The Significance of the Fatigue Threshold for Metallic Materials A

The significance of the fatigue threshold for

metallic materials

A. Hadrboletz, B. Weiss, R. Stickler

Introduction and Basic Considerations

The fatigue threshold may be defined as the critical limit below which cracks

cease to propagate under cyclic loading. As the fatigue threshold represents an

important parameter in design and failure analysis much research effort has been devoted during the last decades to reveal underlying principles.

The first case of a fatigue threshold referred to in the literature is the

fatigue limit or endurance limit deduced from conventional S-N curves. This fatigue limit is currently still used as an essential parameter in design activities

based on the so-called "total life approach". The high-cycle fatigue limit of polycrystalline metals and alloys has been defined as the threshold stress for

propagation of small cracks which may have nucleated but did not grow further during subsequent cyclic loading at this particular stress amplitude.

In many cases of engineering practice the design against fatigue failure is

based on the concept of admissible defects ("defect tolerant approach"). This concept makes use of the threshold stress for the growth of long fatigue cracks as

the essential parameter. Thus the question arises which criteria govern the threshold condition for the growth of such fatigue cracks.

Several criteria have been proposed to describe the onset of the fatigue damage process which eventually results in catastrophic fatigue failure. These criteria

206 Localized Damage include stress or strain concepts, stress intensity concepts based on linear elastic fracture mechanics, J-integral approach based on elastic-plastic fracture mechanics and crack growth effects related to crack opening displacement. The most widely applied concept to describe the long-crack fatigue threshold is based on linear elastic fracture mechanics (LEFM). Under the assumption of a homogenous isotropic elastic material a stress intensity factor,

K, was defined which describes the elastic stress singularity associated with the crack tip in a stress field. This stress intensity factor is given by equation 1:

K = a \7ia • F with a = external applied stress (1)

a = crack length refined by a geometric correction factor F appropriate for various specimen geometries and different loading modes. The K-concept initially developed for unstable crack propagation under tensile loading was applied for the case of cyclic deformation by Paris [1],

AK = Aa • Vrca • F (2) with AK = Kmax " ^min and the cyclic stress amplitude range ACJ =

With respect to the fatigue threshold considerations under low-amplitude loading

(i.e. high-cycle fatigue conditions) a critical AK value is defined below which practically no crack growth should occur. This value was termed threshold stress intensity range, AK^. The near threshold crack propagation rate is very sensitive to minor changes in AK resulting in large variations of da/dN. Crack growth in this region plays a predominant role since in reality a fatigue crack may spend an overwhelming portion of its live in this regime of low da/dN. The threshold stress intensity range, AK^, was initially assumed to be a material parameter. However, numerous experimental results showed that this AKfli is dependent on several intrinsic and extrinsic parameters. It could be shown that the number of affecting parameters can be reduced by taking into account various mechanisms thought to be responsible for the prevention of a fatigue crack to close under reduced loads (2). Introducing the concept of crack closure (3) the threshold stress intensity was replaced by an effective (or intrinsic) fatigue threshold stress intensity, equation 3:

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AKth,eff = Kthmax - KC! ^h,max - niax.stress intensity (3)

K£\ = closure stress intensity which is considered to be a true material constant.

Extended measurements revealed that the AK-concept can be applied

successfully to long fatigue cracks (longer than several millimeters).

Parameters affecting the fatigue threshold Several extrinsic and intrinsic parameters may affect the AK^h values. The effects

of some of these parameters were interpreted to be related to crack closure as the

most common criteria, Table 1.

Essential features of crack closure

Based on LEFM considerations it is assumed that a fatigue crack is completely open during the tensile portion of the loading cycle. Under these conditions the

crack tip experiences the full stress intensity range computed from the externally applied stress range. In reality a different behavior was observed (3) as shown in

the dependence of the crack opening displacement (COD) on external load, G,

Fig.l. For G < Gel (closure stress) only a minor COD is observed. For a > Gel the crack exhibits an opening behavior as expected from elastic

considerations. This means that only above Gel the crack tip experiences the full

applied load. This crack closing (synonymously also called crack opening) behavior was explained in part as a result of plasticity effects in the wake of the

crack tip. During the growth of the fatigue crack the material at the crack tip is plastically strained. Due to the restraint of the surrounding elastic material on this

residual stretch, a retardation of crack growth due to closing of the crack faces is

observed. Since a fatigue crack cannot propagate while it remains closed, the actual AK value experienced at the crack tip is reduced by this closure value to an

effective value, AKeff. That a fatigue crack may be closed over a part of its loading range was pointed out earlier (4), Fig.2.

Various closure mechanisms are listed in the literature (2, 5), the most important being: plasticity induced, fracture surface roughness induced,

geometrical (crack deflection and bifurcation), oxide induced and viscous liquid induced. These mechanisms have been more or less successfully invoked to explain the dependence of AKth on different parameters, Table 1. For details see

Ref.(2).

208 Localized Damage

geometrica l induce d closur e gradua l buil d u p o f closur e durin g transitio n fro m ther e i s n o functio f it volum fractio shor t crack s o lon g leadin g t o pronounce d roughnes s induce closur e roughnes s induce d closur e an a meanderin g fractur e surfac roughnes s varie wit h th crack - orientatio n (roughness-induce d closure) , larg e change s i n crac k pat h orientatio caus crac k pat h (geometrica l closure) . AKth.ef f depend s onl y o n th e mea particl size ; plasti c deformation ) formatio n o f oxide s th e crac k surface frettin g corrosio n (loos e particles) , viscou s induce d closur e coarse r structure s exhibi t roughe fractur e surface fractur e surfac asperitie s | influenc e eliminate d b y closur (inhomogeneou s increase d closur e du t o th fonnatio n f predominantl y roughness - an d oxid e induce closur e lac k o f oxid e induce d closure , absenc roughnes s induce d closur e whic h i enhance b y n o R-dependen c i f Kmin>Kc l plasticit y induce d closur e (crac k ha s t o transmi a n 1 reduce d crac k closur e du t o flattenin g f th 1 corrosio n product s (mainl y oxides) , roughnes s induce d closure : smoot h fractur e surfac a t highe r R-vaiue s enlarge d plasti c zone ) | roug h fractur e surfac a t R = 0

•9

lon g crack s Microstructurall y an d physicall shor t crack s hav e smalle r highe r AKth-valuc s compare d t o single-phas e material AKt h value s an d large r near-threshol propagatio n rate tha o f th e reinforcin g har d particle s crystallographi c orientatio n AKt h increase s wit increasin g grai n siz e AKt h varie s bot wit volum e fractio n an d th averag siz AKt h depend s o n th e angl betwee crac k plan an d highe r AKt h i n bendin g compare d t o tension-compressio mino r increas e o f AKt h wit increasin g frequenc y increase d AKt h i n oi l compare t o inert , gaseou s acceleratio n o f nea r AKt h fatigu e crac k growt AKt h increase s wit risin g temperature environmen t independen t o f R fo r fin e graine d materia l reductio n o f nea r threshol d fatigu e crac k growt h AKt h i s t o b e almos independen f R n a iner environmen t compare d o ai r Max . value s o f AKt h a t R= 0 decreasin g AKth-value s wit h increasin positiv e an d increasin g negativ e R-value s significan t influenc e fo r coars graine d material ; AKt h almos

•g 3 JZ CO c. u "3 shor t crack s textur e grai n siz e test-temperatur e environmen t Extrinsi c 1 metal-matri x | composite s | test-frequenc y 1 compressiv e I tensil e overloa d 1 overloa d

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a, K

COD TIME

Figure 1. Stress (or K) -crack opening Figure 2. Definition of fatigue (COD)

curve; schematic threshold parameters (4)

Critical evaluation of crack closure In general, the AKth,eff value called also the "intrinsic threshold value" should

more correctly describe the crack growth behavior under cyclic loading. An

accurate determination of the amount of crack closure is therefore essential to calculate AKth,eff according to equation (3).

Various procedures have been proposed to determine the closure

contribution (2). Idealized crack opening curves have the form as shown schematically in Fig. 3a. This ideal curve consists of two straight-lines. The branch between K^^x and K^ reflects the closing of the crack with diminishing external load.

At KCI the ideal crack is completely closed and further unloading corresponds to the elastic behavior of a crack-free specimen. The compliance of a specimen with an open crack is larger than that of a specimen containing a closed crack, the slope of the branch below K^ is steeper. In this idealized case, for Kmin < %d ê AKêff corresponds to the difference between Kâx and K^, while AKtfi is usually defined as K^^ - ^min-

In a real material the change in closure behavior is not as abrupt as drawn in Fig.3a but rather gradual as shown schematically in Fig.3b. By reducing the external load from K^ax && crack opening is reduced proportionally along the straight line down to K^, the value at which the crack faces first come into contact at asperities. On further unloading an increasing number of asperities come into contact until at K^osed the crack faces are in tight contact and the COD

210 Localized Damage

T AKth.eff [5]J

n ""AKth.eff [6]

Kcl ^shielding Kmin i COD ^ 'CODmin COD Kclosed

a. idealized material b. realistic material

Figure 3: Load (or K)-COD curves, schematic

is practically zero. On further unloading the compliance of the specimen is again nearly that of a defect free material.

For a real material the values of K^j and K^g^ ^ay differ significantly. For the evaluation of the closure contribution the magnitudes of both

Kmin are essential. In Fig.Sb K^in was selected so that K^n > the crack never fully closes over the whole applied K- range. Although the recording of the K-COD curve is relatively simple, the interpretation of the significance of the various points along the curve is at present controversial. Conventionally K^ is taken as the point of deviation from linearity of the upper branch of the curve. In view of the asymptotic path of the lower branch other investigators proposed to reveal the K^% value from the intersection of the tangent of the lower portion with the extrapolation of the straight line of the upper branch (5), as also indicated in Fig.3b.

Recently, several investigators pointed out that the adopted evaluation procedure may not yield the correct closure values so that the calculated AK^^ff values may be afflicted by a considerable error. It was pointed out by Chen (6) that, in the closure evaluation, the effect of the lower portion of the load-COD curve down to Kjnin (or to K^Qg^j) must be taken into account.

The proposed procedure to measure crack closure (6) involves, for the case Closed < %min < ^cb the determination of the residual crack opening

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n &t Kmin, as indicated in Fig. 3b. From this it follows

AKth,eff = AKth - AKshielding (4)

with AKshielding (taking into account the difference between the closure-free

and the closure-affected crack) as defined in Fig.Sb. For the case Kmin < ^closed ™e obtain AKth,eff = Kmax-

Applying this procedure to experimental results obtained for cold-worked Cu it could be shown that the R-dependence of AKth is related to closure since

AKth,eff remained constant over the whole range of -1

AKth,eff. This was verified by experimental results (7) using the proposed

evaluation procedure (6). The value of AK^eff may also be deduced from a diagram as proposed by Vasudevan (8).

At present three experimental procedures are available for the determination

(i) At high positive R-ratios at which closure is absent (Kmax-method; 9, 10).

The method is limited to low ductility materials since under the required

high tensile stresses ductile materials will be subjected to considerable plastic deformation

(ii) Testing at large negative R-ratios (-3

(iii) Determination of AK^ and the closure contribution from which

can be calculated as described above.

Conclusions, Critical assessment of fatigue threshold

In evaluating the significance of the high-cycle fatigue threshold, experimental procedures and materials' as well as engineering aspects should be taken into account:

* At present the complicated nature of the threshold behavior with respect to a long fatigue crack, in view of the numerous intrinsic and extrinsic

variables, does not permit in all cases a quantification of the threshold value

on basic principles. * Most investigations were concerned with a clarification of the influence of

the stress-ratio on AK^ in particular for R>0. It was pointed out that the R-

212 Localized Damage

dependence of AKth is primarily related to several types of closure. If closure is taken into account, an effective value, AK^eff' ^ obtained

which may resemble a true material parameter, (see Table 1). Extrinsic

factors, such as test temperature and environment, cannot be explained quantitatively at present, since in addition to closure other effects such as

hydrogen embrittlement, fatigue-creep interactions etc., must be taken into

account. * In view of the importance of closure a standardized procedure for its

determination and evaluation is needed. * Various relationships for the dependence of AKth,eff on Young's modulus,

microstructure, yield strength, stress-ratio etc. based on theoretical

assumptions have been proposed. Up to date they appear limited to special conditions. For further work cyclic plasticity in the crack tip region should

be taken into account. The results indicate that as a guide for design work

and for selection of materials resistant to high-cycle fatigue failure, AKth,eff~values may be estimated from the Young's modulus, since most

of the theoretical derivations are characterized by a linear relationship between AKth,eff and E.

* In view of the loading conditions experienced by components during

service, the knowledge pertaining mainly to simple loading and crack opening modes must be supplemented to include more realistic conditions

(e.g. underloads or overloads, mixed modes and variabel amplitudes). When AK{h values are not known for particular loading situations, it

appears reasonable to use AK^,eff && a conservative design guide line.

* The applicability of the threshold concept to microstructurally short cracks is often questioned. Using AK^eff the LEFM may be extended to

physically short cracks, e.g. cracks as short as 1 mm. Furthermore, AK{h,eff has been successfully used to calculate the influence of internal

defects (inclusions and pores) or external defects (surface pits or small

notches) on fatigue life.

Localized Damage 213

Literature

1. Paris, P.C., Gomes, M.P & Anderson, W.E. A rational analytic theory of

fatigue, The Trend in Engng., 1961, 13, 9-14. 2. Hadrboletz, A, Weiss, B. & Stickler, R. Fatigue threshold of metallic

materials - a review, Handbook of Fatigue Crack Propagation in Metallic

Structures, ed A.Carpinteri, pp.847-882, Elsevier Science B.V., Amsterdam, 1994.

3. Elber, W. Fatigue crack closure under cyclic tension, Engng.Fract.Mech., 1970, 2, 37-45.

4. Schmidt, R.A. & Paris P.C. Threshold for fatigue crack propagation and the

effects of load ratio and frequency. Progress in Flaw Growth and Fracture Toughness Testing, ASTM STP 536, pp. 79-94, Philadelphia, Pa., 1973.

5. Newmann, J.C. & Elber, W. (ed). Mechanics of Fatigue Crack Closure,

ASTM STP 982, American Society for Testing and Materials, Philadelphia, Pa, 1988.

6. Chen, D.L., Weiss, B. & Stickler, R. Effect of stress ratio and loading condition on the fatigue threshold, Int.J.Fatigue, 1992, 14, 325-329.

7. Kemper, H., Weiss, B. & Stickler, R. An alternative presentation of the

effects of the stress-ratio on the fatigue threshold, Engng.Fract.Mech., 1989, 32, 591-600.

8. Bailon, J.P. & Dickson, J.J (ed). Proc.Fatigue 93, Quebec, Canada, EMAS, Warley, UK, 1993.

9. Hermann, W.A., Hertzberg, R.W. & Jaccard R. A simplified laboratory

approach for the prediction of short crack behavior in engineering structures, Fat.Fract.Engng.Mat.Struct., 1988, 11, 303-320.

10. Backlund, J., Blom, A. & Beevers, C.J. (ed). Proc.Fatigue Thresholds, Stockholm, EMAS, Warley, UK, 1982.