High cycle fatigue life of metals

Materials Science and Engineering, 15 (1974) 239--245 © Elsevier Sequoia S.A., Lausanne - - P r i n t e d in The Netherlands

High Cycle Fatigue Life o f Metals

P. LUK.~S, M. KLESNIL and J. POL~,K

Institute of Physical Metallurgy, Czechoslovak Academy of Sciences, Brno (Czechoslovakia) (Received October 9, 1973)

SUMMARY

In eight different materials with U.T.S. ranging from 5 to 80 k p m m - 2 the fatigue life curves were determined both for cycling with constant stress amplitude and for cycling with constant plastic strain amplitude in the high cycle region (number of cycles to fracture > 105 ). The results made it possible (1) to check directly the validity of the MansonCoffin law in the high cycle region; (2) to show h o w far the different fatigue life curves can be mutually transformed via the cyclic stress-strain curve; (3) to determine the threshold values of the plastic strain amplitude corresponding to the endurance limit.

1. INTRODUCTION

It is generally accepted that the fatigue process is controlled by the plastic strain amplitude. Therefore, the plastic strain amplitude should also be one of the most important experimental parameters. In spite of that, fatigue life data in the high cycle, low amplitude region have been measured and expressed almost w i t h o u t exception on the basis of stress amplitude, i.e. in the form of S - N curves. This situation is obviously caused by difficulties of measurement and control of the small plastic strain amplitudes which correspond to this region. Data concerning the cyclic stress- strain curves, their relation to the fatigue life and the interrelation of the fatigue life curves for different regimes of cycling have also been unavailable. In the low cycle, high amplitude region many such data are available. Rare, relevant high cycle experiments, in

which the cycling was performed with constant stress amplitude and the plastic strain amplitude was measured [1 - 4 ] , suggested that the Manson - Coffin law is valid also for the high cycle region. This was checked in our earlier work [5] for two materials by plastic strain amplitude controlled experiments. It was shown as well that the cyclic stressstrain curve is meaningful also in the high cycle region. In none of these papers were different types of cycling used, so that no comparison of different regimes is possible. The aim of the present paper is to determine on a variety of materials the fatigue life characteristics and their relation to the plastic stress - strain characteristics for both constant stress amplitude cycling and plastic strain amplitude cycling in the high cycle region roughly corresponding to the number of cycles to fracture from 105 to 107 .

2. EXPERIMENTAL

The materials, specimens and testing techniques used here are identical with those reported in our paper immediately preceding this one [6]. Monotonic parameters are reported in Table 1 of that paper.

3. RESULTS AND DISCUSSION

The cyclic stress- strain curves of all the materials used were reported in the companion paper [6]. It was shown that they obey the power law relation Oa = k

e an'p '

(1)

where On is the stress amplitude and eap is the

240

I

4

~

At ~o

~ ~o" const

_

x

T

• %-const

•

34

,~

?2060

26

•

~

2

• cap"c°nst x ~ "const 22

i 10 s

10 "~

10 s

10 4

10 z

e--

J

h

i

10 5

10 e

10 7

Nt

Nt

Fig. 1. o a vs. N f fatigue life curve for AI; x - - directly measured values; o -- values transformed from plastic strain controlled cycling.

Fig. 2. o a vs. N~ fatigue life curve for carbon steel 12060. x - - directly measured values; o -- values transformed from plastic strain controlled cycling.

plastic strain amplitude. The parameters of this relation are presented in Table 2 of the above-mentioned paper [6].

shown in Figs. 3 and 4. Experimental points in the log- log plot can be again fitted to straight lines. It is valid for all materials with the exception of the austenitic steel 17246, where the approximation by a straight line (in log - log plot) does not represent the best fit; a slightly concave curve would do better. But even in this exceptional case the power law representation is still acceptable. The dependence of the plastic strain amplitude on the number of cycles to fracture is thus of the

3.1. Fatigue life curves for two regimes o f cycling Cycling with the constant stress amplitudes yields the curves Oa versus Nf (Nf = number of cycles to fracture). Two examples are shown in Figs. 1 and 2. In all cases this dependence in log- log plot can be well approximated by a straight line. It means that the o~ versus Nf dependence is of a power law type. This dependence can be written in the form i

o = o~ (2 N f ) b,

(2)

I

where of is the fatigue strength coefficient. Parameters of and b for all the materials are listed in Table 1. Two examples of results from the cycling with the constant plastic strain amplitudes are t

form e a p = e~ (2 N f ) c,

(3)

w h e r e e l ' is t h e f a t i g u e d u c t i l i t y c o e f f i c i e n t . P a r a m e t e r s e l ' a n d c a r e a l s o i n c l u d e d in T a b l e 1. T h e g e n e r a l v a l i d i t y o f e q n . ( 3 ) in t h e high cycle region means that the MansonC o f f i n l a w is f u l f i l l e d . The first question which can be answered o n t h e b a s i s o f t h e s e r e s u l t s is, h o w f a r c a n

TABLE 1 Parameters of the fatigue life curves Material

Parameters of the Oa--N f curve f

A1 Cu Cu - - Zn (12013 ¢~ ~=/ 1~ 2 0 1 0 ~|12060 [14140 Austenitic steel 17246

Parameters of the Cap--Nfcurve i

of (kpmm - 2 )

b

ef

c

10.0 31.4 41.0 32.9 46.9 76.5 38.0 24.0

-0.11 -0.075 -0.085 -0.05 --0.06 --0.07 -0.02 -0.01

0.43 2.18 181.5 3.09 3.09 3.09 0.008 0.13

--0.55 --0.66 --0.91 --0.75 --0.75 --0.75 --0.25 --0.43

241

Cu - Zn

I ld 4

,< O'o - const

Ids

I 104

,

~

L 106

10 5

Nt

107

Fig. 3. Gap vs, N f f a t i g u e life c u r v e f o r C u - Z n . o - directly measured values; x -- values transformed from stress controlled cycling.

the different fatigue life curves (o,, versus N f and eap versus N~) be mutually transformed with the help of the cyclic stress- strain curves? For cyclically fully stable materials, which do not exhibit any hardening and/or softening during the cycling, this transformation should be perfect (a trivial case). With real materials, the agreement between the transformed and the directly measured fatigue life curves can be expected to be good for materials with a short and limited (in the sense of the total change of the stress or strain amplitude) stage of hardening and/or softening. The longer and more marked the hardening and/or softening, the worse agreement between the transformed and the directly measured curves is to be expected. Examples can be seen in Figs. l - 4 . In Figs. 1 and 2 the points a~ = const, are the directly measured values of the dependence oa versus N f , the points e~p = const, were transformed from the dependence e~p versus N f using the respective cyclic stress - strain curve. Analogous-

lids.

,,,,...,..~

ly, in Figs. 3 and 4 the points e a p = const, are the directly measured values and the points oa = const, are values transformed via the cyclic stress - strain curve from the oa versus N~ curve. Figures 1 and 3 represent the worst cases, in which the transformed and the directly measured curves are markedly shifted. Such a clear shift can be stated only for Cu - Zn (very long hardening stage owing to a very difficult cross slip [6] and for A1 (a short, b u t very marked hardening stage). In all other cases the agreement is much better and usually both the directly measured and the transformed points can be covered b y one experimental scatter band (Figs. 2 and 4). Mathematically, the mutual transformation of the fatigue life curves via the cyclic stress- strain curve means that their parameters are n o t independent. In our case, where all these dependences can be expressed by power law relations, the combination of eqns. (1) to (3) leads to b = n' . c

k

(4)

o; =

_

(5)

_

Figures 5 and 6 show h o w far these relations are fulfilled. These diagrams also give an insight into h o w far the transformations are generally justified. The closer the agreement between the curves representing eqns. (4) and (5) and the experimental points, the more justified the transformation. It can be seen that this agreement is good in all cases. Therefore the relations given by eqns. (4) and (5) may be considered as good approximations in the high cycle region.

/

12060

=?~'C

0.10

.'x.

;;2060 0.05

~.,

Cop = cor~st

(Jo "const 104

i

i

L

105

fO6

107

Nt Fig. 4. Gap

-

vs. N f f a t i g u e life c u r v e f o r c a r b o n s t e e l 12060. o -- directly measured values; x -- values transformed from stress controlled cycling.

/4b'/4140

000

Y17246

, I

0.00

0.05

0.10

-r)'C

Fig. 5. R e l a t i o n b e t w e e n e x p o n e n t s o f c y c l i c s t r e s s - s t r a i n c u r v e n ' a n d f a t i g u e life c u r v e s C(eap vs. N f ) a n d b ( o a vs. N f ) .

242

Y

I

14140.f 0 6 0

E

'E60

100

12010

40

,12060 Cu Z n ~ u u 12013 20 "

/

0 0

50

\17246

I

I

I

I

20

40

60

80

=12010 Cu-Zn 12013• .14140 ICu

e17246

I

J

i

50

100

150

I

200 [kp.~ 21

Fig. 6. Relation between the parameter o f cyclic stress- strain curve k and fatigue fracture parameters P Of ~, Ef .

Fig. 8. F a t i g u e s t r e n g t h c o e f f i c i e n t a s' vs. t r u e m o n o t o n i c f r a c t u r e s t r e s s af.

3.2. Cyclic stress - strain response and fatigue life

for Cu and Cu - Zn that the principal assumption of Morrow, relating the total hysteresis losses with the fourth power of the stress amplitude, is not fulfilled in the high cycle region. The same argumentation may n o w be extended to other materials. The proposed empirical relations between the at' and e~', respectively, and the at and el, respectively [7 - 10], are always simple functions of the type o l = f(of) and et ' = f(e~). Figures 8 and 9 show the plots of experimentally determined values (see Table 1 ). Without going into details, it is quite clear that there is no systematic trend of the experimental points. The proposed relations can therefore be considered as invalid in the high cycle region ° From the fact that the relations between the monotonic and cyclic stress- strain para-

For the low cycle, high amplitude region there are semiempirical relations between both the cyclic and monotonic stress- strain parameters on one side and the fatigue life on the other side. These relations represent a reasonable approximation of the experimental data in the low cycle fatigue. Morrow [7] derived a relation between the exponent of the cyclic stress- strain curve n' and the exponent of the eap versus N~ curve c, which is of the form 1 1 + 5n' "

c=

(6)

Figure 7 shows quite clearly that this relation is not acceptable in the studied high cycle region. In our earlier work [5] we have shown

~ -z.

1~ = ~

~2"o6o ~ o ~13

3

1.0

Cu- Zn

r

_~2010o12060

o i

0.5

• 17246 •14140 0

0.11

c----1 1+5n' At 0

0.12 n I

0

L

Fig. 7. E x p o n e n t o f fatigue life curve c vs. of cyclic stress - strain curve n'.

exponent

Jl

i

I

1

2

3

Fig. 9. F a t i g u e d u c t i l i t y c o e f f i c i e n t e l ' vs. t r u e m o n o t o n i c f r a c t u r e d u c t i l i t y el.

243

Ct/-Zn

I 10.3

g

C auo ~•

10"*

t,

o

•

C o 12010

c..o.-

o 12060

,lff 5

I

10 "

10 5

j

I

10 s

I

10 ~

Fig. 10. eap v s . N f d e p e n d e n c e f o r Cu, Cu - Z n a n d c a r b o n steels 1 2 0 1 3 , 1 2 0 1 0 , 1 2 0 6 0 .

meters are generally reasonably fulfilled in the low cycle region and n o t fulfilled in the high cycle region, it follows that either the cyclic stress- strain curve or fatigue life curve (or both) are n o t given by one straight line in the log - log plot over the whole region of fatigue lives (say from 10 to 10 7 cycles). The only m e t h o d of clarifying this discrepancy is to measure all these curves over the whole (both low and high) region of amplitudes keeping all other conditions constant. 3.3. Endurance limit

Figure 10 shows once more some directly measured eap versus N f curves. For the sake of the clarity of the picture n o t all of the measured curves are drawn. It can be seen that the p o w e r law relation between the plas-

tic strain amplitude and the number of cycles to fracture is valid down to the Nf of a b o u t 10 7 cycles. Somewhere near this value the p o w e r law relation ceased to be valid. This picture suggests that there is a threshold value of the plastic strain amplitude, below which the fatigue fracture does n o t appear either at all (carbon steels exhibiting a sharp knee on the fatigue life curves) or n o t until after 10 v cycles (other materials exhibiting rather a smooth asymptotic approach to the threshold level). As it is possible to transform mutually the Oa versus N~ of the eap versus N~ curves, there is a possibility of also transforming the threshold levels. These threshold values of the plastic strain amplitude then correspond to the stress fatigue limit (carbon steels) or to the stress endurance limit (other materials). Table 2 summarizes the threshold values of the plastic strain amplitude from b o t h the directly measured and the transformed eap versus N~ curves. For the regime e a p = const. these threshold values lie within the limits 3 × 10 - 5 to 1.2X 10 - 4 , for the regime oa--const, within the limits 1 × 10 - 5 to 1,2 × 10 - 4 . With a certain approximation it can therefore be said that this threshold level is always of the order of 1 0 - 5. The existence of a threshold plastic strain amplitude is in agreement with some scarce results obtained from stress controlled cycling [1 - 4 ] . As for the absolute values (comparing n o w results for uniaxial cycling only), the

TABLE 2 T h r e s h o l d values o f t h e plastic s t r a i n a m p l i t u d e Material

Threshold v a l u e o f Cap Regime eap= const,

Regime o a = const.

A1

4--5X10

-5

1--2x

10 - 5

Cu

4--5x10

-5

1--2x

10 - 5

Cu--Zn

4--5X10

-5

1--2x

10 - 5

3--4X10 3--4x10

-5 -5

2--3x

10 - 5

3-4×

10-5

~

t 12013 12010

12060 ( 14140

--

3-4×10-5 1.1--1.2x

10 _ 4

1.1--1.2x

10 _ 4

Austenitic steel 1 7 2 4 6

1.0 - - 1.1 x 10 - 4

1.0 - - 1.1 x 10 - 4

TotaIrange

3 × 1 0 - - 5 - - 1 . 2 × 10- 4

1 × 1 0 - - 5 - - 1 . 2 × 10-.4

244 agreement is n o t always good. Kawamoto and Tanaka [1] and Troscenko [2] reported for some materials much lower (of the order of 10 - 6 and 10 - 7 ) threshold values. This might be due to the inhomogeneities of the cyclic plastic deformation, which are especially probable when a softenable material is cycled under constant stress amplitude. One case of such an inhomogeneity is discussed in detail in our foregoing paper [6]. The threshold plastic strain amplitudes lie in a relatively narrow interval for all the materials (Table 2). This offers the possibility of estimating the stress endurance (or fatigue) limit from the cyclic stress - strain curve just by reading the stress interval corresponding to the strain interval of 1 X 10 - 5 to 1.2 × 10 - 4 . Figure 11 shows the results. It can be seen that there is a reasonable agreement between the estimated and the directly measured values of the endurance limit o w . This procedure could be used as a quick method of the endurance limit estimation. Especially when using shortened ways of the cyclic stress- strain curve determination (such as low amplitude modification of the high ampli. rude procedures reported in ref. 11), this method could prove to be advantageous. In the literature there are enough experiments which are relevant to the interpretation of the threshold plastic strain amplitude. Lazan [12] measured the specific damping (which is proportional to the hysteresis loop area) in dependence on the stress amplitude for a number of metals. The measured curves (log damping versus log stress) exhibit a sharp knee at a stress amplitude slightly lower than the endurance limit. Lazan interpreted the results by the transition from the reversible processes for higher amplitudes. The same inter-

30 ~2C

~

"/40

~c~-zo 0

FAt

10

I

I

20

30

Directly measured

Fig. 11.

O'~fkpmm"2]

Estimated values o f endurance limit o w in d e p e n d e n c e o n directly measured endurance limit o w .

pretation is fully applicable to the results of Feltner and Morrow [ 1 3 ] , who measured the relevant stress- strain curve after cycling for sufficiently low strain values. This curve also exhibits a sharp knee (in log - log plot) corresponding to the plastic strain amplitude of the order of 1 0 - 5 . The transition from the reversible to the irreversible processes was detected also b y the measurement of the hysteresis loops in the microstrain region of cyclically deformed metals [ 1 4 ] . This experiment yielded the value of the anelasticity limit (first n o t completely closed hysteresis loop in a series of loops with increasing stress) corresponding to the plastic strain amplitude again of the order of 1 0 - 5 . The quoted and also further experiments in fact forecasted the existence of the threshold e~p values corresponding to the endurance limit. As for the interpretation of this threshold, all the results indicate that the fatigue is necessarily associated with dislocation multiplication -- with irreversible dislocation processes. The stress endurance limit and the corresponding eap threshold are then near (slightly higher) to the transition from the reversible to the irreversible processes.

4. CONCLUSIONS On eight different materials low amplitude testing was performed with both constant stress amplitude and constant plastic strain amplitude. The results can be summarized as follows: 1. The Manson - Coffin law is valid also in the high cycle region. 2. The fatigue life curves (aa versus N~ and ear versus Nf) can be mutually transformed via the cyclic stress - strain curve. 3. Morrow's relation between the e x p o n e n t o f the cyclic stress- strain n' curve and the exponent of the fatigue life curve c and also the relations between the monotonic fracture parameters (of, e~) and the cyclic fracture parameters (or', el') are n o t valid in the investigated high cycle region. 4. There is a threshold value o f the plastic strain amplitude corresponding to the endurance limit. As this threshold value for all eight materials lies in a relatively narrow interval 1 × 10 - 5 to 1.2 × 10 - 4 , it is possible to estimate the endurance limit from the cyclic

245 s t r e s s - strain curve o n l y . This t h r e s h o l d is c o n n e c t e d with t h e t r a n s i t i o n f r o m t h e reversible t o t h e irreversible processes o f t h e plastic deformation.

REFERENCES 1 M. Kawamoto and T. Tanaka, Mem. Fac. Eng., Kyoto Univ., 27 (1965) 65. 2 V.T. Troscenko, Fatigue and Inelasticity of Metals (in Russian), Naukova Dumka, Kiev, 1971. 3 A. Ferro and G. Montalenti, Phil. Mag., 24 (1971) 619. 4 L.P. Karjalainen, Metal Sci. J., 6 (1972) 195. 5 P. Luk~ and M. Klesnil, Mater. Sci. Eng., 11 (1973) 345. 6 J. Pol~k, M. Klesnil and P. Luke, to be published.

7 J. Morrow, Am. Soc. Testing Mater., Spec. Tech. Publ. 378, 1965, p. 45. 8 L.F. Coffin, Appl. Mater. Res., 1 (1962) 129, quoted from C.E. Feltner and L.W. Landgraf, J. Basic Eng., 93 (1971) 444. 9 S.S. Manson and G.R. Halford, Inst. Metals Monograph and Rept. Set., No 32, 1967, p. 154, quoted from C.E. Feltner and R.W. Landgraf, J. Basic Eng., 93 (1971) 444. 10 C.E. Feltner and R.W. Landgraf, J. Basic Eng., 93 (1971) 444. 11 R.W. Landgraf, J. Morrow and T. Endo, J. Mater., 4 (1969) 176. 12 B.J. Lazan, Damping and Resonance Fatigue Properties of Materials Considering Elevated Temperatures, McGraw-Hill, New York, 1961, p. 477. 13 C.E. Feltner and J.D. Morrow, Trans. ASME, Set. D, 83 (1961) 15. 14 P. Luk~ and M. Klesnil, Phys. Status Solidi, 21 (1967) 717.