Applying fracture mechanics to fatigue strength determination – Some basic considerations

International Journal of Fatigue 126 (2019) 188–201

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Applying fracture mechanics to fatigue strength determination – Some basic considerations

T

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Uwe Zerbsta, , Mauro Madiaa, Michael Vormwaldb a b

Bundesanstalt für Materialforschung und -prüfung (BAM), D-12205 Berlin, Germany Technische Universität Darmstadt, Materials Mechanics Group, D-64287 Darmstadt, Germany

A R T I C LE I N FO

A B S T R A C T

Keywords: Fatigue strength Fracture mechanics Fatigue crack propagation Initial crack size Multiple cracks

A discussion is provided on demands that must be met in order to apply fracture mechanics to the determination of overall fatigue lifetime and strength, i.e., S-N curves and fatigue limits. These comprise the determination of the cyclic crack driving force for all stages of fatigue crack propagation, in particular for the short crack stage where the crack driving force has to be determined for elastic-plastic deformation and the gradual build-up of the crack closure phenomenon. Special emphasis is put on a fatigue damage relevant specification of the initial crack size. Different approaches in the literature are discussed. Another important aspect is the adequate treatment of multiple crack propagation. Finally, the discussion is illustrated by an example of a butt weld made of a medium strength steel.

1. Fatigue life and damage tolerance concepts – An introduction Fatigue life concepts for engineering components are commonly based on the S-N curve which describes the dependency of the number of loading cycles up to fracture or up to the formation of an incipient crack on the stress amplitude or stress range. The stress amplitude at and below which, independent of the number of loading cycles, no failure occurs is the fatigue or endurance limit. A principle distinction is made between (i) the plain fatigue strength of the material which is usually obtained on smooth, unnotched specimens and a stress ratio R = σmin/σmax of −1 and (ii) the fatigue strength of structures with designed notches. Effects of influencing factors such as the notch geometry, surface roughness or an R ratio deviating from −1 are considered by correction factors given in guidelines such as [1]. Note that not any material shows an endurance limit, and that mechanisms, e.g., local corrosion, might exist which cause the endurance limit to disappear [2,3]. The major difference of the common fracture mechanics-based damage tolerance concept, when compared with the S-N curve, is that it determines a residual instead of the total lifetime. This is the time or number of loading cycles a pre-existent crack needs to grow to its critical size. The original crack sizes are defined by the detection limits of non-destructive testing (NDT) which are in the order of millimeters. Cracks or crack-like defects too small to be found by NDT are generally considered existent. No need to emphasize that the residual lifetime determined this way will often be substantially shorter than the total

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lifetime. Both the total and the residual lifetime are schematically illustrated in Fig. 1. In principle, fracture mechanics can also be applied to the total lifetime, if certain conditions are fulfilled: (a) The NDT based initial crack size is replaced by a much smaller “fatigue life relevant” one. The meaning of this will still have to be discussed. (b) The early stages of fatigue crack propagation, when the crack is still a short one, must be adequately described. (c) Frequently, multiple crack initiation (at various highly stressed regions in the component), multiple crack growth and coalescence have to be considered. It is the aim of the present paper to provide a discussion of these demands and on how they could be satisfied. The argumentation will be limited to what the authors think are the essentials. With respect to a number of nonetheless important details they will refer to earlier papers which provide in-depth discussions. First, a few basic terms and concepts must be briefly discussed. These refer to the stages of fatigue crack propagation and to the conditions under which cracks arrest. The background to the latter is the realization that the fatigue limit usually does not refer to fatigue crack initiation but to the arrest of cracks that originally have grown [2,3].

Corresponding author.

https://doi.org/10.1016/j.ijfatigue.2019.05.009 Received 27 November 2018; Received in revised form 3 April 2019; Accepted 9 May 2019 Available online 10 May 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.

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Nomenclature a ai a’ a0 b B c d da/dN E E′ f F F FI h J k K′, n′ K Kop Kt ℓ N Nc Pt R ReL Rp0.2 Rm

T U α Δa ΔJ ΔK ΔKeff ΔKp ΔKth ΔKth,eff

crack length (crack depth for surface cracks) initial crack depth (for fracture mechanics analysis) crack depth above which a notch can be treated as part of a crack, Fig. 9 (in relation to the notch root radius ρ) El Haddad parameter, fictitious crack length notch width, Fig. 8 plate with, Fig. 20 half crack length at surface (semi-elliptical crack) arrest crack depth referring to the fatigue limit at the Kitagawa-Takahashi diagram, Figs. 5 and 16 fatigue crack propagation rate modulus of elasticity (Young’s modulus) = E for plane stress and E/(1 − ν2) for plane strain conditions crack closure function, Eqs. (2) and (3) boundary correction or shape factor of K factor solution probability, Fig. 13 F for mode-I loading weld reinforcement J-integral (monotonic loading) depth of a notch or secondary notch coefficients of the cyclic stress-strain curve (RambergOsgood) stress intensity factor (K-factor) K-factor at crack opening (elastic) stress concentration factor section width along the weld toe (Figs. 21 and 23) number of loading cycles number of loading cycles up to fracture roughness parameter used for specifying the secondary notch depth k in the present work loading ratio (=σmin/σmax or Kmin/Kmax) lower yield strength (materials showing a Lüders plateau) 0.2% proof strength (materials without Lüders plateau) uniaxial tensile strength

ΔKth,op Δσe, Δσth Δσw ρ σ σe σw σmax σmin σN σY σY′

plate thickness crack closure factor (=ΔKeff/ΔK) weld flank angle crack extension (monotonic or cyclic) cyclic J-integral (cyclic loading) K-factor range (Kmax – Kmin) effective K-factor range (=Kmax – Kop) (formally) plasticity-corrected ΔK (= ΔJ ·E′ ) fatigue crack propagation threshold intrinsic fatigue crack propagation threshold (no crack closure effect) crack closure component of the fatigue propagation threshold endurance limit range endurance limit range for R = –1 weld toe radius stress endurance limit (stress amplitude), defined for N = 107 in this paper endurance limit (stress amplitude) for R = –1 maximum stress in the fatigue cycle minimum stress in the fatigue cycle nominal net section stress yield strength (in general terms; either ReL or Rp0.2) cyclic yield strength

Abbreviations CPCA eff HAZ LC NASGRO p R-curve SC

2. Stages of fatigue crack propagation and arrest

compression pre-cracking constant amplitude crack closure-free heat affected zone long crack computer program for fatigue crack propagation, provided by NASA plasticity corrected crack resistance curve (monotonic and cyclic version) short crack

the accumulation of irreversible plastic deformation. At a smooth surface this will be accompanied by the development of an intrusion/extrusion pattern. In engineering materials much more common is crack nucleation at local discontinuities such as non-metallic inclusions and pores but also at defects originating from the final steps of manufacturing or from service or maintenance such as scratches, corrosion pits etc. (for a detailed review of this issue see the papers of the authors in [4–7]). As a consequence, the crack nucleation stage will usually be rather short (sometimes one or a few cycles only).

2.1. Crack nucleation The evolution of fatigue damage starts with crack nucleation due to

2.2. Propagation of microstructurally short cracks The first stage of crack propagation is strongly influenced by the defect at which the crack was initiated and its surrounding area and by the characteristics of the microstructure through which it grows. This causes crack acceleration, deceleration or even arrest events and a crack shape far from elliptical or straight. The size of the crack at this stage is designated as “microstructurally short”. Usually, a large number of microstructurally short cracks will be initiated and grow over some distance, but finally almost all of them arrest at microstructural barriers. The stress level at which even the largest of these microcracks loses its capability to grow corresponds to the endurance limit. However, as has been mentioned above, if there are mechanisms such as time dependent local corrosion, the endurance limit will disappear [2]. Whilst there exist other options, the most important microstructural

Fig. 1. Total vs. residual lifetime (fatigue life versus common fracture mechanics concepts). Schematic illustration. 189

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crack. (iii) The oxide debris-induced crack closure mechanism is active in materials susceptible to corrosion. The crack wake surfaces are covered by oxide layers which thicken at low R ratios in that blank rubbing of the crack faces promotes further corrosion (Fig. 2c). Fig. 2d illustrates the definition of the ΔKeff parameter mentioned above. Crack closure is usually quantified by the parameter U = ΔKeff /ΔK.

barriers are provided by the grain boundaries between adjacent grains of different lattice orientation. Since usually not all adjacent grains show significantly different crystal orientations, the barrier effect is not uniform along the crack front and the length of the microstructurally short crack corresponding to the fatigue limit is not identical with one grain size but is roughly in the order of about three grain sizes [3], with some variation from material to material [8]. Note that the specification of the grain size is a matter of its own, see Section 3.3.3. In the absence of geometrical macro-notches from design, the fatigue limit at which the largest microstructurally short crack arrests is the plain fatigue limit of the material.

None of the closure mechanisms exists at the beginning of the lifetime since they all require some crack propagation or, more exactly, some extension of the crack wake. However, when the fatigue crack propagates at its physically short crack stage, the crack closure phenomenon gradually builds-up until it reaches a nearly crack size independent value. What does that mean with respect to crack arrest? It means that there is a further mechanism for a crack to stop. With increasing crack size, ΔKeff might become smaller due to crack closure even if the nominal ΔK becomes larger with increasing crack size. Crack arrest occurs when ΔKeff falls below the crack propagation threshold ΔKth. Since the threshold itself is a crack arrest phenomenon [20], it will also be affected by the crack closure phenomenon in that it gradually increases at the physically short crack stage. This can be described by the cyclic R curve with the capital R standing for “resistance”. This curve is schematically shown in Fig. 3. The fatigue crack propagation threshold consists of two components: (i) an intrinsic one, ΔKth,eff, which refers to the closure-free part of ΔKth (therefore, the index “eff”), and a closureinduced second component ΔKth,op which increases with crack propagation Δa until it reaches its crack size-independent long crack value. The intrinsic threshold seems to be a material parameter which only depends on the lattice and the elastic properties (e.g., [21–23], cf. also [19]). In contrast, the crack closure-caused component ΔKth,op depends on a number of parameters such as the stress ratio R and the grain size of the material (insofar as this influences the roughness-induced crack closure effect), see the discussion in [19]. The experimental determination of the cyclic R curve is described in a number of papers, e.g., in [24] and more recently in [25] and will not be the topic here. Within the frame of a cyclic R curve analysis it can be used to predict crack arrest in a structure and, corresponding to this, the fatigue limit of this structure. The authors will discuss this in more detail in Section 3. An important aspect is that the curve “mirrors” the gradual build-up of the crack closure effect such as is shown in Fig. 4. The correlation with the crack closure parameter U = ΔKeff /ΔK is simply given by

2.3. Propagation of mechanically/physically short cracks If a crack is capable of overcoming the microstructural barriers, e.g. at a higher stress level, it will continue to grow as a mechanically/ physically short crack. Finally, it will be much larger than the microstructurally short one with sizes up to the order of the plastic zone size (as a mechanical length scale) ahead of it. Mechanically and physically short cracks are distinguished with respect to different aspects. (a) As mentioned, the size of a mechanically short crack is usually smaller than the plastic zone which encloses it. As a consequence, it cannot be characterized by the linear elastic ΔK concept and, thus, not by the da/dN-ΔK curve. Instead, a cyclic elastic-plastic parameter such as the cyclic J integral, ΔJ, has to be applied. The adequate numerical and analytical determination of this parameter has been discussed by the authors in a number of papers ([9–14]) and will not be addressed here. (b) Physically short cracks are characterized by the gradual build-up of the crack closure phenomenon. Crack closure means that the crack, within a loading cycle, is only open at a stress level higher than zero (depending on the R ratio and the applied stress range sometimes also below). The concept was first introduced by Elber [15]. Only that part of the K factor range over which the crack is open is thought to contribute to crack propagation. It is designated as the effective K range, ΔKeff = Kmax – Kop. The crack closure phenomenon is realized by various mechanisms (for detailed overviews see, e.g., [16–18]; a rather brief discussion is provided by the authors in [19]). The three most important mechanisms are the plasticity-induced, the roughness-induced and the oxide-debris-induced ones (Fig. 2). In short: (i) The plasticity-induced crack closure mechanism which dominates the Paris regime of the da/dN-ΔK curve is due to the plastic zone ahead of a growing crack. When the crack tip extends, plastically stretched material remains at the surfaces of the crack wake (Fig. 2a) where it causes the geometrical mis-match of the latter. (ii) The roughness-induced crack closure mechanism is due to the roughness of the crack faces which causes asperity contact (Fig. 2b), an effect which can be enhanced by mixed mode loading components (shear) due to kinking or branching of the

ΔKth, SC (a) − ΔKth, eff 1 − USC (a) = 1 − ULC ΔKth, LC − ΔKth, eff

(1)

with ΔKth,LC and ΔKth,eff being the long crack and intrinsic or openingfree fatigue crack propagation thresholds, the parameter a is the crack length or depth, and the indices SC and LC stand for short and long crack.

Fig. 2. Mechanisms of the crack closure phenomenon; (a) Plasticity-induced mechanism; (b) Roughness-induced mechanism; (c) Oxide debris-induced mechanism; (d) Definition of the ΔKeff parameter. 190

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f=

σopen σmax

A0 + A1 R + A2 R2 + A3 R3 for R⩾0 =⎧ ⎨ A A R for - 2 < R<0 + 0 1 ⎩

(2)

and used in

ΔK eff = U ·ΔK =

(1 − f ) ΔK (1 − R)

(3)

where the parameters A0, A1, A2 and A3 are the outcome of the approximation. 2.5. Crack arrest at notches By far the most important practical case is crack initiation at designrelated notches. In addition to what has just been discussed, notches add a further element: the decreasing stress with increasing distance from the notch root. Examples for the latter are provided in Fig. 6. For many years authors have used the term “anomalous” crack growth behavior at sharp notches, for an overview see [33]. This means that the crack first propagates at a high growth rate, then might decelerate, and finally accelerates again or arrests. “Self-arresting cracks” predominantly occur at sharp notches with steep stress gradients into the interior. The rapid crack propagation at the beginning is due to the high stresses (and strains) at (respectively slightly below) the notch root and it is further promoted by the fact that no crack closure phenomenon exists at that stage because the fatigue crack starts as a short one. When the crack propagates, it grows into a zone of lower stresses and strains (when referred to the non-cracked state – of course, the crack generates its own stress field) and the crack closure phenomenon is gradually build-up. Both effects (the decreasing stress and crack closure) lead to its deceleration. Whether the crack finally arrests or not depends on whether the cyclic crack driving force falls below the fatigue crack propagation threshold. Note that the physically short crack is also a mechanically short one and this the more as the notch, above a certain stress level, will generate its own notch plastic zone. As a consequence, a cyclic elastic-plastic crack driving force parameter such as ΔJ has to be used in the analysis [34,35]. In Section 2.3 it has been mentioned that the cyclic R curve can be used in the framework of a cyclic R curve analysis to determine the fatigue limit of a structure. The principle which is illustrated in Fig. 7 has first been introduced by Tanaka & Akinawa [37], for more recent examples see [11,12,38–40]. The cyclic R curve analysis can be compared with the better known monotonic R curve analysis where the crack driving force curve of the crack in the component, e.g., in terms of the J-integral, is plotted as a function of the crack size, a, along with the monotonic R-curve, J-Δa. Adding the third “vertex” in the “fracture mechanics triangle” which is the length or depth of a known or assumed pre-existent crack and fixing the starting point of the R curve by this crack size, a crack driving force curve can be determined which fulfils the tangency condition with the R curve. The corresponding applied load is that load which just leads to unstable extension of the till stably

Fig. 3. Schematic representation of the cyclic R curve.

Note that the information for the R-curve and hence for the function Usc(a) is obtained at stress and strain levels at and near the threshold or endurance limit. However, although it is state-of-the-art, see, e.g., Section 5, the implicit assumption that the R-curve is independent of the stress and strain levels must be challenged. To that purpose an experimental technique is required which would allow to control an applied ΔK for any pre-defined and constant crack growth rate. Besides this, a numerical approach based on a finite-element based node-release technique [26] or a strip yield model [27] seems to be promising. An important side aspect is that the cyclic R curve corresponds with the Kitagawa-Takahashi diagram [28] such as is illustrated in Fig. 5. 2.4. Propagation of long cracks For the sake of completeness, the propagation of the long crack as (usually) the last stage before failure has still to be briefly addressed. The definition of a long crack is complementary to that of the mechanically/physically short one. A crack is “long” if it (i) is large enough to allow the application of the linear elastic K concept and if (ii) the crack closure phenomenon is quasi stabilized, i.e., independent of the crack size for constant amplitude loading. The transition will usually occur at a crack depth in the order of a few tenths to a few millimeters depending on the material but also on the environmental conditions if the oxide-debris-induced crack closure mechanism comes into play (as an example for the latter see [29]). Long crack propagation can be described by the da/dN-ΔKeff curve. A common approach for determining the (plasticity-induced) long crack closure effect, and thus ΔKeff, is provided by the Newman’s equation [30] which is an approximation to results obtained by his modified strip yield model [31]. It became part of the NASGRO approach [32] and is widely used today. In the Newman’s approach, a crack closure function f is determined by

Fig. 4. Parallel development of the crack closure factor U and the fatigue crack propagation threshold with increasing crack depth, schematic view. 191

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Fig. 5. Correspondence between the Kitagawa-Takahashi diagram (left) and the cyclic R-curve (right). sections A, B and C refer to the crack growth stages of microstructurally short (A), mechanically/physically short (B) and long cracks (C).

Fig. 7. Schematic view of a cyclic R-curve analysis. The transition from crack arrest to crack propagation is given by that crack driving force curve (referring to load 2 in the example) which tangentially touches the cyclic R-curve.

Fig. 6. Examples of finite element based through-thickness stress profiles (normalized) at a butt weld toe; according to [36].

3. The initial crack size

growing crack. Likewise, the tangency criterion in the cyclic R curve marks a transition point, but now between crack arrest and crack propagation. As in the monotonic analysis, an initial crack depth ai is needed. Note that this does not refer to a cyclic ΔK equal to zero but to the intrinsic crack propagation threshold, ΔKth,eff, as the smallest possible threshold value. The parameter ΔKp at the ordinate is a “plasticity-corrected” ΔK which is obtained from the cyclic J integral ΔJ by ΔKp = ΔJ ·E′ in [11,12]. Since Load 2 in Fig. 7 just marks the transition from crack arrest to crack propagation, it refers to the fatigue limit of the tension loaded butt weld in the example. Note that the cyclic R curve can also be applied to the prediction of the finite life branch of the S-N curve as is demonstrated in [12], see also Section 5, when it is used for determining the gradual build-up of the crack closure phenomenon in terms of the parameter U at the physically short crack stage (Fig. 4 and Eq. (1)), for details see [11,12].

3.1. The problem When the (elastic-plastic) cyclic crack driving force in the component is determined in terms of ΔJ (Section 2.3a) or ΔKp (Fig. 7) and the cyclic R curve is available (Section 2.3), there is still one missing parameter when fracture mechanics shall be applied to describe the total lifetime and the fatigue strength of a structure: the initial crack size ai. Discussing this in more detail, another question has to be addressed beforehand: when must a notch be treated mechanically as a notch, i.e. as a stress/strain concentrator, and when should it be treated as part of a crack emanating from its root? This is a difficile issue insofar as it not primarily refers to designed notches but to (unintentional) secondary notches such as undercuts, roughness, dents, scratches, corrosion pits. Micro-notches at material defects such as non-metallic inclusions or pores have also a significant effect on fatigue crack initiation and

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notch to be treated as part of the crack. Note, however, that the studies are limited to small-scale yielding conditions since they are based on the K factor concept, i.e., plasticity effects at the notch have to be negligible, cf., the discussion in Section 8.3.2 in Ref. [18]. An elasticplastic consideration of the problem performed by the authors is in progress.

propagation. Instead of a detailed discussion, which the authors provide in [4–6] (see also Section 3.2) only some remarks are added at this place. It is known that defects lose their harmfulness for fatigue when their size falls below a certain limit which is usually reported to be in the order of 10–25 μm for inclusions in steel and aluminum, see, e.g., [7,41–46]. Commonly, this limit is designated as “critical defect size” which corresponds to what is called the “initial crack size” in this paper with the background that a defect will not be harmful when it (i) will not cause crack initiation or (ii) when it will cause the initiation and growth of a crack which will arrest later on. Note that this statement, although generally true, primarily refers to perfect plain surface conditions. In the presence of secondary notches, things become more complex as will be discussed in Section 3.2 which, therefore, will precede the discussion of the various proposals of initial (or critical) crack size determination. It is known that the “critical defect size” depends on the hardness or (ultimate tensile) strength of the material in that it becomes smaller with higher strength. The contribution of the defects to crack initiation is not just an additional stress concentration but comprises further aspects such as stiffness mis-match, thermal expansion mis-match and sometimes also chemical mis-match as well as poor bounding between inclusions and matrix, etc. [5]. Frequently, defects such as inclusions or pores act as clusters or stringers and not as single defects [5,6]. This is similar for scratches which are usually not constant in depth along their length extension this way promoting multiple crack initiation and propagation [6].

3.3. Methods proposed for specifying the initial crack size ai 3.3.1. Brief overview In the following sub-sections six proposals for specifying the initial crack size for fracture mechanics analyses are briefly described and discussed. These comprise: (i) an approach suggested by the authors which is based on the cyclic R curve method for plain specimens; (ii) the specification of ai based on microstructural features; (iii) its definition by the size of the fatigue damage process zone; (iv) specifications as critical distances in the ligament and (iv) an approach based on the assumption of a constant secondary plastic zone size at endurance stress level of structures with small cracks. In order to provide some quantitative inter-comparability the methods are applied to a data set of tension loaded plates of S355NL steel the data of which are reported in a recent special issue of the journal Engineering Fracture Mechanics [51]. Whilst these papers have their focus on weldments, only the base metal is considered here. The monotonic stress-strain data of this are: modulus of elasticity E = 210 GPa, yield strength ReL = 378 MPa, ultimate tensile strength Rm = 555 MPa, elongation at fracture: 39%. The coefficients of the stabilized cyclic stress-strain curve are K′ = 1200 MPa and n′ = 0.2, the ′ = Rp0.2′ = 330 MPa. The plain fatigue limit (of cyclic yield strength σYc the material) at R = −1 (defined for 107 loading cycles) is Δσw = 550 MPa [52]. The secondary notches (see Section 3.2) are specified by the surface roughness what immediately raises the question of how (i.e. by which parameters) to characterize the secondary notch depth k since different roughness measures are common. As a rule, measures relating to the maximum surface irregularity and not averaged values seem to be adequate in the context of fatigue analyses [53]. Applied to the S355NL steel plate specimens used as the comparative example in the sections below, the roughness-related k values of Fig. 10 are obtained by scanning the plate surface. As can be seen, the mean value of the roughness was in the order of 55 μm, with lower and upper bounds between about 30 and 100 μm.

3.2. Secondary notches and cracks Secondary notches, i.e. notches which are not designed, e.g. roughness or undercuts in weldments but also dents, corrosion pits, scratches etc., are often treated like cracks in fracture mechanics analyses [4–6]. At the background of the discussion in the previous sections that means that their depth would have to be added to the size of the short crack arresting at a plain surface. But it is also understandable that a mild notch should not be treated as part of the crack but as a stress concentrator. That the treatment as a crack even in those cases might be effective lies in the fact that micro-cracks which are usually not detectable under realistic conditions could rapidly develop or could even be pre-existent before loading. Murakami [3], based on linear-elastic considerations, concluded that a notch with a small crack at its tip may usually be treated as a crack. Fig. 8 provides the boundary correction functions FI of the stress intensity factor KI of cracks emanating from an elliptical hole. Values larger than FI = 1 are an indication of a notch-induced stress concentration effect on the stress intensity factor. As can be seen, besides the ratio of the crack depth a and the notch root radius ρ, the notch geometry, i.e., its b/k ratio, plays a role. FI becomes larger for shallow, wide notches (large b/k) and smaller for deep and narrow notches (small b/k) and it finally converges to FI = 1 above a certain crack depth to notch root radius ratio a/ρ. Fig. 9 shows a comparable analysis elaborated by Dowling [49]. Combining the solution for a (short) crack, a, affected by the notch and a (long) crack, k + a (including the notch depth), which is not affected by the latter and comparing these to finite element results, the author obtained a limit crack size, a′, above which the notch effect disappears, i.e. the notch can be treated as part of the crack:

a′ =

k (1.12·Kt / FI )2 − 1

(4)

Similar conclusions were reached in [50]. In [12] the authors argue that this equation can also be applied to multiple notches (e.g., an undercut at a weld toe) when the combined effect of the notches on the stress concentration Kt is considered. Figs. 8 and 9 indicate that already a rather short crack emanating from a moderate to sharp notch root is usually sufficient to allow the

Fig. 8. Effect of the secondary notch depth and geometry as well as of the depth of the crack emanating from the notch root on the geometry function FI of the K factor; extract; according to Nisitani and Isida [47], see also [48]. 193

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of the strength of the material is implicitly considered since the cyclic stress-strain curve is included in the determination of ΔJ (and respectively ΔKp). That the effect shows up in the correct way (i.e., the initial crack size becomes smaller with higher strength) has been demonstrated by the authors, e.g. in [11], see also [56,57]. An example is provided in Fig. 12. The figure also gives us an impression of the order of the initial crack depth in these two steels. For the S355NL steel, it is between 10 and 27 μm (expected value about 17 μm). Perhaps the greatest challenge of the approach is the comparatively high effort with respect to the input data. Besides the cyclic stress-strain curve also the cyclic R curve is needed. Although the determination and application of the latter has successfully been demonstrated in [12] even for heat affected zone material, no test guideline for the cyclic R curve exists to date, see the discussion in [25]. In [55] the authors have proposed, in addition to the experimental determination, an estimate based on the Kitagawa-Takahashi diagram which, as is demonstrated in Fig. 5, is complementary to the cyclic R curve. Note that El Haddad’s approximation of the ΔKth-Δa dependency based on his parameter ao [58] has to be modified by a further summand, designated as a* by the authors, in order to fulfil the condition that the threshold referring to Δa = 0 is ΔKth,eff and not zero.

Fig. 9. Variation of the stress intensity factor with crack depth, a, in the presence of a notch of depth k; according to [49].

ΔKth (Δa) = ΔKth, LC ·

Δa + a* Δa + a* + a0

(5)

The term can simply be determined by [55]

a ∗ = a0 ·

(ΔKth, eff /ΔKth, LC )2 1 − (ΔKth, eff /ΔKth, LC )2

(6)

Whilst this is a rather simple task, the much bigger challenge is the adequate determination of ao. This is a more general problem which will be discussed in Section 3.3.5.1. The cyclic R curve analysis and the determination of ai, sensitively depend on the quality of the cyclic R curve, particularly in the curve section which tangentially touches the crack driving force curve of the component. This brings up demands with respect to the scatter of the experimental data as well as to the correct approximation of the data points [25]. Note that a throughout investigation of this issue is still pending. It should be mentioned that the problem of a correct ai reduces when the depth of a secondary notch k is treated as part of the initial crack (Section 3.2), particularly when the depth of the secondary notch is much larger than the plain surface ai. In [12] the authors have added the surface roughness in terms of the maximum irregularity at the weld

Fig. 10. Statistical distribution of the secondary notch depth k provided by the surface roughness (maximum irregularity), S355NL steel; according to [54].

3.3.2. Determination using the cyclic R curve approach, proposal of the present authors The approach proposed by the authors in [55], see also [11,12], is based on the schematics of Fig. 7, where ai is that crack size which fixes the cyclic R curve in the diagram such that it just meets the tangency criterion with a certain applied cyclic crack driving force curve. When this crack driving force corresponds to the plain fatigue limit of the material, ai will be the size of the crack that, after some growth, just would arrest at this stress level. The initial crack size is regarded as a material parameter (which also represents its maximum possible value). Its determination is schematically illustrated in Fig. 11. The cyclic crack driving force ΔKp (derived from ΔJ, see Section 2.3a) is determined for the stress level of the plain fatigue limit of the material. The simulated specimen is a plain sheet with a semi-circular surface crack subjected to tension. The size of the crack is stepwise increased in the analysis with the aspect ratio of the crack being constant at a/c = 1, e.g. the crack is semi-circular. Next, the cyclic R curve, known from the experiment, is plotted in the diagram. It is fixed at ΔK = ΔKth,eff as the ordinate and shifted along the abscissa until the tangency criterion between crack driving force and R curve is met. The initial crack size ai then refers to the origin of the cyclic R curve what is in line with the discussion provided in Section 2.3. Note that the effect

Fig. 11. Cyclic R-curve analysis for determining the initial crack size as a material parameter according to Zerbst and Madia [55]. 194

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the problem becomes less severe when the secondary notch depth (in the present case, the roughness) is added to the initial crack size. 3.3.3. Specification by microstructural features Comparable to the method in Section 3.3.2 the definition of the initial crack size is related to the plain fatigue limit of the material, however, it is thought to be controlled by the arrest of the largest microstructurally short crack at a microstructural barrier (Section 2.2). In the Kitagawa-Takahashi diagram of Fig. 5 this refers to a crack depth d. Chapetti et al. [59] define d as the position of the strongest microstructural barrier from the surface. This can be the ferrite size, the bainite or martensite lath length, etc. Features such as the distance between inclusions, precipitates or second phases can play a role as well. For typical low carbon steel, the authors report values for d in the order of 50 μm. Back calculation from Table 1 in [60] provides values of d = 7.8 to 64 μm for 7 different steels. The determination of d is undoubtedly a major challenge of the approach. In [59] examples are provided for ferrite, ferrite-bainite and bainite-martensite materials based on optical measurements of arrested cracks at the specimen surfaces comparable to what is seen in Fig. 2 of the present paper. The initial crack is then assumed to be semi-circular in shape. For ferritic steel, Chapetti, et al. [59] assume the grain boundaries to be the principal crack growth barriers and, as a consequence, they equate d with the mean grain size (in that case 38 μm). For comparison: the mean grain size of the S355NL steel (base material) has been determined in the order of about 11 μm based on comparative images. An impression of the grain structure of the material is provided in Fig. 14a. Note, however, that the mean grain size determined this way is meaningless since the grain size generally tends to be underestimated on a metallographic section. Assuming the grains as spheres, the real size will be obtained when the grain is exactly cut off in the middle. If it is cut above or below this plane, it appears smaller than it really is, Fig. 14b. As a consequence, only the larger grains on the metallographic image should be used for sizing. For the S355NL steel, this points to a real grain size in the order of 13–21 μm or 15–21 μm when only the 10% or 5% largest grains (out of 860) are measured and the area is converted to an equivalent circle diameter.

Fig. 12. Initial crack sizes and crack sizes at crack arrest obtained by the crack arrest model of [12] for low strength and high strength steels (here: base metals of S355NL and S960QL; according to [56]). In the present example, the fatigue strength has been estimated from the ultimate tensile strength as σw = 0.45 Rm.

toe (Fig. 12) to the initial crack depth ai obtained from the R curve analyses at the plain specimen. Comparing the ai of the material S355NL in Fig. 12 with the depth of the secondary notch (the roughness) of the same plate in Fig. 10, it turns out that the latter is more than three times larger. In other words: When the notch is added to ai it contributes three quarters to the initial crack depth finally used for the cyclic R curve analysis. Although a secondary notch “crack” might be in the order of a few tenths of a millimeter or even more, the micro-crack at its root will behave like a physically short crack in that it shows gradual build-up of the crack closure phenomenon. This brings up a last point: The definition of ai by crack arrest just provides a lower bound. One can easily imagine that this loses its significance in the presence of a much larger defect, e.g., macroscopic lack of fusion in a weldment. This is a trivial statement and the analysis for such a large defect belongs to a classic damage tolerance investigation (see Section 1) rather that to a total lifetime or fatigue strength approach. However, there will also be examples at a much smaller length scale where the initial crack size should exceed the crack size at arrest. As is mentioned above, the critical initial crack size based on the arrest argument becomes smaller for higher strength of the material (see Fig. 12), and at a certain point it will become smaller than the size of pre-existing defects, e.g. non-metallic inclusions or pores. In such a case, the defect size will take over the role of the initial crack depth. This explains the observation that non-metallic inclusions strongly affect the plain fatigue strength of high-strength steels whilst this influence disappears for low-strength ones [3]. However, besides the ability to explain the mentioned empirical observation, these considerations create a serious problem since the defect size, e.g., the size of inclusions (or inclusion stringers) has experimentally to be determined and it certainly will show some scatter not only with respect to its size but also with respect to its location in the highly stressed region. The size distribution of non-metallic inclusions in the weld toe regions of butt welds of S355NL steel is provided in Fig. 13. As can be seen, even for this moderate strength steel, the size of the non-metallic inclusions scatters around the ai values obtained for plain specimens, in fact it is even larger, particularly when the upper bound is considered based on a weakest link argumentation. On the other hand, no information is provided on the statistics of the location of the defects with respect to the surface of the weld toe, except that all data have been taken from the square areas marked in the figure. Again,

3.3.4. Definition by the process zone size Another philosophy is followed by Ostash and Panasyuk [61,62] who proposed that the microstructurally short crack transforms into a mechanically/physically short one (note that the authors themselves use another nomenclature) when it overcomes the boundary of the process zone. The authors’ definition of that zone, d*, which is consequently assumed to be identical to the initial crack size ai for plain specimens and the way they determined it is illustrated in Fig. 15. They showed the distance d* to be independent of the notch root radius ρ, and

Fig. 13. Non-metallic inclusions in the weld toe areas of butt welds of S355NL steel. (a) Metallographic section; (b) histogram of the area equivalent diameter of a circle; (c) extreme value statistics of inclusion diameters; according to [54]. 195

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Table 1 Initial crack sizes ai (of semi-circular cracks, a/c = 1) obtained by the different proposals discussed in Section 3. Proposal of

section

Background

ai in μm

Present authors Chapetti Ostash & Panasyuk Yates & Brown Tanaka & Akinawa Usami & Shida

3.3.2 3.3.3 3.3.4 3.3.5.2 3.3.5.3 3.3.6

Crack arrest consideration Grain size as microstructural obstacle for crack propagation Process zone size El Haddad’s a0 as upper bound As above, but modified for ΔKth,eff Constant cyclic plastic zone size at σw

10–27 (exp. value: 17) 13 (15)−21 14.75 ± 0.75 133–218 13.7–15.2 10.1–21.4

Fig. 14. Microstructure of S355NL steel (base material); (a) Metallographic cut: B. Schork, MPA Darmstadt; (b) Metallographic grains sizes (in the figure pictured as spheres) tend to be too small compared with reality.

Fig. 17. Fatigue crack propagation thresholds determined by different methods at different R ratios, steel S355NL (base metal), according to [11,50].

Fig. 15. Process zone definition d* according to [60,61].

Fig. 18. Definition of a modified fititious crack length a0* in the KitagawaTakahashi diagram according to Tanaka and Akinawa [78].

estimated it by

ΔKth, eff ⎞2 d* = 1.25·β 2·⎛ ⎝ Δσw ⎠ ⎜

⎟

(7)

with Δσw being the plain fatigue limit of the material, and the constant β is given as 0.7 for steels and 1 for aluminium alloys. In the case of the S355NL base plate steel, ΔKth,eff = 2.7 MPa m1/2 and Δσw (R = −1) = 550 ± 14 MPa [52] such that d* = ai ≈ 14.75 ± 0.75 μm. As in Sections 3.3.2 and 3.3.3, this value has still to be added to the secondary notch depth k. Fig. 16. Definition of the fictitious crack length a0 in the Kitagawa-Takahashi diagram.

3.3.5. Critical distance methods 3.3.5.1. Smith’s and El Haddad’s fictitious crack length a0. Further approaches for determining the initial crack size are based on the socalled critical distance methods. The basic idea is taken over from pioneering works of Neuber [63] and Peterson [64] who hypothesized 196

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apply Eq. (8) to limited surface cracks, some authors (e.g. [59,71,72]) modify it by introducing a boundary correction factor F which, in the case of a semi-circular crack in a plate subjected to tension is F = 0.728 [73]: 2

a0 =

an average stress over a characteristic structural length or a punctual stress at a given distance from the notch root, respectively, as the parameter controlling fatigue damage. In 1983, Tanaka [65], for the case of a sharp crack, proposed that the critical distance in Neuber’s approach equals 2a0 with a0 being the fictitious crack length introduced by Smith [66] and El Haddad et al. [58] for the empirical description of short fatigue crack propagation. Others, e.g. Taylor [67,68] and Lazzarin et al. [69], followed this approach. In the frame of a Kitagawa-Takahashi diagram, a0 is determined as the crack size at the intersection of the straight lines for the fatigue limit and the long crack propagation threshold in double-logarithmic scaling, Fig. 16. Although it is possible to determine the Kitagawa-Takahashi diagram and, based on this, a0 in a direct way by testing specimens with defined cracks (or narrow notches), e.g., [70] it is usually determined by

(9)

3.3.5.3. The modified proposal of Tanaka and Akinawa. Note that a0, as commonly defined, empirically includes the elastic-plastic magnification of the cyclic crack driving force as well as the gradual build-up of the crack closure phenomenon because it belongs to the transition range of the Kitagawa-Takahashi diagram. In order to avoid this, Tanaka and Akinawa [78] proposed the determination of a modified parameter a0′ based on the intrinsic threshold ΔKth,eff, Fig. 18: 2

a0′ =

2

⎜

⎟

3.3.5.2. The proposal by Yates and Brown. Yates and Brown [76], performing a simplified R curve analysis, came up with the assumption of a0 being used as the maximum depth of a nonpropagating crack or, in other words, as an upper bound of the initial crack size ai. This is realized, e.g., in [77]. According to Fig. 16, a0 is always larger than d, i.e., it will, at the very most, provide a conservative estimate of the initial crack size d = ai and there is, of course, the question of the extension of the conservatism. Applying the proposal to the S355NL data in line with the CPCA long fatigue crack propagation thresholds in Fig. 17, the result is ai ⩽ a0 = 133 to 218 μm which still would have to be added to the secondary notch depth k. Compared to the other proposals discussed in this section, this is significantly larger by a factor of about ten, a conservatism that seems not to be acceptable.

Fig. 20. Variation of the weld toe radius ρ and the flank angle α along the toe of a butt weld of steel S355NL; according to [54], see also [11].

1 ⎛ ΔKth, LC ⎞ π ⎝ Δσw ⎠

⎜

It is certainly a problem that, frequently without explicit labelling, both Eq. (8) and Eq. (9) are used for a0 determination. Although the values can easily be converted into each other, care has to be exercised when literature data or compendia (e.g. [74]) are applied. Besides this, there exists, however, a more fundamental difficulty in the context of a0 determination which refers to the input parameters Δσw and ΔKth,LC in Eqs. (8) and (9). Particularly the latter might be a problem. Although there exist test standards, a large variety of fatigue crack propagation threshold values is found in the literature as complained, e.g, in [75]. In [19] the authors have proposed a coefficient of variation for the long crack threshold ΔKth,LC in the order of 0.15 based on a limited volume of literature data. Furthermore, for materials prone to corrosion, the standard load reduction method may yield no-conservative threshold values as is shown in Fig. 17 for S355NL steel. The reason is oxide debris-induced crack closure (Section 2.3) acting at the low load reduction gradients which are required in order to avoid premature plasticity-induced crack closure [19]. Note that, for the comparative example of S355NL steel, only the ΔKth values obtained by the CPCA method have been used. The acronym stands for “compression pre-cracking constant amplitude” The initial crack has been generated in compression pre-cracking, i.e. the maximum as well as the minimum K factor in the loading cycle were below zero this way suppressing the build-up of any crack closure effect before the test. The three ΔKth,LC values at R = −1 obtained this way were between 8.4 and 10.2 MPa m1/2. In general, it has to be stated that any uncertainty, scatter as well as systematic differences, in the fatigue crack propagation threshold has a significant effect on a0 since this correlates with the square of the latter.

Fig. 19. Multiple crack propagation along the toe of a butt weld, steel S355NL; according to [54], see also [11].

a0 =

1 ⎛ ΔKth, LC ⎞ π ⎝ F ·Δσw ⎠

1 ⎛ ΔKth, eff ⎞ π ⎝ F ·Δσw ⎠ ⎜

⎟

(10)

⎟

(8)

For the S355NL steel the a0′ value is found to be between 13.7 and 15.2 μm.

for R = −1. For arbitrary R values, Δσw is replaced by Δσe. In order to 197

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Fig. 21. Stochastic determination of the finite life S-N curve and the fatigue limit such as that carried out in [11,12].

21.4 μm. 4. Summary of the results for the comparative example In the following, the ai values for the comparative example (plate of S355NL steel subjected to tension, semi-circular crack) obtained by the different proposals discussed in Section 3.3 are briefly summarized, Table 1. Note that the secondary notch depth k (mean value: 55 μm, range between 30 and 100 μm; in the present case from the surface roughness, Section 3.3.1) would have to be added to this in a statistical way (such as will be described in Section 4) for a fatigue life or strength analysis. Indeed, this will be done in the example provided in Section 5. It is evident that, with the exception of the a0 value used as an upper bound to ai by Yates and Brown (Section 3.3.5.2), all results point to the same order of ai = 10 to 25 μm with a mean value in the order of 15 μm which is also in line with what is empirically known about the “critical defect size” below which defects (in that case, non-metallic inclusions) will show no effect any more on the fatigue limit, see Section 3.1. Fig. 22. Modelling multiple crack propagation and coalescence along the toe of a butt weld, steel S355NL, according to [12].

5. Multiple crack propagation In engineering structures, multiple cracks are common features not only at the micromechanically short but also at the mechanically/ physically and sometimes even at the long crack stages. Typical examples are cracks along weld toes, cracks originating at corrosion pits, cracks initiated at pores and shrinkages, e.g., in cast alloys and multiple cracks along scratches, see the overview of the authors in [5,6]. Usually the reason is an interplay of (i) a varying stress concentration and local stress gradient with multiple peaks at different sites of the component and (ii) scattering material properties. An example is provided in Fig. 19 where fatigue cracks along the toes of butt welds of S355NL steel are visualized by heat tinting after 23–33% of the overall lifetime. The widths of the plates, i.e., the lengths of the weld toes, B, were 50 mm. As can be seen, the number of cracks increases with higher stress level. At a lower stress amplitude of 100 MPa there is the transition between none and just one propagating crack. This is in line with

3.3.6. The proposal of Usami and Shida Usasmi and Shida [79] proposed a fatigue limit criterion based on the cyclic plastic zone ahead of the crack tip (the size rpc). They postulate rpc to be constant at the stress level of the fatigue limit in structures with small cracks of different sizes, and determine it by

rpc =

π ′ ]2 [Kmax, th/ σyc 32

(11)

They obtain an equivalent (initial) crack depth ae as

π − 1⎤ ae = rpc (w)/ ⎡sec ⎢ ⎥ ′ 2(2 σ / σ max, w + R ) YC ⎣ ⎦

(12)

Applying Eqs. (11) and (12) to the present comparative example of S355NL steel, the initial crack depth ai = ae ranges between 10.1 and 198

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Fig. 23. Example for the determination of the fatigue strength by fracture mechanics using the approach of the authors [11,12]. The parameters μ and σ are the expected value and the standard distribution of a log-normal distribution of the fatigue limit.

all specimens fail. Between the two limit states, the number of run outs decreases with increasing stress level. This information is then used for generating the statistical distribution of the fatigue limit. In the procedure developed by the authors, a log-normal distribution is used for this.

the remarks about crack arrest and the fatigue limit in Section 2 because the 100 MPa stress level corresponds to the fatigue limit of these structures. The varying local stress concentration and stress gradient into wall thickness direction along the weld toe is caused by a varying local geometry mainly in terms of toe radius, weld flank angle and weld reinforcement, Fig. 20. Multiple crack propagation must be considered when fracture mechanics shall be applied to fatigue strength determination. Exceptions are cases with rather large initial crack sizes which dominate fatigue damage. Multiple cracks require a stochastic consideration. An example for this, which is realized in [12], see also [11], is provided in Fig. 21. The weld toe is split into equidistant sections. A local geometry is stochastically assigned to each of these sections in terms of toe radius, flank angle, weld reinforcement and secondary notch depth. In addition, each section contains an initial crack the size of which is also given by a statistical distribution. The sections are then fitted to specimens and the “residual” lifetime is determined by fracture mechanics. Depending on the stress level, some cracks will grow faster, others slower and others will even arrest. When the tips of two adjacent cracks touch, crack coalescence is assumed as is shown in the example of Fig. 22. Finally, one (edge) crack is left. Failure can be defined by monotonic fracture or another criterion, e.g., a certain size of the final crack with respect to the wall thickness. At each stress level the analysis is repeated several times. Since the specimens have been generated by merging ever new sections, each time another fatigue life is obtained. Repeating the same procedure at other stress levels, a scatter band of the finite life branch of the S-N curve is finally obtained. The determination of the fatigue limit (in the present case defined for 107 loading cycles) follows a slightly different philosophy in that the question to be answered at each stress level is whether the specimen fails within the 107 loading cycles or whether there will be a run out. Below a certain stress level there are only run outs. Above another one

6. Example An example which comprises all the aspects discussed above is provided in Fig. 23. For a detailed discussion see [11] and [12]. (a) The cyclic crack driving force is determined as an elastic-plastic parameter (ΔJ respectively ΔKp). (b) The gradual build-up of the crack closure phenomenon is modelled as is described in Section 2.3 based on an experimentally obtained cyclic R curve. (c) The depth of the secondary notch, k, (in the present case given by the roughness of the base plate near the weld toe, Fig. 10) is treated as part of the initial crack. The second component of ai is obtained by a crack arrest analysis such as is described in Section 3.3.2. (d) Both the finite life S-N curve and the fatigue limit defined for 107 cycles have been obtained by stochastic analyses such as described in Section 4, i.e., a multiple crack analysis has been carried out. Statistical input parameters were the initial crack size, the parameters of the weld toe geometry and the secondary notch depth in terms of the surface roughness. Failure was assumed for a depth of the final (edge) crack equivalent to one half of the wall thickness. Since most of the lifetime is spent at the short crack stage where the fatigue crack grows within the heat affected zone (HAZ), the input material data have been obtained for this material using thermally simulated all-HAZ microstructure specimens. With respect to this, see [52]. Note that these data are not identical to those of the base metal 199

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given in Section 3.3.1. Not any of the 31 validation examples of the study partly presented in [51] was of the same quality as that in Fig. 23. Particularly specimens tested at R ratios ≥ 0 and specimens with internal restraint (longitudinal gussets) showed some conservatism probably due to residual stress redistributions which were not adequately modelled. For a more in-depth discussion the reader is again referred to [11,12] and, specifically with respect to the residual stress issue to [80].

[4] Zerbst U, Madia M, Klinger C, Bettge D, Murakami Y. Defects as the root cause of fatigue failure of metallic components. Part I: Basic aspects. Eng Failure Anal 2019;97:777–92. [5] Zerbst U, Madia M, Klinger C, Bettge D, Murakami Y. Defects as the root cause of fatigue failure of metallic components. Part II: Non-metallic inclusions. Eng Failure Anal 2019;98:228–39. [6] Zerbst U, Madia M, Klinger C, Bettge D, Murakami Y. Defects as the root cause of fatigue failure of metallic components, Part III: Cavities, dents, corrosion pits, scratches. Eng Failure Anal 2019;97:759–76. [7] Zerbst U, Klinger C, Madia M. Material defects as cause of the fatigue failure of metallic components. Int J Fatigue 2019. [submitted for publication]. [8] Vincent M, Nadot Y, Nadot-Martin C, Dragon A. Interaction between a surface defect and grain size under high cycle fatigue loading: experimental approach for Armco iron. Int J Fatigue 2016;87:81–90. [9] Madia M, Thoffo Ngoula D, Zerbst U, Beier HTh. Approximation of the crack driving force for cracks at notches under static and cyclic loading. Struct Integrity Proc 2017;5:875–82. [10] Tchoffo Ngoula D, Madia M, Beier HTh, Vormwald M, Zerbst U. Cyclic J-integral: numerical and analytical investigations for surface cracks in weldments. Eng Fract Mech 2018;198:24–44. [11] Zerbst U. et al. Fatigue and Fracture of Weldments. The IBESS approach for the determination of the fatigue life and strength of weldments by fracture mechanics analysis. Book, Springer Nature, Cham, Switzerland; 2019. [12] Madia M, Zerbst U, Beier HTh, Schork B. The IBESS model – elements, realization and validation. Eng Fracture Mech 2018;198:171–208. [13] Zerbst U, Madia M. Analytical flaw assessment. Eng Fracture Mech 2018;187:316–67. [14] Vormwald M. Elastic-plastic fatigue crack growth. In: Radaj D, Vormwald M, editors. Advanced methods of fatigue assessment. Springer, Heidelberg et al. 2013:391–481. [15] Elber W. Fatigue crack closure under cyclic tension. Eng Fract Mech 1970;2:37–45. [16] Tanaka K. Fatigue crack propagation. In: Ritchie RO, Murakami Y, editors. Comprehensive Structural Integrity. Volume 4: Cyclic loading and Fracture. Elsevier 2003:95–127. [17] Ritchie RO. Mechanisms of fatigue crack propagation in metals, ceramics and composites: role of crack tip shielding. Mater Sci Eng 1988;A103:15–28. [18] Suresh S. Fatigue of materials. 2nd ed. Cambridge University Press; 2003. [19] Zerbst U, Vormwald M, Pippan R, Gänser H-P, Sarrazin-Baudoux C, Madia M. About the fatigue crack propagation threshold of metals as a design criterion – a review. Eng Fract Mech 2016;153:190–243. [20] Forth SC, Newman CJ, Forman RG. On generating fatigue crack growth thresholds. Int J Fatigue 2003;25:9–15. [21] Hertzberg RW. On the calculation of closure-free fatigue crack propagation data in monolithic metal alloys. Mat Sci Eng 1995;A190:25–32. [22] Pippan R, Riemelmoser FO. Modelling of fatigue growth: Dislocation models. In: Ritchie RO, Murakami Y, editors. Comprehensive Structural Integrity. Volume 4: Cyclic loading and Fracture. Elsevier; 2003. p. 191–207. [23] Pokluda J, Pippan R, Vojtek T, Hohenwarter A. Near-threshold behavior of shearmode fatigue cracks in metallic materials. Fatigue Fract Eng Mat Struct 2014;37:232–54. [24] Tabernig B, Pippan R. Determination of the length dependence of the threshold for fatigue crack propagation. Eng Fract Mech 2002;69:899–907. [25] Maierhofer J, Kolitsch S, Pippan R, Gänser H-P, Madia M, Zerbst U. The cyclic Rcurve – determination, problems, limitations and application. Eng Fract Mech 2018;198:45–64. [26] Herz E, Hertel O, Vormwald M. Numerical simulation of plasticity induced fatigue crack opening and closure for autofrettaged intersecting holes. Eng Fract Mech 2011;78:559–72. [27] Schlitzer T, Vormwald M, Rudolph J. Strip yield model application for thermal cyclic loading. Comput Mat Sci 2012;64:265–9. [28] Kitagawa H, Takahashi S. 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7. Summary The paper provides a general discussion on conditions which have to be fulfilled for fracture mechanics application to the determination of S-N curves and fatigue limits. Starting with an introduction into the various stages of fatigue crack propagation, different mechanisms of crack arrest are discussed: (i) arrest of microstructurally short cracks at microstructural barriers, (ii) crack arrest due to the gradual build-up of the crack closure phenomenon during the physically short crack propagation stage and (iii) crack arrest at notches. This is important at the background that the fatigue limit usually is understood as a crack arrest phenomenon whereby it has to be distinguished between the plain fatigue limit of the material and the fatigue limit of (usually notched) components. The description of the mechanically/physically fatigue crack propagation requires both the determination of an elastic-plastic cyclic crack driving force and the modelling of the gradual build-up of the crack closure phenomenon during that stage. Whilst the authors refer to earlier work with respect to the first aspect, they discuss the application of cyclic crack driving force curves to the latter. Cyclic R curve analyses can be used for the determination of the fatigue limit as well as for the determination of the initial crack size such as is demonstrated in Section 3.3.3. Some space is given to the initial crack size ai required in a fracture mechanics analysis where the authors, besides their own approach, discuss a variety of proposals in the literature. All methods were subjected to a critical discussion. In addition, they all were applied to a comparative example (a plate of S355NL steel subjected to tension). It showed up that, except of one approach which, however, aimed at providing an upper bound solution only, all results pointed to the same order of ai = 10 to 25 μm with a mean value of about 15 μm. These numbers were also in line with what is empirically known about the “critical defect size” below which non-metallic inclusions are known not to be harmful any more for the fatigue limit. In practical application, secondary notch depths (such as undercuts, roughness, dents, scratches, and corrosion pits) usually have to be added to the initial crack size ai. The size of this has also been provided for the comparative example and was used for a S-N curve determination of a butt welded joint in the present paper. The analysis was preceded by a discussion of the question when a notch should be treated as a stress concentrator and when its depth can simply be added to the initial crack depth. The last aspect refers to multiple crack propagation which usually has to be considered when fracture mechanics shall be applied to fatigue strength determination. This is only possible on a stochastic basis. The authors present a proposal on how this can be done, and use the mentioned example for illustration. References [1] Rennert D, Kullig E, Vormwald M, Esderts A, Siegele D. Analytical strength assessment of components made of steel, cast iron and aluminium materials in mechanical engineering. FKM Guideline, 6th ed. Forschungskuratorium Maschinenbau (FKM), Frankfurt/Main, Germany; 2013. [2] Miller KJ. The two thresholds of fatigue behaviour. Fatigue Fract Eng Mat Struct 1993;16:931–9. [3] Murakami Y. Metal fatigue. Effects of small defects and nonmetallic inclusions. Elsevier. Oxford; 2002.

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[40] [41] [42] [43] [44] [45] [46] [47]

[48] [49] [50] [51]

[52]

[53]

[54]

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