634 Structures Under Shock and Impact - WIT Press · & Endo ^ is proposed. However the rainflow counting technique has two disadvantages; a) It neglects the sequence effect of the

The influence of the crack tip plastic zone

strain hardening on the metal High Cycle

Fatigue behavior

D.G. Pavlou

Hellenic Airforce Academy, 16501 Dekelia, Attiki,

Greece.

Abstract

A damage tolerance analysis methodology for cracks subjected to real loading ispresented. A new retardation model, based on material hardening within theoverload plastic zone, is developed for irregular stress histories. For thetransformation of the irregular stress history into a stress history containing onlydistinguished cycles, a technique based on the Rainflow counting method is used.The methodology is applied to real aircraft loading histories for the conventional2024-T3 aluminum and the advanced 8090-T8 aluminum-lithium alloy.Predictions are in encouraging agreement with the experimental results.Limitations of the prediction method are also discussed.

1 Introduction

Aerospace structures are operating under highly variable fatigueconditions. Up to present, fatigue analysis of aerospace structures is

mostly based on test results ; Blom. *To support fatigue crack growth analysis by crack growth models is

not a small task. Real loading histories are consisted of distinguished loadpeaks with high or small stress values. On the other hand, fatigue crackpropagation is a path dependent process, strongly influenced by load

Transactions on the Built Environment vol 32, © 1998 WIT Press, www.witpress.com, ISSN 1743-3509

634 Structures Under Shock and Impact

sequence. Therefore, fatigue crack propagation models that support crackgrowth analysis commonly used under service spectra have been modifiedto account for load interaction phenomena. In most cases the

modifications aim to consider crack growth retardation; Wheeler,

Willenborg, Engle & Wood. This is primarily attributed to thesignificance of the phenomenon for structural life assessment.

It is common belief to attribute the mechanics of the retardationprocess to localized plastic deformation ahead of the crack and to relateretardation to compressive residual stresses developed after overloads.Amongst them, the models proposed by Wheeler and Willenborg havefound wide spread utilization. The two models differ on the empiricalfitting parameter to account for fatigue crack growth retardation. InWillenborg's model the fitting parameter is introduced to formulate aneffective stress intensity factor which varies within the plastic zone. InWheeler's model, the corresponding constant amplitude fatigue crackgrowth rate is multiplied by a position depended retardation factor. Bothmodels are very popular because of their simplicity. However a series of

modifications; e.g. Johnson, * aiming to increase accuracy as well as toaccount for effects related to multiple overloads or underloads have beenmade.

The crack closure retardation models; Elber, are based on Fiber'sexperimental observation that crack propagation caused by a tension stressis followed by a partial closure of the crack lips during unloading. Thisphenomenon has been associated to a variety of micromechanisms;

Elber. Subsequent crack propagation can occur, when the crack is fullyopen. Therefore, a part of the applied stress will be used for crackopening. For the crack propagation only the remaining part of the appliedstress can be utilized. The magnitude of the required opening stressdepends on the previous stress history. Models developed on the basis ofthe Elber's concept differ in the derivation of the opening stress; e.g.

Newman, De Koning, * Aliaga, Davy & Schaff Latter discussedefforts have found significant industrial utilization. Correlation betweenexperimental and computed results has been reported to be satisfactoryparticularly, when the crack length becomes large or the stress level is

high; Broek & Smith, ™ Suresh, Ohrloff, Journet, Pavlou, Pantelakis,

Sera, Poole & Smart. To improve fatigue crack growth prediction for small initial cracks,

models that take into account retardation mechanisms predominant for theinitial fatigue stages are required. In this work, a model suitable for this


Structures Under Shock and Impact 635

case, recently presented; Pavlou and Pantelakis, Kermanidis & Pavlou is applied. The model takes into account the influence of the yield stresschanging within the overload plastic zone on the fatigue crack growthrate. The latter model is limited to cyclic loading with variable amplitude;

Pantelakis, Kermanidis & Pavlou. In this work an extension of the saidmodel for random loading conditions is attempted. For this reason, theirregular stress history is transformed into a history of full distinguishedcycles. For the derivation of the distinguished cycles the application of the

Rainflow counting technique; Richards, La Pointe & Wetzel, Matsuishi

& Endo is proposed. However the rainflow counting technique has twodisadvantages; a) It neglects the sequence effect of the derived loadingcycles, and b) when applied to long spectra it increases the computingcost. To overcome the shortcomings, a linear approximation technique isproposed.

2 Fatigue Crack Growth Prediction

Methodology

2.1 Linear approximation technique

For a given load spectrum the proposed linear approximation techniqueproceeds by considering load steps (Fig. 1). For every

Figure 1: Discretized irregular stress history

load step the rainflow method is applied separately. Because of the load-interaction effects on the crack growth behavior, the sequence of the loadcycles within the load step is very important. Hence, the cycles derived bythe rainflow method are ordered based on the same order of the peaks asin the original load trace (Fig.2). Using the latter criterion, the consistencybetween the sequence of the plastic zones of the irregular history and the



sequence of the plastic zones of the derived cycles is satisfied. The loadcycles resulting from this

A

Figure 2: Classification of the distinguished cycles

counting procedure will, thus, form the load spectrum to be used in crackgrowth analysis.

2.2 Load Interaction Model

In the vicinity of the crack tip the fatigue damage accumulated within eachmaterial element is mainly consequence of plastic deformation. Thereforecan be assumed that the material elements in the overload plastic zone aresubjected to LCF conditions. Figure 3 provides a schematic of theproposed stripplastic zone model. The length of the overload plastic zone

is symbolized by r while a is the crack length after an overload. The

value of the yield stress S is variable for the points along the overload

plastic zone because of the different hardening degree. For these points a

linear distribution of the value S is assumed. The yield stress of the

material ahead of the overload crack tip is assumed to be

c* — /?c (\ \by — poy (i;

where S is the cyclic ultimate stress, and {3 a parameter lying between

S^ISf and 1. The yield stress at the elastic-plastic boundary is

considered to be equal to the cyclic yield stress S of the material. The

current crack length within the plastic region is a. Distinguished from



fa is the plastic zone size co, created by a fatigue load cycle following

an overload. This zone it is very small

Ac,,

Figure 3: Schematic representation of the retardation model

compared to fa such that w, can be taken approximately to be the size

of a material element. Each element is subjected to a constant average

strain amplitude A£ that approximates the strain amplitude distribution

in a>s. Failure is assumed after a certain number of cycles AN. Low cycle

fatigue crack growth is assumed such that the relation

da co, (2)

applies, where the length of the plastic zone #, can be estimated as



AK V

The number of cycles up to failure AN can be determined by the Coffin-

Manson's rule ; Manson, Coffin & Tavernelli: **

= 2s f (4)

According to the works; Pavlou and Pantelakis, Kermanidis & Pavlou

the value Af can be evaluated as :

An estimate of AN can thus be made

AN-V2K,;

21 m

AK

With aid of eqns. (3) and (6), eqn.(2) becomes:

da

(6)

dN 322K^ S]

Low cycle fatigue tests; Fuchs & Stephens, have shown that m is notsensitive to material type ; it can be taken to be 0,5.

2.3 Retardation Factor

Let X be defined as a retardation factor given by :



The retarded crack growth rate (da I dN) corresponds to the situation

in fig. 4(a) where the crack growth takes place in the overload plastic

zone. The nominal crack growth rate (da I

Figure 4: Crack growth under cyclic loading : (a) overload andb) without overload.

corresponds to fig. 4(b) in which the crack growth occurs in the absenceof any overloads. In view of eqn.(7) the retardation factor take the form :

(9)

Recall that S and Sy are respectively the cyclic yield stress without

and with overload plasticity effect (fig.3). It is the relationship between

Sy and Syo that determines the change in material response due to

overload. Invoked is a linearly varying function for Sy :

a / 2) - aOL(10)

'OL

Once 1 is calculated by the eqns (9),( 10) the retarded crack growth ratecan be determined by eqn.(8):



da __ .~

da(11)

where the nominal fatigue crack growth rate (da/dtf) can be

derived as usual by constant amplitude loading tests. Application of themodel to constant amplitude loading containing single overload has

resulted in satisfactory predictions (e.g. Figs 5(a),(b)).

<DK

O ExperimentPavlou's model [13, 14]

0.2 0.4 0.6 0.8

Normalized crack length (a-cto) / roi_

co+3

1.5

to

0.5

AI-6061-T6KoL/Kmax"2R=0.05

O ExperimentPavlou's model [13,14]

0.4 0.8

Crack length a (mm)

1.2

Figure 5: Retardation factor versus crack length for A1-2024-T3

a) Knr I K^ = 1.5 and R=0, b) K^ I K^ = 2.0 and R=0.05



2.4 Crack Growth Accumulation Procedure

As mentioned previously, a large class of engineering structures aresubjected to service loads that vary randomly. The influence of loadinteraction on the fatigue crack growth rate depends on the interaction ofthe produced plastic zones. Treating the random loading plastic zoneinteraction is an extremely difficult task. Therefore, a series ofsimplifications on the existing non-linear models have been introduced. In

the proposed methodology, it has been assumed that each loading cycle i

produces crack increment Aa, = 1 x (da / dN) Therefore, the cycle-

by-cycle crack length summation is carried out by the following equation:

The retardation factor /I . of the proposed methodology has absolutely

different physical meaning than the Wheeler's one. In Wheeler's modelthe retardation factor 1 contain a constant fitting parameterexperimentally derived. In the proposed model, factor 1 is variable anddepends on the hardening degree of the material element ahead of thecrack tip. However, the crack growth computation procedure is the same.The latter is performed block-to-block. The input stress data

O" , er • , required for the crack growth calculations of each block,maXy ' rniiij •* *-'

have been obtained previously by the rainflow method.

3. Results and Discussion

The validity of the proposed model has been examined with experimentalresults. The tests have been performed on bare sheet material of theconventional 2024-T3 aluminum alloy and the advanced 8090-T8aluminum-lithium alloy. The tests were performed by the Deutche Airbuson standard CCT specimens with 1.6 mm thickness; Ohrloff, Journet,

Pavlou, Pantelakis, Sera, Poole & Smart. The Deutche Airbus



standardized load sequence for the upper fuselage shell (reference 12) ofthe aircraft A-300 was used for the tests.

The cyclic properties for evaluating eqns. (4-6) are given in Table 1,and the data were taken from the references 12,13,14,19. To calculate

A1-2024-T3

A1-8090-T8

8, (MPa)

1100

423

S (MPa)

427

302

Table 1. Cyclic properties of the used alloys

the cyclic yield stress S * = ft. Sj> of the material element at crack tip,

the approximation p=l was used. For aluminum alloys such as Al-2024-T3 and A1-6061-T6 the validity of the latter simplification has been

checked ; Pavlou and Pantelakis, Kermanidis & Pavlou. The nominal fatigue crack growth rate was calculated by the Paris rule:

03)

where the parameters n and C has been fitted to n=4.305-1.316 R andO-2.503 E-15 + 4.365 E-13 R for the A1-2024-T3 alloy and, n-3.837-1.184 R, C=-1.08 E-14 + 1.489 E-12 R for the A1-8090-T8. Theabove data were obtained by CAL fatigue tests performed by DRA;

Ohrloff, Journet, Pavlou, Pantelakis, Sera , Poole & Smart. Using thedescribed damage accumulation procedure, a computer program wasformulated. This program was used to predict crack propagation underrandom loading conditions on the mentioned aluminum alloys. Both theaccuracy of the calculations and the required computing time depend onthe choice of the loading steps (Fig.l). The number of reversals assumedto build one step should be a compromise between the gain on computingtime and the loss on accuracy. In the present work, for simplicity it isassumed that the number of peaks in each block is n=60. By executingthis program, the prediction of the crack length versus the number offlights was carried out for the A1-2024-T3 and A1-8090-T8 alloys. Latterpredictions are compared with the experimental results in Fig.6a,b.



E

D£Z0)_g

•82O

JU

25-

20-

15-

10-

5-

c

AI-20;o '/

///

o //

Jo /0 /'

- """ Propnspd n~KI ^** Willenboro'sr~~ O Experimental

i I i t i i i i40 60' '150 170 190 210 230 250

! 1 I ! I I ! 1 ! I0 20 40 60 80 100 120 140 160 180 200 220

24-T3

)delmodelresults

1240 26

Number of flights (x 1000)

Figure 6. a) Theoretical and experimental results for the alloyA1-2024-T3 under real aircraft loading conditions

30

_0>JX.o

25-

20-

10 -

0 -

AI-8090-T8

Proposed modelWillenborg's model

O Experimental results

o-o-'64 80 96 180 196 212 228

T T0 16 32 48 64 80 96 112 128 144 160 176

Number of flights (x 1000)

Figure 6. b) Theoretical and experimental results for the alloyA1-8090-T8 under real aircraft loading conditions



For the alloys A1-2024-T3 and A1-8090-T8 the predictions were veryencouraging, only within the areas a=2-7 mm and 2-16 mm, respectively.The predictions of the Willenborg's model are also presented at the samefigures. The calculations using the latter model, however, were startedfrom larger initial crack lengths, i.e. 5.0 mm for the A1-2024-T3 and4.0mm for the A1-8090-T8. The reason of this decision is the following:for small initial crack lengths the Willenborg's model leads to very largeunderestimation of the fatigue crack increment per cycle. Therefore, the3 2-bit computer used for the calculations was unable to treat such crack

increments of the order of 1E-11 mm (i.e. (ZQ+ A#. = a§ when

Aa, < 10" mm ). The calculations, however, were carried out in order

to obtain some conclusions concerning the validity range of the mentionedretardation models.

Figures 6a,b clearly indicate that the proposed methodologyoverestimates significantly the crack growth for larger crack lengths (e.g.>7.0 mm for the 2024-T3 and >16.0 mm for the 8090-T8)significantly. A possible reason of the above result is the following: forlarger crack lengths, more predominant retardation mechanisms than themechanism of the material's hardening take place. These mechanisms arerelated with the development of compressive self-stresses around theoverload plastic zones. Fig.6a,b shows that the results of the Willenborg'smodel are in good agreement with the experimental results only for largecrack lengths.

The results lead to the following conclusions :a) Models based only on one retardation mechanism are not able to

provide reliable fatigue crack growth prediction for the total fatigue lifeperiod.

b) All the known load interaction models take into account onesingle mechanism. Therefore, the models are able to accurately predictthe fatigue crack growth only within the area where this mechanism ispredominant.

c) Improvement of fatigue crack growth prediction for long fatiguelife periods requires the development of hybridic models that take intoaccount larger number of mechanisms. The basic problem that needsfurther investigation is the definition of the predominant load-interactionmechanisms for every fatigue stage.



4 Conclusions

a) A transformation technique of the irregular stress history to aregular one was used. The technique is based on a modification of therainflow counting method. The advantage of the modified technique is thesuitability to use non-linear damage accumulation models and long stress

histories.b) A new retardation model was applied to the transformed irregular

real spectra. The model is based on the plastic zone strain hardeningfatigue mechanism that takes place during all the fatigue life stages.However for the investigated alloys 2024-T3 and 8090-T8 themechanism seems to be predominant in the early fatigue crackpropagation stages. In respect of these alloys and stress histories, themodel led to extremely encouraging predictions for the initial crackpropagation stages.

c) To improve the fatigue crack growth predictions, it is necessary toinvestigate the boundaries where every load-interaction mechanism ispredominant as well as to develop load interaction models that take intoaccount more load-interaction mechanisms.

5 Acknowledgments

The present research work was carried out at the Hellenic MilitaryAirforce Research Center. It is a continuation of previous work carried outin cooperation with Aerospatiale, Deutsche Airbus, Dassault Aviation,British Aerospace, Defence Research Agency and NLR, in theframework of the Brite/Euram BE-3250 project, supported by theEuropean Union.

References

[1] Blom A.F., Fatigue and damade tolerance verification ofaircraft structures, Proc. of the 4th Int. Conf. On Fatigue and FatigueThresholds, Honolulu Hawaii, Vol. I, pp. 13-22,1990.

[2] Wheeler O.E., Spectrum loading and crack growth, J.Basic Engng.94, pp. 181-186, 1972.

[3] Willenborg J., Engle R.M., Wood R.A., A crack growth retardationmodel using an effective stress concept, AFFDL-TM-71-1 -FBR. 1971.



[4] Johnson W.S., Multi-Parameter-Yield-Zone model forpredicting spectrum crack growth, ASTM STP 748, pp. 85-102,

1981.

[5] Elber W., The significance of fatigue crack closure, ASTM STP486, pp. 230-242, 1971.

[6] Elber W., Fatigue crack closure under cyclic tension, Eng.FractureMechanics, Vol. 2, pp.37-45, 1970.

[7] Newman J.C., A crack closure model for predicting fatigue crackgrowth under aircraft spectrum loading, ASTM STP 748,pp. 53-84,

1981.

[8] De Koning A.U., A simple crack closure model for predictionof fatigue crack growth rates under variable amplitude loading,ASTM STP 743, pp. 63-85, 1981.

[9] Aliaga D., Davy A., Schaff H., A simple crack closure model forpredicting fatigue crack growth under flight simulation loading, ASTMSTP 982, pp. 491-504, 1988.

[10]Broek D., Smith Sh., The prediction of fatigue crack growth underflight-by-flight loading, Engng. Fract. Mech., Vol. 11, pp. 123-141,1979.

[1 IJSuresh S., Micromechanisms of fatigue crack growthretardation following overloads, Eng. Fract. Mechanics, Vol. 18,No. 3, pp. 577-593, 1983.

[12]Ohrloff N., Journet B., Pavlou D., Pantelakis S., Sera L., Poole D.,Smart J., Investigation on Al-Li alloys for damage tolerantapplications, Brite/Euram BE-3250 project, Task 4 (FlightSimulation), Final Report DCR/M-60365/F-93, Suresnes, France,1993.

[13]Pavlou D., Fatigue crack growth prediction under irregular stresshistories, Ph.D. Thesis, University of Patras, Patras,1994.



[14]Pantelakis S., Kermanidis Th., Pavlou D., Fatigue crack growthretardation assessment of 2024-T3 and 6061-T6 aluminum specimens,Theor. and Appl. Fract. Mech., Vol. 22, pp.35-42, 1995.

[15]Richards F., La Pointe N., Wetzel R., A cycle countingalgorithm for fatigue damage analysis, paper No.740278, AutomotiveEngineering Congress, Detroit, Michigan, Feb. 1974.

[16]Matsuishi M., Endo T., Fatigue of metals subjected to varyingstress, Japan Society of Mechanical Engineers, Fukuoka Japan,1968.

[17]Manson S.S., Fatigue: A complex subject-Some simpleapproximations, Exp. Mech. 5,193-202 (1965).

[18]Coffin L.F., Tavernelli J.F., Experimental support for generalizedequation predicted low cycle fatigue, Transaction ASME, J.BasicEngng. 84, 533-542 (1962).

[19]Fuchs H.O., Stephens R.I., Metal fatigue in engineering,J. Wiley &Sons, 1980.


Documents

634 Structures Under Shock and Impact - WIT Press · & Endo ^ is proposed. However the rainflow counting technique has two disadvantages; a) It neglects the sequence effect of the