Fatigue Tanaka

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Enginmfng Fwctun Mechanics, 1974, Vol. 6, pp. 493-W.

Pergamon Press. Printed in Great Britain

FATIGUE CRACK PROPAGATION FROM A CRACK INCLINED

TO THE CYCLIC TENSILE AXIS

KEISUKE TANAKA

Department of Mechanical Engineering, Kyoto University, Kyoto, Japan

Ah&act-Cyclic stresses with stress ratio

R =

0.65 were applied to sheet specimens of ahuninium which

have an initial crack inclined to the tensile axis at angles of 30”, 45”, 72” or 90”. The threshold condition for the

non-propagation of the initial crack was found to be given by a quadratic form of the ranges of the stress

intensity factors of modes I and II. The direction of fatigue crack extension from the inclined crack was

roughly perpendicular to the tensile axis at stress ranges just above the threshold value for non-propagation.

On the other hand, at stress ranges 1.6 times higher than the threshold values the crack grew in the direction

of the initial crack. The rate of crack growth in the initial crack direction was found to be expressed by the

following function of stress intensity factor ranges of mode I, K,, and mode II,

K2:

dc/dN = C(K..)“, where

K,=

[K, + 8K,4]‘“. This law was derived on the basis of the fatigue crack propagation model proposed by

Weertman.

INTRODUCTION

FRACTUREechanics has been established as an important principle dealing with the

growth of fatigue cracks. Since Paris [ l] successfully correlated the rate with the stress

intensity factor, a number of investigators have reported the data on the relation

between the propagation rate and the stress intensity factor. The condition for the

non-propagation of fatigue cracks has also been expressed in terms of the threshold

value of the stress intensity factor[2,3]. Most of their experiments have been

concerned with crack growth under cyclic tensile loading of simple opening type,

mode I.

In practical situations, we sometimes meet the growth of fatigue cracks under

simultaneous application of cyclic loads of types of opening, mode I, in-plane sliding,

mode II, and anti-plane sliding, mode III. For example, a fatigue crack grows along slip

bands for a certain period after nucleation. The growth during this period is identified as

the growth under mode II or III cyclic stress, combined with mode I stress. The

combined mode growth is realized when a fatigue crack is nucleated along inclusions or

welded defects located making an angle with the axis of a tensile load. Fatigue crack

growth under applied multi-axial stress is, of course, of combined mode. In corrosion

fatigue, numerous cracks are usually formed throughout the specimen [3]. Each crack

grows under mutual interaction. The stress system at the crack tip become combined

mode in this situation.

The growth of fatigue cracks under combined mode of I and II was first studied by

Iida and Kobayashi[4]. They used a sheet of 7075T6 aluminium alloy with an initial

crack inclined to the axis of cyclic tensile loading. Their results showed that the initial

crack grew rapidly in the direction which caused the mode II component of the applied

load to go to zero. They also noticed that the presence of even a small cycling of mode

II stress increased the propagation rate significantly. Later, Roberts and Kilber[5] re-

ported the results of experiments on fatigue crack growth under in-plane, mode I,

extensional loads and transverse, mode II, bending loads. Their results indicated that in

certain cases the fatigue crack grew in a manner which did not reduce the mode II

493


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494 KEISUKE TANAKA

component of the load to zero and that the growth rate was accelerated by the mode II

component of stress applied simultaneously.

The original purpose of the present study was to establish the threshold condition of

fatigue crack growth under combined mode cycling of I and II. Like Iida and

Kobayashi, an initial crack was made in sheet material orientated at various angles to

the longitudinal direction of the plate. In the present experiments, it was found that in

certain situation the fatigue crack propagated in the direction of the initial crack. The

growth law of fatigue cracks under mode I and II stress cycling is discussed on the basis

of the theories of fatigue crack growth proposed by Weertman [6] and Lardner [7], and

a new propagation law under combined mode stressing is proposed. The threshold

condition for the non-propagation of fatigue cracks is discussed comparing the data

with the theories of maximum tangential stress criterion [8] and of strain energy density

criterion [9].

EXPERIMENTAL PROCEDURE

Specimens were cut out from a commercially pure aluminium plates of 3*2mm

(l/8 in.) thickness. The preliminary specimens of a wide plate was fatigued after getting

a mechanical slit with band saw and razer blade perpendicular to the tensile axis. A

fatigue crack was grown to a length of about l-5 mm from each end of the mechanical

slit. Then fatigue specimens of final shapes were cut out from cracked preliminary

specimens in the manner that the initial slit and crack were orientated at angles of 30”,

45”, 72” and 90” with respect to the longitudinal direction of the final specimen. The final

dimensions of the fatigue specimen is shown in Fig. 1, where the rolling direction of the

sheet is the direction of the initial slit and crack. For each inclined angle, (Y, he initial

slit-crack lengths adopted are

CYdeg) = 90,72,45,30

2c(mm) = 21 and 10 11,15,21.

All the specimens were annealed at 270°C for 2 hr before subjecting them to fatigue

testing. Table 1 gives the mechanical properties of a strip which was prepared following

the same heat treatment as in the case of fatigue specimen preparation. As can be seen

in the table, the mechanical properties of the material is almost isotropic.

The fatigue tests were conducted in Losenhausenwerk operated at a rate of

.5

saw-cut

Fatigue

-- -_

c crock

3-T

-Ql

y- 1.5

Thickness f =32

Fig. 1.Shapes and dimensions of fatigue specimen (dimension mm).


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Fatigue crack propagation

495

Table 1. Mechanical properties of aluminium sheet.

Rolling direction

Yield stress

Tensile True

Reduction

vs

(0.2% off-set) strength strength of area

stress axis

(kg/mm’)

(kg/mm2) hdmm2)

(%I

Parallel 10.0

10.5 27 77

45” 9.6

9.7 25 78

Perpendicular 10.2

10.4 24 72

1000 cpm. The ratio R of the minimum stress omin o the maximum stress a,, was kept

at a constant value of 0.65 for all fatigue tests. The growth of fatigue cracks was

monitored with a travelling microscope attached to the fatigue testing machine.

NON-PROPAGATION CONDITION AND CRACK GROWTH DIRECTION

Several specimens were fatigued under stress ranges near the situation of the

non-propagation of an initial crack which was estimated from the data of preparatory

experiments. The results are summarized in Table 2. In some experiments, the stress

range was raised step-wise when the initial crack was detected not to grow with a

microscope after applying 3

x

lo5 stress cycles. Since the limit of detection of crack

growth length is about 0.05 mm, a growth rate higher than l-7 x lo-‘mm/cycle can be

detected by this method.

The elastic stress near the tip of a crack which is inclined to the tensile axis is

characterized as mixed mode of opening, mode I, and in-plane sliding, mode II. The

stress intensity factors k, and

kz

of modes I and II due to a tensile stress u are given by

k, =

k

sin’cu (la)

k = kA sina COW

(lb)

with

where c is the half crack length and (Y is the angle between the crack and the axis of

Table 2. Experimental results for determining the threshold condition of fatigue crack

propagation.

Initial

Crack

crack Stress intensity factor

Crack growth growth

Specimen

angle

KI

K,

rate, dcldN angle

No.

(deg)

(kg/z”“) (“)

(“)

(mm/cycle)

Udeg)

4

90

5 90

7

90

7

90

10 72

10 72

13 45

13 45

17 30

17 30

17 30

19 30

19 30

6.52

5.54

6.00

6.38

5.93

6.29

7.24

764

9.49

997

10.9

9.55

10.0

6.52 0

5.8 x lo-’

5.54 0 Nogrowth

6.00 0

Nogrowth

6.38 0

7.9 x lo-’

5.35

1.74 Nogrowth

5.67

1.85 7.4 x lo-’

3.62

3.62 Nogrowth

3.82

3.82

7.2 x lo-’

2.37

4.11

Nogrowth

2.49

4.32 No growth

2,74

4.74

1.4 x lo+

2.39

4.14 Nogrowth

2.50

4.33 2-l x lo-’

0

-

-

0

-

-28

-

-49

-

-23

-

-52

EFM. Vol. 6 No. 3-F


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4%

KEISUKE TANAKA

tension[4, lo] [Fig. 2(a)]. The factor Z is a correction coefficient depending on the

specimen width W. The value of Z was given by Brown and Srawley [I l] as

z= 1-0*1(2c/w)+(2c/w)*

(2b)

for a range of (2c /

W) = 0

0.6. The stress intensifications corresponding to the stress

range Au = u-- ami, are calculated by substituting Au for (T in equation (2). The

ranges of stress intensity factors thus computed are denoted by K, and K, in Table 2.

The specimens in which the crack extended during the period of 3-Ox 10’ cycles were

removed after 3-Ox 10’ cycles from the testing machine and the distance of crack

extension and the average direction & of crack extension from the initial crack were

measured with another microscope at seventy magnification. The values of the rate

dc/dN and angle 8,, obtained are tabulated in the right hand two columns, where 8,, is

the angle of the growth direction with respect to the initial crack direction measured

counterclockwise.

Erdogan and Sih[%] investigated the extension of fracture in a sheet of brittle

material with an inclined crack as in the present experiments, and they found that the

initial crack extended in the direction of the maximum tangential stress. The singular

parts of elastic stresses u,, ue and u

r. near the crack tip under opening and in-plane

sliding loads are

a;. =-&rcos$k,(l+sin’$+$sin0-2k,tan:]

ue = &rcos f k, cos2f-- ik2 sin0

[ I

T* = -&co,

k,

sin0 +

k2(3

ine - l)]

(34

(3b)

where (r, 0) are polar coordinates at the crack tip [Fig. 2(b)]. The direction & of crack

extension which is assumed to be the direction of maximum tangential stress is

obtained by differentiating ue V% with respect to 8 and setting the derivative to zero:

kl

sin&, +

k2(3 os& - 1) = 0

(4)

Substituting equations (la) and (lb) in equation (4) gives

sir& + (3 cOseo - 1) cota = 0

(5)

Fig. 2. The stress state near the tip of a crack inclined to the tensile axis.


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Fatiguecrack propagation

497

The condition of fracture at a constant tangential stress is expressed as

cos (0,/2)[k, cos2(&/2) - (3/2)k2 sin&] = const

(6)

They concluded that the condition of fracture given by equation (6) should be regarded

as a practical design criterion which gave a conservative value. Later, Sih[9] introduced

the concept of strain energy density to explain the fracture condition under mixed

mode. The strain energy density A U/A V has a singularity of inverse r near the crack

tip:

AUIAV = S/r

The energy density factor S in equation (7) is given by

intensity factors k, and k2:

S =

aIlk

2

+ 2a12klk2+ a22k22

with

(7)

a quadratic form of stress

(8)

all = (1/16rG)[(3-4~ - cos e)(l+ cos e ]

@a

al2 = (1/16nG)2 sine[cose -

1 -

2~)]

(8b)

a22 = (1/16&)[4(1- v) l - cos 0) + 1 + cos 8)(3 cos 8 - l)]

(8~)

where G is shear modulus and Y is Poisson’s ratio. Equation (8) is rewritten from

equations (la, b) as

S =

kAz[al l

sin* (Y+

2a12

sin’ CYos Q + a22sin2 QL os’ a]

(9)

The angle tIOof crack growth direction is given by the direction of the minimum S

value:

2(1-2v)sin($-2~~)-2sin[2(&-a)]-sin28~=0

(10)

The onset of fracture is determined when the minimum value of S reaches a critical

value.

The experimental data on growth direction given in Table 2 are plotted by open

circles in Fig. 3. The direction of fracture predicted by maximum stress criterion,

equation (4), and by strain energy density criterion, equation (lo), are indicated by the

- Stmin energy density

criteri on (c =1/3)

0

20 40

60 80

lnitiol

crock angle , (1. deg

Fig.3. The angle of crack extensionfrom initialcrack.


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498

KEISUKE TANAKA

dashed and solid lines, respectively. The value of Poisson’s ratio is taken as l/3 for

calculation in the latter case. As can be seen from the figure, the measured value

increases with decreasing angle (Yfollowing the solid line down to 45”. For the case of

30”, the observed direction is seen to be much smaller than the value estimated based on

the solid or dashed line, especially at the higher rate the point approaches to zero.

As to apply the criteria of maximum stress and energy density to the non-

propagation of fatigue cracks, we assume that the equations which are derived by

substituting the ranges of stress intensity factors, K1 and IL, for stress intensity

factors,

k,

and

k2,

n equations (6) and (9) for 8 = & give the conditions. The effect of

the stress intensity factors at the maximum applied load on the crack growth is

neglected, considering the data reported by Frost [ 121which showed a rather insensitive

character of the non-propagation condition to the mean stress in pulsating tensile

fatigue. The corresponding equations for non-propagation condition are

cos (&/2)[K, cos* (&/2) - (3/2)K, sin &] = T,

(11)

a,,K,‘+ 2a12K1K2+ az2K2’ = SC

(12)

where

a,,, a,2

and

az2 are

values obtained by substituting 8 = & in equations @a, b, c).

The values of constants, T, and SC,of equations (11) and (12) are now evaluated from

the threshold value of

K,

for the case of perpendicular initial crack. The conditions of

equations (11) and (12) thus determined are indicated by the dashed and solid curves in

Fig. 4. The experimentally measured values for the non-propagation condition corres-

ponding to angles CY 30”, 45” and 72” lie between two curves, more precisely speaking,

nearer the curve of strain energy density criterion. This means that equation (11) of

maximum stress criterion provides a conservative law for mixed mode loading while

equation (12) for strain energy density criterion gives a dangerous law for design

purposes. We assume that the condition for the non-propagation of fatigue cracks is

given in a positive quadratic form of K, and K2 similar to equation (12) such as

A,,K,‘ + 2AnKI Kz + AzzK,Z =

1

(13)

and determine the coefficients,

AlI, Al2

and Az2, from experimentally obtained

-.- Experimental curve ;

ell ipse approximation

2, 2-

-Strain energy

.zj

density criterion b -t/31

--- Maximum stress

w

0 I 2 3

4

5

6 7

Stress intemi ty

factor of mode I,K,. I,~/~ ’

Fig. 4. The condition for the non-propagation of fatigue

cracks under cyclic stressing of

combined

mode I and II.


7/16

Fig. 5. Photograph of fractured specimen with (I = 30”and Au = 2.9 kg/mm’.

[Facingpage 498


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threshold values of

equation becomes


499

K, and Kz for angles (Y= 90”, 72” and 45” given in Table 2. The

0.0262K: + OXKl81K~Kz 0.0381K22 = 1

(14)

The dot-dash line in Fig. 4 corresponds to the above equation. It should be noticed that

the measured threshold value for (Y= 30” falls on this line. Although we need at least

three experimental points of threshold condition to determine the coefficients of

equation (13), the condition for the non-propagation of fatigue cracks under combined

mode cycling is expressed by equation (13) more precisely than by equation (11) or (12).

CRACK PROPAGATION LAW UNDER CYCLIC TENSILE AND TRANSVERSE

SHEAR LOADING

For the cases of inclined angles (Y= 45” and 30”, a fatigue crack was found to extend

in the direction of the initial crack when the applied stress range was about 1.6 times

larger than the threshold value. Figure 5 shows a photograph of the specimen with

(Y= 30” fatigued under stress range Au = 2.9 kg/mm*. The arrows in the figure indicate

the ends of the initial crack. In the experiments the crack length was measured along

the growth direction. The rate dc/dN of fatigue crack growth is correlated to the stress

intensity factor of mode I, K1, or mode II, K2, in Fig. 6. The rate becomes much higher

when the mode I component of cycliu stress is accompanied by the Mode II component

as noticed comparing rates at the same K, value. The line in the figure is the

experimental curve obtained by ffitting two straight line segments through the

experimental points for the case of perpendicular initial crack, i.e. cy = 90”. The

equation of the line at higher rates is

dc /dN = 1.8 x 10-‘°K,“~4

(15)

Weertman [6] proposed a theory of fatigue crack propagation on the basis of a model

with a rigid plastic strip extending collinearly to the crack which was given by Bilby,

Cottrell and Swinden (BCS model)[13]. He assumed that a fatigue crack grows when

the sum of the absolute values of the displacement in a strip reaches a critical value a’,.

When a sheet specimen with a straight crack of length 2c is subjected to simple opening

or in-plane shear stress cycling, two types of plastic deformation zones are formed

16’

5xlc7 Id6 5x10* lo-”

5x1d5

Fatigue crock pmpogoiicn rate, d IN, mm/cycle

Fig. 6. Relation between crack propagation rate and stress intensity factor of mode I, K,, or of

mode II, K2.


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SO

TANAKA

10) Plastic yie d strip s

oheod of crock tip.

1 b) Applied cyclic

stress.

Fig. 7. BCS analysis of plastic yield under cyclic stress. (a) Plastic yield strips ahead of the

crack tip. (b) Applied cyclic stress.

ahead of the crack tip as shown in Fig. 7(a), where w and o * are monotonic and reverse

yield zones respectively. For small scale yielding cases, the rate of crack growth in the

crack direction is given by (see Appendix)

dc/dN = (210,) r* Ir$(t’, o*)l dt’

(16)

= [(l - v)?r/3 x 43G@,][K4/Y3]

(17)

where

+(t, w*) = the reverse component of plastic displacement at a point t from

the crack tip

G = shear modulous

Y= Poisson’s ratio

K = the range of stress intensity factor for mode I or II deformation

Y = yield stress of material in the strip.

In order to apply this model to combined mode growth, we assume that the plastic

deformations due to cyclic tension (mode I) and transverse shear (mode II) are not

interactive. Then the material ahead of the crack tip is subjected to mode I dis-

placement, &(t, UT), and mode II displacement, el(t, UT). These displacements are

calculated from the same equation [equation (37) in Appendix] except that the yield

stress Y is replaced by the yield stresses Y, and YZ for mode I and II deformations,

respectively. Further we assume that the displacement in the integrant of equation (16),

called effective displacement and denoted by c&, is the sum of two displacements +1

and & for combined mode cases:

&ff = l&I + 144

18)

According to these assumptions, the rate becomes a function of stress intensity factors

of mode I, K, , and of mode II, K2 , as

dc/dN =

[ l -

u)~/3 x 43G@,

J [ K :/Yl ’ ) + K ;/Y;)]

The value of yield stress Y2 may be assumed to be a half of the Y1 value. Then the

growth law is

dc/dN = C1 K,a )“ ‘ l

(19)


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502

KEISUKE TANAKA

E

0 . =45”

A .=30”

I

‘G

E d

I I

I

I

5xlo-7 IO”

5xd Id5

5xlo-5

Fatigue

Crack plcpogarcn mte, dc~~.mm/cyc~e

Fig. 8. Relation between crack propagation rate and effective stress intensity factor under cyclic

tension transverse shear.

which results in fracture, is the sum of displacements of two modes, and (iii) the yield

stress for shear deformation is one half of that in tension. When the anti-plane shear

mode (mode III) cycling is coupled with the above two mode loadings, the effective

stress intensity factors K,, in equation (20) will be

I&* = [K,4 + 8K,4 + 8K,4/( 1 - V)]“’

where v is Poisson’s ratio, and K1, Kt and K, are the ranges of stress intensity factors

for mode I, II and III deformations, respectively. The exponent of equation (26) is given

as 4 in Weertman’s analysis, while the experimental value was 4.4. This slight

disagreement might come from the departure of plastic deformation from small scale

yielding. It should be remarked that the above extension of the propagation law,

originally proposed for single mode deformation cases, to the case of combined made

growth is valid as far as the further extension of the crack takes place in the direction of

the existing crack.

The direction of extension of an initial crack was roughly perpendicular to the axis

of the applied tensile load under stress ranges just above the threshold values. This

results on growth direction agrees with the result reported by Iida and Kobayashi141,

while at stress ranges 1.6 times larger than the threshold ranges it was collinear with the

initial crack. The region of plastic deformation, detected as a depressed surface zone,

was observed to be of the form of a strip extending collinearly with the crack at higher

stress ranges. The plastic deformation tends to concentrate along a strip ahead of the

crack tip as the applied stress increases. This concentration may be caused by the

prevailing plane stress state under higher stresses. The reverse and monotonic yield

zone sizes, wt and 02, were calculated using small scale yielding equations [equations

(29) and (36) in Appendix]:

ot = Kz2132Y2=

132 = K:,A Y,’

where k2_ is the stress intensity factor of mode II at the maximum stress. Let the yield

stress for shear, Y,, be half the tensile yield stress Y, = 9.9 kg/mm’.

The

sizes


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calculated for an initial crack are;

503

Inclined Applied stress 0: 02

angle

intensity factor

(mm) (mm)

45” KW 0.05 1.7

45 1*6K,, 0.14 4.6

30” K2P 0.07 2.5

30 1.6K,, 0.20 6.2

where KZF is the threshold stress intensity factors of mode II for angles OL 45” and 30”,

and 1*6KZF is l-6 times larger than that. When we compare plastic zone sites of and w

with the specimen thickness t = 3.2 mm, the monotonic yield zone size is found to be

larger than the specimen thickness for the cases of the higher stress intensity factors,

while those for lower stresses are smaller than the thickness. Therefore, in the former

cases plane stress deformation ahead of the crack tip may happen in monotonic yielding

at maximum loads. These results seems to suggest that the transition of the crack

growth direction is realized when the monotonic yield zone size is larger than the

specimen thickness. Further studies are needed to clarify the condition of the transition

of the crack growth direction.

The condition for the non-propagation of an initial crack was expressed by a

positive definite quadratic form of stress intensity factors K, and K,, equation (13),

similar to the condition for brittle fracture extension derived based on energy density

criterion proposed by Sih[9]. However, the coefficients in the quadratic form deter-

mined by the present experimental results are different from those calculated from

energy density criterion. The condition of equation (13) may be explained on the basis

of the nucleation model given by Weertman [6] (see Appendix) as follows: He assumed

that an existing crack extends when the total displacement a* accumulated at the crack

tip reaches a critical value Cp,.The number of cycles NC required for the crack to grow

is given by the following equation in small scale yielding cases:

a,, = a,” = 2N,(f$(O, o*)]

where Cp(0,w*) is the reverse component of displacement at the crack tip. It seems

reasonable to assume that no nucleation, i.e. no crack extension, takes place when the

reverse displacement +(O, o*) is lower than a certain value c$*. This value means the

disability of continuum mechanics treatment of minute plastic deformation, so it may

be about an order of the lattice spacing or several orders higher than that?. This

criterion provides the condition:

c$* =

[ l -

v)/4GY]KF”

(26)

where Kp is the threshold value of K = (am_ - CT,,,&& . The equation holds for

plastic deformation strip extending collinear with the crack line. If we regard the yield

strip as formed in a certain direction under general in-plane loadings, the condition of a

critical displacement at the crack tip will be expressed in a quadratic form of K, and

K2

like equation (13). If we assume that the yield strip is formed in the direction which

yields the maximum value of the crack tip displacement, the ratio of the coefficients in

equation (13) can be calculated mathematically. Then we need only one experimental

data of the threshold value in order to determine the exact form of equation (13).

tsubstitutingG = 2 4 0 0 g/mm’, Y = l/3, Kp = K , , = 6 . 2kg/mm”* and Y, = 9.9 kg/mm’ of the present ex-

periments in equation (26), one obtains 2 1O’A for the value of c$*.


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504

KEISUKE TANAKA

CONCLUSIONS

Specimens of aluminium plate with an initial crack inclined by various angles, 30”,

45”, 72” and QO“, o the tensile axis were fatigued under pulsating cyclic tensile loads

with the stress ratio R = O-65. The conclusions of the present study are summarized as

follows:

(1) The threshold condition for the non-propagation of the initial crack under cyclic

tensile (mode I) and transverse shear (mode II) loadings was given by a quadratic form

of the ranges of stress intensity factors of mode I, K,, and of mode II, &, as

A,&,2 + 2A,zK,Kz + A,&&’ = 1

where

A,, =

O-0262, 2A,* = 0.0081 and A,, = O-0381. The equation derived from max-

imum stress criterion provided a conservative law, while that from strain energy

density criterion a dangerous law for design purposes, if the critical value of each

equation is determined from the result of the perpendicular crack case.

(2) The direction of fatigue crack extension from the initial crack was roughly

perpendicular to the applied tensile axis at stress ranges just above the threshold

values. On the other hand, the crack grew in the direction of the initial crack under

stress ranges 1.6 times larger than the threshold values. This collinear crack growth

may be explained by the predominance of plane stress deformation at higher stress

ranges.

(3) The law of collinear crack growth was derived using Weertman’s model under

further assumptions: (i) plastic displacements due to yield under cyclic tension and

transverse shear are not interactive, (ii) the effective displacement, the total sum of

which determines the fracture process,

is the sum of the absolute values of

displacements of two modes of deformation, and (iii) the yield stress for shear is one

half of the yield stress in tension. The propagation law is given in the form:

de /drV = C(K,)m

where

K, = (K1“ + 8K24)“4. he

exponent m was found to be 4.4 in the experiment.

Acknowledgements-This work was carried out while the author was on leave at the Department of

Engineering, Cambridge University, and at the Department of the Theory of Materials, ShefIietd University.

The experimentaI part was conducted at the former place. A grateful acknowledgement is presented to Dr. K.

J. Miller for his kind permission to use experimental facihties at the Department and for his en~~ment

during the progress of the study. The author appreciates the assistance and suggestions offered by Mr. R. J.

Brand in the procedure of experiments. He wishes to express his gratitude for discussions with, and criticism

given by the members of Professor B. A. Bilby’s and Dr. Miller’s research groups. Especially, he is indebted

to Mr. L. R. F. Rose, Sheffield University, who read the manuscript and helpfuUy commented upon it.

REFERENCES

[l] P. C. Paris and F. Erdogan, A critical analysis of crack propagation laws. J. has. Engng,

Trans.

ASME,

Ser. D 85, 528-533 (1%3).

[2] H. W. Liu, Fatigue crack propalgation and the stress and strains in the vicinity of a crack. Appl. Mat.

Res. 2, 229-237 (1964).

[3] H. Kitagawa, A fracture mechanics approach to ordinary corrosion fatigue of ~notch~ steel specimens.

Presented at Conf. Corrosion

Fatigue,

Manchester (1973).

[4] S. Iida and A. S. Kobayashi, Crack propagation rate in 7075T6 plates under cyclic tensile and transverse

shear loadings. .I.

bas. Engng, Tr ans. ASME, Ser. D

91,764-769 (1%9).

[S] R. Roberts and J. J. Kilber, Mode II fatigue crack propagation. J. bus. Engng, Trans. ASME Ser. D

[6]

J. Weertman, Rate of growth of fatigue cracks calculated from the theory of infiniterimal dislocations

distributed on a plane.

in t. J. Fracture Mech. 2, 460-467

(1966).

[7] R. W. Lardner, A dislocation model for fatigue crack growth in metals. Phil. M sg. 17, 71-82 (196t3).


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505

[S] F. Erdogan and G. C. Sih, On the crack extension in plates under plane bending and transverse shear. J.

bas. Engng Trans. ASME, Ser. D 85, 519-527 (1%3).

[9] G. C. Sih, A special theory of crack propagation. Methods of

Analysis

and Solutions of

Crack

Problems.

(edited by G. C. Sih) XXII-XLV, Noordhoff (1973).

[lo] L. P. Pook, The effect of crack angle on fracture toughness. Engng Fracture Mech. 3,205-218 (1971).

[ll] W. F. Brown and J. E. Srawley, Plane strain crack toughness testing of high strength metallic materials.

ASTM Speci . Tech. Pub. No. 410 (1966).

[12] N. E. Frost and D. S. Dugdale, Fatigue tests on notched mild steel plates with measurements of fatigue

cracks. J. Mech. Phys. Solids, 5, 182-192 (1957).

[13] B. A. Bilby, A. H. Cottrell and K. H. Swinden, The spread of plasticity from a notch. Proc. Roy.

Sot. A 272, 304-314 (1%3).

[14] R. 0. Ritchie and J. F. Knott, Brittle cracking processes during fatigue crack propagation. 3rd ht . Congr.

Fracture, Munich, V-434/A (1973).

[15] P. T. Heald, T. C. Lindley and C. E. Richards, The influence of stress intensity and microstructure on

fatigue crack propagation in a 1% carbon steel. Mat. Sci. Engng 10, 235-240 (1972).

[16] N. I. Muskhelishvili, Singular integral equations. Translated by J. M. Radok Groningen (1953).

[17j B. A. Bilby and P. T. Heald, Crack growth in notch fatigue. Proc.

Roy . Sot .

A 305,429-439 (1968).

[18] J. R. Rice, Plastic yieldingat a crack tip. Proc. 1st. Inter. Congr. Fracture, Sendai, Vol. I, 283-308 (1%5).

(Receiued 19October 1973)

APPENDIX

WEERTMAN’S AND LARDNER’S ANALYSES OF

FATIGUE CRACK EXTENSION USING BCS MODEL

The theory of continuously distributed dislocations was used by Bilby, Cottrell and

Swinden for an approximate analysis of plastic yielding ahead of the crack tip under a

monotonic load (BCS model) [13]. The crack and plastic yield zone are represented by a

collinear array of infinitesimal dislocations. Along the strip of plastic yield, a constant

yield stress is distributed, which is acting against the motion of dislocations. The

equilibrium distribution is obtained, using Muskhelishvili’s method[l6], by letting the

dislocation density vanish at the top of the plastic strip. For mode I and II loading

dislocations distributed are of edge type which have Burgers vector b 010) and b(lOO),

respectively.

This model was extended by Weertman as to explain plastic yield under cyclic

stressing. Weertman considered the propagation of a fatigue crack for the case when

pulsating tensile cyclic stress was applied to a sheet which had a crack located

perpendicular to the stress axis (Fig. 7). Bilby and Heald[17] analysed fatigue crack

growth from a notch using Weertman’s model. In the following, we use the procedure

presented by Bilby and Heald. At the time of first application of the maximum load,

indicated by “0” in Fig. 7(b), the yield zone w and the distribution of dislocations Do(x)

are given by the same equations derived by Bilby, Cottrell and Swinden for monotonic

loading. The equation for w is

w/c = set (7ru,,/2Y) - 1

(27)

and, for 1x1< c + w, Q(x) is

D (x) = 2(1- V)Y x(a2- c2)‘“+ c(u2- x2)“*

0

wbG I n

(a * - * - c (a * - x * ) ‘ I *

(28)

where a = c + o, and Y is the yield stress of the material in the yield strip. In the

following discussion, we confine our interests to the cases of small scale yielding, i.e.

the cases of a,,,, 4 Y and o e c. Expanding set ( rum,/2 Y) in a series of (a-/ Y) and


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506

KEISUKE TANAKA

neglecting higher order terms, we obtain the plastic

o/c = lr=&$gY=

zone size as

(29)

Substituting a = c + o and x = t + c in equation (28), and neglecting the terms with

order of o and t give

where

Do(t) = [2(1- ~)Y/?rbG]f(t, o)

(30a)

f(t, 0) = ln[{l + (1 - t/w)“*}/{1 - (1 -

t/w)“‘}]

Wb)

for o < ItI c w. The displacement Q,,(t) between the upper and lower surfaces of the

yield strip is given by carrying out the integration:

as

where

I

-

,(t) = - b

D,, ( t ’ ) dt’

f

@o(t) = [4(1- v)Yo/wG]g(t, o)

(31)

(32a)

g(t, w) = (1- t/o)‘“-iiln

[

1 + (1- t/W)“2

1 - (1 - t/o)‘”

Wb)

From equations (29) and (32a, b), the displacement at the crack tip is

Q,,(O)= 1 - ~)7rca~,,/2GY

(33)

When the stress is reversed to omin, he distribution of dislocations changes to III(t)

which is given by

D,(t) = Do(t) + A(t)

(34)

where

A(t) = - [4(1- ~)Y/rrbG]f(t, w”) (35)

and o* is the reverse yield zone size given by

W*/C = W’ Um, -

U,in)‘/32YZ

(36)

Equations (35) and (36) are obtained by substituting (CT,,,., a,& and (- 2Y) for CT,,_

and Y in equations (31) and (32), respectively. The size of reverse yield zone is a

quarter of that of monotonic yield zone. The displacement 4(t) corresponding to A(t) is

4(t, w*) = - [4(1-

v)Yo* /TG]g( t , o*)

(37)

During the period of repetition of the applied stress, the value of dislocation distribu-

tion and displacement are Do(t) and Q,(t) at the maximum load, and [D,,(t) + A(t)] and

[@,(t) + a(t)] at the minimum load, respectively.


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Fatiguecrack propagation

507

Weertman assumed that fracture takes place when the sum of absolute value of

displacement reaches a critical value aC.

At the tip of a stationary crack, the

accumulated displacement @* is given by

a* = @,(0)+2N~~(O, w*)l (38)

For small scale yielding cases, %(O) is negligible against (DC.The number of cycles N,

required to crack nucleation is obtained by equating @* with Cp, and using equations

(32), (36) and (37) as

N, = 2GYQ,/[(l- v)~c(u,,,ax- u,,,i,,)‘]

(39)

When the crack grows in each cycle, the rate of crack growth dc /dN can be calculated

approximately by

.

dc

2

140, w*)l dt

-=

dN

O@C @Cl(O)

(40)

Substituting equation (37) in (40) and neglecting higher orders of and the term of QO(0),

one obtains

dc/dN = 16(1- v)Yw**/37TG@~

(41)

Eliminating o* from equations (36) and (41) gives

where

dc/dN = [ l - v)w/3 x 43G@,][K4/Y3]

(42a)

K = (a,, - f&“)Vz

(42b)

In the papers of Weertman and Bilby-Heald, starting with equation (28), they

calculated the exact solutions of equations (31) and (40) and then derived equations

(42a, b) applicable to small scale yielding cases. The above procedure to derive

equations (42a, b) is essentially the same as given by Rice[l8].

Lardner gave the propagation law of fatigue cracks under the assumption that the

rate is equal to the amount of the reverse component of

tip[7]. By substituting t = 0 in equation (37), one gets the

dc/dN = [ l - v)/4G][K2/Y3

where K is given by equation (42b).

displacement at the crack

law:

(43)

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