Int. J. Rock Mech. Min. $ci. & Geomech. Abstr. Vol. 29. No. 5. pp. 469--477, 1992 0148-9062,'92 $5.00 + 0.00 Printed in Great Britain. All rights reserved Copyright ~ 1992 Pergamon Press Ltd

Fatigue Crack Growth in Brittle Sandstones G. Ltt K. H. R. MOELLEt J. A. LEWISI

This study offers critical evidence for fatigue crack propagation in brittle sandstones. The subcritical crack propagation rate in the sandstones tested demonstrated a dependence on the range of the stress intensity factor AK and was highly sensitive to small variations in AK. This phenomenon merits serious consideration in the solution of rock instability and fragmentation problems, where cyclic loading is involved. The rate of fatigue crack propagation in one of the tested sandstones was also very sensitive to variations in the stress ratio R.

INTRODUCTION

Cyclic loading is a significant loading condition in certain geotechnical situations. Results of previous studies--either in terms of strength reduction of rock specimens subjected to cyclic loads, revealed by the conventional S - N approach [1-8], or in terms of subcrit- ical crack propagation by linear elastic fracture mech- anics (LEFM) [9]--have demonstrated that brittle rocks can be strongly affected by cyclic loading. However, there have been no reports to date on actual rock failures, outside well-controlled laboratory conditions, that can be attributable to cyclic loading. This is in distinct contrast to the occurrences of metal fatigue that have caused at least half of the reported mechanical failures of metal structures [10]. It indicates that the understanding of mechanical behaviour of brittle rocks under cyclic loading is still very limited.

The phenomenon of fatigue damage in brittle rocks must be critically examined, in view of the general perception that brittle materials are not susceptible to fatigue. Mechanisms established for fatigue crack growth in ductile metals are based on dislocation activi- ties in the crack-tip region [11-14], leading to a view that brittle materials are insensitive to fatigue damage [15, 16]. Subcritical crack growth observed in brittle ceramics under cyclic loading has been interpreted as a manifestation of stress corrosion cracking, although fatigue effects, that is, enhancement of damage due to stress cycling, have been observed [15, 17, 18].

The definition of fatigue refers to the effects of cyclic loading on damage of materials [10, 19, 20], and not to the effects of environmentally assisted cracking, notably

tlnstitute of Coal Research, The University of Newcastle, University Drive, Callaghan, Newcastle, NSW 2308, Australia.

~Department of Mechanical Engineering, The University of Newcastle, University Drive, Callaghan, Newcastle, NSW 2308, Australia.

stress corrosion cracking that has been termed as static fatigue [21]. Stress corrosion may also significantly con- tribute to damage development in rocks under cyclic loading [22-24]. It follows that, to identify the existence of fatigue, any damage effects must be unambiguously correlated with stress cycling. In the tests performed during the previous studies of rocks, using either the S - N approach [1-8] or LEFM [9], the maximum stress Sma~ or the maximum stress intensity factor K,,~, in a load cycle was a variable. Consequently, the damaging effects of stress cycling were not distinguished from those of the maximum stress of the cycle. The reported strength loss or subcritical crack propagation is thus an inadequate indicator of intrinsic fatigue effects in rocks.

This paper discusses attempts to identify evidence of fatigue damage in brittle sandstones. The method of investigation is based on the LEFM theory, by exper- imentally establishing the crack propagation rate da/dN as a function of the stress intensity factor range AK, which specifies the stress cycling in the crack-tip region. The use of LEFM allows crack growth rates caused by different values of AK to be quantitatively compared within a single specimen. The influence of heterogeneities in rocks on data scatter will thus be much reduced, as compared to that associated with the S - N approach.

ROCKS TESTED AND SPECIMEN GEOMETRY

Three fluvially deposited fine-to-medium grained sandstones have been tested in the present study: the Barrier Spit sandstone from the upper Permian German Greek formation in the Bowen Basin, Queensland, the Seahampton sandstone from the Boolaroo subgroup of the upper Permian Newcastle Coal Measures, and the Wyong sandstone from the Triassic Terrigal formation of the Gosford subgroup. The Seahampton sandstone and Wyong sandstone occur in the Sydney Basin of New

469

470 LI et al.: FATIGUE CIL-~CK GROWTH

Table 1. Mineral composition and the mean grain sizes of phenoclasts in the tested sandstones

Phenoclast grains (%) Matrix and Silicious and

Lithic clay components carbonate Mean grain Rock type Quartz Feldspar fragment Muscovite (%) cement (%) size (mm)

Seahampton 9.4 38.8 11.4 21.5 2.6 0.37 sandstone

Wyong 30.8 25.2 14.2 1.0 16.6 7.4 0.17 sandstone

Barrier Spit 42.4 26.8 5.6 6.2 9.6 8.4 0.11 sandstone

South Wales. Table 1 lists the mineral composition of the tested sandstones, which was assessed by modal analysis according to the procedure suggested by the ZSRM [25].

Fatigue crack growth tests were conducted using short rod specimens with the ISRM proposed standard dimen- sions [26]. The diameters, D, of specimens prepared from the Seahampton sandstone, Wyong sandstone and Bar- rier Spit sandstone were 51.38 ___ 0.18, 56.33 + 0.24 and 60.49 + 0.14 mm, respectively. The other dimensions of the test specimens were in proportion to the specimen diameter, according to the specimen dimension specifica- tions by the ISRM [26].

Table 2 gives the values of fracture toughness Kc of the three tested sandstones, which were obtained by the ISRM suggested test procedure using the standard short rod specimen geometry [26]. The tested sandstones ex- hibit significant variations in their fracture toughness values, as evidenced by the large standard deviations pertaining to the respective sandstones (Table 2).

T H E CRACK-TIP S T R E S S INTENSITY F A C T O R C A L I B R A T I O N OF T H E S T A N D A R D S H O R T

ROD SPECIMENS

Bubsey et al. [27] have given the crack-tip stress intensity factor calibration for short rod specimens as:

P K = ~ Y*, (1)

D , /~V

where K is the crack-tip stress intensity factor in Mode I (opening mode), D and W are the specimen's diameter and length, respectively, P is the applied load and Y* is the dimensionless stress intensity factor coefficient that relates the applied load P and the specimen geometry to the resulting crack-tip stress intensity factor K. Y* is expressed as:

Y*=[~dC'~'-~°l": 'dr , a ~0A (2)

Table 2. Fracture toughness K¢ of the tested sandstones (unit: kN/m t5)

Rock type Mean SD Minimum Maximum Count

Seahampton 770 404 187 1988 23 sandstone

Wyong 1577 565 1010 2665 37 sandstone

Barrier Spit 1590 346 1240 2115 7 sandstone

where

dC' • = 1595.42 - 2939.82~ - 4198.57~ 2 + 10,900.83~ 3,

dzc (3)

(0.40 < ~ < 0.80)

In equations (2) and (3), u and C' are the normalized crack length and compliance, respectively, as defined by equation (AI) in the Appendix. Parameters ~o and uj are the dimensionless chevron notch parameters ~ = ao/W and :q = a~/W, with a0 being the initial crack length (the distance from the load line to the tip of chevron notch), and a~ the distance from the load line to the point of chevron emergence at the specimen surface [27]. With the standard short rod specimen dimensions used in this study, the chevron notch parameters at and ~0 in equation (2) are 1 and 0.33 I, respectively. The derivative dC'/d~ in equation (3) has been obtained from the previously established compliance calibration data per- taining to the standard short rod specimen geometry (see Appendix).

T E S T S O F F A T I G U E CRACK P R O P A G A T I O N

The tests were conducted in load control using an MTS model 810.03 closed-loop electro-hydraulic testing facility at a frequency of 0.5 Hz. The temperature and relative humidity in the laboratory were 20-25°C and 50-60%, respectively. The test set-up followed the ISRM suggested method for deriving fracture toughness using short rod specimens [26]: a test specimen was placed with its axis perpendicular to the loading direction in the test device, and tested in tension by applying a load (P) to open the specimen's crack mouth; a double-cantilever displacement clip gauge was used to measure the crack mouth opening displacement (CMOD) at the load line of the test specimen, and output signals were P vs CMOD curves plotted on an X - Y record.

Nine specimens from the three sandstones were tested. Figure 1 shows the experimental program; the maximum stress intensity factor K,,~ was maintained constantly throughout the testing of a specimen, while the values of AK were changed consecutively by varying the minimum stress intensity factor Kmi,. Each specific AK-value was kept constant until the induced crack propagated over a length of An. The values of stress ratio R (=Kmin/Km~) also changed with the variation of AK-values. The experimental configuration, as shown in Fig. 1, was set up primarily for studying the effects of the stress cycling

LI et al.: FATIGUE CRACK GROWTH

K Ignain Knmx = Constant < Kc R = Igmax

. . . I .

(

a

C

Tune

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

d .

'

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

, (da/a~ 2--222 I I

(da/dN)~ , ~ ', ....... X ~ I I _ I ...................... ii - ................... ............................

I I

', ~

N AN 2 ~;= AN 3 ---I= AN 4

Frequency = 0.5 Hz

Aa > 1.5 mm

Fig. I. The experimental programme.

471

in the crack-tip region, as specified by AK. By keeping the magnitude of Kin,, constant, the application of different AK-values to a single specimen can reveal the influence of AK on crack growth without the confound- ing effects of K~, .

A crack growth rate (da/dN)i, caused by a particular AK~ (Fig. l), is critically determined by the pre-specified crack length increment Aa and the corresponding cyclic number AN,.. If Aa is comparable to the mean size of phenoclasts in the sandstones, the fabric heterogeneities of the rocks would significantly affect the observed crack growth rates. Consequently, the value of Aa must be not less than the respective mean sizes of the phenoclasts in the sandstones. The minimum value of Aa was set at 1.5 mm, a dimension 4-13 times larger than the respect- ive average sizes of the phenoclast grains of the sand- stones (Table l). At growth rates in the order of l0 -8 m/cycle, usually 2 or 3 days were needed to grow the specified crack increment Aa.

In order to identify the occurrence of fatigue damage in the tested sandstones, the magnitude of K,,,, must be sufficiently below the rocks' fracture toughness value to make AK a dominant factor in control of crack growth. According to the studies on fatigue crack growth in ductile metals [10, 16, 28], fatigue cracking will be super- seded by rapid crack growth usually in excess of 10 -~ m/cycle, as K ~ approaches the specimen's fracture toughness. The significant variations in the values of the tested sandstones' fracture toughness (Table 2)

posed a difficulty in the study, as the actual fracture toughness value for a particular test specimen was unknown prior to a test. This made the specification of an appropriate value of Km~--presumably as a certain percentage of the specimen's fracture toughness--for a particular specimen impossible. This problem was solved by growing the pre-crack at a pre-determined growth rate in the order of 10-Tm/cycle; this procedure is described below.

Pre-cracking induced by cyclic loading is a standard procledure in fatigue testing [29]. In the present study, acquisition of crack growth rate data commenced at the crack length ct = 0.45, as compared with the initial crack length 0t0=0.331. A relatively large proportion of the total crack length, namely, Act = c t - ~t0 = 0 .450- 0.331 = 0.119, was thus created during the pre-cracking stage. The frequency of cyclic loading for pre-cracking was also 0.5 Hz. The minimum stress intensity factor Kmi, used in this process remained a constant, ranging from 70 to 80 kN/m LS, a small value which was needed to secure the specimens' correct position at the lowest load level during a load cycle. A constant growth rate in pre-cracking for a test specimen was maintained in the order of l0 -7m/cycle. After the termination of pre- cracking at the crack length ~t = 0.45, the Km~-value used in the pre-cracking of the specimen remained unchanged for the subsequent test. Consequently, by pre-cracking the specimen at a rate in the order of 10-7m/cycle, the corresponding maximum stress

,172 LI et al.: FATIGUE CRACK GROWTH

intensity factor that was also used for the subsequent fatigue testing was low enough to avoid fast crack growth, and thus to ensure a dominant fatigue con- dition. Creation of pre-cracks following the described procedure has been critical for the success of the fatigue tests.

CRACK GROWTH MONITORING AND A C Q U I S I T I O N OF CRACK G R O W T H RATE D A T A

The difficulties in determining crack lengths in the tested sandstones have led to an extensive experimental study aimed at the development of a technique suitable for accurate detection of crack lengths in highly hetero- geneous sandstones. A detailed discussion of this study is given elsewhere [30]. The technique developed in that study uses a parameter called the "second slope ratio" r 2 defined as follows:

tan 0 r, = - - - - s , (4)

" tan 00

where tan 00 and tan 0 are the slope of the unloading-line "ab" in the linear elastic region, and the slope of the unloading-line " 'HL" pertaining to a subsequent load cycle, respectively (Fig. 2). Ouchterlony [26] has given the rigorous procedure for finding the "unloading-line" pertaining to an unload-reload cycle in a P - C M O D record.

Determination of the crack length at any elapsed cycle number during a fatigue test is as follows: at the commencement of the test, the specimen is cycled once in the linear region to record tan 0~ (Fig. 2). The test is then interrupted at regular intervals, and the specimen is cycled once during each interruption at a much lower frequency, for instance, 0.01 Hz, as used in the present study, allowing the P - C M O D record of the load cycle to be plotted. The "second slope ratio" r 2 is then calculated, based on equation (4), from the measured unloading-line slope, tan 0, pertaining to the recorded load cycle, and the tan 0~ obtained at the beginning of the fatigue test. The normalized crack length ~e is then derived from the measured r2-value

according to the following equation given by Li et al. [301:

:e = 0.9491 - 0.9180r., + 0.3368r i, (5)

(0.307 < r, < 0.880).

The derived crack length ce is then used to determine the stress intensity factor K by using equation (1).

To determine a crack growth rate da/dN, the normal- ized crack length a derived from equation (5) is initially converted to the actual crack length " a " using the equation (AI) in the Appendix: a = ~W. The obtained actual crack length " a " is then plotted against its corresponding cyclic number N. Figure 3 shows a typical " a - N " curve obtained in the present study, that is, under a constant AK and a constant stress ratio R over the specified crack increment Aa (Fig. 1). The crack growth rate da/dN was determined as the slope of the straight line fitting the " a - N " data by regression (Fig. 3). Crack growth rate data in the present study were generated in a limited crack length range in the test specimens from ~e = 0.45 to :e = 0.55.

EXPERIMENTAL RESULTS AND DISCUSSION

Subcritical crack growth behaviour of the tested sand- stones under cyclic loading

The crack growth rates da/dN observed on specimens of Wyong sandstone (four specimens) and Seahampton sandstone (one specimen) are plotted as a function of stress intensity factor range AK, on a log-log graph as shown in Fig. 4. The linear log(da/dN) vs log(AK) data distribution with positive slopes, as depicted by Fig. 4, indicates a "da/dN~zAK" relation, and characterizes the crack growth behaviour of the two sandstones. Subcritical crack growth characterized by such a relation is undoubtedly a fatigue phenomenon; with the contri- bution of Km,~ to crack growth being fixed, increasing or decreasing AK-values having caused the crack velocities to increase or to decrease, respectively. This result thus unambiguously demonstrates that the two tested sand- stones are susceptible to fatigue damage.

P1 H a

c o

CMOD

Fig. 2. Procedures for the determination of r 2 from a P~CMOD record of the sandstone specimens.

LI et al.: FATIGUE CRACK GROWTH 473

E

t _

37

36

35

/ W

34 22000

i

i

!

i

Y i i i

24000 26000 28000

Fig. 3. A typical a - N curve recorded under a constant

i i

i

i ! i i

30000 32000

Cyclic Number

AK and constant R over the specified crack increment Aa.

Apart from AK, the stress ratio R may also affect crack growth rates. For both ductile and brittle ma- terials, a higher R-value may produce higher growth rates [9, 10, 31-33]. In the present experimental set-up, a decrease of AK leads to an increase of R (Fig. l). For all the specimens from the two sandstones, decreasing AK- values during the tests has invariably caused a reduction of crack velocities, however, without obvious effects of the associated R increase on enhancing crack propa- gation. This is depicted by the linear data distribution in Fig. 4. The influence of R on crack growth is thus negligible, and AK is shown to be the dominant factor controlling the crack growth in the Wyong and Sea- hampton sandstones. The dominant influence of AK on crack growth was further tested by subjecting one of the specimens from Wyong sandstone to sustained tension

10 .4

( A K = 0 and R =Kmi,/Km,x= 1); after 22.3hr, only 0.12 mm of crack extension was detected. As soon as full cyclic loading (R = 0.08) resumed, crack propagation at a rate of 8.2 x 10 -8 m/cycle in this specimen was im- mediately noticed, suggesting that under the present test condition, noticeable crack extension would not occur unless cyclic loading is applied.

The dominant influence of stress cycling on crack growth, observed on the specimens from Wyong and Seahampton sandstones, suggests that mechanical test data generated under non-cyclic conditions are insuffi- cient to provide a comprehensive insight into damage development in brittle rocks under cyclic loading. Design procedures and concepts ignoring fatigue phenomena may thus be seriously flawed, if the rock structures concerned are loaded cyclically. It is suggested that an

10" S.

10 . 6 .

da/dN

IO" 7

10 . 8

10 . 9

I 0 0

ta Seahampton Sandstone o Wyong Sandstone

0 r~ 0

0 0

D 0

0

. . . . . . i

1000 AK

10000

Unit: AK: kNm -1'5

da/dN: m/cycle

Fig. 4. Fatigue crack growth in Seahampton and Wyong sandstones.

474 LI et al.: FATIGUE CRACK GROWTH

10" 5

10" 6.

da/dN

10 - 7 -

10" 8

100

0

[]

Specimen Ba.31

Specimen Ba.33

Specimen Ba.39

! in %

O

O

O

O

. . . . . . i

1000 AK 1 0 0 0 0

Unit: AK: kNm "1"5

da/dN: m/cycle

Fig. 5. Fatigue crack growth in Barrier Spit sandstone.

assessment of fatigue damage should be made an essen- tial test aspect for rocks under cyclic loading. When fatigue cracking in rocks is found to be principally responsible for the crack development, design pro- cedures for the prevention of rock failure or for the enhancement of rock fragmentation should primarily consider the cyclic nature of loads and the resulting fatigue damage.

Tests on four specimens from the Barrier Spit sand- stone were conducted after completion of the experiment of the Wyong and Seahampton sandstones. Figure 5 shows the data distribution of crack growth rates for the individual specimens from the Barrier Spit sandstone. The first tested specimen, which is not shown in Fig. 5, fractured "unexpectedly" during testing under a reduced AK-value. At first, this fracture incident was thought to be the result of flaws existing in the specimen. However, subsequent tests on the other three specimens have shown that the fracture incident occurred because the crack growth behaviour of the Barrier Spit sandstone was quite different from that of the Wyong and Sea- hampton sandstones. Initially, as shown in Fig. 5, a reduction of AK led to a reduced crack velocity da/dN, indicative of a "da/dNocAK" relation, not unlike that observed on the other two sandstones. With further reduction of AK, however, the situation was reversed; the crack velocities started to increase (Fig. 5). The effect of the AK reduction on intensifying crack growth was further tested by loading Specimen Ba.31 (Fig. 5) under sustained tension (AK = 0 and R = 1). Over a crack length increment Aa = 2.0 mm, the crack propagated at

tThis crack velocity (da/dt) is equivalent to a cyclic crack velocity (da/dN) of 2.18x 10-rm/cycle, according to the relation: f . da ,'dN = da/dt, wherefis the frequency of cyclic loading. In the present study, f = 0.5 Hz.

a rate of i.09 x 10 -7 m/sect, a significantly fast velocity comparable to the growth rate (2.78 x 10-Tm/cycle) when the same specimen was fully cycled.

The crack velocity increase observed on specimens from the Barrier Spit sandstone in response to the reduced AK-values is contrary to the crack growth behaviour of the Wyong and Seahampton sandstones. The increase in crack growth rates achieved by reducing AK (Fig. 5) represents the effects of the increased stress ratio R on controlling the crack growth. In contrast to the Wyong and Seahampton sandstones that exhibited the dominant effects of AK, the Barrier Spit sandstone has demonstrated a much more complex pattern of crack growth behaviour, being affected by both AK and R variations.

In metallic materials, changes in crack growth rate associated with changes in stress ratio R can be ex- plained in terms of such phenomena as plasticity- induced fatigue crack closure [28, 31, 34] or as a result of the participation of a corrosion component of crack growth, according to Hertzberg [16], T6rr6nen [35] and E647-88a [29]. In the case of the Barrier Spit sandstone, however, the mechanisms of the observed R effects require additional special studies, since there is a funda- mental difference in the sandstones' texture and compo- sition compared with those of metals. Furthermore, the complexity of crack growth behaviour of brittle rocks under cyclic loading, suggested by the two distinctly different crack growth patterns in Figs 4 and 5 is a significant aspect that also warrants systematic investi- gation into crack growth behaviour of brittle rocks under various cyclic loading conditions.

Characteristics of fatigue crack growth in brittle rocks Fatigue crack propagation is portrayed by crack

growth curves with positive slopes, as shown in Fig. 4.

LI et al.: FATIGUE-CRACK GROWTH 475

Table 3. The m-values of different materials

Material type m-value Ref.

Ductile metals 3.0-5.0 Brock [32] Brittle ceramics 21.0-42.0 Dauskardt et al. [33] Westerly granite 11.8 Kim and Mubeen [9] Plain concrete 1 3 . 1 - 1 4 . 9 Perdikaris and Calomino [36] Plain concrete 3.1 Baluch et al. [37]

The mos t significant aspect o f fatigue cracking in the tested sandstones is the extreme sensitivity of the crack growth rates d a / d N to AK variat ions. The da ta distri- but ion o f fatigue crack growth in the tested sandstones, when not affected by the var ia t ion o f R-values, is nearly vertical (Fig. 4). The slopes o f straight lines fitting the data pertaining to the Wyong and Seahampton sand- stones (Fig. 4) are 31.0 and 14.7, respectively. Conse- quently, small var ia t ions o f AK are capable of changing crack velocities by orders o f magni tude.

The sensitivity of crack growth rates d a / d N to AK variat ions is character ized by the slope value m of a straight line that fits the l o g ( d a / d N ) - l o g ( A K ) data. Table 3 lists the m-values for ductile metals as well as for various brittle materials , obta ined f rom crack velocity da ta generated under constant R test conditions. The m-values o f brittle materials are considerably larger than those o f ductile metallic materials, with the exception of the result given by Baluch et al. [37]. Fat igue crack growth in brittle materials thus appears distinguishable f rom that in ductile metals by its extremely high sensi- tivity o f d a / d N to AK variat ions. Al though the presently obta ined slope values (14.7 and 31.0) pertain to the data (Fig. 4) generated under variable R condit ions, the above impor tan t conclusion is considered applicable to the tested sandstones, as the crack growth rates in the Wyong and Seahampton sandstones were insensitive to R variat ions.

The p h e n o m e n o n that fatigue crack growth in brittle rocks is extremely sensitive to AK variat ions merits serious considerat ion in the solution o f rock instability as well as rock f ragmenta t ion problems, where cyclic loading is involved.

CONCLUSIONS

1. Brittle sandstones are susceptible to fatigue crack- ing ana logous to that found in metallic materials. Fat igue crack growth in the tested sandstones is, however, considerably more sensitive to var ia t ions of AK-values than that in ductile metallic materials. The d ramat ic changes o f crack growth rates in response to small var ia t ions o f AK merit serious considerat ion in the solution o f rock instability and f ragmenta t ion problems.

2. The exper iments have revealed two different crack growth patterns. The s t rong dependence of crack growth rates on AK exhibited by the W y o n g and Seahampton sandstones indicates that fatigue cracking is principally responsible for the observed crack extension in these two rocks. The Barrier Spit sandstone has demons t ra ted a much more complex crack growth pat tern affected by

both AK and R variat ions. Identification o f the two different crack growth pat terns is indicative o f the complexi ty of crack growth behaviour o f rocks under cyclic loading.

3. Mechanical test da ta generated under non-cyclic condit ions are insufficient to provide a comprehensive insight into the damage deve lopment in brittle rocks under cyclic loading. Design procedures ignoring fatigue phenomena may be seriously flawed, if the rock struc- tures concerned are loaded cyclically.

Acknowledgement--The Road Traffic Authority of New South Wales and Capricornia Coal Management Pry Ltd, Queensland, provided test specimens, this is gratefully acknowledged. Mrs J. C. Pitts kindly arranged the manuscript.

Accepted for publication 31 March 1992.

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33. Dauskardt R. H., Marshall D. B. and Ritchie R. O. Cyclic fatigue-crack propagation in magnesia-partially-stabilized zirconia ceramics. J. Am. Ceram. Soc. 73, 893-903 (1990).

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

35. T6rrSnen K. Environmentally assisted fatigue crack growth of pressure vessel and piping steels in nuclear power plants. In Subcritical Crack Growth due to Fatigue, Stress Corrosion and Creep (Edited by Larsson L. H.), pp. 331-363. Elsevier, Amster- dam (1984).

36. Perdikaris P. C. and Calomino A. M. Kinetics of crack growth in plain concrete. SEM/RILEM Int. Conf. on Fracture o f Concrete and Rock, Houston, TX, pp. 96-101 (1987).

37. Baluch M. H., Qureshy A. B. and Azad A. K. Fatigue crack propagation in plain concrete. SEM/RILEM Int. Conf. on Fracture of Concrete and Rock, Houston, TX, pp. 112-119 (1987).

38. Bubsey R. T., Fisher D. M., Jones M. H. and Srawley J. E. Compliance measurements. In Experimental Techniques in Frac- ture Mechanics (Edited by Kobayashi A. S.), pp. 76-95. Society for Experimental Stress Analysis Monograph I, the Iowa State Uni- versity Press, Westport, CT, and Society for Experimental Stress Analysis, Cambridge (1973).

39. Barker L. M. Compliance calibration of a family of short rod and short bar fracture toughness specimens. Engng Fract. Mech. 17, 289-312 (1983).

40. Raju I. S. and Newman J. C. Jr. Three-dimensional finite-element analysis of chevron-notched fracture specimens. In Chevron- Notched Specimens: Testing and Stress Analysis. ASTM STP 855 pp. 32-48 (1984).

41. Ingraffea A. R., Perucchio R., Han T. Y., Gerstle W. H. and Huang Y. P. Three-dimensional finite and boundary element calibration of the short-rod specimen. In Chevron-Notched Speci- mens: Testing and Stress Analysis, ASTM STP 855, pp. 49-68 (1984).

42. Newman J. C. Jr. A review ofchevron-notched fracture specimens. In Chevron-Notched Specimens: Testing and Stress Analysis, ASTM STP 855, pp. 5-31 (1984).

A P P E N D I X

The specimen's compliance C and crack length "'a'" are usually normalized in a compliance calibration study according to equation (AI ) , so that the effects of the elastic constants of any particular test materials and of the specimen sizes are eliminated. Consequently, the crack length-compliance expression established by the compliance calibration is applicable to any specimen sizes, and test materials, provided that the specimen's geometry remains unchanged [38, 39]:

=a/W, (Al)

C" = E'CD,

where ~, and C' are the normalized crack length and specimen's compliance, respectively, "a" and C are the actual crack length and compliance, and W and D represent the specimen length and diameter. Under a plane stress condition, E" = E, whereas under plane strain, E '= E / ( I - v2). E and v are Young's modulus and the Poisson's ratio, respectively. The term ( I - v 2) has, however, been omitted [27], irrespective of the stress state, because the use of compliance calibration to establish the stress intensity factor K, introduces some uncertainty regarding the usage of the plane strain or plane stress concept [38, 40]. The state of stress throughout a specimen is neither purely plane stress, nor purely plane strain, but the induced error in the calculated K by using E' = E is probably less than 2°/, [40].

The compliance (C') - crack length (¢) data for the standard short rod specimens are obtained from four well-established compliance calibration studies [27, 39-41], and are given in Table AI. In Table AI, the experimentally obtained compliance data given by Bubsey et al. [27] and Barker [39] have been modified from the original data to account for the differences in specimen geometry and Poisson's ratio, respectively, from the analytical results of Raju and Newman [40] and Ingraffea et al. [41]. Details of the data modification are discussed in Newman [42].

Table AI. Compliance vs crack length data from the literature

BMPS BARKER RN IPHGH

C' a C' • C' " C' 0.400 96.36 0.424 108.10 0.400 93.80 0.448 110.56 0.500 129.92 0.503 139.70 0.500 127.40 0.482 123.76 0.550 156.17 0.546 158.50 0.550 151.00 0.517 138.44 0.600 185.89 0.585 183.70 0.600 180.40 0.552 155.24 0.650 223.93 0.627 209.80 0.700 270.20 0.586 174.60 0.700 284.20 0.684 263.70 0.621 197.36 0.750 359.59 0.734 327.70 0.662 230.24 0.800 509.94 0.789 446.00 0.724 299.12

0.759 354.34

BMPS: Bubsey et aL [27]. BARKER: Barker [39]. RN: Raju and Newman [40]. IPHGH: lngraffea et al. [41].

LI et al.: FATIGUE L=RACK GROWTH 477

C,

6 0 0

500

4 0 0 ¸

3 0 0 ¸

2 0 0

1 0 0

[] C'.BMPS

• C'.B o

A C'.RN •

o C./PHGH

• 0 o I~ °

~ s

E] o

o"

O 0

0

0 i t ! i

0.40 0.50 0.60 0 .70 0.80 a/W

Fig. AI. Compliance vs crack length data obtained from the literature.

The C' vs a data in Table AI from the four different sources were treated here as a single data set because of the agreement between them as evidenced by Fig. Al. A fourth-order polynomial has been found to be satisfactory, fitting the data in Fig. AI, resulting in the crack length-compliance expressions as follows:

C ' = -293.69 + 1595.42a - 1469.91a 2 - 1399.52ct3 + 2725.21a +

(0.40 < a < 0.80)

Differentiation of equation (A2) then gives equation (3).

(A2)

RMMS 29/~----C