The formation and propagation of mode I branch cracks in mixed mode fatigue failure

Engineering Fracture Mechanics, Vol. 56, No. 2, pp. 213-231, 1997

Pergamon

Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0013-7944/97 $17.00 + .00

Plh S0013-7944(96)00053-7

THE FORMATION AND PROPAGATION OF MODE I BRANCH CRACKS IN MIXED MODE FATIGUE FAILURE J. TONG, f J. R. YATES and M. W. BROWN SIRIUS, Department of Mechanical and Process Engineering, University of Sheffield, PO Box 600, Sheffield, SI 4DU, U.K.

Abstract--Fatigue crack propagation behaviour under mode II and mixed mode I and II Ioadings has been investigated in a weldable structural steel BS4360 50D, using an asymmetrical four point bend arrangement. The results show that the conditions for the formation and propagation of mode I branch cracks are of decisive importance for fatigue failures under combined loading conditions. Various aspects of the subject, including fatigue thresholds, branch angle, path and branch crack growth rate, have been studied and the results are reported in this paper. While cracks have been observed to propagate in the maximum AKI direction as widely reported, complications associated with the conditions to initiate such branch cracks are far less well understood. In addition, simple tools are in demand to measure and predict branch cracking in this well tried test system. Attempts towards these objectives have been made in this work. Copyright ©1996 Elsevier Science Ltd.

NOMENCLATURE a0 ab hi, h2

K~, Ku

K~q K{~,K~I kl, kll AKIth m p.d. R W

vt, v2 A Oo

mode I precrack length branch crack length equivalent straight crack length active and reference p.d. probe spacings mode I and II stress intensity factors equivalent mode I stress intensity factor mode I and II stress intensity factors for an equivalent straight crack stress intensity factors at branch crack tip intrinsic mode I threshold ratio of AKI and AKn potential drop load ratio PrmnlPmax specimen width active and reference potential drops range of K initial branch crack angle.

1. INTRODUCTION FOR ESSENTIALLYelastic conditions, cracks under both quasi-static and fatigue loading in isotropic materials tend to grow perpendicular to the maximum principal applied tensile stress, or in fracture mechanics terms, mode I. When cracks grow in non-uniform stress fields, the direction of crack growth will normally be curved or twisted such that the local stress field at the crack tip is symmetric. This result has been an observation from numerous investigations [1-14] and is consistent with the various proposed mixed mode fracture criteria [15-17]. Presented in Fig. 1 is a typical crack path observed in a single edge cracked specimen of high strength structural steel under combined cyclic bending and shear loads; a branch crack formed at t Now at the Department of Mechanical and Manufacturing Engineering, University of Portsmouth, Anglesea Road, Portsmouth POI 3DJ, U.K. 213

J. TONG et al.

214

the precrack tip and propagated until failure. The condition for the formation and propagation of this mode I branch crack is of significant engineering importance. A mode I branch crack will not grow unless a threshold stress or stress intensity is exceeded. Furthermore, if a branch crack is indeed mode I, its crack growth rate and crack path should be predictable from existing criteria. However, since many materials are not purely linear elastic, results describing such behaviour in the literature are far from consistent. Various mixed mode fracture criteria have been proposed to predict the formation of mode I fatigue branch cracks, yet no satisfactory criterion appears to exist. Despite results pertaining to the growth of local mode I branch cracks, the field parameters used to date to correlate crack growth data yield apparently contradictory results [7-9,11]. Moreover, the prediction of crack paths has been made difficult by the uncertainty associated with the stress intensity factors for curved cracks. Since the curvature is not known a priori, the K solution must follow the crack step-by-step. It has become clear only recently [18,19] that a major complication in characterizing the fatigue crack behaviour under combined mode loading is the interference between mating crack surfaces, which makes the measured thresholds geometry-dependent. By minimizing the effect of surface abrasion, however, a lower bound may be constructed for design purposes. While tedious step-bystep finite element analyses may always be required to obtain detailed stress field information near an extending crack tip, approximate methods of reasonable accuracy may always be acceptable remedies for engineering applications. In a previous paper [14], the onset condition for mode I branch cracks in a structural steel BS4360 50D under combined mode I and II loading was presented. The branch crack fatigue threshold was considered to be the highest stress intensity factor range that failed to lead to branch crack formation, following the definition of Pook [20]. Two factors in particular were found to be critical in influencing the branch crack threshold, namely: the mean stress or R-ratio, and the mixed mode ratio m(=AKi/AKn). The initial branch angles, conversely, appear to be influenced solely by the mixed mode ratios. The purpose of this paper is to examine the threshold behaviour of the branch crack, with specific reference to the fractography of such cracks. The experimental results on crack path and growth rates of branch cracks in BS4360 50D steel are presented, and the analysis extended to characterize such mode I crack growth behaviour. In addition, the effects of mean stress and mixed mode loading on branch crack path and crack growth rate are examined and discussed.

2. E X P E R I M E N T A L P R O C E D U R E

The combined mode I and II fatigue crack growth characteristics of a weldable structural steel BS4360 50D were examined at varying load ratios. Notched four-point bend specimens were first precracked using a decreasing AK procedure under symmetrical four point bending in a computer controlled Instron servo-hydraulic testing machine with load ratio of 0.5. The rationale to adopt such an approach was detailed in ref. [32]. The precracked specimens were then tested under asymmetrical four-point bending following an increasing load sequence at a fixed AKI/AKu ratio for each test. Load ratios R of 0.1, 0.5 and 0.7 were used. Crack extension was monitored using a d.c. potential drop technique. Presented in Fig. 2 is a schematic illustration of the asymmetric four-point bend specimen geometry and the p.d. monitoring system. The intrinsic mode I threshold AKlth, which is independent of load ratio, was obtained from a series of mode I tests at different R-ratios. Based on such tests, Aglth was determined to be 4.00 MPax/-m. Full details of the experimental procedures, as well as the stress intensity factor solutions for the asymmetric four-point bend specimen, are described elsewhere [14]. In order to use p.d. technique to monitor the growth of stable branch cracks, a modified Gilbey and Pearson calibration equation [21] was developed. Two pairs of leads were placed symmetrically across the crack mouth (Fig. 2); the p.d. of vl across the leads, with spacing hi, is given by:

(rrab~=(l+k2v2 I seC\2w]

-2-+

k4v4~

(rrh,'~

24 ] / c o s h \ 2 w ] ,

(1)

Formation and propagation of mode I branch cracks

Fig. 1. A typical crack path in a four-point bending specimen under combined bending and shear loads.

215

Formation and propagation of mode I branch cracks

InputCurrent~

Specimen

217

OutputCurrent

~ht

/2't

SteelWireI

Computer

CopperWire ConstantTemperatureChamber

Fig. 2. Schematic of asymmetric four-point bending arrangement and p.d. monitoring system setup.

30

EE o m

25

O) C

o x A • + v D o

TestB092 Test B142 Test B182 TestBlt2 Test B202 Test B212 TestB192 Test B 2 2 2

"

"

" ~

/ v

/

~

J

J ×J " o

o

2O

"0 OJ u} J~ 0

15

10 10

15

20

25

30

Calculated crack length a c m m

Fig. 3. P.d. calibration, comparison of optical observation and the modified Gilbey and Pearson's method.

and for a more widely spaced pair of leads with a p.d. of v2, with spacing h2,

rrab

sec ( - ~ - )

=

( l+--f-+--~}/F. k2v~ k4v~

(2)

The constant F was found from the initial crack length value, by solving eq. (1) with the initial voltage readings for k, and then substituting into eq. (2). Lead spacings were hi = 2 mm for Vl, = 40 mm for v2. For every voltage pair (v], v2), a quadratic equation for k 2 can be solved, so that the crack length is then given in a closed form by eq. (1). The branch crack length was monitored simultaneously with a pair of travelling microscopes on both front and back surfaces of the specimen. Good agreement was found between the calculated and observed branch crack lengths (Fig. 3); based on this comparison, overall accuracy of _0. 25% was found on absolute crack lengths in the range of = 0.35 to 0.70. The branch crack paths were measured from an average value of the projections on to the X and Y axes on both front and back surfaces of the specimen.

h2

ab/W

J. TONG et al.

218 3.5[ 30

.

^

• ~..

£ v

/m

~

b "~'~"":~'~ 2 5 [ " "' "]~" "~':':~'" • i-,. '"'-.."-:~:~.~ ...... ....... :~:~. I ..... - ......

....

2.0 '-~

= 0.000

//m-0.167

""-

~g

........

/ / ~ //I,'//m / / / /

~"~./ -....~::~.:~.....:.:....

"'" ""-

"'-..~.

1.o

m = 0.426 = 0.826 m=1.716

.... "~'-~::~.'~.,:~.

/

--8

/

....""-'--'-

mode Y

0.5 0 0.2

0.4

0.6

0.8

1.0

R

Fig. 4. Branch crack threshold data correlated with the MTS criterion.

80

u)

70

'1o

v, o

"6

50

t,.Q

40

~

3o -

-

MTS caledon

20 o

lO

20

30

40

5o

60

70

80

tanI(AKI/AKI,)

Fig. 5. Initial branch crack angles correlated with the MTS criterion.

3. EXPERIMENTAL RESULTS Presented in Fig. 4 is the branch crack threshold condition, expressed as normalized equivalent mode I fatigue threshold vs R-ratio. The equivalent mode I threshold was calculated from the Maximum Tangential Stress (MTS) criterion [15]: AK~ = AK1 cos 3(00 / 2) - 3AKII cos 2(00/2) sin(00/2),

(3)

where 0o is the initial branch crack direction given by the MTS criterion, such that AKI sin 00 + AKn(3 cos 00 - 1) = 0.

(4)

It is clear that mode I fatigue threshold values are the lowest in all situations of mode I/II loading. When the mode II component increases (m -- 0) the nominal equivalent stress intensity factor ranges necessary to grow the cracks become higher. On the other hand, a significant mean stress effect on the branch crack fatigue threshold has been found. The branch crack threshold decreases as the R-ratio is increased from 0.1 to 0.7.

Formation and propagation of mode I branch cracks

219

10 (a) R = 0 . 1

o.........~m=0

E E

v . . . . . .'p A-----A ........ e x---)<

D.

m m m m

: : = =

~f

0.165 0.426 0.826 1.716

/" / // ,,~ /

,,/

j/

/

O rt

/

E

/~

..."" . j . .o'"" .7.~ "~ .jf~r

/

I.. I v

/

8r .

~"

....o ~ . - . -

0

2

0

4

6

Distance from precrack tip x

mm

10 (b) R = 0 . 5

E E

[ ] .........o m = 0 v . . . . . -~ m = 0 , 1 6 5 &------,~. m = 0 •4 2 6

,/ /"

<,....... -e. m = 0.S2S x---x

/

/

m - 1.716

/~ .o / / i ..

~/ /

.A o,."

/-

K / . . . ,"

///

~./.""

e~

E

8¢ m

0

2

4

6

8

10

Distance from precrack tip x m m 10 (c) R = 0.7

E e-

A------A m = 0.426 ........ o m = 0 . 8 2 6 x - - - - - ) < m = 1.716

8

e~

/

/ 6

/

/

// E

/2 <"

/~

x

..~%.A

/ +~J" / :,//x/1..~ f

8¢ ¢}

0 ~ I " "~".

0

.

.

.

1.5

.

.

.

.

.

.

3.0

.

.

.

.

.

4.5

.

.

.

.

.

6.0

7.5

Distance from pmcrack tip x m m

Fig 6 Observed branch crack paths at different mixed mode loading combinations. (a) R=0.1, (b) R=0.5, (c) R=0.7.

The dependence of the initial branch crack angle on AKI/AKn at R = 0.1-0.7 is presented in Fig. 5, together with the predicted results using the MTS criterion. Within the scatter of the test results, the branch angles are essentially independent of R-ratio for the range studied. The direction of initial branch crack extension appears insensitive to the load ratios.

J. T O N G et al.

220

k,,

k,,

,K:.; K,,

aeqi+l

aeqi

ao

Fig. 7. An equivalent crack defined in the analysis.

The measured branch crack paths are presented in Fig. 6. There appears to be a deviation in branch crack trajectories around m ~ 1, where the branch crack paths at m = 1.716 deviated dearly from the rest of the cases for m < 1. On the other hand, there seems to be no consequence of Rratio for the observed branch crack path (figures are omitted for simplicity), a fact consistent with previous observations [22]. For conventional mode I loading, fatigue crack growth data are usually correlated with the stress intensity factor range AKI. Under combined mode loading conditions, however, a complication arises due to the fact that cracks grow along a curved path. The stress intensity factors for a curved crack, therefore, have to be known. The knowledge of the stress intensity factors at an extending branch crack tip is of decisive importance to achieve a relation between the crack growth rate and the loading parameter. Finite element analyses are usually required, in those cases, to calculate the stress intensity factors by a step-by-step approach. It has been suggested [23] that the directions of the crack tip and the crack length projected in the direction normal to the tensile axis give the predominant influence on the stress intensity factor. As a first approximation a simplified method may be used to assume the branch crack as a projected straight crack with a small finite kink (Fig. 7), similar to the methods adopted in refs [11,24]. The stress field at an equivalent straight crack with a length of aeqi is characterized by the stress intensity factors K~, K~i, which can be determined from the stress intensity factor calibrations for the asymmetric four-point bend arrangement [25-27]. The direction of the infinitesimally small branch may be estimated from the MTS criterion:

tan

- 4Kill

4 ~ \ glli l +8.

(5)

The stress intensity factors at the assumed tip of the branch crack a r e kli, klli. klli will be assumed to drop immediately to zero, while kli may be estimated from the MTS criterion accordingly: kli = K~ii c o s 3 ( 0 i / 2 ) - 3KIWi c o s 2

(0i/2)sin (0i/2).

(6)

The analysis was then carried out on a step-by-step basis, with the next equivalent straight crack length aeqi+1 = aeqi + abi COS ( -- Oi), and the branch crack increment abi --< 0. 5 mm was taken at each step. The local mode I stress intensity factor kl was therefore estimated and the complete branch crack path was predicted. The measured branch crack growth data, d a b / d N , are plotted as a function of Akl at various mixed mode combinations in Fig. 8 and for different R ratios in Fig. 9. Pure mode I data are shown for comparison. In general, the branch crack growth data for various m ratios coincide with the pure mode I data, consistent with previous studies [24]; however, the effect of R ratios on the branch crack growth behaviour seems still to exist, but to a lesser extent compared to the mean stress influence on thresholds.

Formation and propagation of mode I branch cracks

221

(a) R = 0 . 1 104

o t04

E E

z 10= m '10

o° 10"I

Ak=

(b) R = F

x

rn = 1 . 7 1 6

o o v

m 0.826 m = 0.426 m =0.165

o

m = 0.000

MPa~/m

=:

104

104

E E

Z ~

t04

v

g

lO-7

o v

~

m = 0.426 m = 0.165 m = 0.000

v

ilk=

MPa,/m

Fig. 8. Branch crack growth behaviour, the effect of mixed mode loading. (a) R=0.1, (b) R=0.5.

4. DISCUSSION 4.1. The role of sliding mode crack closure The branch crack threshold results show that the contributions of both load ratio and mode II loading are substantial. It is believed that the observed behaviour is largely attributed to the "sliding mode crack closure" or crack surface interference, as discussed in refs [29,28] and quantified in refs [30,31] for mixed mode I and II loading conditions. Indeed, recent scanning electron microscopy revealed significant fractographic evidence of abrasion and wear associated with different loading combinations and R-ratios, as shown in Fig. 10. Generally, the crystallographic features generated during the precracking were smeared to various degrees resulting in a smooth featureless appearance for mode II dominant loading and low Rratios. While rough and irregular wear debris are evident in all cases, less severe damage of the fracture surface was observed in mode I dominant loading [Fig. 10(a) and (c)] compared to mode II loading [Fig. 10(b) and (d)]. Under the same loading combination, a rougher smeared fracture topography was found for tests at R=0.1 compared to R=0.5; this is shown in Fig. 10(a) and (b) for R=0.1 and in Fig. 10(c) and (d) for R=0.5. The facetted precrack surfaces introduced during the precracking process tend to wedge open under predominant mode II loading, generating compressive normal stresses along the crack flanks and thereby frictional shear stresses opposing slip. As a result the nominal shear stresses will be

222

J. TONG et al.

(a) ,,~/,% = o

(b)

,o° v

AK]/AK. -'="0.165

v

=-

Q

10 ¸

"0

104

E E Z

v

v

10 ¸

o

v

m -o v

104

#v!

o

R=O,1

v

R:0.5

v

10

14

R=0.1 R=0.5 r

v

v

10'

v

18

12

16

20

Ak= MPaqm

(c)

(d)

AK,/,% = 0 4 ~

,,

10.4

0

A ~ / ~ , = o.626

0 0

10"4

~'~

10-=

Z

104

104

~ ~

(11 "0

° g

10 4

10 a

o v

R=0.1 R=0.5

v R=0.5 o R=0.7

15

10

30

20

Ak I MPa4m (e)

(f)

1041 AK~/AK1, = 1.716

00 o

10-4 0 0

0 104 104

z 104

! I

o

10n

o

RffiO.1 J v RffiO.5

o

o

10

R "0.7

1o"

R=0.1 1 v. RffiO.5 o R-0.7

I

20

5

10

20

50

Ak I MPa4m

Fig. 9. Branch crack growth behaviour, the effect of mean stress. (a) m=0, (b) m=0.165, (c) m=0.426, (d) m=0.826, (e) m=1.716, (0 m = co.

Formation and propagation of mode I branch cracks

223

attenuated to various extents, depending on loading combinations and R-ratios. Extensive wear debris and smeared surfaces for mode II dominant loading or at low R-ratios indicate that the effective stress intensity values in these cases must be smaller than those in mode I dominant loading or at high R-ratios for identical nominal stress intensity values; this trend is revealed in the experimental data. Mechanically the magnitude of cyclic slip at low R-ratios is higher than at high R-ratios; accordingly, the surface wear is likely to be greater. On the other hand, any external mode I component will tend to oppose the local compressive normal stresses and therefore reduce the frictional attenuation, leading to better preserved fracture surfaces, as shown in Fig. 10. Not surprisingly, no criterion exists for the formation of mode I branch cracks in the presence of such a complication as crack surface interference. It is shown, however, that with carefully controlled precracks [32] variations in branch crack fatigue threshold values may be evaluated in response to changes in testing variables, such as R-ratio. The MTS criterion may be used to construct a lower bound for design purposes, although it may be unduly conservative if crack closure effects are not taken into account. 4.2. Prediction of initial branch crack angle and path The current results indicate that under mixed mode loading, the initial branch crack direction is dependent on mixed mode ratio; furthermore, the branch crack angle decreases with mixed mode ratio m (Fig. 5). The MTS criterion can be used to predict the initial branch crack direction within a reasonable scatter band. Such a prediction is naturally extended to cover the branch crack paths, as shown in Fig. 11 for various mixed mode ratios m and R-ratios. Again, good agreement is found between the observed and predicted paths. Deviations tend to increase when branch cracks become longer, which may be due to the attraction of the nearest free surfaces [4], or the increasing influence of the interactions between the crack tip stress field and loading point stress field [24]. The situations are slightly worse for dominant mode I loading cases, compared to dominant mode II loading cases. Further examination of Fig. 11 reveals that branch crack fronts generally experience curvature during the extension, although the precrack fronts are fairly straight (as a result of precautions taken during the precracking) [32]. Variations from ideal quasi-two-dimensional crack shape can present problems as such deviations introduce unwanted mode III displacements [20]. Generally more "twisted" branch crack fronts were observed in mode II dominant cases (m < 1) compared to mode I dominant cases (m > 1), which may facilitate branch crack propagation. The average crack paths, however, appear to be insensitive to the variation of branch crack shapes. In general, the MTS criterion can be used to predict the curved branch crack path satisfactorily, irrespective of the complications due to the branch crack curvature. 4.3. Branch crack growth rate It has been shown earlier that the correlation of branch crack growth rate using local mode I stress intensity factor may be the most appropriate among various proposed characterizing parameters [7-9,1 l, 12,22], as it reflects the physical characteristics of a naturally grown branch crack. Indeed, the current analysis confirms that branch crack growth rate is generally comparable to the mode I crack growth rate, given the same stress intensity factor range, consistent with previous results [24]. Some complications, however, arise as a result of crack surface interference, as noted above. For the present study, the initial part of the branch crack growth appears retarded under mode II dominant loading and at R = 0.1 [Fig. 8(a)]. This effect may be attributed to friction on the surfaces of mode I precracks. Nevertheless these retardations diminish gradually as the branch cracks extend. For cases of dominant mode I loading or at high R-ratios, the frictional interference becomes less effective, and a better correlation is obtained with pure mode I data [Fig. 8(b)]. Branch crack growth rate may be predicted by a Paris type of equation: dab = A (Akl) 'n , dN

(7)

224

J. T O N G

et al.

where Aki is the branch crack-tip local mode I stress intensity factor, calculated from the method described above. The experimental results show that in the Paris region, branch crack growth rates appear insensitive to the mixed mode ratios, but depend on R-ratio. A statistical study shows that for R >_ 0.5, scaling parameters A and m are: A = 1.8 × 10 -8 [mm/cycle][MPa,f-m-]-" and m = 2.94 in mixed mode cases, compared to A = 1.4 × 10 -8 [mm/cycle][MPav/-~-]-m and m = 2.84 in pure mode I; for R = 0. 1, A = 5.0 x 10 -I° [mrrdcycle][MPax/m---]-m and m = 3.83 in mixed mode cases, corresponding to A = 4.0 × 10 -9 [mm/cycle][MPa,/-~-]-'n and m = 3.11 in pure mode I. Compared to mode I cases, much retardation on the branch crack growth rate was observed for cases of m < 1 and R = 0. 1, where crack surface interaction seems to be most active. However, the branch crack growth rate results at different R-ratios tend to converge at larger crack lengths, when crack surfaces become more separated and interference is minimized. 4.4. Simulation of branch crack growth The stress intensity factors at the branch crack tip were calculated from a finite element model using the finite element program ABAQUS [33], to assess the accuracy of the approximate method. The branch crack paths observed in the experiments were used in the model. Eight node isoparametric plane strain elements were adopted with the mid-side nodes in the elements surrounding the crack tips moved to a quarter point of each element side, whereby a square root singular deformation field at the crack tip is simulated. Crack tip triangular elements were formed by collapsing one side of quadrilateral elements. Two cases were examined, namely for dominant mode I loading and for mode II. The finest elements used immediately ahead of the crack tip were 0.5% of the specimen width in length. The continued crack growth was modelled in consecutive static steps. For the mode II case eight steps were modelled, whereas ten steps were used with mode I case. At each step re-meshing was performed to maintain the crack tip triangular elements and to simulate the square root singularity of the crack tip field. The stress intensity factors KI, KII were calculated from the crack surface displacements at the nodes closest to the crack tip [34]. Considering two nodes i and j on the opposite crack surfaces, symmetrically situated with respect to the x-axis, i.e. rj = r;, qbj = -qbi = rr, the displacements relative to the branch crack tip may be written as: -l

(uy, i - Uy,j) = ~ gl

( 2 / r r i ) 1/2 ( k + 1 ) ,

(8)

l(ux, i - Ux,j ) = ~Kll (2~ri) 1/2 (k + 1), "~

(9)

2

where G

=

E w 2(1 + v)

and k = ~" 3 - 4v in plane strain (3 - v)/(1 + v) in plane stress, where (x', y') and (r, qb) refer to the local coordinate system at the crack tip. The variations of stress intensity factors at the branch crack tip with ab are shown in Fig. 12, together with the estimated values using the approximate method. The overall agreement between the two methods is surprisingly good. However, deviations tend to increase with longer branch cracks. The largest error is about 12% for mode I case and 14% for mode II case. In most of the steps, a small mode II component appears to remain up to 10% of the corresponding KI value, consistent with an earlier finite element analysis on a centre inclined crack in a tensile panel [1]. The results from the approximate analysis are encouraging in providing a good quantitative measure of branch crack trajectories. However, errors in K estimation can present problems when ki is used to correlate with crack growth rate data. Firstly an error of 10% with respect to KI may alter the crack growth rate for up to a factor of 1.5 in the present cases. Corrections of this error

Formation and propagation of mode I branch cracks

Fig. 10(a) and (b). See caption on page 226.

225

226

J. TONG et al.

Fig. 10. Fraetographic evidence associated with different loading combinations and R-ratios. For R=0.1, (a) dominant mode I loading, m=1.716, (b) dominant mode II loading, m=0.165; for R=0.5, (c) dominant mode I loading, m=1.716, (d) dominant mode II loading, m=0.165.

Formation and propagation of mode I branch cracks

227

would increase the branch crack growth rate, which may lead to a better correlation of mixed mode data with that of mode I for R=0.1 and an increase of branch crack growth rate for R=0.5, though the variation may still be within a factor of two for the crack growth rate data [35]. Secondly, if/(ix still exists when the crack extends, though only a small portion of Kh the correlation with the local KI seems to be less appropriate, and a parameter which represents the nature of mixed mode crack growth would be more desirable. The present analysis may therefore overestimate the branch crack growth rate, as no contribution of mode II component has been taken into account. Nevertheless, it is felt that as no convenient analytical solution exists, the approximate method provides an effective measure to evaluate changes in crack growth rate in response to test variables, such as R-ratio, under a given mixed mode loading combination. 4.5. The effect of R-ratio A significant influence of R-ratio can be observed from the test data at the same mixed mode ratio, as shown in Fig. 9. It is interesting to note that the influence decreases with the increase of AK, giving log (dab/dN) vs AK curves which are not parallel. Since the branch crack growth is predominantly mode I, comparisons are now attempted between the current results and the mode I data for the same material in the literature [36]. The effect of the R-ratio on the crack growth rate reported by Radon et al. [36] appears to be very similar to the results from the present tests, in that both A and m are influenced by R-ratio. It was suggested [37--40] that for steel weld metal and low alloy steel, the R-ratio dependence on growth rate behaviour was associated with a change in fracture mechanism. Griffiths et al. [37] examined the influence of mean stress in a steel weld metal; they noted that the dependence of da/dN on Kmaxwas attributed to the occurrence of fracture by void-coalescence. Ritchie and Knott [38] studied fatigue crack propagation in low alloy steel EN30A; they found that the occurrence of either void-coalescence or intergranular fracture made the rate of crack propagation mean stress dependent. On another occasion they found cleavage also led to R-ratio effect [39]. Whereas the occurrence of such "static" modes clearly leads to a strong dependence of load ratio on growth rates, this effect only predominates at high growth rates as Kmaxapproaches KIc. The work by Elber [41] provided a plasticity induced closure argument to rationalize the effect of load ratio. It was suggested that premature contact could be made possible by residual stresses due to crack tip plasticity left in the wake of an extending crack under plane stress conditions. However, it falls short of a general explanation to cover cases like the present ones, where plane strain conditions were maintained until failure. Further experimental evidence shows that roughness-induced and oxide-induced crack closure [42] may be more relevant to near-threshold fatigue crack behaviour. The R-ratio effect in steel was rationalized by oxide thickness [43]; and by the mismatch between the crack face asperities [44]. Crack closure effects are more pronounced in mixed mode situations than in mode I. It is believed that the development of roughness induced closure and possible oxide layers may be the primary causes for the discrepancies observed in the low growth rate results at different R-ratios for the current cases. The SEM studies show that intergranular fracture facets dominate a large part of branch crack fracture surfaces. The morphology generated during the fracture course may facilitate the roughness induced closure and/or prompt oxide induced closure. An observation on the same material revealed [45] extensive oxide bands on the specimen surfaces tested at low Rratio. As the crack grows longer, the mating surfaces become more separated such that the effect of closure is reduced. Eventually the crack growth rate data converge and the effect of R-ratio becomes minimized.

5. CONCLUSIONS Mixed mode I and II fatigue crack growth results in BS4360 50D steel indicate the conditions for the formation and propagation of mode I branch cracks are critical for fatigue failures of the material. The following conclusions may be made from this investigation:

228

J. TONG et al. 10 mffil.716

/

m = 0.426

m ffi 0.826

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(c) Fig. 11. Branch crack path prediction using the MTS criterion. (a) R=0.1, (b) R=0.5, (c) R=0.7.

Formation and propagation of mode I branch cracks (a)

229

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Fig. 12. Variation of stress intensity factors for branch crack, comparison of the results from

finite element analysis and the approximate method. (a) For mode II case; (b) for dominant mode I case.

(1) There exists no criterion for the formation of mode I branch cracks due to the complications of crack surface interference. However, the MTS criterion may be used to construct a lower bound for design purposes, if crack closure effects are taken into account. The significant effect of load ratio and mixed mode ratio may be rationalized by sliding mode crack closure considerations, which appear to be substantiated by evidence from scanning electron micrographs. (2) Under combined mode I and II loading, cracks immediately propagate in the maximum Kt direction when the branch crack threshold is exceeded. The initial branch angle and the branch crack path are influenced by the ratio of mixed mode loading, but insensitive to the changes of load ratio. The MTS criterion may be used to predict branch crack paths in conjunction with the approximate method, as well as the initial branch crack direction. (3) Branch crack growth rates in mixed mode loading appear to coincide with the crack growth rates in pure mode I, provided the local (crack tip) stress intensity factor range is used to characterize growth rates. Branch crack growth rates appear to be sensitive to the load ratio, even in the midrange of crack growth, but insensitive to the mixed mode ratio. Acknowledgements--This project was financially supported by the Science and Engineering Research Council of the

U.K.

The material and specimens were provided by the Welding Institute and the National Engineering Laboratory, as part of

230

J. TONG et al.

a European Structural Integrity Society investigation into multiaxial fatigue crack propagation. Helpful discussions from Prof. R. O. Ritchie of Lawrence Berkeley Laboratory, University of California, are much appreciated.

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(Received 10 May 1995)