A Fracture Mechanics

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A.1 Introduction

The applied mechanics framework for study of the behaviour of cracked bodies under load is known as fracture mechanics. The application of fracture mechanics to fatigue crack propagation is well established, and most modern books on metal fatigue include an introduction to the topic. The account below is based on Frost et al. (1974) and Pook (2000a, 2002a). These books include numerous references. The book by Frost et al. was the first on metal fatigue in which a fracture mechanics approach was used throughout. It was reprinted in 1999. Fracture mechanics does not provide any information about the processes involved in fatigue crack propagation. It does provide the descriptive and ana- lytic framework needed for their characterisation, and for the application of fatigue crack propagation data to practical engineering problems. Simplifying assumptions have become conventional in much present day fracture mechanics, and these are satisfactory for many purposes. The material is assumed to be a homogeneous isotropic continuum, and its behaviour is 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 for the actual behaviour of real materials. The basic ideas in linear elastic fracture mechanics are straightforward. The mathematics involved is often for- midable, but does lead to the useful and easily applied key concept of stress intensity factor, which describes the elastic stress and displacement fields in the vicinity√ of a crack tip. A stress intensity factor has√ the dimensions of × stress length. The most widely used units are MPa m. These units√ appear 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 make the 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 the crack tip is the origin of the coordinate system and the z axis lies along the crack tip. Displacements of points within the cracked body when the body is loaded are u, v, w in the x, y, z directions. The terms crack tip and crack front are synonymous. Crack tip tends to be used for two dimensional situations and crack front in three dimensions.

A.2.1 CRACK SURFACE DISPLACEMENT

A fundamental fracture mechanics concept is that of crack surface displacement. 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 displacement. If a load is applied to a cracked body, then the crack surfaces move relative to each other. For points on opposing crack surfaces that were initially in contact there are three possible modes of crack surface displacement (Figure A.2); Mode I where opposing crack surfaces move directly apart in directions parallel to the y axis; Mode II where crack surfaces move over each along the x axis, that is, perpendicular to the crack tip; and Mode III where crack surfaces move over each other in directions parallel to the z axis, that is, parallel to the crack tip. By superimposing the three modes, it is possible to describe the most general case of crack surface displacement. The terms Mode I, Mode II and Mode III are usually capitalised, and are often used in the metal fatigue literature without explanation. The descriptive terms; opening mode, edge sliding mode,andshear mode are sometimes used for Modes I, 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 may also be used to characterise crack propagation. A particular type of elastic crack tip stress field is associated with each mode of crack surface displacement (Paris and Sih 1965). These stress fields are characterised by stress intensity factors, symbol K. Subscripts I, II and III are used to denote mode. Where there is no subscript, Mode I is usually implied; this convention is sometimes used in the text. Corresponding displacement 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 propagate in Mode I, so attention is largely confined to this mode (see Sections 3.4.2, 170 Appendix A: Fracture Mechanics

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 different 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 of crack surface displacement (Figure A.2) provide an adequate description of the movements of crack surfaces when a load is applied. However, if the surface is regarded as consisting of infinitesimal elements, then element rotations must 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 multiply connected body, that is a cylinder with a central hole. An account in English is 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 the cylinder is elastic. The cut surfaces may be moved relative to each other in six different ways, so there are six distinct Volterra distorsioni. The conventional approach to Volterra distorsioni needs to be modified for the description of crack tip surface movements (Pook 2000a). Consider a pair of infinitesimal elements, A and B, which are in the xy plane and are situated on the upper and lower surface of an unloaded crack respectively (Figure A.3). Their initial coordinates are (r, 180◦) and (r, −180◦). The elements A and B are connected by a ring element of infinitesimal width. For clarity, this ring element is shown as having a square cross section and the elements 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 of crack 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 Crack Paths. 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 distance −v (Figure A.4 left). For consistency with crack tip surface displacement notation this is called a Mode I dislocation. The ring element remains within the xy plane, that is, initially plane sections, perpendicular to the crack tip 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 direction, 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 plane sections 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 sections perpendicular to the crack front do not remain plane. Distortion of the ring element has rotational symmetry about the x axis. The remaining three Volterra distorsioni involve rotation of the elements A and B, and are usually called disclinations. The notation used here is based on the idea that each mode of dislocation and disclination is associated with the same coordinate axis (Pook 2002a). Hence in a Mode I disclination, element A rotates through an angle β about an axis parallel to the y axis, and element B 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 axis and element B rotates through an angle −α (Figure A.5 centre). In a Mode III disclination, element A rotates through an angle γ about an axis parallel to the z axis and element B rotates through an angle −γ (Figure A.5 left). Modes II and III disclinations cannot exist in isolation because of interference between elements A and B. Any dislocation mode may be decomposed into a dipole of equal and opposite disclinations of either of the other two modes. 172 Appendix A: Fracture Mechanics

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

If movements of the elements A and B are identical, then a dislocation becomes a translation and a disclination a rotation. For convenience, continuous 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 to express this numerically. The presence of a crack dominates the stress field in the 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 and distortions are assumed to be small, and conditions of equilibrium and com- patibility must be satisfied (Gere and Timoshenko 1991). The use of a two dimensional analysis simplifies the mathematics, and also simplifies some descriptions. For example, what is meant by crack length is unambiguous. Mode III is not possible in two dimensions, so for this mode a quasi two dimensional anti plane analysis is used. Stress intensity factors may be used to characterise the mechanical properties of cracked specimens in just the same way that stresses are used to characterise the mechanical properties of uncracked specimens. They help to quantify the rather elusive concept of a material’s toughness. One convenient definition is this: resistance to crack propagation (including fatigue crack propagation). For example the fracture toughness, Kc, of a metallic material 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 fracture 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 term of a series expansion of the stress field. For a particular mode of crack surface displacement this leading term is always of√ the same general form. Individual stress 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 intensity factor is a singularity of order −1/2; as the crack tip is approached, stresses tend to infinity. A formal definition of the Mode I stress intensity factor is √ KI = lim(r → 0)σy 2πr , (A.1) σ x where y is√ the stress perpendicular to the crack along the axis. The numerical factor, 2π, in the equation is the usual convention. Other conventions are occasionally encountered, especially in early work, leading to numerically different values for stress intensity factors. There are corresponding equations, in terms of shear stresses, for the other two modes. Stress intensity factors of the same mode may be combined by algebraic addition. Once K is known, stress and displacement fields in the vicinity of the crack 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). For example, for Mode I on the x axis in front of the crack

KI σy = √ . (A.2) 2πr Also, for the upper crack surface

K r v = 2 I 2 , E π (plane stress) (A.3) where v is the displacement in the y direction, that is, perpendicular to the crack, and E is Young’s modulus. The equation implies that for Mode I a crack opens up into a parabola (see Section A.4). Displacements are also parabolic for Modes II and III. Equation (A.3) also implies that Mode I crack surface displacement is a combination of a Mode I dislocation and a Mode III disclination (Figures A.4 and A.5). Similarly, Mode III crack surface displacement is a combination of a Mode III dislocation and a Mode I disclination. However, Mode II crack surface displacement is just a Mode II dislocation. 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 174 Appendix A: Fracture Mechanics either positive or negative, and the sign used is a matter of convention. It is usually taken as positive, but careful attention to sign is needed when calculating stresses and displacements. It is possible to obtain stress ﬁeld equations, using Kirchoff plate bending theory,intermsofabending mode stress intensity factor, KB (Paris and Sih 1965). At a plate surface KB is equivalent to KI. Corresponding crack surface displacements are a Mode II disclination (Figure A.5 centre). Because of crack surface interference this is not physically realistic so KB is not now used.

A.3.1 STRESS INTENSITY FACTOR SOLUTIONS

Stress intensity factors are available for numerous configurations, for example Murakami (1987, 1992b, 2001) and this facilitates practical applications. Solutions for test specimens are included in appropriate standards, for example Anon. (2003a). A solution for a particular configuration is sometimes called a K-calibration or a compliance function. Where a solution is presented as an equation fitted to numerical results, care must be taken not to use it outside its specified range. Conventionally, two dimensional solutions are used for sheets and plates of constant thickness subjected to in plane loads. This is usually satisfactory. To illustrate the general form of solutions for 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 crack have no effect.

A.3.1.1 Two Dimensional Solutions

For a centre crack, length 2a (it is conventional to take the length of an internal 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, shown in 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 stress is used (Figure A.8), and to a crack at a sharp notch if a is taken as the crack length 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. To facilitate 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 mechanics based standards. For example, for the three point bend single edge notch specimen shown in Figure A.10 (Anon. 2003b), FY K = √ , (A.7) I B W where F is force, B specimen thickness and W specimen width. Y is given by √ 6 α [1.99 − α(1 − α)(2.15 − 3.93α + 2.7α2)] Y = , (A.8) (1 − 2α)(1 − α)3/2 where α = a/W and the equation is valid for 0 ≤ α ≤ 1. In Anon. (2003b) 1.5 Equation (A.7) includes a factor 10 so that with force measured√ in kN, and specimen dimensions measured in mm, KI values are in MPa m. 176 Appendix A: Fracture Mechanics

Figure A.7. Small edge crack in a sheet under uniaxial tension. Reprinted from Linear Elastic Fracture 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 on the crack faces (Figure A.11) F K = √ . (A.9) I πa

KI usually increases with increasing crack length. This is one of the few cases in 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

a K = σ , I 2 π (A.10) where a is the crack radius. Hence Y in Equation (A.6) is 2/π. With central point loads F on the crack faces (cf. Figure A.11) F K = . (A.11) I (πa)3/2 For an elliptical crack in an inﬁnite body under uniaxial tension the maximum stress intensity factor is at the ends of the minor axis and is given by Metal Fatigue 177

Figure A.8. Small edge crack at a blunt notch in a sheet under uniaxial tension. Reprinted from Linear Elastic Fracture Mechanics for Engineers. Theory and Applications. LP Pook, WIT Press, ISBN 1-85312-703-5, 2000. √ σ πa K = , (A.12) I E(k) where a is the semi minor axis of the elliptical crack and E(k) is the complete elliptic integral of the second kind. This is given by π/2 E(k) = 1 − k2 sin2 d, (A.13) 0 where a2 k = 1 − (A.14) c2 and c is the semi major axis of the ellipse. An approximation for E(k) is a 1.65 (k) = + . ( ≤ a/c ≤ ). E 1 1 464 c 0 1 (A.15) For c = a Equation (A.12) reduces to Equation (A.10) and for c a to Equation (A.4), this is also the solution for a tunnel crack, width 2a,inan inﬁnite body. Many of the cracks observed in service are surface cracks (Figure A.12). They are often called part through cracks.Theaspect ratio of a surface crack is the ratio of crack surface length to crack depth. Other deﬁnitions of aspect ratio are sometimes used (see Sections 7.6.1 and 8.3.1). Surface cracks with 178 Appendix A: Fracture Mechanics

Figure A.9. Small edge crack at a sharp notch in a sheet under uniaxial tension. Reprinted from 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 elliptical cracks (see Section A.5.1). For such a semi elliptical surface crack in a semi inﬁnite body under uniaxial tension the maximum stress intensity factor is at the deepest point of the crack, and its value is dominated by the crack depth rather than the crack surface length. For a semi inﬁnite body approximate values 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 (Frost et al. 1974). √ 1.12σ πa K = . (A.16) I E(k) For c a Equation (A.16) reduces to Equation (A.5), for c = 2.89a to Equation (A.4), and for a semi circular surface crack (a = c)itbecomes √ KI = 0.713σ πa. (A.17) 180 Appendix A: Fracture Mechanics

A.3.1.3 Effect of Residual Stresses

The effect of introducing a crack into a body containing large scale residual stresses is to relieve the residual stresses on the crack plane. Provided that the crack is not too large the corresponding stress intensity factor is the same as if stresses, equal in magnitude but opposite in sign, were applied to the crack surfaces. 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 residual stresses of unknown magnitude can be a serious limitation in the practical application of fracture mechanics.

A.3.2 VALIDITY OF STRESS INTENSITY FACTORS

The application of stress intensity factors to practical engineering problems involving cracks has been a spectacular success over the past 40 years or so. Nevertheless there are some limitations on their validity that arise from the linearly elastic stress analyses on which they are based. Accumulated exper- ience has shown how these limitations can be managed, and what steps need to be taken to mitigate their effects. A stress intensity factor provides a reasonable description of the crack tip stress field in a K-dominated region at the crack tip, radius r ≈ a/10, where a is the crack length, Figure A.13 (Pook 2000a). An apparent objection to the 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, in that strains and displacements are not small. However, as the assumptions are violated only in a small core region,radius r, the general character of the K-dominated region is, to a reasonable approximation, unaffected. Similarly, by this small scale argument, small scale nonlinear effects due to crack tip yielding, microstructural irregularities, internal stresses, irregularities in the crack surface, the actual fracture process, etc., may be regarded as within the core region. If a crack is too short then it may not be possible to use this small scale argument (see Section 7.4.5). Elastic stress fields at the tips of sharp notches are, with due attention to detail, similar to those for cracks. By the small scale argument, stress intensity factors for cracks can be used to describe the elastic stress field at the tips of sharp notches. The elastic stress at the tip of a sharp V-notch can be described by 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 Elastic Fracture 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 open cracks, that is, cracks where the surfaces are separated by a small amount. When yielding is not small scale, stress intensity factors do not provide a reasonable description of the crack tip stress ﬁeld, and other less versatile fracture mechanics parameters become appropriate. Failure to check whether large scale yielding might be occurring is the commonest error in the practical application of stress intensity factors. One early approach was to ensure that the 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 metallic material test standards, minimum acceptable test specimen dimensions are speciﬁed in order to avoid large scale yielding. Actual minimum values depend 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 and sheets implicitly assumes that the crack front is straight through the thickness and also perpendicular to the surfaces. In practice, crack fronts are usually curved. For example Figure A.14 shows the fracture surface of a 19 mm thick aluminium alloy fracture toughness test specimen, the fatigue precrack front is clearly curved. Standards place limits on the permissible amount of fatigue crack front curvature, so that a two dimensional stress intensity factor solution can be used, for example Anon. (2003b, 2005g). 182 Appendix A: Fracture Mechanics

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

Extension of the essentially two dimensional concept of stress intensity factor to three dimensions makes two implicit assumptions. The first is that the line defining a crack front is smooth so that stress intensity factors do not change abruptly along the crack front. This is generally, but not always, true for cracks observed in practice when these are viewed on a macroscopic scale, and does not usually cause difficulties. The second assumption is that a crack front is continuous, that is, it is a closed curve. This is sometimes true, for example for an internal elliptical crack. It is not true at a corner point where a crack front intersects a free surface (Figure A.15). The nature of the crack tip singularity changes in the vicinity of a corner point. This can usually be neglected on the basis of the small scale argument, but sometimes has to be taken 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 practical consequences. First, it leads to a practical deﬁnition of plane strain in the presence of a crack which differs from that usual in the theory of elasticity. Secondly, the relaxation of stresses within the crack tip plastic zone means that, to maintain equilibrium, stresses outside the plastic zone increase, and the effective crack length is increased. Metal Fatigue 183

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

A.3.3.1 Definition of Plane Strain From a theory of elasticity viewpoint an uncracked plate loaded in uniaxial tension is in a state of plane stress, that is the stress in the thickness direction is 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 less highly stressed surrounding material, and stresses are induced in the thickness direction 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 material adjacent to the crack tip does not affect the situation in the interior of a thick plate (Figure A.12). In this context a plate is said to be thick if 2 the thickness is at least 2.5(KI/σY) where KI is the Mode I stress intensity factor and σY is the yield stress. This practical definition of plane strain arose from 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 if 2 the specimen thickness was at least 2.5(Kc/σY) (see Section 7.4.2.2). This minimum value can be regarded as a material constant, and is known as the 184 Appendix A: Fracture Mechanics

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

plane strain fracture toughness, KIc. In the special case of this symbol, the subscript I denotes both a Mode I crack and plane strain. The minimum speci- 2 men thickness requirement of 2.5(Kc/σY) is included in plane strain fracture toughness test standards, for example Anon. (2005g). When a crack front is curved, 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 such cracks can usually be regarded as being in plane strain. 2 When the plate thickness is very much less than 2.5(K/σY) , then the crack tip plastic zone size becomes comparable with the thickness, and yielding 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 symbol Kc is sometimes reserved for the plane stress fracture toughness which is then regarded as a material constant. At intermediate thicknesses the stress state 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 to yielding at the crack tip is shown schematically in Figure A.18. A ﬁrst estimate of the plastic zone size may be obtained by substituting von Mises’ criterion 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

2 Figure A.17. Plane stress, plate thickness 2.5(KI/σY) (Frost et al. 1974). 2 1 KI rY = (plane stress), (A.19) 2π σY where rY is the plastic zone size measured in the crack direction, KI is the Mode I stress intensity factor, and σY is the yield stress. The actual size and shape of a crack tip plastic zone depends on the ﬂow properties of the metal, 2 but its dimensions are always proportional to (KI/σY) . Typically, a plastic zone size is about twice that given by Equation (A.19), so rY is interpreted as the plastic zone radius. The effective crack length becomes a + rY, as indicated in Figure A.18, and the corresponding stress intensity factor is calculated iteratively. Under plane strain conditions the plastic zone radius is about one third of that given by Equation (A.19), and 2 1 KI rY = (plane strain). (A.20) 6π σY The plastic zone corrections given by Equations (A.19) and (A.20) are often very small and hence unnecessary. At one time they were quite popular, but are now rarely used. Plastic zone corrections do not appear to be speciﬁed in any standards. 186 Appendix A: Fracture Mechanics

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 sometimes observed in thin sheets under both static and fatigue loading, as shown schematically in Figure A.19, is an exception to the observation that fatigue cracks in 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 Mode III, 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 finite element analysis (Pook 1993). It is sometimes called shear crack propagation, on the grounds that it takes place on planes of maximum shear stress in an uncracked sheet, but this is a misnomer. In the calculation of stress intensity factors it is usual to treat slant crack propagation, and crack propagation in the transition region, as if they were Mode I crack propagation, and to use a two dimensional stress intensity factor solution. This is difficult to justify by the small scale argument (see Section A.3.2), but it does not cause difficulties in practice (see Section A.5.2).

A.4 Corner Point Singularities

The analyses on which the concept of stress intensity factor is based are essentially 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 shows the direction of fatigue crack propagation (Pook 1983a). Reproduced under the terms of the Click-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 corner point, where a crack front intersects a free surface. The crack front shown by the dashed line in Figure A.15 intersects the surface at two corner points. As is well known, the nature of the crack tip singularity changes in the vicinity of 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 the corner point. The angle φ is measured from the crack front. The stress intensity measure, Kλ, is used to characterise corner point singularities, where λ is an exponent deﬁning the corner point singularity. λ 1−λ Stresses are proportional to Kλ/r and displacements to Kλr ,wherer is measured from the corner point. For a crack surface intersection angle, γ of 90◦, deﬁned as in Figure A.20, there are two modes of stress intensity measure. These are the symmetric mode, KλS, where crack tip surface displacements are Mode I (Figure A.2), and the antisymmetric mode, KλA,whichisa combination of Modes II and III displacements. In other words, the presence of one of these modes of crack tip surface displacement always induces the other. For the special case of λ = 0.5, stress intensity factors are recovered. KλS becomes KI,andKλA a combination of Modes II and III stress intensity factors, KII and KIII. For the symmetric mode, and Poisson’s ratio, ν = 0.3, 188 Appendix A: Fracture Mechanics

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 plane stress, by Equation (A.3), and the crack opens up into a parabola (Figure A.22 centre). The radius at the tip of the loaded crack, r,isgivenby 4K2 r = I , (A.21) πE2 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 calculated in 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 to zero as a corner point is approached. Conversely, when λ>0.5 r =∞and stress 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 stress intensity measure, although it must be associated with some characteristic dimension, such as sheet thickness. As with stress intensity factors, an apparent objection to the use of the stress 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 = 0forλ<0.5. For λ = 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 (see Section A.3.2). However, as the assumptions are violated only in a small core region, the general character of the corner point singularity dominated region in the vicinity of the crack tip is unaffected, as is shown for stress intensity factors in Figure A.13. Similarly, small scale nonlinear effects may be regarded as within the core region inside a corner point singularity dominated region. In turn the corner point singularity dominates only within a limited region, so in some circumstances a corner point singularity dominated region may lie within a K-dominated region, as shown schematically for a surface plane in Figure A.23 (Pook 2002a). For practical engineering purposes the use of stress intensity measures is usually unnecessary. They do not appear in standards which make use of stress intensity factors, for example Anon. (2005f). However, they are two situations 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 190 Appendix A: Fracture Mechanics

Figure A.23. K-dominated, corner point singularity dominated and core regions at a surface plane. 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 factors then have ﬁnite values in the corner point region. For β<βc, λ<0.5, and for β>βc, λ>0.5. For the symmetric mode β is given approximately by (Pook 1994a) c (ν − 2) β = −1 , c tan 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 value may be observed for Mode I cracks, and in consequence crack fronts in plates of constant thickness are often curved, for example Figure A.14. For the antisymmetric mode β is given approximately by c (1 − ν) β = −1 . c tan 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 displacements. For any given value of γ there are two possible values of βc.For γ = 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 Modes I, 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 in three dimensional bodies, explicitly make the assumption that stress intensity factors provide a description of the crack tip stress field. This is true only for a continuous crack front, for example an internal crack, and for a crack front which intersects the surface at the critical crack front intersection angle, βc. In consequence, such schemes cannot adequately reproduce the behaviour of stress intensity factors in the vicinity of a corner point. In practice values obtained are finite, and usually of the same order as elsewhere along the crack front. At a corner point, values are, in effect, extrapolations which depend on details of the numerical scheme used (Pook 1994a, 2000c). At a corner point Mode II and Mode III displacements cannot exist in isolation. The presence of one of these modes always induces the other (see Sec- tion A.4). Numerical calculations usually show induced values of the Mode II stress intensity factor, KII, or the Mode III stress intensity factor, KIII,as would be expected, in the vicinity of a corner point. Two dimensional numerical schemes are widely used in the determination of stress intensity factors for quasi two dimensional specimens of constant thickness. Only Mode I and Mode II stress intensity factors are possible in two dimensions, so such schemes cannot reveal induced KIII values at corner points. Many three dimensional finite element calculations are carried out for a Poisson’s ratio of about 0.3, and crack front and crack surface intersection ◦ angles of 90 (Figures A.20 and A.21). Hence theoretically KI should tend to zero, 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 limiting value, which is a function of the crack front inclination angle, β,andPois-√ son’s ratio, ν.Forβ = 90◦ and ν = 0.3itis0.5andforν = 0.5itis 0.5. From an engineering viewpoint the finite stress intensity factor values actually obtained at corner points do not matter provided both that results are reasonably consistent, and also that difficulties do not arise in practical applications. In practice, the situation for Mode I is indeed satisfactory, and numerous standards make use of Mode I stress intensity factors without any mention of corner point singularities, for example Anon. (2005f). The situation appears to be reasonably satisfactory for Mode II, but it is not satisfactory for Mode III where there are inconsistencies in reported KIII values. 192 Appendix A: Fracture Mechanics

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

A.4.2.1 Stress Intensity Factors for Square Models

Figures A.24–A.28 show some typical results for through thickness distributions 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 mm square model, with a crack extending from the middle of one side of the square to its centre. Loadings and boundary conditions were chosen to give large K-dominated regions, and Poisson’s ratio was taken as 0.3. For Mode I and Mode II loadings, the results are normalised by stress intensity factors obtained from two dimensional calculations, whereas for the Mode III loading they are normalised by centre line values. The results for the Mode I loading of a 4 mm thick plate and a 40 mm long bar (Figure A.24) show the well known increase in the Mode I stress intensity factor, KI, at the centre line compared with the corresponding two dimensional 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 conﬁrmed by estimating values of λ from crack surface displacements at the model surface. These estimates gave λ = 0.452 for both the plate and the bar, which agrees 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 II loading (Pook 2000c). ated that the size of the corner point singularity dominated region (boundary layer) 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 in Figure A.25. At the centre line, KII for the 40 mm long bar is lower than for the 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. They also 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 nearly constant 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. These are zero at the centre line (by symmetry) for both the plate and the bar, and increase towards the surface, except that for the plate there is a sharp decrease at the surface. The Mode III crack surface displacements at the surface did not deﬁne corner point singularity dominated regions, and it was not possible to 194 Appendix A: Fracture Mechanics

Figure A.26. Through thickness variation of KIII for 20 mm square models under Mode II loading (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 the ratio KIII/KII. The through thickness distribution of KIII for Mode III loading of the 40 mm long bar is shown in Figure A.27. KIII is constant over the central part 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 surface displacements at the surface indicated a corner point singularity dominated region size of about 0.2 mm, but it was not possible to derive a value for λ. Induced values of KII are zero at the centre line, decrease slightly, and then increase towards the surface (Figure A.28). The Mode II crack surface displacements at the surface indicate a corner point singularity region size of about 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 obtained from finite element analyses. Scatter increases in the vicinity of a corner point, and what must be regarded as nominal values of KII and KIII are strongly dependent on details of numerical calculation methods. The inconsistencies 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 because shear stresses perpendicular to a free surface must be zero. This effect shows 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 (Pook 1986). In general, cracks three dimensional cracks observed in service cannot be approximated by two dimensional stress intensity factor solutions. Naturally occurring cracks, and crack like ﬂaws, are often irregular in shape, for example the casting defects in Figures 3.17 and 7.2. The three 196 Appendix A: Fracture Mechanics

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,andthrough the thickness cracks. Methods of estimating stress intensity factors for irregular cracks have been of interest for many years (Paris and Sih 1965, Anon. 1980b, Chang 1982, Pook 1982a, Murakami 2002). Normally, stress intensity factors vary along a crack front, and it is the largest value of the Mode I stress intensity factor, KI, that is usually of interest. Adjacent cracks, as in Figure 7.1, can increase stress intensity factors through interaction effects. These are difﬁcult to assess when cracks are not coplanar. In the analysis of laboratory and service failures, the size and shape of cracks can usually be ascertained in detail by the examination of fracture surfaces. However, the results of non destructive testing will usually supply only a limited amount of information. For example, only the surface length and maximum depth of a surface crack might be available (see Appendix C). Initially, an intuitive approach was used by various authors to estimate stress intensity factors for irregular cracks. This approach was based on general knowledge of stress intensity factors for similarly shaped regular cracks (Paris and Sih 1965). Fifteen years later standardised estimation procedures for various situations started to appear (Anon. 1980b), and are now in wide- spread use (Anon. 2005f). Standardised procedures should be used whenever possible. Some of the ideas used in estimating stress intensity factors for irregular cracks are described below. The ideas sometimes have a sound theoretical basis, but the main justiﬁcation for their use is that they have been found 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 modelled by a semi ellipse of the same surface length and depth. The Mode I stress 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 a containment rectangle. are close to this shape, as shown schematically in Figure A.29 left, and in Figure A.30. The approach does not work well for some more irregular surface cracks such as shown in Figure A.31. What is sometimes done is to ﬁrst construct a containment rectangle around the crack, and then inscribe a semi ellipse in the rectangle (Figure A.32). This results in a longer semi ellipse with a concomitantly higher value of KI at the deepest point. A similar idea is sometimes used for internal cracks in bodies of rectangular cross section, as shown in Figure A.33. The containment rectangle sides are parallel to the body surfaces. A containment rectangle may be used as it stands for a through the 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 they are in Mode I. One approach is to project them onto a plane so that they become equivalent Mode I cracks. The plane chosen is a plane of maximum principal tensile stress in the uncracked body. After projection, a crack can then be modelled as in the previous section. As an example, the method works quite well for the specimen shown in Figure A.20 when this is loaded in three 198 Appendix A: Fracture Mechanics

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 branch crack (Pook 1989a). Reproduced under the terms of the Click-Use Licence. point bending (Pook and Crawford 1990). Another example is treating slant crack 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 projected onto the plane indicated by dashed lines in the ﬁgure. One theoretical justiﬁcation is that under mixed mode loading a small Mode 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 appropriate 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 particular, the elastic stress fields associated with a crack front and with a dislocation are both singularities. The associated energy means that a dislocation has a line tension, which controls its shape under an applied stress field (Cottrell 1964). Similarly, a crack front may be regarded as having a line tension which controls its shape, but with the important difference that the motion of a crack front 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 fatigue crack front is smooth and any initial sharp corners rapidly disappear as the crack propagates. Overall, an initially irregular Mode I crack rapidly becomes convex, as shown by the dashed line in Figure A.35: at a re-entrant region (x on the figure) the Mode I stress intensity factor, KI, is much higher than elsewhere on the crack front, leading to rapid fatigue crack propagation towards a convex shape. Under fatigue loadings, stress intensity factors for initially irregular cracks may be approximated by first enclosing them by a convex outline, as in the figure, and then using the methods in the previous section.

A.5.4 USE OF CRACK AREA

For irregular cracks that are small compared with other dimensions it is possible to use the crack area, A, as a characteristic crack dimension (Chang 1982, 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 very slender cracks the crack length is truncated to 10 times the width before calculating A. 200 Appendix A: Fracture Mechanics

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 stress intensity factor along the crack front, K , is given approximately by I √ ∼ KI = 0.5σ π A, (A.24) where σ is the stress perpendicular to the crack. The equation applies to cracks 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 10 times the crack depth before calculating A, and for a deep surface crack the crack 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 equivalent two dimensional solution.

A.5.5 INTERACTION BETWEEN CRACKS When two cracks are close to each other the interaction between them increases their stress intensity factors compared with those for isolated cracks. Unfortunately, this interaction effect cannot be expressed by a simple equation, 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 degrees of conservatism. The usual approach is first to define the crack separation between two cracks below which interaction occurs, and then in some way to define an equivalent single crack for which stress intensity factors are calculated. For example, the following rules are suggested for two adjacent semi circular surface cracks of different sizes (Murakami 2002). If there is enough space between the cracks to insert an additional crack of the same size as the smaller crack (Figure A.36(a)) then the interaction effect is negligibly small, and A in Equation (A.25) is taken as A1. If the space between the two cracks is too small to insert a crack of the same size as the smaller crack then under fatigue loading the cracks coalesce rapidly. An imaginary semi circular crack is inserted 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 used only in this appendix. a,b constants in two parameter Weibull distribution f frequency G(f ) power spectral density H wave height H1/3 signiﬁcant wave height I irregularity factor m0, m2, m4 moments of spectral density function N number of cycles, return period P(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 function S random process s instantaneous value of S S/σ¯ expected value of S/σ Sc/σ clipping ratio Sm mean value of S So value of S below which peaks are omitted T total time, wave period t time γ Euler’s constant = 0.5772 ... 204 Appendix B: Random Load Theory and RMS

ε spectral bandwidth σ standard deviation (random process theory), root mean square (fatigue) σp root mean square of peaks σp,c root mean square of peaks after clipping of high peaks σp,o root mean square of peaks after omission of low peaks σp,t root mean square of peaks after truncation of high peaks σr root mean square of ranges σ 2 variance τ time interval φ 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 engineering viewpoint, it might appear that some of the points made are unimportant and pedantic. However, lack of attention to detail can result in difﬁculty in interpreting fatigue test data. In reporting random loading fatigue data it is important that the precise conventions used in calculations be clearly stated. No one set of conventions is of universal applicability. Some equations and ﬁgures which appear in the main text are repeated in order to make the appendix 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),whereS is a random process and t 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 means that statistical parameters characterising the process are independent of time. Ergodic means, broadly, that different samples of the same process yield the same values for statistical parameters. Only stationary random processes can be ergodic, and in practice most are. Considering the time interval 0 to T the mean value of S, Sm is given by 1 T Sm = lim(T →∞) S(t)dt (B.1) T 0 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 of the Click-Use Licence. and the mean square value φ2 by 1 T φ2 = lim(T →∞) S2(t) dt. (B.2) T 0 206 Appendix B: Random Load Theory and RMS

Hence the root mean square (RMS) value, φ,isgivenby 1 T φ = lim(T →∞) S2(t) dt. (B.3) T 0 The positive square root is understood in Equation (B.3) and subsequent equations. The RMS can equally well be calculated for periodic processes such as a sine wave. The use of RMS ﬁrst became popular in electrical engineering because it can be used directly in calculations involving power. For convenience it is sometimes used in metal fatigue (Pook 1987a). Carrying out calculations from the mean rather than from zero gives the variance, σ 2,where T 2 1 2 σ = lim(T →∞) {S(t) − Sm} dt (B.4) T 0 and the standard deviation, σ ,isgivenby 1 T 2 σ = lim(T →∞) {S(t) − Sm} dt. (B.5) T 0 The quantities given by Equations (B.1)–(B.5) are related through the expression φ2 = σ 2 + S2 . m (B.6) Hence, for zero mean the RMS and standard deviation are numerically equal. Instantaneous values of S(t) may be characterized by probability distribution functions. The exceedance, P(S), is the probability that a value exceeds S.Thecumulative probability,1− P(S), is the proportion of values up to S.Theprobability density, p(S), is the derivative of the cumulative probability. For convenience, S is often normalised by σ . The instantaneous values of many ‘naturally occurring’ random processes are statistically stationary, at least in the short term, and approximate to the Gaussian distribution (or Nor- mal distribution), which theoretically extends from −∞ to +∞. The probability density of a Gaussian distribution (Figure B.2(a)) for a process with zero mean is given by S 1 −S2 p = √ exp (B.7) σ 2π 2σ 2 and the exceedance (Figure B.2(b)) by S 2 ∞ −S S P = √ exp d . (B.8) σ 2π S/σ 2σ 2 σ 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 therefore twice those given by Equation (B.8). P(S/s)is the area under the curve of p(S/σ) between (S/σ ) and inﬁnity, as indicated by the shaded area in Figure B.2(a).

B.2.2 FATIGUE LOADING

Conventions used in the metal fatigue literature sometimes differ from those used in random process theory. The random processes encountered in metal fatigue are usually symmetrical, in a statistical sense, about the mean value, Sm, and all calculations are then carried out using values of S measured from Sm. 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- 208 Appendix B: Random Load Theory and RMS ation is not normally used in the characterisation of random processes. (Al- ternatively it might be said that RMS is used where standard deviation is meant.) This convention is used in what follows unless the context indicates otherwise. In particular, references to zero are understood to include the mean value of a process with non zero mean. Retaining the symbol σ for what was called standard deviation and is now called RMS Equation (B.5) becomes 1 T σ = lim(T →∞) S2(t) dt. (B.9) T 0 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 random input is applied to a sharply tuned resonant system (Papoulis 1965, Pook 1983b, 1984, Bendat and Piersol 2000). Individual sinusoidal cycles appear whose frequency corresponds to the centre frequency of the resonant system. They have a slowly varying random amplitude. The probability density function for the occurrence of a positive peak of amplitude S (Figure B.4(a)) is givenbytheRayleigh distribution S S −S2 p = exp . (B.10) σ σ 2σ 2 As the process is statistically symmetrical, corresponding negative peaks also appear. The exceedance (Figure B.4(b)) is given by S −S2 P = exp . (B.11) σ σ 2 Equations (B.10) and (B.11) become exact only as the bandwidth tends to zero (see Section 4.2.2). Used in its general sense Rayleigh distribution does not imply the existence of a corresponding narrow band random process, and parameters in Equations (B.10) and (B.11) may differ. A narrow band random process is Gaussian, so instantaneous values do follow the Gaussian distribution (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 Figures A2.4, 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. Theoretically the Rayleigh distribution extends to inﬁnity, but in practice peaks do not exceed a cut off value of S/σ, known as the clipping ratio. Clipping implies that 210 Appendix B: Random Load Theory and RMS higher peaks are reduced to the level given by the clipping ratio; truncation that they are omitted altogether. The clipping ratio does not usually exceed four or ﬁve. As fatigue damage depends on the peak values of cycles, and is largely independent of waveform (see Section 3.2), the root mean square (RMS) σ value of√ peaks, p, is sometimes used. For narrow band random loading: σ = σ p 2 . 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 engineering applications; some of its general properties are discussed by Lipson and Sheth (1975). For metal fatigue purposes it is convenient to write its exceedance in the form (Pook 1984) S −b S a P = , σ exp a σ (B.12) where a and b are adjustable constants (parameters) used to fit the equation as needed. A functional relationship between b and a can be obtained by assuming that Equation (B.12) gives the distribution of peaks of a sinusoidal process which is symmetrical about zero. There is no closed form relationship between b and a.Valuesofb 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 = 1anda = 1, b = 1 gives as special cases the Rayleigh distribution (Equation (B.11)) and the Laplace distribution (or Exponential distribution) S −S S P = = p . σ exp σ σ (B.14) Exceedances for a range of a values are shown in Figure B.5. A logarithmic scale is used for exceedances in order to emphasise detail at low values. As a result the curve for the Rayleigh distribution has a different appearance from Figure B.4(b) where a linear scale is used. The peaks of the C/12/20 load history, shown in Figure 4.9, can be fitted approximately by the two parameter Weibull equation with a = 1.2715 (Pook 1987a). Differentiating Equation (B.12) gives the probability density of the two parameter 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. S b S a−1 −b S a p = . σ a σ exp a σ (B.15) If only the distribution of peaks is speciﬁed by a probability distribution, then in a computer generated process both the order in which peaks are applied and the waveform connecting them need to be speciﬁed. Usually, a negative peak is made arithmetically equal to the preceding positive peak. The value of 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 to give a sinusoidal process, σp = 2 σ and σr = 2 2 σ . In general, computer generated processes are do not follow the Gaussian distribution. However, for the special case of a sinusoidal process with a Rayleigh distribution of peaks, 212 Appendix B: Random Load Theory and RMS and with constant frequency, the process is Gaussian, irrespective of the order in which peaks are applied.

B.3.3 INCOMPLETE DISTRIBUTIONS AND PROCESSES

Distributions and processes encountered in metal fatigue practice are always incomplete in some way. In consequence root mean square (RMS) values differ from those of theoretical forms which extend to inﬁnity. Provided that a difference in RMS values is less than about 1 per cent it can usually be neglected. 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. This is because, in a physical system involving a narrow band random process, it is nonlinearities rather than clipping or truncation that limit the peaks which appear, and it may not be possible to maintain a clear distinction between truncation and clipping (see Section B.3.1). However, in a process generated by ﬁrst generating positive, and corresponding negative peaks, which follow some distribution, then joining the positive peaks and adjacent negative peaks with 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 to be 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,isgivenby ∞ S 2 S S σp = p d . (B.16) 0 σ σ σ

Hence, if the process is truncated at a clipping ratio, Sc/σ , the RMS of the truncated distribution of peaks, σp,t,isgivenby S 2 S S Sc/σ p d σ = σ σ σ , p,t S (B.17) − P c 0 1 σ where the term {1 − P(Sc/σ )} corrects for the reduction in the total number of 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,isgivenby Metal Fatigue 213

σp,c = (B.18) 2 Sc−σ 2 Sc Sc Sc S S S P + 1 − P p d , σ σ σ 0 σ σ σ where the ﬁrst term under the square root sign represents the peaks that that have been reduced to Sc/σ . Clipping has less effect on RMS than truncation. Equations (B.17) and (B.18) may be used to calculate the change in the RMS of peaks due to truncation and clipping of different versions of the two parameter Weibull distribution (Equation (B.12)). Results show that for the Rayleigh distribution (a = 2 in Equation (B.12)) the effects are negligible when 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 symmetrically. As an example of what can happen, consider the construction of a sinusoidal process, whose peaks follow the two parameter Weibull distribution, with a taken as 0.5. Assume that the complete process will be truncated to give a desired clipping ratio of 5. From Equation (B.17), taking S/σ = 5, σp,t = 0.7704σ , and the clipping ratio for the truncated process is 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. Reducing instantaneous 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 to be reshaped, as in form (c). Truncation or clipping of a process that follows the Gaussian distribution renders it non Gaussian. However, it can reasonably be 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 fatigue testing times, on the grounds that they cause negligible fatigue damage. The root mean square (RMS) value of the remaining peaks, σ , ,isgivenby p o 2 ∞ S p S d S σ = σ σ σ . p,o S (B.19) P o So/σ σ 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. 214 Appendix B: Random Load Theory and RMS

Figure B.6. Clipping a half cycle. (a) Original half cycle. (b) Clipped. (c) Clipped and reshaped (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 loading from which negative peaks have been removed, to give a one sided process (Sherratt and Edwards 1974). Three ways of doing this are shown, for a constant amplitude sinusoidal process, in Figure B.7. Part (a) of the ﬁgure shows the original process. In Figure B.7(b) the negative half cycles have been reduced to zero height, whereas in Figure B.7(c) they have been removed altogether. In Figure B.7(d) the negative peaks have been removed altogether and the positive peaks joined to zero by sine waves. Usually, root mean square (RMS) values are calculated for the original complete process. Parameters can be calculated for a one sided process, but different results are S sometimes obtained for√ the three methods. For example, mean values ( m) 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 methods σ of removal, and√ is equal to the RMS of peaks, p, for the original complete process, 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 process 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 contain a ﬁnite number of cycles, N.Forapseudo random process, N is the return period after which it repeats exactly. One consequence is that the maximum peak size, and hence the clipping ratio, are restricted (see Sections B.3.1 and B.3.3.1). An intuitive approach is to set the exceedance, P(S/σ), equal to 1/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 half cycles removed and positive half cycles reshaped (Pook 1987a). Reproduced under the terms of the Click-Use Licence. as the clipping ratio. However, in narrow band random loading, large cycles occur in groups and the expected maximum value of S/σ, S/σ¯ , which will 216 Appendix B: Random Load Theory and RMS

Table B.1. COLOS 7 level load history. Level number Number of cycles in level RMS of level 7 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¯ √ γ 1 ≈ ln N + , (B.20) σ 2 ln N 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 root mean square (RMS) values, and perhaps other parameters, are a slowly varying function of time. In the short term they can usually be regarded as statistically stationary. For a succession of narrow band random loadings whose RMSs follow the positive half of a Gaussian distribution (Equation (B.7)) the peaks sum to the Laplace distribution (Equation (B.14)) (Pook 1983b). Corresponding load histories for fatigue testing also need to be non stationary. A procedure was developed (Pook 1984) which made it possible to approximate a wide range of probability distributions as the sum of several Rayleigh distributions and hence produce load histories which consist of a sequence of narrow band random loadings. In one example a 7-level approximation of the Laplace distribution was used as the basis of an agreed standard load 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 the of the overall RMS, σ . The water surface elevations of ocean waves are an example of a process which often approximates to a non stationary narrow band random process (Pook and Dover 1989). In oceanography the primary parameter used in the characterisation of sea state is the wave height, H , which is measured peak to 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, s 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5 12.5 13.5 0.31440348000000 0.916415913540400000 1.5218103164782420000 2.13653126953371000 2.74019103724182000

3.350946713131000 3.960123633861000 4.470062031102100 5.180051315121000 5.790019463000

6.400002242002 7.010000137000 7.620001104020 8.230001200000 8.840000022000

9.450000000002 Parts per 1924. a sea state to be regarded as statistically stationary, the usual measure of its severity is the significant wave height, H1/3, which is the average height of the 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 data can be usefully presented is by means of a scatter diagram that gives the relative occurrences of sea states within specified small intervals of H1/3 and wave period, T , which is the reciprocal of the wave passing frequency (Pook 1987b). Table B.2 is an example of a scatter diagram obtained by the MV Famita (Holmes and Tickell 1975). The M V Famita is one of a number of ships that have been stationed in the North Sea to collect oceanographic data. Long term records show that variations in sea state have the appearance of a random process (Anon. 1985), but examination of data from four different sources 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 the Gumbel 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) 218 Appendix B: Random Load Theory and RMS

For S>4, P(S) ≈ exp(−S),andforS<2, P(S) ≈ 1. Taking S as the signiﬁcant wave height, H1/3, the sea state data were well ﬁtted by the expression 1.9 − H1/3 P(H / ) = 1 − exp − exp , (B.22) 1 3 1.06 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 metal fatigue 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 common one in metal fatigue is the irregularity factor, I, which is the ratio of mean crossings to peaks (see Section 4.3.3). It lies in the range 0 to 1. The irregularity factor has the advantages that is easily understood, and is not restricted to processes whose instantaneous values follow the Gaussian distribution.

B.4.1 SPECTRAL DENSITY FUNCTION

In random process theory a process which is statistically stationary and ergodic, and whose instantaneous values follow the Gaussian distribution, is usually 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 process S(τ) at times t and t + τ, and is given by (Papoulis 1965, Bendat and Piersol 2000) 1 T R(τ) = lim(T →∞) S(t)S(t + τ)dt. (B.23) T 0 The SDF, G(f ),wheref is frequency, is the Fourier transform of R(τ),and is given by ∞ G(f ) = 2 R(τ)exp(−iπfτ)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 domain to the frequency domain. The SDF is sometimes plotted on a logarithmic scale and sometimes on a linear scale. Figure B.1(b) shows the SDF for the broad band random loading shown in Figure B.1(a). In practice it is calculated using an algorithm known as the Fast Fourier Transform (FFT) (Bendat and Piersol 2000). Physically, the SDF gives the frequency content of the random process, and in the narrow band random process case is sharply peaked at the centre frequency (resonant frequency). An alternative method of determining the SDF is to pass the process of interest through a bandpass filter of very narrow bandwidth 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, as given by Equation (B.2). In electrical engineering the SDF is usually known as the power spectral density (PSD) because it provides a measure of the electrical power, which may be ascribed to the various frequency components. Some useful results depend only on the spectral bandwidth, ε,whichisa measure 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 m2 ε = 1 − 2 , (B.25) m0m4 where m0, m2 and m4 are the zeroth, second and fourth moments of the SDF about 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 peak distribution 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 random process are illustrated by the example shown in Figure B.3. Inevitably, only a finite length process can be examined, so decisions are needed on how to deal with the beginning and end of the process. Also, the mean value of the process has to be determined. The horizontal line in Figure B.3 is intended to be the mean value. In the figure there are 44 upward going zero crossings of this line and 45 positive peaks, giving an irregularity factor of 44/45 ≈ 0.98. 220 Appendix B: Random Load Theory and RMS

Figure B.8. Spectral density functions for a 0.76 m diameter horizontal member immersed 10.8 m, signiﬁcant wave height 4.75 m. (a) Water surface elevation; (b) bending stress; (c) axial stress (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 should not be counted, giving an irregularity factor of 43/45 ≈ 0.96. However, if the horizontal line were slightly lower there would be 45 upward going crossings, and the irregularity factor would be 45/45 = 1. What should be taken as the correct value is not easily resolved. In principle, the irregularity factor could be determined by first finding the spectral bandwidth and then using Equation (B.26) but there are corresponding difficulties in determining the spectral bandwidth of a narrow band random process. As an example of the sort of information that can be derived from spectral density functions, Figure B.8 shows data for a tubular welded tall platform in the North Sea (Pook 1989b). A 0.76 m diameter horizontal member, immersed 10.8 m, was strain gauged so that bending and axial stresses could be derived. In practice, although sea states have a dominant wave passing frequency, they are not particularly narrow band so energy may be available to excite structural resonances (Pook 1987b). The SDF for the water surface elevation (Figure B.8(a)) shows this. There is a clearly defined peak corresponding to the dominant wave passing frequency, but there is significant energy at other frequencies. To avoid structural resonances, offshore platforms are designed so that resonant frequencies are substantially greater than the dominant wave passing frequency. This has been successful for the axial stresses since, as might be expected, the SDF (Figure B.8(b)) is of similar form, with no structural resonances exited. However, the SDF for the bending stress (Figure B.8(c)) does show two peaks corresponding to structural resonances. The point of collecting data of this sort is to permit comparison of actual structural behaviour with theoretical calculations. C Non Destructive Testing

C.1 Introduction

Non destructive testing (NDT) is not a clearly defined concept (Halmshaw 1991). NDT has a wide range of applications in the detection and evaluation of flaws in materials. Many different methods are used, and a wide range of commercially available instruments has been developed. These are often automated under computer control. The key feature of NDT is that it has no deleterious effect on the item tested. In the context of metal fatigue the usual meaning of non destructive testing is the detection and sizing of cracks and crack like flaws in components, structures and laboratory specimens. This includes monitoring of fatigue crack propagation in service and in laboratory specimens; the advantages and disadvantages of various methods are summarised by Richards (1980). The accuracy of crack sizing that can be achieved varies widely. Some of the non destructive testing techniques used in metal fatigue work are described briefly in this appendix, together with the important statistical concepts of probability of detection and probability of sizing. In order to make the appendix more self contained some figures in the main text are repeated here.

C.2 Visual Inspection

Visual inspection is the simplest method of detecting surface cracks, usually called surface breaking cracks in the non destructive testing literature. This term is used in this appendix. The utility and importance of visual inspection are often underestimated. Under good conditions fatigue cracks with a surface length 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 portable lighting are useful. For a permanent record photographs may be taken or replicas of the surface made. If the surface is irregular, as in welds, surface breaking cracks are difficult to detect by visual methods. Regulatory authorities often call for periodic visual inspection of structures for defects, including cracks. For example, Figure C.1 shows unexpec- ted fatigue cracking found in an aircraft engine nacelle during a routine inspection (Pook 2004). Another example is the cracking in a burner from a domestic central heating boiler shown in Figure 8.13. Routine visual inspection is tedious, and fatigue cracks are sometimes missed. For example, one of the concerns at the official inquiry into the catastrophic fatigue failure of a 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 length measurement is often aided by markings etched or scribed onto the specimen 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 crack propagation rate tests, scribed marks were used to collect the data shown in Figure C.3. The use of guide markings does not meet resolution accuracy requirements in modern fatigue crack propagation rate testing standards such as 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 equipment needed is relatively inexpensive, but they are labour intensive when used to monitor 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 is 0.1 inch (2.54 mm). National Engineering Laboratory photograph. Reproduced under the terms 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 a plate of constant thickness a fatigue crack front of a through the thickness crack is often curved (Figure C.4). This curvature can affect the calculation of stress intensity factors (see Section A.3.2). With a surface breaking crack 226 Appendix C: Non Destructive Testing

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 breaking crack.

(Figure C.5) it would not be possible to calculate stress intensity factors because 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 breaking cracks. Any cracks found are then sized using an appropriate technique such as ultrasonics (see Section C.6) or alternating current potential drop (see Section C.7.2).

C.3 Magnetic Particle Inspection

Magnetic particle inspection (MPI) is a well established technique for the detection of surface breaking cracks. The method can be used only on ferromagnetic 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 electromagnetic fields. These can be represented as lines of magnetic force through space which form a magnetic flux. The principal of magnetic particle inspection is shown in Figure C.6. A magnetic flux is established in the material by placing the poles of a magnet (usually an electromagnet) in contact with the material. If the magnetic flux encounters a transverse surface breaking crack the flux becomes distorted. Some of the magnetic flux passes through the crack, some passes around the crack tip, and some leakage flux passes around the crack at the surface. This leakage flux attracts ferromagnetic particles to the crack mouth,andthe resulting visible concentration of particles marks the crack. The magnetic particles are applied as a suspension in a carrier liquid, such as light oil or water, at a concentration by volume of about 2 per cent. If water is used, a wetting agent and a corrosion inhibitor are incorporated. The suspension is normally supplied in an aerosol, and is sometimes called a magnetic ink. The particles are usually black iron oxide of around 1–25 µmin size, and may be dyed to improve visibility. Florescent dyes are sometimes used, and the particles are then viewed under ultra violet light Magnetic particle inspection is the most widely used non destructing testing method for detecting surface breaking cracks in welded joints. MPI is easily carried out using portable equipment, but expertise is needed for satisfactory results, and it can be a messy procedure. The major disadvantages are that only the surface length of a crack can be determined, and the accuracy of crack sizing is low. An advantage is that, with special equipment, magnetic particle inspection can be used under water. 228 Appendix C: Non Destructive Testing

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

C.4 Dye Penetrant

The dye penetrant method is used to detect surface breaking cracks. The method can be applied to any material that has a non absorbent surface. Most of the cracks found by dye penetrants can be seen visually in good conditions, but dye penetrants make them much easier to detect. The principle of the method is shown in Figure C.7). After the surface has been cleaned a penetrant, which contains a dye in solution, is applied. The penetrant is chosen so 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 thin layer of a porous developer is applied. Penetrant is drawn out of cracks by the developer, 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 range of techniques is available, and for good results the technique chosen must be carefully matched to the intended application (Halmshaw 1991). Unfortu- nately, much published information on the results of dye penetrant non destructive testing is of little value because full details of techniques used are not included. The dye penetrant method is widely used for aluminium alloys and other metallic materials which cannot be magnetised so that magnetic particle inspection is impossible (see previous section). The main advantage of dye penetrant is that it is simple to use, and particularly suitable for ﬁeld work. The main disadvantage, as with visual inspection and magnetic particle, is that only the surface length of a crack can be determined (see Sections C.2 and C.3). If fatigue crack propagation is being monitored, a potential disadvantage is that dye penetrant remaining in a fatigue crack could affect its subsequent propagation behaviour. Metal Fatigue 229

Figure C.8. Fatigue cracking from shrinkage cavity in 30 × 35 mm cast steel bar. National Engineering 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 voluminous defects including internal cavities, porosity and inclusions. An example is the shrinkage cavity in a cast steel bar shown in Figure C.8. This has an irregular boundary and can be regarded as a crack like flaw. The general shape of the cavity was obtained by radiography, and it is shown schematically in Figure 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 towards a sheet of photographic film, which acts as a detector. The X-rays are partly absorbed as they pass through the specimen, and then strike the photographic film to produce a radiograph. There is less absorption when the X-rays pass through a cavity, and a two dimensional image of the three dimensional cavity is formed on the film. To get three dimensional information on the cavity, 230 Appendix C: Non Destructive Testing

Figure C.10. Principle of radiography in its original form. including its location within the specimen, radiographs have to be taken from more 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 with very much shorter wavelengths. They are produced when a beam of high energy electrons strikes a metal anode in a vacuum. Wavelengths of X-rays used in radiography range from about 10−4 nm to about 10 nm. Long wavelength X-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 can penetrate up to about 50 cm thick steel. Gamma rays are sometimes used in radiography. They are also a form of electromagnetic radiation and are produced by the decay of a radioactive isotope such as cobalt-60. Radiography is a very versatile and easily used technique. It is probably the oldest non destructive testing technique used for the quality control of welded joints. Radiographs provide a convenient, permanent record. The major disadvantage of radiography is that stringent safety precautions have to be taken to protect operators, and the public at large, from radiation. From a metal fatigue viewpoint its major disadvantage is that it is difﬁcult to detect and size tight cracks, that is cracks whose opposite surfaces are either close together or touching. Metal Fatigue 231 C.6 Ultrasonics

In metal fatigue the main use of ultrasonics is crack sizing, and a wide range of techniques is available. The method is based on the propagation of sound waves through the material at frequencies above the audible range, hence the term ultrasonics. Sound waves are mechanical vibrations. Hence the velocity of propagation is different in different materials, and also depends on the type of wave. Frequencies used are of the order of a MHz, and the resulting wavelengths are of the order of a mm. The essential feature of the waves used in ultrasonics is that they propagate through the material in the same way that ocean waves move across the water surface. This contrasts with the standing waves observed in the vibrations of a tuning fork. Inadvertent standing waves can be a problem in the use of ultrasonics. Two main types of wave are used in ultrasonics. One is compressional waves, also known as longitudinal waves, where particles vibrate in the direction of wave propagation. The other is shear waves, also known as transverse 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 in pulses, not continuously. Ultrasonic wave pulses are generated by applying short electric pulses to a suitable probe. One type of probe uses a piezoelectric disc, which resonates at 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 into the specimen under test. Ultrasonic waves emerging from the specimen are received by a suitably positioned probe, and processed to give information about defects within the specimen. The same probe is sometimes used for both transmission and reception. The arrangement for ultrasonic testing of a cracked specimen, using a compressional probe, is shown schematically in Figure C.11. Ultrasonic wave pulses are reﬂected from the crack, and also from the top and bottom surfaces of the specimen. These echoes are displayed on an oscilloscope using what is known as an A-scan, shown schematically in Figure C.12. The time base of the oscilloscope is triggered as each ultrasonic pulse is transmitted. Hence, the positions of the echoes on the time axis provide information on the crack location. In practice, dispersion effects within the specimen mean that subsidiary echoes also appear. These subsidiary echoes can complicate interpretation of the display. Other methods of display may be used when a probe is scanned across a specimen surface. A B-scan is obtained from a scan along a line on the 232 Appendix C: Non Destructive Testing

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

Figure C.12. A-scan showing crack. surface. The display is arranged so that it shows the sizes and positions of flaws on a cross section perpendicular to the surface of the specimen. A C- scan produces a radiograph like display (see previous section). It is obtained from a series of scans along parallel lines, and the display shows a plan view of the specimen in which defects appear in their correct positions, but with no information on their through the thickness locations. A D-scan is similar to a B-scan, but is obtained from a series of scans along parallel lines. The terms A-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 schematically in Figures C.13 and C.14. In the end on technique a high intensity compressional 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 determined directly from an A-scan. A different approach is used in the time of flight diffraction (TOFD) technique, shown schematically in Figure A.14. In this technique there are separate transmitter and receiver compressional probes. The principle is that when a compressional ultrasonic beam meets a crack tip some of the energy is diffracted. The diffracted waves spread over a large angular range, and may be detected by a suitably placed receiver probe. If the transmitter and receive probes are symmetrically placed about the crack tip, then a simple calculation gives the position of the crack tip, and hence the crack depth. To ensure symmetry about the crack tip the two probes may be linked mechanically, and traversed across the crack. The probes are symmetrically positioned when the time of flight is minimised. Shear waves are sometimes used in the TOFD technique; these are more suitable for deeper cracks. Both the end on technique and the TOFD technique are compatible with methods of approximation of stress intensity factors since, in effect, an oblique 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 the echo on an A-scan is inversely proportional to the square of the distance of the flaw from the probe. A distance amplitude correction curve, usually known as a DAC curve, is used to correct for this effect. The term DAC level refers to the heights of echoes, relative to background noise, that are regarded as significant. (Background noise is known as grass, because of its appearance on an oscilloscope screen.) Thus, 50 per cent DAC (level) means that only 234 Appendix C: Non Destructive Testing

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 grass are considered significant. The choice of DAC level is important when the probability of detection and probability of sizing are being determined (Visser 2002) (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 readily adaptable to specific applications. The major disadvantages of ultrasonics are cost and that it is not suitable for small, thin specimens. It is also difficult to use on austenitic steels.

C.7 Electromagnetic Fields

In metal fatigue electromagnetic field methods are used both for crack detection and for crack sizing. They are based on the injection of a uniform alternating current field into the surface of a specimen, such as the plate shown in 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) 1 δ = √ , (C.1) πµσf Metal Fatigue 235 where µ is magnetic permeability, σ is electrical conductivity, and f is the frequency of the alternating current. Some typical skin depths shown in Table 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 only be used as a guide to the selection of an appropriate frequency.

Table C.1. Typical skin depths. Material Frequency 1 kHz 10 kHz 100 kHz 1 MHz 18/8 stainless steel 13.1 mm 4.14 mm 1.31 mm 0.414 mm Brass 3.90 mm 1.23 mm 0.390 mm 0.123 mm Aluminium 2.65 mm 0.838 mm 0.265 mm 0.084 mm Copper 2.00 mm 0.632 mm 0.200 mm 0.063 mm Mild steel 0.148 mm 0.047 mm 0.015 mm 0.005 mm

For satisfactory results the skin depth must be small compared with the crack depth. Typically δ is about 0.1 mm so, in general, electromagnetic ﬁeld methods cannot be used for cracks less than about 1 mm deep. The response of a crack being interrogated by an alternating current depends on the value of the dimensionless parameter m, which is given by (Lewis et al. 1988) µ a m = 0 , µδ (C.2)

−7 −1 where µ0 is the magnetic permeability of a vacuum (= 4π ×10 Hm ). For non magnetic materials of high electrical conductivity, such as aluminium, µ ≈ µ0, m is large because δ/a is small, and eddy current testing is appropriate. For magnetic materials, such as ferritic steels, µ µ0, m is small, and ACPD testing is appropriate. Typical values of m are 12 for aluminium and 0.6 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 and sizing 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 skin depth (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 specimen at a small ﬁxed lift off distance. The alternating current in the coil produces 236 Appendix C: Non Destructive Testing

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 induced eddy currents in turn produce an alternating magnetic flux. This opposes the flux produced by the current carrying coil, and changes the impedance of the coil. As the probe passes over a crack the eddy currents are distorted and the presence of a crack is detected, using suitable instrumentation, by changes in the coil impedance. In some systems the effect of the magnetic flux produced by the eddy currents is monitored by voltages induced in a second coil, similar to the current carrying coil. An important practical advantage of eddy current testing is that physical contact between the probe and the specimen is not necessary. Eddy current testing is a very versatile method of crack detection and sizing, and a wide range of techniques is available. It is possible to detect cracks as small as 10 µm deep. However, for good results the technique and instrumentation chosen must be carefully matched to the intended application. Its major disadvantages are cost and, at times, interpretational difficulties. It is possible to monitor fatigue crack propagation by using a system in which the probe is traversed by a motor, and locks onto the crack tip. When used on magnetic materials, eddy current testing is sometimes called alternating current field measurement (Lewis et al. 1988). Lower frequencies 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 order to ensure a small skin depth (Table C.1). The principle of ACPD testing is shown in Figure C.16. The current is impressed through two contacts some distance apart. It flows along the specimen skin from one contact to the other, passing down one side of the crack and up the other. The voltage (potential drop) is measured using a probe with known contact spacing, , placed across the crack. In automatic systems for monitoring fatigue crack propagation, current impression and potential drop measurement measuring wires are spot welded to the specimen (Austin 1999). Calibration is straightforward. A voltage, V1, is first measured by placing the 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 the current is constant for the two measurements, then the crack depth, a,isgiven by V a = 2 − 1 . (C.3) V1 2 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 configurations. Because the technique is self calibrating, the impressed current field does not have to be completely uniform. This makes the technique suitable for irregularly shaped specimens such as welded joints. The technique is particularly suitable for automatic collection of fatigue crack propagation data during structural fatigue tests. For fatigue crack propagation rate testing on 238 Appendix C: Non Destructive Testing

Figure C.17. Alternating current potential drop calibration. plate specimens, in which a crack front may be curved (Figure C.4), contact points can be arranged so as to measure an average crack length through the thickness. The main advantages of the ACPD method are its versatility and its ease of calibration. Voltages are low, so no safety precautions are needed, and the method can be used under water. Care has to be taken in arranging the impressed current and probe leads so as to avoid spurious interactions, and also interference in electrically noisy environments. Crack bridging by metallic particles is not usually a serious problem. Various systems are available commercially. Techniques are still evolving. It is moderately expensive. A disadvantage is that for an oblique crack the method gives the crack length, not the depth of the crack tip below the surface (Figure C.17 right). The ACPD method is therefore not compatible with methods of approximation 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, as is the concomitant approximated stress intensity factor.

C.8 Probability of Detection

Traditionally, it has been assumed that a particular non destructive testing technique is capable of detecting all cracks larger than a critical size, and that no cracks smaller than the critical size will be detected. What is meant by 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. In practice a critical crack size is not clearly deﬁned, and the probability of detection (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 is clearly 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 crack is present but one is apparently detected. In a relative operating characteristic (ROC) curve the probability of detection is plotted against the false call probability (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 probability of detection of ay least 80 per cent combined with a false call probability of at most 20 per cent. 240 Appendix C: Non Destructive Testing

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) detected 2.2 No 8.8 Yes 21.5 Yes 2.5 No 9.5 No 21.7 Yes 3.0 No 10.7 Yes 21.8 Yes 3.4 Yes 11.5 Yes 22.2 No 3.8 No 13.4 Yes 22.4 Yes

4.2 Yes 13.9 No 22.9 Yes 5.1 Yes 16.0 Yes 23.9 Yes 5.8 No 17.6 Yes 24.4 Yes 6.5 Yes 17.9 No 26.0 Yes 7.1 Yes 19.4 Yes 26.2 Yes

7.3 Yes 19.7 Yes 28.2 Yes 8.0 No 19.9 Yes 28.9 Yes

C.8.1 DETERMINATION OF PROBABILITY OF DETECTION

Probability of detection (POD) curves are determined experimentally by blind trials on a set of specimens containing cracks of known sizes. The objective is to see what can be achieved by a skilled inspector using a particular non destructive testing technique, not to test the inspector. It is good practice to include 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 the specimens, or on the results of the trials. There are two particular difficulties in carrying out POD trials. The first is producing specimens with cracks of the desired size range. Cracks are usually produced by fatigue loading but fatigue cracks are, in general, difficult to control. The second is ensuring that crack sizes, especially crack depths, are accurately known. The most satisfactory way of determining crack depths is to section specimens, but this destroys them so that they cannot be used for further trials. What is sometimes done is to size the cracks using an accurate method, such as time of flight diffraction (see Section C.6) and, from time to time, section a few specimens in a library as a cross check. The sectioned specimens are replaced by new specimens containing similar cracks. Typical data obtained from a POD trial are shown in Table C.2. In order to obtain point estimates of POD, specimens are grouped according to crack size, as shown in Table C.3. It is good practice to have about the same number of 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 POD range (mm) group (per cent) 0–5 6 33.3 5–10 8 62.5 10–15 4 75 15–20 6 83.3 20–25 8 87.5 25–30 4 100

Figure C.20. Probability of detection point estimates. detections in each group divided by the number of specimens in the group is a point estimate of POD, and is usually expressed as a percentage. The point estimates are plotted at the upper limit of each size range, and are usually joined by straight lines, as shown in Figure C.20. The probabilities of detection shown in Table C.3 are point estimates based on small samples of 4 to 8 cracks. However, point estimates obtained from small samples may not accurately reflect probabilities of detection obtained from large samples. Point estimates obtained from small samples are sometimes analysed statistically to provide lower bound estimates of probabilities of detection at, say, the 95 per cent confidence level. However, the small sample sizes typically used in practice for cost reasons make rigorous statistical analysis difficult, and lower bound estimates of probabilities of detection are not in general use (Visser 2002). 242 Appendix C: Non Destructive Testing

Table C.4. Typical data obtained from a probability of sizing trial. Actual crack Measured crack Actual crack Measured crack size (mm) size (mm) size (mm) size (mm) 1.5 Not detected 13.1 13.7 1.9 1.9 13.5 15.1 2.3 Not detected 14.7 15.4 2.7 3.7 16.1 16.2 3.1 Not detected 16.2 18.4

3.3 5.5 16.5 17.3 3.9 5.0 17.7 19.7 4.5 Not detected 17.9 17.8 5.2 4.7 18.1 19.9 5.5 7.9 18.3 Not detected

5.9 5.4 18.5 20.2 6.5 Not detected 18.9 21.0 7.1 7.4 19.7 21.2 7.7 9.6 20.1 19.4 7.9 Not detected 21.5 21.6

9.4 9.4 21.6 22.0 11.1 10.7 23.3 23.6 11.4 Not detected 23.9 26.1

Table C.5. Point estimates of probability of sizing (POS) using 90 per cent accuracy criterion compared with probability of detection (POD). Size range (mm) Number in group POS (per cent) POD (per cent) 0–5 8 12.5 50 5–10 8 50 75 10–15 5 60 80 15–20 10 60 90 20–25 5 80 100

C.8.2 PROBABILITY OF SIZING

The probability of sizing (POS) is a reﬁnement of probability of detection in which a crack is counted as detected only if its size is measured to within an accuracy criterion. An accuracy criterion of x per cent means that the measured 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, and Table C.5 point estimates of probability of sizing, using a 90 per cent accuracy criterion. 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 probability of detection. table. These point estimates are plotted in Figure C.21; probabilities of sizing are signiﬁcantly lower than probabilities of detection. References

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K-calibration, 174 blind trial, 240 K-dominated region, 141, 144, 180, 192 block fatigue loading, 42, 127 S/N curve, 16 block loading, 42 T -stress, 143 block programme, 53 T -stress criterion, 143 boundary layer, 188, 193 T -stress ratio, 144 branch crack, 138, 155, 198 branch crack formation, 140, 149 A-scan, 231 branch crack propagation, 140 absolute liability, 78 branch crack propagation angle, 141, 148 acceptance testing, 76 branch point, 93 accuracy criterion, 242 brittle fracture, 103, 110, 136, 138, 154 additive distribution, 31 broad band random loading, 35, 50, 218, 219 Almen strip, 99 alternating current field measurement, 236 C-scan, 232 alternating current potential drop, 226, 234, carburising, 98 236 centre cracked tension specimen, 106 alternating stress, 15, 16 analytical approach, 72, 75, 162 centre frequency, 208, 219 antisymmetric mode, 187, 190 centre line average roughness, 84 aspect ratio, 153, 177 chaos theory, 141 attractor, 143, 156 chaotic event, 141 autocorrelation function, 218 characteristic crack dimension, 175, 199 autofrettage, 99 civil liability, 78 cliff, 156 B-scan, 231 clipping, 209, 212, 213 bandpass filter, 219 clipping ratio, 42, 59, 209, 212, 214 bandwidth, 40, 208, 218 code, 70, 73 basic situation, 69 code design, 70 Basquin equation, 18 COLOS, 216 bending mode, 174 compliance function, 174 biaxial fatigue loading, 37 compressional probe, 232 biaxial loading, 37, 59 compressional wave, 231 biaxiality ratio, 143 conditional distribution, 32 bimodal distribution, 31 constant amplitude loading, 15 260 Index constant amplitude fatigue loading, 15, 39, D-scan, 232 42, 96, 105, 106, 117, 120, 122, 124, DAC curve, 233 127 DAC level, 233 Consumer Protection Act, 78, 79 damage density curve, 49 core region, 141, 180, 189 damage density function, 49 corner point, 151, 182, 187, 190, 191 damage tolerance, 71, 103 corner point singularity, 152, 187 defective, 69, 77, 78, 80, 164 corner point singularity dominated region, design defect, 69, 78 189, 192 Det Norske Veritas, 132 corner region, 188 detector, 229 couplant, 231 developer, 228 crack area, 199 differential geometry, 11, 155 crack bridging, 238 diffracted wave, 233 crack closure, 114, 127 directionally stable crack, 142 crack front, 168 directionally unstable crack, 142 crack front curvature, 138 disclination, 171, 173 crack front inclination angle, 191 dislocation, 152, 171, 173, 199 crack front intersection angle, 152, 188, 189 dispersion effect, 231 crack front shape, 129 distance amplitude correction curve, 233 crack initiation dominated, 83, 88, 92, 98 dye penetrant, 228 crack mouth, 227 Eastabrook’s theorem, 125 crack opening, 113 eddy current, 234, 235 crack path prediction, 149, 151, 158 edge sliding mode, 169 crack path stability, 142 effective crack length, 182, 184 crack propagation, 138, 172, 173, 186 effective value, 113 crack propagation dominated, 92, 101 effects of thickness, 110 crack sizing, 223, 227, 231 electromagnetic field method, 234 crack surface displacement, 168, 170, 192 electromagnetic radiation, 230 crack surface intersection angle, 187, 190, end on technique, 232 191 endurance, 17 crack tip, 168 endurance limit, 20 crack tip plastic zone, 105, 112, 113, 125, enforcement authority, 79, 80 137, 150, 182 enforcement officer, 79 crack tip surface displacement, 169, 170, equivalent Mode I crack, 197 183, 187, 190 equivalent constant amplitude stress range, cracked situation, 101 47 criminal liability, 77, 79 equivalent cycle, 50, 124 critical crack front intersection angle, 190, equivalent single crack, 201 191 equivalent stress, 63, 64, 89, 96 critical length, 27 equivalent stress intensity factor, 159 critical plane, 63 equivalent tensile stress, 62 critical plane approach, 64 ergodic, 204, 218 crystal dislocation, 170, 199 escape clause, 75 cumulative damage, 44 exceedance, 40, 206, 208, 210, 217 cumulative probability, 40, 206 exponential distribution, 210 cycle, 15 cycle counting, 50, 125 facet, 156 cyclic load, 1 fail safe, 70, 154 Metal Fatigue 261 failure, 2, 4 Gaussian distribution, 21, 39, 206, 208, 211, failure analysis, 68, 83, 97, 102, 109, 122, 213, 214, 216, 218 130, 162 geometric correction factor, 175 failure mechanism map, 139 Gerber diagram, 86 false call, 239 Gerber parabola, 85 false call probability, 239 gigacycle fatigue, 19, 58 FALSTAFF, 54 Goodman diagram, 86, 96 Fast Fourier Transform, 219 good practice, 73 fatigue, 7 grass, 233 fatigue assessment, 67, 69, 71–74, 83, 93, Gumbel distribution, 217 97, 109, 117 fatigue crack growth, 26 hard X-rays, 230 fatigue crack initiation, 25, 26, 31, 83, 98, high cycle fatigue, 17 132 fatigue crack path, 10, 75, 135, 138 ideal crack path, 142 fatigue crack propagation, 11, 26, 105, 109, in house tool, 70 114, 117, 118, 122, 124, 128, 129, induction hardening, 98 135, 138, 155, 159, 164, 165, 167, initial crack, 139, 154, 155, 158, 198 223, 224, 228, 236, 237 initial crack size, 101, 102, 122 fatigue crack propagation life, 102, 122, 124 integral approach, 64 fatigue crack propagation rate, 105, 106, interaction effect, 125, 127, 196, 200 internal crack, 142, 196, 200 109, 110, 114, 116, 119, 122, 124, 127, 133, 224, 237 internal defect, 32 intrinsic fatigue strength, 84 fatigue cycle, 15, 105 irregular crack, 196 fatigue design, 67, 109 irregularity factor, 40, 218 fatigue fracture toughness, 104 fatigue life, 83, 122, 124, 130 JOSH, 54 fatigue limit, 18, 120 fatigue load, 1, 55 Kirchoff plate bending theory, 174 fatigue loading, 15, 38, 61, 94, 98, 138, 149, Kitagawa diagram, 120 154, 156, 199, 204, 240 knee, 19 fatigue strength, 90, 162 KoNoS hypothesis, 65 fatigue strength reduction factor, 90 fatigue testing, 8, 9 Laplace distribution, 49, 210, 213, 216 final crack size, 101, 103, 122 LBF Normal distribution, 53 fish eye, 29 leak before break, 5, 154 flame hardening, 98 leakage flux, 227 Forsyth’s notation, 26, 137, 148 library, 240 Foucault current, 236 life, 17 fractals, 11 lift off, 235 fractography, 11, 129 limit load, 103 fracture criterion, 64 line tension, 152, 199 fracture mechanics, 12, 164, 167 linear damage rule, 45 fracture toughness, 104, 110, 155, 172, 183 linear elastic fracture mechanics, 167 frequency dependence, 116 load cycle, 16 frequency domain, 219 load history, 12, 35, 44, 49, 53, 56, 58, 65, frequency independent, 18, 76, 116 127, 130, 135 load spectrum, 13, 44 gamma rays, 230 long crack, 118 262 Index longitudinal wave, 231 narrow band random process, 41, 59, 208, low cycle fatigue, 17, 65 219 negative peak, 42 macrocrack, 27, 136 negligence, 78, 79 magnetic flux, 227, 236 nitriding, 98 magnetic ink, 227 no fault liability, 78 magnetic particle inspection, 226 non destructive testing, 103, 164, 196, 223, magnetic permeability, 235 238 main crack, 139 non metallic material, 4 major project, 70 non propagating crack, 93, 116, 120, 142 manufacturing defect, 69, 78 non proportional fatigue loading, 59, 64 mass product, 70 non proportional loading, 59 mathematical description, 37 non proportional random loading, 62 maximum normal stress criterion, 64 non stationary random processes, 42 maximum principal stress dominated crack nonlinear dynamics, 143 propagation, 137 Normal distribution, 21, 39, 206 maximum stress, 15, 20 notch, 83, 89, 180 mean stress, 10, 15, 85, 88 notch insensitive, 92 mean stress insensitive, 110 notch sensitive, 92 mean stress sensitive, 110, 114 notch sensitivity index, 92 mechanical description, 1, 7, 161, 165 number of cycles, 16 metal fatigue, 1, 7, 37, 161–163, 165, 204, 207, 210, 212, 218, 223, 229, 231, ocean wave, 216, 231 235, 236 omission dilemma, 58 metal fatigue damage, 10 omission level, 58, 213 metal fatigue mechanism, 24 one sided process, 214 metallurgical description, 1, 161, 165 open crack, 181 microcrack, 25, 27 opening mode, 169 micromechanisms, 165 overload, 127 Miner’s law, 45 oxide induced crack closure, 115 Miner’s rule, 13, 45, 49, 50, 97, 126 minimum stress, 15 P-S-N curves, 23 Mises criterion, 63 Palmgren–Miner law, 45 mixed mode, 26, 138, 155, 156, 158, 169, Palmgren–Miner rule, 45 198 Paris equation, 105 mixed mode threshold for fatigue crack Paris law, 105 propagation, 139 Paris region, 117, 122 Mode I, 169, 170, 172, 183, 184, 186, part through crack, 152, 153, 177 190–192, 196, 197, 199 peak counting, 51 Mode II, 169, 171, 172, 186, 191, 192 penny shaped crack, 176 Mode III, 169, 171, 172, 186, 191, 192 periodic processes, 206 modified Goodman diagram, 86 philosophies of design, 70, 154 multiaxial failure criterion, 62 physical crack length, 184 multiaxial fatigue loading, 35, 39, 60, 63, 65, plane strain, 109, 182, 183, 185 88 plane strain fracture toughness, 104, 154, multiaxial loading, 35, 59 184 plane stress, 109, 154, 183, 184, 188 narrow band random loading, 35, 56, 58, plane stress fracture toughness, 154, 184 210, 214–216 plastic collapse, 103, 154 Metal Fatigue 263 plastic wake, 113, 119, 125, 127 safety factor, 72 plastic zone, 128, 182, 184 safety regulation, 78, 79, 80 plastic zone correction, 185 scalar criterion, 63 plastic zone radius, 185 scatter, 20, 31, 72, 96, 97, 107, 122, 133 point estimate, 240, 242 scatter band, 106 positive peak, 41 scatter diagram, 217 postulated cracks, 103 sea state, 216, 221 power spectral density, 219 service loading testing, 72, 75, 76, 163 probability density, 21, 40, 206, 208, 210 shakedown, 31 probability of detection, 223, 234, 238, 240, shear crack propagation, 138, 186 242 shear dominated crack propagation, 137 probability of failure, 23, 96, 97 shear lip, 111 probability of sizing, 223, 234, 242 shear mode, 169 product liability, 67, 77, 78 shear wave, 231, 233 programme loading, 42, 129 short crack, 118 programme marking, 129 shot peening, 88, 99 proof loading, 99 shrinkage, 101 proportional fatigue loading, 59, 64 sigmoidal, 117 proportional loading, 59 significant wave height, 217 pseudo random, 35, 214 skin depth, 234, 235 skin effect, 234 quality system, 80 slant crack propagation, 137, 186, 198 slant fatigue crack propagation, 110 radiograph, 229 slip, 25 radiography, 229 slip line, 10 rainflow counting, 51 small scale argument, 180, 186 random process, 204, 207 Soderberg line, 88 random process theory, 35, 39, 204, 207, 218 soft X-rays, 230 random walk, 142 spectral bandwidth, 219 range counting, 51 spectral density function, 40, 56, 218 range of scales, 24 stable state, 114 ratchetting, 30 Stage I crack, 26, 137, 140, 158 Rayleigh distribution, 41, 208, 210, 213, 216 Stage II crack, 26, 101, 138, 148 re-assessment, 73 Stage III, 27 redundant, 71 standard deviation, 21, 206, 207 reference stress, 38 standard load history, 13, 53–56, 62, 216 regulatory authority, 75, 80, 224 standard procedure, 13, 70, 73, 162 relative operating characteristic, 239 standard test method, 12, 107 residual stress, 88, 98, 99, 124, 133, 180 standing wave, 231 resonant frequency, 219 static failure, 101 return period, 35, 42, 58, 214 static failure region, 118, 122 root mean square, 39, 131, 206, 207, 210, static strength, 1 212–214, 216, 218 statistically non stationary, 75, 216 rotating bending, 16 statistically stationary, 39, 204, 216, 218 rotation, 172 stochastic process, 38 roughness induced crack closure, 115 stress concentration factor, 91, 150 stress criterion, 28, 117 safe life, 70 stress cycle, 15, 124 safe product, 77, 79, 80 stress history, 15 264 Index stress intensity factor, 12, 104, 113, 116, truncation dilemma, 58 119, 122, 125, 132, 138, 143, 150, truncation level, 56 155, 158, 159, 164, 167, 169, 172, twist crack, 156, 158 180, 186, 190–192, 195, 196, 199, 200, 225, 233, 238 ultrasonics, 226, 231 stress intensity factor range, 105 uncracked situation, 83, 96 stress intensity measure, 187, 189 underload, 127 stress range, 10, 15, 105 uniaxial fatigue loading, 59 stress ratio, 15, 45, 110 uniaxial loading, 35, 59, 90 stress relief, 124 unsafe product, 80 stress state, 154 striation, 27, 129 validity corridor, 142 strict liability, 78, 79 variable amplitude fatigue loading, 35, 37, subsidiary echo, 231 96, 124 surface breaking crack, 223, 226, 228, 232, variable amplitude loading, 35 238 variance, 206 surface crack, 25, 151, 196, 200, 223 viscous fluid induced crack closure, 115 surface factor, 85 visual inspection, 223 surface finish, 84 Volterra distorsioni, 170 surface hardening, 98 von Mises criterion, 63, 88, 92, 184 surface irregularity, 84 symmetric mode, 187, 190 Wöhler curves, 17 Wöhler’s laws, 9 thermal loading, 31, 150 WASH, 54 thermodynamic criterion, 28, 117, 137 water surface elevation, 216, 221 threshold for fatigue crack propagation, 11, wave height, 216 29, 116, 117, 119, 122, 132, 138, 158 wave loading, 55, 56, 58 threshold region, 117, 122 wave passing frequency, 56, 59, 217, 221 through the thickness crack, 154, 196, 225 wave period, 217 tight crack, 230 waveform, 18, 76, 211 time domain, 219 Weibull distribution, 210, 213 time history, 40, 52 weighted average stress range, 47, 124 time of flight diffraction, 233, 240 welded joint, 124, 130, 227, 230, 237 toughness, 172 Wheeler’s model, 128 trading standards officer, 79 white noise, 42 transformation induced crack closure, 115 transition region, 111, 186 X-rays, 229 translation, 172 transverse wave, 231 yield criterion, 63 Tresca criterion, 63 Young’s modulus, 109 trough, 42, 209 truncation, 42, 58, 210, 212, 213 zero crossing, 40