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A

Fracture Mechanics

A.1 Introduction

The applied mechanics framework for study of the behaviour of cracked bod-ies under load is known as fracture mechanics. The application of fracturemechanics to fatigue crack propagation is well established, and most mod-ern books on metal fatigue include an introduction to the topic. The accountbelow is based on Frost et al. (1974) and Pook (2000a, 2002a). These booksinclude numerous references. The book by Frost et al. was the first on metalfatigue in which a fracture mechanics approach was used throughout. It wasreprinted in 1999.

Fracture mechanics does not provide any information about the processesinvolved in fatigue crack propagation. It does provide the descriptive and ana-lytic framework needed for their characterisation, and for the application offatigue crack propagation data to practical engineering problems.

Simplifying assumptions have become conventional in much present dayfracture mechanics, and these are satisfactory for many purposes. The mater-ial is assumed to be a homogeneous isotropic continuum, and its behaviouris assumed to be linearly elastic. Crack surfaces are assumed to be smooth,although on a microscopic scale they are generally very irregular. Modifica-tions are made to basic linear elastic fracture mechanics theory to allow forthe actual behaviour of real materials. The basic ideas in linear elastic frac-ture mechanics are straightforward. The mathematics involved is often for-midable, but does lead to the useful and easily applied key concept of stressintensity factor, which describes the elastic stress and displacement fields inthe vicinity of a crack tip. A stress intensity factor has the dimensions ofstress ×√

length. The most widely used units are MPa√

m. These units ap-pear in many standards and are therefore to be preferred. The use of MPa

√m

168 Appendix A: Fracture Mechanics

Figure A.1. Notation for crack tip stress field (Frost et al. 1974).

is not particularly convenient since crack sizes are normally measured in mm,and N/mm3/2 units are sometimes used. 1 MPa

√m = √

1000 N/mm3/2 ≈31.62 N/mm3/2.

Some figures which appear in the main text are repeated in order to makethe appendix self contained.

A.2 Notation for Stress and Displacement Fields

The conventional notation for the position of a point relative to the crack tip,and for the stresses at this point, is shown in Figure A.1. The point on thecrack tip is the origin of the coordinate system and the z axis lies along thecrack tip. Displacements of points within the cracked body when the body isloaded are u, v, w in the x, y, z directions. The terms crack tip and crack frontare synonymous. Crack tip tends to be used for two dimensional situationsand crack front in three dimensions.

A.2.1 CRACK SURFACE DISPLACEMENT

A fundamental fracture mechanics concept is that of crack surface displace-ment. In fracture mechanics the interest is in what happens in the vicinity

Metal Fatigue 169

Figure A.2. Notation for modes of crack surface displacement (Frost et al. 1974).

of the crack tip, so it is sometimes referred to as crack tip surface displace-ment. If a load is applied to a cracked body, then the crack surfaces moverelative to each other. For points on opposing crack surfaces that were ini-tially in contact there are three possible modes of crack surface displacement(Figure A.2); Mode I where opposing crack surfaces move directly apart indirections parallel to the y axis; Mode II where crack surfaces move over eachalong the x axis, that is, perpendicular to the crack tip; and Mode III wherecrack surfaces move over each other in directions parallel to the z axis, thatis, parallel to the crack tip. By superimposing the three modes, it is possibleto describe the most general case of crack surface displacement. The termsMode I, Mode II and Mode III are usually capitalised, and are often used inthe metal fatigue literature without explanation. The descriptive terms; open-ing mode, edge sliding mode, and shear mode are sometimes used for ModesI, II and III respectively. The term mixed mode means that at least one mode,other than Mode I, is present. The modes of crack surface displacement mayalso be used to characterise crack propagation.

A particular type of elastic crack tip stress field is associated with eachmode of crack surface displacement (Paris and Sih 1965). These stress fieldsare characterised by stress intensity factors, symbol K. Subscripts I, II andIII are used to denote mode. Where there is no subscript, Mode I is usuallyimplied; this convention is sometimes used in the text. Corresponding dis-placement fields permit calculation of crack tip surface displacements.

It is matter of observation that, when viewed on a macroscopic scale,and under essentially elastic conditions, cracks in metals tend to propagatein Mode I, so attention is largely confined to this mode (see Sections 3.4.2,


Figure A.3. Square section ring element around crack tip. Reprinted from Linear Elastic Frac-ture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312-703-5, 2000.

7.1 and 8.1). At smaller scales, cracks are generally very irregular and differ-ent modes of crack surface displacement may be observed.

A.2.2 VOLTERRA DISTORSIONI

If a crack surface is considered as consisting of points then the three modes ofcrack surface displacement (Figure A.2) provide an adequate description ofthe movements of crack surfaces when a load is applied. However, if the sur-face is regarded as consisting of infinitesimal elements, then element rotationsmust also be described, and Volterra distorsioni (distortions) are appropriate.Volterra distorsioni are also applicable in the theory of crystal dislocations(Nabarro 1967).

In his analysis of distorsioni Volterra considered the simplest multiplyconnected body, that is a cylinder with a central hole. An account in Englishis given in Zastrow (1985). The cylinder, free of body forces, surface forces,or any initial stress, is made simply connected by a cut along a radial plane.The cut surfaces are regarded as completely rigid, but the remainder of thecylinder is elastic. The cut surfaces may be moved relative to each other insix different ways, so there are six distinct Volterra distorsioni.

The conventional approach to Volterra distorsioni needs to be modifiedfor the description of crack tip surface movements (Pook 2000a). Considera pair of infinitesimal elements, A and B, which are in the xy plane and aresituated on the upper and lower surface of an unloaded crack respectively(Figure A.3). Their initial coordinates are (r, 180◦) and (r,−180◦). The ele-ments A and B are connected by a ring element of infinitesimal width. Forclarity, this ring element is shown as having a square cross section and theelements are separated in the figure. Such ring elements correspond to Vol-terra’s cylinders.

Three of the six Volterra distorsioni, correspond to the three modes ofcrack tip surface displacement, Figure A.4 (Pook 2002a). Under a Mode I

Metal Fatigue 171

Figure A.4. Volterra distorsioni, modes of crack surface dislocation. Reprinted from CrackPaths. LP Pook, WIT Press, ISBN 1-85312-927-5, 2002.

loading element A moves a distance v in the y direction, and element B a dis-tance −v (Figure A.4 left). For consistency with crack tip surface displace-ment notation this is called a Mode I dislocation. The ring element remainswithin the xy plane, that is, initially plane sections, perpendicular to the cracktip remain plane. Distortion of the ring element is symmetrical about the xz

plane. Under a Mode II loading element A moves a distance u in the x dir-ection, and element B a distance −u, and this is a Mode II dislocation (Fig-ure A.4 centre). The ring element remains within the xy plane, that is planesections remain plane, but its distortion is not symmetrical about the xz plane.Under a Mode III loading element A moves a distance −w in the z direction,and element B a distance w (Figure A.4 right). This is a Mode III dislocation.The ring element does not remain within the xy plane, that is plane sectionsperpendicular to the crack front do not remain plane. Distortion of the ringelement has rotational symmetry about the x axis.

The remaining three Volterra distorsioni involve rotation of the elements Aand B, and are usually called disclinations. The notation used here is based onthe idea that each mode of dislocation and disclination is associated with thesame coordinate axis (Pook 2002a). Hence in a Mode I disclination, elementA rotates through an angle β about an axis parallel to the y axis, and elementB rotates through an angle −β (Figure A.5 right). In a Mode II disclination,element A rotates through an angle α about an axis parallel to the x axisand element B rotates through an angle −α (Figure A.5 centre). In a Mode IIIdisclination, element A rotates through an angle γ about an axis parallel to thez axis and element B rotates through an angle −γ (Figure A.5 left). Modes IIand III disclinations cannot exist in isolation because of interference betweenelements A and B. Any dislocation mode may be decomposed into a dipoleof equal and opposite disclinations of either of the other two modes.


Figure A.5. Volterra distorsioni, modes of crack surface disclination. Reprinted from CrackPaths. LP Pook, WIT Press, ISBN 1-85312-927-5, 2002.

If movements of the elements A and B are identical, then a dislocationbecomes a translation and a disclination a rotation. For convenience, con-tinuous distributions of dislocations are usually referred to as dislocations,and similarly for disclinations, translations and rotations.

A.3 Stress Intensity Factors

It is intuitively obvious that a large crack is more severe than a small crack.A basic requirement for the study of the behaviour of cracked bodies is toexpress this numerically. The presence of a crack dominates the stress field inthe vicinity of the crack tip, and some results are not intuitively obvious.

The key concept of stress intensity factor, for Mode I and for Mode II,arises from a two dimensional linearly elastic analysis for a straight crack.This follows the usual methods of elastic stress analysis in which strains anddistortions are assumed to be small, and conditions of equilibrium and com-patibility must be satisfied (Gere and Timoshenko 1991). The use of a twodimensional analysis simplifies the mathematics, and also simplifies somedescriptions. For example, what is meant by crack length is unambiguous.Mode III is not possible in two dimensions, so for this mode a quasi twodimensional anti plane analysis is used.

Stress intensity factors may be used to characterise the mechanical prop-erties of cracked specimens in just the same way that stresses are used tocharacterise the mechanical properties of uncracked specimens. They help toquantify the rather elusive concept of a material’s toughness. One conveni-ent definition is this: resistance to crack propagation (including fatigue crackpropagation). For example the fracture toughness, Kc, of a metallic mater-ial may be defined as the value of the Mode I stress intensity factor, KI, for

Metal Fatigue 173

failure under a static load. In practice, failure is not abrupt, and the frac-ture toughness is defined for a small, specified amount of crack propagation(Anon. 2005g).

The stress field in vicinity of a crack tip is dominated by the leading termof a series expansion of the stress field. For a particular mode of crack surfacedisplacement this leading term is always of the same general form. Individualstress components are proportional to K/

√r where K is the stress intensity

factor and r is the distance from the crack tip (Figure A.1). A stress intensityfactor is a singularity of order −1/2; as the crack tip is approached, stressestend to infinity. A formal definition of the Mode I stress intensity factor is

KI = lim(r → 0)σy

√2πr , (A.1)

where σy is the stress perpendicular to the crack along the x axis. The numer-ical factor,

√2π , in the equation is the usual convention. Other conventions

are occasionally encountered, especially in early work, leading to numericallydifferent values for stress intensity factors. There are corresponding equa-tions, in terms of shear stresses, for the other two modes. Stress intensityfactors of the same mode may be combined by algebraic addition.

Once K is known, stress and displacement fields in the vicinity of thecrack tip are given by standard equations (Pook 2000a). In Modes I and II,stresses are independent of the stress state, but displacements are a factor(1 − ν2) less for plane strain than for plane stress (ν is Poisson’s ratio). Forexample, for Mode I on the x axis in front of the crack

σy = KI√2πr

. (A.2)

Also, for the upper crack surface

v = 2KI

E

√2r

π(plane stress), (A.3)

where v is the displacement in the y direction, that is, perpendicular to thecrack, and E is Young’s modulus. The equation implies that for Mode I acrack opens up into a parabola (see Section A.4). Displacements are alsoparabolic for Modes II and III. Equation (A.3) also implies that Mode I cracksurface displacement is a combination of a Mode I dislocation and a ModeIII disclination (Figures A.4 and A.5). Similarly, Mode III crack surface dis-placement is a combination of a Mode III dislocation and a Mode I disclina-tion. However, Mode II crack surface displacement is just a Mode II disloca-tion.

KI must be positive, since a compressive load simply holds a crack closed.However, the Modes II and III stress intensity factors, KII and KIII can be


either positive or negative, and the sign used is a matter of convention. It isusually taken as positive, but careful attention to sign is needed when calcu-lating stresses and displacements.

It is possible to obtain stress field equations, using Kirchoff plate bendingtheory, in terms of a bending mode stress intensity factor, KB (Paris and Sih1965). At a plate surface KB is equivalent to KI. Corresponding crack sur-face displacements are a Mode II disclination (Figure A.5 centre). Becauseof crack surface interference this is not physically realistic so KB is not nowused.

A.3.1 STRESS INTENSITY FACTOR SOLUTIONS

Stress intensity factors are available for numerous configurations, for ex-ample Murakami (1987, 1992b, 2001) and this facilitates practical applica-tions. Solutions for test specimens are included in appropriate standards, forexample Anon. (2003a). A solution for a particular configuration is some-times called a K-calibration or a compliance function. Where a solution ispresented as an equation fitted to numerical results, care must be taken notto use it outside its specified range. Conventionally, two dimensional solu-tions are used for sheets and plates of constant thickness subjected to in planeloads. This is usually satisfactory. To illustrate the general form of solutionsfor the Mode I stress intensity factor, KI, some examples are given below.These are all for loads perpendicular to the crack. Stresses parallel to a crackhave no effect.

A.3.1.1 Two Dimensional Solutions

For a centre crack, length 2a (it is conventional to take the length of an in-ternal crack as 2a) under a remote uniaxial tension σ (Figure A.6)

KI = σ√

πa . (A.4)

For a small edge crack, length a, in a sheet under uniaxial tension, shownin Figure A.7 (Pook 2000a).

KI = 1.12σ√

πa . (A.5)

Equation (A.5) also applies to a crack at a blunt notch if the local stressis used (Figure A.8), and to a crack at a sharp notch if a is taken as the cracklength plus the notch depth shown in Figure A.9 (Pook 2000a).

Solutions are sometimes presented in the form

KI = σY√

πa , (A.6)

Metal Fatigue 175

Figure A.6. Centre crack in an infinite sheet under uniaxial tension (Frost et al. 1974).

where Y is a geometric correction factor, usually of the order of 1, and a

is a characteristic crack dimension. Hence in Equation (A.5) Y = 1.12. Tofacilitate calculations other definitions of Y are sometimes used as in Equa-tion (A.7), below.

Test specimens of standard design are included in various fracture mech-anics based standards. For example, for the three point bend single edge notchspecimen shown in Figure A.10 (Anon. 2003b),

KI = FY

B√

W, (A.7)

where F is force, B specimen thickness and W specimen width. Y is givenby

Y = 6√

α [1.99 − α(1 − α)(2.15 − 3.93α + 2.7α2)](1 − 2α)(1 − α)3/2

, (A.8)

where α = a/W and the equation is valid for 0 ≤ α ≤ 1. In Anon. (2003b)Equation (A.7) includes a factor 101.5 so that with force measured in kN, andspecimen dimensions measured in mm, KI values are in MPa

√m.


Figure A.7. Small edge crack in a sheet under uniaxial tension. Reprinted from Linear ElasticFracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312-703-5, 2000.

For a centre crack, length 2a, with point forces F per unit thickness onthe crack faces (Figure A.11)

KI = F√πa

. (A.9)

KI usually increases with increasing crack length. This is one of the few casesin which KI decreases with increasing crack length.

A.3.1.2 Three Dimensional Solutions

A circular crack in an infinite body is often called a penny shaped crack.Under uniaxial tension, σ , perpendicular to the crack

KI = 2σ

√a

π, (A.10)

where a is the crack radius. Hence Y in Equation (A.6) is 2/π . With centralpoint loads F on the crack faces (cf. Figure A.11)

KI = F

(πa)3/2. (A.11)

For an elliptical crack in an infinite body under uniaxial tension the maximumstress intensity factor is at the ends of the minor axis and is given by

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Figure A.8. Small edge crack at a blunt notch in a sheet under uniaxial tension. Reprintedfrom Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook,WIT Press, ISBN 1-85312-703-5, 2000.

KI = σ√

πa

E(k), (A.12)

where a is the semi minor axis of the elliptical crack and E(k) is the completeelliptic integral of the second kind. This is given by

E(k) =∫ π/2

0

√1 − k2 sin2 � d�, (A.13)

where

k =√

1 − a2

c2(A.14)

and c is the semi major axis of the ellipse. An approximation for E(k) is

E(k) =√

1 + 1.464(a

c

)1.65(0 ≤ a/c ≤ 1). (A.15)

For c = a Equation (A.12) reduces to Equation (A.10) and for c a toEquation (A.4), this is also the solution for a tunnel crack, width 2a, in aninfinite body.

Many of the cracks observed in service are surface cracks (Figure A.12).They are often called part through cracks. The aspect ratio of a surface crackis the ratio of crack surface length to crack depth. Other definitions of aspectratio are sometimes used (see Sections 7.6.1 and 8.3.1). Surface cracks with


Figure A.9. Small edge crack at a sharp notch in a sheet under uniaxial tension. Reprintedfrom Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook,WIT Press, ISBN 1-85312-703-5, 2000.

Figure A.10. Three point single edge notch bend specimen (Frost et al. 1974).

an aspect ratio of more than two are usually approximated as semi ellipticalcracks (see Section A.5.1). For such a semi elliptical surface crack in a semiinfinite body under uniaxial tension the maximum stress intensity factor is atthe deepest point of the crack, and its value is dominated by the crack depthrather than the crack surface length. For a semi infinite body approximatevalues of the stress intensity factor at the deepest point are given by

Metal Fatigue 179

Figure A.11. Centre crack in an infinite sheet with point forces on crack surfaces (Frost et al.1974).

Figure A.12. Semi elliptical surface crack in a semi infinite body under uniaxial tension (Frostet al. 1974).

KI = 1.12σ√

πa

E(k). (A.16)

For c a Equation (A.16) reduces to Equation (A.5), for c = 2.89a toEquation (A.4), and for a semi circular surface crack (a = c) it becomes

KI = 0.713σ√

πa . (A.17)


A.3.1.3 Effect of Residual Stresses

The effect of introducing a crack into a body containing large scale residualstresses is to relieve the residual stresses on the crack plane. Provided that thecrack is not too large the corresponding stress intensity factor is the same asif stresses, equal in magnitude but opposite in sign, were applied to the cracksurfaces. For example Equation (A.5), for a small edge crack (Figure A.7)becomes

KI = 1.12P√

πa , (A.18)

where P is the pressure on the crack surfaces. The presence of residualstresses of unknown magnitude can be a serious limitation in the practicalapplication of fracture mechanics.

A.3.2 VALIDITY OF STRESS INTENSITY FACTORS

The application of stress intensity factors to practical engineering problemsinvolving cracks has been a spectacular success over the past 40 years or so.Nevertheless there are some limitations on their validity that arise from thelinearly elastic stress analyses on which they are based. Accumulated exper-ience has shown how these limitations can be managed, and what steps needto be taken to mitigate their effects.

A stress intensity factor provides a reasonable description of the crack tipstress field in a K-dominated region at the crack tip, radius r ≈ a/10, wherea is the crack length, Figure A.13 (Pook 2000a). An apparent objection tothe use of the stress intensity factor approach is the violation, in the imme-diate vicinity of the crack tip, of the initial linearly elastic assumptions, inthat strains and displacements are not small. However, as the assumptions areviolated only in a small core region, radius � r, the general character of theK-dominated region is, to a reasonable approximation, unaffected. Similarly,by this small scale argument, small scale nonlinear effects due to crack tipyielding, microstructural irregularities, internal stresses, irregularities in thecrack surface, the actual fracture process, etc., may be regarded as within thecore region. If a crack is too short then it may not be possible to use this smallscale argument (see Section 7.4.5).

Elastic stress fields at the tips of sharp notches are, with due attention todetail, similar to those for cracks. By the small scale argument, stress intensityfactors for cracks can be used to describe the elastic stress field at the tips ofsharp notches. The elastic stress at the tip of a sharp V-notch can be describedby stress intensity factors, provided that the included angle does not exceed

Metal Fatigue 181

Figure A.13. K-dominated and core regions at a crack tip. Reprinted from Linear ElasticFracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312-703-5, 2000.

30◦. Negative values of KI are possible for sharp notches and also for opencracks, that is, cracks where the surfaces are separated by a small amount.

When yielding is not small scale, stress intensity factors do not providea reasonable description of the crack tip stress field, and other less versatilefracture mechanics parameters become appropriate. Failure to check whetherlarge scale yielding might be occurring is the commonest error in the practicalapplication of stress intensity factors. One early approach was to ensure thatthe nominal net section stress did not exceed 80 per cent of the yield stress,σY, and this is still a useful check. In stress intensity factor based metallicmaterial test standards, minimum acceptable test specimen dimensions arespecified in order to avoid large scale yielding. Actual minimum values de-pend on values of the stress intensity factors and the material’s yield stress,for example Anon. (2003b, 2005g).

Use of a two dimensional stress intensity factor solution for plates andsheets implicitly assumes that the crack front is straight through the thicknessand also perpendicular to the surfaces. In practice, crack fronts are usuallycurved. For example Figure A.14 shows the fracture surface of a 19 mm thickaluminium alloy fracture toughness test specimen, the fatigue precrack frontis clearly curved. Standards place limits on the permissible amount of fatiguecrack front curvature, so that a two dimensional stress intensity factor solutioncan be used, for example Anon. (2003b, 2005g).


Figure A.14. Fracture surface of 19 mm thick aluminium alloy fracture toughness test speci-men (Pook 1968). Reproduced under the terms of the Click-Use Licence.

Extension of the essentially two dimensional concept of stress intensityfactor to three dimensions makes two implicit assumptions. The first is thatthe line defining a crack front is smooth so that stress intensity factors do notchange abruptly along the crack front. This is generally, but not always, truefor cracks observed in practice when these are viewed on a macroscopic scale,and does not usually cause difficulties. The second assumption is that a crackfront is continuous, that is, it is a closed curve. This is sometimes true, forexample for an internal elliptical crack. It is not true at a corner point wherea crack front intersects a free surface (Figure A.15). The nature of the cracktip singularity changes in the vicinity of a corner point. This can usually beneglected on the basis of the small scale argument, but sometimes has to betaken into account (see Section A.4).

A.3.3 EFFECTS OF SMALL SCALE YIELDING

Small scale yielding in the high stress region at a crack tip has two main prac-tical consequences. First, it leads to a practical definition of plane strain in thepresence of a crack which differs from that usual in the theory of elasticity.Secondly, the relaxation of stresses within the crack tip plastic zone meansthat, to maintain equilibrium, stresses outside the plastic zone increase, andthe effective crack length is increased.

Metal Fatigue 183

Figure A.15. Semi elliptical surface crack under uniaxial tension showing corner points (Frostet al. 1974).

A.3.3.1 Definition of Plane Strain

From a theory of elasticity viewpoint an uncracked plate loaded in uniaxialtension is in a state of plane stress, that is the stress in the thickness directionis zero. This is still so in the bulk of a plate when a Mode I crack is introduced,but highly stressed material adjacent to the crack tip is constrained by the lesshighly stressed surrounding material, and stresses are induced in the thicknessdirection in the interior of the plate in the vicinity of the crack tip.

This situation is often referred to as plane strain in fracture mechanics.It must not be confused with the conventional theory of elasticity definition,which is used in the theoretical analysis of crack tip surface displacements(see Sections A.2.1 and A.3). Limited plastic flow due to yielding of the ma-terial adjacent to the crack tip does not affect the situation in the interiorof a thick plate (Figure A.12). In this context a plate is said to be thick ifthe thickness is at least 2.5(KI/σY)2 where KI is the Mode I stress intensityfactor and σY is the yield stress. This practical definition of plane strain arosefrom consideration of the results of tests to determine the fracture toughness,Kc. It was observed that, in general, Kc decreased as the thickness increased,but that for most metallic materials it reached a constant minimum value ifthe specimen thickness was at least 2.5(Kc/σY)2 (see Section 7.4.2.2). Thisminimum value can be regarded as a material constant, and is known as the


Figure A.16. Plane strain, plate thickness ≥ 2.5(KI/σY)2 (Frost et al. 1974).

plane strain fracture toughness, KIc. In the special case of this symbol, thesubscript I denotes both a Mode I crack and plane strain. The minimum speci-men thickness requirement of 2.5(Kc/σY)2 is included in plane strain fracturetoughness test standards, for example Anon. (2005g). When a crack front iscurved, as in a semi elliptical surface crack (Figure A.12) there is a high de-gree of constraint along the crack front, except at the corner points, and suchcracks can usually be regarded as being in plane strain.

When the plate thickness is very much less than 2.5(K/σY)2, then thecrack tip plastic zone size becomes comparable with the thickness, and yield-ing can take place on 45◦ planes. This relaxes the through thickness stresses,so that the whole plate is in a state of plane stress (Figure A.17). The sym-bol Kc is sometimes reserved for the plane stress fracture toughness whichis then regarded as a material constant. At intermediate thicknesses the stressstate is uncertain and Kc is a function of thickness.

A.3.3.2 Effective Crack Length

The increase in effective crack length over the physical crack length due toyielding at the crack tip is shown schematically in Figure A.18. A first estim-ate of the plastic zone size may be obtained by substituting von Mises’ cri-terion of yielding (see Section 4.5.2.1) into the elastic crack tip stress fields.For Mode I, plane stress this leads to

Metal Fatigue 185

Figure A.17. Plane stress, plate thickness � 2.5(KI/σY)2 (Frost et al. 1974).

rY = 1

2π

(KI

σY

)2

(plane stress), (A.19)

where rY is the plastic zone size measured in the crack direction, KI is theMode I stress intensity factor, and σY is the yield stress. The actual size andshape of a crack tip plastic zone depends on the flow properties of the metal,but its dimensions are always proportional to (KI/σY)2. Typically, a plasticzone size is about twice that given by Equation (A.19), so rY is interpreted asthe plastic zone radius. The effective crack length becomes a + rY, as indic-ated in Figure A.18, and the corresponding stress intensity factor is calculatediteratively. Under plane strain conditions the plastic zone radius is about onethird of that given by Equation (A.19), and

rY = 1

6π

(KI

σY

)2

(plane strain). (A.20)

The plastic zone corrections given by Equations (A.19) and (A.20) are oftenvery small and hence unnecessary. At one time they were quite popular, butare now rarely used. Plastic zone corrections do not appear to be specified inany standards.


Figure A.18. Physical and effective crack length, rY is the plastic zone radius.

A.3.3.3 Slant Crack Propagation in Thin Sheets

The transition from square (Mode I) to slant crack propagation sometimesobserved in thin sheets under both static and fatigue loading, as shown schem-atically in Figure A.19, is an exception to the observation that fatigue cracksin metals tend to propagate in Mode I (see Sections 7.4.2.2, 8.1.1 and A.2.1).Slant crack propagation is sometimes stated to be mixed Mode I and ModeIII, but this is true only for the sheet centre line. Away from the centre line,it is mixed Mode I, Mode II and Mode III. This has been confirmed by finiteelement analysis (Pook 1993). It is sometimes called shear crack propaga-tion, on the grounds that it takes place on planes of maximum shear stress inan uncracked sheet, but this is a misnomer. In the calculation of stress intens-ity factors it is usual to treat slant crack propagation, and crack propagation inthe transition region, as if they were Mode I crack propagation, and to use atwo dimensional stress intensity factor solution. This is difficult to justify bythe small scale argument (see Section A.3.2), but it does not cause difficultiesin practice (see Section A.5.2).

A.4 Corner Point Singularities

The analyses on which the concept of stress intensity factor is based are es-sentially two dimensional in nature, and the crack front is a point (see Sec-tion A.3.2). When analysis is extended to three dimensions, the crack front

Metal Fatigue 187

Figure A.19. Transition from square to slant crack propagation in thin sheets. The arrow showsthe direction of fatigue crack propagation (Pook 1983a). Reproduced under the terms of theClick-Use Licence.

becomes a line. Derivations then include the implicit, and usually unstated,assumption that a crack front is continuous. This is not the case at a cornerpoint, where a crack front intersects a free surface. The crack front shown bythe dashed line in Figure A.15 intersects the surface at two corner points. Asis well known, the nature of the crack tip singularity changes in the vicinityof a corner point. For corner point singularities, the polar coordinates (r, θ)

in Figure A.1 are replaced by spherical coordinates (r, θ, φ) with origin at thecorner point. The angle φ is measured from the crack front.

The stress intensity measure, Kλ, is used to characterise corner pointsingularities, where λ is an exponent defining the corner point singularity.Stresses are proportional to Kλ/rλ and displacements to Kλr

1−λ, where r ismeasured from the corner point. For a crack surface intersection angle, γ of90◦, defined as in Figure A.20, there are two modes of stress intensity meas-ure. These are the symmetric mode, KλS, where crack tip surface displace-ments are Mode I (Figure A.2), and the antisymmetric mode, KλA, which is acombination of Modes II and III displacements. In other words, the presenceof one of these modes of crack tip surface displacement always induces theother. For the special case of λ = 0.5, stress intensity factors are recovered.KλS becomes KI, and KλA a combination of Modes II and III stress intensityfactors, KII and KIII. For the symmetric mode, and Poisson’s ratio, ν = 0.3,


Figure A.20. Definition of crack surface intersection angle, γ . Reprinted from Crack Paths.LP Pook, WIT Press, ISBN 1-85312-927-5, 2002.

Figure A.21. Definition of the crack front intersection angle, γ . Reprinted from Pook (1994a),Copyright 1994, with permission from Elsevier.

the theoretical value of λ is 0.452 for a crack front intersection angle of 90◦,Figure A.21 (Pook 2002a), whereas for the antisymmetric mode λ is 0.598.

For λ = 0.5, Mode I crack surface displacements are given, for planestress, by Equation (A.3), and the crack opens up into a parabola (Figure A.22centre). The radius at the tip of the loaded crack, r, is given by

r = 4K2I

πE2, (A.21)

where KI is the Mode I stress intensity factor and E is Young’s modulus.When λ �= 1/2 stresses and displacements cannot, in general, be calculatedin detail because of lack of information. However, when λ < 0.5 r = 0(Figure A.22 top) and, from Equation (A.21), stress intensity factors tend tozero as a corner point is approached. Conversely, when λ > 0.5 r = ∞ andstress intensity factors tend to infinity (Figure A.22 bottom).

There is only limited information available on the size of the corner region(boundary layer) in which the crack tip stress field is dominated by the stressintensity measure, although it must be associated with some characteristicdimension, such as sheet thickness.

As with stress intensity factors, an apparent objection to the use of thestress intensity measure approach is the violation, in the vicinity of the crack

Metal Fatigue 189

Figure A.22. Crack profiles for loaded Mode I crack. Crack tip radius r = 0 for λ < 0.5. Forλ = 0.5 r is finite and for λ > 0.5 r = ∞. Reprinted from Pook (1994a), Copyright 1994,with permission from Elsevier.

tip, of the initial assumption on which linearly elastic analyses are based (seeSection A.3.2). However, as the assumptions are violated only in a small coreregion, the general character of the corner point singularity dominated regionin the vicinity of the crack tip is unaffected, as is shown for stress intensityfactors in Figure A.13. Similarly, small scale nonlinear effects may be re-garded as within the core region inside a corner point singularity dominatedregion. In turn the corner point singularity dominates only within a limitedregion, so in some circumstances a corner point singularity dominated regionmay lie within a K-dominated region, as shown schematically for a surfaceplane in Figure A.23 (Pook 2002a).

For practical engineering purposes the use of stress intensity measuresis usually unnecessary. They do not appear in standards which make use ofstress intensity factors, for example Anon. (2005f). However, they are twosituations in which corner point singularities have an important influence,and these are discussed in the next two sections.

A.4.1 CRACK FRONT INTERSECTION ANGLE

The coefficient defining a stress intensity measure, λ, is a function of Pois-son’s ratio, ν, and the crack front intersection angle, β (Figure A.21). At a


Figure A.23. K-dominated, corner point singularity dominated and core regions at a surfaceplane. Reprinted from Crack Paths. LP Pook, WIT Press, ISBN 1-85312-927-5, 2002.

critical crack front intersection angle, βc, λ = 0.5 and stress intensity factorsthen have finite values in the corner point region. For β < βc, λ < 0.5, andfor β > βc, λ > 0.5.

For the symmetric mode βc is given approximately by (Pook 1994a)

βc = tan−1

[(ν − 2)

ν

], (A.22)

where ν is Poisson’s ratio. For ν = 0.3, βc = 100.4◦.It has been argued, from energy and other considerations, that the crack

front intersection angle must be βc. Intersection angles of about this valuemay be observed for Mode I cracks, and in consequence crack fronts in platesof constant thickness are often curved, for example Figure A.14.

For the antisymmetric mode βc is given approximately by

βc = tan−1

[(1 − ν)

ν

]. (A.23)

For ν = 0.3, βc = 67.0◦.When the crack surface intersection angle γ �= 90◦, a crack at a corner

point is always in a combination of Modes I, II and III crack tip surface dis-placements. For any given value of γ there are two possible values of βc. Forγ = 45◦ and ν = 0.3 these are 108◦ and 60◦. There is a corresponding value

Metal Fatigue 191

of the ratio KI : KII : KIII for each value of βc. KI, KII and KIII are the ModesI, II and III stress intensity factors.

A.4.2 DERIVATION OF STRESS INTENSITY FACTORS

Numerical schemes for the calculation of stress intensity factors for cracks inthree dimensional bodies, explicitly make the assumption that stress intensityfactors provide a description of the crack tip stress field. This is true onlyfor a continuous crack front, for example an internal crack, and for a crackfront which intersects the surface at the critical crack front intersection angle,βc. In consequence, such schemes cannot adequately reproduce the behaviourof stress intensity factors in the vicinity of a corner point. In practice valuesobtained are finite, and usually of the same order as elsewhere along the crackfront. At a corner point, values are, in effect, extrapolations which depend ondetails of the numerical scheme used (Pook 1994a, 2000c).

At a corner point Mode II and Mode III displacements cannot exist in isol-ation. The presence of one of these modes always induces the other (see Sec-tion A.4). Numerical calculations usually show induced values of the ModeII stress intensity factor, KII, or the Mode III stress intensity factor, KIII, aswould be expected, in the vicinity of a corner point. Two dimensional numer-ical schemes are widely used in the determination of stress intensity factorsfor quasi two dimensional specimens of constant thickness. Only Mode Iand Mode II stress intensity factors are possible in two dimensions, so suchschemes cannot reveal induced KIII values at corner points.

Many three dimensional finite element calculations are carried out for aPoisson’s ratio of about 0.3, and crack front and crack surface intersectionangles of 90◦ (Figures A.20 and A.21). Hence theoretically KI should tend tozero, and KII and KIII should tend to infinity, as corner points are approached.As the corner point is approached, the ratio KIII/KII tends to a finite limitingvalue, which is a function of the crack front inclination angle, β, and Pois-son’s ratio, ν. For β = 90◦ and ν = 0.3 it is 0.5 and for ν = 0.5 it is

√0.5.

From an engineering viewpoint the finite stress intensity factor values ac-tually obtained at corner points do not matter provided both that results arereasonably consistent, and also that difficulties do not arise in practical ap-plications. In practice, the situation for Mode I is indeed satisfactory, andnumerous standards make use of Mode I stress intensity factors without anymention of corner point singularities, for example Anon. (2005f). The situ-ation appears to be reasonably satisfactory for Mode II, but it is not satisfact-ory for Mode III where there are inconsistencies in reported KIII values.


Figure A.24. Through thickness variation of KI for 20 mm square models under Mode Iloading (Pook 2000c).

A.4.2.1 Stress Intensity Factors for Square Models

Figures A.24–A.28 show some typical results for through thickness distribu-tions of stress intensity factors for Mode I, Mode II and Mode III loadings(Pook 2000c). These were obtained from finite element analysis of a 20 mmsquare model, with a crack extending from the middle of one side of thesquare to its centre. Loadings and boundary conditions were chosen to givelarge K-dominated regions, and Poisson’s ratio was taken as 0.3. For Mode Iand Mode II loadings, the results are normalised by stress intensity factors ob-tained from two dimensional calculations, whereas for the Mode III loadingthey are normalised by centre line values.

The results for the Mode I loading of a 4 mm thick plate and a 40 mm longbar (Figure A.24) show the well known increase in the Mode I stress intens-ity factor, KI, at the centre line compared with the corresponding two dimen-sional solution. The decrease in KI towards the surface is also well known,and suggests the existence of a corner point singularity dominated region.The presence of a corner point singularity dominated region was confirmedby estimating values of λ from crack surface displacements at the model sur-face. These estimates gave λ = 0.452 for both the plate and the bar, whichagrees with the theoretical value (see Section A.4). The displacements indic-

Metal Fatigue 193

Figure A.25. Through thickness variation of KII for 20 mm square models under Mode IIloading (Pook 2000c).

ated that the size of the corner point singularity dominated region (boundarylayer) was about 0.2 mm for the plate, and about 1 mm for the bar.

Through thickness distributions of KII for Mode II loading are shown inFigure A.25. At the centre line, KII for the 40 mm long bar is lower than forthe corresponding two dimensional solution. It increases towards the surface,again suggesting the existence of a corner point singularity dominated region.This was confirmed by the Mode II crack surface displacements at the surface,which gave λ = 0.560 compared with the theoretical value of 0.598. Theyalso indicated a corner point singularity dominated region size of about 3 mm.The results for the 4 mm thick plate show a different trend. KII is nearlyconstant through the thickness, except for a sharp decrease at the surface,the corner point singularity dominated region size is about 3 mm, and λ =0.535. Figure A.26 shows the distribution of induced values of KIII. Theseare zero at the centre line (by symmetry) for both the plate and the bar, andincrease towards the surface, except that for the plate there is a sharp decreaseat the surface. The Mode III crack surface displacements at the surface did notdefine corner point singularity dominated regions, and it was not possible to


Figure A.26. Through thickness variation of KIII for 20 mm square models under Mode IIloading (Pook 2000c).

Figure A.27. Through thickness variation of KIII for 20 mm square bar under Mode III loading(Pook 2000c).

derive values of λ. Also, it was not possible to derive a limiting value for theratio KIII/KII.

The through thickness distribution of KIII for Mode III loading of the40 mm long bar is shown in Figure A.27. KIII is constant over the centralpart of the bar, but there is a marked decrease as the surface is approached,

Metal Fatigue 195

Figure A.28. Through thickness variation of KII for 20 mm square bar under Mode III loading(Pook 2000c).

rather than the theoretical increase towards infinity. The Mode III crack sur-face displacements at the surface indicated a corner point singularity domin-ated region size of about 0.2 mm, but it was not possible to derive a valuefor λ. Induced values of KII are zero at the centre line, decrease slightly, andthen increase towards the surface (Figure A.28). The Mode II crack surfacedisplacements at the surface indicate a corner point singularity region size ofabout 0.3 mm, and λ = 0.570.

The inconsistencies in the results for Modes II and III loadings (Fig-ures A.25–A.28) in the vicinity of corner points are typical of results ob-tained from finite element analyses. Scatter increases in the vicinity of acorner point, and what must be regarded as nominal values of KII and KIII arestrongly dependent on details of numerical calculation methods. The incon-sistencies arise partly from the use of linearly elastic finite element analyses.In principle, in a linearly elastic analysis KIII at a surface must be zero be-cause shear stresses perpendicular to a free surface must be zero. This effectshows up clearly in Figure A.27, but not in Figure A.26.

A.5 Stress Intensity Factors for Irregular Cracks

In three dimensions there are numerous possible crack configurations (Pook1986). In general, cracks three dimensional cracks observed in service cannotbe approximated by two dimensional stress intensity factor solutions.

Naturally occurring cracks, and crack like flaws, are often irregular inshape, for example the casting defects in Figures 3.17 and 7.2. The three


Figure A.29. Crack types.

main types of cracks encountered in practice are shown schematically in Fig-ure A.29. These are surface cracks, internal cracks, and through the thicknesscracks. Methods of estimating stress intensity factors for irregular cracks havebeen of interest for many years (Paris and Sih 1965, Anon. 1980b, Chang1982, Pook 1982a, Murakami 2002). Normally, stress intensity factors varyalong a crack front, and it is the largest value of the Mode I stress intensityfactor, KI, that is usually of interest. Adjacent cracks, as in Figure 7.1, canincrease stress intensity factors through interaction effects. These are difficultto assess when cracks are not coplanar.

In the analysis of laboratory and service failures, the size and shape ofcracks can usually be ascertained in detail by the examination of fracturesurfaces. However, the results of non destructive testing will usually supplyonly a limited amount of information. For example, only the surface lengthand maximum depth of a surface crack might be available (see Appendix C).

Initially, an intuitive approach was used by various authors to estimatestress intensity factors for irregular cracks. This approach was based on gen-eral knowledge of stress intensity factors for similarly shaped regular cracks(Paris and Sih 1965). Fifteen years later standardised estimation proceduresfor various situations started to appear (Anon. 1980b), and are now in wide-spread use (Anon. 2005f). Standardised procedures should be used wheneverpossible. Some of the ideas used in estimating stress intensity factors for ir-regular cracks are described below. The ideas sometimes have a sound the-oretical basis, but the main justification for their use is that they have beenfound to work well in practice, and also that they do not lead to unconservat-ive results.

A.5.1 USE OF SEMI ELLIPSES AND ELLIPSES

A flat irregular surface crack under Mode I loading is nearly always mod-elled by a semi ellipse of the same surface length and depth. The Mode Istress intensity factor, KI, is greatest at the deepest point of the semi ellipse(see Section A.3.1.2). This intuitive approach is satisfactory for cracks that

Metal Fatigue 197

Figure A.30. Modelling of an irregular surface crack as a semi ellipse.

Figure A.31. Modelling of a more irregular surface crack as a semi ellipse.

Figure A.32. Modelling of a more irregular surface crack as a semi ellipse inscribed in acontainment rectangle.

are close to this shape, as shown schematically in Figure A.29 left, and inFigure A.30. The approach does not work well for some more irregular sur-face cracks such as shown in Figure A.31. What is sometimes done is to firstconstruct a containment rectangle around the crack, and then inscribe a semiellipse in the rectangle (Figure A.32). This results in a longer semi ellipsewith a concomitantly higher value of KI at the deepest point. A similar ideais sometimes used for internal cracks in bodies of rectangular cross section,as shown in Figure A.33. The containment rectangle sides are parallel to thebody surfaces. A containment rectangle may be used as it stands for a throughthe thickness crack (Figure A.29 right).

A.5.2 PROJECTION ONTO A PLANE

Cracks are not necessarily flat, and are not necessarily oriented so that theyare in Mode I. One approach is to project them onto a plane so that they be-come equivalent Mode I cracks. The plane chosen is a plane of maximumprincipal tensile stress in the uncracked body. After projection, a crack canthen be modelled as in the previous section. As an example, the method worksquite well for the specimen shown in Figure A.20 when this is loaded in three


Figure A.33. Modelling of an internal crack as an ellipse inscribed in a containment rectangle.

Figure A.34. Quasi two dimensional mixed Modes I and II crack with a small Mode I branchcrack (Pook 1989a). Reproduced under the terms of the Click-Use Licence.

point bending (Pook and Crawford 1990). Another example is treating slantcrack propagation in thin sheets (Figure A.19) as if it were Mode I (see Sec-tion A.3.3.3). In effect, the slant crack and the transition region are projectedonto the plane indicated by dashed lines in the figure.

One theoretical justification is that under mixed mode loading a smallMode I branch crack may form at the initial crack tip, as shown in Figure A.34(see Section 8.2.1). Projection of the initial mixed mode crack onto an appro-priate plane can provide a method of estimating KI for such a branch crack(Pook 1989c).

Metal Fatigue 199

Figure A.35. Crack front smoothing.

A.5.3 CRACK FRONT SMOOTHING

A crack has some analogies with a crystal dislocation (Pook 2002a). In partic-ular, the elastic stress fields associated with a crack front and with a disloca-tion are both singularities. The associated energy means that a dislocation hasa line tension, which controls its shape under an applied stress field (Cottrell1964). Similarly, a crack front may be regarded as having a line tension whichcontrols its shape, but with the important difference that the motion of a crackfront is irreversible; that is a crack can propagate, but in general cannot con-tract. The line tension concept explains why, on a macroscopic scale, a fa-tigue crack front is smooth and any initial sharp corners rapidly disappearas the crack propagates. Overall, an initially irregular Mode I crack rapidlybecomes convex, as shown by the dashed line in Figure A.35: at a re-entrantregion (x on the figure) the Mode I stress intensity factor, KI, is much higherthan elsewhere on the crack front, leading to rapid fatigue crack propagationtowards a convex shape. Under fatigue loadings, stress intensity factors forinitially irregular cracks may be approximated by first enclosing them by aconvex outline, as in the figure, and then using the methods in the previoussection.

A.5.4 USE OF CRACK AREA

For irregular cracks that are small compared with other dimensions it is pos-sible to use the crack area, A, as a characteristic crack dimension (Chang1982, Murakami 2002). A is calculated after projection onto a plane, fol-lowed by crack front smoothing (see Sections A.5.2 and A.5.3). For veryslender cracks the crack length is truncated to 10 times the width before cal-culating A.


Figure A.36. Interaction between two semi circular surface cracks. (a) No interaction. (b)Imaginary third crack inserted.

For an internal crack (Figure A.29 centre) the maximum Mode I stressintensity factor along the crack front, KI, is given approximately by

KI∼= 0.5σ

√π

√A , (A.24)

where σ is the stress perpendicular to the crack. The equation applies tocracks whose length is up to about 5 times the width.

For a surface crack

KI∼= 0.65σ

√π

√A . (A.25)

For a very shallow surface crack the crack surface length is truncated at 10times the crack depth before calculating A, and for a deep surface crack thecrack depth is truncated at 2.5 times the crack surface length.

For a very shallow surface crack Equation (A.25) becomes

KI∼= 1.16σ

√πa , (A.26)

where a is crack depth, and it is close to Equation (A.5), which is the equiva-lent two dimensional solution.

A.5.5 INTERACTION BETWEEN CRACKS

When two cracks are close to each other the interaction between them in-creases their stress intensity factors compared with those for isolated cracks.Unfortunately, this interaction effect cannot be expressed by a simple equa-tion, partly because of the numerous possible configurations. Various approx-imations have been proposed for a wide range of configurations but these tend

Metal Fatigue 201

to be inconsistent, partly because different authors introduce different degreesof conservatism.

The usual approach is first to define the crack separation between twocracks below which interaction occurs, and then in some way to define anequivalent single crack for which stress intensity factors are calculated. Forexample, the following rules are suggested for two adjacent semi circularsurface cracks of different sizes (Murakami 2002). If there is enough spacebetween the cracks to insert an additional crack of the same size as the smal-ler crack (Figure A.36(a)) then the interaction effect is negligibly small, andA in Equation (A.25) is taken as A1. If the space between the two cracks istoo small to insert a crack of the same size as the smaller crack then under fa-tigue loading the cracks coalesce rapidly. An imaginary semi circular crack isinserted between the two cracks and the areas of all three cracks are summed.That is insert in Equation (A.25) A = A1 + A2 + A3 (Figure A.36(b)).

B

Random Load Theory and RMS

Notation

A separate notation is included because many of the symbols listed are usedonly in this appendix.

a, b constants in two parameter Weibull distributionf frequencyG(f ) power spectral densityH wave heightH1/3 significant wave heightI irregularity factorm0, m2, m4 moments of spectral density functionN number of cycles, return periodP(H1/3) exceedance of H1/3

P(S) exceedance of S

P (S/σ ) exceedance of S/s

p(S) probability density of S

p(S/σ ) probability density of S/σ

R(τ) autocorrelation functionS random processs instantaneous value of S

S/σ expected value of S/σ

Sc/σ clipping ratioSm mean value of S

So value of S below which peaks are omittedT total time, wave periodt timeγ Euler’s constant = 0.5772 . . .

204 Appendix B: Random Load Theory and RMS

ε spectral bandwidthσ standard deviation (random process theory), root mean square

(fatigue)σp root mean square of peaksσp,c root mean square of peaks after clipping of high peaksσp,o root mean square of peaks after omission of low peaksσp,t root mean square of peaks after truncation of high peaksσr root mean square of rangesσ 2 varianceτ time intervalφ root mean square (random process theory)

B.1 Introduction

In this appendix the application of random process theory (Papoulis 1965,Bendat and Piersol 2000) to fatigue loading is discussed. From an engineer-ing viewpoint, it might appear that some of the points made are unimportantand pedantic. However, lack of attention to detail can result in difficulty ininterpreting fatigue test data. In reporting random loading fatigue data it isimportant that the precise conventions used in calculations be clearly stated.No one set of conventions is of universal applicability. Some equations andfigures which appear in the main text are repeated in order to make the ap-pendix self contained.

B.2 Basic Definitions

B.2.1 RANDOM PROCESS THEORY

Figure B.1(a) shows a random process in which load is plotted against time.This may be described by the function S(t), where S is a random process andt is time. In metal fatigue S will be a quantity such as stress or load.

Assume that S(t) is statistically stationary and ergodic. Stationary meansthat statistical parameters characterising the process are independent of time.Ergodic means, broadly, that different samples of the same process yield thesame values for statistical parameters. Only stationary random processes canbe ergodic, and in practice most are. Considering the time interval 0 to T themean value of S, Sm is given by

Sm = lim(T → ∞)1

T

∫ T

0S(t) dt (B.1)

Metal Fatigue 205

Figure B.1. Broad band random process, irregularity factor 0.410, spectral bandwidth 0.912.(a) Time history. (b) Spectral density function (Pook 1987). Reproduced under the terms ofthe Click-Use Licence.

and the mean square value φ2 by

φ2 = lim(T → ∞)1

T

∫ T

0S2(t) dt. (B.2)


Hence the root mean square (RMS) value, φ, is given by

φ = lim(T → ∞)

√1

T

∫ T

0S2(t) dt . (B.3)

The positive square root is understood in Equation (B.3) and subsequent equa-tions.

The RMS can equally well be calculated for periodic processes such as asine wave. The use of RMS first became popular in electrical engineering be-cause it can be used directly in calculations involving power. For convenienceit is sometimes used in metal fatigue (Pook 1987a).

Carrying out calculations from the mean rather than from zero gives thevariance, σ 2, where

σ 2 = lim(T → ∞)1

T

∫ T

0{S(t) − Sm}2 dt (B.4)

and the standard deviation, σ , is given by

σ = lim(T → ∞)

√1

T

∫ T

0{S(t) − Sm}2 dt . (B.5)

The quantities given by Equations (B.1)–(B.5) are related through the expres-sion

φ2 = σ 2 + S2m. (B.6)

Hence, for zero mean the RMS and standard deviation are numerically equal.Instantaneous values of S(t) may be characterized by probability distribu-

tion functions. The exceedance, P(S), is the probability that a value exceedsS. The cumulative probability, 1 − P(S), is the proportion of values up toS. The probability density, p(S), is the derivative of the cumulative probab-ility. For convenience, S is often normalised by σ . The instantaneous valuesof many ‘naturally occurring’ random processes are statistically stationary, atleast in the short term, and approximate to the Gaussian distribution (or Nor-mal distribution), which theoretically extends from −∞ to +∞. The probab-ility density of a Gaussian distribution (Figure B.2(a)) for a process with zeromean is given by

p

(S

σ

)= 1√

2πexp

(−S2

2σ 2

)(B.7)

and the exceedance (Figure B.2(b)) by

P

(S

σ

)= 2√

2π

∫ ∞

S/σ

exp

( −S

2σ 2

)d

(S

σ

). (B.8)

Metal Fatigue 207

Figure B.2. Gaussian distribution. (a) Probability density. (b) Exceedance (Frost et al. 1974).

This integral does not have an explicit solution. The values shown in Fig-ure B.2(b) are for the positive half of a Gaussian distribution and are there-fore twice those given by Equation (B.8). P(S/s) is the area under the curveof p(S/σ ) between (S/σ ) and infinity, as indicated by the shaded area inFigure B.2(a).

B.2.2 FATIGUE LOADING

Conventions used in the metal fatigue literature sometimes differ from thoseused in random process theory. The random processes encountered in metalfatigue are usually symmetrical, in a statistical sense, about the mean value,Sm, and all calculations are then carried out using values of S measured fromSm. Mathematically, this is equivalent to treating only cases where Sm is zero.It follows that there is no numerical difference between root mean square(RMS) and standard deviation, and in metal fatigue the term standard devi-


ation is not normally used in the characterisation of random processes. (Al-ternatively it might be said that RMS is used where standard deviation ismeant.) This convention is used in what follows unless the context indicatesotherwise. In particular, references to zero are understood to include the meanvalue of a process with non zero mean. Retaining the symbol σ for what wascalled standard deviation and is now called RMS Equation (B.5) becomes

σ = lim(T → ∞)

√1

T

∫ T

0S2(t) dt . (B.9)

The apparent lack of rigour is justified because RMS, as given by Equa-tion (B.9), has no physical significance in metal fatigue (see Section 4.3.3).

B.3 Some Sinusoidal Processes

B.3.1 NARROW BAND RANDOM LOADING

In general a narrow band random process (Figure B.3) results when a randominput is applied to a sharply tuned resonant system (Papoulis 1965, Pook1983b, 1984, Bendat and Piersol 2000). Individual sinusoidal cycles appearwhose frequency corresponds to the centre frequency of the resonant system.They have a slowly varying random amplitude. The probability density func-tion for the occurrence of a positive peak of amplitude S (Figure B.4(a)) isgiven by the Rayleigh distribution

p

(S

σ

)= S

σexp

(−S2

2σ 2

). (B.10)

As the process is statistically symmetrical, corresponding negative peaks alsoappear. The exceedance (Figure B.4(b)) is given by

P

(S

σ

)= exp

(−S2

σ 2

). (B.11)

Equations (B.10) and (B.11) become exact only as the bandwidth tends tozero (see Section 4.2.2). Used in its general sense Rayleigh distribution doesnot imply the existence of a corresponding narrow band random process, andparameters in Equations (B.10) and (B.11) may differ. A narrow band randomprocess is Gaussian, so instantaneous values do follow the Gaussian distribu-tion (Equations (B.7) and (B.8)).

Conventionally, in discussion of the Rayleigh and related distributions,only positive peaks are described and shown in diagrams such as FiguresA2.4, it being understood that the negative peaks, with due attention to sign,

Metal Fatigue 209

Figure B.3. Narrow band random process, frequency ≈100 Hz, irregularity factor ≈0.99(Pook 1987a). Reproduced under the terms of the Click-Use Licence.

Figure B.4. Rayleigh distribution. (a) Probability density. (b) Exceedance (Frost et al. 1974).

are also included. Negative peaks are sometimes called troughs. Theoreticallythe Rayleigh distribution extends to infinity, but in practice peaks do not ex-ceed a cut off value of S/σ , known as the clipping ratio. Clipping implies that


higher peaks are reduced to the level given by the clipping ratio; truncationthat they are omitted altogether. The clipping ratio does not usually exceedfour or five.

As fatigue damage depends on the peak values of cycles, and is largelyindependent of waveform (see Section 3.2), the root mean square (RMS)value of peaks, σp, is sometimes used. For narrow band random loading:σp = √

2 σ . The RMS of the ranges between positive and negative peaks,σr, is also in use for narrow band random loading, σr = 2σp = 2

√2 σ .

B.3.2 TWO PARAMETER WEIBULL DISTRIBUTION

The two parameter form of the Weibull distribution has a variety of engin-eering applications; some of its general properties are discussed by Lipsonand Sheth (1975). For metal fatigue purposes it is convenient to write its ex-ceedance in the form (Pook 1984)

P

(S

σ

)= exp

{(−b

a

) (S

σ

)a}, (B.12)

where a and b are adjustable constants (parameters) used to fit the equationas needed.

A functional relationship between b and a can be obtained by assum-ing that Equation (B.12) gives the distribution of peaks of a sinusoidal pro-cess which is symmetrical about zero. There is no closed form relationshipbetween b and a. Values of b for a in the range 0.5 to 3 are tabulated in Pook(1984). The expression

b = (1 − 0.076(a2 − 3a + 2) (B.13)

provides a satisfactory fit for a in the range 0.71 to 2.36. Putting a = 2,b = 1 and a = 1, b = 1 gives as special cases the Rayleigh distribution(Equation (B.11)) and the Laplace distribution (or Exponential distribution)

P

(S

σ

)= exp

(−S

σ

)= p

(S

σ

). (B.14)

Exceedances for a range of a values are shown in Figure B.5. A logarithmicscale is used for exceedances in order to emphasise detail at low values. Asa result the curve for the Rayleigh distribution has a different appearancefrom Figure B.4(b) where a linear scale is used. The peaks of the C/12/20load history, shown in Figure 4.9, can be fitted approximately by the twoparameter Weibull equation with a = 1.2715 (Pook 1987a).

Differentiating Equation (B.12) gives the probability density of the twoparameter Weibull distribution as

Metal Fatigue 211

Figure B.5. Exceedances for the two parameter Weibull distribution (Pook 1987a). Repro-duced under the terms of the Click-Use Licence.

p

(S

σ

)=

(b

a

) (S

σ

)a−1

exp

{(−b

a

) (S

σ

)a}. (B.15)

If only the distribution of peaks is specified by a probability distribution, thenin a computer generated process both the order in which peaks are appliedand the waveform connecting them need to be specified. Usually, a negativepeak is made arithmetically equal to the preceding positive peak. The valueof σ depends on the waveform used to connect the peaks, but both σp and σr

are independent of waveform. When peaks are connected by sine waves togive a sinusoidal process, σp = √

2 σ and σr = 2√

2 σ . In general, computergenerated processes are do not follow the Gaussian distribution. However, forthe special case of a sinusoidal process with a Rayleigh distribution of peaks,


and with constant frequency, the process is Gaussian, irrespective of the orderin which peaks are applied.

B.3.3 INCOMPLETE DISTRIBUTIONS AND PROCESSES

Distributions and processes encountered in metal fatigue practice are alwaysincomplete in some way. In consequence root mean square (RMS) valuesdiffer from those of theoretical forms which extend to infinity. Provided thata difference in RMS values is less than about 1 per cent it can usually beneglected. Examples illustrating some of the issues involved are given below.

B.3.3.1 Truncated and Clipped Distributions

The term clipping ratio is used to cover both truncation and clipping. Thisis because, in a physical system involving a narrow band random process, itis nonlinearities rather than clipping or truncation that limit the peaks whichappear, and it may not be possible to maintain a clear distinction betweentruncation and clipping (see Section B.3.1). However, in a process generatedby first generating positive, and corresponding negative peaks, which followsome distribution, then joining the positive peaks and adjacent negative peakswith an appropriate waveform, a distinction has to be made to avoid ambigu-ity. In any process used for fatigue testing the maximum load applied has tobe limited to meet physical limitations.

If the distribution of peaks is known in terms of the root mean square(RMS) value of the process, then the RMS of the peaks, σp, is given by

σp =√∫ ∞

0

(S

σ

)2

p

(S

σ

)d

(S

σ

). (B.16)

Hence, if the process is truncated at a clipping ratio, Sc/σ , the RMS of thetruncated distribution of peaks, σp,t, is given by

σp,t =√√√√∫ Sc/σ

0

(Sσ

)2p

(Sσ

)d

(Sσ

)1 − P

(Scσ

) , (B.17)

where the term {1 − P(Sc/σ )} corrects for the reduction in the total numberof peaks, and σ is the RMS of the complete process.

If the process is clipped, then the RMS of the clipped distribution of peaks,σp,c, is given by

Metal Fatigue 213

σp,c = (B.18)√√√√P

(Sc

σ

)(Sc

σ

)2

+{

1 − P

(Sc

σ

) ∫ Sc−σ

0

(S

σ

)2

p

(S

σ

)d

(S

σ

)},

where the first term under the square root sign represents the peaks that thathave been reduced to Sc/σ . Clipping has less effect on RMS than trunca-tion. Equations (B.17) and (B.18) may be used to calculate the change in theRMS of peaks due to truncation and clipping of different versions of the twoparameter Weibull distribution (Equation (B.12)). Results show that for theRayleigh distribution (a = 2 in Equation (B.12)) the effects are negligiblewhen the clipping ratio exceeds about 3.5, and for the Laplace distribution(a = 1) when it exceeds about 8. Further terms appear in Equations (B.17)and (B.18) if positive and negative peaks are not truncated or clipped sym-metrically.

As an example of what can happen, consider the construction of a si-nusoidal process, whose peaks follow the two parameter Weibull distribu-tion, with a taken as 0.5. Assume that the complete process will be trun-cated to give a desired clipping ratio of 5. From Equation (B.17), takingS/σ = 5, σp,t = 0.7704σ , and the clipping ratio for the truncated processis 5/0.7704 = 6.490. For the clipping ratio of the truncated process to be 5,the clipping ratio applied to the complete process would have to be 3.243.

For a process to remain sinusoidal, clipping has to be carried out correctly,as shown in Figure B.6. Form (a) is an original unclipped half cycle. Reducinginstantaneous values of the process to the clipping ratio results in form (b),which is not sinusoidal. For the process to be sinusoidal the half cycle has tobe reshaped, as in form (c). Truncation or clipping of a process that followsthe Gaussian distribution renders it non Gaussian. However, it can reasonablybe regarded as Gaussian if the percentage change in RMS is negligibly small.

B.3.3.2 Omission of Low Loads

Peaks below an omission level, σo, are sometimes omitted to reduce fatiguetesting times, on the grounds that they cause negligible fatigue damage. Theroot mean square (RMS) value of the remaining peaks, σp,o, is given by

σp,o =√√√√∫ ∞

So/σ

(Sσ

)2p

(Sσ

)d(

Sσ

)P

(Soσ

) . (B.19)

In practice, large numbers of cycles are omitted, so there is always a signific-ant effect on RMS. Omission is always combined with truncation or clipping.


Figure B.6. Clipping a half cycle. (a) Original half cycle. (b) Clipped. (c) Clipped and re-shaped (Pook 1987a). Reproduced under the terms of the Click-Use Licence.

B.3.3.3 One Sided Narrow Band Random Loading

Tests are sometimes carried out using a modified narrow band random load-ing from which negative peaks have been removed, to give a one sided pro-cess (Sherratt and Edwards 1974). Three ways of doing this are shown, fora constant amplitude sinusoidal process, in Figure B.7. Part (a) of the figureshows the original process. In Figure B.7(b) the negative half cycles havebeen reduced to zero height, whereas in Figure B.7(c) they have been re-moved altogether. In Figure B.7(d) the negative peaks have been removedaltogether and the positive peaks joined to zero by sine waves. Usually, rootmean square (RMS) values are calculated for the original complete process.Parameters can be calculated for a one sided process, but different results aresometimes obtained for the three methods. For example, mean values (Sm)are σ/π , 2σ/π and σ/

√2 respectively, where σ is the RMS of the original

complete process The RMS of ranges, σr, is the same for all three methodsof removal, and is equal to the RMS of peaks, σp, for the original completeprocess, that is

√2 σ , and would appear to be a good choice. The original

complete process is Gaussian, but instantaneous values of a one sided pro-cess do not follow the Gaussian distribution (Equations (B.7) and (B.8)).

B.3.3.4 Finite Random Processes

Any practical random sinusoidal process must be of finite length and containa finite number of cycles, N . For a pseudo random process, N is the returnperiod after which it repeats exactly. One consequence is that the maximumpeak size, and hence the clipping ratio, are restricted (see Sections B.3.1 andB.3.3.1). An intuitive approach is to set the exceedance, P(S/σ ), equal to1/N and then take the corresponding value of S/σ from Equation (B.11)

Metal Fatigue 215

Figure B.7. One sided constant amplitude sinusoidal processes. (a) Original cycle. (b) Neg-ative half cycles reduced to zero height. (c) Negative half cycles removed. (d) Negative halfcycles removed and positive half cycles reshaped (Pook 1987a). Reproduced under the termsof the Click-Use Licence.

as the clipping ratio. However, in narrow band random loading, large cyclesoccur in groups and the expected maximum value of S/σ , S/σ , which will


Table B.1. COLOS 7 level load history.

Level number Number of cycles in level RMS of level7 1,000 4.07σ

6 4,000 3.46σ

5 40,000 2.90σ

4 180,000 2.27σ

3 575,000 1.68σ

2 1,250,000 1.10σ

1 2,950,000 0.426σ

be the expected clipping ratio, is somewhat less. It is given approximately by(Pook 1978)

S

σ≈ √

ln N + γ

2

√1

ln N, (B.20)

where γ is Euler’s constant = 0.5772. . . . For example, for N = 105,P(S/σ ) = 10−5 and from Equation (B.11) S/σ = 4.80, whereas Equa-tion (B.20) gives S/σ = 3.48.

B.3.4 NON STATIONARY NARROW BAND RANDOM LOADING

In service, random loadings are usually statistically non stationary so that rootmean square (RMS) values, and perhaps other parameters, are a slowly vary-ing function of time. In the short term they can usually be regarded as stat-istically stationary. For a succession of narrow band random loadings whoseRMSs follow the positive half of a Gaussian distribution (Equation (B.7)) thepeaks sum to the Laplace distribution (Equation (B.14)) (Pook 1983b).

Corresponding load histories for fatigue testing also need to be non sta-tionary. A procedure was developed (Pook 1984) which made it possible toapproximate a wide range of probability distributions as the sum of severalRayleigh distributions and hence produce load histories which consist of asequence of narrow band random loadings. In one example a 7-level approx-imation of the Laplace distribution was used as the basis of an agreed standardload history known as the COmmon LOad Sequence (COLOS) (Anon. 1985).The numbers of cycles and load levels are listed in Table B.1 in terms of theof the overall RMS, σ .

The water surface elevations of ocean waves are an example of a processwhich often approximates to a non stationary narrow band random process(Pook and Dover 1989). In oceanography the primary parameter used in thecharacterisation of sea state is the wave height, H , which is measured peakto trough. Over a period of time short enough (conventionally 20 min) for

Metal Fatigue 217

Table B.2. Scatter diagram for M V Famita for full year.

H1/3 m Zero crossing wave period, s4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5

0.3 14 40 34 8 0 0 0 0 0 00.91 64 159 135 40 4 0 0 0 0 01.52 18 103 164 78 24 2 0 0 0 02.13 6 53 126 95 33 7 1 0 0 02.74 0 19 103 72 41 8 2 0 0 0

3.35 0 9 46 71 31 3 1 0 0 03.96 0 1 23 63 38 6 1 0 0 04.47 0 0 6 20 31 10 2 1 0 05.18 0 0 5 13 15 12 1 0 0 05.79 0 0 1 9 4 6 3 0 0 0

6.40 0 0 0 2 2 4 2 0 0 27.01 0 0 0 0 1 3 7 0 0 07.62 0 0 0 1 1 0 4 0 2 08.23 0 0 0 1 2 0 0 0 0 08.84 0 0 0 0 0 2 2 0 0 0

9.45 0 0 0 0 0 0 0 0 0 2Parts per 1924.

a sea state to be regarded as statistically stationary, the usual measure of itsseverity is the significant wave height, H1/3, which is the average height ofthe highest one third portion of the waves, and is approximately equal to 4σ

(Sarpkaya and Isaacson 1981). One of the ways in which wave height datacan be usefully presented is by means of a scatter diagram that gives therelative occurrences of sea states within specified small intervals of H1/3 andwave period, T , which is the reciprocal of the wave passing frequency (Pook1987b). Table B.2 is an example of a scatter diagram obtained by the M VFamita (Holmes and Tickell 1975). The M V Famita is one of a number ofships that have been stationed in the North Sea to collect oceanographic data.

Long term records show that variations in sea state have the appearance ofa random process (Anon. 1985), but examination of data from four differentsources showed that the distribution of H1/3 is not Gaussian (Pook 1987b).A detailed examination of some data for five years (Pook and Dover 1989)showed that the distribution of H1/3 was more accurately represented by theGumbel distribution (Gumbel 1958). In its simplest form the exceedance,P(S), of the Gumbel distribution is given by

P(S) = 1 − exp{− exp(−S)}. (B.21)


For S > 4, P(S) ≈ exp(−S), and for S < 2, P(S) ≈ 1. Taking S asthe significant wave height, H1/3, the sea state data were well fitted by theexpression

P(H1/3) = 1 − exp

{− exp

(1.9 − H1/3

1.06

)}, (B.22)

where P(H1/3) is the exceedance of H1/3.

B.4 Broad Band Random Loading

In a broad band random loading (Figure B.1) individual cycles cannot be dis-tinguished (see Section 4.3.3). When such a process is encountered in metalfatigue it is usually characterised by σ , that is the root mean square (RMS)value of the whole process. A measure of bandwidth is also required; a com-mon one in metal fatigue is the irregularity factor, I , which is the ratio ofmean crossings to peaks (see Section 4.3.3). It lies in the range 0 to 1. Theirregularity factor has the advantages that is easily understood, and is notrestricted to processes whose instantaneous values follow the Gaussian dis-tribution.

B.4.1 SPECTRAL DENSITY FUNCTION

In random process theory a process which is statistically stationary and er-godic, and whose instantaneous values follow the Gaussian distribution, isusually regarded as adequately described statistically if its root mean square(RMS) value and spectral density function (SDF) are known. Mathematic-ally, the SDF is obtained by first calculating the autocorrelation function,R(τ). This describes the relationship between the values of the random pro-cess S(τ) at times t and t + τ , and is given by (Papoulis 1965, Bendat andPiersol 2000)

R(τ) = lim(T → ∞)1

T

∫ T

0S(t)S(t + τ) dt. (B.23)

The SDF, G(f ), where f is frequency, is the Fourier transform of R(τ), andis given by

G(f ) = 2∫ ∞

−∞R(τ) exp(−iπf τ) dτ

= 4∫ ∞

−∞R(τ) cos(2πf τ) dτ. (B.24)

Metal Fatigue 219

Determining the SDF in this way is called transforming from the time domainto the frequency domain. The SDF is sometimes plotted on a logarithmic scaleand sometimes on a linear scale. Figure B.1(b) shows the SDF for the broadband random loading shown in Figure B.1(a). In practice it is calculated usingan algorithm known as the Fast Fourier Transform (FFT) (Bendat and Piersol2000). Physically, the SDF gives the frequency content of the random process,and in the narrow band random process case is sharply peaked at the centrefrequency (resonant frequency). An alternative method of determining theSDF is to pass the process of interest through a bandpass filter of very narrowbandwidth and plot the amplitude of the resulting signal against frequency.The area under the PSD is equal to the mean square value of the process, asgiven by Equation (B.2).

In electrical engineering the SDF is usually known as the power spectraldensity (PSD) because it provides a measure of the electrical power, whichmay be ascribed to the various frequency components.

Some useful results depend only on the spectral bandwidth, ε, which is ameasure of the RMS width of the SDF (Pook 1978, Bendat and Piersol 2000).It lies in the range 0 to 1 and is given by

ε =√

1 − m22

m0m4, (B.25)

where m0, m2 and m4 are the zeroth, second and fourth moments of the SDFabout the origin. It is related to the irregularity factor by

ε2 = 1 − I 2. (B.26)

As ε → 0 the distribution of peaks tends to the Rayleigh distribution (Equa-tions (B.10) and (B.11)) and as ε → 1 to the Gaussian distribution (Equa-tions (B.7) and (B.8)).

There is no generally accepted definition of what is meant by narrow band,partly because of the physical difficulties of measuring bandwidth as ε → 0.In metal fatigue a random process is usually called narrow band if the peakdistribution approximates to the Rayleigh distribution; this is generally so,provided that I ≥ 0.99, corresponding to ε ≤ 0.14 (see Section 4.2.2).

Difficulties in determining the irregularity factor for narrow band randomprocess are illustrated by the example shown in Figure B.3. Inevitably, onlya finite length process can be examined, so decisions are needed on how todeal with the beginning and end of the process. Also, the mean value of theprocess has to be determined. The horizontal line in Figure B.3 is intended tobe the mean value. In the figure there are 44 upward going zero crossings ofthis line and 45 positive peaks, giving an irregularity factor of 44/45 ≈ 0.98.


Figure B.8. Spectral density functions for a 0.76 m diameter horizontal member immersed10.8 m, significant wave height 4.75 m. (a) Water surface elevation; (b) bending stress; (c) axialstress (Pook 1989b). Reproduced under the terms of the Click-Use Licence.

Metal Fatigue 221

It could be argued that the mean crossing at the end of the process shouldnot be counted, giving an irregularity factor of 43/45 ≈ 0.96. However, if thehorizontal line were slightly lower there would be 45 upward going crossings,and the irregularity factor would be 45/45 = 1. What should be taken asthe correct value is not easily resolved. In principle, the irregularity factorcould be determined by first finding the spectral bandwidth and then usingEquation (B.26) but there are corresponding difficulties in determining thespectral bandwidth of a narrow band random process.

As an example of the sort of information that can be derived from spectraldensity functions, Figure B.8 shows data for a tubular welded tall platformin the North Sea (Pook 1989b). A 0.76 m diameter horizontal member, im-mersed 10.8 m, was strain gauged so that bending and axial stresses couldbe derived. In practice, although sea states have a dominant wave passingfrequency, they are not particularly narrow band so energy may be avail-able to excite structural resonances (Pook 1987b). The SDF for the watersurface elevation (Figure B.8(a)) shows this. There is a clearly defined peakcorresponding to the dominant wave passing frequency, but there is signific-ant energy at other frequencies. To avoid structural resonances, offshore plat-forms are designed so that resonant frequencies are substantially greater thanthe dominant wave passing frequency. This has been successful for the axialstresses since, as might be expected, the SDF (Figure B.8(b)) is of similarform, with no structural resonances exited. However, the SDF for the bend-ing stress (Figure B.8(c)) does show two peaks corresponding to structuralresonances. The point of collecting data of this sort is to permit comparisonof actual structural behaviour with theoretical calculations.

C

Non Destructive Testing

C.1 Introduction

Non destructive testing (NDT) is not a clearly defined concept (Halmshaw1991). NDT has a wide range of applications in the detection and evaluationof flaws in materials. Many different methods are used, and a wide rangeof commercially available instruments has been developed. These are oftenautomated under computer control. The key feature of NDT is that it has nodeleterious effect on the item tested.

In the context of metal fatigue the usual meaning of non destructive test-ing is the detection and sizing of cracks and crack like flaws in compon-ents, structures and laboratory specimens. This includes monitoring of fatiguecrack propagation in service and in laboratory specimens; the advantages anddisadvantages of various methods are summarised by Richards (1980). Theaccuracy of crack sizing that can be achieved varies widely.

Some of the non destructive testing techniques used in metal fatigue workare described briefly in this appendix, together with the important statisticalconcepts of probability of detection and probability of sizing. In order to makethe appendix more self contained some figures in the main text are repeatedhere.

C.2 Visual Inspection

Visual inspection is the simplest method of detecting surface cracks, usuallycalled surface breaking cracks in the non destructive testing literature. Thisterm is used in this appendix. The utility and importance of visual inspectionare often underestimated. Under good conditions fatigue cracks with a surfacelength of 3 mm can be detected by the naked eye, but in general 25 mm is a

224 Appendix C: Non Destructive Testing

Figure C.1. Fatigue cracks in an aircraft engine nacelle.

more realistic detection limit. A low power lens (say ×3) and additional port-able lighting are useful. For a permanent record photographs may be taken orreplicas of the surface made. If the surface is irregular, as in welds, surfacebreaking cracks are difficult to detect by visual methods.

Regulatory authorities often call for periodic visual inspection of struc-tures for defects, including cracks. For example, Figure C.1 shows unexpec-ted fatigue cracking found in an aircraft engine nacelle during a routine in-spection (Pook 2004). Another example is the cracking in a burner from adomestic central heating boiler shown in Figure 8.13. Routine visual inspec-tion is tedious, and fatigue cracks are sometimes missed. For example, oneof the concerns at the official inquiry into the catastrophic fatigue failure ofa fairground ride was why fatigue cracks, which should have been detected,were missed during routine visual inspections (Pook 1998).

When fatigue crack propagation is being monitored visually, crack lengthmeasurement is often aided by markings etched or scribed onto the speci-men surface, for example the grid shown in Figure C.2 (see Section 8.2.2).Visual methods have been widely used to collect data during fatigue crackpropagation rate tests, scribed marks were used to collect the data shown inFigure C.3. The use of guide markings does not meet resolution accuracy re-quirements in modern fatigue crack propagation rate testing standards suchas Anon. (2003b). A common technique, which does meet the requirements,is to use a micrometer thread travelling microscope with a magnification of×20 to ×50.

Visual methods of inspection have the advantage that the equipmentneeded is relatively inexpensive, but they are labour intensive when used tomonitor fatigue crack propagation, and are not amenable to automation. The

Metal Fatigue 225

Figure C.2. Fatigue crack path in a Waspaloy sheet under biaxial fatigue load. The grid is0.1 inch (2.54 mm). National Engineering Laboratory photograph. Reproduced under theterms of the Click-Use Licence.

Figure C.3. Fatigue crack propagation curve for a central crack, length 2a, in a 0.76 m wide× 2.5 mm thick mild steel specimen. Nominal stress 108 ± 31 MPa (Frost et al. 1974).

major disadvantage is that only crack surface lengths can be measured. In aplate of constant thickness a fatigue crack front of a through the thicknesscrack is often curved (Figure C.4). This curvature can affect the calculationof stress intensity factors (see Section A.3.2). With a surface breaking crack


Figure C.4. Fracture surface of 19 mm thick aluminium alloy fracture toughness test specimen(Pook 1968). Reproduced under the terms of the Click-Use Licence.

Figure C.5. Uniform alternating current on the surface of a plate containing a surface breakingcrack.

(Figure C.5) it would not be possible to calculate stress intensity factors be-cause these are largely dependent on crack depth (see Section A.3.1.2).

In practice, the major use of visual inspection is to detect surface breakingcracks. Any cracks found are then sized using an appropriate technique suchas ultrasonics (see Section C.6) or alternating current potential drop (seeSection C.7.2).

C.3 Magnetic Particle Inspection

Magnetic particle inspection (MPI) is a well established technique for thedetection of surface breaking cracks. The method can be used only on fer-romagnetic materials which can be strongly magnetised. These include irons

Metal Fatigue 227

Figure C.6. Principle of magnetic particle inspection.

and ferritic steels, but not all steels. Magnetic effects arise through electro-magnetic fields. These can be represented as lines of magnetic force throughspace which form a magnetic flux.

The principal of magnetic particle inspection is shown in Figure C.6. Amagnetic flux is established in the material by placing the poles of a magnet(usually an electromagnet) in contact with the material. If the magnetic fluxencounters a transverse surface breaking crack the flux becomes distorted.Some of the magnetic flux passes through the crack, some passes aroundthe crack tip, and some leakage flux passes around the crack at the surface.This leakage flux attracts ferromagnetic particles to the crack mouth, and theresulting visible concentration of particles marks the crack.

The magnetic particles are applied as a suspension in a carrier liquid, suchas light oil or water, at a concentration by volume of about 2 per cent. If wateris used, a wetting agent and a corrosion inhibitor are incorporated. The sus-pension is normally supplied in an aerosol, and is sometimes called a mag-netic ink. The particles are usually black iron oxide of around 1–25 µm insize, and may be dyed to improve visibility. Florescent dyes are sometimesused, and the particles are then viewed under ultra violet light

Magnetic particle inspection is the most widely used non destructing test-ing method for detecting surface breaking cracks in welded joints. MPI iseasily carried out using portable equipment, but expertise is needed for satis-factory results, and it can be a messy procedure. The major disadvantages arethat only the surface length of a crack can be determined, and the accuracy ofcrack sizing is low. An advantage is that, with special equipment, magneticparticle inspection can be used under water.


Figure C.7. Schematic view of dye penetrant in crack after removal of excess penetrant fromthe surface.

C.4 Dye Penetrant

The dye penetrant method is used to detect surface breaking cracks. Themethod can be applied to any material that has a non absorbent surface. Mostof the cracks found by dye penetrants can be seen visually in good condi-tions, but dye penetrants make them much easier to detect. The principle ofthe method is shown in Figure C.7). After the surface has been cleaned a pen-etrant, which contains a dye in solution, is applied. The penetrant is chosenso that it wets the material being inspected, and it is drawn into cracks by ca-pillary action. Excess penetrant is then removed from the surface, and a thinlayer of a porous developer is applied. Penetrant is drawn out of cracks by thedeveloper, thus making cracks visible.

The dye and developer colours are chosen to provide good contrast. Pen-etrant dyes and developers are usually supplied in aerosols. A wide rangeof techniques is available, and for good results the technique chosen mustbe carefully matched to the intended application (Halmshaw 1991). Unfortu-nately, much published information on the results of dye penetrant non de-structive testing is of little value because full details of techniques used arenot included.

The dye penetrant method is widely used for aluminium alloys and othermetallic materials which cannot be magnetised so that magnetic particle in-spection is impossible (see previous section). The main advantage of dye pen-etrant is that it is simple to use, and particularly suitable for field work. Themain disadvantage, as with visual inspection and magnetic particle, is thatonly the surface length of a crack can be determined (see Sections C.2 andC.3). If fatigue crack propagation is being monitored, a potential disadvantageis that dye penetrant remaining in a fatigue crack could affect its subsequentpropagation behaviour.

Metal Fatigue 229

Figure C.8. Fatigue cracking from shrinkage cavity in 30 × 35 mm cast steel bar. NationalEngineering Laboratory photograph. Reproduced under the terms of the Click-Use Licence.

Figure C.9. Schematic view of shrinkage cavity in 30 × 35 mm cast steel bar.

C.5 Radiography

The main use of radiography in metal fatigue is the detection of voluminousdefects including internal cavities, porosity and inclusions. An example isthe shrinkage cavity in a cast steel bar shown in Figure C.8. This has anirregular boundary and can be regarded as a crack like flaw. The general shapeof the cavity was obtained by radiography, and it is shown schematically inFigure C.9 (Pook et al. 1981).

The principle of radiography in its original form is shown in Figure C.10.X-rays are emitted from a small source, and travel in straight lines towardsa sheet of photographic film, which acts as a detector. The X-rays are partlyabsorbed as they pass through the specimen, and then strike the photographicfilm to produce a radiograph. There is less absorption when the X-rays passthrough a cavity, and a two dimensional image of the three dimensional cav-ity is formed on the film. To get three dimensional information on the cavity,


Figure C.10. Principle of radiography in its original form.

including its location within the specimen, radiographs have to be taken frommore than one direction. This was done to produce the sketches shown in Fig-ure C.9. As an alternative to using film, X-rays can be captured electronically,and images displayed in real time on a monitor.

X-rays are a form of electromagnetic radiation, similar to light, but withvery much shorter wavelengths. They are produced when a beam of high en-ergy electrons strikes a metal anode in a vacuum. Wavelengths of X-rays usedin radiography range from about 10−4 nm to about 10 nm. Long wavelengthX-rays are sometimes called soft X-rays and will penetrate only small dis-tances. Short wavelength X-rays are sometimes called hard X-rays, and canpenetrate up to about 50 cm thick steel. Gamma rays are sometimes used inradiography. They are also a form of electromagnetic radiation and are pro-duced by the decay of a radioactive isotope such as cobalt-60.

Radiography is a very versatile and easily used technique. It is probablythe oldest non destructive testing technique used for the quality control ofwelded joints. Radiographs provide a convenient, permanent record. The ma-jor disadvantage of radiography is that stringent safety precautions have tobe taken to protect operators, and the public at large, from radiation. From ametal fatigue viewpoint its major disadvantage is that it is difficult to detectand size tight cracks, that is cracks whose opposite surfaces are either closetogether or touching.

Metal Fatigue 231

C.6 Ultrasonics

In metal fatigue the main use of ultrasonics is crack sizing, and a wide rangeof techniques is available. The method is based on the propagation of soundwaves through the material at frequencies above the audible range, hence theterm ultrasonics. Sound waves are mechanical vibrations. Hence the velocityof propagation is different in different materials, and also depends on thetype of wave. Frequencies used are of the order of a MHz, and the resultingwavelengths are of the order of a mm. The essential feature of the waves usedin ultrasonics is that they propagate through the material in the same way thatocean waves move across the water surface. This contrasts with the standingwaves observed in the vibrations of a tuning fork. Inadvertent standing wavescan be a problem in the use of ultrasonics.

Two main types of wave are used in ultrasonics. One is compressionalwaves, also known as longitudinal waves, where particles vibrate in the dir-ection of wave propagation. The other is shear waves, also known as trans-verse waves, where vibration is at right angles to the propagation direction.Several other types of wave are used for special purposes (Halmshaw 1991).Ultrasonic waves used to interrogate the specimen under test are generated inpulses, not continuously.

Ultrasonic wave pulses are generated by applying short electric pulses toa suitable probe. One type of probe uses a piezoelectric disc, which resonatesat a selected frequency (Figure C.11). A couplant, such as thin layer of oil,is used to ensure good transmission of ultrasonic waves from the probe intothe specimen under test. Ultrasonic waves emerging from the specimen arereceived by a suitably positioned probe, and processed to give informationabout defects within the specimen. The same probe is sometimes used forboth transmission and reception.

The arrangement for ultrasonic testing of a cracked specimen, using acompressional probe, is shown schematically in Figure C.11. Ultrasonic wavepulses are reflected from the crack, and also from the top and bottom sur-faces of the specimen. These echoes are displayed on an oscilloscope usingwhat is known as an A-scan, shown schematically in Figure C.12. The timebase of the oscilloscope is triggered as each ultrasonic pulse is transmitted.Hence, the positions of the echoes on the time axis provide information onthe crack location. In practice, dispersion effects within the specimen meanthat subsidiary echoes also appear. These subsidiary echoes can complicateinterpretation of the display.

Other methods of display may be used when a probe is scanned acrossa specimen surface. A B-scan is obtained from a scan along a line on the


Figure C.11. Arrangement for ultrasonic testing of a cracked specimen using a compressionalprobe.

Figure C.12. A-scan showing crack.

surface. The display is arranged so that it shows the sizes and positions offlaws on a cross section perpendicular to the surface of the specimen. A C-scan produces a radiograph like display (see previous section). It is obtainedfrom a series of scans along parallel lines, and the display shows a plan viewof the specimen in which defects appear in their correct positions, but with noinformation on their through the thickness locations. A D-scan is similar to aB-scan, but is obtained from a series of scans along parallel lines. The termsA-scan, etc., are often used in the ultrasonics literature without explanation.

Various ultrasonic techniques are used to size cracks and crack like flaws.Two of these, which are used for surface breaking cracks, are shown schem-atically in Figures C.13 and C.14. In the end on technique a high intensitycompressional probe, located at the opposite surface, is used to find the tip

Metal Fatigue 233

Figure C.13. End on technique for measuring the depth of a surface crack.

of the crack (Figure C.13). The depth of the crack can then be determineddirectly from an A-scan. A different approach is used in the time of flightdiffraction (TOFD) technique, shown schematically in Figure A.14. In thistechnique there are separate transmitter and receiver compressional probes.The principle is that when a compressional ultrasonic beam meets a cracktip some of the energy is diffracted. The diffracted waves spread over a largeangular range, and may be detected by a suitably placed receiver probe. Ifthe transmitter and receive probes are symmetrically placed about the cracktip, then a simple calculation gives the position of the crack tip, and hencethe crack depth. To ensure symmetry about the crack tip the two probes maybe linked mechanically, and traversed across the crack. The probes are sym-metrically positioned when the time of flight is minimised. Shear waves aresometimes used in the TOFD technique; these are more suitable for deepercracks. Both the end on technique and the TOFD technique are compatiblewith methods of approximation of stress intensity factors since, in effect, anoblique crack is projected onto a plane perpendicular to the surface (see Sec-tion A.5.2).

Simple theory suggests that, for a flaw of a given size, the height of theecho on an A-scan is inversely proportional to the square of the distance of theflaw from the probe. A distance amplitude correction curve, usually knownas a DAC curve, is used to correct for this effect. The term DAC level refersto the heights of echoes, relative to background noise, that are regarded assignificant. (Background noise is known as grass, because of its appearanceon an oscilloscope screen.) Thus, 50 per cent DAC (level) means that only


Figure C.14. Time of flight diffraction technique for measuring the depth of a surface crack.

echo heights that are at least 50 per cent greater than the height of the grassare considered significant. The choice of DAC level is important when theprobability of detection and probability of sizing are being determined (Visser2002) (see Section C.8).

Ultrasonics is a very versatile and well established method of crack sizing,and a wide range of techniques is available. Technique details are readilyadaptable to specific applications. The major disadvantages of ultrasonics arecost and that it is not suitable for small, thin specimens. It is also difficult touse on austenitic steels.

C.7 Electromagnetic Fields

In metal fatigue electromagnetic field methods are used both for crack detec-tion and for crack sizing. They are based on the injection of a uniform al-ternating current field into the surface of a specimen, such as the plate shownin Figure C.5. Eddy current and alternating current potential drop (ACPD)methods are usually regarded as distinct methods of non destructive testing,but they are actually limiting cases of general electromagnetic field methods .Due to the skin effect the alternating current density is greatest at the surface,and decreases exponentially with depth below the surface. The skin depth, δ,is usually defined as the depth at which the alternating current density is 1/e

(36.8%) of its surface value, and this depth is given by (Lewis et al. 1988)

δ = 1√πµσf

, (C.1)

Metal Fatigue 235

where µ is magnetic permeability, σ is electrical conductivity, and f isthe frequency of the alternating current. Some typical skin depths shown inTable C.1. Magnetic permeability is sensitive to precise material composition,and it is also a function of current density, Hence data in the table should onlybe used as a guide to the selection of an appropriate frequency.

Table C.1. Typical skin depths.

Material Frequency1 kHz 10 kHz 100 kHz 1 MHz

18/8 stainless steel 13.1 mm 4.14 mm 1.31 mm 0.414 mmBrass 3.90 mm 1.23 mm 0.390 mm 0.123 mmAluminium 2.65 mm 0.838 mm 0.265 mm 0.084 mmCopper 2.00 mm 0.632 mm 0.200 mm 0.063 mmMild steel 0.148 mm 0.047 mm 0.015 mm 0.005 mm

For satisfactory results the skin depth must be small compared with thecrack depth. Typically δ is about 0.1 mm so, in general, electromagnetic fieldmethods cannot be used for cracks less than about 1 mm deep.

The response of a crack being interrogated by an alternating current de-pends on the value of the dimensionless parameter m, which is given by(Lewis et al. 1988)

m = µ0a

µδ, (C.2)

where µ0 is the magnetic permeability of a vacuum (= 4π×10−7 H m−1). Fornon magnetic materials of high electrical conductivity, such as aluminium,µ ≈ µ0, m is large because δ/a is small, and eddy current testing is appropri-ate. For magnetic materials, such as ferritic steels, µ µ0, m is small, andACPD testing is appropriate. Typical values of m are 12 for aluminium and0.6 for mild steel (Lewis et al. 1988).

C.7.1 EDDY CURRENT

In metal fatigue the main uses of eddy current testing are the detection andsizing of cracks in non magnetic materials, especially aluminium alloys. Fre-quencies of the order of one MHz are used in order to ensure a small skindepth (Table C.1).

The principle of eddy current testing is shown schematically in Fig-ure C.15. A probe with a current carrying coil is scanned across the specimenat a small fixed lift off distance. The alternating current in the coil produces


Figure C.15. Principle of eddy current testing of a cracked specimen.

an alternating magnetic flux, which induces eddy currents in the specimen.These eddy currents are sometimes called Foucault currents. The inducededdy currents in turn produce an alternating magnetic flux. This opposes theflux produced by the current carrying coil, and changes the impedance of thecoil. As the probe passes over a crack the eddy currents are distorted and thepresence of a crack is detected, using suitable instrumentation, by changesin the coil impedance. In some systems the effect of the magnetic flux pro-duced by the eddy currents is monitored by voltages induced in a second coil,similar to the current carrying coil. An important practical advantage of eddycurrent testing is that physical contact between the probe and the specimen isnot necessary.

Eddy current testing is a very versatile method of crack detection and siz-ing, and a wide range of techniques is available. It is possible to detect cracksas small as 10 µm deep. However, for good results the technique and instru-mentation chosen must be carefully matched to the intended application. Itsmajor disadvantages are cost and, at times, interpretational difficulties. It ispossible to monitor fatigue crack propagation by using a system in which theprobe is traversed by a motor, and locks onto the crack tip.

When used on magnetic materials, eddy current testing is sometimescalled alternating current field measurement (Lewis et al. 1988). Lower fre-quencies are used, and methods of data reduction are different.

C.7.2 ALTERNATING CURRENT POTENTIAL DROP

In metal fatigue the main uses of alternating current potential drop (ACPD)techniques are the detection and sizing of cracks in magnetic materials, espe-

Metal Fatigue 237

Figure C.16. Principle of alternating current potential drop testing of a cracked specimen.

cially steel. Frequencies of the order of around 1–10 MHz are used in orderto ensure a small skin depth (Table C.1).

The principle of ACPD testing is shown in Figure C.16. The current is im-pressed through two contacts some distance apart. It flows along the specimenskin from one contact to the other, passing down one side of the crack and upthe other. The voltage (potential drop) is measured using a probe with knowncontact spacing, �, placed across the crack. In automatic systems for monit-oring fatigue crack propagation, current impression and potential drop meas-urement measuring wires are spot welded to the specimen (Austin 1999).

Calibration is straightforward. A voltage, V1, is first measured by placingthe probe near the crack, as shown in Figure C.17 (left). The voltage, V2,across the crack is then measured (Figure C.17 (centre)). Assuming that thecurrent is constant for the two measurements, then the crack depth, a, is givenby

a =(

V2

V1− 1

)�

2. (C.3)

This one dimensional solution has to be modified for a two dimensional crack,such as that shown in Figure C.5. Modifiers are available for various config-urations.

Because the technique is self calibrating, the impressed current field doesnot have to be completely uniform. This makes the technique suitable forirregularly shaped specimens such as welded joints. The technique is par-ticularly suitable for automatic collection of fatigue crack propagation dataduring structural fatigue tests. For fatigue crack propagation rate testing on


Figure C.17. Alternating current potential drop calibration.

plate specimens, in which a crack front may be curved (Figure C.4), contactpoints can be arranged so as to measure an average crack length through thethickness.

The main advantages of the ACPD method are its versatility and its easeof calibration. Voltages are low, so no safety precautions are needed, and themethod can be used under water. Care has to be taken in arranging the im-pressed current and probe leads so as to avoid spurious interactions, and alsointerference in electrically noisy environments. Crack bridging by metallicparticles is not usually a serious problem. Various systems are available com-mercially. Techniques are still evolving. It is moderately expensive.

A disadvantage is that for an oblique crack the method gives the cracklength, not the depth of the crack tip below the surface (Figure C.17 right).The ACPD method is therefore not compatible with methods of approxima-tion of stress intensity factors where an oblique crack is projected onto a plane(see Section A.5.2). In effect, the projected crack length is overestimated, asis the concomitant approximated stress intensity factor.

C.8 Probability of Detection

Traditionally, it has been assumed that a particular non destructive testingtechnique is capable of detecting all cracks larger than a critical size, andthat no cracks smaller than the critical size will be detected. What is meantby crack size depends on the application; for a surface breaking crack (Fig-ure C.5) this is usually either the surface length or the maximum depth. Inpractice a critical crack size is not clearly defined, and the probability of de-tection (POD) increases with the crack size as shown schematically in Fig-ure C.18, which also shows the ideal situation where a critical crack size isclearly defined.

Metal Fatigue 239

Figure C.18. Actual and ideal probability of detection (POD) curves.

Figure C.19. Schematic relative operating characteristic (ROC) curve.

Another possible outcome of an inspection is a false call where no crackis present but one is apparently detected. In a relative operating characteristic(ROC) curve the probability of detection is plotted against the false call prob-ability (FCP) as shown schematically in Figure C.19. A good performance,shown by dashed lines on the figure, is defined by Visser (2002) as a probab-ility of detection of ay least 80 per cent combined with a false call probabilityof at most 20 per cent.


Table C.2. Typical data obtained from a probability of detection trial.

Crack size Whether Crack size Whether Crack size Whether(mm) detected (mm) detected (mm) detected2.2 No 8.8 Yes 21.5 Yes2.5 No 9.5 No 21.7 Yes3.0 No 10.7 Yes 21.8 Yes3.4 Yes 11.5 Yes 22.2 No3.8 No 13.4 Yes 22.4 Yes

4.2 Yes 13.9 No 22.9 Yes5.1 Yes 16.0 Yes 23.9 Yes5.8 No 17.6 Yes 24.4 Yes6.5 Yes 17.9 No 26.0 Yes7.1 Yes 19.4 Yes 26.2 Yes

7.3 Yes 19.7 Yes 28.2 Yes8.0 No 19.9 Yes 28.9 Yes

C.8.1 DETERMINATION OF PROBABILITY OF DETECTION

Probability of detection (POD) curves are determined experimentally by blindtrials on a set of specimens containing cracks of known sizes. The objectiveis to see what can be achieved by a skilled inspector using a particular nondestructive testing technique, not to test the inspector. It is good practice toinclude some uncracked specimens so that data on false calls can be obtained.Sets of specimens kept for blind trials are sometimes called a library. Inspect-ors carrying out blind trials are not given any information on the cracks in thespecimens, or on the results of the trials.

There are two particular difficulties in carrying out POD trials. The first isproducing specimens with cracks of the desired size range. Cracks are usuallyproduced by fatigue loading but fatigue cracks are, in general, difficult tocontrol. The second is ensuring that crack sizes, especially crack depths, areaccurately known. The most satisfactory way of determining crack depths isto section specimens, but this destroys them so that they cannot be used forfurther trials. What is sometimes done is to size the cracks using an accuratemethod, such as time of flight diffraction (see Section C.6) and, from timeto time, section a few specimens in a library as a cross check. The sectionedspecimens are replaced by new specimens containing similar cracks.

Typical data obtained from a POD trial are shown in Table C.2. In orderto obtain point estimates of POD, specimens are grouped according to cracksize, as shown in Table C.3. It is good practice to have about the same numberof groups as there are specimens in each group. The number of successful

Metal Fatigue 241

Table C.3. Point estimates of probability of detection (POD).

Crack size Number in PODrange (mm) group (per cent)

0–5 6 33.35–10 8 62.5

10–15 4 7515–20 6 83.320–25 8 87.525–30 4 100

Figure C.20. Probability of detection point estimates.

detections in each group divided by the number of specimens in the group isa point estimate of POD, and is usually expressed as a percentage. The pointestimates are plotted at the upper limit of each size range, and are usuallyjoined by straight lines, as shown in Figure C.20.

The probabilities of detection shown in Table C.3 are point estimatesbased on small samples of 4 to 8 cracks. However, point estimates obtainedfrom small samples may not accurately reflect probabilities of detection ob-tained from large samples. Point estimates obtained from small samples aresometimes analysed statistically to provide lower bound estimates of probab-ilities of detection at, say, the 95 per cent confidence level. However, the smallsample sizes typically used in practice for cost reasons make rigorous statist-ical analysis difficult, and lower bound estimates of probabilities of detectionare not in general use (Visser 2002).


Table C.4. Typical data obtained from a probability of sizing trial.

Actual crack Measured crack Actual crack Measured cracksize (mm) size (mm) size (mm) size (mm)

1.5 Not detected 13.1 13.71.9 1.9 13.5 15.12.3 Not detected 14.7 15.42.7 3.7 16.1 16.23.1 Not detected 16.2 18.4

3.3 5.5 16.5 17.33.9 5.0 17.7 19.74.5 Not detected 17.9 17.85.2 4.7 18.1 19.95.5 7.9 18.3 Not detected

5.9 5.4 18.5 20.26.5 Not detected 18.9 21.07.1 7.4 19.7 21.27.7 9.6 20.1 19.47.9 Not detected 21.5 21.6

9.4 9.4 21.6 22.011.1 10.7 23.3 23.611.4 Not detected 23.9 26.1

Table C.5. Point estimates of probability of sizing (POS) using 90 per cent accuracy criterioncompared with probability of detection (POD).

Size range (mm) Number in group POS (per cent) POD (per cent)0–5 8 12.5 505–10 8 50 75

10–15 5 60 8015–20 10 60 9020–25 5 80 100

C.8.2 PROBABILITY OF SIZING

The probability of sizing (POS) is a refinement of probability of detection inwhich a crack is counted as detected only if its size is measured to withinan accuracy criterion. An accuracy criterion of x per cent means that themeasured crack size is within ±(100 − x) per cent of the actual crack size.Table C.4 shows typical data obtained from a probability of sizing trial, andTable C.5 point estimates of probability of sizing, using a 90 per cent accuracycriterion. Point estimates of probability of detection are also shown in the

Metal Fatigue 243

Figure C.21. Comparison of probability of sizing (90 per cent accuracy criterion) with prob-ability of detection.

table. These point estimates are plotted in Figure C.21; probabilities of sizingare significantly lower than probabilities of detection.

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Index

K-calibration, 174K-dominated region, 141, 144, 180, 192S/N curve, 16T -stress, 143T -stress criterion, 143T -stress ratio, 144

A-scan, 231absolute liability, 78acceptance testing, 76accuracy criterion, 242additive distribution, 31Almen strip, 99alternating current field measurement, 236alternating current potential drop, 226, 234,

236alternating stress, 15, 16analytical approach, 72, 75, 162antisymmetric mode, 187, 190aspect ratio, 153, 177attractor, 143, 156autocorrelation function, 218autofrettage, 99

B-scan, 231bandpass filter, 219bandwidth, 40, 208, 218basic situation, 69Basquin equation, 18bending mode, 174biaxial fatigue loading, 37biaxial loading, 37, 59biaxiality ratio, 143bimodal distribution, 31

blind trial, 240block fatigue loading, 42, 127block loading, 42block programme, 53boundary layer, 188, 193branch crack, 138, 155, 198branch crack formation, 140, 149branch crack propagation, 140branch crack propagation angle, 141, 148branch point, 93brittle fracture, 103, 110, 136, 138, 154broad band random loading, 35, 50, 218, 219

C-scan, 232carburising, 98centre cracked tension specimen, 106centre frequency, 208, 219centre line average roughness, 84chaos theory, 141chaotic event, 141characteristic crack dimension, 175, 199civil liability, 78cliff, 156clipping, 209, 212, 213clipping ratio, 42, 59, 209, 212, 214code, 70, 73code design, 70COLOS, 216compliance function, 174compressional probe, 232compressional wave, 231conditional distribution, 32constant amplitude loading, 15

260

constant amplitude fatigue loading, 15, 39,42, 96, 105, 106, 117, 120, 122, 124,127

Consumer Protection Act, 78, 79core region, 141, 180, 189corner point, 151, 182, 187, 190, 191corner point singularity, 152, 187corner point singularity dominated region,

189, 192corner region, 188couplant, 231crack area, 199crack bridging, 238crack closure, 114, 127crack front, 168crack front curvature, 138crack front inclination angle, 191crack front intersection angle, 152, 188, 189crack front shape, 129crack initiation dominated, 83, 88, 92, 98crack mouth, 227crack opening, 113crack path prediction, 149, 151, 158crack path stability, 142crack propagation, 138, 172, 173, 186crack propagation dominated, 92, 101crack sizing, 223, 227, 231crack surface displacement, 168, 170, 192crack surface intersection angle, 187, 190,

191crack tip, 168crack tip plastic zone, 105, 112, 113, 125,

137, 150, 182crack tip surface displacement, 169, 170,

183, 187, 190cracked situation, 101criminal liability, 77, 79critical crack front intersection angle, 190,

191critical length, 27critical plane, 63critical plane approach, 64crystal dislocation, 170, 199cumulative damage, 44cumulative probability, 40, 206cycle, 15cycle counting, 50, 125cyclic load, 1

D-scan, 232DAC curve, 233DAC level, 233damage density curve, 49damage density function, 49damage tolerance, 71, 103defective, 69, 77, 78, 80, 164design defect, 69, 78Det Norske Veritas, 132detector, 229developer, 228differential geometry, 11, 155diffracted wave, 233directionally stable crack, 142directionally unstable crack, 142disclination, 171, 173dislocation, 152, 171, 173, 199dispersion effect, 231distance amplitude correction curve, 233dye penetrant, 228

Eastabrook’s theorem, 125eddy current, 234, 235edge sliding mode, 169effective crack length, 182, 184effective value, 113effects of thickness, 110electromagnetic field method, 234electromagnetic radiation, 230end on technique, 232endurance, 17endurance limit, 20enforcement authority, 79, 80enforcement officer, 79equivalent Mode I crack, 197equivalent constant amplitude stress range,

47equivalent cycle, 50, 124equivalent single crack, 201equivalent stress, 63, 64, 89, 96equivalent stress intensity factor, 159equivalent tensile stress, 62ergodic, 204, 218escape clause, 75exceedance, 40, 206, 208, 210, 217exponential distribution, 210

facet, 156fail safe, 70, 154

Index

Metal Fatigue 261

failure, 2, 4failure analysis, 68, 83, 97, 102, 109, 122,

130, 162failure mechanism map, 139false call, 239false call probability, 239FALSTAFF, 54Fast Fourier Transform, 219fatigue, 7fatigue assessment, 67, 69, 71–74, 83, 93,

97, 109, 117fatigue crack growth, 26fatigue crack initiation, 25, 26, 31, 83, 98,

132fatigue crack path, 10, 75, 135, 138fatigue crack propagation, 11, 26, 105, 109,

114, 117, 118, 122, 124, 128, 129,135, 138, 155, 159, 164, 165, 167,223, 224, 228, 236, 237

fatigue crack propagation life, 102, 122, 124fatigue crack propagation rate, 105, 106,

109, 110, 114, 116, 119, 122, 124,127, 133, 224, 237

fatigue cycle, 15, 105fatigue design, 67, 109fatigue fracture toughness, 104fatigue life, 83, 122, 124, 130fatigue limit, 18, 120fatigue load, 1, 55fatigue loading, 15, 38, 61, 94, 98, 138, 149,

154, 156, 199, 204, 240fatigue strength, 90, 162fatigue strength reduction factor, 90fatigue testing, 8, 9final crack size, 101, 103, 122fish eye, 29flame hardening, 98Forsyth’s notation, 26, 137, 148Foucault current, 236fractals, 11fractography, 11, 129fracture criterion, 64fracture mechanics, 12, 164, 167fracture toughness, 104, 110, 155, 172, 183frequency dependence, 116frequency domain, 219frequency independent, 18, 76, 116

gamma rays, 230

Gaussian distribution, 21, 39, 206, 208, 211,213, 214, 216, 218

geometric correction factor, 175Gerber diagram, 86Gerber parabola, 85gigacycle fatigue, 19, 58Goodman diagram, 86, 96good practice, 73grass, 233Gumbel distribution, 217

hard X-rays, 230high cycle fatigue, 17

ideal crack path, 142in house tool, 70induction hardening, 98initial crack, 139, 154, 155, 158, 198initial crack size, 101, 102, 122integral approach, 64interaction effect, 125, 127, 196, 200internal crack, 142, 196, 200internal defect, 32intrinsic fatigue strength, 84irregular crack, 196irregularity factor, 40, 218

JOSH, 54

Kirchoff plate bending theory, 174Kitagawa diagram, 120knee, 19KoNoS hypothesis, 65

Laplace distribution, 49, 210, 213, 216LBF Normal distribution, 53leak before break, 5, 154leakage flux, 227library, 240life, 17lift off, 235limit load, 103line tension, 152, 199linear damage rule, 45linear elastic fracture mechanics, 167load cycle, 16load history, 12, 35, 44, 49, 53, 56, 58, 65,

127, 130, 135load spectrum, 13, 44long crack, 118

262

longitudinal wave, 231low cycle fatigue, 17, 65

macrocrack, 27, 136magnetic flux, 227, 236magnetic ink, 227magnetic particle inspection, 226magnetic permeability, 235main crack, 139major project, 70manufacturing defect, 69, 78mass product, 70mathematical description, 37maximum normal stress criterion, 64maximum principal stress dominated crack

propagation, 137maximum stress, 15, 20mean stress, 10, 15, 85, 88mean stress insensitive, 110mean stress sensitive, 110, 114mechanical description, 1, 7, 161, 165metal fatigue, 1, 7, 37, 161–163, 165, 204,

207, 210, 212, 218, 223, 229, 231,235, 236

metal fatigue damage, 10metal fatigue mechanism, 24metallurgical description, 1, 161, 165microcrack, 25, 27micromechanisms, 165Miner’s law, 45Miner’s rule, 13, 45, 49, 50, 97, 126minimum stress, 15Mises criterion, 63mixed mode, 26, 138, 155, 156, 158, 169,

198mixed mode threshold for fatigue crack

propagation, 139Mode I, 169, 170, 172, 183, 184, 186,

190–192, 196, 197, 199Mode II, 169, 171, 172, 186, 191, 192Mode III, 169, 171, 172, 186, 191, 192modified Goodman diagram, 86multiaxial failure criterion, 62multiaxial fatigue loading, 35, 39, 60, 63, 65,

88multiaxial loading, 35, 59

narrow band random loading, 35, 56, 58,210, 214–216

narrow band random process, 41, 59, 208,219

negative peak, 42negligence, 78, 79nitriding, 98no fault liability, 78non destructive testing, 103, 164, 196, 223,

238non metallic material, 4non propagating crack, 93, 116, 120, 142non proportional fatigue loading, 59, 64non proportional loading, 59non proportional random loading, 62non stationary random processes, 42nonlinear dynamics, 143Normal distribution, 21, 39, 206notch, 83, 89, 180notch insensitive, 92notch sensitive, 92notch sensitivity index, 92number of cycles, 16

ocean wave, 216, 231omission dilemma, 58omission level, 58, 213one sided process, 214open crack, 181opening mode, 169overload, 127oxide induced crack closure, 115

P-S-N curves, 23Palmgren–Miner law, 45Palmgren–Miner rule, 45Paris equation, 105Paris law, 105Paris region, 117, 122part through crack, 152, 153, 177peak counting, 51penny shaped crack, 176periodic processes, 206philosophies of design, 70, 154physical crack length, 184plane strain, 109, 182, 183, 185plane strain fracture toughness, 104, 154,

184plane stress, 109, 154, 183, 184, 188plane stress fracture toughness, 154, 184plastic collapse, 103, 154

Index

Metal Fatigue 263

plastic wake, 113, 119, 125, 127plastic zone, 128, 182, 184plastic zone correction, 185plastic zone radius, 185point estimate, 240, 242positive peak, 41postulated cracks, 103power spectral density, 219probability density, 21, 40, 206, 208, 210probability of detection, 223, 234, 238, 240,

242probability of failure, 23, 96, 97probability of sizing, 223, 234, 242product liability, 67, 77, 78programme loading, 42, 129programme marking, 129proof loading, 99proportional fatigue loading, 59, 64proportional loading, 59pseudo random, 35, 214

quality system, 80

radiograph, 229radiography, 229rainflow counting, 51random process, 204, 207random process theory, 35, 39, 204, 207, 218random walk, 142range counting, 51range of scales, 24ratchetting, 30Rayleigh distribution, 41, 208, 210, 213, 216re-assessment, 73redundant, 71reference stress, 38regulatory authority, 75, 80, 224relative operating characteristic, 239residual stress, 88, 98, 99, 124, 133, 180resonant frequency, 219return period, 35, 42, 58, 214root mean square, 39, 131, 206, 207, 210,

212–214, 216, 218rotating bending, 16rotation, 172roughness induced crack closure, 115

safe life, 70safe product, 77, 79, 80

safety factor, 72safety regulation, 78, 79, 80scalar criterion, 63scatter, 20, 31, 72, 96, 97, 107, 122, 133scatter band, 106scatter diagram, 217sea state, 216, 221service loading testing, 72, 75, 76, 163shakedown, 31shear crack propagation, 138, 186shear dominated crack propagation, 137shear lip, 111shear mode, 169shear wave, 231, 233short crack, 118shot peening, 88, 99shrinkage, 101sigmoidal, 117significant wave height, 217skin depth, 234, 235skin effect, 234slant crack propagation, 137, 186, 198slant fatigue crack propagation, 110slip, 25slip line, 10small scale argument, 180, 186Soderberg line, 88soft X-rays, 230spectral bandwidth, 219spectral density function, 40, 56, 218stable state, 114Stage I crack, 26, 137, 140, 158Stage II crack, 26, 101, 138, 148Stage III, 27standard deviation, 21, 206, 207standard load history, 13, 53–56, 62, 216standard procedure, 13, 70, 73, 162standard test method, 12, 107standing wave, 231static failure, 101static failure region, 118, 122static strength, 1statistically non stationary, 75, 216statistically stationary, 39, 204, 216, 218stochastic process, 38stress concentration factor, 91, 150stress criterion, 28, 117stress cycle, 15, 124stress history, 15

264

stress intensity factor, 12, 104, 113, 116,119, 122, 125, 132, 138, 143, 150,155, 158, 159, 164, 167, 169, 172,180, 186, 190–192, 195, 196, 199,200, 225, 233, 238

stress intensity factor range, 105stress intensity measure, 187, 189stress range, 10, 15, 105stress ratio, 15, 45, 110stress relief, 124stress state, 154striation, 27, 129strict liability, 78, 79subsidiary echo, 231surface breaking crack, 223, 226, 228, 232,

238surface crack, 25, 151, 196, 200, 223surface factor, 85surface finish, 84surface hardening, 98surface irregularity, 84symmetric mode, 187, 190

thermal loading, 31, 150thermodynamic criterion, 28, 117, 137threshold for fatigue crack propagation, 11,

29, 116, 117, 119, 122, 132, 138, 158threshold region, 117, 122through the thickness crack, 154, 196, 225tight crack, 230time domain, 219time history, 40, 52time of flight diffraction, 233, 240toughness, 172trading standards officer, 79transformation induced crack closure, 115transition region, 111, 186translation, 172transverse wave, 231Tresca criterion, 63trough, 42, 209truncation, 42, 58, 210, 212, 213

truncation dilemma, 58truncation level, 56twist crack, 156, 158

ultrasonics, 226, 231uncracked situation, 83, 96underload, 127uniaxial fatigue loading, 59uniaxial loading, 35, 59, 90unsafe product, 80

validity corridor, 142variable amplitude fatigue loading, 35, 37,

96, 124variable amplitude loading, 35variance, 206viscous fluid induced crack closure, 115visual inspection, 223Volterra distorsioni, 170von Mises criterion, 63, 88, 92, 184

Wöhler curves, 17Wöhler’s laws, 9WASH, 54water surface elevation, 216, 221wave height, 216wave loading, 55, 56, 58wave passing frequency, 56, 59, 217, 221wave period, 217waveform, 18, 76, 211Weibull distribution, 210, 213weighted average stress range, 47, 124welded joint, 124, 130, 227, 230, 237Wheeler’s model, 128white noise, 42

X-rays, 229

yield criterion, 63Young’s modulus, 109

zero crossing, 40

Index

A Fracture Mechanics - Home - Springer978-1-4020-5597-3/1.pdf · 172 Appendix A: Fracture Mechanics...

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