Crack Paths 2006

Proceedings of Crack Paths 2006, Parma, Italy

International Conference on

C R A CPKA T H S(CP 2006)

Parma(Italy), 14th-16th September, 2006

University of Parma, Via Università, 12


SomeExperiences with CrackPath Issues

L. P. Pook1

1 University College London

ABSTRACT.As is well known many engineering structures and components contain

cracks or crack-likeflaws. It is widely recognised that crack growth must be considered

both in design and in the analysis of failures. The complete solution of a crack growth

problem includes determination of the crack path. Macroscopic aspects of crack paths

have been of industrial interest for a very long time. At the present state of the art the

factors controlling the path taken by a crack are not completely understood. The

purpose of this introductory paper is to set the scene for the more detailed papers which

follow. Eight brief case studies are presented. These are taken from the author’s professional and personal experienc of macroscopic crack paths over the past 50

years. They have been chosen to illustrate some of the more important aspects of crack

paths. Many more examples are included in the invited and contributed papers

presented during the Conference.


As is well-known, manyengineering structures and components contain cracks or flaws

and, therefore, crack growth must be considered both in design and in the analysis of

failures. The complete solution of a crack growth problem includes the determination of

the path taken by the crack. The path taken by a crack in a critical component or

structure can determine whether failure is catastrophic or not. Knowledge of potential

crack paths is also needed for the selection of appropriate non-destructive testing

procedures. Muchcurrent work is concerned with crack growth viewed on macroscopic

scale. However, crack tip features associated with the growth of a crack, maybe viewed

at different scales [1], as shown for metals in Table 1. All these scales are of interest in

the consideration of crack paths. The International Conference on Fatigue Crack Paths,

held in Parma in 2003 [2, 3], was devoted to consideration of fatigue crack paths at

various scales.

From a theoretical viewpoint the complete solution of a crack growth problem

includes determination of the crack path. It is often assumed that the crack path is

known, either from theoretical considerations, or from the results of laboratory tests.

However, at the present state of the art, the factors controlling the path taken by a crack

are not completely understood [4] and, in practice, macroscopic crack paths in

structures are often determined by large scale structural tests [5, 6].

In order to set the scene for various crack path issues discussed in the detailed

papers which follow some brief case studies are presented. These are taken from the

author’s professional and personal experience of macroscopic crack paths over the past

50 years.

Table 1. Fracture process scales

Scale (mm) Feature

Ions, electron cloud




Subgrain boundary precipitates


Subgrain slip band


Grains, inclusions, voids


Large plastic strains


Elastic-plastic field


Stress intensity factor



Componentor specimen

Figure 1. Cracks in undercarriage bay

Figure 2. Surface of crack in an

undercarriage bay bracket.



The relationship between modeof fatigue loading and paths taken by fatigue cracks has

been of interest for a long time [7, 8]. This information can be useful in failure analysis

and Figure 1 shows an example from 1961. It is a bracket from an aircraft undercarriage

bay which showed unexpected cracking at rivet holes. The bracket was a formed 18 swg

(1.2 m mthick) aluminium alloy angle, 10 u 0.8 u 0.8 in (254 u 20.3 u 20.3 mm). The

figure shows a general view of typical cracks observed after the bracket was removed

from the bay. Examination of the fracture surfaces of the cracks showed that fatigue

cracks had originated at both surfaces of the bracket at the rivet hole corners and then

propagated inwards on elliptical crack fronts, with the two cracks intersecting at or near

the centre line of the sheet. Figure 2 shows the fracture surface of a typical crack. This

indicates that failure was caused by out of plane alternating bending fatigue loads,

which were not anticipated by the designer. Examination of the fracture surfaces at high

magnification showed the presence of striations and hence confirmed that cracking was

due to fatigue. This is an example of the useful crack path information which can be

obtained from simple examination of a failed component with the naked eye.


By 1965 plane strain fracture toughness testing using ModeI specimens, in which crack

growth is perpendicular to the applied load, was well established [9] but little was

knownabout fracture toughness behaviour under mixed mode loading, where loads are

applied at an angle to the crack. Sometests were therefore carried out in 1966 [10, 11]

to investigate the mixed mode fracture toughness of D T D5050, a 5 ½ %Zn aluminium

alloy with KIc = 28.8 M P a — m[10]. A 19 m mthick angle notch specimen was used, with

E of 75q, 60q and 45q, as in Figure 3. Specimens

the initial notch inclined at an angle

were precracked in fatigue. Figure 4 shows the fracture surface of one of the specimens

with the initial notch inclined at E = 45q. The fatigue precrack (bright area at the notch

root) is of nearly constant depth, and at the end of the precrack

E | 48q. A feature of the

test is that under the static loading to determine the fracture toughness the specimen

failed very abruptly, but the macroscopic crack path features followed on from the

fatigue precrack. At the time the fracture surface appearance was puzzling, but is easily

interpreted from a modern viewpoint [4], in that that there is a tendency to ModeI crack

growth on two scales. On a scale of 1 m minitially crack growth was mixed mode. As

the crack grew the crack front rotated until it was perpendicular to the specimen

surfaces, and crack growth was in Mode I, with the exception of shear lips at specimen

surfaces. On this scale the crack follows a curved path which tends towards a plane of

symmetry. This is in accordance with the well knownobservation [4] that the tendency

to ModeI crack growth means that cracks tend to grow perpendicular to the maximum

principal tensile stress. On a smaller scale of 0.1 m mthe tendency to ModeI fatigue

crack growth results in the production of what is knownas a twist crack [4] containing

individual ModeI facets connected by cliffs. The ModeI facets gradually merge as,

viewed on the 1 m mscale, the crack growth surface becomes perpendicular to the

specimen surfaces. Merging of Mode I facets shows up more clearly under fatigue


Some fatigue tests were carried out in 1989 on 20 m mthick medium strength

E values of 75q, 60q and 45q.

structural steel angle notch specimens [12] with initial

Figure 5 shows the fracture surface of one of the specimens, initial

E = 60q. The light

area at the top is where the specimen was broken open in liquid nitrogen.

These examples illustrate the strong tendency to ModeI crack growth in isotropic

materials under essentially elastic conditions.

Figure 3. Angle notch Charpy specimen, Figure 4. Fracture surface of D T D5050

crack initiation along notch tip.

5 ½Zn aluminium angle notch fracture

toughness test specimen, initial

E = 45q.

Figure 5. Fracture surface of medium Figure 6. Crack path in a Waspaloy

strength structural steel angle notch

sheet under biaxial fatigue load.

E = 60q.

fatigue test specimen, initial

The grid is 2.54 mm.


The question of the stability of a crack path had been of interest for some time [13] but

in general it wasn’t possible to predict crack paths under biaxial fatigue loading.

Therefore, in 1974 some tests [14], were carried out at room temperature on Waspaloy,

a nickel based gas turbine material, in order to determine the conditions under which a

fatigue crack path became unstable under biaxial loading. The specimens were 254 m m

square and 2.6 m mthick. The material had been cross rolled during production to

ensure that its properties were reasonably isotropic. Tests were carried out using

sinusoidal constant amplitude loading at a stress ratio (ratio of minimumto maximum

load in fatigue cycle), R, of 0.1. In each test the fatigue load perpendicular to the crack

was kept constant. Cracks were first grown from each end of an initial slit under

uniaxial loading. An in phase load was then applied parallel to the crack, and crack path

behaviour observed. Figure 6 shows the crack path for a load parallel to the crack of

twice the load perpendicular to the crack. The crack path became unstable and deviated

from its initial path as soon as the load parallel to the crack was applied. At the time the

tests were carried out it wasn’t possible to do more than describe the results. However,

reanalysis of these and other results in 1997 [4, 15] showed it was possible to correlate

crack path stability in terms of a parameter called the T-stress ratio.


Figure 7. Plastic domestic tap.

Figure 8. Crack surface of plastic

domestic tap.

In 1991 a plastic domestic tap in the author’s utility room was observed to be leaking

where it was screwed into a fitting on the supply pipe. The tap had a fitting for a hose

pipe, and appeared to be a replacement for the original brass tap. Whenan attempt was

made to unscrew the tap it failed completely. The two parts of the broken tap are shown

in Figure 7 and a close up of the fracture surface in Figure 8. The dark area is fatigue

and the light area the final static failure. The age of the tap at the time of failure is

unknown, but as one fatigue cycle is applied each time a tap is turned on and off it is

likely that thousands of cycles had been applied. Safety critical pressure containing

components are often designed to leak-before-break [6] in order to avoid catastrophic

failure. It is fortunate that the tap did so otherwise the utility room would probably have

been flooded. The failed tap was replaced with a brass tap, and it was observed that the

detail design in the vicinity of the threads was exactly the same. The replacement tap is

still in use. The episode is an example of the danger of using a different material for a

component without making appropriate changes to detail design.


Before the days of quartz clocks, spring driven wall clocks were widely used in public

buildings. The example shown in Figure 9 was originally used in a school, but since

1968 it has been in use in the author’s kitchen. In 1994 the mainspring failed while the

clock was being wound. Examination showed that this was the final failure following

fatigue crack growth. A general view of the failed mainspring is shown in Figure 10.

Fatigue has been a problem in clock mainsprings for centuries, and traditionally they are

designed using rules of thumb based on experience [16], rather than by detailed

analysis. The total fatigue life is not known, but the clock is wound weekly so it must be

thousands of cycles.

Figure 9. Wall clock by John Davidson, Figure 10. Failed wall clock mainspring.


A clock mainspring is loaded in bending, with loading and unloading moving along

the spring as it is woundand unwinds. Whena mainspring breaks in fatigue the crack is

usually straight across the spring, with crack growth predominantly through the

thickness. However, in this particular mainspring crack path behaviour is unusually

complicated, and details are shown in Figure 11. A fatigue crack initiated at a corner at

one edge of the 27 m mwide mainspring. Initially, crack growth was across the spring

(downwards in the picture) but after about 9 m mof growth the crack turned sharply

towards the outer end of the spring (right in the picture), and then grew in a spiral

fashion towards the other edge of the spring until the final failure took place. During

this crack growth two secondary cracks initiated, and then joined so that a small

triangular piece of spring became detached. The joined secondary crack then grew in a

spiral fashion towards the centre of the spring, but did not contribute to the final failure.

This is an example of a nuisance fatigue failure which did not have serious

consequences. Such failures are not normally investigated at all. The offending

component is simply replaced. In this particular case the replacement mainspring is still

intact after 12 years.

Figure 11. Centre portion of failed wall Figure 12. Fracture appearance of mild

steel Charpy specimens tested at 10q C.

clock mainspring.

E = 90q.

Top, standard specimen,

E = 45q.

Bottom, angle notch specimen,


Somepreliminary tests [17] were carried out in 1971 on angle notch Charpy specimens,

but crack paths were not investigated in detail. Specimen design was based on the

standard Charpy V-notch specimen with E values (Figure 3) of 90q (standard specimen),

75q, 60q, and 45q. The true notch tip radius was reduced so that the notch tip radius

measured in a plane parallel to the specimen sides was the same as in the standard

Charpy specimen (0.25 mm). Figure 12 shows the appearance of specimens tested at

10q C. More detailed tests were carried out in 1997 using EN6amild steel (0.36% C)

specimens [18]. All specimens were tested in the normalised condition (tensile strength

550 MPa, yield stress 280 MPa). Tests were carried out in a 300 J Charpy machine

equipped with a 2 m mradius striker. They are an example of the complexity often

observed in crack path behaviour under dynamic loading. The fracture surface

appearance of the standard Charpy specimens (E = 90q) is typical of mild steel. In the

lower shelf region, that is at below about -15q C, fracture surfaces are crystalline, and in

the upper shelf region, above about 30q C, they are ductile. In the transition region

fracture surfaces are initially ductile, and the amount of crystalline crack growth

decreases with increasing temperature. Shear lips appear at above about -15q C, and

increase in size with increasing temperature. The fracture appearance transition

temperature (50 per cent crystalline) is about 25q C. In the upper shelf region fracture

surfaces are ductile.

Figure 13. Angle notch Charpy specimen, abrupt transition to crystalline crack growth.

The fracture surface appearance of the angle notch specimens is controlled by a

tendency towards square (Mode I) crack growth, but modified by plasticity and by crack

path constraint due to the initial notch. The value of E has little effect on either the 50

per cent crystalline transition temperature, or on the temperature below which fractures

are crystalline. Shear lips for E = 75q and 60q are similar to those on standard Charpy

specimens, but could not be distinguished for E = 45q. In the transition region fracture

surfaces are initially ductile. The amount of initial ductile crack growth increases with

increasing temperature. Crack initiation is along the notch tip, and in the notch plane, so

the initial crack growth is mixed mode. For E = 75q and 60q a crack twists as it grows,

becoming ModeI as it approaches the striker position (Figure 3). For E = 45q there is an

abrupt transition to ModeI crack growth (Figure 13). This ModeI growth is at least

initially crystalline. At below about -15q C fracture surfaces of the angle notch

specimens are fully crystalline. Crack origins are ModeI. For E = 75q and 60q there are

a number of individual ModeI crack origins along a notch tip, linked by vertical cliffs

(apparently ModeIII). The initial ModeI cracks link up as a crack grows, and overall a

crack twists as it approaches the striker position. For E = 45q the tendency to ModeI

crack growth is so marked that the crack path is not constrained by the notch. At

intermediate absorbed energy levels there is one crack origin at the centre of a notch,

and crack growth is ModeI throughout (Figure 14). At high absorbed energy levels

there are crack origins at both notch corners. The cracks follow curved, apparently

ModeI paths, as shown schematically for a single crack in Figure 15. The two paths

merge as they approach the striker position.

Figure 14. Angle notch Charpy specimen,

Figure 15. Angle notch Charpy specimen,

crack origin at centre of notch.

crack origin at notch corner.


During routine maintenance in 2002 one of the two burners in the gas fired domestic

central heating boiler installed in the author’s house was found to be cracked due to

thermal fatigue. A general view of the burner is shown in Figure 16, and the crack is

shown in Figure 17. The boiler was about 12 years old so, assuming it fired about 10

times per day, about 44,000 thermal fatigue cycles had been applied. The burner

consists of a steel box with a series of small and large holes on top to distribute the gas

to the flame above the box. The larger holes have reinforced perimeters. An internal

wire mesh, just visible in Figure 17, helps to distribute the gas evenly. Cracking appears

to have initiated at three places on the perimeter of a smaller hole, grown into two larger

holes with a small triangular piece becoming detached, and then two cracks grew across

most of the width of the box, resulting in improper combustion. The designer did not

appear to have appreciated the point that stress concentration factors are largely

independent of hole size. The reinforcement had prevented crack initiation at the large

holes but its absence had allowed cracking at a small hole. This is another example of a

nuisance fatigue failure. Annual inspection was recommended by the boiler

manufacturer. This ensured that the cracking was detected before it became dangerous,

and the burner was replaced..

Figure 16. Burner from domestic central heating boiler.

Figure 18. Cracks in sole of walking shoe.

Figure 17. Crack in burner from

domestic central heating boiler.


In 2005 the author found that the plastic soles of pair of walking shoes had become

badly cracked and one no longer fitted properly. This more severely damaged shoe is

shown in Figure 18. The sole of a shoe is subjected to repeated bending. Going uphill a

sole is also subjected to repeated tension as the rearward force applied by the wearer’s

heel is transferred to the ground. This particular pair of shoes had covered several

hundred kilometres, which is equivalent to around 3 u 105 cycles. In the shoe shown

two separate cracks had initiated in grooves near the toe, grown past each other and then

curved together, in a well known crack path behaviour [19], so that a piece of sole

became detached. The heel had also cracked and, in what appears to have been the final

event that reduced the stiffness of the shoe so much that it became unusable, the sole

separated from the upper at the end of this crack. The use of a plastic, instead of rubber,

for the soles has reduced the rate of wear but led to fatigue failure. This is another

example where a change of material has resulted in fatigue cracking.


Paths taken by cracks have been of industrial interest for a very long time [7, 8]. A large

amount of empirical knowledge has been accumulated, but at the present state of the art

the factors controlling the path taken by a crack are not completely understood.

The numerous possible crack configurations [20] mean that a systematic approach to

the determination of crack paths isn't feasible, so particular practical problems need to

be tackled on an ad hoc basis. In carrying out analyses care has to be taken to view

crack paths at an appropriate scale.

The examples given have been chosen from the author’s experience to illustrate

some of the more important aspects of crack paths. Many more examples are included

in the invited and contributed papers presented during the Conference.


[1] McClintock, F.A. and Irwin, G.R. (1965) In: Fracture Toughness Testing and its

Applications. ASTMSTP 381, pp. 84-113, American Society for Testing and

Materials, Philadelphia, PA.

[2] Carpinteri, A. and Pook, L. P. (Ed). (2003). Proceedings (on CD) of the

International Conference on Fatigue Crack Paths (FCP2003) Parma (Italy), 18

20 September 2003. University of Parma.

[3] Carpinteri, A. and Pook, L. P. (2005) Fatigue Fract. Engng. Mater. Struct., 28, 1.

[4] Pook, L. P. (2002) Crack Paths.: W I TPress, Southampton.

[5] Marsh, K. J. (Ed).( 1988) Full-Scale Testing of Components and Structures.

Butterworth Scientific Ltd,. Guildford:

[6] Pook, L. P. (2000) Linear Elastic Fracture Mechanics for Engineers. Theory and

Applications. W I TPress, Southampton.

[7] Cazaud, R. (1953) Fatigue of metals. Chapman& Hall Ltd,. London.

[8] Longson, J. (1961) A photographic study of the origin and development of fatigue

fractures in aircraft structures. RAE Report No. Struct 267. Royal Aircraft

Establishment, Farnborough.

[9] Srawley, J. E and Brown, W. F. (1965) Fracture toughness testing methods. In

Fracture toughness testing and its applications. A S T MSTP 381. American

Society for Testing and Materials, Philadelphia, PA, pp. 133-198.

[10] Pook, L. P. (1968) Brittle Fracture of Structural Materials Having a High

Strength Weight Ratio. PhDthesis, University of Strathclyde, Glasgow.


Pook, L. P. (1971) The effect of crack angle on fracture toughness. Eng. fract.

Mech., 3, 205-218.


Pook, L. P. and Crawford, D. G. (1991) The fatigue crack direction and threshold

behaviour of a medium strength structural steel under mixed Mode I and III

loading. In: Kussmaul, K., McDiarmid, D. L. and Socie, D. F. (Ed). Fatigue

Under Biaxial and Multiaxial Loading. ESIS 10. pp. 199-211. Mechanical

Engineering Publications, London.


Cotterell, B. (1966) Notes on the paths and stability of cracks. Int. J. fract. Mech.,

2, 526-533.


Pook, L. P. and Holmes, R. (1976) In: Proc. Fatigue Testing and Design Conf.

Vol. 2, pp. 36.1-36.33. Society of Environmental Engineers Fatigue Group,

Buntingford, Herts:


Pook, L. P. (1998). An alternative crack path stability parameter. In: Brown, M.

W., de los Rios, E. R. and Miller, K. J. (Eds). Fracture from Defects. E C F12.

Vol. I, pp. 187-192. E M A SPublishing, Cradley Heath, West Midlands.


Britten, F. J. (1978) The watch & clock makers' handbook, dictionary and guide. 16th Edition. Revised by Good, R. Arco Publishing CompanyInc, N e wYork.


Pook, L. P. (1972) The effect of notch angle on the behaviour of Charpy

testpieces. Eng. fract. Mech., 483-486.


Pook, L. P. and Podbury, M. J. (1998) Failure mechanism map for angle notch

Charpy tests on a mild steel. Int. J. Fract., 90, L3-L8.


Melin, S. W h ydo cracks avoid each other? Int. J. Fract. 1983, 23(1), 37-45.


Pook, L.P. (1986) Keyword Scheme for a Computer Based Bibliography of Stress

Intensity Factor Solutions. NEL Report 704. National Engineering Laboratory,

East Kilbride, Glasgow.

Progress in Identifying the Real 'Keffective

in the Threshold Region and Beyond

Paul C. Paris 1, Diana Lados 2, and Hiroshi Tada1

University, St. Louis, M O ,USA.

1 W a s h i n g t o n

2 W o r c e s t e r Polytechnic Institute, Worcester, MA,USA.

A B S T R A C The use of the crack tip stress intensity factor, K, has survived almost 50

years as the key parameter correlating fatigue crack growth. As time past the range of

the stress intensity, 'K, was recognized as causing alternating plasticity at the crack

tip. The threshold level for ' Kwas discovered. Further the occurrence of crack closure

was noted which effected the 'Kfor different load ratios, R, of cyclic loading. The

A S T Mmethod of counting the linear part of the load displacement for determining



was found to understate the

which correlates data for different

load ratios. One approach to adjust for this problem is the “Partial Closure Model”,

where the closure only occurs away from the crack tip. Here it will be discussed that

such a model leads to a universal growth law. Moreover, this law shows application in

estimating the order of magnitude of crack growth life (>107cycles) for example with

very high cycle fatigue (>109cycles). Some advances in this application will also be



The use of the elastic crack tip stress intensity factor, K, was submitted for publication

in 1959 [1] and was promptly rejected by 3 major journals (ASME,AIAAand a U K

journal). In all three cases the reviewers argued that an elastic parameter could not

correlate fatigue crack growth data because plasticity must be involved. Figure 1 shows

the original plots of data from three independent sources on 2 aluminum alloys showing

the correlation of data ignored by those reviewers. Further discussion appears in a

subsequent paper [2], comparing earlier suggested parameters based on more limited

data. The wide range of data provided by McEvily [3] settled this search for K as the

leading parameter of interest. It is acknowledged that McEvily introduced a stress

concentration type parameter, which was a less popular but correct approach.

Figure 1 The original 1959 correlation of data on 2024 and 7075 aluminum alloys [1].

In this later paper [3] the power law of crack growth was presented in terms of the

range of the stress intensity, 'K,with a constant, C, dependant on the load ratio, R, to

express the growth rate as:

daN C ' K n

where C=C(R)

This form was merely an empirical fit of McEvily’s data over a wide range of growth

rates (5+ log cycles). It was observed by Hertzberg that this law failed at rates below

one Burger’s vector, b, per cycle by leveling to a threshold ' K (private communication

1964). Even earlier Anderson [4] noted that growth rates were similar for all metal

alloys if the stress intensity range was normalized by dividing by elastic modulus, E.

It was later in the 1960’s that Elber [5] drew attention to crack closure in fatigue,

although closure was noted by Christensen [6] much earlier. Thereafter, [7] Hertzberg

noticed that for load ratios, R, above 0.7, where no closure occurs, that the preceding

law herein can be madeuniversal for all metal alloys as:


§ E' Kb © ¨ · ¹ ¸


' K



where n = 3 and threshold occurs for

E b


Indeed this empirical law works for a wide variety of steels; aluminum, titanium,

magnesium, and copper-beryllium alloys [7]. It remains to develop this law to an even

more universal form by finding a 'Keffective

so that it may be applied to all load ratios,

R, by including the effects of crack closure.

T H ES E A R CFHO R 'Keffective


There is no analytical method of calculating the crack closure (or opening) level during

cyclic loading. For variable amplitude there is also no method. The A S T Mhas tried to

develop a method (see A S T ME 647-00) of measuring the opening load by determining

the load level for which the load displacement record becomes linear as the crack peals

open. Data in terms of load vs. displacement is analyzed to obtain the point at which the

deviation from subsequent linearity is a certain small % of that slope. This load is used



to compute

which along with the maximumload for

is used to

compute a stress intensity range as:

' K K max K open


' K causing fatigue crack growth.

This was at one time regarded as the relevant

However, precise computer controlled load-displacement data from Donald [8] covers a

wide range of load ratios, R. It shows that the A S T Mmethod does not well correlate the

data of widely differing load ratios. It improves correlation at high stress intensities but

worsens correlation near threshold. This effect is shown on Figures 2 and 3. Donald [9]

Figure 2 Data on 7055 aluminum alloy using applied stress intensity range,'K , [10].

proposed the “Adjusted Compliance Ratio Method” and also noted [10] a minor effect

of Kmax

in the data. See Figures 4 and 5. After several years of consideration there is

no known model or theory to justify this A C Rmethod. On the other hand the “Partial

Closure Model” [11] will be revisited here, which does have a physical and analytical

basis. With it we shall show that the preceding normalized power law can be made

universal for all load ratios.

Figure 3 Data on 7055 using the A S T M' K method, [10]. opening

Figure 4 Data on 7055 using Donald’s Adjusted Compliance 'KACR method, [10].

Figure 5 Data on 7055 using ' K A C R and with Donald’s adjustment for K , [10]. max


The doctoral dissertation of Bowles [12] noticed that with cyclic fatigue crack closure a

region near the crack tip stays open at minimumload. Whether closure is due to

plasticity, asperities on the surface, or fragments etc it can be modeled as a rigid layer of

height, 2h, extending into the crack a distance, d, from the tip. Figure 6 shows the

Figure 6 The computational model for the partial closure method [11].

model for (a) minimumload and (b) at opening load when crack closure occurs. For the

condition at full unloading, (a), the crack tip stress intensity is found to be:


Eh 2Sd V nom min

Keff min


For (b) at opening load the stress intensity is:


K open



Combining these gives:


nom min

2 SKopen

K eff min



Where Vnom min

is the nominal tensile stress perpendicular to the crack with the

crack absent. Since the term is quite small because d is also small, it can be neglected.

Consequently it is seen that the minimumeffective stress intensity is very nearly:

K # 2K eff min S


As follows from this we have called this the “Partial Closure Model” or 2/Pi0 – method

where the effective stress intensity range is:

2 K K #Kmax S K

' K


max eff min


This implies that the A S T Mopening stress intensity should be reduced by

2 S to correctly compute the real stress intensity range. Figure 7 shows


the same preceding data of Donald from Figures 2, 3, 4, and 5 where the data is

correlated quite closely into a single curve. Though this is data on a single material the

reader will find many other materials with comparative correlations in the references

cited herein.

Figure 7 Data on 7055 using the partial closure model (2/Pi0) 'Keff.

The Partial Closure Model is emphasized here with some reservation. All physical

models are crude approximations of reality and this one is no exception. However it

happens to helpfully correlate data for considerations of whether the data is well

founded and whether the material is not an oddity. The A C Rmethod of Donald serves

this same purpose in general. At least for one material he has tested, Donald has

acknowledged (private communication) that the Partial Closure Model provides tighter

correlation. The disadvantage of both of these models is that closure load levels must be

measured experimentally, which make the data difficult to use in practical applications

to life prediction. In any case these correlations do help to show that ' K as modified

for closure is the primary and dominant variable causing fatigue cracking.

It is of further interest to also revisit the preceding cubic power law using the

effective stress intensity range developed here.


In order to make the previous third power law herein into a universal law for all load

ratios, R, it is only necessary to substitute the effective stress intensity factor. It is

acknowledge that a small effect of the maximumstress intensity factor is present, as

illustrated in Figure 5. Since this effect is minor it shall be ignored in further discussion.

Consequently, the “Universal Law”is stated as:

da dN #b ' K eff § E b


da dN d b



where for threshold



© ¨

¹ ¸

This Universal Law is a good approximation for all data on metal alloys known to

these authors but is only an approximation. Figure 8 shows the results of the plotted

lines of the law as compared to data 7055 aluminum (a very good fit) and for 2324

aluminum (a good fit except this alloy exhibits a superior threshold or larger Burger’s

vector). These are extremes in the precision of fit and again the reader will find further

supporting evidence in the references herein, especially [7]. The Universal Law is

suggested to provide a maximumgrowth rate limit for data not influenced by aggressive

environments. It applies equally well to “small cracks” as a maximumgrowth rate. As

such it can be used in estimates of minimumand order of magnitude estimates of crack

growth lives for many applications.

Figure 8 Data on both 7055 and 2324 with predicted lines from the Universal Law.

For example in a series of applications to Very High Cycle Fatigue, >108 cycles,

exhibiting failure initiation from internal metallurgical discontinuities, this law can be

used to show that the accompanying crack growth life is much smaller, <106 cycles.

Therefore, V H C Flife is dominated by initiation of cracking, see [13-17].

Dimensional Considerations of the Universal Law

The immediately above power law is noted to be dimensionally correct. If only the

effective stress intensity range, the maximumstress intensity, the elastic modulus, and

the Burger’s vector are present in the growth rate law, then the non-dimensional

parameters involve are: dN, dba , 'EKeff b. and EKmax b

. Restricting the parameters to these

items is strongly supported by the preceding data. A general form of the law can then be

written as:



da b˜ F 'Keff , K max

© ¨

¹ ¸

E bE b


It is acknowledged that b could be a micro-structural characteristic of the material of the

order of the Burger’s vector (such as micro-constituent phase size, etc.). However the

Universal Law applied to data in all cases strongly supports the third power effect, i.e. a




growth rate proportional to ' K eff

. As a consequence the law becomes:

© ¨

¹ ¸

E b

da dN b§ 'EKeff b ·


© ¨

¹ ¸

¸ A˜KmaxEb§©¨·¹¸


Donald [10] in his work chooses: F 1 K max §


, (with m = 1) in an attempt

© ¨

¹ E b

to fit the data even better and where A is a non-dimensional constant. This choice might

be subject to further investigation. However, with that choice the law becomes:





' K eff § E b ·

da dN A˜b



© ¨

¹ ¸

¹ ¸

© ¨

E b

where threshold occurs at:

§ 'EKeffb ,EKmax b ·



© ¨


and where B is also a dimensionless constant.

It is noted that the Universal Lawas previously stated above is within the restrictions

of these dimensional considerations. Other attempts to formulate laws of mechanical

fatigue crack growth incorporating other factors (such as yield stress, etc.) are contrary

to the broad trends of data used in implying and developing the Universal Law through

the analysis here.

It remains for someone to give a full physical explanation of the fact that stress

intensity divided by elastic modulus times square root of Burger’s vector is show by all

the data on metal alloys to be the universal normalizing factor. Further, the influence of

environment remains another effect requiring attention as well.


(1) The power law of stress intensity factor range, 'K, has withstood almost 50

years of exploration and remains the most dominant parameter causing fatigue

crack growth.

(2) Crack closure effects the stress intensity range.

(3) The A S T Mmethod of determining open load and thereby'K does not open

adequately express the full stress intensity range with closure.

(4) Following the work of Bowles, the Partial Closure Model shows a 'Keff greater

than the A S T Mmethod. Donald’s A C Rmethod also correlates data better but

lacks an analytical model’s justification.

(5) All fatigue crack growth data strongly show that dividing the stress intensity by

elastic modulus times square root of Burger’s vector normalizes that data.

(6) From the previous conclusions a Universal Power Law of mechanical fatigue

crack growth for all metal alloys has been reviewed and presented herein.

(7) This Universal Lawmay be affected in a minor way by the maximumapplied

stress intensity and sometimes in major ways by environmental influences.

(8) Applications of this Universal Law are only good for order of magnitude

estimates of minimumcrack growth lives (for example for very high cycle

fatigue >108 applications).


The encouragement of the Washington University (St. Louis) Dean of Engineering,

Christopher Byrnes and Dr. A. K. Vasudevan of the U.S. Office of Naval Research

in producing this work is due great thanks. Effective help in developing the

manuscript by Nancy Rubin is also acknowledged with thanks.


1. Paris, P. C., Gomez, M. P., And Anderson, W. E., (1961) The Trend in

Engineering, 1, 9-14.

2. Paris, P. C. and Erdogan, F., (1963) Trans. of ASME, J. of Basic

Engineering 85, 528-534.

3. McEvily, A. J. Jr. and Illg, W. (1958) N A C AT N4394.

4. Donaldson, D. R. and Anderson, W. E. (1961) Proceedings of the Crack

Propagation Symposium,2, Cranfield, England.


Elber, W.(1970) Engin. Fracture Mech., 2, 37-45.


Christensen, R. H. (1963) Appl. Mat. Res. October 207-210.

7. Hertzberg, R. W. , (1996) Deformation and Fracture Mech. of Eng. Mat. – 4th Ed., John Wiley & Sons, N e wYork.

8. Donald, J. K. (2003) Private communication of 7055 and 2324 data, Fracture

Technology Associates, Bethlehem, Pa.

9. Donald, J. K., Bray, G. H. and Bush, R. W., (1997) A S T M –STP1332.

10. Donald, J. K., Bray, G. H. and Bush, R. W. (1997) High Cycle Fatigue of

Struct. Mat., T M S123-141.

Paris, P. C., Tada, H. and Donald, J. K. (1999) Int. Jour Fatigue. 21.


Bowles, Q. (1972) Doctoral Dissertation, Delft University, The Netherlands.


13. Bathias, C. and Paris, P. C.,(2005) Gigacycle Fatigue in Mechanical

Practice, Marcel Dekker, N e wYork.


Paris, P. C., Marines-Garcia, I., Hertzberg, R. W. and Donald, J. K. (2004) Proc. of the 3rd Int. Conf. on Very High Cycle Fatigue, Ritsumeikan Univ.,

Kusatsu. Japan, 1-13.

15. Marines-Garcia, I., Paris, P. C., Tada, H., Bathias, C. and Lados, D., (2006) Proceedings of the T M SSymp. To Honor the 80th Birthday of A. J. McEvily,

to be published in Int. Jour. Fatigue.


Marines-Garcia, I., Paris, P. C., Tada, H. and Bathias, C. (2006) Proceedings

of Fatigue 2006 Conference, Atlanta, Ga., to be published.


Marines-Garcia, I., Paris, P. C., Tada, H. and Bathias, C. (2006) Proceedings

of the Internat. Conf. on Fatigue Damageof Struct. Mat. VI, Hyannis, Ma.

To be published.

Fatigue Crack Path and Threshold in ModeII and ModeIII


Y. Murakami1,Y. Fukushima1, K. Toyama2and S. Matsuoka1

1 Department of Mechanical Engineering Science, Kyushu University, 744 Motooka,

Nishi-ku, Fukuoka, 819-0395, Japan,

2 Forensic Science Laboratory, Fukuoka Prefectural Police Headquarters, 7-7 Higashi

Koen, Hakata-ku, Fukuoka, 812-8576, Japan.

ABSTRACTI.n order to investigate the crack path under ModeII or ModeIII loadings,

reversed torsion tests were carried out on SAE52100and ModeII fatigue crack growth

tests were carried out on 0.47 % carbon steel specimens. In the torsional fatigue test

(SAE52100), the type of inclusion in the torsional fatigue fracture origin was slender

MnS inclusions which are elongated in the longitudinal direction. The cracks first

propagated by ModeII up to crack length 2a = 100 ~ 200 Pm (which are almost equal

to the length of MnS inclusion) in the longitudinal direction, and then branched by

Mode I to the direction (~ ± 70.5 deg.) perpendicular to the local maximumnormal

stress (VTmax) at the crack tip.

In the ModeII fatigue crack growth test (0.47 % carbon steel) in air and in a vacuum,

the cracks first propagated by ModeII. After the ModeII fatigue crack growth stopped,

the crack branched to the direction perpendicular to the local maximumnormal stress

(VTmax) at the crack tip, and finally branched to the angle close to the direction

perpendicular to the remote maximumprincipal stresses.

A fibrous pattern on the ModeII fatigue fracture surface tested in a vacuum was clearer

than that in air. The ModeII threshold stress intensity factor ranges, 'KIIth

= 10.2 M P a

m (Longitudinal crack) and 'KIIth

= 12.5 M P a (mTransverse crack) in a vacuum


= 9.4 M P a m(Longitudinal crack) and 'KIIth =

were higher than those in air,

m (Transverse crack). Both in a vacuum and in air, the values of 'KIIth for

10.8 M P a

crack growth perpendicular to the rolling direction were higher than those for crack

growth parallel to the rolling direction.

The values of KII and KIII at a 3D elliptical crack tip under shear stress were analyzed

to investigate the shear crack growth pattern in materials. The 3D crack analysis shows

that the most stable aspect ratio b/a of a small planar elliptical crack under cyclic shear

stress is b/a = 0.49 in absence of friction at crack surfaces. The aspect ratio b/a = 0.49

can be explained by the equal resistance against fatigue crack growth both in ModeII

However, the aspect ratio b/a for the failure of a real


= 'KIIIth.

and ModeIII, i.e.

railway wheel did not stay at the stable aspect ratio b/a = 0.49 and continued

decreasing. The cause for the decrease in the aspect ratio b/a smaller than 0.49 was

revealed to be the friction between crack surfaces.


ModeII fatigue failure occurs in several components such as bearings, gears, rails, rolls,

etc., as the damage types of shelling, spalling and pitting. The origins of the ModeII

fatigue crack are surface or subsurface of components. ModeII fatigue crack starting

from surface propagates in air or with lubricant. ModeII fatigue crack initiating from

subsurface inclusions is thought to propagate in a vacuum. It has been reported that in

ModeI fatigue crack growth, the crack growth behaviour in air is different from that in

a vacuum [1-5]. Kikukawa et al. [2] and Jono et al. [3] reported that the ModeI crack

growth threshold 'KIth in a vacuum was higher than that in air and the crack growth

resistance was increased in a vacuum. McEvily et al. [4] reported that the crack tip

opening displacement (CTOD)in a vacuum is larger than that in air due to the lack of

oxidation. Thus, the ModeII fatigue crack growth behaviours in a vacuum can also be

different from those in air.

In this study, the Mode II fatigue crack path and the threshold value 'KIIth under

ModeII loading and ModeII + III crack growth under torsional fatigue loading were

studied. The influence of a vacuumenvironment on the ModeII fatigue threshold and a

3D shear crack growth behaviours under Mode II and Mode III loading were also



Torsional fatigue test

Table 1 shows the chemical composition of a bearing steel, SAE52100. The Vickers

hardness, measured with a load 0.98 N, is H V= 797. The scatter of H Vmeasured at 12

points is within 11%.

Figure 1 shows the shape and dimensions of the torsional fatigue test specimen. After

finishing the specimen surface with emery paper, the surface layer was removed by

electro-polishing. A hydraulically controlled biaxial fatigue testing machine was used.

Tests were conducted under load control at a frequency of 0.1 ~ 0.2 Hz. Crack paths

were measured by using replica method.

Table 1. Chemical composition of SAE52100(wt. %)

C Si

M n P




M o Cu Al


O (ppm)

0.992 0.27

0.39 0.015 0.005 0.08



0.11 0.008 0.030


85.1R2 R02





Figure 1. Torsional fatigue test specimens (mm).

ModeII fatigue crack growth test in a vacuum

In comparison with the ModeII fatigue crack growth behaviours in air [10-12], ModeII

fatigue crack growth tests in a vacuumwere conducted by using 0.47% carbon steel.

Table 2 shows the chemical composition of the 0.47 % carbon steel. Specimens were

machined after annealing at 844°C, for 1h. After turning specimen, the specimens were

annealed in a vacuumat 600°C for 1h to relieve the residual stress introduced by turning.

The Vickers hardness after vacuum annealing is H V= 196. The scatter of H Vmeasured

at 4 points is within 5 %.

Figure 2(a) shows the shape and dimensions of the ModeII fatigue crack growth test

specimen. The ModeII fatigue crack growth tests were conducted by using the specially

designed double cantilever (DC) type specimen designed by Murakamiand Hamada [6

8]. The specimen has a chevron notch and side grooves in the slit. This specimen

enables 'KII-decreasing test under a constant load amplitude. Twoseries of specimen

were prepared. One is for ModeII fatigue crack growth parallel to the rolling directions

(Longitudinal crack) and another is for ModeII fatigue crack growth perpendicular to

the rolling directions (Transverse crack).

Figure 2(b) shows the Mode II fatigue test system in a vacuum. A conventional

tension-compression fatigue testing machine was used with a pair of the specimens.

Test method details are given in Ref. [6-8]. Tests were conducted at a constant load

amplitude, 'P= 11.8 kNwith R = -1 and at a frequency of 6 Hz. In order to conduct the

ModeII fatigue crack growth test in a vacuum, the vacuumchamber was installed to the

fatigue testing machine. An oil sealed rotary pump and turbo-molecular pump were

used to make a vacuum condition. The maximumvacuum pressure in the chamber is

about 2×10-3 Pa.

Table 2. Chemical composition of 0.47 % carbon steel (wt. %)



M n






ModeII specimen









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