Engineering Fracture Mechanics xxx (2017) xxx–xxx Contents lists available at ScienceDirect Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech The cyclic R-curve – Determination, problems, limitations and application Jürgen Maierhofer a,⇑, Stefan Kolitsch a, Reinhard Pippan b, Hans-Peter Gänser a, Mauro Madia c, Uwe Zerbst c a Materials Center Leoben Forschung GmbH, A-8700 Leoben, Austria Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, A-8700 Leoben, Austria c Bundesanstalt für Materialforschung und -prüfung (BAM), Division 9.1, D-12205 Berlin, Germany b a r t i c l e i n f o Article history: Received 19 April 2017 Received in revised form 28 August 2017 Accepted 26 September 2017 Available online xxxx Keywords: Cyclic R-curve Fatigue crack propagation threshold Crack closure mechanisms Cyclic R-curve analysis Kitagawa-Takahashi diagram a b s t r a c t The so-called cyclic R curve, i.e. the crack size dependence of the fatigue crack propagation threshold in the physically short crack regime, is a key parameter for bringing together fatigue strength and fracture mechanics concepts. Its adequate determination is of paramount importance. However, notwithstanding this relevance, no test guideline is available by now and only very few institutions have spent research effort on cyclic R curves so far. The aim of the present paper is to give an overview on the state-of-the-art. Besides an introduction into the basic principles, the discussion will concentrate on the experimental determination on the one hand and questions of its application on the other hand. Ó 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction The fatigue crack propagation threshold DKth is a function of the crack extension Da of physically short cracks. For this, the term ‘‘cyclic R-curve” has been introduced by Tanaka et al. [1,2] in the context of cracks growing from notches. Other authors prefer the term ‘‘fatigue threshold R-curve”, e.g. [3]). The cyclic R-curve is the key for linking together fatigue strength (and life) and fracture mechanics approaches; this is why it has a high potential both for better understanding fatigue phenomena and for improving fatigue life predictions. Beginning with a brief discussion of some basic aspects of the fatigue crack propagation threshold, the present paper will concentrate on the methodology for determining the cyclic R-curve, its relationship with the well-known KitagawaTakahashi diagram and aspects of its application to components. In parallel to giving a comprehensive overview of the basic methods for determining and applying the cyclic R-curve, some related problems and yet unresolved issues of these methods are discussed. Based on this overview and analysis of open issues, we finally suggest possible future research directions which could pave the path for a more widespread application of the cyclic R-curve in engineering practice. ⇑ Corresponding author. E-mail address: juergen.maierhofer@mcl.at (J. Maierhofer). https://doi.org/10.1016/j.engfracmech.2017.09.032 0013-7944/Ó 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 2 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx Nomenclature a ai a0 a⁄ b k li l90 y A E J K KCP Kmax Kmin Kop R U U(a) ULC DU/U0 W Y Da DaLC DJ DJeff,l DK DKeff DKth DKth,eff DKth,LC DKth,op Dr Drth Drth,0 mi qc r r0 crack length (crack depth for surface cracks) initial (closure-free) crack depth (for fracture mechanics analysis) El Haddad parameter (intrinsic length scale) in Eq. (8) correction term in the modified El Haddad model, Eqs. (14) and (15) exponent in the R-cure power law fit Eq. (6) constant in the R-curve fit Eq. (5) characteristic length for crack closure mechanism i in Eq. (7) characteristic length of crack closure build-up probe tip distance in DCPD measurements constant in the R-cure power law fit Eq. (6) modulus of elasticity (Young’s modulus) J-integral stress intensity factor (SIF, K-factor) maximum applied stress intensity factor in compression maximum stress intensity factor in a loading cycle minimum stress intensity factor in a loading cycle stress intensity factor at crack opening load ratio (stress ratio) (=Kmin/Kmax or rmin/rmax) crack closure ratio (=DKeff/DK) crack closure ratio for mechanically short cracks crack closure ratio for long cracks (independent on crack length) normalized electrical potential in DCPD measurements specimen width geometry function in Eq. (9) crack extension crack extension at the transition to the long crack regime in Eq. (6) cyclic J-integral effective local cyclic J-integral K-factor range (=Kmax Kmin) effective K-factor range (=Kmax Kop) fatigue crack propagation threshold intrinsic fatigue propagation threshold fatigue propagation threshold in the long crack regime contribution of crack closure effects (extrinsic effects) to DKth stress range (=rmax rmin) endurance limit stress range endurance limit stress range of a smooth specimen weighting factor for crack closure mechanism i in Eq. (7) plastic zone size stress material yield strength Abbreviations CP compression pre-cracking DCPD direct current potential drop eff effective i initial l local LC long crack op opening p plasticity corrected SIF stress intensity factor (K-factor) th threshold Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx 3 Fig. 1. Cyclic R-curve for the threshold of the stress intensity factor range DKth as a function of the crack extension Da, schematic diagram. 2. The threshold against fatigue crack propagation DKth 2.1. General aspects The fatigue crack propagation threshold DKth is the stress intensity factor at which a crack, depending on the point of view, arrests or starts to grow under cyclic loading.1 Insofar it can be compared with the fatigue endurance limit, albeit in the presence of a crack. No detailed discussion on the threshold concept as such shall be provided here since this has already been given in [4]. What is important in the present context is the partition hypothesis, by which the parameter DKth can be split into two components: an intrinsic one, DKth,eff, which is independent of the crack length and crack extension, and an extrinsic one, DKth,op, describing all extrinsic (or closure) effects stemming from mechanisms acting via the crack flanks2: DK th ¼ DK th;eff þ DK th;op : ð1Þ Based on Eq. (1), the intrinsic component DKth,eff can also be defined as the threshold which is obtained in the absence of any extrinsic (or closure) effects from the crack flanks, i.e. by testing at high load ratios R (=Kmin/Kmax), usually at R P 0.7 or higher [5]. Since the crack closure phenomenon requires a certain size of the crack wake, it will not be existent and DKth,op will be zero when a crack starts to grow from a small pre-existing crack-like defect (typical metallurgical or processing defects) in a nominally defect-free component, as it is frequently the case in conventional fatigue of engineering materials. When the crack propagates, DKth,op will gradually increase with the crack extension Da until it reaches a stabilized value at which the closure effects become independent of the crack size a or the crack extension Da, respectively. This is illustrated in Fig. 1. 2.2. Crack closure effects ‘‘Crack closure effect” means that the opposite faces of a crack come into contact during unloading. There are various mechanisms responsible for this [6–8]: (a) The plasticity-induced crack closure effect stems from the fatigue crack propagation through the plastically deformed zone ahead of the crack tip. While the crack tip leaves the initial plastic zone behind itself, it continually creates new plastic deformation in front of itself. In this way, a continuous plastically deformed zone behind the crack tip – the socalled ‘‘plastic wake” – is generated. (b) The oxide-debris-induced effect is due to an oxide layer which ‘‘clogs” the crack. At low load ratios the crack faces, again and again, are locally ‘‘furbished” due to friction with the effect that the broken-up oxide layer grows to a debris layer the thickness of which is in the order of the crack tip opening displacement at loads near the threshold DKth. Of course, this effect strongly depends on the oxidation behaviour of the material and the environmental conditions. (c) The roughness-induced crack closure effect, by its nature, is a geometric mismatch effect between the corresponding crack faces due to microscopic mixed mode crack extension when the crack is deflecting from its plane as a consequence of local shearing. The effect is strongly influenced by different crystallographic orientations of the grains and by the microstructure in general, e.g., larger grains can contribute to rougher crack surfaces. It might be magnified by effects such as crack kinking and branching. 1 In the present context, it is always assumed that the crack is loaded in Mode I. Sometimes the extrinsic component is also designated by DKshielding, which is somewhat more general since it contains also crack bridging, e.g. in case of composite materials or lamellar microstructures, and geometric shielding effects, e.g. in case of intergranular/interlamellar crack fronts. 2 Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 4 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx Note that, whilst plasticity-induced crack closure is observed throughout all three regimes of fatigue crack growth regardless of the applied SIF DK, the oxide debris and roughness-induced effects dominate the near-threshold range due to the small crack tip displacements. Note further that, in addition to the aforementioned crack closure mechanisms, there might be further phenomena from case to case, e.g., trapped viscous liquids in the crack wake or deformation-induced phase transformations. 2.3. The intrinsic and extrinsic threshold components DKth,eff and DKth,op According to our present knowledge, the intrinsic threshold component DKth,eff almost completely depends on the crystal lattice of the metallic material and the elastic properties which, on the macro scale, are characterized by the modulus of elasticity E as an indirect measure of the strength of the metal bonding and the lattice distortion in terms of the Burgers vector. Most notably, it is independent of the microstructure and the parameters controlled by this such as the yield strength, see, e.g. [9]. For many materials a meaningful correlation is provided by the simple equation DK th;eff 1:6 105 E ð2Þ with DKth,eff in MPa m1/2 and E in MPa [9], see also [10]. Based on their experience, the authors of the present paper would prefer a slightly lower factor of 1.3 105 instead of 1.6 105 in Eq. (2), cf. [11]. Several mechanisms governing the intrinsic threshold have been proposed. They are: (1) no dislocation is emitted from the crack tip [12]; (2) new fracture surfaces generated due to crack blunting by dislocation emission during loading do not reweld during unloading when the dislocation moves back to the crack tip [13,14]; and (3) accumulated dislocation structures do not give rise to internal stresses high enough to break the bond [15,16]. All these model hypotheses lead to a threshold value proportional to Young’s modulus or shear modulus. In contrast to the intrinsic threshold, the extrinsic threshold component DKth,op is strongly affected by the microstructure of the material as well as by external parameters such as load ratio, type and sequence of loading, and environmental issues. Most of these influences can be explained by crack closure effects. For instance, the correlation between DKth and the yield strength is an indirect rather than a direct relation, as both roughness-induced crack closure and yield strength depend causally on the grain size. Besides the crystal orientation and the microstructure of the material, other factors which control DKth,op are the load ratio R, mixed mode and variable amplitude loading, residual stresses and environmental factors such as corrosive environments and temperature (for a detailed discussion see [4]), and – most importantly within the present context – the crack extension as shown in Fig. 1. 3. Crack size dependence of the fatigue crack growth threshold – cyclic R-curve 3.1. Experimental determination 3.1.1. Basic procedure The method described below was mainly developed at the Erich Schmid Institute of Materials Physics in Leoben [17–20]. In a first step, fatigue pre-cracks are generated at a sharp notch by compression pre-cracking, i.e., by loading the specimen completely in the compressive stress range. The background to this is described in a number of papers [21–25]. In principle, Fig. 2. Development of the plastic zone and residual stresses near the notch root in cyclic compression loading according to [20]: (a) first loading cycle; (b) after small crack extension; (c) at crack arrest. Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx 5 there would be no crack extension when both minimum and maximum load in a cycle are below zero. However, because of the gap width of the notch in the specimen (the notch is usually machined by electric discharge machining (EDM)) there will be a kind of a ‘‘scissors effect” ahead of its tip. Due to this, a spatially confined tensile residual stress field will be built up during unloading so that the crack is kept open, which enables subsequent crack propagation. This is illustrated in Fig. 2 (a) for the case of elastic-perfectly plastic material behaviour. As under monotonic loading, a plastic zone forms ahead of the crack; however, the monotonic plastic zone is now accompanied by a smaller cyclic plastic zone formed due to the load reversals. The size of the monotonic plastic zone controls the depth of the residual stress field in the ligament direction. Since it is roughly inversely proportional to the square of the yield strength, there will be marked differences with respect to materials of different strengths. As will be discussed in Section 3.1.2 (b), this can constitute problems for applying the method to high strength materials, as the crack resistance will be built up within a very short crack extension (cf. Fig. 15) such that the crack tip still might be influenced by the notch strain field. Note that at constant load amplitude and when no cyclic creep takes place, the initial monotonic plastic zone stays unaffected during the pre-cracking process, and so does the spatial extension of the residual stress field. In contrast, the cyclic plastic zone becomes increasingly smaller with increasing crack extension and shifts toward the outer edge of the monotonic zone, Fig. 2(b). Crack propagation stops when the crack tip approaches that position where the residual stresses disappear Fig. 2(c). In the usual cyclic compression pre-cracking method, which exclusively aims at the determination of long crack threshold values [26,27], the crack, after it has been generated this way, is propagated to a certain crack extension by constant amplitude loading at that stress ratio at which further testing will take place. This is done in order to eliminate any potential transient crack propagation behaviour due to the prior compressive loading, and to guarantee stabilized crack-opening stresses in the near-threshold regime. Another procedure is adopted when the cyclic R-curve has to be determined. Instead of performing the ‘‘stabilization phase”, the tests immediately follow the compression pre-cracking. This principle is illustrated in Fig. 3. At a chosen stress ratio R the load is stepwise increased, with the cyclic stress intensity factor DK (=Kmax Kmin) being in between the intrinsic (closure-free) and the long crack value (with fully developed crack closure effects). No crack advance will be noticed as long as DK 6 DKth,eff. For DKth,eff < DK 6 DKth,LC (with ‘‘LC” standing for ‘‘long crack”) the crack will initially grow, but after some propagation Da, it will arrest due to the gradual build-up of closure and/or crack bridging effects [17,18,20]. Finally, when DK > DKth,LC, the long crack threshold is exceeded; no crack arrest will occur any more, and the crack will propagate until failure. The cyclic R-curve is finally generated by connecting the points (Da, DK) of the consecutive arrest events. Whilst the basic principle is straightforward, a number of problems arise in practical application. The most important of these will be in the focus of the next section. 3.1.2. Details, problems and demands regarding the experimental technique Provided that specimen orientation, material microstructure, heat treatment and manufacturing processes are comparable, one has to take three points into account in order to determine a material specific and reproducible cyclic R-curve – or, in other words, to keep load history and geometry effects on the R-curve as small as possible. These points are related to (a) crack initiation, (b) notch geometry, and (c) testing procedure. Fig. 3. Loading scheme of cyclic R-curve determination according to [20]. Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 6 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx From the authors’ experience, varying specimen geometries and/or testing rigs are expected to have no influence on the determination of a cyclic crack resistance curve – provided one takes appropriate care of these issues, as will be discussed in detail in what follows. (a) Crack initiation One problem is that there still exists a limited region of tensile residual stresses at the crack tip after compression precracking such as schematically illustrated in Fig. 2(c). Since it cannot be excluded that these tensile stresses will affect the final cyclic R-curve [20], they have either to be eliminated, e.g. by heat treatment after pre-cracking [24] – which is only possible in very few materials without changing the microstructure – or kept as small as possible. For cyclic R-curves not to be affected by the pre-cracking conditions, it is necessary to keep the cyclic stress intensity factor DK during compression precracking as small as possible [28], which in turn requires a very sharp notch root radius. In [20] the authors obtain notch tip radii in the order of 10–20 lm by a razor blade grinding technique. An example of a crack tip obtained in this way is shown in Fig. 4. This technique was also applied in [29]. Note that usually a commercial razor blade made of stainless steel is used to this purpose, which may represent a limitation in case of hard materials. In this case, ceramic blades may be adopted. Note, however, that due to wear the blades should be often replaced to guarantee a sharp notch tip. As the razor blade must grind the complete crack front, it is always aligned with the notch direction and exactly perpendicular to the specimen loading direction. The pressure force must be kept as low as possible to ensure that the notch root is ground by the diamond polishing compound, and not plastically deformed by the razor blade; cf. Fig. 4. This, usually, requires some equipment. A systematic study the aim of which is to formulate rules is currently in progress at BAM. Fig. 5 demonstrates the effect of the pre-cracking stress intensity factor range DK on the cyclic R-curve for a 359 T6 cast aluminium alloy. The specimens were prepared in compression pre-cracking with a load ratio of R = 20 and the pre-cracking p SIF range DK varying from 4 to 16 MPa m. As can be seen, the measured R-curves become steeper with lower pre-cracking p DK due to the reduced plastic zone from pre-cracking. For pre-cracking at and below 8 MPa m the curves coincide, which indicates that pre-cracking has no noticeable influence on the R-curve any more. (b) Notch geometry It has already been mentioned that the determination of cyclic R-curves can be problematic for high strength materials. The reason is that the monotonic plastic zone due to the first loading cycle in compression pre-cracking, which controls the spatial expansion of the tensile residual stress field and hence the size of the crack until arrest, might become as small as some tens of lm for yield strengths in the order of 1000 MPa and above, following the equation p jK CP j 2 qc ¼ ; 8 r0 ð3Þ where qc is the extension of the plastic zone, KCP is the maximum applied stress intensity factor in compression and r0 is the material yield strength. The crack must reach a certain minimum extension Da to avoid notch effects on the stress intensity factor, i.e., for the standard SIF solution to be valid; in [26], for C(T) specimens with a specimen width of W = 76 mm and a notch flank distance of 2.5 mm, Da = 0.5 mm for a notch angle of 60° (and a somewhat smaller value for 45°) were reported. Note that the angle between the notch flanks generated by the razor blade grinding method is much smaller (see Fig. 4). ASTM standard E647-15 [30] requires a notch envelope starting at the tip of the fatigue pre-crack tip with an angle of 30° and a maximum notch flank distance of W/16. So, as a simple rule complying with the standard it can be recommended that the compression pre-crack is Fig. 4. Notch tip generated by razor blade grinding. The specimen was moved with respect to the razor blade, which was loaded by a force as low as possible (ca. 0.2 N). The removal of the material should be achieved by lubrication, usually provided by a 1 lm diamond polishing compound; according to [20]. Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx 7 Fig. 5. The effect of different loading conditions DK in cyclic pre-cracking on the resulting cyclic R-curve, material: 20 vol% SiC particle reinforced 359 T6 cast aluminium alloy, test at R = 1 after compression pre-cracking at R = 20; according to [20]. Fig. 6. Geometrical consideration of the notch and the (compression) pre-crack. The pre-crack should be large enough to avoid interference between the crack tip and notch strain fields. For this, a 30° envelope should not intersect the contours of the notch [30]. The gap width of the notch should be as small as possible, e.g., in the order of 0.3 mm. large enough such that a 30° envelope from its tip does not intersect the contours of the spark erosion and razor blade notches at any point, Fig. 6 [30]. In addition, the notch width should not exceed one sixteenth of the specimen width [30], whereas the minimum pre-crack length requirements from [30] do clearly not apply to the determination of the Rcurve. Also the notch depth can influence the determination of the cyclic crack resistance curve. For constant load tests (where DK increases with the crack depth), the appropriate choice of the initial crack depth ai decides whether a particular region of the crack resistance curve can be determined at all. As the total crack depth a consists of the initial crack depth ai and the crack extension Da, higher stress ranges must be applied for specimens with smaller initial crack length ai to reach a desired SIF range DKth at a given crack extension Da, and the resulting load curves in the DKth vs. Da diagram are steeper. This is illustrated in Fig. 7, where load curves for SE(B) specimens with three different initial crack lengths are plotted; for the purpose of demonstration, two different specimen dimensions were used, one with a height of 20 mm (small SE(B)) and one with a height of 50 mm (big SE(B)). It is evident from this figure that the measurable regime of DKth vs. Da is determined by the gradient dDK/da in a stepwise increasing constant load amplitude test. The maximum measurable Da is determined by the osculation point of the load curve and the R-curve, and hence depends on the specimen geometry (mainly on the initial crack depth ai). It can be seen that for a short initial crack (small SE(B), ai = 1 mm) the maximum reachable point on the p p cyclic R-curve is around 12 MPa m, whereas for a longer crack with ai = 6.64 mm (small SE(B)) about 13 MPa m can be reached. To determine reliably the flat ‘‘tail” of the cyclic R-curve near the long crack threshold at large crack extensions Da, the specimen dimensions must be increased. Using a big SE(B) specimen with an initial crack length ai = 12.64 mm, it p is possible to reach a threshold SIF range DKth of 14 MPa m, as shown in Fig. 7. For this reason, the probably best method to avoid the influence of initial crack depths ai on the determination of a cyclic R-curve is to use constant DK tests instead of constant load tests. A very common method for crack propagation measurement is the direct current potential drop technique (DCPD), where due to crack propagation the electrical resistance in the investigated specimen increases; the resulting change in the voltage is used to calculate the current crack length by using, for example, Johnson’s formula [31]. ( " py !#) cosh 2W DU 2W a0 a ¼ arccos cosh arccosh cosh 2W U0 p cos p2W py 1 ð4Þ Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 8 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx Fig. 7. Influence of notch depth on the determination of a cyclic R-curve. When using this method for crack length measurements, attention should be paid to the geometry of the notch and the arrangement of the probe tips, because inaccurate crack length measurements will unavoidably lead to inaccurate cyclic Rcurves. To avoid influences from the notch geometry, the width of the notch should be as small as possible in comparison to the notch depth. To achieve a notch width as small as possible, the probably best method is to create the notch by electric discharge machining (EDM). The distance 2y of the probe tips measuring the electrical potential drop DU also influences the accuracy of Johnson’s formula. To show the influence of different probe tip distances, finite element (FEM) simulations of the stationary electric potential field were performed for a 4 mm deep and 0.3 mm wide notch with half-distances of the probe tips y = 1.5, 1 and 0.5 mm, and the potential difference between the probe tips was compared with the prediction from Johnson’s formula. The investigation shows that there exists a significant difference between FEM results and Johnson’s formula for probe tip distances of 2y = 3 mm whilst the difference becomes smaller with decreasing y and is negligible for tip distances 2y 1 mm (Fig. 8). Fig. 8. Determination of crack extension for an EDM notch (0.3 mm width) from DCPD measurements. Comparison between finite element (FEM) results and Johnson’s formula for different probe tip distances 2y. Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx 9 (c) Testing procedure Finally, also the execution of the testing procedure itself can have significant influence on the determination of a cyclic Rcurve. So load history effects like overloads (leading to an extended monotonic plastic zone, and hence to additional residual stresses, in front of the crack tip) or large cycle numbers at small loads (leading to oxide debris) should be avoided during specimen testing as well as unintended clamping faults, because these can significantly influence the shape and position of the cyclic R-curve in the DKth-Da diagram. In what follows, some specific issues regarding the start of the experiment as well as subsequent ongoing testing are addressed. (i) Start of an experiment During clamping a specimen in a testing rig it is crucial to avoid any additional bending moments and misalignment (e.g., of the test rig axes/cylinders); this is especially important – and difficult to avoid – at negative load ratios, where the loading changes its direction during each load cycle. Such undesirable additional stresses will influence the crack growth behaviour. In the near-threshold regime, where the R-curve experiments are conducted, the influence of clamping stresses is much stronger than in the Paris regime and will lead to wrong, possibly non-conservative, results. In the Paris regime, on the contrary, the applied load is much higher than potential bending stresses due to clamping; therefore, their influence in that regime is negligible. After carefully clamping the specimen, the next step to take care about is the power-up of the testing machine. Especially when using a resonant testing machine, load amplitudes significantly higher than the pre-set value can occur during that phase. Such overloads at the start of an experiment lead to an increase in plasticity-induced crack closure, which may cause significantly decreased crack growth rates or even premature crack arrest. To avoid such overloads due to transient overshoot of the control loop in the ramp-up phase of the experiment, it is recommended to start cyclic crack growth experiments by first applying the mean load and then applying a load amplitude which is quite a few times smaller than the targeted load amplitude. Subsequently, the load amplitude may be stepwise increased until its target value is reached. (ii) Ongoing experimental procedure When the cyclic R-curve is determined by a single specimen technique, all load increment steps are subsequently applied within one test that consequently can last then for days – probably time enough to fully develop the oxide debris layer of the oxide-debris-induced crack closure effect. Alternatively, the cyclic R-curve may be obtained by a multiple specimen method as proposed in [29]. The principle is identical to static R-curve testing, which can also be realized as single and multiple specimen testing. For the latter, one constant DK is applied for each test (respectively specimen) and the corresponding Da value is optically determined on the fracture surface after the specimen is broken open after testing. The cyclic R-curve is finally derived from the arrest points of all tests. Fig. 9 provides a comparison of single and multiple test results of S355NL steel at R = 1 performed within the IBESS cluster project, see [29,32]. As it can be seen, both curves coincide. Note that oxide formation did not play much a role for the material investigated. When applying the procedure to a material susceptible to oxide formation, the results could be different. Fig. 9. Comparison of cyclic R-curves obtained by single and multiple specimen methods for S355 steel. The tests have been performed on four-point bending specimens (cross section 10 20 mm2; notch + pre-crack depth: ca. 5 mm); according to [29,32]. Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 10 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx A further problem in the context of the extremely low crack propagation rates during the cyclic R-curve tests, when realized by the single specimen method, might be what could be called rest period effects which are otherwise known to affect the fatigue threshold as well as slow crack propagation, see e.g. [4]. During the stages of slow propagation or even nonpropagation, additional oxide could be built up with the consequence that the crack would not propagate in the next loading step when the load increase is not high enough. The effects would be overcome by higher load increments or by the multiple specimen technique, which is not affected by this kind of test history effects. A very important aspect in measuring the crack length by potential drop methods is the compensation of side effects, one of which is given by temperature variations during the tests. This is especially important during short crack growth, where a variation of few tens of microns in crack length due to temperature effects could yield significant differences due to the high gradient of the R-curve at small crack extensions (cf., e.g., Fig. 9). According to the experience of the authors of this paper, an oscillation of about 20–30 lm is very common due to variation of the electric potential given by temperature, which is very relevant for crack lengths of about tenth of mm (at and right after pre-cracking). The sources of measurement errors due to temperature are [33]: (i) thermo-electric voltage related to differential temperature of the potential probes; (ii) temperature variations of the test specimen. In both cases a compensation is possible in order to achieve an acceptable accuracy. In particular, thermo-electric effects can be avoided by reversing the current source, whereas temperature variations of the test specimen can be excluded by measuring a reference electric potential on the specimen remote from the crack. Alternatively, the temperature can be measured and its influence on the potential difference can be corrected by an analytical correction function. 3.2. Fitting the cyclic R curve To the aim of carrying out a so-called R-curve analysis for assessing component behaviour (see Section 5), the experimental cyclic R-curve has to be fitted by an appropriate equation. It turns out to be of high relevance to describe the initial curve section, i.e. the range of small Da values, with high accuracy [29]. Here, slightly different curve shapes can be of large effect on the crack arrest stress as the target parameter of the analysis. The authors found widely applied approaches such as the exponential law by McEvily et al. [34] DK th;op ¼ ½1 expðk DaÞ ðK op;LC K min Þ ð5Þ to provide a poor fit of the initial slope of the cyclic R-curve; this holds also with respect to the power law correlation p between DKth and the square root of the defect area for crack-like defects ( area approach) proposed by Murakami [35]. One problem is certainly that those approaches use one fit parameter only. Instead, the authors in [29] propose a simple two-parameter power law DK th ¼ DK th;LC ( A Dab for Da < DaLC 1 for Da P DaLC ð6Þ with A and b being fit parameters and DaLC setting the transition to the long crack regime. A further approach giving even more detailed control about the shape of the R-curve was proposed in [36], where the fit of the cyclic R-curve " DK th ¼ DK th;eff þ ðDK th;LC DK th;eff Þ 1 n X mi exp i¼1 # Da li ð7Þ is based on the physical idea that each closure mechanism requires a certain crack extension length li to build up completely (see Fig. 10). 3.3. Factors which affect the cyclic R-curve The factors which affect the cyclic R-curve are essentially those which influence the extrinsic or crack closure related components of the threshold DKth like the load ratio, the microstructure and the environment. In Fig. 11 the influence of different load ratios (R = 3, 1, 0.5) is shown for the steel 25CrMo4. The SIF range threshold for long cracks DKth,LC decreases with increasing load ratio until, for very high load ratios (R > 0.8), DKth,LC approaches the effective threshold SIF range DKth,eff. Also the initial slope of the R-curve is much steeper for smaller load ratios, which is a rather desirable condition for damage tolerant design, see Section 5. Also the microstructure of a material can have significant influence on the shape of the cyclic R-curve, as shown for different grain size in Fig. 12. Here two R-curves of ARMCO iron with different grain sizes (10 mm and 500 mm, respectively) were determined. Smaller grain size leads to higher strength, a smaller plastic zone and thereby to plasticity-induced crack closure which increases faster than in low strength materials. Therefore, the initial slope of the R-curve for ARMCO-Fe with 10 mm grain size is steeper than for ARMCO-Fe with 500 mm grain size. On the contrary, the influence of roughness and also of oxide-debris-induced crack closure (the rougher a surface is, the more fretting corrosion can occur) is almost negligible for the material with 10 mm grain size. However, for ARMCO-Fe with 500 mm grain size the effect of roughness- (and oxidePlease cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx 11 Fig. 10. Illustration of the build-up of different closure mechanisms, see [36]. Fig. 11. Example of cyclic R-curves for different load ratios R according to [36]. Fig. 12. Grain size dependence of the cyclic R-curve of Armco iron according to [37]. debris-) induced crack closure is enormous, so that finally the threshold SIF range for long cracks DKth,LC is significantly higher than for ARMCO-Fe with 10 mm grain size. Also the environmental conditions may influence the cyclic R-curve. Different gaseous or aqueous environments can lead to different thicknesses of crack flank oxide deposit and thereby to higher oxide-induced crack closure [38,39]. Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 12 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx 3.4. Selected examples of cyclic R-curves An example of a cyclic R-curve for the steel S355NL streel has already been provided in Fig. 9. In Fig. 13 the cyclic R-curve of a common quenched and tempered 25CrMo4 steel is shown for two different load ratios [40]. The crack arrest points were determined up to 4 mm crack extension in order to meet the short/long crack transition. As this crack length is much higher than the size of the crack tip plastic zone, this leads to the conclusion that the roughness- and oxide-induced crack closure mechanisms play a major role within this material; further investigations on these effects are reported in [41]. Fig. 14 shows the influence of particle reinforcement on the cyclic R-curve of an aluminium alloy [42]. In comparison with the unreinforced alloy the long crack threshold DKth,LC is significantly higher for the reinforced Al-alloy. Also the effective threshold SIF range DKth,eff and the initial slope of the R-curve are higher for the reinforced alloy; the effective threshold because of the increased Young’s modulus, cf. Eq. (2), and the extrinsic threshold contribution – visible in the slope of the R-curve and the long crack threshold – due to increased roughness-induced closure. After a crack extension of 0.5 mm the cyclic R-curve is built up completely. Finally, in Fig. 15 the cyclic R-curve of a high strength tool steel (S390 microclean; 60.8 HRC) is shown, where a long crack p threshold of approximately 7 MPa m is reached already at 0.3 mm crack extension. 3.5. Towards a master R-curve A proposal for a master R-curve describing the build-up of crack closure in a dimensionless diagram was ventured in [4] on the basis of experimental results for nine different steel grades and alloys at two different load ratios; a statistical evaluation of such a curve based on experiments for four different QT steels was proposed in [44]. Meanwhile, the database has Fig. 13. Cyclic R-curves measured on SE(B) specimens (material 25CrMo4 (EA4T), dimensions 50 mm 6 mm 250 mm, diameter of raw material 190 mm) [40]. Fig. 14. Cyclic R-curves of reinforced aluminium alloy 2124 + 17% SiC and of the unreinforced alloy 2124 in L-LT orientation at room temperature (load ratio R = 0.1) [42]. Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx 13 Fig. 15. Cyclic R-curve of a high strength tool steel (S390 microclean) – single specimen method [43]. Fig. 16. Master curve for cyclic R-curves of different materials obtained at different load ratios. been extended and is visually summarized in Fig. 16. The closure contribution to the threshold SIF, DKth,op = DKth DKth,eff, is normalized by the final closure contribution of the long crack, DKth,LC DKth,eff; the crack extension is normalized by that crack extension l90 of each individual R-curve where 90% of the closure have been built up. As the materials are vastly different, the contributions of plasticity-, roughness- and oxide- debris-induced closure will be also very different. Furthermore, the pre-cracking conditions of some experiments are not documented, which leads to some additional uncertainty for small crack extensions. The resulting large width of the scatter band makes the derivation of statistical tolerance bands appear a quite meaningless exercise at the moment; rather, further work is necessary to reduce the scatter. This can be done by trying to get master curves for each mechanism separately, which requires a more detailed understanding of the individual mechanisms. While plasticity-induced closure is already well understood [19,45,46], roughness-induced and, above all, oxide-debris-induced closure are still subject of ongoing research [41]. A second unresolved issue is the relation of the normalization parameters to other, more easily available, material parameters. For plasticity-induced closure this is certainly the yield strength and the resulting size of the plastic zone; for roughness-induced closure, typical microstructural sizes (e.g., grain size, pearlite packet size, inclusion size and spacing) appear as promising candidates; for oxide-induced closure, more research is urgently required. Overall, it must be concluded that the concept of a master R-curve, while theoretically appealing, is – for all the aforementioned reasons – still far from being practically applicable. Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 14 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx 4. Cyclic R-curve and Kitagawa-Takahashi diagram 4.1. Empirical description of the Kitagawa-Takahashi diagram The Kitagawa-Takahashi diagram [47] empirically combines the endurance limit stress range Drth of the S-N curve concept with the fracture mechanics based fatigue crack propagation threshold DKth when plotted against the depth a of a crack emanating directly from the surface (Fig. 17). The curve separates the ranges of crack arrest (enclosed) and non-arrest (above and right of it). The empirical fit of the midsection is provided by a term a0 which was introduced by El Haddad et al. [48], Drth ¼ Drth;0 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ a0 : a þ a0 ð8Þ This equation follows from the fundamental equation of LEFM DK th ðaÞ ¼ Drth ðaÞ Y pﬃﬃﬃﬃﬃﬃ pa ð9Þ if, instead of taking the actual threshold SIF DKth(a) – which depends on the crack size a due to closure effects, one introduces a fictitious intrinsic length a0 such that the long crack threshold SIF DKth,LC is obtained also for short cracks, DK th;LC ¼ Drth ðaÞ Y pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pða þ a0 Þ: ð10Þ For a technically smooth surface without a macroscopic crack, a = 0, the threshold SIF range is computed from the endurance limit stress range of the smooth specimen Drth,0 and the fictitious intrinsic length a0 as DK th;LC ¼ Drth;0 Y pﬃﬃﬃﬃﬃﬃﬃﬃ pa0 : ð11Þ Relating Eqs. (10) and (11) finally gives Eq. (8). Although attempts have been made to attribute a physical meaning to a0 (for a review see [5]) it is first of all simply a fit parameter obtained as the intersection point of the smooth specimen endurance limit Drth,0 and the LEFM threshold SIF, Eq. (9), leading to (cf. Fig. 19). a0 ¼ 1 p DK th;LC Y Drth;0 2 : ð12Þ In the present context, it is important to note that both the Kitagawa-Takahashi diagram and the cyclic R-curve describe the same phenomenon, namely the build-up of crack closure as described by Eq. (6). Therefore, by combining Eqs. (9) and (10) and replacing the crack depth a by the crack extension Da in order to facilitate application to cracks emanating from notches (see Section 4.2), also the Kitagawa-Takahashi diagram may be written as a cyclic R-curve DK th sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Da : ¼ DK th;LC Da þ a0 ð13Þ Fig. 17. Schematic illustration of the parameter to describe the effective driving force in different fatigue regimes [49]. Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx 15 However, this needs a little correction in that the threshold DKth is not zero at Da = 0 such as indicated by Eq. (13), but equals DKth,eff instead [50,51]. The correction can easily be done by introducing a further term a⁄ into Eq. (13), DK th ¼ DK th;LC sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Da þ a ; Da þ a þ a0 ð14Þ which is chosen such that it satisfies the condition DKth (Da = 0) = DKth,eff: 2 DK th;eff =DK th;LC 2 : 1 DK th;eff =DK th;LC a ¼ a0 ð15Þ This offers the possibility to estimate the cyclic R-curve from Kitagawa-Takahashi data from Eq. (14) such as proposed in [50]. The input parameters needed for this are the length scale a0 or, alternatively, the material’s endurance limit stress range Drth,0 and the long crack threshold DKth,LC, by which a0 can be determined from Eq. (11), a0 ¼ 1 p DK th;LC Y Drth;0 2 ; ð16Þ and the intrinsic threshold DKth,eff needed for the determination of a⁄. However, when a0 is taken from the literature, e.g. from sources such as [52], it must be taken into account that those values frequently refer to a value of the boundary correction function Y = 1 in Eq. (16) in order to make the parameter formally geometry-independent. Nevertheless, dealing with short cracks, the assumption of a semi-circular crack shape under tension loading is usually more meaningful, and for this Y = 0.728 [2]. The discrepancy has also to be observed when empirical estimates of a0 based on other parameters such as the yield strength for steels are used [53]. A comparison of cyclic R-curves estimated by Eq. (14) and experimentally determined values is provided in Fig. 18. Note that the application to load ratios other than 1 requires a mean stress correction. In the present example a Goodman-type approach [54] was used to this purpose. 4.2. Kitagawa-Takahashi diagram based on cyclic R-curve What is important in this context is that both the Kitagawa-Takahashi diagram and the cyclic R-curve describe whether a current crack under given loading conditions is able to propagate or not. However, as it has been shown in [42,55,56], the Kitagawa-Takahashi diagram is in many cases not conservative for cracks which have not built up their crack closure completely (short cracks). That implies that the region of non-propagating cracks in the Kitagawa-Takahashi diagram for physically short cracks is smaller than the prediction based on the El Haddad approximation (see Fig. 19). Instead of modifying the empirical length scale approach of Eq. (8) as done in Eq. (14), it has been proposed in [57] to rederive the Kitagawa-Takahashi diagram from its physical basis, i.e., from the build-up of crack closure. To this purpose, it is indispensable to depart from an exact mathematical representation of the R-curve. The DKth(Da) curve according to Eq. (6), describing the various mechanisms, has been used as a reasonably accurate approximation [36,57]. In addition, one has to account for the depth ai of the initial notch – where the fracture surfaces are not in contact and therefore no build-up of crack closure can occur – and the subsequent crack extension Da due to fatigue crack growth, giving a total crack length of a = ai + Da (this has not been an issue in the original work by Kitagawa & Takahashi [47] and El Haddad et al. [48], who have dealt only with cracks initiating at smooth surfaces, i.e., ai = 0 and a = Da). This leads to Fig. 18. Comparison of experimental cyclic R-curves of S355NL steel and estimates by Eq. (13). The tests have been performed on four-point bending specimens (cross section 10 20 mm2; notch + pre-crack depth: ca. 5 mm) [50]. Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 16 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx Fig. 19. Kitagawa–Takahashi diagram showing the areas of non-propagating cracks according to El Haddad and Chapetti, respectively [42,55]. Drth ðai ; DaÞ ¼ DK th ðDaÞ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ; pðai þ DaÞ Yðai þ DaÞ ð17Þ which can be visualized as a three-dimensional Kitagawa-Takahashi diagram (see Fig. 20) [51,57]. Obviously, ai and Da have different individual influences on the threshold stress range; i.e., it depends not only on their sum, the total crack length, whether a crack is in the region of non-propagating cracks or in the finite life region. This hypothesis has been substantiated by experimental results from constant load DK increasing tests (single specimen method), Fig. 21 [57]. 5. Application fields of the cyclic R-curve The major application field of the cyclic R-curve is cyclic R-curve analysis of components. This can be compared to Rcurve analysis in monotonic loading, in which the monotonic R-curve, e.g. in terms of K-Da or J-Da, is compared to crack driving force curves K-a or J-a. The aim is to determine those applied loads or stresses at which cracks growing hitherto in a stable manner become unstable. Likewise, in a cyclic R-curve analysis the limit between crack arrest and crack growth is determined. Note that, as in a monotonic analysis, besides the applied loading and the cyclic R-curve as a material property the initial crack size is a third parameter of the analysis. A more detailed discussion on the topic is found in [50,58] within this issue where cyclic R-curve analyses are carried out for different types of weldments. As mentioned in Section 3.3, the long crack threshold as well as the initial slope of the cyclic R-curve increase markedly with decreasing load ratio. For damage tolerant design, it is clearly desirable to arrive at high thresholds already at small crack extensions. If it is, due to given loading conditions, not possible to choose the load ratio of the external loads freely, the effective load ratio can be lowered by thermo-mechanical post-treatment processes reducing the tensile macroscopic Fig. 20. Three-dimensional Kitagawa–Takahashi diagram (threshold stress range Drth plotted against crack extension Da and notch depth ai) [57]. Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx 17 Fig. 21. Predicted crack extension compared with experimental results for different notch depths ai [57]. residual stresses (e.g., stress relief annealing) or introducing compressive macroscopic residual stresses (e.g., induction hardening, peening, deep rolling) and thereby leading to a higher resistance against crack propagation. In combination with R-curve analyses, further applications are conceivable. Materials could be evaluated with respect to their notch sensitivity, i.e. the ratio of the reduction of the fatigue limit due to surface roughness or notches to the according stress concentration factor. Indeed, the authors in [59] were able to simulate the dependence of the endurance limit on the ultimate tensile strength, taking into account different surface roughness values (notch sensitivity). Furthermore, the method could be applied to specify a limit defect size, e.g., a limit diameter of non-metallic inclusions, beyond which the fatigue strength will be a function of this and no longer be controlled by crack arrest. Note that the crack size at arrest becomes smaller with increasing material strength (as the authors demonstrate in [50]) whilst the difference in the inclusion particle size does not necessarily follow this trend. When the arrest crack size becomes smaller than the inclusion size (or its upper bound, since it has to be considered as a statistical parameter), the latter will define the initial crack size for fatigue growth. On the other hand, when the arrest crack size is larger than the inclusion size, it will take over this function. Quantifying these correlations could provide a target inclusion size for material development, i.e., an inclusion size below which no further improvement of the fatigue strength is to be expected and vice versa. Finally, there could be some potential in modelling loading sequence effects near the threshold. For instance, compression overloads should have the effect of flattening the crack faces, thereby reducing the plasticity- (and perhaps also the roughness-) induced crack closure effect. Its gradual build-up again should follow the same characteristics as during early crack propagation, which means that it could also be described by the cyclic R-curve, UðaÞ 1 DK th ðaÞ DK th;eff ; ¼ U LC 1 DK th;LC DK th;eff ð18Þ see [50]. The principle is schematically illustrated in Fig. 22. Fig. 22. Re-buildup of the (plasticity-induced) crack closure effect following a compression overload event, schematic view. (a) Effect on the crack closure parameter U, (b) as mirrored by the cyclic R-curve. Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 18 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx 6. Recommendations, open issues and possible directions for further research Based on this review, some conclusions can be drawn, leading to recommendations for the experimental procedure as well as possible directions for further research. 1. The techniques for specimen preparation are quite mature. Manufacturing of the starter notch by EDM, leading to very small notch flank distances and therefore to almost crack-like notches, has largely become the state of the art. Subsequent razor blade grinding is necessary for milled V-notches; it may be not strictly necessary for EDM notches that are subsequently subjected to compression pre-cracking, but is still recommended for a more defined starter notch tip to facilitate pre-cracking. 2. Compression pre-cracking is mandatory for obtaining an open fatigue pre-crack. Pre-cracking in tension would lead to compressive stresses in front of the crack tip and therefore to undefined closure, thereby effectively precluding the determination of an R-curve. However, too large compressive stresses during pre-cracking lead to extended zones with tensile residual stresses, shifting the R-curve to lower threshold values in the initial region of small crack extensions, cf. Fig. 5. Although such an R-curve is conservative, the question remains to find an optimum pre-cracking compressive stress intensity factor which leads to an open crack with only negligible residual stresses – whether tensile or compressive – remaining. In a certain sense, this parallels the situation in determining the FCG threshold with the load shedding technique, where the threshold depends on the load reduction rate. Similar to prescribing an allowed range for the load shedding rate, one could give an allowed range for the pre-cracking compressive stress intensity factor; however, for this it will be necessary to gain still more data from experience. In this context, it should be noted that the R-curve method gives an interval for the long crack threshold; its lower limit is given by the last point of crack arrest, its upper limit by the subsequent load at which no arrest occurs anymore (see Section 3.1). The load reduction method typically gives thresholds near the upper end, and even sometimes outside this interval; this is suspected – but not yet fully proven – to be the result of oxidic crack closure occurring near the end of the load shedding experiment, where low loads and high crack extensions combine to give favourable conditions for oxide buildup. In short, compared to the results of the load shedding method, the last point of crack arrest in the load increasing method gives a conservative estimate of the long crack threshold, whereas the next point – where the crack does not stop anymore – gives a non-conservative estimate. This must always be kept in mind when comparing results from the two different methods. 3. The experimental determination of the cyclic R-curve itself relies on careful measurements. Here, either the multiple specimen technique with fractographic crack length measurements or the single specimen technique with DCPD crack length measurements (validated by fractography) is recommended. The caveats to be observed are documented in this review, which should allow successful implementation in any suitably equipped laboratory. 4. It has been shown that the Kitagawa-Takahashi diagram can be constructed from the cyclic R-curve and the endurance limit of the polished sample, see Eq. (17). This compares favourably against some of the simpler empirical approximate formulations for this diagram; the latter are still useful in their own right for engineering estimates where no detailed Rcurve data are available. However, the accuracy of this representation, Eq. (17), depends crucially on capturing the influence of the mean stress (or stress ratio, respectively) on both the endurance limit and the R-curve. Hence, to obtain the full benefit of this method, a large amount of detailed accurate data is necessary. 5. It is thus tempting to postulate a master R-curve – a typical shape of an R-curve in a dimensionless formulation – from which the R-curve for an individual material can be obtained via a characteristic length and the long crack threshold as scaling parameters, cf. Fig. 16. However, as there are typically several closure mechanisms at different length scales present in a material, the assumption of just one characteristic length is a gross simplification leading to a large scatter in the master curve. Still, a master R-curve could serve as a useful engineering approximation just the same way as there are approximate formulations for the Kitagawa-Takahashi diagram. Again, more data are urgently needed to pursue this concept. 6. Finally, some useful applications of several varieties of the R-curve concept in damage tolerant design and in design for materials with defects (such as weldments) have been highlighted. Of course, the practical usefulness depends crucially on a favourable trade-off between the experimental cost and the associated gain in predictive accuracy. This is where – as for many other engineering concepts – the R-curve concept should be available at various levels of accuracy; for example, in a basic level using only basic material properties and simple approximations such as the modified El Haddad equation or the master R-curve together with Goodman’s mean stress correction, or an advanced level using the detailed R-curve description with contributions from the various closure mechanisms at their length scales. 7. Summarizing these considerations in a nutshell, the cyclic R-curve concept stands a reasonable chance of becoming a useful engineering tool if it is possible to start a concerted effort in the fatigue and fracture community to create a commonly experimental standard similar to ASTM and ISO standards for fracture toughness and fatigue crack growth, and to compile unambiguous R-curve data for developing the method further and implementing it in everyday engineering practice. We hope that the present review may serve as an initial point for such future activities. Please cite this article in press as: Maierhofer J et al. The cyclic R-curve – Determination, problems, limitations and application. Engng Fract Mech (2017), https://doi.org/10.1016/j.engfracmech.2017.09.032 J. Maierhofer et al. / Engineering Fracture Mechanics xxx (2017) xxx–xxx 19 7. Summary and outlook A short review has been given on the state-of-the-art of the determination and application of the cyclic R-curve describing the crack size dependence of the fatigue crack propagation threshold in the physically short crack regime. Topics that have been addressed are: basic questions of the cyclic R-curve; its experimental determination including critical and unsolved points; its relation to the Kitagawa-Takahashi diagram, including an estimation method based on a modified Kitagawa-Takahashi approach on one hand and a method of constructing Kitagawa-Takahashi diagrams for notched components on the other hand; and, finally, aspects of its application (and potential additional application fields) in component assessment and material development. With respect to the experimental determination, the focus was on requirements for the notch geometry, the conditions for generating the pre-crack and other features such as specimen clamping, potential oxide effects and others. Emphasis was also put on adequate curve fits at the background of application within damage tolerance concepts. Parameters which influence the cyclic R-curve are briefly discussed and illustrated by examples. In conclusion, it has been demonstrated that the cyclic R-curve is a valuable tool for component and materials design and assessment. However, the many parameters influencing its experimental determination call for a concerted effort to create a commonly accepted experimental standard similar to ASTM and ISO standards for fracture toughness and fatigue crack growth. Once such a standard is established and unambiguous R-curve data will become available, related assessment methods such as cyclic R-curve analysis and the modified Kitagawa-Takahashi diagram are expected to gain widespread acceptance in engineering application. Acknowledgements Financial support by the Austrian Federal Government (in particular from Bundesministerium für Verkehr, Innovation und Technologie and Bundesministerium für Wirtschaft, Familie und Jugend) represented by Österreichische Forschungsför derungsgesellschaft mbH and the Styrian and the Tyrolean Provincial Government, represented by Steirische Wirtschaftsför derungsgesellschaft mbH and Standortagentur Tirol, within the framework of the COMET Funding Programme is gratefully acknowledged. The authors also gratefully appreciate the funding by the Forschungsvereinigung Automobiltechnik e.V. within the AiF network (Arbeitsgemeinschaft industrieller Forschungsvereinigungen). References [1] Tanaka K, Nakai Y. Propagation and nonpropagation of short fatigue cracks at a sharp notch. Fatigue Fract EngMater Struct 1983;6:315–27. [2] Tanaka K, Akiniwa Y. Resistance-curve method for predicting propagation threshold of short fatigue cracks at notches. Eng. Fract Mech 1988;30:863–76. 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