Application of the cyclic R-curve method to notch fatigue analysis

International Journal of Fatigue xxx (2015) xxx–xxx

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Application of the cyclic R-curve method to notch fatigue analysis Mauro Madia ⇑, Uwe Zerbst BAM-Federal Institute for Materials Research and Testing, Division 9.1, D-12205 Berlin, Germany

a r t i c l e

i n f o

Article history: Received 8 February 2015 Received in revised form 12 June 2015 Accepted 13 June 2015 Available online xxxx Keywords: Notch fatigue Notch sensitivity Cyclic R-curve Non-propagating cracks Critical distance

a b s t r a c t The paper deals with the prediction of the fatigue limit of notched specimens. Some authors in the literature have shown that this issue can be solved by employing an approach based on critical distances, others used elastic fracture mechanics. In this paper the fatigue limit of notched specimens is given by the non-propagating condition of mechanically small surface cracks. According to the so-called cyclic R-curve method, the crack driving force of a growing small crack is compared to its resistance force, which incorporates the gradual build-up of crack closure. The fatigue limit is determined by that applied nominal stress, for which the tangency condition of crack driving and resistance force is satisfied. The approach has been modified to incorporate plasticity effects in the mechanically short crack regime and it has been applied to a mild and a high-strength steel, in case of an infinite plate with circular hole under remote tensile stress. The method has been successfully applied to the evaluation of fatigue notch-sensitivity in case of surface roughness. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Notch fatigue analysis is of primary importance in the fatigue design of many engineering components, as fatigue damage takes place often at stress raisers (for instance shoulder fillets in shaft and axles, threaded connections, welded joints, etc.). Many authors dealt with this issue and proposed different approaches to the aim of predicting the fatigue limit of a given material in presence of notches. The methodologies can be classified under two main headings: (i) critical distance or stress based approach and (ii) fracture mechanics based approach. In general, the critical distance method is based on the concept that it is not the maximum stress at the notch root the important parameter controlling the fatigue limit, but a certain stress value beyond the notch root. Already the pioneering works by Neuber [1] and Peterson [2] hypothesized that the controlling damage parameter is an average stress over a characteristic structural length or a punctual stress at a given distance from the notch root, respectively. A major work about critical distances has been presented by Taylor in his unifying theoretical model [3,4]. The underlying assumption of his work has been that the average elastic stress must exceed the fatigue limit over a certain critical volume beyond the notch tip, in order for failure to occur, the size of which correlates to the El Haddad parameter a0 [5]. It is of some importance to ⇑ Corresponding author. Tel.: +49 30 8104 4166; fax: +49 30 8104 1539. E-mail address: [email protected] (M. Madia).

underline that the critical distance method, though known and spread in the formulation by Taylor, was previously developed by Tanaka in its point and line version [6]. Taylor in [7] tried to link the critical distance approach to the mechanism of growth and arrest of mechanically short crack at notches (non-propagating cracks), worked well in case of crack-like (sharp) notches, whereas in case of short and blunt notches the predictions were poor. The existence of non-propagating cracks at the notch root was already found by Frost in the fifties during fatigue tests on cylindrical and plate notched specimens (see for instance [8]). A major work on the prediction of non-propagating cracks at notches has been published by El Haddad et al. [5], who proposed a model which is able to predict fairly well the length of non-propagating cracks at notches, as well as the dependence of the fatigue limit on crack length. Furthermore they demonstrated the effect of the notch geometry on the initiation and propagation of small cracks. A further important contribution for the understanding of the influence of notch geometry on fatigue strength has been presented by Lukáš et al. [9]. The authors proposed an equation for assessing the critical size of non-damaging notches depending on the tensile strength of steels. An alternative methodology based on fracture mechanics for the prediction of fatigue limits and corresponding non-propagating cracks is the so-called cyclic R-curve method. The cyclic R-curve represents the resistance force of a fatigue crack to advance, which is described in terms of a threshold stress intensity factor range DKth depending on the crack advance Da. To the best knowledge of the authors of this paper, the first work using

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Please cite this article in press as: Madia M, Zerbst U. Application of the cyclic R-curve method to notch fatigue analysis. Int J Fatigue (2015), http:// dx.doi.org/10.1016/j.ijfatigue.2015.06.015

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Nomenclature ai aR a0 a⁄ b f(Lr) x A A5 Kf Kt Lr R ReL Rm Rm,min Rz W

q q

rðxÞ

initial crack length constant in the definition of the roughness-induced fatigue factor El Haddad parameter material constant in Peterson’s formula exponent in the cyclic R-curve law ligament yielding correction function coordinate from notch root parameter in the cyclic R-curve law permanent elongation at fracture notch fatigue factor theoretical stress concentration factor ligament yielding parameter stress ratio lower yield stress ultimate tensile strength constant in the definition of the roughness-induced fatigue factor surface roughness parameter semi-width of the plate notch-tip radius material constant in Neuber’s formula local stress distribution normal to the crack plane

this kind of approach has been published by Yates et al. [10], who approximated the cyclic R-curve by means of the well-known Kitagawa–Takahashi diagram and use it for predicting the occurrence and length of non-propagating cracks. Sometime later Tanaka et al. [11], not only estimated the length of the non-propagating cracks, but provided also a method to evaluate their propagation threshold. A similar approach has been proposed by Akiniwa et al. [12], in which for the first time the development of the plasticity-induced crack closure has been used to explain the arrest condition of nucleated cracks. An application of the cyclic R-curve method can be traced also in the work of Taberning et al. [13]. The present work is aimed to address the influence of the surface roughness on the prediction of the fatigue limit for two different steel grades, a medium and a high strength steel. To this purpose, a procedure previously developed by the authors has been employed [14] and modified in order to deal with notches. The calculations are based on the cyclic R-curve method, in which the crack driving force has been calculated according to elastic– plastic fracture mechanics in order to deal with mechanically short cracks. An overview on the cyclic R-curve method and the developed procedure is given in the following, then the model is applied to a simple infinite plate with different circular holes under tensile loading. Finally the calculations in case of specimens with different roughness profiles are presented and the results compared with simple approximation formulas given in the literature.

ra re0 rref rY r0 Da DaLC DK DKp DKth DKth,eff DKth,LC DKth,op DLr Dra  an Dr Dre Dren DreR Dre0 Drref

applied stress amplitude fatigue limit for R = 1 (stress amplitude) reference stress yield strength reference yield stress crack advance crack advance at the transition to the long crack regime stress intensity factor range plasticity-corrected stress intensity factor range threshold stress intensity factor range intrinsic fatigue crack propagation threshold fatigue crack propagation threshold for long cracks extrinsic component of DKth ligament yielding parameter range applied stress range stress range averaged over 2a0 from the notch fatigue limit range fatigue limit range for the notched specimen fatigue limit range in presence of surface roughness fatigue limit range for R = 1 reference stress range

temperature and then cooled in air. The final microstructure is characterized by a fine one-way striping in the rolling direction with fine perlite and ferrite bands. The S960QL is a high-strength steel which is delivered in a quenched and tempered condition, i.e. the material is reheated above the austenite recrystallization temperature followed by water cooling. The resulting microstructure is then fine martensite. The basic mechanical properties are summarized in Table 1. It is of some importance to see that both materials do not exhibit any anisotropic behavior in the hot-rolled condition, the response in the longitudinal (rolling direction) and transverse direction is identical. No experimental tests have been performed in order to derive the fatigue limits. Nevertheless good and reliable estimates for smooth specimens made of steel in tension–compression tests at stress ratio R = 1 can be provided employing common engineering formulas like re0 ¼ 0:4  0:5Rm [15]. In particular the fatigue limits have been estimated as follows: (i) for the steel grade S355NL re0 ¼ 0:45Rm  250 MPa; (ii) for the steel grade S960QL re0 ¼ 0:45Rm  460 MPa. Note that the values agree well with those which can be found in the literature [16–18]. 2.2. Experimental derivation of the cyclic R-curve The experimental determination of the cyclic R-curve represents one of the major issues of this work, as it is the fundamental tool for the description of the resistance force for a fatigue crack to propagate and eventually arrest, thus becoming non-propagating. The cyclic R-curve is meant to describe the evolution of the crack propagation threshold DKth with crack advance Da. In general,

2. Material properties 2.1. Basic properties Two different kind of structural steels have been investigated, namely the S355NL and the S960QL, both have been delivered as hot-rolled plates with 10 mm thickness. The S355NL is a medium-strength steel which undergoes a normalizing heat treatment, i.e. it is reheated above the austenite recrystallization

Table 1 Basic mechanical properties. (source: P. Kucharczyk, RWTH Aachen.)

S355NL, longitudinal S355NL, transverse S960QL, longitudinal S960QL, transverse

ReL (MPa)

Rm (MPa)

A5 (%)

371 373 968 970

547 549 1016 1016

32.3 32.6 16.5 15.2

Please cite this article in press as: Madia M, Zerbst U. Application of the cyclic R-curve method to notch fatigue analysis. Int J Fatigue (2015), http:// dx.doi.org/10.1016/j.ijfatigue.2015.06.015

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Fig. 1. Schematic diagram of the cyclic R-curve.

the contribution to the overall DKth is given by two components (see Fig. 1): an intrinsic one, named DKth,eff, which can be regarded as a material characteristic, strongly dependent on the microstructure, but independent of the crack size; the second component is the extrinsic part DKth,op which describes the evolution of the fatigue crack opening level with crack advance. The increase in the threshold values is related to the development of crack closure, that depends on many diverse phenomena which can take place at the crack tip and flanks during the propagation, among which the most important are plasticity-induced, roughness-induced and oxide-induced crack closure (for a complete overview the reader may refer to [19]). In Fig. 1 it is also illustrated how the crack propagation threshold stabilizes to a value DKth,LC for long cracks after some crack estension. There are until now essentially two methodologies for determining the cyclic R-curve. The first one has been developed by Tanaka et al. [11] who adopted a multiple specimen technique, i.e. a single cyclic R-curve is obtained interpolating the results of several tested samples. Each specimen has been tested at a given threshold range and the corresponding non-propagating crack has been measured with a scanning electron microscope, after braking the sample in liquid nitrogen. It must be remarked that, though such a testing procedure could be really time consuming, it yields the highest precision and reliability as the depth of the non-propagating crack can be measured optically all along the crack front. The second experimental procedure has been developed by Taberning et al. [13,20] and it is based on a single test technique, as depicted in Fig. 2. In the first stage the notched specimen is cycled under full compressive loading (both maximum and minimum load are negative), which enables the nucleation and growth of a short closure-free pre-crack from the notch root. The length of this crack is strictly correlated to the estension of the plastic region generated in the first loading cycle (monotonic plastic zone), that means essentially it depends on the mechanical properties of the material and on the entity of the stress intensity factor range used during the compression pre-cracking phase. It is important to put in evidence that, in order to minimize the plastification of the ligament during pre-cracking, the notch machined into the specimen should be characterised by a sharp edge and small tip radius, which enables the use of a lower DK for obtaining the desired pre-crack. After the pre-cracking stage, the test is performed at a given stress ratio. The pre-crack does not propagate until DK > DKth,eff. The value of DK is increased stepwise during the test in the range DKth,eff < DK < DKth,LC and for each step the crack propagates until the closure takes over and the crack nearly stops. Each crack advance Da and the corresponding DKth represent a point of the cyclic R-curve as depicted in Fig. 2. When DK > DKth,LC, then the propagation threshold cannot increase any longer and the crack

finds no obstacle to propagation until the final fracture of the specimen. In contrast to the multi-specimen technique described above, in the single specimen technique the crack length cannot be measured with direct methods, as the specimen cannot be broken up. Therefore indirect methodologies must be adopted, such as compliance measurements or potential drop techniques. The potential drop is known to be a very reliable method, but it is questioned by some researcher to provide poor or even not reliable values during test conducted at negative stress ratios. The author of this paper provided in a previous work [14] some comparison between single-specimen tests, in which the potential drop technique has been used, and the multi-specimen technique for SENB specimen tested at R = 1. No evidence of a strong dependence of the experimental values on the test methodology has been found. The only clear difference has been stated in the measurements of the pre-crack length, which in fact did not affect the values in the cyclic R-curve as the crack estension Da is used. In the present study the multi-specimen technique has been used. The SENB specimens have been tested in a RUMUL resonant fatigue testing machine by means of a 4-point bending test device. The SENB specimens have been notched in two different stages: firstly a main notch of about 5 mm depth has been machined by means of electro discharge machining, then the final sharper notch of about 100 lm depth has been created by a razor polishing technique initially introduced by Taberning et al. [20]. A sketch of the specimen used in the test is provided in Fig. 3a. The crack length has been measured optically with a KEYENCE digital microscope, after the specimen has been broken open. In particular, 50 optical measurements have been taken over the specimen thickness and the crack length has been provided as the average value. Fig. 3b shows an example in case of a S355NL specimen tested at p DK = 5 MPa m after compression pre-cracking. It is possible to observe onto the surface a first mark given by the arrest of the pre-crack, followed by a second line which marks the crack advance for the given initial DK. The experimental points for both materials are depicted in Fig. 4. The data points had to be fitted by a proper law to be used in a resistance curve analysis. The existing works in the literature dealing with this issue, propose a fitting with an exponential law [21,22]. The use of these laws in case of our data points did not provide a satisfactory approximation in the mechanically short crack regime, which is the most important to the goal of the present work, as it is the region in which non-propagating cracks occur. A much better fitting has been obtained by the following power law (see also Fig. 4):

(

DK th ¼

A  Dab þ DK th;eff DK th;LC

for Da < DaLC for Da P DaLC

ð1Þ

The values of the material parameters in Eq. (1) are reported in Table 2. 3. Analytical modeling 3.1. Resistance curve analysis In the following the principles of a resistance curve analysis are illustrated (see Fig. 5). An initial crack length ai is given, which is assumed to be a newly nucleated crack and therefore closure-free, i.e. the crack is fully open and its propagation threshold is DKth,eff. At this point it must be clear that the origin of the nucleated crack can be the presence of defects or inhomogeneities in the material (non-metallic inclusions, pores, etc.), as well as the irreversible processes associated to cyclic loading. The initial crack ai is also the origin of the cyclic R-curve, as it is the point from

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Fig. 2. Schematic diagram of the cyclic R-curve determination according to [20].

(ii) The applied stress is too low and the resistance force takes over after a very short propagation stage which results in a non-propagating crack (ra,1 in Fig. 5). (iii) There exists a stress level (ra,2 in Fig. 5) at which the tangency of the crack driving force curve to the resistance curve is fulfilled. This stress represents the transition between crack arrest and propagation, i.e. the fatigue limit for the initial crack ai. The authors of this paper took advantage of this property of the R-curve analysis to propose a methodology for the determination of the initial crack length ai and the subsequent prediction of the fatigue limit in case of weld joints [14]. 3.2. Determination of the initial crack length and fatigue limit The application of the cyclic R-curve method to the calculation of the endurance limit of notched components consists of two steps. The first one deals with the determination of a so-called intrinsic crack length, which has to be understood as the largest non-propagating crack in case of perfectly smooth (polished) specimens tested at the fatigue limit (ra = re0). It is assumed that this crack length is a material characteristic, and it should be interpreted as an initial intrinsic length from which a fatigue analysis

Fig. 3. Experimental procedure for the determination of the cyclic R-curve by means of a multi-specimen technique: (a) SENB specimen; (b) optical measurement p of the crack length for a S355NL specimen tested at DK = 5 MPa m after compression pre-cracking.

which the crack closure starts developing. Once the resistance force has been determined, the resistance curve analysis can be carried out by plotting in the same graph the crack driving force associated to a growing crack for a given constant applied stress amplitude ra. Three different scenarios can occur, as already stated by Yates et al. [10]: (i) The applied stress is too high and the crack propagates till fracture (ra,3 in Fig. 5).

Fig. 4. Cyclic R-curves, test data points and interpolating curves for the test at R = 1.

Please cite this article in press as: Madia M, Zerbst U. Application of the cyclic R-curve method to notch fatigue analysis. Int J Fatigue (2015), http:// dx.doi.org/10.1016/j.ijfatigue.2015.06.015

M. Madia, U. Zerbst / International Journal of Fatigue xxx (2015) xxx–xxx Table 2 Parameters for the cyclic R-curve in Eq. (1).

S355NL S960QL

A

b

p DKth,eff (MPa m)

p DKth,LC (MPa m)

DaLC (mm)

6.25 5.58

0.43 0.25

2.8 2.8

9.5 9.9

1.16 2.7

Regarding the first issue, the authors of this paper proposed in different publications the use of a plasticity-corrected cyclic K-factor, DKp, determined on the basis of the modified R6 equations [23] such as proposed by McClung [24]:

DK p ¼ based on fracture mechanics can be started. The procedure is schematically depicted in Fig. 6: given the crack driving force calculated for ra = re0, the cyclic R-curve can be shifted progressively (dashed lines) toward the crack driving curve until the tangent point is reached (full line), thus determining the initial crack length ai. It is then understood that ai represents the largest non-propagating crack at the fatigue limit for smooth specimens. Once the initial crack length has been determined, then the fatigue limit for a notched geometry can be predicted. As ai has been denoted as an intrinsic (characteristic) length of the material, the hypothesis holds to consider ai as the initial crack length for the resistance curve analysis at a given notch. The analysis follows then the philosophy which has been explained in the Section 3.1. In fact ai and the cyclic R-curve are known, therefore the fatigue limit for the notch can be calculated as that applied remote stress fulfilling the tangency condition of the crack driving force to the cyclic R-curve. It is of some importance to note that the calculation of the driving force have to account for the presence of the notch (stress gradient). 3.3. Elastic–plastic crack driving force A very important issue in this work is the calculation of the crack driving force, mainly for two reasons: (i) Linear elastic fracture mechanics concepts are not applicable in the mechanically short crack regime because the crack length is comparable to the extension of the plastic zone. (ii) The stress field about the notch is characterized by a gradient, which can be very steep in case of sharp notches, so that the procedure must deal with both sharp and large notches.

Fig. 5. Schematic description of the resistance curve method. Given an initial crack depth ai and the corresponding cyclic R-curve, the lower stress limit to crack propagation is determined by the tangency of the crack driving force curve to the resistance curve.

5

DK f ðDLr Þ

and DLr ¼

Drref : 2r Y

ð2Þ

ð3Þ

The function f(Lr) is the so-called ligament yielding correction function, whereas the parameter Lr is a ligament yielding parameter comparable to those defined in the R6 procedure [23]. The authors proposed the following formulation of DLr

DLr ¼

Dra 2r 0

ð4Þ

where Dra is the cyclic applied stress and r0 the reference yield stress of the structure containing cracks [25,26]. The reference yield stress is not a limit load, but it is rather an alternative to the limit load and it has been developed in order to obtain a better description of the crack driving force as a function of the applied load. The approach allows a wider application range and the solutions demonstrated to provide more accurate crack driving force estimates compared with the ones based on common limit load solutions, especially in case of large cracks [25]. The foregoing procedure has been developed and adopted in case of unnotched components under remote tensile membrane and bending stresses. Dealing with notches, the effect of the notch stress field must be included, as the stress distribution on the perspective crack ligament is deeply influenced by the presence of the notch and the reference yield stress r0 is a local parameter by nature. If in principle no difficulties arise in the calculation of the linear elastic stress intensity factor thanks to the existence of approximation formulas based on the weight function method for non-uniform stress distributions (in this work the approaches by Shiratori et al. [27] and Wang et al. [28] have been used), the definition of the plastic correction has to be changed in order to obtain estimates of DKp for mechanically short cracks, which can account for the presence of the notch. The idea has been to define the plastic correction over a critical distance from the notch. In fact, for sharp notches the steep gradient is limited to a very small region about the notch, whereas for

Fig. 6. Schematic view of the resistance curve method applied to the determination of the initial crack length for the S355NL. Given the crack driving force calculated for ra = re0, the cyclic R-curve can be shifted progressively (dashed lines) toward the crack driving curve until the tangent point is reached (full line), thus determining the initial crack length ai.

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larger (blunt) notches the region controlled by Kt is in general so large that non-propagating cracks are well within it and the fatigue limit is controlled by Kt. In general, it has been demonstrated that the stress solution is controlled only by the notch in a region of estension 0.3q [29], where q is the notch-tip radius. According to the theory of critical distance, in particular the work by Taylor [3,7], the plastic correction has been identified over a distance 2a0, where a0 is the El Haddad parameter defined as

a0 ¼

1

p

 

DK th;LC Dr e

2 ð5Þ

The so-defined distance gives a reasonable estension of the process zone which affects the propagation of a mechanically short crack (see also the schematic depicted in Fig. 7). This means that, if the notch is sharp (q << a0), its influence zone is very small and the corresponding gradient will not affect the fatigue limit; on the contrary, in presence of blunt notches (q >> a0), the fatigue limit is fully controlled by Kt. In principle this method should work also in case of those notches that lie inbetween sharp and blunt notches, which have been denoted short notches by Taylor [7]. Accordingly, the qualitative explanation of the procedure must be followed by the quantitative assessment of the plastic correction. This has been done by modifying Eq. (4) as follows:

DLr ¼

 an Dr 2r0

ð6Þ

 an represents the average stress range over the critical diswhere Dr tance 2a0 from the notch root:

 an ¼ Dr

1  2a0

Z

2a0

rðxÞdx

ð7Þ

applied to the prediction of the fatigue limits in case of specimens characterized by different levels of surface roughness. 4.1. Comparison with existing models The model and the working hypotheses have been checked for an infinite plate with circular hole and remote membrane tensile stress. The advantage of using such a model is that it has been used in the literature (see for instance [3]), so that other solutions for notch fatigue analysis are available and can be used for comparison with the procedure developed in this work. Another nice property is that the local stress distribution is given in closed form [30]:

rðxÞ ¼

4. Application of the model Two different analyses are presented in the following. The first one has been carried out on an infinite plate with a circular hole and remote membrane tensile stress; then the procedure has been

2

"  2þ



2

q

xþq

þ3



4 # :

q

ð8Þ

xþq

Note that Eq. (8) refers to the stress distribution normal to the crack plane and x has its origin at the base of the notch (hole). It is important at this stage to point out that the predictions have to show the same general features, whatever the model employed: the solution must tend to the plain fatigue limit as the hole size tends to zero, whereas for blunt notches the solution must tend to Dre =K t as the fatigue limit is controlled by the notch field. Four models have been taken from the literature in order to compare their predictions with the ones provided by the present procedure. The first simple model goes back to the notch theory by Neuber [1]:

Dren ¼

Dr e Dre ¼ t 1 ffiffiffiffiffi Kf 1 þ Kp 2q  1þ

0

where r(x) is the local stress distribution normal to the crack plane (see Fig. 7). Note that in the following the calculations carried out by Eq. (6) will be denoted as local model, whereas the ones employing the complete stress distribution over the thickness, i.e. Eq. (4), will be denoted as global model.

ra

ð9Þ

q

where q⁄ is a material constant which can be approximated by

q ¼ ð140=Rm Þ2 . A similar approach is given by the Peterson’s formula [2]:

Dren ¼

Dr e Dre ¼ Kf 1 þ K1þt 1 a

ð10Þ

q

in which the material constant a ¼ 0:0254  ð2070=Rm Þ1:8 . The last two models are the application of the critical distance method by Taylor to the specific problem of the infinite plate with circular hole [3]. In particular, the point method writes as follows:

Dren ¼ Dre  2þ



q

2

qþa0 =2

2 þ3



ð11Þ

4

q qþa0 =2

whereas the line method is given in the following form:

Dren ¼ Dre 

Fig. 7. Schematic representation of the local stress field at notches in relation to the calculation of the plastic correction.

2  a0 2

ð12Þ

4

q q 2  a0 þ q   qþ2a  12  ðqþ2a 0 1 2

0Þ

3

It is important to remark that the all the previous approaches are stress-based, in the sense that they rely on the calculation of the stress over a certain critical distance. The point and line method by Taylor are partially fracture mechanics based, as the critical length is defined by the El Haddad parameter, even though no crack length is introduced. The method developed within this work is based completely on a fracture mechanics analysis. The first step of analysis has been the calculation of the intrinsic initial crack length ai according to the procedure explained in the Section 3.2. The values of ai at the fatigue limit have been 17 lm for the S355NL and 10 lm for the S960QL. Already this result is somehow important, because it speaks in favor of a higher sensitivity of the S960QL (high strength steel) compared to the S355NL (medium strength steel). The initial crack length for which the fatigue process is initiated is almost double in the medium strength

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steel. Another interesting point of view is that the decreasing trend of ai with increasing values of the tensile strength is similar to the observations of Neuber and Peterson in case of the material constants q⁄ and a⁄ respectively. Given the values of ai, the fatigue limits for different dimensions of the circular hole can be calculated by the procedure presented in Section 3.2. Note that for all the analyses the assumption of an infinite plate has been realized by assuming a fixed high ratio between plate estension and radius of the hole, namely W = 100q, where W is the semi-width of the plate. Another really important assumption is that non-propagating cracks are not through-thickness cracks, but semi-circular surface cracks. This point is confirmed by experimental observations and clearly stated in [9]. Therefore the elastic stress intensity factor range has been calculated by means of weight functions reported in [27] and [28]; the comparison of the values provided by the two different formulations showed a negligible difference. The results of the fatigue limit calculations is depicted in Fig. 8 for the S355NL and in Fig. 9 for the S960QL. Some general conclusions can be drawn, independently of the material. The global and local model yield in general similar results for sharp and short notches, whereas the global model fails to recover the theoretical solution for q ? 1, i.e. it does not tend asymptotically to the value Dre0 =K t . It can be concluded that the local plastic correction, rather than the global one, is the right approach for dealing with notch fatigue analysis. The other general comment is that the approach presented in this work tends to yield conservative estimates compared to the critical distance method; unfortunately no experimental values have been available, so to judge the results against experimental values. Note however that the experimental comparison published by Taylor in [3] shows that the point and line methods tend to be non-conservative (at least referring to that particular case) and also in the more extensive validation proposed in [4] it is stated that the critical distance approach predictions were accurate within 20% of the experimental data. The nature of the difference between the present model and the others mentioned in this paper could be that the proposed model is based on fracture mechanics, for which the intrinsic crack length ai is about few microns for the material under investigation. This means that even the small and short notches can affect the loss of fatigue strength. There are some considerations which can be made regarding the different behavior of the two materials. Firstly all the models show the higher notch sensitivity of the high strength steel. By

Fig. 8. Predictions of the fatigue limits for different notch dimensions in case of an infinite plate with circular hole and remote tensile stress. The values refer to the steel grade S355NL under alternate symmetric stress (R = 1).

7

Fig. 9. Predictions of the fatigue limits for different notch dimensions in case of an infinite plate with circular hole and remote tensile stress. The values refer to the steel grade S960QL under alternate symmetric stress (R = 1).

the comparison of the predictions in Figs. 8 and 9, it can be seen how the S960QL (high strength steel) is characterized by a larger loss of fatigue strength in the small notch regime, say for q 6 0:1 mm. Furthermore, the difference between global and local model results to be more pronounced in the case of the mid strength steel grade (S355NL) than in case of the high strength steel (S960QL). This has to be attributed to the different weight of the plastic correction in the predictions and can be explained as follows. The plastic correction depends mainly on the ligament yielding parameter given in Eqs. (4) and (6), namely the higher is the ratio of the applied stress to the reference yield stress, the higher is the plastic correction. Being the reference yield stress much higher in case of the high strength steel (about 2.4 times the one of the medium strength steel for plain specimens), then it follows that the S960QL is less sensitive to the use of a local or global plastic correction, as in both cases the applied stresses, averaged whether over 2a0 or over the whole ligament, will be always much lower than the reference yield stress.

4.2. Predictions in case of surface roughness The model has been applied to the prediction of the notch sensitivity in case of specimens with different surface roughness and subjected to a remote membrane tensile stress. A reference to the influence of surface irregularities can be found in [3] and [9]. In particular it is stated that in principle the surface roughness can be treated as a series of small notches or cracks and the effect of the size of such irregularities depends on the material strength, namely the higher the strength of the material, the smaller the non-damaging (critical) dimension. In this work the surface roughness has been idealized as a series of notches and three different characteristic roughness dimensions have been simulated by finite elements in order to derive the stress profile at the notch root. To this aim, 2D models with quadratic plane-stress elements have been employed. In order to catch the steep stress gradient at the notch, the minimum element size has been fixed to 5 lm. Fig. 10 shows the two different modeling strategies employed in the calculations; for the same nominal roughness dimension (Rz), the irregularity has been modeled as rounded or spline profile (Fig. 10a and b respectively). As previously stated, three different level of surface roughness Rz have been taken into account, namely 50, 100 and 200 lm, which are

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M. Madia, U. Zerbst / International Journal of Fatigue xxx (2015) xxx–xxx

Fig. 10. Modeling strategies for the finite element calculations with surface roughness: (a) rounded profile; (b) spline profile.

typically obtained after surface scrubbing in the fabrication of hot-rolled plates. In the notch fatigue analysis the same intrinsic crack length have been used, as for the analysis of the plate with circular holes. As stated in Section 3.2, ai is a material characteristic which does not depend on geometry. The local stress profile r(x) has been calculated via finite elements with a really refined mesh at the notch root in order to catch the stress gradient (Fig. 10). The results of the analyses have been compared with empirical formulations of the roughness-induced fatigue factor which can be easily found in the literature for different materials and surface finishing. In particular we referred to the FKM guidelines [31], in which the fatigue strength reduction due to the surface roughness is approximated by:

  DreR 2  Rm ¼ 1  aR  logðRz Þ  log Dre0 Rm;min

ð13Þ

where aR and Rm,min are constants which depend on the material. In case of steel no distinction is made between different grades, but it is generally prescribed to use aR = 0.22 and Rm,min = 400 MPa. The values predicted by means of the local model are compared with the FKM recommendations in Fig. 11 (here the predictions for the rounded profile are shown). The results show that there is a reasonable agreement between the values calculated by Eq. (13) and the prediction of the local model. Qualitatively, the local model is able to predict correctly the loss of fatigue strength with increasing roughness dimension, furthermore the higher notch sensitivity of the high strength steel is also predicted by the model. Quantitatively, the predictions of the model are characterized by a higher dispersion, which means that the differences between the minimum and maximum roughness size is larger than the ones calculated by Eq. (13). In particular the values for Rz = 50 lm are

Fig. 11. Comparison between the local model prediction (rounded profile) and the values obtained given by the FKM guidelines.

over-predicted, whereas the ones for Rz = 200 lm are under-predicted, nevertheless the worse prediction differs by less tha 15% from the FKM values. It has to be remarked at this point that the FKM guidelines refer to experimental evidences with a real roughness profile, whereas in this work the roughness profile has been idealized by a simple geometry. A further interesting outcome of the calculations is that a negligible difference has been found between the predictions obtained using the stress profile from a rounded or spline roughness model (see Table 3). The reason can be explained in terms of the local stress distribution at the notch. Fig. 12 shows the stress profile r(x) normalized by the applied stress amplitude ra for the three roughness values and for both rounded and spline profile. Even though the stress concentration factor at the notch root is very different for rounded and spline profile, the distributions tend to coincide within few microns in depth. In a fracture mechanics perspective, at the depth of the intrinsic initial crack ai (17 lm for the

Table 3 Fatigue limit reduction in presence of surface roughness: Comparison between calculations carried out using a stress distribution provided by a rounded and spline roughness finite element model. Rz (lm)

DreR =Dre0 Rounded

DreR =Dre0 Spline

Error%

S355NL

50 100 200

0.89 0.78 0.66

0.88 0.77 0.64

1 1 3

S960QL

50 100 200

0.77 0.65 0.55

0.76 0.62 0.48

1 5 13

Fig. 12. Stress distribution ahead of the notch tip, normal to the crack plane. The stresses r(x) are normalized by the applied stress amplitude ra.

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S355NL and 10 lm for the S960QL) a vanishingly small effect of the roughness shape type on the stress profile can be noticed. In fact the greatest discrepancy in the values provided in Table 3 is found for the S960QL in case of Rz = 200 lm, where the intrinsic initial crack depth lies in a region in which there is still a noticeable difference between the stress distributions for rounded and spline profile. 5. Summary This work presented an approach for notch fatigue analysis, based on the cyclic resistance curve method. Compared to other similar approaches in the literature, the present method introduces the concept of an initial intrinsic crack length, which is a material characteristic, independent of the geometry. Moreover the calculation of the crack driving force to be used in the cyclic R-curve analysis is based on a plasticity-corrected stress intensity factor range. The plastic correction is based on a modification of the R6 procedure in order to account for cyclic loading. A new strategy for the calculation of the plastic correction is proposed, which is better suited in case of notches, as the notch stress field is characterized by a gradient. In particular, instead of averaging the stress over the whole ligament (global model), the new correction is based on the average stress over a critical distance identified by 2a0 from the notch root, where a0 is the El Haddad parameter. The procedure has been compared with existing models in case of an infinite plate with circular hole under remote tensile stress. Then the application to the problem of surface roughness has been proposed. The main results of the comparison are: (i) The local model works better then the global one in case of notches, as it is able to predict correctly the asymptotic trend of the fatigue limit for blunt notches. (ii) The present approach provides conservative estimations compared to other models given in the literature, especially respect to the critical approach method. (iii) The higher notch sensitivity in case of the high strength steel is correctly predicted. The application of the local model to the calculation of the fatigue limit in presence of surface roughness provided also promising results, though the geometry of the roughness profile has been idealized: (i) The predictions show a loss of fatigue strength with increasing roughness dimension. (ii) A higher loss is correctly predicted in case of the high strength steel. (iii) The comparison with the values prescribed by the FKM guidelines shows a reasonable good agreement, being the largest difference less than 15% in the worse case. Further work has been planned in order to check the model predictions against experimental values for other notch typologies, which can be found in the literature. Furthermore a real roughness profile has been scanned and can be now simulated by finite elements in order to check whether the geometrical simplification is responsible for the differences pointed out in the fatigue analysis in presence of surface roughness. Acknowledgements This work is part of the DFG/AiF research cluster ‘‘IBESS’’. The authors gratefully appreciate the funding by the AiF network

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Please cite this article in press as: Madia M, Zerbst U. Application of the cyclic R-curve method to notch fatigue analysis. Int J Fatigue (2015), http:// dx.doi.org/10.1016/j.ijfatigue.2015.06.015