Combined resolved shear stresses as an alternative to enclosing geometrical objects as a measure of shear stress amplitude in critical plane approaches

International Journal of Fatigue xxx (2014) xxx–xxx

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Combined resolved shear stresses as an alternative to enclosing geometrical objects as a measure of shear stress amplitude in critical plane approaches F.C. Castro ⇑, J.A. Araújo, E.N. Mamiya, P.A. Pinheiro Department of Mechanical Engineering, University of Brasilia, 70910-900 Brasilia, DF, Brazil

a r t i c l e

i n f o

Article history: Received 31 October 2012 Received in revised form 10 March 2014 Accepted 31 March 2014 Available online xxxx Keywords: Multiaxial fatigue Critical plane Shear stress amplitude

a b s t r a c t A measure of shear stress amplitude based on a combination of resolved shear stress amplitudes on two perpendicular directions of a material plane is investigated in this paper. This measure is very fast to calculate. Hence, it turns unnecessary numerical schemes to accelerate the critical plane search, as well as it enables to significantly reduce the processing time of finite element based fatigue calculations, even when small angle increments are used. Findley’s relationship with the proposed shear stress amplitude provided estimates within a ±15% error interval for published fatigue limits obtained under proportional and non-proportional multiaxial loadings. The accuracy and computational cost of the approach are compared with those obtained with other measures of shear stress amplitude available in the literature. Ó 2014 Published by Elsevier Ltd.

1. Introduction High cycle fatigue failure has been a critical concern in industry for more than a century. One of the difficulties engineers have encountered when designing components and structures under cyclic loading is the multiaxial non-proportional stress history experienced at critical regions. To estimate fatigue strength under such condition, critical plane [1–9], integral [8,10–12], stress invariants [13–20] and multiscale stress-based models [21] have usually been considered. Critical plane models for multiaxial high cycle fatigue assessment are focused in this paper. The differences among them are related to the definition of the shear stress amplitude, the relationship between shear stresses and normal stresses, and the method used to determine the critical plane. Reviews and comparative studies of critical plane and other multiaxial fatigue models can be found in Refs. [22–25]. There are various methods to evaluate the amplitude and mean value of the shear stresses acting on a material plane. Earlier proposals known as the Longest Projection Method and the Longest Chord Method have been criticized by Papadopoulos due to drawbacks related to the determination of the mean shear stress [23,26]. In order to avoid such inconsistencies, Papadopoulos has

⇑ Corresponding author. E-mail address: [email protected] (F.C. Castro).

proposed the use of the Minimum Circumscribed Circle (MCC) to the shear stress path as the proper way to calculate the amplitude and mean value of the shear stresses. This method, however, provides the same amplitude for proportional and non-proportional shear stress paths circumscribed by the same minimum circle [9]. To quantify the non-proportionality of stress paths, the authors [17] as well as Freitas and co-workers [14,15] have proposed stress amplitudes based on a combination of the radii of minimum circumscribing ellipses, instead of the radius of the MCC. More recently, Araújo et al. [9] proposed an alternative method to evaluate the non-proportionality of the shear stress path, called Maximum Circumscribed Rectangle (MCR), which was shown to be a simple and fast approach to calculate the shear stress amplitude. In parallel to the efforts to improve the accuracy of the fatigue models, there has been an increasing interest in reducing the processing time of computer-based fatigue calculations. This trend has emerged from the widespread use of finite element-based fatigue analysis and the industrial demand for computational schemes to search for optimal design of components under fatigue-related constraints [27–29]. Methods to speed up the computation of the MCC were proposed by Weber and co-workers [30] and Bernasconi [31]. Later, the computational cost of these two and other wellknown methods were studied by Bernasconi and Papadopoulos [32] who suggested the use of the Randomised Algorithm as the most efficient one among the investigated methods. Some techniques have also been developed to accelerate the critical plane search [30,33].

http://dx.doi.org/10.1016/j.ijfatigue.2014.03.025 0142-1123/Ó 2014 Published by Elsevier Ltd.

Please cite this article in press as: Castro FC et al. Combined resolved shear stresses as an alternative to enclosing geometrical objects as a measure of shear stress amplitude in critical plane approaches. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.03.025

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This paper investigates the accuracy and computational efficiency of a new measure of shear stress amplitude. The results which will be presented here show that the use of Findley’s relationship with the proposed shear stress amplitude is capable to provide accurate estimates of endurance limits under multiaxial proportional and non-proportional stress histories. Another attractive feature of the proposal is that it is very fast to compute, so that its use in critical plane fatigue models does not need any kind of technique to speed up the calculation of the shear stress amplitude or of the critical plane search. 2. Basic concepts and definitions Critical plane models are based on the mechanisms of crack growth during their early stages of propagation. In the high-cycle fatigue regime, experimental observations have shown that small cracks occur on planes where a combination of shear and normal stresses is most severe [1–3,22,34]. Therefore, stress-based critical plane models require the evaluation of shear and normal stresses acting on the material planes. Expressions for such stresses are provided in this section, for both three-dimensional and plane stress states. The stress state at a material point is described with respect to a xyz coordinate system as

0

1

rx sxy sxz ry syz C A sxz syz rz

r¼B @ sxy

ð1Þ

Fig. 1(b) illustrates the shear stress vector and its components. To compute the normal and shear stresses given by Eqs. (3) and (5), respectively, the vectors ny0 ; nz0 and n can be described in terms of the angles h and / defined in Fig. 1 as follows:

0

1 sin h B C ny0 ¼ @ cos h A

0

cos / cos h

1

B C nz0 ¼ @ cos / sin h A sin /

0

0

1 sin / cos h B C n ¼ @ sin / sin h A cos / ð6Þ

Expressions for the calculation of normal and shear stresses for a state of plane stress are useful in practical applications, as most fatigue cracks initiate at the free surface of a component. In this case, the stress state is described as

0

1

rx sxy 0 C r¼B @ sxy ry 0 A 0

0

ð7Þ

0

with respect to a xyz coordinate system such that the x- and y-axes lie on the free surface and the z-axis points outward from the surface, as shown in Fig. 2(a). Substitution of Eqs. (6) and (7) into Eqs. (3) and (5) then leads to

rn ¼

r þ r  rx  ry x y 2 þ cos 2h þ sxy sin 2h sin / 2 2 

sx0 y0 ¼ 

rx  ry 2

 sin 2h þ sxy cos 2h sin /

r þ r  sin 2/ rx  ry x y þ cos 2h þ sxy sin 2h 2 2 2

ð8Þ ð9Þ

where rx ; ry ; rz and sxy ; sxz ; syz are the normal and shear stress components, respectively. The stress vector, t ¼ rn, acting on a material plane with unit normal vector n can be split into shear and normal components as

sx0 z0 ¼ 

t ¼ s þ rn n

For proportional stress histories, the path described by the shear stress vector on a material plane is a line segment, hence its amplitude is unambiguously defined as half of its length. On the other hand, for non-proportional stress histories various measures of shear stress amplitude, sa , are possible, as the shear stress path may be a more general curve [9,23,26,31,32,35]. However, it should be noticed that there are experimental data which shows that non-proportional loading may have a detrimental effect on the fatigue endurance [36], hence a measure of sa which is capable of distinguishing between proportional and non-proportional loading is not only an essential feature in the modeling of the fatigue phenomenon, but it has an influence on the quality of the fatigue limit estimates [9]. In this setting, the following measure of shear stress amplitude is investigated in this work:

ð2Þ

where s is the shear stress vector and

rn ¼ ðrnÞ  n

ð3Þ

is the normal stress. We now introduce a coordinate system ðny0 ; nz0 Þ lying on a material plane as shown in Fig. 1(a), with ny0 parallel to the intersection of the material plane with the xy-plane and nz0 pointing to the z axis. With respect to this coordinate system, the shear stress vector can be written as

s ¼ sx0 y0 ny0 þ sx0 z0 nz0

ð4Þ

with shear stress components given by the expressions

sx0 y0 ¼ rn  ny0

sx0 z0 ¼ rn  nz0

ð5Þ

ð10Þ

3. Measure of shear stress amplitude

Fig. 1. (a) Material plane with angles h and / and (b) illustration of a shear stress path.

Please cite this article in press as: Castro FC et al. Combined resolved shear stresses as an alternative to enclosing geometrical objects as a measure of shear stress amplitude in critical plane approaches. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.03.025

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Fig. 2. (a) Shear stresses acting along and inwards the surface,

sa ¼

ffi 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Ds2x0 y0 þ Ds2x0 z0 2

ð11Þ

where the shear stress ranges are calculated along directions y0 and z0 and then combined into an equivalent shear stress amplitude. The choice of such particular directions is motivated by the observation of crack initiation mechanisms at free surfaces [34]. In such plane stress conditions, crack formation is mainly governed by shear stresses along and/or inwards the surface, which correspond to the directions y0 and z0 defined in the previous Section (see Fig. 2). Hence, the ranges of shear stresses, Dsx0 y0 and Dsx0 z0 , can be interpreted, respectively, as driving forces for tearing (Mode III) and sliding (Mode II) modes of crack advance [37]. It is important to remark that directions y0 and z0 were deliberately chosen so that one can establish a physical connection between the directions of crack formation and their associated/governing shear stresses. Notice that the choice of such directions does not depend on the load path, but turns the measure of sa , as defined in Eq. (11), a frame-dependent problem (i.e. the choice of different directions would provide a different measure of sa ). It is important to note that the proposed measure is able to distinguish a proportional from a non-proportional shear stress path. Consider, for instance, the stress paths shown in Fig. 3. The proportional path W1 has a length equal to 2R and is enclosed by a circular path W2 with diameter equal to 2R. Thus, for the proportional path, pffiffiffi Dsx0 y0 ¼ Dsx0 z0 ¼ 2R and Eq. (11) yields sa ¼ R, while in the case of pffiffiffi the circular path sa ¼ 2R, as Dsx0 y0 ¼ Dsx0 z0 ¼ 2R. This distinction cannot be obtained by means of the MCC, for which in both cases sa ¼ R.

sx0 y0 and sx0 z0 , respectively, and (b) illustration of their ranges.

Another appealing feature of the proposed measure is that it requires very basic mathematical operations. As a consequence, it is straightforward to implement and provides very fast fatigue calculations. A final remark concerns the computation of Eq. (11) for an elliptical path, say with pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi semi-radii a and b. In this case, sa is always 2 equal to a2 þ b no matter the orientation of the ellipse, as a consequence of the theorem proved in Ref. [38]. 4. Comparison with experimental data Assessment was carried out based on Findley’s relationship

maxðsa þ jrn max Þ ¼ k

ð12Þ

h;/

where sa is the proposed shear stress amplitude given by Eq. (11), rn max is the maximum normal stress, while j and k are material parameters. For comparative purposes, three other measures of shear stress amplitude were also considered: (i) the MCR method [9] defined as

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 2

ðDsx0 y0 ðuÞÞ2 þ ðDsx0 z0 ðuÞÞ2 sa ¼ max u

ð13Þ

where Dsx0 y0 ðuÞ and Dsx0 z0 ðuÞ are the ranges of shear stresses resolved in the directions y0 ðuÞ and z0 ðuÞ, which are obtained by rotating around n the original coordinate system y0  z0 by an angle u; (ii) the measure

1 2

sa ¼ maxðDsx0 y0 ; Dsx0 z0 Þ

ð14Þ

reported in Ref. [35], which takes the maximum value between the shear stress amplitudes resolved along directions y0 and z0 and (iii) the radius of the MCC [26] defined as



sa ¼ min maxksðtÞ  sI mk I t sm

Fig. 3. Shear stress paths which illustrate the capability of the proposed measure to distinguish a proportional for a non-proportional pffiffiffi path. For the proportional path W1 ; sa ¼ R, while for the circular path W2 ; sa ¼ 2R.



ð15Þ

In this expression, the radius of the circle circumscribing the I shear stress  path, with  an arbitrary center sm , is given by the quanI  tity maxt sðtÞ  sm . Therefore, the radius of the MCC will be the minimum value of this quantity among all circles circumscribing the path. As a preliminary assessment of the resulting fatigue models, thirty long-life fatigue strength data sets taken from the literature were chosen, as listed in Table 1. Fatigue strengths were obtained with specimens made of different types of steels subjected to proportional and non-proportional loadings. Only data without superimposed mean stresses were considered, as the aim was to evaluate the influence of different measures of shear stress amplitude on the predictive model. The number of cycles at which

Please cite this article in press as: Castro FC et al. Combined resolved shear stresses as an alternative to enclosing geometrical objects as a measure of shear stress amplitude in critical plane approaches. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.03.025

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Table 1 Long-life fatigue strength data sets. Data set

Steel

Loading

N0 (cycles)

Ref.

1

0.1% C (normalized)

PB + T, H, IP, S

107

[46]

2

0.4% C (normalized)

PB + T, H, IP, S

107

[46]

3

0.4% C (spheroidized)

PB + T, H, IP, S

107

[46]

4

0.9% C (pearlitic)

PB + T, H, IP, S

107

[46]

5

3% Ni (30/35 ton)

PB + T, H, IP, S

107

[46]

6

3/3.5% Ni (45/50 ton)

PB + T, H, IP, S

107

[46]

7

CrVa (45/50 ton)

PB + T, H, IP, S

107

[46]

8

PB + T, H, IP, S

107

[46]

PB + T, H, IP, S

107

[46]

10

3.5% NiCr (normal impact) 3.5% NiCr (low impact) NiCrMo (60/70 ton)

PB + T, H, IP, S

7

10

[46]

11

NiCrMo (75/80 ton)

PB + T, H, IP, S

107

[46]

12

NiCr (95/105 ton)

PB + T, H, IP, S

[46]

13 14 15 16

Hard Mild C20 annealed (XC18) 39NiCrMo3

PB + T, H, IP, OP, S PB + T, H, IP, OP, S PB + T, H, IP, OP, S Ax + T, H, IP, S, AS

107 – – –

17 18 19 20 21

25CrMo4 – Kaniut 25CrMo4 – Mielke (a) 25CrMo4 – Mielke (b) 34Cr4 (a) 0.35% C

22 23 24

S65A steel 42CrMo4 SAE 52100 (DIN 100Cr6) 34Cr4 (b)

Ax + T, H, IP, OP, AS Ax + T, H, IP, OP, S Ax + T, NH Ax-T, NH Ax-IP-EP, H, IPh/OPh, S PB-T, H, IPh, S PB-T, H, IPh/OPh, S Ax-T, H, IPh/OPh, S

9

25 26 27

30NCD16 (a) 30NCD16 (b)

3  106 – – – – – – – 107

Table 2 Fully reversed axial (or bending) and torsional long-life fatigue strengths and parameters of Findley’s relationship.

[47] [47] [48,49] [40] [41,50] [39,43] [50,43] [50,39,44] [39,51] [52] [53,39] [54,55]

Steel

r1 (MPa)

s1 (MPa)

j

k (MPa)

0.1% C (normalized) 0.4% C (normalized) 0.4% C (spheroidized) 0.9% C (pearlitic) 3% Ni (30/35 ton) 3/3.5% Ni (45/50 ton) CrVa (45/50 ton) 3.5% NiCr (normal impact) 3.5% NiCr (low impact) NiCrMo (60/70 ton) NiCrMo (75/80 ton) NiCr (95/105 ton) Hard Mild C20 annealed (XC18 steel) 39NiCrMo3 25CrMo4 – Kaniut 25CrMo4 – Mielke 0.35% C S65A 42CrMo4 SAE 52100 (DIN 100Cr6) 34Cr4 (b) 30NCD16 (a) 30NCD16 (b) 30NCD16 (c) 30NCD16 (d)

269 332 275 352 343 445 429 541 510 602 661 772 314 235 332 368 340 361 216 584 398 866 410 585 690 660 560⁄ 658  690] 423

151 207 156 241 205 267 258 352 324 338 343 453 196 137 186 265 228 228 137 371 260 540 256 405 428 410 428 428 428 286

0.12 0.25 0.14 0.40 0.20 0.20 1.66 0.32 0.28 0.12 0.04 0.18 0.26 0.17 0.12 0.49 0.36 0.27 0.28 0.28 0.32 0.26 0.26 0.42 0.25 0.25 0.62 0.32 0.25 0.38

152.1 213.6 157.4 259.3 209.0 272.5 263.5 369.2 336.6 340.6 343.2 460.0 202.3 138.9 187.4 295.1 242.6 263.3 142.2 385.4 273.1 557.3 264.3 438.8 441.0 422.6 625.7 474.0 444.1 305.6

XC48 Ax-T, H, IPh/OPh, S

2  106

RB-T, H, IPh, S/AS PB-T, H, IPh/OPh, S

106 6

28

30NCD16 (c)

PB-T, H, IPh/OPh, S

29

30NCD16 (d)

30

XC48

Ax-RB-PB-T, H, IPh/ OPh, S Ax-IP-T, H, IPh/OPh, S

–

[39,56– 58] [42] [59]

]   ⁄

Plane bending. Rotating bending. Push–pull.

[42]

10 –

[49,60]

105

[39,61]

PB = plane bending, RB = rotating bending, Ax = axial, T = torsion, IP = internal pressure, EP = external pressure, H = harmonic, NH = nonharmonic, IP = in-phase, OP = out-of-phase, S = synchronous, AS = asynchronous, N 0 = number of cycles at which long-life fatigue strength was obtained.

with conservative and non-conservative estimates. Material parameters j and k provided in Table 2 were obtained with the following expressions:

1  0:5r

j ¼ pffiffiffiffiffiffiffiffiffiffiffi r1

fatigue strengths were obtained are also reported, when such information was available. Monotonic and cyclic properties of the materials are listed in Table 2. For a detailed description of the materials and fatigue strengths considered in this work, the reader may refer to Papuga’s FatLim Database [39]. Multiaxial stress paths selected for this study are shown in Fig. 4. All cases describe combined normal and shear stresses. Stress paths generated under in-phase and out-of-phase synchronous loadings are shown in Fig. 4a. Those corresponding to asynchronous loadings are shown in Fig. 4b–d and were taken from experimental work conducted respectively by Bernasconi et al. [40], Kaniut [41] and Froustey [42]. Cross and box stress paths investigated by Mielke [43] and Heidenreich et al. [44] are shown in Fig. 4e and f, respectively. As frequently considered in the literature [23,25], the error defined as

maxðsa þ jrn max Þ  k e¼

h;/

k

ð%Þ

ð16Þ

was chosen for the assessment of the models, where sa and rn max are quantities calculated with observed long-life fatigue strengths. Positive and negative values of the error are associated respectively

r1

k ¼ pffiffiffiffiffiffiffiffiffiffiffi 2 r1

ð17Þ

where r1 and s1 are respectively the fully reversed axial (or bending) and torsional fatigue strengths at a given life and r ¼ r1 =s1 . Fig. 5 shows the errors of the models for each data set. Results were identical for both the method of Ref. [35] and the MCC, with errors within ±15% bandwidth, except for two data. Estimates using the proposed method and MCR were almost identical, except for a few tests of data sets 16, 17 and 18, and all errors fell within a ±15% bandwidth. 5. Computational cost To assess the computational cost of the proposed shear stress amplitude, a non-proportional shear stress path having a butterfly shape was used [31]. Results were compared with the ones obtained with shear stress amplitudes defined by the MCR and MCC methods, as well as with the method reported in Ref. [35]. The procedures for the computation of the shear stress amplitudes were implemented in MATLAB [45]. Simulations were performed on a machine with Intel Core i7 processor running at 1.87 Hz and with 8 GB RAM. To compute the MCC, the incremental algorithm proposed by Dang Van [21] was employed: The initial center was defined as the centroid of the points, the initial radius and the coefficient of expansion were set to 102 and 0.05, respectively, and iterations stopped when the relative difference between

Please cite this article in press as: Castro FC et al. Combined resolved shear stresses as an alternative to enclosing geometrical objects as a measure of shear stress amplitude in critical plane approaches. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.03.025

F.C. Castro et al. / International Journal of Fatigue xxx (2014) xxx–xxx

5

(a)

(b)

(c)

(d)

(e)

(f)

Fig. 4. Multiaxial stress paths.

the new and old radii was smaller than 104 . In the MCR algorithm, the circumscribing rectangle was rotated in steps Du ¼ 10 , as recommended in Ref. [9]. The benchmark test evaluated in Ref. [31] considers a shear stress history given as

sx0 y0 ðtÞ ¼ 100 sin ðxtÞ

sx0 z0 ðtÞ ¼ 100 sin ð2xt  p=4Þ

ð18Þ

which provides the butterfly path shown in Fig. 6. Fig. 7 shows the processing times of the algorithms for loading paths made of 30, 60, 120 and 240 points. Each processing time was defined as the mean value of the elapsed times over 104 executions. As can be seen, the processing time of the proposed shear stress amplitude does not depend on the number of points. The

same trend was observed for the measure of Ref. [35] and the MCR. On the other hand, the MCC algorithm has a linear dependence between processing time and number of points. It can also be seen that the computational cost of the proposed measure, and the one given in Ref. [35], is respectively about 10 and 100 times smaller than that of the MCR and MCC algorithms.

6. Discussion and conclusions From an engineering point of view, perhaps the simplest approach to take the non-proportionality of the shear stress vector path into account is to combine the resolved shear stress

Please cite this article in press as: Castro FC et al. Combined resolved shear stresses as an alternative to enclosing geometrical objects as a measure of shear stress amplitude in critical plane approaches. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.03.025

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Ref. [37]

Proposed

Fig. 5. Prediction errors of Findley’s relationship with the investigated shear stress amplitudes.

Fig. 6. Shear stress path defined by Eq. (18) for 30 discretization points.

Ref. [37]

Proposed

Fig. 7. Computational cost of the investigated shear stress amplitudes for the butterfly path defined by Eq. (18).

amplitudes on two perpendicular directions of a material plane. As a consequence, such measure becomes extremely simple to calculate and to implement, while providing results as accurate as the ones obtained by other measures, however at a much lower computational cost. At a first glance, such approach could be thought of as a simplification of the MCR method where only one rectangle with specific directions is required to calculate sa . However, as it is further detailed below, the proposed measure does not require the definition of a geometric object enclosing the shear stress path in order to calculate the shear stress amplitude, but only resolved stresses. Due to such feature, different shear stress vector paths,

Fig. 8. Shear stress paths havingpdifferent ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi shapes, but the same resolved shear 2 stress ranges. In both cases, sa ¼ a2 þ b .

but which have the same resolved shear stress ranges, will provide the same amplitude of equivalent shear stress, according to the proposed measure, as illustrated in Fig. 8. This figure contains circular and rectangular paths. For these two different paths, it is pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 clear that Eq. (11) provides the same sa ¼ a2 þ b . The method of Ref. [35] also provides the same sa ¼ a (assuming a > b) for the two paths, but it is only sensitive to one of the resolved shear stress ranges. On the other hand, shear stress amplitudes based on the concept of enclosing geometric objects are dependent on the shape of the stress path. For instance, both the MCC and MCR methods would compute different values of sa for the two paths. The fact that this new measure of sa does not require the definition of a geometric object enclosing the shear stress path in order to calculate the shear stress amplitude, but only resolved stresses, has another interesting consequence from the physical standpoint. For plane stress conditions, the mechanisms of crack formation can be directly associated with shear stresses along directions y0 and z0 , which are driving forces for tearing and sliding modes of crack advance. On the other hand, for more general (3D) state of stress, the use of a frame dependent sa as proposed here may lack the physical interpretation discussed for plane stress conditions. However, it should be emphasized that most of the fatigue problems occur in free surfaces of components. Based on the analysis and results of this paper, the following conclusions can be stated: 1. For steels, multiaxial fatigue limit estimates based on Findley’s relationship with the proposed measure of shear stress amplitude fell within a ±15% error interval. For the MCC, MCR and the measure reported in Ref. [35] similar results were obtained. 2. The processing time of the proposed shear stress amplitude, as well as the one from Ref. [35], are approximately 10 and 100 times smaller than those of MCR and Dang Van’s MCC algorithms, respectively. 3. The proposed measure proved to be much easier to calculate and to implement than approaches based on enclosing geometric objects. 4. The very small processing time of the proposed shear stress amplitude, as well as the one reported in Ref. [35], turns fatigue calculations based on critical plane models very fast, even when small angle increments are used. Further, there is no need of numerical schemes to accelerate the critical plane search.

Acknowledgements The authors would like to acknowledge the financial support of the National Counsel of Technological and Scientific Development

Please cite this article in press as: Castro FC et al. Combined resolved shear stresses as an alternative to enclosing geometrical objects as a measure of shear stress amplitude in critical plane approaches. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.03.025

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Please cite this article in press as: Castro FC et al. Combined resolved shear stresses as an alternative to enclosing geometrical objects as a measure of shear stress amplitude in critical plane approaches. Int J Fatigue (2014), http://dx.doi.org/10.1016/j.ijfatigue.2014.03.025