Studies on cyclic crack path and the mixed-mode crack closure behaviour in Al alloy

Int J Fatigue 14 No 1 (1992) pp 21-29

Studies on cyclic crack path and the mixed-mode crack closure behaviour in AI alloy A.M. Abdel Mageed and R.K. Pandey

The cyclic crack path has been studied on angled centre-cracked specimens of 2024T3 AI alloy under uniaxial cyclic tensile loading at a stress ratio of 0.2. The crack closure has been measured for the mixed-mode loading at a distance of 1 mm behind the crack tip using a compliance gauge and the results are compared with mode-| closure. The fatigue crack path is predicted using the maximum tensile stress (MTS) and the strain energy density (SED) criteria and compared with the experimental path for different crack angles (13 = 15, 30, 45 and 60°). A better agreement is noted between the experimental crack path and the one based on the MTS criterion by using the effective value of AK instead of the applied value of AK. The mixed-mode fatigue life has been predicted using the MTS and the SED criteria and compared with the experimental results. The results have been discussed. Key words: mixed mode; opening mode; crack path; effective crack length; fatigue life; equivalent and effective equivalent AK

Notation at1, at2, a22 functions of ~, v and 0 in the expression aeff a0, at C, n ct,r/r

da/dN, ( da/ dN)eff K. Kn K~ov, K~tov K,eq lop_

k[, kn R R* U 130, 13

for AS half the effective crack length half-crack lengths Paris constants constants for equations of da/dN against AS' fatigue crack growth rate, effective crack growth rate mode-I and mode-II stress intensity factors mode-I and mode-II components of crack opening K--under mixed mode equivalent value of K~ov under mixed mode mean values of k] and kit stress ratio plastic zone size in mixed mode ratio of Kiefr to AK crack angle

Fatigue cracks initiated at microstructural inhomogeneities, inclusions, welding or casting defects and corrosion cracks generally have an arbitrary orientation, and a mixed mode (combined modes I and II) prevails under the uniaxial tensile loading at the tip of such cracks. The growth of fatigue cracks under a combined mode is not self-similar. The direction of crack propagation under mixed-mode loading has been predicted based on the maximum tensile stress (MTS) criterion, the strain energy density (SED) criterion, the total strain energy release rate criterion, the maximum tangential strain criterion etc. The above criteria have been employed for static loading and their applicability to fatigue loading is not taken

AK[, ~ I 1 ~eff, * /~kKIIeff ~'] q

Akl, Akii AK~op Act, AS, AS' 0, 00 tz v o". . . . 0rmin C, op,

~y

(K~op)i

crack length increment stress intensity ranges for mode-I and mode-II, respectively effective values of Ki' and Ki'l taking into account crack closure equivalent ARt for mixed mode functions of ~kKI and ~kKIl difference between the mixed mode and the opening mode Klop stress range = O'max -- O'min, mean stress strain energy density range crack initiation angle shear modulus Poisson's ratio maximum and minimum stress in a fatigue loading cycle crack opening stress and opening stress intensity factor for pure mode-I yield strength

for granted. In fact, controversial results have been produced by different investigators with regard to the application of the above criteria. 1-z The ultimate objective of the above studies is to estimate or predict the realistic life of an engineering component that is subjected to fatigue either under mixed-mode loading or the opening mode case. Furthermore, it is a well recognized fact that the crack closure parameter should also be duly incorporated in a realistic estimate of the life. The phenomenon of crack closure has been fully investigated for the opening mode case; however, to the best of our knowledge, study of crack closure under a mixed mode and

0142-1123/92/010021-09 (~) 1992 Butterworth-Heinemann Ltd Int J Fatigue Januacy 1992

21

its correlation with the opening mode is rarely available in the literature. Such a correlation would help in obtaining an appropriate estimate of the fatigue life in the presence of the mode-II component along with the mode-I component for a mixed-mode loading. The present investigation undertakes to study the mixedmode crack growth with two objectives. (1) (2)

To compare the experimental cyclic crack path and fatigue life with the predicted values based on the MTS and the SED criteria. To investigate the crack closure under a mixed mode and relate the mixed-mode crack closure parameter with the opening mode one.

Over the years, the MTSs and the SED9 criteria have gained prominence because of their predictive capability for the crack path and fatigue life, and also because they can be easily applied. Therefore, the MTS and SED criteria have been employed in the work presented and their predictive capability for the crack path and fatigue life has been examined. The investigation has been conducted in 2024-T3 AI alloy sheets of thickness 1.27 mm.

Crack growth study The chemical composition and tensile properties of the alloy are given in Table 1. The tensile properties were obtained by testing plain strip specimens in an Instron machine with a crosshead speed of 0.5 mm min-L For the study of fatigue crack growth, centre crack specimens of 180 mm width (W) and 250 mm length were employed. A central crack starter notch was made by drilling a 2 mm diameter hole at the centre. The hole was extended by a jeweller's saw at the desired angle up to a total length of 16 mm (ie 8 mm on each side) and finally by fatigue loading to a length of 20 mm (ie 10 mm on each side). Details of the preparation of the specimens are reported elsewhere. 1° A typical specimen is shown in Fig. l(a). The crack angles (13) employed in the present work are 15, 30, 45, 60 and 90°. The tests were conducted using a vibrophore machine with a frequency of 90 Hz under constant-amplitude loading. A stress ratio of R = 0.2 and a load range (Act) corresponding to 20% of the proof stress (gross) were used. The crack length was monitored using a travelling microscope. The crack initiation angle and the angles at different crack increments were measured by a toolmakers's microscope after removal of the specimens from the machine. The crack path was traced

Table 1. Chemical composition (wt. %) and tensile properties of AI alloy (2024-T3)

composition (%)

Cu 4.5

Tensile

O-ps

Mg 1.5

Crack closure measurement A clip gauge of gauge length 2.5 mm was used for detection of crack closure. The gauge consisted of a thin strip of spring steel mounted by four strain gauges each of 125 1"/resistance. The gauge was precalibrated to confirm a linear behaviour and mounted across the crack with the help of rubber bands. The closure load was obtained by stopping the machine at the selected crack length, mounting the gauge and monotonically loading and unloading the specimen to obtain the load-displacement diagram. The closure load was taken as the load level during the unloading cycle at which nonlinearity started. The closure load was found to be identical to the crack opening load obtained during the loading cycle. All the closure measurements were made at 1 mm behind the crack tip for different crack lengths and crack inclinations. A typical load-displacement curve is shown in Fig. l(b).

Results Determination of crack path using MTS and SED criteria

Experimental programme

Chemical

from the measured instantaneous angles of the crack extension. The path traced was employed to obtain the effective crack lengths and the effective (instantaneous) crack angles, 13, at different crack increments.

Mn 0.6

Si Zn Cr 0.5 0.25 0.1

The growth of a fatigue crack is taken as a number of discrete incremental steps. After each increment of crack growth, the effective crack angle changes from the original value of 13 and so the effective length of the crack also changes. For the next increment of crack growth, one has to consider the new crack length al and crack angle 131as shown in Fig. 1(c). A vectorial method can be used to characterize the effective, instantaneous crack length in the present case as described earlier. 4-6 The method provides a fair approximation to a bent crack. The experimental findings n have also shown a good agreement with the vectorial method. For a bent crack OAB (Fig. lc), from geometrical considerations, 4,s 131 = ~o + and a,=a0+

Aa sin 00 ao + Aa cos Oo

(1)

(+,,ocoS0o) ~ o s

aa

where, terms a0, 130 and 00 are the initial crack length, crack angle and crack initiation angle, respectively. According to the MTS criterion, s the crack will grow in a direction perpendicular to the largest tension. The angle of maximum tensile stress is given by the following equation, s AKI sin 0 + Agn sin 0 (3 cos 0 - 1) = 0

properties*

(MPa) (MPa) 320

440

Elongation n (%) 11 O.10

*(rps, p r o o f stress; (r u, tensile strength; n, strain hardening exponent.

22

(3)

where the ranges of stress intensity factors of mode I and II, ~ r ~ I and AKn, are defined as /~kKI ~- A O- ~

sin 2 13

AKI, = Act V ' ~ sin 13cos 13 aru

(2)

(4)

where Act is the stress range, O'max - - O'min. The crack initiation angle 00 was computed using equation (3) by applying Newton-Raphson algorithm to find the root of the equation. With a crack growth increment of 0.25 mm and knowledge of the crack extension angle the entire crack path was obtained.

Int J Fatigue January 1992

Y

I Ao Actual crack profile

12--

p

~

00

A

m

250

X

o

6 --

Po_ff //

I~ I

m

180 Ifio

°

Assumed c

3-

a

B

',-~7,

OA= G0

OB = a1

!BI=B+AB

§ 0

Dimensions(ram)

Displacement (mm)

b

C

Fig. 1 (a) Specimen geometry for mixed-mode crack growth studies. (b) Typical load-displacement curve. (c) Incremental crack growth from the inclined crack

The strain energy density range, AS in the SED criterion is expressed as: 9 AS = at1 A/e~ + 2at2 AkiAkli + a22 Ak~t

(designated Ki*v, KI*Iov for the mixed mode to distinguish mixed mode from the opening mode) are determined as follows:

(5)

where a l t, a l2 and a22 are functions of only the shear modulus, 1~, Poisson's ratio, v, and 0; and kx, kn are functions of KI and KII. The criteria for crack initiation is given by 0AS/a0 = 0 at which 0 = 00. The optimization of the function aAS/00 in the expected range was done in the present work by using the 'golden section' and the 'successive parabolic interpolation' method. 12 The experimental crack path for different crack angles and the predicted path using the MTS and the SED criteria are shown in Fig. 2.

/fi{op = O'op~

sin 2 ~3

(6a)

K~I1op= atop ~

sin 13cos 13

(6b)

Variations of the effective crack length, 2aeff, and the opening stress intensity factor, K]*p, with instantaneous crack angle, J3i.... are shown in Figs. 3(a) and (b), respectively. The effect of crack length o n g~o p and the equivalent mode-I opening K, •gv'* eq are shown in Figs 4(a) and (b), respectively, where - lop, g~o p is defined by Equation (7) following the MTS criterion,

Crack closure

K~'o~q = g~op c o s 3 ½ 00 - 3g~iop c o s 2 ½ 00 sin ½00

The crack opening (or the closure stress) ~op was obtained from the experiments described earlier. Mode-I and mode-II components of the crack opening stress intensity factors

(7)

The effective stress intensity factors ~kgI*ef f and AK~'lefffor the mixed mode may be determined using the closure parameters as,

E] SED A MTS - - Exp

.j 0

I

I

I

I

~

10

20 t8=15°

30

g0

0

~

~ 10

~

~

~

20

1

iA 30

40

30

40

[3

13=45o

a

C

A 0

b

10

20 13= 30°

30

q0

Distance (mm)

0

10

20 B = 60°

d

Distance (mm)

Fig. 2 Experimental and predicted cyclic crack paths

Int J Fatigue January 1992

23

60-

flini(deg) • 15 A 30 o 45 Q 60

O

zl

O 16

121

50-•

0

q0

r'l

O

o 45

~"

o

A

8in i Cdeg) • 15 Zl 30

i:1

A

E

-

•

rl 60

12-

A

P4

/'I r'l

A

OO

[3

O

•

30--

0

A

[]

• 8

A

30

0

0

20--

I

I

I

I

I

I

40

50

60

70

80

90

I 30

flinst (deg)

a

o

A

-

•

40

I

I

I

I

I

50

60

70

8O

90

8inst(deg)

b

Fig. 3 Variation of (a) effective crack length and (b) /~]op with instantaneous crack angle

(deg) 12 rl 13

12

A O

.¢~

%

Jl

~0 ¢o

~.

Bin i (deg) ,,15 A 30 O 45 " 60

o A

#.

O

8

o

c]

o

A

,o 4 0.

0.4

8

0.2

i 13" QO

• A

eA_

4

0.0

1

I

I

0.2

0.3

0.4

a

~ 30 o 45

_/7

0.6

0.

I

I

I

0.2

0.3

0.4

b

2 aeff/w

2aeff/w

Fig. 4 Change of K~{op, R* and K~o', q as a function of crack length, where w is the specimen width

~leff

= KImax - K~op

(8a)

and

a n d l~[/l'ii. The effective crack growth rate ( d a / d N ) e f f may be

expressed as a function of AKyq as follows: (da/dN)eff = c(AKyq)"

~Ieff

(8b)

= KIImax -- K~iop

The variations of AK*I=. and AKI*Ie. with crack length are presented in Fig. 5. As expected AKI*. increases and AK~'ieff decreases with increasing crack length.

Predicted fatigue experiments

life and comparison

with

Mq~q = AKI cos 3 ½0o - 3 M ( . cos 2 ½00 sin ½0o

following Patel and Pandey, 15 in which da/dN is expressed as da_ dN

The fatigue life using the MTS criterion is predicted in the following way. Because of the extension of the fatigue crack Mql, MqI. and M(i q change continuously. The AKiq is a function of AK1 and M ( . as follows, .3 (9)

The value of fia~lq may be obtained by taking the instantaneous crack growth direction 00 (based on the MTS criterion), AKI 24

where c and n are the Paris constants for mode I. With the known values of c, n and the AKiq, the value of (da/dN)e. may be computed and can then be used to determine the number of fatigue cycles, that is the fatigue life. The prediction using the SED approach is made in two ways: (i)

Discussion

(10)

(ii)

( 4'rr~ )./2 C \ l ~ v As

(11)

where c and n are the Paris constants from mode-I loading, and v and p~ are the Poisson's ratio and shear modulus, respectively; following Sih and Barthelmy, 6 d a / d N = c'(AS')"'

(12)

where c', n' are the constants from da/dN against AS' plots for mode-I loading and AS' is a function of Int J Fatigue

January

1992

x oO

O

12-

12

°

o o xo

8o <3

x o Z~

×

•15

•

A

0

Z~ 30

•

A

o A

Bin i (deg)

o

#.

8

•

n 60

• 15 A

q-

oJ

30

4

O 45

<3

0.1

A

ell

o0

•

o

¢

[]

l

I

I

0.2

0.3

0.4

a

A •o

a 60 X 90

0

o 45

8 ini (deg)

0 0.

2 oeff/w

o

o

f

I

0.2

0.3

o

I 04

2aefflw

b

Fig 5 Variation of mode I and mode II components of effective opening stress intensity factor with crack length, where w is the specimen width

stress amplitude A(~ as well as mean stress (~ and is

given as AS' = 2 [a11(0o)/~t Aki "t- at2(00)

(kIl Ak~

+ ]~1AklI) + a22(0o) kli Akll]

(13)

In the above equation, Akl and Akli are the stress intensity ranges as defined in Ref. 9, and/e and/eli are the mean stress intensity factors. A comparison between the experimental fatigue life and the predicted life is now made on the equal crack length basis based on (i) the MTS criterion, (ii) the SED criterion, Equation (11), and (iii) the SED criterion, Equation (12), as shown in Fig. 6 for different crack angles. The predicted fatigue life based on the MTS criterion is in better agreement 70

50 ~o

40

30

a

20

@ SED-based

70

g

~

It may be noted from Fig. 3(b) that the initial crack inclination has a marked effect on K~op for the mixed mode. As the crack angle decreases, Ki~op tends to increase. In addition K~op also increases as 13 increases. The latter is, however, an indirect effect of the increasing effective crack length, 2aeff, (Fig. 3(a)), which would subsequently increase Kmax as well as AK and enhance the size of the plastic enclave. The effect of 2aeff on K~op is shown distinctly in Fig. 4(a), where K~'op appears to be uniquely related to the crack length within a reasonable scatter irrespective of the (initial) crack angle 13. The scatter, in fact, becomes even smaller when Ki~op is replaced by ~'*°q 11, lop in Fig. 4(b). The plastic wake along the crack path as well as the instantaneous crack tip plastic zone are known to govern the K opening for mode-I loading. With increasing plastic zone (and plastic wake length), the Kop is known to increase. Hence an attempt was made to relate the ix w.eq lop with the equivalent plastic zone length for the mixed mode. The latter was estimated using the following relationship. 14 (14)

where O'y is the yield strength. The monotonic plastic zone, R*, is shown in Fig. 4(b) as a function of the effective crack length. The value of R*, in fact, either remains constant or has a mild tendency to decrease with 2ae,. Thus, w*~q 1~, lop for the mixed mode, unlike mode-I loading, does not exhibit a direct correlation with monotonic plastic zone size.

8 = 30 ° 40

30

b

mode

Crack opening stress intensity factor as a function of crack ang/e [3 and crack/ength

2~cr~

5o

20

F a t i g u e c r a c k c l o s u r e in m i x e d

R* - 3[~AK~ (1-2v) 2 + AK2I]

MTS-based prediction { proposed ) Experimental results

• 60

prediction 5

with the experimental values as compared with values that are based on other criteria. Between Equations (11) and (12), the former appears to be in better agreement with the experiments than the latter, except for 13 = 15°. In general, Equation (11) overestimates the fatigue life whereas it is underestimated by Equation (12) in most cases.

M i x e d m o d e Kzop as a function of AKz and AK~I I

I

I

I

I

I

I

4

8

12

16

20

2q

28

32

Number of c y c l e s (10 4 )

Fig 6 Comparison between experimental and predicted fatigue lives for ( a ) 13 = 15 a n d 4 5 ° a n d ( b ) f o r 13 = 3 0 a n d 6 0 °

Int J F a t i g u e J a n u a r y

1992

It is of interest to know how the presence of mode-II influences the value of Klop in angled cracks and how the mixed-mode K~'op compares with the pure mode-I opening K, (Klop) 1. Fig. 7(a) shows Ki~opas a function of AKI for different crack angles. It may be noted from the figure that for a given 25

13 (deg) • 15 /X 30 045 o 60 X90

16

12

O

A • 0o x •

--

~00

~O

F

~0

A K I ( M p a /-~)

0 O.

• o O. o

A O

Q

n

X

15

A

I

o ,0r

n

12

crack a n g l e

_

ack an ,e

5

o

18.5

I

I

I

I

8

12

16

20

a

B="'°

2q

0

b

AKI(MPa ¢~)

3

6

9

12

15

18

21

Crack path difference (~)

Fig. 7 (a) Dependence o f / ~ o p on crack angle and AKI. (b) Relation between difference in crack path and K=op for the mixed mode and the opening mode

value of AKI, K~op is higher for smaller 13 so that the opening mode (13 = 90°) shows the minium Klop value. The difference between K~op and (K1op)b however, decreases with increasing AKI. It is believed that the effect of crack angle t3 on the opening K1 has been caused by an effect of [3 on the size of the plastic wake, which in turn influences the value of g~op. The effect of plastic wake, on the other hand, would be related to the length of the actual crack path, a . . . . . t ( i e the zig-zag path traversed by the crack) for a given crack inclination 13 at the specified ~kgI level. To investigate the '[3 effect' on Kl*p, the dimensionless quantity AKi~opis shown against the dimensionless crack path difference (Aa ..... 1) in Fig. 7(b) (for crack angles of 15 and 45% where the two terms are defined as ~op

---- K~op - (Klop)t

Aa,~ = a ..... ,03) - a(90 °)

K~xop= -3.895+0.857 (~K'I) + 3.406

(15b)

(AKII//~EKI)

(16) The opening mode parameter (Kiop)i may be obtained by substituting AKn = 0 in Equation (16). A negative value of K~op indicates 'no closure'. In addition, Ki~op is also shown as a function of M¢yq/~r(1 in Fig. 8(b). K~'opmay be expressed as follows: K~'op = -10.17 + 0.889 (AKI) + 5.99

(~q/~kKi)

(17)

(15a)

Superimposed on the diagram are the values of the corresponding quantity ~ff(i. Interestingly, the percentage difference in Kiop decreases with diminishing path difference with respect to mode-I cracks and acquires a low or negligible value for almost 'nil path difference'. This correlation shows that the difference between the mixed-mode value, KI*op, and the mode-I value, Klop, for a given AKI level is apparently the result of their difference along actual crack path. As the path difference for a given initial angle 13 decreases with increasing crack growth, the AK~'op also decreases. The larger effect of the smaller crack angle on AK~op is thus the result of its greater path difference• From a comparison of the diagram for 13 = 15 and 45 °, it may be noticed that for a given path difference the AKi~op is higher for 13 = 15° as compared with 45 °. This is explained by the fact that the AKI level at a given path difference is not the same for 13 = 15 and 45 °. The value of AKI is higher for the smaller angle ([3 = 15°) than that for the larger one 03 = 45 °) at a given path difference. This is reflected in AKi~opbeing greatest for smaller values of [3. It would be of immense practical interest to predict the value of Ki~op for the mixed mode in terms of AKI and AK,.

26

An attempt has been made in Fig. 8(a) where Klop is shown as a function of AKn/AKI ratio for different levels of AK1. Using a multiple-regression technique, K~'opmay be described by the following equation:

The multiple-correlation coefficient for Equations (16) and (17) is about 0.970 and both these equations may be used to predict the value of Ki~op in the combined-mode loading.

Mixed-mode combined K-opening The K-opening parameter for the mixed mode has an additional component Ki~iop in addition to Ki'op. Hence, the use of 'equivalent' K-opening, ,v*°qlopwould perhaps be more meaningful to characterize the total effect under the mixed mode. The 1~. rr*eq lop should understandably be a function of Kiqm~x as well as of AKTq, along the lines of pure mode-I loading. Indeed, it may be noticed from Figs 9(a) and (b) eq that I~, v.eq lop is uniquely related to K~m~xand AK~q for various ix lop tO crack angles [3. It has been attempted to relate v-.~q KY%x and R as follows: K~'o~ = KIq,x (1-2.43R)

(18)

It would, however, be of wider interest from a practical • K *eq standpoint to predict lop in terms of the loading parameters AKI and AK.. A dimensionless form would be preferred as it will probably be independent of material. In Fig. 9(c), the *~ lop to the opening mode KI, (Klop)], is plotted ratio of v.eq against AKII/AKI and, interestingly, the data points for

Int J Fatigue January 1992

AKICMPa ~/m) A K I (MPa /m) 12

• J

E

/

• 9.5

~"

0 12.5 A 15.5 [] 18.5

12

x 11.0

9.5

x 11.0 12.5 A 15.5 [] 18.5 O

9 -

y

"~

g.

g.

8

n o

I

I

I

I

0.2

0.4

0.6

0.8

a

0 1.0

1.0

I

I

I

I

I

1.1

1.2

1.3

1.4

1.5

b

AKIIIAK 1

1.6

L~K~qI6K 1

Fig. 8 Variation of K~xopwith (a) AKHIAK~ and (b) /tKTq/AKl

16

16

-

12

--

eL~ •

x

o

12 E

xO

e,

B (deg)

y, A

8

•

o..n o

13 {deg)

15

•

0

w

8

X 30

30

o 45

O 45 60

60

0

I

I

I

I

I

12

16

20

24

28

a

8

b

~K~q (MPa ,I'm) 3.5

o

I

I

I

I

I

12

16

20

24

28

K~qmax(MPa -,/m)

IS (deg) 6 15 X 30

3.0

O.

o 2.5 Cry.

~

2.0

1.5

-

1.0 1.0 C

0.8

I

I

0.6

0.4

•

KII/AKI

Int J Fatigue January 1992

"[ "~t,~ 0.2

Fig. 9 Variation of/~o'~ with (a) /~q and (b) K1'~ox. (c) Opening stress intensity ratio /~o'~/(K=op)x as a function of the ratio

~K./ A KI

27

various angles fall along a single line characterizing a unique relationship. Using multiple-regression analysis, a relation is obtained as follows:

Predicted crack path based on closure

The occurrence of crack closure is believed to have a significant effect on the fatigue propagation path. Figure 11 shows the experimental path of the crack growth, which has been traced by the measured instantaneous value of 00 at different crack increments. An attempt was made to predict the crack path following the MTS criterion by using Equation (3). An obvious discrepancy between the experimental and the predicted paths is noticed. In the next step, the values of ~¢t'i and 5Kix were corrected for the closure effect to give the effective stress intensity parameters zlK*i~ffand Mt'i~laf. The predicted crack path incorporating closure effects is also shown in Fig. 11. A better agreement between the experimental and the predicted crack paths is noticed. The neglect of closure effects may lead to a considerable error in the predicted paths of the kinked cracks especially those with very small crack angles, 13.

K~o~qlKlop = 1.0645-1.737 ( ~ i i / ~ i ) + 2.368 (AK./AK,) 2

(19)

with a multiple-correlation coefficient of 0.99. This equation is indeed very useful as it can be used to predict the mixedmode opening K, ~"**q from mode-I opening data. The effect of closure on ~ is indicated by the parameter U, which is defined as, U -- (Km~ x -- Kop)/AK

(20)

Figure 10 shows the variation of U with AK] and ~¢t'iq for different crack inclinations. Most of the values of U (ie U1 and uIq) lie within a scatterband of 0.55-0.60 for 13 ranging from 15 to 60° over a ~ range of 9-22 MPa Vmm. The values of U are smaller for small 13and tend to increase with [3. A comparison with the opening mode U-values, as shown in Fig. 10(a), indicates that the U-value is higher in the opening mode. In addition, U is more sensitive to variations of ~K" in the opening mode in contrast to the mixed-mode loading.

Conclusions

1)

The SED and MTS criteria are unable to predict accurately the cyclic crack path especially under modeII dominant loading (small 13 angles).

0.7

0.7

•

O 0 13

0.6

•

A

L~ o" ~

0

OO

oAO o

0.6

•

~

fl (deg) • 15 A 30 0 45 O 60

0.4

•

I

I

4

8

I 12

a

~f ~ ' ~ ° ~ ° A A °

stress

o

o

45 13 60

o~

A

A

A

o

^~o

o

I ~ "'

o

o

A

0.5

O.q

90

J 16

I 20

0.3 24

intensity

I

I

I

I

I

4

8

12

16

20

b

AK i (MPa / m )

Fig. 10 Variation of effective

15

A 30 o

O

~o

0.5

0.3

fl (deg)

range

~

24

,',K~q (MPa / m )

ratio with (a) AK~ and (b) AK~q

~

A

a

u~

c z~ MTS O MTS (based on closure) - - Exp

V y I /

b

I

I

I

I

I

10

20

30

qO

50

Distance (mm)

I I

d

I0

20

30

40

50

Distance (mm)

Fig. 11 Predicted crack path using the MTS criterion with and without incorporation of the closure effect

28

Int J Fatigue J a n u a r y 1992

2)

3)

The fatigue life for the mixed mode based on the MTS criterion is in reasonable agreement with the experimental results. A better prediction of crack path may be obtained by incorporating closure effects into the MTS criterion. The mixed-mode value ir*eq I x lop does not exhibit a direct correlation with the monotonic plastic zone size unlike mode-I loading. The mixed mode K~op is higher than the mode-II Kiop for a given AKI level, which appears to be caused by the path difference between the mixedmode crack and the opening mode crack. Equations have been obtained to predict the value of K~op for mixed modes in terms of loading parameters such as AK1, AKII/AK! ratio, R and the mode-I Kiov. The effective stress intensity range ratio, U, is less sensitive to a variation of AK in the mixed mode in contrast to the opening mode loading.

6.

Sih, G.C. and Barthelmy, B. M. 'Mixed mode fatigue crack growth prediction' Eng Fract Mech 13 (1980) pp 439-451

7.

Murakami, Y. 'Prediction of crack propagation path (numerical analysis and experiments)' Proc of Int Conf on Analytical and Experimental Fracture Mechanics, Rome, Italy, Ed G. C. Sih (Sijthoff and Noordhoff, Alphen aan den Rijn, The Netherlands, 1981) pp 871-882 Erdogan, F. and Sih, G.C. 'On the crack extension in plates under plane loading and transverse shear' J Basic Eng D 86 (1963) pp 519-527

I 8i

9.

Sih, G.C. 'Strain energy density factor applied to mixed mode crack problems' Int J Fract 10 (1974) pp 305-321

10.

Abdel Mageed, A.M. 'Study of fatigue crack closure and crack propagation for opening and the mixed mode in AI-alloys' PhD Thesis (lIT, New Delhi, India, 1990)

11.

Chinadurai, R., Pendey, R.K. and Joshi, B.K. 'Fatigue crack propagation from an inclined crack under combined mode loading' Advances in Fracture Research, 1CF 6 Eds S.R. Valluri et al (Pergamon, Oxford, 1984) Vol 3, pp 1703-1710

Meiti, S.K. and Smith, R.A. 'Comparison of the criteria for mixed mode brittle fracture based on the preinstability stress-strain field. Part h Slit and elliptical cracks under uniaxial tensile loading' IntJ Fract21 (1983) pp 281-295

12.

Brent, R.P. 'Algorithms for minimization without derivatives' (Prentice-Hall, Englewood Cliffs, NJ, 1972)

13.

Broek, D. Elementary Engineering Fracture Mechanics (Martinus Nijhoff, The Hague, 1986)

2.

Maiti, S.K. and Smith, R.A. 'Comparison of the criteria for mixed mode brittle fracture based on the preinstability stress-strain field. Part Ih Pure shear and uniaxial compressive loading'. Int J Fract 24 (1984) pp 5-22

14.

Hills, D.S. and Ashelby, D.W. 'Combined mode fatigue crack propagation predictions using mode I data' Eng Fract Mech 14 (1979) pp 589-594

3.

Maiti, S.K. and Smith, R.A. 'Criteria for brittle fracture in biaxial tension' Eng Fract Mech 9 (1984) pp 793-804

4.

Pandey, R.K. and Petel, A.B. 'Mixed-mode fatigue crack growth under biaxial loading' Int J Fatigue 6 (1984) pp 119-123

5.

Patel, A.B. and Pandey, R.K. 'Fatigue crack growth under mixed mode loading' Fatigue Eng Mater Struct 4 (1981) pp 65-77

4)

5)

References 1.

Int J F a t i g u e J a n u a r y 1992

Authors The authors are with the Centre for Materials Science and Technology, Indian Institute of Technology, New Delhi 110016, India. Inquiries should be addressed to Professor R.K. Pandey. Received 19 September 1990; accepted in revised form 12 August 1991.

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