Fatigue Crack Paths 2003

International Conference on

F A T I G UCER A CPKA T H S(FCP 2003)

Parma(Italy), 18-20 September, 2003

University Campus, Faculty of Engineering, Parco Area delle Scienze, 69/A

ISBN:978-88-95940-26-7

Introduction to Fatigue CrackPaths in Metals

L. P. Pook1

1 University College London

ABSTRACT.As is well known many engineering structures and components, made

from metallic materials, contain cracks or crack-likeflaws. It is widely recognised that

fatigue crack growth must be considered both in design and in the analysis of failures.

The complete solution of a fatigue crack growth problem includes determination of the

crack path. Macroscopic aspects of fatigue 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 growing fatigue crack are not completely understood. In the last four decades there

have been substantial advances in the understanding and prediction of fatigue crack

paths, largely through developments in fracture mechanics and in the application of

modern computers and microscopes. The purpose of this introductory paper is to set the

scene for the more detailed papers which follow. To do this some of the more important

ideas are presented and illustrated by examples. These have been chosen to illustrate

some of the more important aspects of fatigue crack paths in metals.

I N T R O D U C T I O N

As is well known [1] many engineering structures and components, made from metallic

materials, contain cracks or crack-like flaws. It is widely recognised that fatigue crack

growth must be considered both in design and in the analysis of failures [2, 3]. The

complete solution of a fatigue crack growth problem includes determination of the crack

path. At the present state of the art the factors controlling the path taken by a growing

fatigue crack are not completely understood. In general, fatigue crack paths are difficult

to predict and, in practice, fatigue crack paths in structures are often determined by large

scale structural tests [2].

Crack tip features associated with the growth of a crack, may be viewed at different

scales [4], as shown in Table 1. All these scales are of interest in the consideration of

fatigue crack paths. To put these scales in perspective, reading this paper is at a scale of

observation of about 0.1 mm, viewing television at a scale of about 1 mm, and walking

about a room at a scale of about 10 mm. Present day fracture mechanics is largely

concerned with macroscopic aspects of crack growth, corresponding to the three largest

scales in the table [2]. This is why crack surfaces are often assumed to be smooth, even

though on a microscopic scale they are generally very irregular. The different results

obtained by viewing at different scales is typical of a wide range of physical phenomena

[5].

T A B L E1. Fracture process scales

Scale (mm) Feature

10-6

Ions, electron cloud

10-5

Dislocations

10-4

Subgrain boundary precipitates

10-3

Subgrain slip band

10-2

Grains, inclusions, voids

10-1

Large plastic strains

1

Elastic-plastic field

10

Stress intensity factor

Component or specimen

100

Figure 1. Network of surface fatigue cracks in a chilled cast iron rolling mill roll [6].

Magnification: x 12.5.

Macroscopic aspects of fatigue crack paths have been of industrial interest for a very

long time. For example Figure 1 shows a network of surface fatigue cracks in a chilled

cast iron rolling mill roll. This was published in 1930 [6]. The same Ref. includes a

primitive example of the use of a critical plane approach in fatigue design calculations.

The book by Cazaud published in 1948 includes an analysis of fatigue crack paths in

both laboratory specimens and industrial components. An English translation of his

book, including additional material, was published in 1953 [7]. At about the same time,

the metallurgical microscope was being used by Forsyth in the examination of fatigue

crack paths at microscopic scales [8, 9]. By 1962 the use of quantitative fractography in

the reconstruction of crack path information was well developed [10].

In the intervening four decades there have been substantial advances in the

understanding and prediction of fatigue crack paths, largely through developments in

fracture mechanics and in the application of modern computers and microscopes. See

for example the handbook edited by Carpinteri [3] and a recent summary [11]. At

microscopic scales there has been interest in the use of fractals [12] and random process

theory [11, 13] in the characterisation of fatigue crack paths. At macroscopic scales

differential geometry has been used in the interpretation of some features of crack paths

[11, 14].

The purpose of this introductory paper on fatigue crack paths in metals is to set the

scene for the more detailed papers which follow. To do this some of the more important

ideas are presented and illustrated by examples.

C R A CPKA T HD E S C R I P T I O N

In describing the path taken by a growing fatigue crack it is necessary to describe a

crack growth surface which contains successive positions of the crack front, and also a

family of lines on this surface which describes successive positions of the crack front.

As a mathematical concept, a crack in an unloaded body is a cut which has zero

width. It is therefore possible, without ambiguity, to speak of a single ‘crack surface’,

and to describe its shape [11]. However, if a load is applied to the body the two

opposing crack surfaces move relative to each other, and it may be necessary to

distinguish between these ‘upper’ and ‘lower’ crack surfaces. It may also be necessary

to make this distinction for a physical crack, which is not necessarily of zero width,

even in a stress free body. The distinction is unnecessary if the scale of observation is

such that it is impossible to distinguish between the two opposing crack surfaces.

The two opposing crack surfaces meet at a crack front. There is usually no ambiguity

in describing its shape, except at a very small scale where the positions of individual

atoms have to be taken into account. Describing a crack growth surface in terms of

successive crack front positions avoids the need to distinguish between the two crack

surfaces.

In practice, experimental crack growth surfaces may be determined by examining

one of the crack surfaces. Irregular surfaces, such as crack surfaces, cannot be

adequately characterised unless the scale at which measurements are made is first

specified in some way [5]. In theory, there are difficulties in describing a crack growth

surface at a particular scale of observation [15], and subjective judgements are needed.

In practice, it is usually reasonably straightforward, if somewhat tedious. Modern image

processing techniques, in conjunction with a stereo pair of scanning electron

micrographs [16] make an automated approach possible. Recent developments in digital

camera technology [17] now permit the extraction of three dimensional information

using a monomicroscope.

Aluminium Alloy Lug

Figure 2 shows the crack surfaces of an 84 m mwide L65 aluminium alloy lug, part of

an aircraft structure, which has been subjected to programme fatigue loading. Both

halves of the lug are shown. The programme included both ground to air and gust loads.

The crack surfaces have the banded appearance characteristic of programme loaded

specimens, with each band corresponding to one load programme. Different features in

the bands correspond to different load levels, and the bands provide a crack front

family.

Figure 2. Crack surfaces of an L65 aluminium alloy lug subjected to programme fatigue

loading.

Figure 3. Modesof crack tip surface displacement.

C R A CGKR O W TM OH D E

A fundamental fracture mechanics concept is that of crack surface displacement [2, 11].

If a load is applied to a cracked body, the crack surfaces move relative to each other,

and there are three possible modes of crack surface displacement (Fig. 3). These are:

Mode I where opposing crack surfaces move directly apart; Mode II where crack

surfaces move over each other perpendicular to the crack front; and Mode III, where

crack surfaces move over each other parallel to the crack front. In fracture mechanics

the interest is in what happens in the vicinity of the crack front, and it is more correct to

refer to the crack tip surface displacement. By superimposing the three modes, it is

always possible to describe a general case of crack tip surface displacement.

The modes of crack tip surface displacement may also be used to characterise crack

paths. It is well known [1, 11] that on a macroscopic scale, and under essentially elastic

conditions, fatigue cracks tend to grow in ModeI, so attention is often confined to this

mode. This applies to macrocracks (Stage II in Forsyth’s notation [9, 18]). A ModeI

crack path is not necessarily flat. On a microscopic scale fatigue crack surfaces in

metallic materials are generally very irregular, and modes of crack tip surface

displacement will usually differ from those observed on a macroscopic scale.

It was pointed out in 1953 [7], and recently re-emphasised [19], that on a

macroscopic scale there are two fundamentally distinct classes of fatigue crack growth;

maximumprincipal stress dominated, and shear dominated. Mode I fatigue crack

growth is maximumprincipal stress dominated. Shear dominated fatigue crack growth

is an important exception to the tendency to ModeI fatigue crack growth. It is often

observed when crack tip plasticity becomes significant, and this applies to microcracks

(Stage I in Forsyth’s notation [9, 19]. A shear dominated fatigue crack grows in ModeII

or ModeIII or in a combination of the two.

Tubular Welded Joints

Fatigue crack path data were collected [20] during a test on a tubular Y-joint of the type

shown in Fig. 4. The chord thickness was 16 mm, and the brace thickness 12.5mm. The

joint was madefrom BS4360: 50Dsteel, and it was tested as welded. During the test the

joint was immersed in artificial sea water, with cathodic protection applied. The brace

was loaded perpendicularly at its end, in the out-of-plane direction, using a variable

amplitude fatigue loading. One side was in tension only, and the other in compression

only. The chord was built in at its ends. Crack growth was in the chord at the toe of the

weld.

Figure 4. Tubular welded Y-joint. Dimensions in mm.

Figure 5 shows the crack front family obtained [21]. Crack surface lengths are

measured along the toe of the weld. Crack depths were recorded automatically using

alternating current potential drop (acpd) equipment. From the nature of the acpd

technique the scale of observation is about 0.1 mm.Individual data points are connected

by straight lines. The crack growth surface is curved so some detail has been lost in

plotting the data on Cartesian coordinates. Figure 6 shows two sections through the

crack growth surface. The crack paths are approximately perpendicular to the chord

surface. The average deviation from the perpendicular, on a number of similar sections,

was about 10°. Crack growth surfaces in tubular welded joints are some times tortuous

and this complicates the interpretation of crack path data. Figure 7 shows an example

[22].

Figure 5. Crack front family for tubular welded joint.

Oscillating Crack Front Families

The crack surfaces produced on many metallic alloys by a fatigue crack growing

continuously under a constant amplitude loading have, in general, a smooth matt

appearance, free from markings visible to the naked eye [23]. However, it has been

known for a long time that some high strength aluminium alloys, exhibit quite distinct

crescent shaped markings in tests on thin sheets at high mean stress values [8]. This is

shown schematically in Fig. 8. The markings become larger and more pronounced as

crack growth proceeds.

Each dark area is an element of brittle fracture occurring in one stress cycle [8]

whereas each light area corresponds to a period of fatigue crack growth. Confirmation

of this composite crack growth behaviour is provided [23] by the fact that ‘clicks’ may

be heard during a crack growth test. The bursts of brittle fracture correspond to the pop

in phenomenon sometimes observed during fracture toughness testing [2]. After each

pop-in fatigue crack growth rates are higher near the surface than in the interior and the

crack front gradually becomes straighter until another pop-in takes place. The result is

that the crack front shape oscillates between extremes corresponding to the boundaries

between the light and dark areas.

Figure 6. Sections through crack growth surface of tubular welded joint.

Figure 7. Tortuous crack growth surface in a tubular welded joint.

Spot Welds

Manyspot welds are lap joints, loaded in tension, as shown in Fig. 9. Whenspot welded

lap joints are loaded in fatigue, there are two distinct types of crack path [24], as shown

in Fig. 10. At low stress levels maximumprincipal stress dominated (Mode I) fatigue

cracks originate at or near a point A in Figs. 9 and 10(a). These link up into a single

crack and grow through the thickness of a sheet. The resulting crack front family is

shown schematically in Fig. 10(a). At high stress levels shear dominated fatigue cracks

originate at points C and remain within the plane of the weld, as shown schematically in

Fig. 10(b). At points C only ModeIII is present, but in general shear dominated fatigue

crack growth is in mixed ModesII and III. A high stress may be defined [25], as one in

which the average shear stress across the weld, at the maximumload in the fatigue

cycle, is greater than about 80 per cent of the shear yield stress.

Figure 8. Oscillating crack front families in high strength aluminium alloy. Crack

growth left to right.

Figure 9. Spot welded lap joint.

Figure 10. Crack paths in spot welded lap joints. (a) Maximumprincipal stress

dominated crack path. (b) Shear dominated crack path.

Aluminium Alloy Specimen

Under microscopic examination the most striking feature of many of the fracture

surfaces created by a growing fatigue crack is the presence of distinct line markings,

parallel to each other, and normal to the local direction of crack growth. Figure 11 is an

example on a fatigue crack surface of an Al-7.5 Zn-2.5 M g specimen tested under

constant amplitude loading [1]. The figure was derived from Fig. 2.4 of Ref. [26]. These

lines are called striations, and each striation corresponds to one load cycle. Hence the

distance between striations is the amount the crack front has movedforward during one

cycle, and striations are examples of crack front families. The scale of observation in

Fig. 11 is about 10-3 mm.In effect, the crack surface has been projected onto a plane, so

some detail has been lost [15].

Figure 11. Crack front family on a fatigue crack surface of an Al-7.5 Zn-2.5 M g

specimen.

Slant Crack Growth in Thin Sheets

The slant crack growth which sometimes takes place in thin sheets is an important

exception [1, 27], to the tendency to ModeI crack growth. It is a special case in that it

takes place in isotropic, homogeneous materials under essentially elastic conditions.

Slant fatigue crack growth is sometimes stated to be mixed Modes I and III, but this is

only true for the sheet centre line. Awayfrom the centre line it is mixed Modes I, II and

III [28].

Slant fatigue crack growth is usually observed following a transition from an initial

maximumprincipal stress dominated crack. The main features of the transition are

shown [29] in Fig. 12. The transition starts with the appearance of shear lips. Shear lips

increase in size as crack growth proceeds until they either reach a maximumsize or in

'thin' sheets they meet completing the transition. After the transition the crack growth

surface is flat, and its inclination to the sheet surface is approximately 45°. A large

amount of detailed, empirical information is available on the transition, for example in

Ref. [27].

Figure 12. Transition to slant crack growth under fatigue loading.

Figure 13. ModeIII crack with element of ModeIbranch crack.

Figure 14. Section through twist crack perpendicular to crack growth direction.

Twist Cracks

In the presence of ModeIII displacements on an initial crack a ModeI branch crack can

only intersect the initial crack at one point. This is shown in Fig. 13 for a pure ModeIII

initial crack. Under a loading which includes Mode III a number of Mode I branch

cracks may be initiated along the initial crack front and, as crack growth proceeds, these

may link up to produce what is called a ‘twist’ crack, [2, 11, 30]. Figure 14 shows a

schematic section through a twist crack at a scale of observation of around 0.1 mm.The

crack growth direction is perpendicular to the paper. The twist crack consists of narrow

ModeI facets connected by irregular, predominantly ModeIII cliffs. It is produced by a

combination of maximumprincipal stress dominated crack growth and shear dominated

crack growth [31].

Figure 15. Three point bend slant notch specimen. Dimensions in mm.

Figure 16. Twist crack in a mediumstrength structural steel.

Some laboratory scale fatigue tests on a medium strength structural steel [32]

illustrate the development of twist cracks. The tests were carried out on three point bend

slant notch specimens (Fig. 15). Figure 16 shows a typical crack surface (β = 60°). The

expected tendency to ModeI fatigue crack growth was observed on two distinct scales.

On a scale of 1 m mcrack fronts were approximately straight, and initially fatigue crack

growth was mixed mode. As a crack grew the crack front rotated until it was

perpendicular to the specimen surfaces, and fatigue crack growth was in ModeI. Onthis

scale the crack follows a curved path which tends towards a plane of symmetry as

shown schematically in Fig. 17. This is in accordance with the well knownobservation

[1], that the tendency to Mode I crack growth means that cracks tend to grow

perpendicular to the maximumprincipal tensile stress. More generally, it has been

observed [33] that fatigue crack paths often tend towards a particular plane. Attractor is

an appropriate modern nonlinear dynamics term for such a plane [34] in a physical

rather than a state space sense.

On a smaller scale of 0.1 m mthe tendency to ModeI fatigue crack growth results in

the production of a twist crack containing individual ModeI facets. The ModeI facets

gradually merge (Fig. 16) as, viewed on the large scale, the crack growth surface

becomes perpendicular to the specimen surfaces (Fig. 17).

Figure 17. Typical crack path from an initial slant crack.

M O D IEC R A CPKA T HP R E D I C T I O N

Attempts to predict the path which will be taken, on a macroscopic scale, by a growing

fatigue crack have a long history. Cazaud [7] states that 'Visual examination of broken

machine parts, and photoelastic studies, have shown that fatigue cracks follow a

direction perpendicular to the applied principal stresses.' In support he cites a 1933

German reference by Oschatz. Figure 18 showing the section through a crack path in a

shaft at an abrupt change of section is taken from this reference. The left hand sketch

shows the crack path predicted assuming that it is perpendicular to maximumprincipal

stress trajectories. The other two sketches show that actual crack paths are close to the

prediction even though the distribution of stress is perturbed.

Figure 18. Crack path in a shaft at an abrupt change of section.

Theoretical prediction of ModeI (principal stress dominated) fatigue crack paths in

two dimensions is relatively simple. The crack path is a line and only Modes I and II

can be present. In each increment of crack growth all that is needed to define the crack

path is a direction, together with the amount of crack growth. Twodimensional ModeI

crack path predictions, for an initial mixed Modes I and II crack, have been carried out

by a number of authors using the same general scheme [11]. Calculations are carried out

numerically using small increments of straight crack growth. The direction taken by

each increment is selected using the criterion that it is pure ModeI. Predicted paths in

general tend to a curve. Agreement between theoretical predictions and experimental

data obtained using thin sheets is variable [33]. This is not surprising because three

dimensional effects are not taken into account.

In three dimensions the situation is muchmore complicated. If fatigue crack growth

is assumed to be in ModeI then, in the general case in which ModeIII displacements

are present on the initial crack, a twist crack (Figs 14, 16, 17) is produced. At the

present state of the art it is not possible to predicted the path of a twist crack on a scale

of observation, say 0.1 mm, at which the individual ModeI facets of a twist crack can

be distinguished. It is merely possible to describe observed twist cracks.

If a larger scale of observation, say 1 mm,is used then a twist crack growth surface

may be regarded as smooth but crack growth is not, in general, in pure ModeI. If an

appropriate criterion were available it would be possible to define the direction of crack

growth at points on the crack front together with amounts of crack growth. This then

determines a new crack front, and the process is repeated to build up a net of crack

fronts and crack trajectories.

In a practical numerical implementation [35] for a slant crack specimen similar to

that shown in Fig. 15, the strain energy rate release criterion was used to determine

crack growth directions and increments. It was pointed out that the predicted crack path

could only be regarded as a qualitative estimate. Nevertheless a plausible crack path was

obtained in which the ModeII stress intensity factor was zero and, as the crack grew,

the ModeIII stress intensity factor tended to zero.

C O N C L U D IRNEGM A R K S

Paths taken by growing fatigue cracks have been of industrial interest for a very long

time. 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 growing fatigue crack are not

completely understood.

The numerous possible crack configurations [36] mean that a systematic theoretical

approach to fatigue 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 fatigue

crack paths at an appropriate scale.

The examples given have been chosen to illustrate some of the more important

aspects of fatigue crack paths in metals.

R E F E R E N C E S

1.

Pook, L.P. (1983) The Role of Crack Growth in Metal Fatigue. Metals Society,

London.

2.

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

Applications. W I TPress, Southampton.

3. Carpinteri, A. (Ed). (1994) Handbook of Fatigue Crack Propagation in Metallic

Structures. Elsevier Science BV, Amsterdam.

4. 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.

5. Mandelbrot, B.M. (1977) Fractals, Form, Chance and Dimension. San Francisco:

W H Freeman and Co.

6. Bacon, F. (1930) Fatigue Stresses with Special Reference to the Breakage of Rolls.

The South Wales Institute of Engineers, Cardiff.

7. Cazaud, R. (1953) Fatigue of Metals. Chapman& Hall Ltd, London.

8.

Forsyth, P.J.E. and Ryder, D.A. (1960) Aircraft Engineering 32, 96-99.

9. Forsyth, P.J.E. (1961) In: Proc. Crack Propagation Symposium, Vol. 1, pp. 76-94,

The College of Aeronautics, Cranfield.

10.

Pook, L.P. (1962) Quantitative Fractography. Examples Showing Information

Derived from Aircraft Components after Fatigue Testing. Laboratory Test Note

LTN212. Coventry: Hawker Siddeley Aviation Ltd.

11. Pook, L.P. (2002) Crack Paths. W I TPress, Southampton.

12. Mandelbrot, B.M. (1983) The Fractal Geometry of Nature. W.H. Freeman and

Company, N e wYork.

13. Pook, L.P. (1976) J. Soc. Env. Eng. 15-4, 3-8.

14. Hull, D. (1999) Fractography: Observing, Measuring and Interpreting Fracture

Surface Topography. Cambridge University Press, Cambridge.

15. Chermant, J.L. and Coster, M. (1979) J. Mat. Sci. 19, 509-534.

16. Minoshima, K., Suezaki, T. and Komai, K. (2000) Fatigue Fract. Engng. Mater.

Struct. 23, 435-443.

17. Schwarz, H. and Scherer, S. (2003) Materials World 11, 18-19.

18. Forsyth, P.J.E. (1969) The Physical Basis of Metal Fatigue. Blackie and Son Ltd,

London.

19.

Miller, K.J. and McDowell, D.L. (1999) In: Mixed-Mode Crack Behavior. ASTM STP 1359, pp. vii-ix, Miller, K J. and McDowell, D.L. (Ed.). Americ n Society for

Testing and Materials, West Conshohocken, PA.

20.

Vinas-Pich, J., Kam, J.C.P. and Dover, W.D. (1996) In: Fatigue Crack Growth in

Offshore Structures, pp. 107-145, Dover, W.D., Dharmavasan, S., Brennan, F.P.

and Marsh, K.J. (Ed.). Engineering Materials Advisory Services Ltd, Solihull.

21.

Pook, L.P. (1998) The Archive ofMechanical Engineering 45, 143-156.

22.

Smith, A.T., Dover, W.D. and McDiarmid, D.L. (1996) In: Fatigue Crack Growth

in Offshore Structures, pp.231-301, Dover, W.D., Dharmavasan, S., Brennan, F.P.

and Marsh, K.J. (Ed.). Engineering Materials Advisory Services Ltd, Solihull.

23.

Frost, N.E., Marsh, K.J. and Pook, L.P. (1974) Metal Fatigue. Clarendon Press,

Oxford. Reprinted (1999). Dover Publications Inc., Mineola, NY.

Radaj, D. and Sonsino, C.M. (1999) Fatigue Assessment of Welded Joints by Local

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Approaches. Abington Publishing, Cambridge.

25.

Pook, L.P. (1975) Int. J. Fract., 11, 173-176.

26.

Stubbington, C.A. (1963) Some Observations on Air and Corrosion Fatigue

Fracture Surfaces of Al, 7.5Zn, 2.5Mg. Report No. CPM4, Royal Aircraft

Establishment: Farnborough.

27.

Zuidema, J. (1995) Square and Slant Fatigue Crack Growth in Al 2024. Delft

University Press, Delft.

28.

Pook, L.P. (1993) In: Mixed-Mode Fracture and Fatigue. ESIS 14, pp. 285-302,

Rossmanith, H.P. and Miller, K.J. (Ed.). Mechanical Engineering Publications,

London.

29.

Schijve, J. (1974) Eng. fract. Mech., 6, 245-252.

30.

Lawn, B.R. and Wilshaw, T.R. (1975) Fracture of Solids. Cambridge University

Press, Cambridge.

31.

Pook, L.P. (1985) In: Multiaxial Fatigue, A S T M S T 8P53, pp. 249-263, Miller, K.J.

and Brown, M.W.(Ed.). American Society for Testing and Materials, Philadelphia,

PA.

32.

Pook, L.P. and Crawford, D.G. (1991) In: Fatigue under Biaxial and Multiaxial

Loading. ESIS 10, pp. 199-211, Kussmaul, K., McDiarmid, D.L. and Socie, D.F.

(Ed.). Mechanical Engineering Publications, London.

33.

Pook, L.P. (1995) Int. J. Fatigue, 17, 5-13.

34.

Pook, L.P. (1997) Int. J. Bifurcation and Chaos, 7, 469-475.

35.

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36.

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Intensity Factor Solutions. NELReport 704. National Engineering Laboratory, East

Kilbride, Glasgow.

CrackPath and Branching from Small Fatigue Crack

under MixedLoading

Y. Murakami1,K. Takahashi2 and K. Toyama1

1 Department of Mechanical Engineering Science, Kyushu University, 6-10-1 Hakozaki,

Higashi-ku, Fukuoka, 812-8581, Japan, ymura@mech.kyushu-u.ac.jp

2 Department of Energy & Safety Engineering, YokohamaNational University, 79-5

Tokiwadai, Hodogaya-ku, Yokohama,240-8501, Japan, ktaka@ynu.ac.jp

ABSTRACT.In order to investigate the crack path of materials containing a small

crack under mixed mode loading, reversed torsion and combined push-pull/torsion

fatigue tests were carried out on 0.47% carbon steel specimens containing an initial

small crack. The initial small cracks were introduced by a preliminary push-pull fatigue

test using a specimen which contained an artificial small hole of 40Pm diameter/depth.

Firstly, the mechanism of fatigue crack growth under reversed torsion and combined

push-pull torsion were investigated. Then, fatigue tests of push-pull followed by

reversed torsion and reversed torsion followed by push-pull were carried out. Fatigue

tests of combined push-pull/torsion followed by push-pull were also carried out to

examine the effect of crack geometry, such as branching and kinking, on cumulative

fatigue damage. Different crack growth behaviours due to different loading modes and

sequences complicatedly influence the fatigue crack path and eventully the cumulative

fatigue damage. Thus, existing fatigue damage theories cannot be applied to the cases

presented in this study. The crack like factory-roof morphology is locally made on the

fracture surface of the specimen having a semi-elliptical crack under cyclic torsion.

Torsional fatigue tests of circumferentially cracked specimens were carried out to investigate

the mechanism of mode III crack growth and formation of the factory-roof morphology. The

factory-roof morphology in torsional fatigue of cracked specimen is formed by mode I crack

branching from small semi-elliptical cracks nucleated ahead of the initial crack tip by shear.

I N T R O D U C T I O N

The behaviours of small fatigue cracks under mixed mode loading have recently been

studied by several researchers [1,2]. It has been recognized in existing literature that

analyzing the path of small cracks is essential to make clear mechanical and

microstructural factors affecting the fatigue strength under mixed mode loadings.

The effect of loading mode sequence has been studied by several researchers. [3-5].

It has been pointed out that the cumulative fatigue damage of torsion followed by push

pull (rotating-bending) and push-pull (rotating-bending) followed by torsion are

different and cannot be predicted by Miner’s rule for carbon steels [3,4] and stainless

1

steel [5]. The reason for deviation from Miner’s rule is presumed to be the complicated

behaviours of small cracks under the different loading modes and sequences.

Harada et al. [3] carried out sequential-fatigue tests of rotating bending and reversed

torsion using a 0.24% C steel and reported that: (i) In rotating bending followed by

reversed torsion, the cumulative damage (D) was in the range of 1.46 to 2.15, and (ii) in

reversed torsion followed by rotating bending, D was approximately unity (D # 1).

Zhang and Miller [4] carried out sequential-fatigue tests of push-pull and reversed

torsion using a 0.45% C steel and reported that: (i) In the sequence of push-pull

followed by torsion (PP–to–T), D was always greater than unity and D #

2 for certain

conditions, and (ii) in the sequence of torsion followed by push-pull (T–to–PP), D was

smaller than unity (D<1). However, these studies were conducted using plain specimens

in which the initiation and growth behaviour of stage I cracks and growth bahaviour of

stage II influence so-called fatigue damage D. Separating the influence of stage I crack

and stage II crack is necessary to understand the deviation of D from 1 under various

conditions.

In this paper, fatigue tests of PP–to–T and T–to–PP were carried out on 0.47% C

steel specimens containing an initial small crack of 400Pm in surface length. Fatigue

tests of combined push-pull/torsion followed by push-pull (PP/T–to–PP) were also

carried out to investigate the effects of crack geometry, such as branching and kinking

from an initial small crack, on cumulative fatigue damage. Excluding the influence of

initiation and growth of stage I crack, cumulative fatigue damage was studied from the

viewpoint of crack propagation.

The factory roof morphology was formed on the fracture surface of the specimen

having a semi-elliptical crack only when the surface length of a semi-elliptical crack

was larger than ~1 mm.It has been reported by several workers that the fracture surface

of mode III fatigue crack growth test specimen shows so-called “factory-roof”

morphology [6,7]. However, the exact formation mechanism of factory-roof has not

been made clear. Torsional fatigue tests of circumferentially cracked specimens were

carried out to study the mechanism of mode III crack growth and formation of the

factory-roof morphology.

E X P E R I M E N TPARLO C E D U R E S

Material

The material used was a rolled bar of 0.47% C steel (JIS S45C) with diameter of 25mm.

The chemical composition of material is (wt.%): 0.47C, 0.21Si, 0.82Mn, 0.018P, 0.018S,

0.01Cu, 0.018Ni and 0.064Cr. Mechanical properties of the material are: 620MPa

tensile strength, 339MPa lower yield strength, 1105MPa true fracture strength and

53.8% reduction of area.

Specimen having a small semi-elliptical surface crack

Figure 1 shows the shape and dimension of test specimen. Specimens were made by

turning after annealing at 844°C for 1h. After surface finishing with emery paper,

2

~25Pm of surface layer was removed by electro-polishing. After electro-polishing, a

hole was introduced onto the surface of each specimen. The diameter of the hole, 40Pm,

is equal to the depth. After introducing a small hole, the specimens were annealed in a

vacuum at 600°C for 1h to relieve residual stress induced by drilling. The Vickers

hardness after vacuum annealing is HV=174, which is a mean value of each specimen

measured at 4 points with load of 0.98N. The scatter of H Vis within 5%.

A hydraulically controlled biaxial testing machine was used for both introduction of

the pre-crack by push-pull and subsequent fatigue tests. Push-pull fatigue tests were

conducted at Va = 230 MPa, in order to introduce pre-cracks of 400 P m in surface length

including a hole. These tests were conducted under load control, at a frequency of 20 Hz

with zero mean stress(R = -1). Specimens were annealed in a vacuum at 600°C for 1 h

again to relieve the prior fatigue history by push-pull.

(a)

(b)

Figure 1. Fatigue test specimen. (a) Shape and dimension of the specimen

for reversed torsion and combined push-pull/torsion fatigue

tests; dimensions in mm.(b) Artificial small hole.

Reversed torsion and combined push-pull/torsion fatigue tests

Fatigue tests under reversed torsion and combined push-pull/torsion were conducted

under load control at a frequency of 12 to 20 Hz with zero shear mean stress. In order to

investigate the effects of axial mean stress on the crack path, torsional fatigue tests with

tensile or compressive mean stress were also carried out. The loading cycle pattern is a

sine wave. In-phase combined push-pull/torsion fatigue tests were carried out at

Wa/Va=2.0.

constant stress amplitude ratio,

Plastic replicas were taken during the tests in

order to monitor crack path. The fracture surfaces of specimens were observed using the

scanning electron microscope (SEM).

Sequential fatigue tests

Sequential fatigue tests of push-pull, reversed torsion and combined push-pull/torsion

were carried out as follows.

(a) Reversed torsion followed by push-pull. (T–to–PP)

(b) Combinedpush-pull/torsion in phase followed by push-pull (PP/T–to–PP)

(c) Push-pull followed by reversed torsion (PP–to–T)

Stress ranges for sequential fatigue tests were chosen so that the fatigue lives (Nf)

under single loading would be in the range of Nf= 3×105 to 4×105.

Va = 191 MPa, Nf,pp = 3.43×105.

x Push-pull:

Wa= 167 MPa, Nf,t = 3.12×105.

x Reversed torsion:

Va = 71 MPa, Wa= 142 MPa, Nf,pp/t = 3.79×105.

x Combinedpush-pull/torsion:

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where Nf,pp, Nf,t and Nf,pp/t denote fatigue life for each loading pattern. In the subsequent

discussion, pp, t and pp/t denote push-pull, reversed torsion and combined push

pull/torsion, respectively.

ModeIII fatigue crack growth test

Figure 2(a) shows the shape and dimensions of the test specimen for the mode III

fatigue crack growth test. The specimen has a circumferential notch as shown in Fig.

2(b). A hydraulically controlled biaxial testing machine was used for the introduction of

the pre-crack by push-pull test and also for the torsional fatigue tests. Push-pull fatigue

MPa, in order to introduce a pre-crack of ~200Pm in

tests were conducted at Va=150

depth. These specimens were annealed in a vacuumat 600°C for 1 h again to relieve the

prior fatigue history introduced in push-pull fatigue test. The torsional fatigue tests were

conducted under a load control condition at a frequency of 5~12Hz with zero mean

stress. The loading cycle pattern is a sine wave.

(b)

(a)

Figure 2. Fatigue test specimen. (a) Shape and dimension of the specimen for

modeIII crack growth test (Torsional fatigue). (b) Detail of notch;

dimensions in mm.

R E S U L TASN DDISCUSSIONS

Crack path under reversed torsion and combined push-pull/torsion

Figure 3 shows the branched cracks emanating from the initial crack tip under reversed

torsion. mode I branched cracks continued propagating and led the specimen to failure.

Figure 4 shows the paths of the branched cracks of broken specimens. The branched

cracks which started from the initial crack tips were illustrated in the same figures,

respectively, for 200Pm, 400Pm and 1000Pmpre-cracked specimens [8]. The line of

qr45is the direction perpendicular to the principal stresses and the line of qr5.70 is the

local maximumnormal stress (VTmax) at the crack tip.

Figure 5 shows cracks kinked by modeI from the tip of initial crack under combined

push-pull/torsion. modeI cracks continued to propagate and led the specimen to failure.

Figure 5(b) shows the shape and angle of kinked cracks [9]. The line of -38.0q is the

direction perpendicular to the principal stresses and the line of -61.4q is the local

maximumnormal stress (VTmax) at the crack tip.

The branched cracks and kinked cracks propagated eventually in a direction

perpendicular to the principal stresses, though the initial branching or kinking angles are

obviously larger than them and close to the direction perpendicular to the local

maximumtangential stress (VTmax) [8,9].

4

Murakami and Takahashi showed that the fatigue limit of pre-cracked specimens

under reversed torsion [8] and combined push-pull/torsion [9] is the threshold condition

for nonpropagation of mode I cracks emanating from the initial crack tip, i.e., the

fatigue limit is determined by the condition for the nonpropagation of branched cracks

for reversed torsion and kinked cracks for combined push-pull/torsion. Based on the fact

that nonpropagating cracks under torsion and combined stress are mode I cracks, the

area parameter model [10,11] could be applied to predict the fatigue limit [8,9].

N=4×105

Figure 3. Propagation of branched cracks. (The 400 P mpre-cracked

specimen, Wa=152 MPa, Nf=7.9×105.)

(b)

(a)

(c)

Figure 4. The shape and angle of branched cracks. (a)The 200Pmpre-cracked

specimen, (b)The 400Pmpre-cracked specimen, (c)The 1000Pm

pre-cracked specimen.

5

(a)

(b)

Figure 5. (a) Propagation of kinked cracks under combined push-pull/torsion

loadings.

Va=71MPa, Wa=142MPa.

(b) The shape and angle of kinked

cracks. The origin of the coordinate is the tip of the initial crack.

Effects of axial meanstress the on the fatigue crack path under torsional loading

In order to examine the effects of axial mean stress on the crack path, torsional fatigue

tests with tensile or compressive mean stress were carried out [12]. Figures 6 and 7

show the branching of the surface cracks from an initial semi-elliptical crack by

torsional fatigue with tensile and compressive mean stress, respectively.

The initial branching angle of cracks that emanated at initial crack tip was close to

±70.5° to the initial crack plane, where the cyclic component of local tangential stress

'VT at the crack tip has the maximumvalue. The branched cracks eventually propagated

perpendicularly to the plane, where the cyclic component of normal stress has the

maximumvalue, i.e. ±45°. The paths of the branched cracks shown in Figs. 3, 6 and 7

are almost same regardless of axial mean stress. Thus, fatigue crack path is determined

by the direction where the cyclic component of nominal stress has the maximumvalue.

In other words, meanstress hardly influence the direction of fatigue crack propagation.

(a) N =7.0×105

(b) Angle ofbranching

Figure 6. Propagation of the branched cracks under torsion with tensile mean

stress. (Vm=98 MPa, Wa =142 MPa, Nf=1.0×106).

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(b) Angle of branching

(a) N = 3.5×105

Figure 7. Propagation of the branched cracks under torsion with compressive

mean stress. (Vm= -98 MPa, Wa =162 MPa, Nf=6.7×105).

Ohji et al. [13] studied the path of a fatigue crack in the residual stress fields of HT80

steel. Ohji et al. indicated that the cyclic components of normal stress determined the

crack path. Tanaka et al. [14] investigated the path of branched cracks under cyclic

torsion with or without tensile mean stress in the mediumcarbon steel tubular specimen

having a pre-crack of 1mm.Tanaka et al. reported that the path of branched cracks was

determined by the cyclic components of normal stress. The present experimental results

are consistent with those of Ohji et al. and Tanaka et al..

Effects of loading sequence on fatigue crack path

Cumulative fatigue damage

Figure 8 shows the results of so-called cumulative damage tests compared to results

predicted by Miner’s rule. The fraction of life in reversed torsion (nt/Nf,t) and combined

push-pull/torsion (npp/t/Nf,pp/t)

is plotted against the fraction of life in push-pull (npp/Nf,pp).

The fraction of life of the first loading is selected from 0.2, 0.4, 0.6 and 0.8. (In order to

show the sequence of stress, the terms “the first loading” and “the second loading” will

be used in this paper.) After completion of the first loading and commencementof the

second loading, the fatigue tests were continued until specimen failure. The cumulative

fatigue damage (D) was calculated as the summation of fractions of fatigue life of the

first and second loadings. In all loading sequences, D is larger than unity (D > 1). In the

sequence of T–to–PP, D was in the range of 1.43 to 2.13. This result of fatigue

accumulation is opposite to that for plain specimens of similar materials [3,4]. In the

sequence of PP/T–to–PP, D is smaller than the value of D obtained in the sequential test

of T–to–PP. Therefore, D is clearly dependent on the first loading, i.e., reversed torsion

or combined push-pull/torsion.

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Figure 8. Fatigue test results compared to Miner’s rule

Crack path in sequential fatigue tests

Figure 9(a) shows the crack path from initial crack tips under torsion. Under the

subsequent push-pull, cracks started from the branched crack tips and propagated in a

direction perpendicular to the specimen axis, leading the specimen to failure. Not all

cracks initiating from the branched crack tips necessarily continued propagating; some

stopped propagating as shown by the arrows in Fig. 9(a). The stress intensity factor at

the tip of branched cracks is smaller than that of the tip of straight crack and this cause

the reduction in crack growth rate whentorsion is switched to push-pull. The evaluation

of stress intensity factor is discussed in the next section.

Figure 9(b) shows the crack path from the initial crack in the sequence of PP/T–to–

PP. Kinked cracks emanated from the initial crack tips under combined push

pull/torsion. Under the subsequent push-pull, cracks extended from the kinked crack

tips. These cracks propagated perpendicular to the specimen axis and led the specimen

to failure.

Figure 9(c) shows the crack path from the initial crack in the sequence of PP–to–T.

Cracks initiated from the initial crack tips under push-pull and naturally propagated

perpendicular to the specimen axis. After changing the loading to reversed torsion, this

crack branched and propagated in the direction perpendicular to the remote maximum

principal stress, i.e., ±45° to the axial direction.

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(a)

(b)

(C)

Figure 9. Crack path from the initial crack with length of 400 Pm: (a)

Reversed torsion with nt/Nf,t=0.4 followed by push-pull, 3.97×105

push-pull cycles, npp/Nf,pp=1.16; (b) Combined push-pull/torsion

with npp/t/Nf,pp/t=0.4

followed by push-pull, 1.72×105 push-pull

cycles, npp/Nf,pp=0.5; (c) Push-pull with npp/Nf,pp=0.4 followed by

torsion, 6.24×104 torsion cycles, nt/Nf,t=0.2.

Crack propagation curves and fracture mechanics evaluation

Although the reality of fatigue damage of a specimen should be related to the size of

crack [15], the term “fatigue damage” will be used in this paper as the value defined

conventionally by Miner’s rule. The fatigue damage calculated by Miner’s rule will be

discussed from the viewpoint of crack propagation. Figure 10 shows crack propagation

curves. Crack propagation under the first loading is shown by a dotted line and crack

propagation under the second loading is plotted by open marks connected with solid line.

Crack length is denoted by the surface length projected onto the axial direction.

The crack propagation curves for pure push-pull or reversed torsion is also shown in

Figs. 10(a) to (c). Immediately after switching from the first loading to the second

loading, a reduction in crack growth rate compared to the single loading occurred, i.e.,

compared to push-pull in Figs.10(a) and (b) and reversed torsion in Fig. 10(c).

Comparing Figs. 10(a) and (b), the reduction in the crack growth rate is larger for the

sequence of T–to–PP than for PP/T–to–PP. Thus, D obtained in the sequence of T–to–

PP was larger than D of PP/T–to–PP.

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