A study on combined torsion and axial load fatigue limit tests with stresses of different frequencies

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International Journalof Fatigue

International Journal of Fatigue 30 (2008) 1430–1440

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A study on combined torsion and axial load fatigue limit tests with stresses of different frequencies A. Bernasconi a, S. Foletti a, I.V. Papadopoulos a

b,*

Politecnico di Milano, Dipartimento di Meccanica, via La Masa 34, 20156 Milano, Italy b European Commission, Joint Research Centre, 21020 Ispra (Varese), Italy

Received 11 April 2007; received in revised form 2 October 2007; accepted 15 October 2007 Available online 22 October 2007

Abstract The aim of this paper is to investigate the high cycle fatigue behaviour of metals in the case of multiaxial loading where the stresses are of different frequencies. In particular, combined axial load and torsion fatigue tests were performed on specimens made of quenched and tempered steel. This material has been chosen because it is commonly employed for the production of mechanical components intensively stressed under lengthy fatigue loads. Two frequency ratios between the shear stress and the normal stress are examined and the resulting fatigue limits from these tests are compared to the results from tests where the shear and normal stress are of the same frequency and inphase. Moreover, different multiaxial fatigue limit criteria, based on the critical plane concept and on the so-called integral approach, are used to analyse the obtained experimental results. In general, better agreement between predictions and experimental results is achieved by the criteria of the integral approach category. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Non-proportional loading; Multiaxial fatigue; High cycle fatigue; Fatigue limit

1. Introduction A limited number of high cycle multiaxial fatigue experiments have been reported in the literature for tests where the applied stresses vary with different frequencies. Tests of this kind have been performed by Mielke and Kaniut; the results are reported in Liu and Zenner [1]. Mielke and Kaniut used stress systems of two normal stresses or of one shear and one normal stress. Similar tests have been conducted by Heidenreich et al. [2]. Tests with two normal stresses of different frequencies have been performed by McDiarmid [3,4], whereas Froustey [5] performed a limited number of bending and torsion tests. Both the effect of the stress waveform and the effect of different frequencies have been addressed by the tests with two normal stresses of Dietmann et al. [6]. *

Corresponding author. Tel.: +39 0332789373; fax: +39 0332786507. E-mail address: [email protected] (I.V. Papadopoulos). 0142-1123/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2007.10.003

The results of Mielke and Kaniut refer to tests conducted on 25CrMo4 steel. Two types of loading were examined; biaxial pulsating normal stresses and combined fully reversed tension–compression and fully reversed torsion. In all cases the stress waveform was sinusoidal. In the tests with two normal stresses the stress system is defined by rx = rxa sin(fxt) + rxm and ry = rya sin(fyt) + rym. The amplitudes of the two normal stresses and the corresponding mean values were equal, i.e. rxa = rya and rxm = rym. Also the stress ratio (min stress over max stress) of each normal stress was the same and equal to R = 0.05. Two frequency ratios ky = fy/fx were considered namely, ky = 1 and ky = 2. Clearly, for ky = 1 the loading is proportional and the principal stress directions remain fixed. For ky = 2 the loading is non-proportional. However, even for the nonproportional loading with ky = 2, the two applied stresses are principal normal stresses and the directions along which they are acting, which are obviously principal directions, remain fixed. The second load system tested by Mielke and Kaniut, was the combined fully reversed

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tension–compression and alternating torsion where the corresponding normal and shear stresses vary with different frequencies, i.e. rx = rxa sin(fxt) and sxy = sxya sin(fxyt). The value of the shear stress amplitude was half the value of the normal stress amplitude sxya = rxa/2. Tests were performed for various frequency ratios kxy = fxy/fx and in particular for kxy = 0.25, 1, 2 and 8. With the exception of the value kxy = 1 which produces proportional loading, all the other frequency ratio values lead to non-proportional loading. A noticeable difference between the present stress system and the previous one consisting of two normal stresses is that a non-proportional torsion and axial loading always leads to varying principal stress directions, whereas as explained above a load system composed of two normal stresses conserves the principal stress directions invariant independently of the proportionality or not of the loading. For the tests of Mielke and Kaniut in both loading cases, i.e. two normal stresses or one shear and one normal stress, the fatigue limit decreased for frequency ratios higher than ky = 1 and kxy = 1, respectively. Similar results were obtained by Heidenreich et al. [2] for a 34Cr4 steel tested with frequency ratios ky = 2 and kxy = 0.25, 1, and 4. McDiarmid [3,4] conducted tests on tubular specimens made of EN24T steel by applying fully reversed biaxial stresses of equal amplitude, either in-phase or in-phase opposition (i.e. 180° out-of-phase), at different frequencies, with ky = 1, 2 and 3. Results showed decreasing fatigue limits for increasing frequency ratios. The data reported by Froustey [5] refer to combined alternating bending and torsion tests on a 30NCD16 steel, with sxya = rxa and frequency ratios kxy = 0.25 and kxy = 4, which showed that the fatigue limits are practically the same irrespectively of which of the stress components varies with higher frequency. However, no results are provided for the case kxy = 1 with sxya = rxa, thus precluding the possibility to draw any conclusion about the effect of increasing or decreasing the frequency ratio between the shear stress due to torsion and the normal stress due to bending. Dietmann et al. [6] reported results for biaxial normal stresses loading, where the stress waveforms were sinusoidal, triangular and trapezoidal, with frequency ratios of ky = 1 and ky = 2, and initial phase difference of either 0° or 90°. Different results were obtained for different waveforms, but in all cases a decrease of the fatigue limit with respect to the in-phase loading was observed. From the above brief account of previous work done in this field, it appears that a ample part of the performed experiments concerns biaxial stress systems made-up of two normal stresses. As already explained this stress system implies the invariance of the principal stress directions, which remain the directions of the applied normal stresses, irrespectively if these stresses are in-phase, out-of-phase, of the same or of different frequencies. A partial set of data refers to combined axial (or bending) load and torsion where the shear and normal stresses are of different frequencies. This non-proportional load system, unlike the

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biaxial normal stress system which conserves the principal stress directions, produces a continuous variation of the principal stress directions in time. Combined axial load and torsion is the loading system used for the tests presented in this work. 2. Experiments The material used for the fatigue experiments is a steel of grade 39NiCrMo3, supplied in quenched and tempered bars of 35 mm diameter, which had already been used for the study of the influence of a mean shear stress upon the fatigue limit in torsion, Davoli et al. [7]. This material had been chosen because it is commonly employed for the production of highly stressed mechanical parts submitted to lengthy fatigue loads. Macro and micro-hardness measurements were performed in order to verify the uniformity of the heat treatment. A constant macro-hardness of HRC 26 was obtained, while a micro-hardness of HV 279 was measured along a diameter from the surface to the core, ensuring that a uniform heat treatment had been applied. Monotonic and cyclic tests on standard specimens, extracted from the inner part of the bars, were carried out and the results are summarized in Table 1. The strength factor K 0 and the strength coefficient n 0 , reported in Table 1, refer to a cyclic stress–strain relationship of the Ramber– n0 Osgood type Dr=2 ¼ K 0 ðDep =2Þ . The specimens used for fatigue tests were machined by turning, followed by conventional polishing with progressively finer emery papers. Specimen shape and dimensions are shown in Fig. 1. A final average roughness Ra = 0.69 lm was measured. After polishing, surface residual stresses were measured on a reduced set taken from the whole batch of the specimens. Residual stresses were measured by means of an X-ray diffractometer, revealing that non-uniform compressive residual stresses were present at the material surface. In order to avoid the possible disturbance of the surface residual stresses on the results of the fatigue tests, all the specimens were submitted to an electrolytic polishing (EP) treatment. The pertinent fatigue properties of the 39NiCrMo3 steel, i.e. axial (fully reversed tension–compression) and shear (fully reversed torsion) fatigue limits, are reported in Table 2. These fatigue limits were evaluated by adopting the short staircase method [8] and the staircase method, respectively (the number of specimens used is reported in Table 2). Since the short staircase method does not allow for an Table 1 Monotonic and cyclic properties of the 39NiCrMo3 steel used in the experiments Young Modulus Tensile strength Tensile yield strength Elongation at failure Cyclic strength coefficient Cyclic strength exponent

E = 20,6000 MPa Rm = 856 MPa Rp0.2 = 625 MPa A% = 18.5% K 0 = 1188 MPa n 0 = 0.1342

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Fig. 1. Specimen shape and dimensions.

Table 2 Fully reversed tension–compression and alternating torsion fatigue limits of the 39NiCrMo3 steel Type of loading

Fatigue limit

Number of specimens

Fully reversed tension–compression Alternating torsion

rW = 367.5 MPa sW = 265.0 MPa

6 13

accurate calculation of the associated scatter, this latter is not reported here. The short staircase method was also applied to determine the fatigue limits under combined torsion and axial load with different values of the frequency ratio k between the shear stress s and the normal stress r. The stress system employed is defined by rx = rxa sin(frt) and sxy = sxya sin(fst). Three different values of k = fs/fr were chosen k = 1, 2 and 3. The multiaxial fatigue tests

were conducted a constant stress amplitude ratio of pffiffiwith ffi ðsxya =rxa Þ ¼ 1= 3. The first two load paths are shown in Fig. 2. Clearly, the load combinations with k = 2 and k = 3, generate non-proportional load paths, characterized by the variation of the principal stress directions with time, as shown in Fig. 3 for k = 2. In the upper part of this figure the two principal stresses time histories are drawn, whereas in the bottom part the variation with time of the first principal direction measured from the longitudinal axis of the specimen is shown. All the tests were performed in load control mode by employing a MTS 809 multiaxial tension/torsion servohydraulic testing system, with a capacity of 250 kN axial force and of 2500 N m torque. Tests were interrupted at specimen failure (appearance of a crack causing a maximum torsional rotation of 2 degrees) or after 3.106 cycles

Fig. 2. Applied shear and axial stresses corresponding to k = 1 and k = 2.

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Fig. 3. Time histories of the principal stresses and of the first principal stress direction for the case of k = 2.

in torsion (run-outs). Run-outs did not show any crack on the surface of the specimens. Orientation of initiated cracks for the failed specimens is discussed in the next section. Additionally, an analysis of a potentially inelastic behaviour of our specimens during testing has been done, to check if the use of nominal stress values in the fatigue calculations shown in this work, is legitimate. It is noticed that the highest level of maximum von Mises equivalent stress is attained in our testing campaign for the in-phase loading case (i.e. k = 1), and for the highest applied stress amplitudes of 320 MPa (axial stress) and 185 MPa (shear stress due to torsion). For these stress conditions the von Mises equivalent stress is equal to 453 MPa, which is well below the cyclic yield limit of 516 MPa of the material. This cyclic yield limit is calculated at 0.2% plastic strain using n0 the cyclic stress–strain curve Dr=2 ¼ K 0 ðDep =2Þ , established under fully reversed axial loading. The values of the strength factor K 0 and the strength coefficient n 0 are reported in Table 1. Since the highest von Mises stress ever attained amongst all the tests we performed is significantly lower than the cyclic yield limit of the material, it is concluded that elastic behaviour prevailed throughout the whole experimental campaign. Hence, use of nominal stress values in the fatigue calculations shown in this work, is correct. 3. Results and evaluation of multiaxial fatigue criteria Table 3 reports the fatigue limits obtained for the three chosen values of the frequency ratio k. As already men-

Table 3 Multiaxial fatigue limits of the 39NiCrMo3 steel under combined torsion and axial load with different frequency ratios Frequency ratio

rxaD [MPa]

sxaD [MPa]

Number of specimens

k=1 k=2 k=3

294.5 259.5 266.0

170 15 153.6

7 7 7

tioned, the fatigue limits were evaluated by applying the short staircase method [8], for each value of k to the full sequence of test results under multiaxial loading. Table 4 shows the tests results along with all the steps of the short staircase method used for determining the fatigue limits. However, the short staircase method does not allow for an accurate calculation of the associated scatter. Hence, values of the scatter are not reported here. It is seen from Table 4 that for k = 2, that is when the frequency of the torsion load becomes twice that of the axial load, the limiting (allowable) normal and shear stress amplitudes are clearly lower than those achieved at k = 1. For k = 3, the allowable shear and normal stress amplitudes achieve values slightly higher than the values observed at k. However, these allowable values remains always lower than those of the proportional (k = 1) loading case. Our test results are plotted in the graph of Fig. 4 together with the axial–torsional test results of Mielke and Kaniut on 25CrMo4 steel and the results of Heidenreich on 34Cr4 steel. These data referring to test run with a ratio of the shear stress amplitude over the axial stress

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Table 4 Multiaxial fatigue test results Specimen number

rxa (MPa)

sxya (MPa)

Number of torsional cycles to failure

k=1 1 2 3 4 5 6 7

320 303 286 303 286 303 286

185 175 165 175 165 175 165

158,507 450,550 Run-out 300,816 Run-out 1,050,903 Run-out

68 69 63

k=2 1 2 3 4 5 6 7

286 269 252 269 252 269 252

165 155 145 155 145 155 145

523,026 1,388,100 Run-out 1,111,022 Run-out 1,331,896 Run-out

124 125 123

k=3 1 2 3 4 5 6 7

234 251 268 251 268 286 268

135 145 155 145 155 165 155

Run-out Run-out 575,964 Run-out Run-out 388,893 582,300

109

w (deg)

63

125

105 104

the fractured specimens the observed cracking behaviour was characterized by the appearance of a crack on the specimen surface, which formed an angle w with the specimen axis. The determination of the angle w required observation of the fracture surfaces at the optical microscope. The angle w formed by the crack profile on the specimen surface was measured at the initiation site. Values of the angle w are reported in the last column of Table 4. The results were compared with the predictions of selected existing multiaxial fatigue criteria. From the class of critical plane approaches the Dang Van [9] and Findley [10] criteria have been maintained, since they are broadly cited in the relevant literature. A variant of the Dang Van criterion, involving the Tresca and hydrostatic stresses, without the necessity of locating a critical plane is also examined, Dang Van et al. [11], Dang Van [12] because it achieved some popularity in recent years. Moreover, the Papadopoulos [13,14] and the Liu–Zenner [1] criteria were chosen as representative of the class of criteria implementing the so-called integral approach. Detailed description of the criteria employed in the present work, as well as the methods of identification of their parameters can be found in the references cited above. 3.1. Critical plane type criteria The comparison of experimental results with the predictions of the critical plane type criterion of Dang Van [9] is shown in Fig. 5. In this figure the following graphical representation of the data is adopted. In the abscissa it is reported the value of the frequency ratio k = fs/fr in a logarithmic scale. In the ordinates it is reported the ratio of the limiting normal stress amplitude (fatigue limit) achieved at a given k value, denoted as rxa(k), normalized over the experimentally determined amplitude of the axial stress at the fatigue limit for k = 1, denoted as rxa(k = 1); i.e. the ratio rxa(k/rxa(k = 1) is reported in the ordinates. Hence,

Fig. 4. Values of the allowable axial stress as a function of the frequency ratio k for our tests on 39NiCrMo3 and tests of Mielke and Kaniut (25CrMo4 steel) and Heidenreich (34Cr4 steel). Allowable axial stress values are normalized by dividing by the fatigue limit in tension– compression of the corresponding material.

amplitude equal to 0.5 are reported in Liu and Zenner [1]. In Fig. 4, for every combined tension and torsion loading test the allowable normal stress amplitude, divided by the fatigue limit in tension–compression rW of the corresponding material, is plotted against the frequency ratio. For all the three steels shown in Fig. 4 it appears that for k 6¼ 1 the observed fatigue strength is lower than for k = 1. As said before specimens which did not fail (run-outs) did not show any cracks at the end of the cycling. For

Fig. 5. Comparison between experimental data and prediction of the critical plane Dang Van criterion.

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the co-ordinates of the test point corresponding to the proportional loading case k = 1 are (1, 1). Two more points in Fig. 5 indicate the test results at k = 2 and k = 3. The experimental points are shown as solid circles. Predictions by the critical plane Dang Van criterion have been calculated for the integer values of the frequency ratio k = 1, 2, 3, . . ., 10, as well as for their reciprocal k = 1/2, 1/3, . . ., 1/10. These predictions are shown as star shaped points in the same graph. For reporting the Dang Van predictions the same normalisation has been used as with our experimental test data. In the sequel the discussion will be limited in the range of integer k values since it is for these values that we have experimental data. Observing the Dang Van predictions in Fig. 5 one can notice that for k = 1,3,5,7 and 9 the Dang Van criterion predicts the same fatigue limit. Thus the Dang Van criterion predicts that the fatigue limit remains unaffected by the different frequencies of the shear and normal stresses provided that the frequency ratio k is an odd integer. Always from Fig. 5 one can notice that if k is an even integer k = 2,4, . . . , the predicted by Dang Van fatigue limits are decreasing, but the predictions remain higher than the limit shown in the Dang Van graph at k = 1. Already, for k = 1, there is a discrepancy between the prediction of the Dang Van criterion and the experimentally determined limit, which lies higher than the corresponding Dang Van point. But what is striking is the fact that for k = 2 the Dang Van criterion predicts not only a limit higher than the experimentally determined value, but also it shows improved fatigue strength with respect to its own prediction for the proportional case k = 1, whereas the opposite trend is verified experimentally. Thus the performance of the Dang Van criterion seems to be in disagreement with the experimental results. The above reported somehow peculiar behaviour of the Dang Van criterion deserves more attention given that this criterion achieved some popularity in recent years. To attempt an explanation a short reminder of the Dang Van model is needed. Let us consider a material plane D passing through a point O of a body submitted to periodic loading. The material plane is defined by its unit normal vector denoted as ~ n. It is noticed that the components of ~ n can be expressed in a O. xyz frame attached to the specimen with the help of the spherical angles u and h, where h is the angle that makes ~ n with the z axis and u is the angle that makes the projection of ~ n onto the xy plane, with the x axis, Fig. 6. Therefore, the stresses acting on a material plane can be considered as functions of u and h. The Dang Van equivalent stress acting on D is defined as req(u, h, t) = sDV(u, h, t) + arH(t) at any time of the load period. The quantity rH is the hydrostatic stress. The quantity sDV(u, h, t) is defined as: sðu; h; tÞ ~ sm ðu; hÞk sDV ðu; h; tÞ ¼ k~

ð1Þ

~ sðu; h; tÞ is the shear stress vector acting on the material plane under consideration and the symbol k  k denotes the length of the enclosed vector. The meaning of

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z

θ Δ

n→

y

→ m

χ

ξ ϕ

x Fig. 6. Material plane D.

~ sm ðu; hÞ will be made clear further on. The stress vector acting on D is given as ~ sn ðu; h; tÞ ¼ rðtÞ  ~ n and the normal stress vector as ~ rn ðu; h; tÞ ¼ ½~ n  rðtÞ  ~ n~ n, where r(t) is the stress tensor. Clearly, the shear stress vector is the difference between the vectors ~ sn and ~ rn , i.e. ~ sn ðu; h; tÞ ¼ rðtÞ  ~ n  ½~ n  rðtÞ  ~ n~ n. For any constant amplitude periodic loading the vector ~ sðu; h; tÞ describes a closed curve on the material plane being considered. For proportional loading this curve is rectilinear; actually, it is reduced to a straight line segment passing through the origin (zero shear stress). This line segment is run through twice by the tip of the vector ~ sðu; h; tÞ during a loading cycle. In this case the constant vector ~ sm ð/; hÞ, appearing in the expression of sDV ðu; h; tÞ ¼ k~ sðu; h; tÞ ~ sm ðu; hÞk, is the vector which points to the middle of the line segment described by ~ sðu; h; tÞ on D in a cycle. Thus, always for proportional loading k~ sm ð/; hÞk, is the usual mean shear stress. For non-proportional loading ~ sm ð/; hÞ is defined as the vector which points to the centre of the smallest circle enclosing the curve described by the tip of the shear stress vector ~ sðu; h; tÞ on D, Papadopoulos [15]. The vector ~ sm ð/; hÞ, thus defined is unique and constitutes a generalisation in the case of non-proportional loading of the mean shear stress concept usually applied for proportional loading. Indeed, for proportional loading the smallest enclosing circle is centred in the middle of the line segment representing the path of ~ sðu; h; tÞ on the material plane under investigation. An assessment of algorithms on how to construct the smallest enclosing (circumscribed) circle to the closed curve described by the tip of ~ sðu; h; tÞ in order to define the unique mean shear stress vector ~ sm ð/; hÞ and hence calculate the

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re-centred shear stress path ~ sDV ðu; h; tÞ can be found in Bernasconi and Papadopoulos [16]. The application of the Dang Van criterion requires determining first the so-called critical plane. For Dang Van it is the plane onto which ~ sDV ðu; h; tÞ reaches its maximum value. Therefore, once determined the mean shear stress ~ sm ð/; hÞ on every plane passing through a point of the body, the determination of the critical plane according to Dang Van requires the solution of the double maximisation problem shown below: max½max k~ sðu; h; tÞ ~ sm ðu; hÞk u;h

t

ð2Þ

Denoting by u* and h* the spherical co-ordinates solution of the above problem, the Dang Van criterion is written as: sDV ðu ; h ; tÞ þ arH ðtÞ 6 sW ; 8t

ð3Þ

The above criterion depicted in the plane sDV  rH with rH in the abscissas and sDV in the ordinates, achieves the form of a straight line of (negative) slope a intercepting the vertical axis at the fatigue limit in fully reversed torsion denoted as sW. For the 39NiCrMo3 steel tested sW = 265 MPa. Moreover, the slope a can be expressed with the help of the fatigue limits in fully reversed torsion sW and fully reversed tension–compression rW as a = 3(sW/rW  1/2), Dang Van [9]. For the material tested one has a = 0.66. This line of negative slope a delimits the safe against fatigue locus from above, this same locus being delimited from below by the rH axis. For the purpose of brevity in the sequel the dependence of sDV on u* and h* is abandoned in our notation; hence sDV(t) is understood as the sDV corresponding to the critical plane, i.e. sDV(u*, h*, t). Eq. (3) states that fatigue crack nucleation takes place if the sDV(t) against rH(t) curve intersects the fatigue limit line of (negative) slope a. When the sDV(t)  rH(t) curve is tangent to the limit line, the point of tangency defines the stress state, which corresponds to the fatigue limit conditions. After this short reminder of the Dang Van approach, let us turn our attention to the analysis of the behaviour of this criterion in the case where the frequency ratio k is equal to 2. It is reminded here that for k = 2 the Dang Van criterion predicts a higher fatigue limit than for, k = 1 whereas the opposite is observed experimentally. In Fig. 6, on the plane of the Dang Van criterion the sDV(t)  rH(t) stress path for k = 1 is drawn. On the same graph it is also reported another sDV(t)  rH(t) path, which is built with the same normal stress and shear stress amplitudes as for k = 1, but now for k = 2. The first sDV  rH stress path (with k = 1) is made up from two straight line segments. The highest point of the right hand line segment is the point of the sDV(t)  rH(t) path closest to the limit line of the criterion (i.e. closest to the line of negative slope a). Let us call this point the critical stress state of the k = 1 loading. For k = 2 the whole path sDV  rH falls well below the limit line; it entirely remains within the boundaries of the safe domain, Fig. 7. In particular its critical

Fig. 7. Application of the critical plane type Dang Van criterion; stress paths on the sDV  rH plane for k = 1 and k = 2.

stress state (i.e. the point of the path closest to the line of the criterion) remains significantly lower than the critical stress state of the k = 1 case. Therefore, according to Dang Van the different frequencies loading with k = 2, is less damaging than the equal frequencies k = 1 loading. Hence, for the different frequencies loading with k = 2 we have to increase the stress amplitudes of the normal and shear stresses in order to reach the limit line. This explains why the fatigue limit predicted by the Dang Van graph for k = 2 in Fig. 5 is higher than the limit predicted by this same criterion for k = 1. Here the case k = 2 is examined. Nevertheless, this conclusion seems to hold true for any even k, see Fig. 7. Conversely, as shown in Fig. 8, when k = 3, the corresponding critical stress state of the sDV  rH path (i.e. the point of the path closest to the line of the criterion) coincides with the critical stress state of the sDV(t)  rH(t) path built for the same stress amplitudes as for k = 3, but with k = 1. This happens for all odd integer values of the frequency ratio k examined in this work. This explains why the Dang Van approach predicts fatigue limit values in Fig. 5, which are all equal for k odd. Having noticed the above curious behaviour of the original Dang Van model [9], it was considered of interest to examine a more recent variant of this model, Dang Van et al. [11]. This version does not necessitate the location of a specific critical plane beforehand as the original Dang Van model. Actually, following [11,12] one has first to construct the smallest hyper-sphere enclosing the path described by the stress deviator s(t) in the corresponding five-dimensional space. Let us denote by s the stress deviator which corresponds to the co-ordinates of the centre of this smallest enclosing hyper-sphere. The proper statement of the problem of the minimum circumscribed (smallest enclosing) hyper-sphere to the path described by the stress deviator can be found in Papadopoulos et al. [17], whereas efficient algorithms are included in Welzl [18]. Then one can calculate a time varying stress deviator ^sðtÞ as follows:

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n  rðtÞ  ~ n rn;max ðu; hÞ ¼ max½~ t

Fig. 8. Application of the critical plane type Dang Van criterion; stress paths on the sDV  rH plane for k = 1 and k = 3.

^sðtÞ ¼ sðtÞ  s

ð4Þ

Clearly, the path of ^sðtÞ is re-centred around the (zerostress) origin. With the help of ^sðtÞ a Tresca stress is further introduced by: 1 ^sT ðtÞ ¼ ðs^I ðtÞ  ^sIII ðtÞÞ ð5Þ 2 ^sI and ^sIII are the instantaneous maximum and minimum principal stresses of the re-centred deviator ^sðtÞ. The present version of the Dang Van criterion is then written as: ^sðtÞ þ arH ðtÞ 6 sW 8t

ð6Þ

It turns out that the predictions of this version of the Dang Van criterion are practically identical to the predictions shown in Fig. 5 of the previously examined critical plane Dang Van model. Hence, further discussion will be omitted here for brevity purpose. The above described peculiarities of the Dang Van approach are not found when comparing the experimental results with the Findley criterion, expressed by: sa ðu ; h Þ þ jF rn;max ðu ; h Þ 6 bF

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ð9Þ

Moreover, for non-proportional loading the shear stress amplitude sa(u, h) acting on a material plane is defined as the radius of the smallest circle enclosing the path described by the tip of the shear stress vector i.e. sa ðu; hÞ ¼ maxt k~ sðu; h; tÞ ~ sm ðu; hÞk, whereas ~ sm ð/; hÞ is defined like in the critical plane version of the Dang Van approach as the vector which points to the centre of this smallest circle. Application of the Findley approach for the load conditions of concern in the present work leads to the fatigue limit values reported in Fig. 9. Here, it can be observed that the Findley criterion is able to predict the decrease of the fatigue limit for increasing frequency ratios, but the predicted limits are as 10% as lower than the experimental results. Since the Findley approach reproduces the general trend of the experimental results (albeit with noticeable discrepancy) one can conclude that the peculiarities encountered with the Dang Van approach do not characterize the whole class of critical plane approaches; they are rather tied to the Dang Van criterion. 3.2. Integral approach criteria Two criteria have been maintained from the class of the so-called integral approaches, namely the criteria proposed by Papadopoulos and Liu–Zenner, respectively. The Papadopoulos criterion is based on a mesoscopic scale approach; however, the final formula is based on the usual (macroscopic) stresses, Papadopoulos [14]. A measure of the intensity of the cyclic shear stresses acting at a point O of a periodically loaded body is introduced as follows. Let us consider first a material plane D defined by the spherical co-ordinates u and h of its unit normal vector ~ n. As explained previously, for a periodic loading the tip of the shear stress vector ~ sðu; h; tÞ describes on the material plane D a closed curve. A direction lying on this plane D

ð7Þ

The parameters of the criterion are given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jF ¼ ½1  rW =ð2sW Þ= rW =sW  1 and bF ¼ sW j2F þ 1, Findley [10]. For the material investigated jF = 0.493 and bF = 295 MPa. In Eq. (7) sa and rn,max are understood as the shear stress amplitude and the maximum normal stress acting on the critical plane that is the plane (u*, h*), which according to Findley is given by: sa ðu ; h Þ þ jF rn;max ðu ; h Þ ¼ max½sa ðu; hÞ þ jF rn;max ðu; hÞ u;h

ð8Þ

Let us remind that the normal stress vector acting on a n  rðtÞ  ~ n~ n; therematerial plane is given as ~ rn ðu; h; tÞ ¼ ½~ fore its algebraic value is equal to rn ðu; h; tÞ ¼ ½~ n  rðtÞ  ~ n such that rn,max is simply obtained as:

Fig. 9. Comparison between experimental data and prediction of the critical plane type Findley criterion.

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can be defined by the angle v that makes this direction with a fixed, arbitrarily chosen reference direction n on the plane under consideration. Let us denote as ~ m a unit vector along the direction characterized by this angle v, Fig. 6. On this line is acting a resolved shear stress given by the projection ~, i.e. given by the inner product of ~ sðu; h; tÞ onto m [~ m ~ sðu; h; tÞ]. Clearly, the resolved shear stress being defined as the inner product of two vectors is a scalar quan~ of can be expressed in the O. xyz tity. The components m frame through the spherical angles u, h and v. Therefore, the resolved shear stress is function of u, h and v (and of time t); hence it is denoted as s(u, h, v, t).The amplitude of the resolved shear stress sa(u, h, v) is calculated as below:   1 max½~ sa ðu; h; vÞ ¼ m ~ sðu; h; tÞ  min½~ m ~ sðu; h; tÞ t t 2 ð10Þ With the help of the resolved shear stress amplitude a measure denoted as Mr, of the intensity of the cyclic shear stress state at a point O of the body is defined as:  1=2 Z 2p Z p Z 2p 5 2 Mr ¼ s ðu; h; vÞ dv sin h dh du 8p2 u¼0 h¼0 v¼0 a

Fig. 10. Comparison between experimental data and predictions of the Papadopoulos and Liu–Zenner criteria belonging in the integral approach.

 sequ;a ¼ sequ;m ¼

ð11Þ Actually, the above Mr integral is an expression of the average value of the shear stress amplitudes evaluated over all directions on a material plane (hence the integration over v) and over all the material planes passing through the point under consideration (hence the integration over u and h). The terminology ‘‘integral approach’’ used for this criterion derives from the introduction of the above Mr integral. Analytical expressions for the Mr integral under in-phase and out-of-phase sinusoidal loading where the stresses are of the same frequency can be found in Papadopoulos [13,14]. For sinusoidal loading where the stresses are of different frequencies, as the loading considered in the present work, a numerical evaluation of the Mr integral is required. The criterion is written as: M r þ aP rH ;max 6 sW

ð12Þ

The parameter pffiffiffi aP is related to sW and rW by aP ¼ 3sW =rW  3; hence for the material examined aP = 0.43. Comparison of the predictions of the Papadopoulos criterion with the test results is shown in Fig. 10; discussion of these predictions is given further on jointly with the discussion of the predictions of the Liu–Zenner criterion. The Liu–Zenner criterion is expressed by the following relationship: as2equ;a þ br2equ;a þ cs2equ;m þ dreqn;m 6 r2W

ð13Þ

In the above formula the quantities sequ,a and sequ,m can be conceived as particular averages of the shear stress amplitude and mean shear stress acting on all material planes passing through a given point of the body. They are defined as:

15 8p

Z

2p u¼0



1 sequ;a

15 8p

Z

p

1=2 s2a ðu; hÞ sin h dh du

ð14Þ

h¼0

Z

2p u¼0

Z

p

1=2 s2a ðu; hÞs2m ðu; hÞ sin h dh du

h¼0

ð15Þ In Liu and Zenner [1] it is not suggested a specific approach to be used for the calculation of the shear stress amplitude sa(u, h) and mean value sm(u, h) appearing in the above integrals, a problem which is of interest especially when dealing with non-proportional loading. It is shown in Bernasconi and Papadopoulos [16] that incorrect proposals exist, which lead to non-uniqueness of sm(u, h). It is essential to understand that all the stress quantities appearing in the formulation of the examined criteria necessitate definitions, which unambiguously assure uniqueness of the calculated values. Lack of this rigour led sometimes in the past to confusion regarding the use of various fatigue criteria, see Papadopoulos et al. [17]. Care has been taken in the present work to fulfil this requirement of rigorousness. Indeed, for the application of the Liu–Zenner criterion, as with the criterion of Findley and the critical plane Dang Van model, the concept of the (unique) smallest circle enclosing the path described by the tip of the shear stress vector is used. Hence, the (unique) mean shear stress sm(u, h) acting on a material plane is defined as the length of the vector which points to the centre of this smallest circle; accordingly, the shear stress amplitude sa(u, h) is defined as the radius of this smallest circle. Moreover, the quantities sequ,a and sequ,m appearing in the Liu–Zenner criterion, Eq. (13), are specific averages of the normal stress amplitude and mean value, respectively, acting on a material plane. They are defined as follows: 

requ;a

15 ¼ 8p

Z

2p u¼0

Z

p

h¼0

1=2

r2n;a ðu; h; Þ sin h dh du

ð16Þ

A. Bernasconi et al. / International Journal of Fatigue 30 (2008) 1430–1440

requ;m ¼



1 r2equ;a

15 8p

Z

2p

Z

u¼0

p

r2n;a ðu; hÞrn;m ðu; hÞ sin h dh du



h¼0

ð17Þ Reminding that the algebraic value of the normal stress acting on a material plane is ½~ n  rðtÞ  ~ n, the corresponding amplitude and mean value are given by: 1 rn;a ðu; hÞ ¼ fmax½~ n  rðtÞ  ~ n  min½~ n  rðtÞ  ~ ng t 2 t 1 n  rðtÞ  ~ n þ min½~ n  rðtÞ  ~ ng rn;m ðu; hÞ ¼ fmax½~ t 2 t

ð18Þ ð19Þ

It is noticed that in the definition of requ,m, Eq. (17), the mean normal stress rn,m appears in the integrand linearly, thus allowing to take into account the fact that tensile and compressive mean normal stresses affect differently the fatigue strength. The four parameters of the criterion Eq. (13) require the knowledge of four fatigue limits, which could be the fatigue limits in fully reversed torsion sW and fully reversed tension–compression rW, along with the fatigue limits in pulsating torsion sSch and pulsating tension rSch. The parameters a and b are expressed with the help of sW and rW alone: "   # 2 1 rW 3 a¼ 4 ð20Þ 5 sW "  2 # 1 rW 62 b¼ ð21Þ 5 sW For fully reversed loading, as the test data examined here, the above two parameters suffice for the application of the Liu–Zenner criterion. For the 39NiCrMo3 steel of the present work one has a = 0.35 and b = 0.43. For loading cases with a mean stress state the knowledge of parameters c and d is requested, which in turn necessitates additionally the knowledge of sSch and rSch: 2  sSch 2 r2W  rsWW 2 ð22Þ c¼  2 s 4 Sch

7

2

 2  4c 2 r2W  rSch 1 þ 21 2 d¼ 5 rSch 7

ð23Þ

The pulsating tension fatigue limit rSch can also be estimated through empirical relationships involving the tensile strength of the metal. Such relationships expressing the sensitivity of the axial fatigue limit of the material to the application of a mean normal stress is suggested in graphical form in Zenner et al. [19]. The fatigue limits predicted by the Papadopoulos and the Liu–Zenner criteria are reported onto the graph of Fig. 10, together with the experimental results. In this case it clearly appears that these two criteria of the class of integral approaches allow for better predictions than the Dang Van and the Findley criteria examined previously. The best correlation with experiments is achieved by the Liu–Zenner criterion. A possible explanation of the globally better performance of the criteria of the integral class for these results can be found considering the more complex mechanism of fatigue crack nucleation, which is likely to activate more than one slip systems when the principal directions rotate during the load cycle. Therefore, the criteria of the integral approach class are likely to describe these phenomena better, since they define equivalent stresses as mean values over all material planes, thus implicitly assuming the activation of multiple slip systems. 4. Concluding remarks The multiaxial fatigue behaviour of the 39NiCrMo3 steel was investigated by means of combined tension–compression and alternating torsion tests. In these tests, the ratio between the frequencies of the axial and the torsion loads was varied. Results were compared with the predictions of the Dang Van and Findley criteria representative of the critical plane approach and the Papadopoulos and Liu–Zenner criteria belonging to the so called integral approaches. A version of the Dang Van criterion, based on the Tresca stress of the re-centred stress deviator and the hydrostatic stress, which does not require the location in advance of a critical plane, is also examined. It is noticed that in the present work it has not been addressed the physical basis of the fatigue models used. The various criteria have been examined solely from the viewpoint of their predictive performance against the experimental results. The following conclusions can be drawn:

2

The four parameters of the Liu–Zenner criterion could perhaps discourage one from using this approach. However, in Zenner et al. [19] it is argued that the judgment on the appropriateness of a proposed criterion has not to be based on the number of parameters used, as far as dedicated experiments or appropriate empirical formulae allow for the determination of the parameters included. Indeed, in the absence of experimentally determined fatigue limit in pulsating torsion the following empirical formula is suggested in Zenner et al. [19] 4sW sSch ¼ 2rW þ1 rSch

1439

ð24Þ

– In general, increasing the frequency of the alternating torsion with respect to the frequency of the push–pull axial load leads to a reduction of the allowable stress amplitudes. Indeed, for both frequency ratios k = 2 and k = 3 examined, the experimentally determined fatigue limits are lower that the limit at k = 1. It is noticed that the limit at k = 3 is slightly higher than the limit at k = 2 although this increase appears negligible for practical purposes. – Among the criteria of the critical plane class, the Findley criterion correctly reproduces the general trend of the test data albeit with significant discrepancies between predictions and test values. The Dang Van approach

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produces somehow peculiar results. In particular, following the Dang Van approach one concludes that the fatigue behaviour remains unaffected by changing the frequency ratio of the torsion over the axial load as far as the frequency ratio k is an odd integer. In addition, for k even the Dang Van criterion predicts an improvement of the fatigue resistance; a prediction which is in disagreement with the overall trend of the test data. – The multiaxial fatigue criteria based on the integral approach yield predictions closer to the experimental results than the criteria of the critical plane class, with the Liu–Zenner formula producing better correlation than the Papadopoulos criterion. – Multiaxial fatigue tests under non-proportional load systems producing continuous variation of the principal stress directions inside each load cycle are more likely to evidence differences between predictions of criteria and experimental results than tests conducted with fixed principal stress directions.

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