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Engineering Fracture Mechanics xxx (2017) xxx–xxx
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Engineering Fracture Mechanics
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The cyclic R-curve – Determination, problems, limitations and
Jürgen Maierhofer a,⇑, Stefan Kolitsch a, Reinhard Pippan b, Hans-Peter Gänser a, Mauro Madia c,
Uwe Zerbst c
Materials Center Leoben Forschung GmbH, A-8700 Leoben, Austria
Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, A-8700 Leoben, Austria
Bundesanstalt für Materialforschung und -prüfung (BAM), Division 9.1, D-12205 Berlin, Germany
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
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 (
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: (J. Maierhofer).
0013-7944/Ó 2017 The Authors. Published by Elsevier Ltd.
This is an open access article under the CC BY-NC-ND license (
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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)
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
material yield strength
compression pre-cracking
direct current potential drop
long crack
plasticity corrected
stress intensity factor (K-factor)
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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 :
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.
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.
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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
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
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.
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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].
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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
SIF range DK varying from 4 to 16 MPa m. As can be seen, the measured R-curves become steeper with lower pre-cracking
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 ;
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].
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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
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
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
a0 a ¼ arccos cosh
cos p2W
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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.
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(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].
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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
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
to provide a poor fit of the initial slope of the cyclic R-curve; this holds also with respect to the power law correlation
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
DK th;LC
A Dab
for Da < DaLC
for Da P DaLC
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
mi exp i¼1
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
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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].
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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
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].
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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.
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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
a þ a0
This equation follows from the fundamental equation of LEFM
DK th ðaÞ ¼ Drth ðaÞ Y pffiffiffiffiffiffi
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:
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 :
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
DK th;LC
Y Drth;0
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
¼ DK th;LC Da þ a0
Fig. 17. Schematic illustration of the parameter to describe the effective driving force in different fatigue regimes [49].
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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
which is chosen such that it satisfies the condition DKth (Da = 0) = DKth,eff:
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
DK th;LC
Y Drth;0
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].
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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
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].
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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
U LC 1
DK th;LC DK th;eff
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.
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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
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
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
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.
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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.
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
The authors also gratefully appreciate the funding by the Forschungsvereinigung Automobiltechnik e.V. within the AiF
network (Arbeitsgemeinschaft industrieller Forschungsvereinigungen).
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