The effect of single overloads upon fatigue cracks in 5083-H321 aluminium

oow7944/86 s3.w + .a0 Q 1986 Pergamon Press Ltd.

THE EFFECT OF SINGLE OVERLOADS UPON FATIGUE CRACKS IN 5083-H321 ALUMINIUM W. M. THOMAS? Queensland Railways, Townsville, Australia Abstract-The application of a single overload cycle to a growing fatigue crack has long been known to delay future growth of that crack. Although much work has been done in this area, considerably less work has been undertaken with a view to examining the effect of secondary influences such as stress ratio upon such retardation. This paper examines the effect of various load related variables upon both the number of delay cycles and the extent of the delay zone. Results are found which seem to indicate that the basic shape of a delay curve is constant-oniy the scale being affected by loading variation. The criterion of optimisation is kept in mind throughout the experiments discussed in this paper and several economised testing techniques are discussed. One of the most significant techniques is the use of overload experiments to evaluate the constant amplitude behaviour of the material under consideration. The material chosen for examination was 5083-H321. This material, although not often used in high technology areas, is a common industrial material of particular interest to the transport industry.

1. INTRODUCTION

THE STUDY of fatigue has been a complex and ongoing struggle since Wohler first recognised the problem in 1860. His study of failures in railway axles has been improved in many ways over the subsequent century but the basic method of analysis remains the same. It is only in relatively recent times that Wohler’s S-N curve technique has been recognised as a procedure which is restricted to the initiation regime of fatigue crack formation ; and indeed, the S-N curve is still considered as a panacea by many designers who have had limited fatigue experience. A mere twenty years ago Paris and Erdogan[l] published their classical experiments which related the rate of growth of propagating fatigue cracks to fluctuations in Westergaards’ stress intensity factor. The fact that there appears to be three classes of crack propagation, i.e. (a) initiation, (b) propagation, and (c) final failure stems from their experiments. One fact which is not brought out strongly enough in the literature is that students of fatigue can only ignore the quasistatic nature of stress intensity factor fluctuation/crack growth rate relations at their own risk. Although it is true that load fluctuation in a time domain is a prerequisite for fatigue crack growth, any variation in the frequency domain can cause severe abnormalities in crack growth behaviour. The transient nature of crack growth behaviour is the underlying theme of this paper. Since the topic is so complex, however, it was found to be expedient to limit this paper’s scope to the effect of mechanical variables upon growth rate transience following a single tensile overload. The simplest cases where fatigue situations occur are in loaded elements of rotating machinery. Elements such as gears, bearings and shafts lend themselves to a basic Wohler type S-N analysis which is both simple and accurate. More complex cases from the aircraft industry have been the subject of many treatises, but there seems to have been a general lack of interest in some of the more common “workhorse” materials. Although 5000 series aluminium alloys do not offer the excellent properties of their age hardening 7000 series brothers, they remain a vital material in general engineering. The excellent weldability of 5000 series aluminium alloys has led to their use in many bulk transport areas such as truck and wagon bodies, where welded aluminium construction offers clear weight advantages. Since the use of aluminium in the transport area is fraught with fatigue related difficulties it seems timely to examine the properties of at least one of the more common alloys (i.e. 5083). Although Tobler and Reed[2], amongst others, have considered the crack growth behaviour of 5083 under room temperature conditions, there is comparatively little literature dealing with the effect of overloads upon medium strength aluminium alloys such as 5083. The high cost of materials testing has traditionally led to rather narrow spans of investigation, t Current address : Queensland Railways, Rockhampton, Australia. 1015

I016

W. M. THOMAS

which only consider the effect of one fundamental parameter on a given material property. The test procedure which is detailed in this paper incorporates a rectangular experimental design which varies the magnitude of the overload along the ordinate and stress ratio along the abscissa. Naturally such an experimental design offers information regarding overload magnitude, stress ratio and their interactive effects upon crack growth rates; but since the effect of overload upon growth rate is transient in nature such a design can also synthesise replications of constant amplitude loading results by overlaying the steady state portions of tests conducted at constant stress ratio. The importance of overload induced crack growth retardation lies in its potential to extend the fatigue lives of existing structures by relatively simple means. Since the extent of such improvements can only be anticipated in terms of empirical evidence, it seems reasonable to expect that the value of a material under fatigue growth situations can only be reflected by examination of both constant amplitude and overload situations. The economised testing regime described above offers just such information.

2. DELAY

THEORY

Since the critical area of a fatigued component must lie at the crack tip, it is instructive to consider the behaviour of crack tip plastic enclaves. Irwin[3] initially proposed that the size of the plastic zone at a crack tip can be estimated by the relation 1 WC--

K2

0 -

n

.

Y

This relation strictly applies to plane stress conditions, but it is relatively easy to show that the diameter of the plastic enclave under plane strain conditions is one-third of the plane stress size. The critical difference between cracks loaded by monotonic loads, and those subjected to cyclic loads, lies in the interpretation of yield stress (Y). Under monotonic loading a particular element of material must be taken from the unloaded to the loaded state before yielding occurs, whereas under cyclic loading the first stress fluctuation will produce a residual compressive plastic zone-subsequently the element of material must be stressed from negative yield stress to positive yield stress before yielding occurs. In other words, under cyclic loading the effective value of yield stress is twice the actual value. Since material yield point is different for monotonic loading and cyclic loading, the monotonic plastic diameter is related to monotonic yield point while the cyclic plastic diameter is related to twice the cyclic yield point. This is an important fact in fatigue situations because tensile overloads obviously produce plastic enclaves (monotonic class) which eclipse plastic encalves formed by cyclic stresses (cyclic class). A number of authors have considered the effect of work hardening upon plastic enclave size but one of the most succinct analyses is due to Rice[4], who showed that the ratio of plastic enclave size in a work hardening material to that in a perfectly plastic material is

where the strain hardening exponent(n) is either the monotonic value or the cyclic value depending on whether loading is monotonic or cyclic. Although this relation strictly relates to mode III cracking it remains a useful approximation in the mode I situations discussed in this paper. Almost all models of delay follow the lead of Wheeler[5] and examine delay in terms of plastic enclave interactions. This approach stems from the syllogism that (a) since material failure (and consequently crack extension) occurs at crack tips, and (b) since the crack tip stress field is determined primarily by the existence and extent of plastic zones, then crack extension is controlled by the extent of the crack tip plastic enclave. Although Wheeler proposes that crack growth rate depends upon the distance that the crack tip has penetrated into the monotonic overload zone, he gives no reason for the effect. Elber[6], on the other hand, hypothesises that the monotonic plastic zone superimposes a residual stress on the component stress field which keeps the crack closed for part of the tensile fluctuation. This in turn reduces the magnitude of the cyclic tensile load and consequently the rate of crack growth is reduced (i.e. delay occurs). More recently it has been shown that crack growth rate

Fatigue cracks in 5083-H321aluminium (KI

I

c TIME

(t)

APPLIED

LOAD

CYCLES

IN1

Fig. 1. The relation between loading and delay.

reductions following an overload are not necessarily related to crack closure[7]. It is with this fact in mind that tests in this paper were conducted under nonclosure conditions. Several researchers have examined parameters which affect delay. One of the first seems to have been Rice[4], who examined the effect of AK/K,, upon delay. The basic form of his results are shown

in Fig. 1. When a constant amplitude loading situation is disturbed by the introduction of a tensile overload then Rice observed that the length of delay experienced is related to AK/K,, in an exponential manner. His results are based on observations on 7075-T6 aluminium alloy. It is interesting to note that the loading is not fully defined by AK/K,, (even assuming that the loading waveform is piecewise sinusoidal). In order to complete the definition it is necessary to specify the stress ratio (R)of the loading. Although Rice is not explicit, the data that he examined relate to a R = 0 condition. Very little work appears to have been undertaken relating stress ratio to delay. Although the concept of delay (NJ simplifies the analysis of crack growth transience it tends to mask some of the more subtle effects. These effects become apparent when the Parisian manner of data presentation is used (Fig. 2). Immediately following an overload application some, but by no means all, materials have been observed to increase their crack growth rate. Subsequently all materials exhibit greatly reduced growth rates for a distance which appears to be related to the size of the crack tip plastic enclave. Eventually the transient disappears and growth rates revert to the values which would have existed had no overload been applied. Since the crack does grow during the retardation process this post-overload rate is not the same as the pre-overload growth rate but is always larger. Although most theories are unable to explain the enhanced growth rates observed immediately subsequent to an overload, the model of Matsuoka and Tanaka[8] does offer an explanation. They suggest that when an overload is applied the crack tip blunts. This causes the faces of the crack to lie further apart until the effect of the overload has passed. Consequently, for cracks which are closed during part of the pre-overload loading cycle, the crack is open for a greater proportion of the loading cycle. Since a greater load fluctuation severity occurs immediately following an overload it is only logical that greater growth rates occur after an overload. The corollary of the Matsuoka and Tanaka

1018

W. M. THOMAS

i/_

STRESS

,NTENSlTY

FACTOR

FLUCTUATION

t AK I

Fig. 2. Typical da/dN * u*AK relation for a cracked component subjected to tensile overload.

hypothesis is that if a crack does not exhibit closure then post-overload crack acceleration would not be anticipated. Although it is generally acknowledged that the magnitude of an overload is the most important factor controlling the length of delay, there are several other factors which can have an important effect. These include (a) the frequency of load application, (b) the duration of the overload, and (c) the delay after the overload before the base loading is applied. All of these factors were held constant in the tests of this paper. 3. EXPERIMENTAL

PROCEDURE

AND TECHNIQUE

The use of 5083-H321 aluminium alloy in this paper stems from its importance in the bulk transport area. 5083-H321 has the composition Si 0.4%

Fe 0.4%

Cu 0.1%

Mn O.l-O.S%

Mg 4.0-4.9%

Cr 0.05-0.25%

Zn 0.25%

Ti 0.15%

Al Rem.

Although this material does not have the excellent strength characteristics of age hardening alloys, its comparatively low cost and excellent weldability make it an attractive proposition for many applications. Crack measurement is an important aspect of any fatigue crack propagation test, but all too often little is said about it. The technique used in this paper is the compliance method, which relies upon changes in specimen stiffness to measure crack length. Although the basic crack mouth opening displacement/crack length relation has been given in closed form by Hudak[9], an extensive set of tests were conducted as a validation procedure. These tests showed some deviation from Hudak’s relation-probably attributable to clip gauge knife edge deflection and crack tip plasticity. The results from compliance crack measurements proved to be both reproducible and of demonstrable accuracy. Further, by noting the linearity of the load/crack face displacement relation it was possible to confirm whether or not closure was occurring. This confirmation was naturally limited to the global scale because closure very close to the crack tip would cause very marginal load redistribution and consequently very little loss of linearity in the compliance relation. Since gross closure was observed at just below 3 kN tension for the specimens used in testing, no closure was observed or anticipated.

Fatigue

cracks in 5083-H321

aluminium

1019

_-

1 Fig. 3. Details of fatigue crack growth

testing specimens.

Details of specimen configuration are given in Fig. 3. In order to randomise effects due to (a) variations in material properties, (b) variations in environmental influences such as temperature and humidity, and (c) variations in specimen configuration all test specimens were manufactured in a single batch and were selected randomly for individual tests. Such a procedure essentially eliminated all nonload related effects from test results-which is precisely why the methodology was selected. In order to maximise the data obtained during experimentation a rectangular design was used incorporating four levels of overload (35,40,45 and 50 kN) and three levels of stress ratio (3-30, 10-30 and 15-30 kN). In addition to the effects of changing overload and stress ratio the absolute effect of load level was examined. As a crack extends across a fatigue specimen both the magnitude of the base stress intensity factor fluctuation and the overload stress intensity factor fluctuation retain a constant ratio. Since the absolute magnitude of stress intensity factor fluctuation increases with crack length, it is possible to evaluate its effect by applying overloads at several crack lengths. In order to prevent interactions between overloads, no overloads were applied until growth rates were equal to the value expected by extrapolation of pre-overload crack growth data (Parisian format) to the current crack length. In reality this procedure only permitted two overloads to be applied to most specimens. Clearly the test procedure described above represents an economic method because 12 tests can yield information about the effects of (a) environment, (b) material variability, (c) stress ratio, (d) stress intensity factor level, (e) overload level (if applicable) upon crack growth rates under constant amplitude and overload conditions. During testing it was clear that negligible delay would result from the minimum overload level (35 kN)/maximum stress range (3-30 kN) combination, so all data were drawn from the remaining eleven tests. The actual procedure used for testing was based on ASTM E647-81 [lo]. An Instron electrohydraulic tensile testing machine was used to apply load to a centre cracked tension specimen (Fig. 3) through fairly standard friction grips. Centre cracked tension specimens were chosen for testing so that future testing could be undertaken under compression conditions. The normalised

W. M. THOMAS

1020

notch length of the specimens was a/W = 0.21 which, with the specimen thickness of 10 mm, offered quite straight crack fronts up to the elastic limit of the specimens (a/W = 0.76). Initial notches were prepared by slotting and were sharpened just prior to testing with a scalpel blade. The initiating cracks were produced from these notches by fatigue. Precracking was achieved by stepping the applied loading from a O-50 kN excursion down to the final base loading in approximately equal load increments. The duration of each precrack load level was such that crack growth could be visually discerned. Precracking and testing frequencies were identical (i.e. 10 Hz). Since crack fronts were not entirely straight it was necessary to evaluate the amount that the centre of the crack front extended ahead of the edges of the crack front. This discrepancy is termed “pop-in” and was found to be almost exactly 0.5 mm regardless of the type of loading. Stress intensity factor levels were evaluated through the formula

The evaluation of crack growth rates was made through an incremental polynomial method. Since crack length measurements were recorded at equal intervals throughout the test procedure it was possible to use a highly efficient version of this method. The analysis of the improved incremental polynomial method is detailed in Appendix 1. In all cases both precrack and final fatigue cracks were symmetric to within 2.0 mm and f 5”. 4. RESULTS

AND DISCUSSION

4.1 Delay cycles

The initial evaluation of test results can be achieved through the use of the delay parameter (N,). The evaluation of N, was undertaken through the method shown in the crack length/applied load cycle relation of Fig. 1. Results of this analysis are shown in Table 1, where the rectangular experimental design is apparent. If these data are plotted in the format due to Rice then the basic form of Rice’s relation is maintained-but the scatter on such a diagram is very large (even considering that Table 1. Effect of varying

LTmx=

R,K,JKmarand AK (MPa ml/‘) on crack growth delay (No)

1.167

1.333

( KOL=35kN) R=

ND

AK No

0.10

( 3-30kN )

1.500

1.667

(KOL=40kN)

(KOL=45kN)

(KOL=50kN)

AK

AK

ND

AK

ND

7.115

13000

7.497

34000

ND

7.088

4500

Tests

9.500

3000

Conducted

12.200

2500 10.410

2000 11.950

9000

5.270

2000

6.369

4000

5.351

16000

6.670

30000

6.323

0

7.823

0

7.911

6000

a.575

14000

7.506

3000

6.748

11000

6.381

11000

6.162

53000

0.33

( lo-30kN )

0.50 ( 15-30kN )

Fatigue

cracks in 5083-H321

1021

aluminium

R = 0.500

I

I

I

#lllll

I

I

I

I

,llll

lo4

103

I

I

lo5

DELAY CYCLES ( ND )

Fig. 4. Relationship

between AK/K,,

and N,

(R = 0.50).

delay is a fatigue related phenomenon). It was found, however, that almost all scatter could be eliminated by assuming that three distinct relations operate at the three R ratios. As is seen from Figs. 4-6 the difference between relations lies solely in the variation of intercept at N, = 1. An appropriate regression of experimental data yields that the relation between applied load and Nn is N, = 22.8 x lo6 exp

1.0

r R = 0.333

103

IO4

105

DELAY CYCLES ( ND )

Fig. 5. Relationship

between AK/I&

and Nn

(R= 0.333).

W, M. THOMAS

1022 1.0,

R = 0.100

0 103

1

a

Illal

4

I

#,,I

104

I

105

DELAY CYCLES ( ND )

Fig. 6. ~elations~p

~tw~n*AK/~~~

and ND (R = 0.100).

The reason Rice did not discover the effect of R appears to be quite simply that he only considered one value of R. There still remains one further level of analysis regarding the effect of mechanical variables upon N,. (This is the effect of the absolute value of AK upon N,.) Since multiple “single overloads” were applied to each specimen, and AK increases with crack length if load is constant [eqn (3)], the variation in N,, in any given specimen can be used to infer the effect of AK upon N,. The final relation was N, = 18.41 x 10’ exp & I.

- AK[~

+ 0.9wll.

Since tests were conducted under random environments, and on random specimens, the difference between Table 1 and eqn (5) must be due to environmental variations, material variations or experimental error. Since all tests were conducted under nonclosure conditions it is clear that the Elberian hypothesis of delay is not a general law.

4.2 Crack growth subsequent to overload Before the effect of mechanical load variables are considered per se it is essential to evaluate the accuracy of crack growth rate calculations. Since the accuracy of crack growth rate data is proportional to the accuracy of crack length measurement and inversely proportional to the number of cycles between crack length measurement (Appendix A), it is clear that for optimum crack length measurement to be achieved (6~ = 0.17 mm) it is necessary to increase (sic) the time between crack measurements in order to improve overall accuracy. This produces a difficulty because the first value of da/dN which may be estimated following an overload is three AN cycles after the overload is applied (assuming a seven point incremental polynomial method is used). This means that in order to examine the behaviour of a crack immediately following an overload application, accuracy must be sacrificed. The compromise reached in this paper is to use AN = 2000 cycles resulting in a growth rate accuracy of + 3.6 x lo-* m/cycle. It is vital to note that this accuracy level is highly conservative because it represents the case where all crack lengths suffer maximum measurement error simultaneously-a highly unlikely event.

1023

Fatigue cracks in 5083-H321 aluminium

By using the above accuracy criterion, no evidence of crack tip acceleration following an overload could be discovered. In addition any bursting phenomena (where the rate of crack growth suffers oscillations under apparently stable loading conditions) were hidden in general system noise. Of the 23 overloads which were applied, only four revealed a period of decreasing crack growth rate. The remaining 19 tests only showed periods of increasing growth rate-indicating that minimum growth rates occurred before 6000 post-overload cycles had elapsed. 4.3 Crack growth under constant amplitude loading

Since materials testing is such an expensive exercise it was decided to attempt to predict constant amplitude data from overload data. This was done by interpolating through overload induced transients (Fig. 2). The result of such manipulation is shown in Figs. 7-9. It is clear that these diagrams show several points, viz.: (a) The relation between stress intensity factor fluctuation and crack growth rate does not follow a linear law (in the log-wise sense) as predicted by Paris. (b) Apart from the 35 kN overload on the R = 0.33 test the scatter of test results is very narrow-in general it is less than one third of an order of magnitude. The reason the 35 kN test shows such deviance is not fully understood. (c) Crack growth rate increases with R ratio. Since no correlation between environment and the scatter in Figs. 7-9 was evident it is inferred that the scatter bands in these diagrams are a function of material variability. All tests were conducted in air in the environmental range of (a) dry bulb temperature 15°C to 31°C (b) relative humidity 50% to 90%.

_ _

___

50 kN overload

-------

45 kN overload

.------.-

40 kN overload

I III, 1 I lo-S1 1 5 10 ma STRESS INTENSITY FACTOR FLUCTUATION ( DK )

1 rn4

Fig. 7. da/dN .v *AK relation for 5083-H321 (R = 0.1).

I 30

1024

W. M. THOMAS 1o-6

- - ---

50 kN overload

--_-*--

45 kN overload

____ [

I i

40 kN overload

-_---

:

-----.e---- 35 kN overload

,,'

10-7 2 :: -il E

!

i

%

10-9

/ I ,ss,* I 5 STRESS INTENSITY FACTOR FLUCTUATION

I 30 ( AK ;0

MYa 4

Fig. 8. da/dN vu*AK relation for 5083-H321 (R = 0.33).

The high rate of increase in growth rates at low stress intensity factor fluctuations is clear in all constant amplitude tests. It is felt that this phenomenon does not reflect the initiation of fatigue cracks : rather that the crack growth relation is nonlinear. This inference is drawn from the fact that the anticipated threshold stress intensity factor fluctuation is 1.1 MPa -ml/2 and that precracking was conducted very carefuliy with a view to preventing such initiation effects. Since Tobler and Reed[2] have observed similar behaviour (and moreover, the results detailed in this paper compare favourably with theirs) it seems that the extraction of constant amplitude information from overload tests is a justifiable technique. 4.4 Retardation zone size

The final point which was examined on all test specimens was the retardation zone size. This is the distance that the crack front must proceed before all delay related effects disappear (CO,,). It is noted that this value is independent of, but related to, the distance that the crack propagates in N, cycles (a,), which is shown in Fig. 1. Since it has been repeatedly shown in the literature that there is a relation between retardation zone size and the size of the monotonic plastic overload, normalised retardation zone size parameters were defined such that aD

and

=

013(y/K,L~2

(6)

Fatigue cracks in 5083-H321 aluminium

-----

50 kN overload

------.

45 kN overload

.___.-_._

40 kN overload

1025

-..-..--.-- 35 kN overload

1

5 STRESS INTENSITY FACTOR FLUCTUATION

30 ( *I?)

MPa mJI

Fig. 9. da/dN - u*AK relation for 5083-H321 (R = 0.50).

The several values of c(b and aI, observed during experimentation, are shown in Table 2. These are related to stress intensity factor fluctuation, in a least squares sense, by ur, = 2.666 AK-‘.‘**

(8)

a, = 1.200 AK-“.‘80.

(9)

and

Although both of these curves have considerable scatter they are significant at the 0.9 significance level. Clearly both equations have, to all intents and purposes, equal exponents. This leads to the result that a,/a, bears a constant ratio of 0.407. This is independent of both overload level and stress ratio. The reason plastic zone size should be a function of stress intensity fluctuation has been pursued by other researchers. Gan and Weertman[ 1l] propose that secondary stresses, induced by crack closure immediately behind the crack tip, tend to carry the monotonic plastic enclave forward. This may be so, but in the above nonclosure tests it is more likely that the larger cyclic plastic zones that are formed at high stress fluctuations tend to reduce the effective size of an overload plastic zone. Matsuoka and Tanaka’s observations[8], which indicate that a,, N 3n/16, appear to disagree with the results of this paper; actually, however, they reinforce this paper’s results because the constant cubvalue determined by Matsuoka and Tanaka is only constant because a relatively small span of AK is used in their tests. To enable comparison between different materials it is recommended that work hardening effects EM23:6-P

W. M. THOMAS

1026

Table 2. Relation between normalised loading and normaiised delay crack length changes

r

1

K ,167

I.333 a=B

R = 0.10

1.500 a=B

a=_

B

;l=

1.271

&= 0.569

q=

G$

1.815

;g

ag CO.381

G$

-

Uy

0.404

if

0.876

2.325

k_

qf

_

al=1.254 R = 0.33

%=

$=

R = 0.50

1.333

_

1.667

0.173

T; =B

Zf

0.163

E1=0.082

;s

0.204

ZD= 0.218

<=

-

TB= 0.184

EB= 0.072

al= 0.822

Gl= 0.782

Ccl= 0.288

s=

1.316

$)=

zD= 0.696

<=

_

ug= -

G-p 0.015

5=

0.191

$=

0.253

al=

q)= 0.436

$=

0.442

?$= 0.479

$=

$=

_

T$,= -

_

1.933

0.261

$= 0.291

al= 0.216

q= 0.299

al=

0.211

?I=0.397

$=

0.312

j,=

1.223

E$

0.274

$= _

$=

_

s=

_

?+

-

q=

0.522

q=

0.309

il=

0.164

Zl= 0.237

$=

1.043

??= 0.783

$=

0.655

?$= 0.490

be eliminated through the use of eqn (2) and the knowledge that the monotonic strain hardening exponent for 5083-H321 is 0.18, viz. a

D

=

* f_dfc-“.‘88 1+n

2 242

(10)

and x1 = 0.407 oln.

(11)

These equations should now apply to the various tempers of 5083 that are available. All one needs to evaluate the various delay zone sizes is a knowledge of the particular temper’s monotonic strain hardening exponent. The constant exponent in eqns (8) and (9) is of vital importance because it seems to imply that the shape of a crack length versus applied load cycle curve (Fig. 1) retains the same shape regardless of the

Fatigue

cracks in 5083-H321

aluminium

1027

applied load-that is N, and w,, fully describe the delay phenomenon. Provided all growth rate relations retain similarity, then the data from Matsuoka and Tanaka[8] can be used to predict that the distance at which minimum growth rates occur (tla) is clg = 0.2ocrn

(12)

and N, = 0.20 N,, where N, is the number of cycles before minimum growth rate occurs. 5. CONCLUSION It is clear that a great deal of work is yet to be undertaken in the area of fatigue crack delay. This paper examines one particular alloy (viz. 5083-H321) and shows that although both delay cycles and delay zone size are functions of applied loading, the basic shape of the relation between crack length and applied cycles retains a characteristic shape. Although there is some scatter in all cases this is probably attributable to the environmental variations discussed in Section 4.3. By conducting testing under nonclosure conditions results have been obtained which are at variance with contemporary theories of crack delay. This naturally begs the question of “What postulates do ‘save the appearances’ (in the Grecian philosophic sense) of the experiments conducted in this paper?” Such analysis is far beyond the scope of this paper. Several aspects of this work have been concerned with optimum techniques for both testing of materials in general, and the analysis of results therefrom. Since the use of a rectangular design offers such extensive information for very little effort its value is obvious. On the other hand, although the expedient of taking crack length readings at equal intervals may not be as clear, the simplifications that can be achieved more than justify the procedure. A secondary benefit is the ability to precisely define the limits of accuracy of crack growth relations-an important aspect of any testing process. Acknowledgements-The Mechanical Engineering

author is indebted Science thesis.

to Dr. D. Radcliffe

who supervised

an earlier version

of this work as part of a

REFERENCES Ill P. Paris and F. Erdogan, A critical analysis 121R. Tobler and R. Reed, Fracture mechanics

of crack propagation laws. J. Basic. Engng 85,258 (1963). parameters for a 5083-O aluminium alloy at low temperatures.

J. Engng Mat. and Technol., 306312 (October, 1977). G. Irwin, Handbuch der Physik, Vol. 6, p. 551. Springer-Verlag, Berlin (1958). ::; J. Rice, Mechanics of crack tip deformation and extension by fatigue. ASTM STP 415,247-311(1967). in aluminium alloys. Engng c51C. Bathias and M. Vancon, Mechanisms of overload effect on fatigue crack propagation Fracture Mech. lo,409424 (1978). PI W. Elber, The significance of fatigue crack closure. ASTM STP 486,230-242 (1971). c71S. Chu and J. Li, Delayed retardation of overloading effects in impression fatigue, J. Engng Mat. Technol. 102,337-340 (I 9801. and K. Tanaka, Delayed retardation phenomenon of fatigue crack growth resulting from a single PI S. Matsuoka application of overload. Engnu Fracture Mech. 10. 515525 (1978). methods of testing and analysing fatigue crack growth rate data. Westinghouse c91S: Hudak, Development of&ndard Electric Corporation, AFML-TR-78-40 (1978). fatigue crack growth rates above 10-s m/cycle. ASTM WI ASTM E647, Standard test method for constant-load-amplitude Std. (1981). Cl11D. Gan and J. Weertman, Fatigue crack closure after overload, Engng Fracture Mech. l&155-160 (1983).

APPENDIX A. AN IMPROVED METHOD OF EVALUATING The basic incremental

polynomial

method

of crack growth

INCREMENTAL POLYNOMIAL CRACK GROWTH RATES rate evaluation

a = e,+e,N+e2N2

fits a quadratic

equation,

i.e. (Al)

to the available data and then evaluates the rate of growth by differentiating the resulting equation. The least squares process which is used to fit the quadratic equation typically uses 3,5 or 7 points ; but for the sake of brevity only the 7 point process will

1028

W. M. THOMAS

be dealt with here in detail. The evaluation of the constants e,, e, and e, is achieved through the least squares matrix equation

i = -3, -2,..., ~e,~Ni’k=~Nfa,;

3,

j==O,1,2,

i

642)

k = 0,1,2.

Consequently, if data collection is restricted to equal increments of N (i.e. AN), and it is noted that shifting the origin to N = N, does not alter the final result (i.e. da/dN) then eqn (A2) may be transformed to

i = -3, -2 ,..., 3, ~ejANj~ij+*=~ikai;

j=O,1,2,

i

(A3)

k = 0,1,2.

Now consider the special case of k = 1, whence (A4)

Further, forj even, it is clear that c ij+’ becomes zero so

i

i = -3, -2 ,..., 3.

28ANe, = c ia,; By differentiatingeqn

(Al), and producing an origin shift of N = N,again, it is seen that

tw

da/dN = e,, 01

da/dN =&Tiai;

i= -3,-2

,..., 3.

(A7)

-2,-l

,..., 2,

tA8)

Similar anaIyses for the 3 and 5 point methods yield respectively daJdN = &Cin,;

i=

L

WY

daJdN

APPEIWM

R ACCURACY OF CRACK GROWTH RATE GRAPHS

It is often fruitful to assess the scatter band which encloses a given curve. In the case of da/dN *u* AK curves such information indicates the limits of the curve’s accuracy and forms a yardstick from which to determine if phenomena, such as bursting, actually occur. Only the 7 point incremental polynomial method will be considered here in detail. Clearly from eqn (A7) the change in growth rate ~d~/dN due to changes in crack length data of Ga,is &la/dN =&ciSai;

i= -3,-2

i

,..., 3.

Since the largest error in da/dN must occur when the &a,terms maximise the summation in the latter equation, it is clear that &a_, = -&a_, = da-, = --&a, =&a, = -Sa,

032)

and 6a, = 0 produce the largest possible Gda/dN. In the special case of this paper where the accuracy of crack length measurement is constant @a) it is clear that GdafdN = &x(-l)?

1

(R3)

3&a =iz or, for the 5 and 3 point methods respectively, Gda/dN = -$

@4)

Fatigue cracks in 5083-H321 aluminium and &ta/dN = &. Hence, the relative accuracies of the 7,5 and 3 point methods are 7 : 5 : 3. (Receioed 11 February 1985)

UW