proc. Inst. Mech. Eng., 1991, vol. 205, pp. 1-13. 6. M. Robillard and G. Cailletaud: in Fatigue under Biaxial and

Multiaxial Loading, ES1S 10, K. Kussmaul, D. Mc Diarmid, and D. Socie, eds., Mechanical Engineering Publications, London, 1991, pp. 103-30.

7. S. Harada and T. Endo: in Fatigue under Biaxial and Multiaxial

Loading, ESIS 10, K. Kussmaul, D. Mc Diarmid, and D. Socie, eds., Mechanical Engineering Publications, London, 1991, pp. 161-78.

8. L.F. Coffin: in Fracture 77, vol. 1, 1CF4, Waterloo, ON, Canada, 1977, pp. 263-92.

9. J.H. Driver, C. Gorlier, C. Belrami, P. Violan, and C. Amzallag: in Low-Cycle-Fatigue, ASTM STP 942, H.D. Solomon, G.R. Halford, L.R. Kaisand, and B.N. Leis, eds., ASTM, Philadelphia, PA, 1988, pp. 438-55.

10. G.C. Wood: Oxid. Met., 1970, vol. 2, pp. 11-57. 11. C. Levaillant, J. Grattier, M. Monot, and A. Pineau: in Low-

Cycle-Fatigue, ASTM STP 942, HD. Solomon, G R . Halford, L.R. Kaisand, and B.N. Leis, eds., ASTM, Philadelphia, PA, 1988, pp. 414-37.

12. J. Weiss and A. Pineau: Proc. ASTM Symp. on Multiaxial Fatigue, San Diego, CA, Oct. 14-15 1991, in press.

13. T. Nakazawa, H. Abo, M. Tanino, and H. Ramatsu: Proc. 4th Int. Conf. on Creep and Fracture of Engineering Materials and Structures, Swansea, Apr. 1-6, 1990.

14. C. Levaillant and A. Pineau: in Low-Cycle-Fatigue and Life Prediction, ASTM STP 770, C. Amzallag, B.N. Leis, and P. Rabbe eds., 1982, pp. 169-93.

15. B. Jacquelin, F. Hourlier, and A. Pineau: in Multiaxial Fatigue, ASTM STP 853, K.J. Miller and M.W. Brown, eds., ASTM, Philadelphia, PA, 1985, pp. 285-313.

16. J. Weiss: Ph.D. Thesis, Ecole des Mines de Paris, Paris, Sept. 1992.

17. T. Nakazawa, N. Fujita, H. Kaguchi, H. Kaneko, and H. Ueda: in Low-Cycle-Fatigue and Elasto-Plastic Behaviour of Mate- rials-3, K.T. Rie, ed., Elsevier Applied Science, New York, NY, 1992, pp. 88-93.

18. B. Rezgui, P. Petrequin, and M. Mottot: in 1CF 5, D. Franqois, ed., Cannes, France, 1981, vol. 5, pp. 2393-2402.

19. B. Weiss and R. Stickler: MetaU. Trans., 1972, vol. 3, pp. 851-65. 20. R.D.K. Misra and T.V. Balasubramanian: Acta. Metall. Mater.,

1990, vol. 38, pp. 1263-66. 21. A.W. James and C.M. Shepherd: Mater. Sci. Technol., 1989,

vol. 5, pp. 333-45. 22. J. Bourgeot, G. Henry, B. Michaut, and B. Thomas: Mere. Sci.

Rev. Metall., 1977, pp. 491-512. 23. D.A. Mortimer and M.G. Nicholas: Met. Sci., 1976, vol. 10,

pp. 326-32. 24. R. Raj: J. Eng. Mater. Technol., 1976, vol. 98, pp. 132-39. 25. R.P. Skelton and J.I. Bucklow: Met. Sci., 1978, vol. 12, pp. 64-70. 26. M. SchiJtze: Mater. Sci. Technol., 1990, vol. 6, pp. 32-38. 27. L.M. Kachanov: lvz. Akad. Nauk. SSSR, 1958, No. 8, pp. 26-31. 28. L. Bertini and E. Vitale: Materials at High Temperatures, 1991,

vol. 9 (1), pp. 23-29. 29. J. Weiss and A. Pinean: in Low-Cycle-Fatigue and Elasto-Plastic

Behavior of Materials-3, K.T. Rie, ed., Elsevier Applied Science, New York, NY, 1992, pp. 82-87.

30. G.R. Irwin and P.C. Paris: in Fracture, An Advanced Treatise, H. Liebowitz, ed., Academic Press, New York, 1971, vol. 3, pp. 2-44.


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Fatigue and creep-fatigue damage of