The fatigue criterion with mean stress effect on failure

Materials Science and Engineering, A 111 (1989) 63-69

63

The Fatigue Criterion with Mean Stress Effect on Failure K. M. GOLOS Warsaw Technical University, Institute of Machine Design Fundamentals. ul. Narbutta 84, 02-524 Warsaw (Poland) (Received October 28, 1987: in revised form August 30. 1988)

Abstract

In this paper an energy-based fatigue criterion has been developed for tests with cyclic fully reversed and mean stress. The criterion can be used for both Masing- and non-Masing-type descriptions of the material. It is shown that the strain energy density is a consistent damage parameter for low and high cycle fatigue. A thorough investigation of the influence of tensile mean stress on fatigue life has been made. A number of relations describing the effect of mean stress on fatigue resistance are shown to be particular cases of the proposed model. 1. Introduction

Mechanical systems are often subjected to cyclically varying loads or deformations. In addition, many components have design details which involve severe stress concentrations. At these highly stressed locations the stress-strain will frequently be well above the elastic limit. Very often the history of cyclic loading has a mean stress or strain. Conditions of this type are common and can be the cause of premature failure in pieces of equipment intended for longlife applications. Costly failures of this type have occurred in the fields of power generation, aircraft propulsion, chemical plants and pressure vessels. Numerous investigations have been carried out to explain and describe the fatigue resistance of materials subjected to various loads and/or deformation patterns [1-15]. The proposed criteria used to predict fatigue life can be broadly divided into three groups: (1) stress-based criteria, (2) strain-based criteria and (3) energy-based criteria. A correlation between strain energy and fatigue life is desirable since it allows the inclusion of both stress and strain as bulk measures of fatigue life. The significance of the strain energy 0921-5093/89/$3.50

approach also lies in its ability to unify microscopic and macroscopic testing data and to formulate multiaxial life prediction methods [16,

171. The aim of this paper is to formulate a generally valid energy-based procedure for quantitative determination of fatigue life for tests with fully reversed and tensile mean stress. During cyclic loading, energy is dissipated because of plastic deformation. Part of this energy is converted into heat, while the remainder is rendered irrecoverable at every cycle in the form of plastic strain energy density. In this study it is assumed that damage due to cyclic loading is a function of the "total mechanical input energy". The total strain energy density, equal to the sum of the plastic strain energy density and the tensile elastic strain energy, is used as a damage parameter [14, 15]. It is shown that the total strain energy density is a consistent damage parameter in low and high cycle fatigue for fully reversed constant-stress or strain-controlled tests. This parameter can also be used to correlate fatigue strain-controlled tests with prestrain or mean strain. In the case of cyclic load control tests with mean stress, the effect of mean stress is of primary importance [1-3, 18-24]. Tensile mean stress in the case of stress-controlled cycling shortens fatigue life, while compressive mean stress prolongs it. In this study a relationship between total strain energy density, mean stress and number of cycles to failure is proposed for both low and high cycle fatigue. The resulting procedure is used to compare fatigue life predictions with experimental results for three steels. Good agreement is noted between the predicted and experimental results. It is also shown that a number of relations describing the mean stress effect in high cycle fatigue can be obtained as particular cases of the model developed here. © Elsevier Sequoia/Printed in The Netherlands

64

2. Energy-based fatigue criterion

ae =g(N,)

It is generally accepted that fatigue is a result of cyclic deformations. Cyclic plasticity gives rise to persistent slip bands which eventually create microcracks and regions with local stress concentrations. For the initiation of microcracks, a great number of models have been proposed [1-3, 24], discussion of which is beyond the scope of this paper. Owing to the increased stress concentration, the microcracks will link up until a crack the size of several grain diameters is formed. During the crack initiation stage, plastic deformation will occur throughout the small volume of material. Plastic deformation on a microscale consists mostly of generation of dislocations. These dislocations introduce energy in the lattice, so that the change in energy during cyclic deformation is likely to be a reliable measure of the damage. It is worth mentioning that local plastic deformation can occur even when the macroscopic (bulk) response of the material is quasi-elastic. With increasing length, the growing cracks spread from their original location near the 45 ° oriented slip planes and tend to propagate perpendicular to the stress axis. The explanation of this translation phenomenon probably lies in the change from plane strain to plane stress conditions at the crack tip. In the propagation stage, only one crack is usually propagating, all the others stopping well within the initiation stage. In the course of further cyclic loading the damage due to fatigue increases, and the crack size is a physical measure of that damage. The rate of crack propagation increases until fracture occurs, when the remaining uncracked material is unable to sustain the cyclic loading. A number of authors have advanced several combinations of variables, known as damage parameters, with the goal of describing the fatigue process. Since the damage manifests itself as a consequence of plastic deformation, any macroscopic analysis of fatigue should be based on cyclic plasticity. From the point of view of continuity in the analysis, it would also be more convenient if the same damage parameter could be used to describe the crack initiation and propagation stages. Often in practice, data anlayses of experiments conducted to obtain fatigue properties of materials have been oriented towards correlating the strain or stress range with the number of cycles to failure, i.e.

or

(1)

Ao = h ( N f )

(2)

The results of strain-controlled fatigue tests on a material may be written as AE t

2

A6 ~ -

2

Ae p I-

Of' -

2

E

N f b + ef'Nf ~

(3)

and the best-fit line for the uncontrolled stress is given by Ao E A e e 2 2

,

b

= a, Nf

(4)

where o f ' l E and t3f' are the strain amplitudes corresponding to the elastic and plastic intercepts for one cycle. In this type of approach the interrelation between stress and strain (path dependence) is generally overlooked. However, since stress range and strain range are uniquely related to each other and to the number of cycles to failure (at least approximately), there should exist a function of these which will also correlate with the fatigue life. In the present work an energy-based criterion is developed. The key element of this approach is that the damage caused as a result of cyclic loading can be related to the mechanical energy input to the material, i.e. [14-15]

AWt=f(Nf)

(5)

where A W t is the total strain energy density per loading cycle and Nf is the number of cycles to failure. The total strain energy density A W t is defined as follows (Fig. 1 ): A W t = A W ~+ +AWP

(6)

where A Wc+ is the tensile elastic strain energy density and A WP is the plastic strain energy density. The major part of this energy input is dissipated as heat and the remaining energy causes dislocation movements along slip lines, volumetric changes and eventual crack formation. The plastic strain energy density A WP is the area of the hysteresis loop. A specific expression for the loop area can be obtained by the choice of the stress-strain relation along the hysteresis branches. To describe the branches of the hysteresis loop for a Masing material, we can use a

65 AeP /

/

/

,'

/

F

Master Curve

I

L../FLxW e+ ~-~xWP

Ac 6 I

m,-

ee/2

/

// AW t= AWP+ AWe+

o

Strain Range

Ae

r~

Fig. 2. Hysteresis loop plotted with matched upper branches indicating the master curve.

Fig. 1. Description of stable hysteresis loop.

cyclic stress-strain curve magnified by a factor of two, i.e.

[Aol'' Ae:

E +2/]2-K'

(7)

where the origin for determining A o and Ae is at the compressive tip of the hysteresis loop. Usually the hysteresis loop at half-life is taken to be representative and is used to calculate the plastic strain energy density. This energy is given by [8] A WP-

1 -n' l+n

, AoAeP

(8)

For a non-Masing material model description, the prediction of eqn. (8) may not be accurate. Jhansale and Topper [11] proposed an approach for describing the loop shapes of a nonMasing material. This approach was developed in refs. 12 and 1 3 and is adopted herein. In general, a master curve different from the cyclic stress-strain curve is defined. This curve is obtained from matched upper branches of the hysteresis loops by translating each loop along its linear elastic proportional range (Fig. 2). The expression for the master curve with the origin at the smallest hysteresis loop, point O* in Fig. 2, is

Ae* Ao*

[Ao*l

where n* and K* are parameters chosen to fit the experimental results. Note that for a non-Masing material a minimum of two tests are required to specify the master curve eqn. (9), one with the smallest, the other with the largest hysteresis loop. The cyclic plastic strain energy density is then calculated from 1 BH*

A W p - 1 + n* ( A a -

6 a o ) A e P + 6 o , AeP

lO)

where

\z/

(11

is the increase in the proportional stress, the measure of hardening. It is clear that for an ideal Masing material, the master curve and cyclic curve will coincide, i.e. n* = n', K* = K' and 6o0 = 0. In this case eqn. (10) reduces to eqn. (8). The tensile elastic strain energy density (Fig. 1 ) is given by AoA~ + Aa: A W

-

-

8

- -

8E

(12)

66

Therefore the total strain energy density A W t is obtained by combining eqns. (10) and (12) to give 1 -n*

A Wt = -

(Ao-

l+n*

60"O)AeP Ao 2

(13)

+ 60"0AeP -t- - -

8E

It is noted that for a given state of strain range or stress range, the total strain energy density can be calculated if the cyclic stress-strain curve and/ or the master curve are available. To express the fatigue criterion mathematically, the experimental results in particular suggest a power-law-type relation of the form [14, 15] A Wt= kNfa+ C

(14)

The constant C in eqn. (14)is the tensile elastic energy input which does not cause any damage. It is related to the material fatigue strain energy density by 2

(15)

C~-~ A W f = O'f

2E The relation between A W t and the number of cycles, Nf, for A-516 Gr. 70 low carbon steel is shown in Fig. 3 on a log-log scale. The chemical composition, mechanical and cyclic properties of this steel are given elsewhere [12-15]. The tests have been conducted on solid specimens subjected to strain- and stress-controlled loading. The best-fit line, eqn. (14), is also plotted in Fig. 3. It is seen that the scatter of results is small. The unifying nature of the proposed damage parameter, eqn. (14), for low and high cycle fatigue

regions is thus evident. Additionally, the total strain amplitude Aet/2 vs. fatigue life correlation of the same data as in Fig. 3 is shown in Fig. 4. The use of the strain energy density as a damage parameter is also consistent with a nonlinear fracture mechanics approach to fatigue crack propagation through the AJ integral.

3. Mean stress effect

In the previous section the fatigue criterion for fully reversed stress- or strain-controlled loading has been presented. Here we will introduce the effect of mean stress on the fatigue life of materials. The influence of mean stress on fatigue failure is different for compressive mean stress values than for tensile mean stress values. In the tensile mean stress region failure is very sensitive to the magnitude of the mean stress. In this region cycledependent creep occurs and has an important effect on fatigue damage, mainly at short lifetimes. The influence of mean stress in the compressive region is greater for shorter than for longer lifetimes. The basic causes of the effect of mean stress have been the subject of many studies [1-3, 24]. Some of the possibilities are that mean stress affects the stable stress-strain behaviour, the rate of crack initiation, the size of shear crack necessary to start a normal mode crack, the crack propagation rate, etc. On the other hand, a great number of empirical data have made it possible to formulate generally valid models or the quantitative determination of fatigue life [18-23]. An energy-based failure criterion, equation (5), for cyclic loading with mean stress can be expressed in the general form

F(A Wt, om)=f(Nf) i ii I

i

i ii I

i

i ii I

i

i ii I

r

r I11

(16)

i

107i 10-1

!

I011

~

<1 I00

£3 ID G LU

10 "1

o m e a n strain = 0 D m e a n strain > 0 Load control

- ~}'~x

i0-~{~2 i

I Ill J 103

l IJl i 104

r Ill J 10 5

x x~ o~"~- x~P--..~

E <:

Material:

~

t860 N - ' m . 2o--4--S00 f

-2--

i Ill ¢ IO t5

j ,,I 10 7

t

108

vs.

number of cycles to

'

22N -'s°° f

xo Material: A S T M A 516 G r 70

c~

,,

0

10-3

c 0~

A S T M A 516 G r 70

Cycles to Failure, Nf Fig. 3. Total strain energy density failure.

&

10-2

o ~

x m e a n stress = O, p r e s t r a i n = 0 • m e a n stress = O, p r e s t r a i n < 0 • m e a n stress = O, prestrain > 0

.E L

"D

10-4 10 2

I

I Ill

103

i

i i II

l

i i ii

104

105

l

i llI

106

i

i llI

107

i

I i

108

Cycles to Failure, Nf Fig. 4. Total strain amplitude for the same data as in Fig. 3.

vs.

number of cycles to failure

67

where A W t is the total strain energy density for the given A o., which is the same as for the fully reversed tests, and o.m is the mean stress. As a particular form of eqn. (16), the following relation is proposed: A Wty(o.m)= kNf a + C

(17)

where

Y(o'm)=

{ ( o'm/+1'' ÷'+¸:'/'+

l-kilo..'J]

1 + n'(A W~'/4A W~+ ) 1 + A WP/4A W~+

(19)

where A WP and k W ~+ are the plastic and elastic components of the total strain energy density for the fully reversed tests. It is seen that for low cycle fatigue AeP > A e e, k WP >>k W ~÷ and h = n'. For high cycle fatigue AeP<~Ae e, AWP<~AW e+ and r~= 1, and if we neglect the plastic strain energy density, eqn. (17) can be expressed as follows:

{1-/ilo.,,} /

o.~,

o.~

Additionally, if x = 2 and /3= Gerber parabola:

(20)

where o',,m is the stress amplitude at half-life for the test with mean stress and o-a is the fully reversed stress amplitude. A number of attempts have been made to develop empirical methods for determining the effect of mean stress on the allowable alternating stress at a particular endurance. Based on eqn. (20) for x = 1, by varying the parameter/3 we can obtain a number of previously proposed relationships. For example, if/3 = o.Jor', where o~ is the ultimate tensile strength, we obtain the modified G o o d m a n relationship:

Oa

1

(21)

o'u

For il~-o.y/o.f', where o.y is the tensile yield strength, the Soderberg rule is recovered: o',~m+ o'm (7 a

o.y

1

we get the

Equations (17) and (20) are valid when the following condition is satisfied: o....... = O,m + o'm'-
(22)

(25)

However, for notched structures, when the surrounding material will inhibit unrestricted plastic flow at the notched tip, eqn. (17) or (20) can be used even if condition (25) is slightly violated.

4. Comparison with experimental data To be able to make a comparison, we require data on the cyclic stress-strain curve or (and) master curve, amplitudes of stress and strain, and numbers of cycles to failure for each set of tests. In this study we use experimental data for SAE 1045 steel [22], A-201 steel and A-517 steel [23]. A simple and convenient method of presenting the influence of mean stress on failure is the alternating stress-mean stress (A-M) diagram. For simplicity in the comparison we use eqn. (17) with /3 = 1 and x = 1. The comparison is made for lifetimes from 10-' to 10 v cycles. The A - M diagrams for the materials examined are presented in Figs. 5-7. It is seen that the correlation of eqn. (17)

600 [ ~"

500

L

\

, F - Omax=Ou

Experiments[22]

\

~ 400~-

3oo[ __ o'm o'am " { - - - =

o.Jo.r',

(18)

Here o-r' is the fatigue strength coefficient and/3, x and r~ are the material parameters. The value of fl is equal to unity or less and r~

When i l = 1, eqn. (20) reduces the well-known Morrow relationship:

10000

\ .~

~

\

"-%.>...\

~

o ,ooooo []

\

loooooo

• ,0000oo0 +roseo,

~ 2oo

0

I -200

I 0

I 200

I 400

I 600

800

1000

1200

Mean Stress, o m (MPa) Fig. 5. Effect of mean stress for S A E 1045 steel [22],

68 600

TABLE 1 Experiments [23] Nf o 100 ~ 1 0O0

O_ 500 ~

~amax=au

~,.\

o loo00

~ 30o

r- 200 etedal:

A-~I

500

6o0

E

Ratios of o,~/o,m t for materials examined

Model

A-517

A-201

SAE 1045

(a) (b) (c) (d) (e)

1.22/1.24 0.80/-1.12/1.12 1.07/1.10 1.04/1.10

1.28/-0.80/-1.10/-1.10/-1.08/--

1.25/0.80 0.70/-0.96/0.88 0.97/0.87 0.94/0.92

Goodman Gerber Morrow Smith-Watson-Topper present model

The first number of each pair is associated with the tensile mean stress.

• 10O

<

0

10o

2o0

300

4o0

700

mean stress since it includes a squared term. Models (c), (d) and (e) show approximately the same accuracy for the materials examined. For these models the deviations are within about 10%. In the analysis we used the simplest form of eqn. (17), i.e. with x = 1 and fl= 1. If we calculate the individual values of x and fl for each material, the proposed model yields the best description of the mean stress effect.

M e a n Stress, o m (MPa)

Fig. 6. Effect of mean stress for A-201 steel [23].

900

D

~" 80O i1.

Experiments [23] --o ax=O m u

~ f . /

~ z0o

E

~

~-, ~

o

10000

.

1o000o

~

,_~ " % . ~

~ 5o0

Nf 1:3100

,,<. ~ \

~

1,ooo

e

sert Method

400

5. Conclusions

.E 300 t..~CD

200

<

100

Material: A-517

~ , ~ I

-2O0

0

200

400

600

800

1000

Mean Stress, o m (MPa)

Fig. 7. Effect of mean stress for A-517 steel [23].

with the experimental data is fairly good over the entire range of fatigue lives. An alternative comparison of the present method with experimental data as well as with other models can be made on the basis of the ratios of absolute experimental values of stress amplitude with mean stress (o~me) to values estimated by use of the basic damage parameter-life curve (Oam').In the analysis the following methods are included: (a) Goodman relationship [2]; (b) Gerber parabola [2]; (c) Morrow relationship [18]; (d) Smith-Watson-Topper parameter [20]; (e) present model. Mean values of the ratios for each of these models were determined by regression analysis and are given in Table 1. The deviations from unity give a measure of the capability of the models to predict the mean stress effect. Models (a) and (b) show relatively large deviations of accuracy. It is worth mentioning that the Gerber relationship cannot be applied to the compressive

In this study an energy-based fatigue failure criterion for fully reversed constant strain (or stress) has been presented. The criterion can be applied for both Masing and non-Masing types of material behaviour. It is demonstrated that the total strain energy density is a consistent damage parameter for low and high cycle fatigue. Subsequently, the criterion presented was generalized to cyclic load control tests with mean stress. A thorough investigation of the influence of tensile mean stress on long-life fatigue strength has been made. It is shown that a number of earlier models can be derived as particular cases of the proposed criterion. A comparison between experimental data and the predictions of the proposed criterion has been made for three different materials, and the results are found to be in good agreement.

References l E. Krempl, The influence of state of stress on low-cycle fatigue of structural materials, in A S T M Spec. Tech. Publ. 549, 1974, pp. 129-179. 2 H. O. Fuchs and R. I. Stephens, Metal Fatigue, WileyInterscience, New York, 1980. 3 S. Kocanda, Zmeczeniowe niszczenie metali, WNT, Warsaw, 1985. 4 S. S. Manson, Behaviour of materials under conditions of thermal stress, NA CA TN-2933, Cleveland, OH, 1954.

6Y

5 L. F. Coffin, A study of the effects of cyclic thermal stresses in a ductile metal, ASME Truns., Ih (1954) Y31-Y50. 6 J. D. Morrow, Cyclic plastic strain energy and fatigue of metals, in Inten~al Friction, Dumping and Clvclic, Plasticity; ASTMSpec. Tech. Pub/. 378, 1965, pp. 45-84. 7 L. F. Coffin, A note on low cycle fatigue law<. J. Muter., fi I I Y7 I j 388-402. 8 G. K. Halford, The energy required for fatigue. J. .9lrrrer., /(lY66)3-IX. Y B. N. Leis. An energy-based fatigue and creep-fatigur damage parameter, J. I’re.w. Vehsrl Twhnol.. Trum. Am. 50~. ,Mrc,h. Erg.,

YY (I 077) 524-533.

for material under pro10 V. Kliman, Fatigue life prediction grammable loading using the cyclic stress-strain properties, Mafer. Sci. Erg.:., hx j lY84j I-IO. II H. R. Jhansale and T. H. Topper, Engineering analysis of the inelastic stress response of a structural metal under variable cyclic strains, in ASTM .Spec,. Twh. Pub/. 5/Y. I Y73, pp. 246-270. I2 D. Lefchvre and F. Ellyin, Cyclic response and inelastic strain energy in low cycle fatigue, Int. J. Frrtigue, 6 (I 984) Y-15. 13 F. Ellyin and D. Kujawski, Plastic strain energy in fatigue failure. J. Press Ve.ssel Technol., Tram. Am. Sot. Mech. Eng., Iffi ( I YX4) 342-347. 14 K. Golos, Cumulative damage fatigue in St5 medium carbon steel. In J. Nemec and R. Scrucany teds.). I’roc. /.sf C‘brr/: O,I ,\lrchanics. I’rrrSLcc,-Hmti.\lrr~~,, IW7, Vol. IV. 4/111. pp. I I I- I 14. IS K. Golos and F. Ellyin, A total strain energy density theory for fatigue and cumulative damage, J. I’res.x Vesel Techno/., Trams. Am. Sot. Mech. Erg., in the press, to the evaluation of fatigue I6 Y. S. Garud, A new approach under multiaxial loading. in W. J. Q+tergen and J. R. Whitehead (ed,.), Methods of Predicting Mciterial LiJi, in F&igue,

I7

F. Ellyin Lng

press. I8 J. D.

20

21

22

23

hlcuw.

Tec~hnol.. 7nrns. Am.

Morrow.

.Soc,. .Ilrch.

Ey.,

J.

in the

Fatigue properties of metals, Furigur Society of Automotive Engineers, Warrendale. PA, 1968. S. S. Manson, Thermal stresses in design, Part 21 Effect of mean stress and strain on cyclic lift. Muc~hine De.sign, .iZ ( I Y6Y/ 12Y- 135. K. N. Smith. P. Watson and T. H. Topper, A stress-strain function for the fatigue of metals, J. ,2later., 5 (1970) 767-77X. R. W. Landgraf, Cumulative fatigue damage under complex strain histories, in ASTM Spec. Tech. PubI. 5/Y. lY73,pp. 213-228. M. T. Yu, T. H. Topper. D. L. DuQusesny and M. S. Levin, The effect of compressive peak stress on fatigue hehaviour. Inr. J. Frrtigue. 8 (1 YS5) Y-l 5. J. Dubuc. J. R. Vanasse, A. Biron and A. Bazergui, Effect of mean stress in low-cycle fatigue of A-S I7 and A-20 1 Design

19

ASME. New York. I Y7Y, pp. 247-258. and K. Gobs, Multiaxial fatigue criterion,

Handbook,

steels, J. Eq. lnd., Tram. Am. (1970) 35-52. 24 M. Klesnil and P. Lukas, Fatigue Elsevier.

Amsterdam,

.Yoc. ,llec,h. of Metallic

Erg..

Y.?

:\4uteriub.

1980.

Appendix A: Nomenclature

c E k K’ K” 4 n’ n*

AW’+,AW”

AW’ AW’

elastic energy which does not cause damage, eqn. ( 14) Young’s modulus energy coefficient in eqn. [ 14) strength coefficient in eqn. (7) strength coefficient in eqn. (9) number of cycles to failure cyclic strain-hardening exponent of idealized Masing material cyclic strain-hardening exponent of master curve, eqn. (9) tensile elastic and plastic components of the cyclic strain energy density total strain energy density elastic strain energy density at fatigue limit

energy exponent in eqn. ( 14) material parameter maximum strain in the cycle minimum strain in the cycle cyclic strain range, E,,, - E,,, elastic and plastic components of the strain range fully reversed stress (a,,, = 0) alternating stress amplitude fully reversed fatigue stress at fatigue limit fatigue strength coefficient mean stress, ( omax+ 0,,,)/2 maximum stress in the cycle minimum stress in the cycle ultimate tensile strength tensile yield strength increase in the proportional stress limit, Fig. 2 cyclic stress range, a,,, - urn,*, cyclic stress range with origin at O*, Fig. 2