Modelling of fatigue crack growth following overloads

Int J Fatigue 13 No 5 (1991) pp 423-427

Modelling of fatigue crack growth following overloads H : J . C . Voorwald, M.A.S. Torres and C . C . E . Pinto J 6 n i o r

In variable-amplitude loading there are interaction effects between the loading history and the crack propagation rate. The most important of these effects is the retardation in the crack propagation, which may raise the life of the cracked structure considerably. The main objective of this research is to analyse and quantify the retardation of crack propagation in a thin plate of the high-resistance aluminium alloy 2024-T3, comparing the results obtained from the mathematical models proposed to account for the retardation effect. The specimens were tested under high-low loading sequences, for different crack sizes and overload ratios. A simplified model was developed, based on crack closure theory, that could represent the crack behaviour during retardation very well.

Key words: crack growth; retardation

The aim of all the numerous methods described in the literature 1-5 is to express the correlation between the crack propagation rate and the stress intensity factor under constantamplitude loading as precisely as possible. Predicting the rate at which fatigue cracks will grow under variable-amplitude loading is a major problem in the design and reliability assurance of a variety of structures if there are interaction effects betw.een the loading history and the crack propagation rate. The effect of load sequence on crack growth showed that the application of high-to-low load sequences could cause a greater crack propagation life than would have been predicted on the basis of the summation of crack growth for each cycle using constant-amplitude growth rate data. 6'7 This phenomenon has been termed crack retardation. Accelerated growth because of underloads, or low-high loading sequences, is also known to occur; however, this phenomenon is much less pronounced than crack growth retardation. 7.8 The influence of the variety of variables that contribute to fatigue crack growth retardation is most often characterized in terms of the number of delay cycles, ND. Increasing the magnitude of the overload cycle 9 or the number of overload cycles 1° can cause ND to increase; however, this effect often saturates as the number of overload cycles increases. 7 It was observed that an underload immediately after a single overload can reduce crack growth retardation,8 although the application of an underload immediately preceding the overload has little or no influence on ND.i* The number of delay cycles is related to the overload plastic zone size and therefore changes with the specimen thickness. 12 For variable-amplitude loading several semiempirical models, based on the concept of an effective crack-tip stress intensity factor, have evolved in an attempt to predict fatigue crack growth retardation. 13.14 Modern crack growth prediction models employ the crack closure concept. On the basis of crack propagation tests on AI 2024-T3 and of crack opening displacement

measurements, Elber proposed the following equation for constant-amplitude loading: 15 da/ d N = C (AKeff)"

(1)

with AKef~ = UdiK, where U is determined from the empirical expression U = 0.5 + 0.4R. Here R is the ratio of the minimum to the maximum stress. The two-dimensional elastic-plastic finite-element methods that are used to analyse the crack closure phenomenon have been shown to be quite accurate in explaining many of the types of behaviour observed during both constantamplitude crack growth and simple overload histories. 16-18 However, they were very complicated and require large computing facilities. There have been several attempts to develop simple models of crack closure 19-23 based on extensions of Dugdale's strip yielding the approach described in Ref. 24. The crack closure concept is also employed through models that do not explicitly evaluate the elastic displacement fields around the propagation cracks, but introduce simple expressions for the determination of the opening stress, Sop. The aim of the present study is to analyse and quantify the retardation of fatigue crack propagation following a descending loading sequence, for different crack sizes and overload ratios. This paper also evaluates the effect of the larger overload plastic zone sizes associated with increasing overload magnitude on the number of delay cycles. Special block load sequences were selected for this investigation. They include overloads with different numbers of high-load cycles, all of which are greater than the limiting saturation value. For the evaluation of the retardation effects, constantamplitude data were also required. Such tests were performed for different R-ratios.

Experimental data The experimental programme was performed on the highstrength aluminium alloy 2024-T3. The mechanical properties of this material are shown in Table 1.

0142-1123/91/050423-05 © 1991 Butterworth-Heinemann Ltd Int J Fatigue September 1991

423

T a b l e 1. M e c h a n i c a l m i n i u m alloy

properties

of 2 0 2 4 - T 3

alu-

• • • 5000+2500 N o o o 6500+2500 N + + ÷ 8 0 0 0 4"2500 N

40

Yield stress (MPa) Tensile strength (MPa) Elongation (%)

417 516 8.6

0

E

All the test specimens used in this experimental programme were of the single-edge-notched type. The thickness of the specimens was 1.27 mm and the width (w)90 mm. The stress intensity expression for this sample geometry is given by K = (r,~,a [1.99-0.41 (a/w) + 18.70 (a/w) 2

0 0

30 t +

¢D N

44-

O

¢~j

0

•

4o + 4o 4- o 4- o 0 4-

20

(2)

#

- 38.48 (a/w) 3 + 53.85 (a/w) +]

4"

Precracking was performed under constant-amplitude loading cycled at 5000 +- 2500 N. All tests were run at a cyclic rate of 10 Hz at room temperature. Cyclic crack growth measurements were obtained using visual optics. The experimental data used in the analysis of the retardation in fatigue crack growth arising from consecutive overloads were obtained from the load sequence shown in Fig. 1. A load of 5000 -+ 2500 N was used as a reference and the overloads were 6500 +- 2500 N, 7000 +- 2500 N, 7500 ± 2500 N and 8000 -+ 2500 N. The ratios of the overload maximum stress intensity, Kob to the maximum stress intensity of the subsequent constant-amplitude loading, K,+~, were 1.20, 1.27, 1.33 and 1.40, respectively. It was observed that a combination of K,,jK~a< 1.20 no retardation and K,,]/K~> 1.60 produces temporary arrest.

0

I0

d0

°""

o O0 • • •

"

I

0

50

I

I

I00 150 Number of cycles x 103

Fig. 2 The effect of cyclic stress range on crack growth

5000_+2500 ~

6500+2500

N

( ~ 5000_+ 2 5 0 0 N--~ 6500_+ 2500 N (~) 6 5 0 0 + 2 5 0 0 N--~ 5 0 0 0 + 2 5 0 0 N

Results and discussions

40

The data for the constant-amplitude cyclic load fluctuations are shown in Fig. 2. An increase in the magnitude of the cyclic load fluctuations results in a decrease of the fatigue life. The constant-amplitude test results can be represented quite well by the following equation: da 3.6 × 10 -~ ( U ~ K ) 3.°t" dN = 9i.1 - K,i~x

E E

-~ ¢o 3 0 J¢

20

(3)

with da/dN in m/cycle, ~K in MPa m 1'+2 and U = 0.5 + 0.4R. Figures 3 and 4 show the effect of the load sequences on crack growth, for ratios of overload maximum stress intensity to the maximum stress intensity of the subsequent constantamplitude loading of 1.20 and 1.40, respectively.

10

0

....

900

i

950

i

I000

i

1050

Number of cycles x 103

Fig. 3 The effect on crack growth of high-low toad sequences, K~Io~>/ K~.~I~. I 1.20 Z

"o o o _j

8OOO_+25OO

7500+2500

7 0 0 0 + 2500 6500+2500 5000_+2500 0

I

I

I

I

I

2

4

6

8

I0

Fig, 1 T h e e x p e r i m e n t a l

424

1

I

I

I

I

l

I

12 14 16 18 20 22 24 Crock size (ram}

load sequence

The amount of crack retardation increases as this ratio increases. Tests with high-low load sequences showed a greater crack propagation life. There are different numbers of high-load cycles for the various block sequences applied to the specimen, but all of them are greater than the limiting saturation value.

Int J Fatigue S e p t e m b e r 1991

5 0 0 0 + 2 5 0 0 N.~-~ 8000_+ 2 5 0 0 N

76

o

C) 5000_+ 2 5 0 0 N"-~8000 + 2500 N 40

(~) 8000+- 2 5 0 0 N " - ~ 5 0 0 0 + - 2 5 0 0

0

0

N

0 o

70

° oo°

E

0 0 u~

20

o 0 o 0

o

(J

0 0

X ¢}

I0

50

@ D

0

I

I

I

I

I

I

I

I00

150

200

250

300

350

o

40 Number of cycles x 105

0 0

J~

Fig. 4 The effect on crack growth of h i g h - l o w load s e q u e n c e s ,

E

0 0

z

Kr.~x(o,)/Kma.
N -I~ N N

0 0 0 0 0 0

0 >,~ ¢3

50

ooo 8 0 0 0 _e.2 5 0 0 5000" 2500 =•= 5000_+2500

0

60'

0o

30 o 0 0 0

The fatigue crack growth immediately after a descending loading sequence is shown in Figs 5, 6 and 7 for a ratio of the overload maximum stress intensity to the maximum stress intensity of the subsequent constant-amplitude loading of 1.40 and for different crack sizes. The results observed in Figs 5, 6 and 7 indicate that the crack propagation following a high-low sequence can be associated with three different types of behaviour related to the stress intensity factor. When the base mean stress intensity is low, the case of shorter cracks, the tensile overloads produce a plastic zone that induces residual compressive stress. The residual stress reduces the effective stress intensity at the crack

2 0 o0 0 0 0 0

I0 o 0

P •

16.00

I

i

16.40

i

.

16.80 17.20 rp ,.j

I

i

i

17.60

18.00

17.36 Crock size (ram) Fig. 6 Crack growth as a function of the number of cycles applied, a = 16.00 mm

0 0

40

0

40

0 0 0

0

0

0

0 0

re)

o

0

0

o

~_°3 0

0

o

)<

0

× 30

o

0

8 0 0 0 ~_2 5 0 0 N - - ~ o o o 5000:1:2500N •• • 5000+2500N

o

0

0

o

o

0

4)

0

0

0

o° o o

~a

0

,. 2 0

o o o 80004"2500 N -I~ 5000+ 2500 N • • • 5 0 0 0 :k 2 5 0 0 N

~ 20

o o

E o

z

I0

I0

2,.o0 I

i

•

2..o

i

2.'.80

r.

25'.o

i

2 6o

2 oo 26.8,j

Crack size (ram)

Fig. 7 Crack growth as a function of the number of cycles applied, a = 24.00 mm |

I

8.00 8.20 8.40

rp

I

i

i

8.60

8.80

~ 8.439

i

900

/

i

i

9.20

9.40

9.60

Crack size (ram)

Fig. 5 Crack growth as a function of the number of cycles applied, a = 8.00 mm

Int J F a t i g u e S e p t e m b e r

1991

tip and this retards crack growth rate during subsequent lower amplitude cycles (Fig. 5). So, the degree of retardation and resistance to crack growth are related to the plastic zone size introduced by the overload. When the base mean stress

425

intensity reaches a critical level, the case of longer cracks, the tensile overloads become high enough to cause coarse intermetallic particles ahead of the crack to separate from the matrix. 2s These incipient cracks reduce the stress intensity at the main crack tip and this increases the magnitude of retardation (Fig. 6). With even higher base mean stress intensities, the case of the longest cracks, the crack advances appreciably during the tensile overloads. As the stress levels ahead of the crack tip are high the retardation does not have a well defined behaviour once the main crack can interact with the subcritical cracks induced in front of it (Fig. 7). The retardation effects arising from the overloads can be associated with Kol/K~. For values of this ratio of 1.20 and 1.27 the fatigue crack growth following a high-low sequence loading is well defined, with the highest number of delay cycles for short cracks, smaller values for intermediate cracks and the lowest number for long cracks. For K,,I/K~a equal to 1.33 the same general tendency could be observed. The test results showed that for a value of the ratio of 1.40 a higher number of delay cycles are observed for intermediate cracks followed by short and long cracks. This means that the experimental behaviour can be represented quite well by equations for the three first cases, once a unique trend can be observed. For the last case, however, some scatter can be expected. In order to demonstrate the effect of compressive residual stress on the fatigue crack growth following a descending loading sequence, Figs 5, 6 and 7 show the crack propagation at the reference load and the plastic zone size created by the overload, which is given on the basis of the proposed equation: rp = ot

(4)

where cl is given by one of the following expressions: 1/65r for t > 2.5

(Km~×/~)2

-or ' f o r t ~< Iv-1 (Kmax/g~)2 and 1 + 5 ( 2 . 5 - t(Kmax/~)2 t 6rr 6 ~ \ 2.-5 - 7r 7 /

1 (Km,,]2
\

Sop(i ) = Sop(H )

--

(Sop(H)

--

Sop(L))

[(A~-A)/rpJ

(5)

where Sop(i) is the crack opening stress (initial), Sop(~f) is the crack opening stress (high), Sop(L) is the crack opening stress (low), Ai is the crack size (initial) and A is the reference crack size. The second method is based on the fact that there is a greater influence of the compressive residual stress on the high-low loading sequence, within approximately a third of the plastic zone size created by the overload. This tendency coincides with observations for other values of the ratio

Koi/Kca. Thus the model is based on the assumption that the crack opening stress level during a variable-amplitude loading history could be represented by the following expression, which gives a proposed correction for a third of the plastic zone size created by the overload: Sop(i ) = Sop(L ) -~- S"e(H)--Sg£'(L)[(r

rp

p)2 _ (Ai_A)2j,/e (6)

where r* is a third of the plastic zone size. Th~ number of delay cycles, both experimental and predicted by the proposed models and the plastic zone size, are represented in Table 2 for all the values of the ratio K,,JK,.~ and the crack sizes for descending loading sequences. From Table 2 it can be seen that the proposed linear model (model I) is non-conservative for all the conditions studied while model II, in which the crack opening stress is given by Equation (6), can correctly represent the number of delay cycles for values of K
Conclusions

for ~ \

method proposed by Nelson and Fuchs, 4 based on constantamplitude data, will be considered in this paper. For the crack closure concept, however, two possibilities are presented. The first method is the following linear equation for the crack opening stress during the total plastic zone size created by the overload:

cr~ /

where r r is the overload plastic zone size. Modern crack growth prediction models employ the crack closure concept. A linear crack propagation prediction

The amount of crack retardation increases as the ratio of overload maximum stress intensity, Kol, to the maximum stress intensity of subsequent constant-amplitude loading, K,.a, increases. It was found that a value of Kol/Jca < 1.20 produces no retardation and a value of K,,I/Kc~, > 1.60 produces temporary arrest.

Table 2. The number of delay cycles 8.00

Crack

12.00

16.00

20.00

24.00

size

~'m) Ko, ~

Kca

1.20

X Exp. 5.000

Model

Model

I

II

9.681

5.005

Exp. 4.000

Model

Model

I

II

8.533

1.27

11.000

17.763

11.216

10.800

15.258

1.33

18.000

39.224

32.071

18.800

33.052

1.40

426

25.000 135,480 160.250

4.451 9.711 27.187

25.000 112.608 135.398

Model

Model

Exp.

I

II

3.800

7.017

8.700 12.431

Model Model

Model Model

Exp.

I

II

Exp.

I

II

3.701

4.000

5.585

3.000

3.000

4 . 1 8 1 2.309

8.015

7.000

9.760

6.350

6,500

7.162

4.904

22.183

17.500 20,812 17.400

14.900 15,364 13.450

35.000 91,199 110.102

63.750 70.309 85.290

26.250 53.157 66.871

20.000 26.766

Int J F a t i g u e S e p t e m b e r

1991

The experimental results indicate that the crack propagation following a high-low load sequence can be associated with three different types of behaviours related to the stress intensity factor. The number of delay cycles arising from the overloads can be associated with Kol/Kca. A correction was proposed for the plastic zone size created by the overload in the case that the thickness of the specimens used did not guarantee a totally plane stress plastic zone size. For crack growth predictions, two models were developed and compared to describe the observed fatigue crack propagation. Equation (5) is non-conservative for all conditions studied and the crack opening stress given by Equation (6) can correctly represent the number of delay cycles for values of Kol/Kca of 1.20 and 1.27, with the predicted values being in very good agreement with the experimental data. For the ratio 1.33 some overestimation in the crack propagation occurred for small cracks and in the case of 1.40, the predictions deviate from the test results.

2024-T3 aluminum' Progress in Flaw Growth and Fracture Toughness Testing, ASTM STP 536 (American Society for Testing and Materials, 1973) pp 115-146 11.

12. 13. 14.

15.

16.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Paris, P. and Erdogan, F. 'A critical analysis of crack propagation laws' J Basic Eng, Trans ASME 85 (1963) pp 528-534 Forman, R.G., Kearney, V.E. and Engle, R.M. 'Numerical analysis of crack propagation in cyclic loaded structures' J Basic Eng, Trans ASME (1967) pp 459-464 Elber, W. 'The significance of fatigue crack closure. Damage tolerance in aircraft structures' ASTM STP 486 (American Society for Testing and Materials, 1971) pp 230-242 Nelson, D.V. and Fuchs, H.O. 'Prediction of fatigue crack growth under irregular loading' ASTM STP 595 (American Society for Testing and Materials, 1976) pp 267-286 Miller, M.S. and Gallagher, J.P. 'An analysis of several fatigue crack growth rate (FCGR) descriptions' ASTM STP 738 (American Society for Testing and Materials, 1981) pp 205-251 Zhang, A., Marissen, R., Schulte, K., Trautmann, K.K., Nowach, H. and Schijve, J. 'Crack propagation studies on AI 7475 on the basis of constant amplitude and selective variable amplitude loading histories' Fatigue Fract Eng Mater Struct 1 (1987) pp 315-332 Voorwald, H.J.C. 'Fatigue crack propagation in high strength thin sheet aluminum alloy subjected to variable amplitude loading' PhD thesis (FEG/UNESP, Campinas, Brazil, 1988) Jacoby, G.H., Nowack, H. and van Lipzig, H.T.M. 'Experimental results and a hypotheses for fatigue crack propagation under variable amplitude loading' Fatigue Crack Growth Under Spectrum Loads, ASTM STP 595 (American Society for Testing and Materials, 1976) pp 172-183 Von Euw, E.F.J., Hertzberg, R.W. and Roberts, R. 'Delay effects in fatigue crack propagation, stress analysis and growth of cracks' Proc of 1971 National Syrup on Fracture Mechanics, Part I, ASTM STP 513 (American Society for Testing and Materials, 1972) pp 230-259 Trebules, V.M., Jr, Roberts, R. and Hertzberg, R.W. 'Effect of multiple overloads on fatigue crack propagation in

Int J F a t i g u e S e p t e m b e r 1991

Elber, W. 'The significance of fatigue crack closure' ASTM STP 486 (American Society for Testing and Materials, 1971 ) pp 230-242 Newman, J.C., Jr and Armen, H., Jr 'Elastic-plastic analysis of a propagation crack under cyclic loading' AIAA J 13 (1975) p 1017

17.

Obji, K. and Ohkubo, Y. 'Cyclic analysis of a propagating crack and its correlation with fatigue crack growth' Eng Fract Mech 7 (1975) pp 457-464

18.

Newman, J.C., Jr 'A finite-element analysis of fatiguecrack closure' Mechanics of Crack Growth, ASTM STP 590 (American Society for Testing and Materials, 1976) p 281

19.

Fuhring, H. and Suger, T. 'Acceleration and retardation effects with fatigue crack growth and their calculation based on fatigue fracture mechanics' Proc 2nd Int Conf on Mechanical Behaviour of Materials (1976) p 721 Fuhring, H. and Seega, T. 'Dugdale crack closure analysis of fatigue cracks under constant amplitude loading' Eng Fract Mech 11 (1979) p 99

References 1.

Corbly, D.M. and Packman, P.F. 'On the influence of single and multiple peak overloads and fatigue crack propagation in 7075-T6511 aluminum' Eng Fract Mech 5 (1973) p 497 Wei, R.P. and Shih, T.T. 'Delay in fatigue crack growth' Int J Fract 10 (1974) pp 77-85 Wheler, O.E. 'Spectrum loading and crack growth' J Basic Eng, Trans ASME 94 (1972) p 181 Willenborg, J., Engle, R.M. and Wood, H.A. 'A crack growth retardation model using an effective stress intensity concept' Technical Report AFFDL-TM-71-1-FBR (Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio, 1971)

20.

21.

22.

23.

24. 25.

Dill, H.D. and Salt, C.R. 'Spectrum crack growth prediction method based on crack surface displacement and contact analyses' Fatigue Crack Growth Under Spectrum Loads, ASTM STP 595 (American Society for Testing and Materials, 1976) p 306 Newman, J.C., Jr 'A crack-closure model for predicting fatigue crack growth under aircraft spectrum loading' Methods and Models for Predicting Fatigue Crack Growth Under Random Loading, ASTM STP 748 (American Society for Testing and Materials, 1981 ) p 53 Newman, J.C., Jr 'Prediction of fatigue crack growth under variable-amplitude and spectrum loading using a closure model' Design of Fatigue and Fracture Resistant Structures, ASTM STP 761 (American Society for Testing and Materials, 1982) p 255 Dugdale, D.S. 'Yielding of steel shutters containing slits' J Mech Phys Solids 8 (1960) Sanders, T.H., Jr and Staley, J.T. 'Review of fatigue and fracture research on high strength aluminum alloys' ALCOA Report (Aluminum of America, 1985)

Authors

H.J.C. Voorwald and C.C.E. Pinto Jfinior are with the Department of Materials and Technology, FEG/UNESP, Campinas, SP, Brazil. M.A.S. Torres is with the Department of Mechanics, FEG/UNESP, Campinas, SP, Brazil.

427