A multiaxial fatigue criterion for various metallic materials under proportional and nonproportional loading

International Journal of Fatigue 28 (2006) 401–408 www.elsevier.com/locate/ijfatigue

A multiaxial fatigue criterion for various metallic materials under proportional and nonproportional loading Ying-Yu Wang*, Wei-Xing Yao Department of Aircraft Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Received 5 January 2005; received in revised form 19 May 2005; accepted 11 July 2005 Available online 2 September 2005

Abstract Proportional and nonproportional tension–torsion fatigue tests were conducted on LY12CZ aluminum alloy. Two types of tubular specimens were used, one is smooth and the other is notched. The experimental data are analyzed. A new critical plane criterion including the strain and stress parameters is proposed. The capability of fatigue life prediction for the proposed fatigue damage model is checked against the experimental data of LY12CZ aluminum alloy and two other metals under proportional and nonproportional loading, and the predicted results are compared with results from common multiaxial fatigue model. It is demonstrated that the proposed criterion gives better satisfactory results for all the three checked materials. q 2005 Elsevier Ltd. All rights reserved. Keywords: Multiaxial fatigue; Fatigue criteria; Critical plane approach; Nonproportional loading

1. Introduction Many engineering components usually undergo complex multiaxial loadings, which lead to changing of the principle stresses and strains directions during a cycle of loading. The additional hardening of material, which is caused by the rotation of the principle stress and strain axes, is considered to have tight relation to the reduction of fatigue life under nonproportional loading compared with that under proportional loading [1–4]. Although many multiaxial fatigue criteria suitable to different materials and different loading conditions have been proposed, those capabilities to correlate the experimental fatigue life under multiaxial loading do not reach a satisfactory level. Reviews of available multiaxial fatigue life prediction methods are presented by Garud [5], Brown and Miller [6], You and Lee [7], Papadopoulos [8], Macha and Sonsino [9], Wang and Yao [10]. Fatigue life prediction approaches using the concept of a critical plane have been found very effective because the critical plane concept is based on the physical observations that * Corresponding author. Tel.: C86 25 84892177; fax: C86 25 84891422. E-mail address: [email protected] (Y.-Y. Wang).

0142-1123/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2005.07.007

cracks initiate and grow on preferred planes. According to the parameter used in the fatigue criteria, the critical plane approaches can be classified into three categories, namely stress critical plane criteria [11–15], strain critical plane criteria [16–21] and energy critical plane criteria [22–25]. However, the critical plane approaches have a shortcoming due to the fact that the critical plane does not always coincide with the plane where the fatigue damage parameter takes its maximum value [26–28]. Successful models should be able to predict both the fatigue life and the dominant failure plane(s) [29]. In the present study, a series of proportional and nonproportional tension–torsion fatigue tests were conducted on LY12CZ aluminum alloy. Two types of tubular specimens were used, one is smooth and the other is notched. The objective of the fatigue tests on the notched specimen is to study fatigue crack nucleation angle under multiaxial loading. A multiaxial fatigue parameter based on the critical plane concept is proposed. Two methods to define the critical plane (one is the maximum shear strain range plane and the other is the maximum fatigue damage plane) are compared with the statistical experimental results of the crack nucleation angle. The predictive capabilities of the proposed parameter and one popular multiaxial fatigue model proposed by Kandil, Brown and Miller [17] are

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Nomenclature D3n Dg Dgmax sn E sy su sf 3f K

n q D sf0 b 3f0 c Nf N50

normal strain range on the critical plane shear strain range on the critical plane maximum shear strain range normal stress amplitude on the critical plane Young’s modulus yield stress (0.2%) ultimate strength true fracture strength true fracture strain strength coefficient

checked against the experimental data of LY12CZ aluminum alloy, and some other experimental data on 1045 steel [30] and 6061 aluminum alloy [31] under multiaxial loading are also compared.

2. Experiment 2.1. Material and specimen The material tested was a common aeronautic material, LY12CZ aluminum alloy. The material has the following chemical compositions (wt%): Cu4.34; Mg1.48; Mn0.77; Fe0.29; Si!0.15; Zn!0.1; Ni!0.05 and its mechanical and cyclic properties are shown in Tables 1 and 2, respectively. The mechanical and cyclic properties of two other metals, which will be employed to check the proposed models for fatigue life prediction, are also listed in Tables 1 and 2, respectively. Two types of tubular specimens were used, the first one is a thinwalled tubular specimen (in Fig. 1) while the other is a notched thin-walled tubular specimen (in Fig. 2), the notch is a transverse circular hole with 2 mm diameter on semi-circle of the tubular specimen.

strain hardening exponent crack nucleation angle damage parameter axial fatigue strength coefficient axial fatigue strength exponent and axial fatigue ductility coefficient axial fatigue ductility exponent fatigue life fatigue life at survival probability 50%

2.2. Experimental procedure and results Stress controlled tension–torsion biaxial fatigue tests were carried out on a servo-hydraulic MTS Model 858 axial-torsion testing system. The experiments were conducted at room temperature. A fully reversed sinusoidal stress wave was used (i.e. the stress ratio is equal to K1.0). 2.2.1. Experiment of the thin-walled tubular specimen The stress paths employed in this study were: (a) inphase, (b) 458 out-of-phase, (c) 908 out-of-phase loading. Three maximum Mises’ equivalent stress amplitudes for each stress path were chosen, that is 350, 300 and 250 MPa, respectively. The testing frequencies were 10 Hz. Five specimens were tested for each loading condition. The failure was defined as the first observation of a surface crack 1.0 mm long. The experimental results are summarized in Table 3. 2.2.2. Experiment of the notched tubular specimen The stress paths employed on the notched tubular specimens were: (a) in-phase, (b) 458 out-of-phase and (c) 908 out-of-phase loading. The maximum Mises’ equivalent stress amplitude used in each test was 161 MPa. The testing

Table 1 Mechanical properties of LY12CZ, 1045 steel and 6061 aluminum alloy Material

E (GPa)

sy (Mpa)

su (MPa)

n

sf (MPa)

3f

K (MPa)

n

LY12CZ 1045HR 6061Al

73 202 80

400 380 337

545 621

0.33 0.29 0.32

643 985

0.18 0.71

850 1185

0.158 0.23

Table 2 Material cyclic properties Material

sf0 (MPa)

b

3f0

c

K (MPa)

n0

LY12CZ 1045HRa b 6061Al

724 948 528

K0.063 K0.092 K0.089

0.137 0.260 0.225

K0.654 K0.445 K0.629

870 1258 470

0.097 0.208 0.11

a b

Fatigue properties are given by referenced papers. Fatigue properties are calculated from test data.

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Fig. 1. Smooth tubular specimen (mm).

frequencies were 5 Hz. Three tests were performed for each stress path. The crack nucleation position around the hole edge on the specimen surface is described by q as shown in Fig. 2. The angle measured anticlockwise from the specimen’s axis is defined as positive, and it is assumed to range in the interval K908 to C908. Each loading case has shown the following number of cracks on the edge of the hole: (a) inphase:6, (b) 458 out-ofphase:3, (c) 908 out-of-phase:7. Table 4 shows the distribution of the crack nucleation positions, in which the relative frequency of crack nucleation position represents, for each interval range of 108 range, the number of experimental tests whose crack nucleation position falls in the interval considered, normalized with respect to the total number of cracks found in the considered loading case.

3. Multiaxial fatigue damage model 3.1. One existing popular multiaxial fatigue model The most popular strain critical plane criterion was proposed by Brown and Miller [16]. They proposed that the shear strain governs crack initiation and growth, and that the normal strain assists crack growth. The general form is

expressed as: Dgmax C f ðD3n Þ Z C 2

(1)

A convenient expression for Eq. (1) was suggested by Kandil, Brown and Miller (KBM) [17]: Dgmax C SD3n Z C 2

(2)

where Dgmax is the maximum shear strain range, D3n is the normal strain range acting on the Dgmax plane and S is a material constant. 3.2. A new multiaxial fatigue damage parameter The relationship between the fatigue life and the maximum shear strain range Dgmax, the normal strain range D3n acting on the Dgmax plane and the amplitude of the maximum normal stress sn acting on the Dgmax plane are illustrated in Fig. 3(a)–(c), respectively. From Fig. 3(b) and (c), it can be observed that under the same equivalent stress loading, D3n and sn acting on the Dgmax plane become larger when the phase delay between axial and torsional loading increases. The experimental fatigue lives presented in Table 2 and Refs. [30,31] are usually reduced when the phase delay between axial and torsional loading increases. Therefore, it is rational

Fig. 2. Notched tubular specimen (mm).

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Table 3 Fatigue lives of LY12CZ under in-phase and out-of-phase loading (Cycles) Stress path

a

sa (MPa)

ta (MPa)

f (deg)

Nf (cycles)

N50 (cycles)

247.52

142.91

0

13,824 13,329 12,173 24,725 24,360 51,494 33,391 44,203 81,622 53,386 405,312 381,483 348,696 424,156 336,097

16,831

212.16

122.50

0

176.81

102.08

0

247.52

142.91

45

212.16

122.50

45

176.81

102.08

45

247.52

142.91

90

212.16

122.50

90

176.81

102.08

90

104,32 7900 11,487 12,715 9305 18,800 34,334 29,048 –a 51,813 198,490 619,530 109,071 623,110 618,572 7297 13,947 11,190 6658 21,898 21,911 30,558 43,640 33,002 25,934 10,4315 74,855 88,715 64,209 103,859

50,585

377,694

10,229

31,340

348,899

11,067

30,173

85,684

Accidental break.

to use the D3n and sn acting on the Dgmax plane as the fatigue damage parameters under multiaxial loading, because they reflect the fact that the fatigue lives are reduced when the phase delay between axial and torsional loading increases. Based on the critical plane concept, a multiaxial fatigue damage parameter, including the stress and strain acting on the critical plane, is proposed. The maximum shear strain plane is taken as the critical plane.
 Dgmax sn C 1K (3) D3n Z f ðNf Þ 2 2sy

Table 4 Distribution of crack nucleation positions Angle interval (deg)

Relative frequency of crack nucleation positions (%)

K95 to K85 K85 to K75 K75 to K65 K65 to K55 K55 to K45 K45 to K35 K35 to K25 K25 to K15 25 to 35 35 to 45 45 to 55 55 to 65 65 to 75 75 to 85 85 to 95

0 0 25 75 0 0 0 0 0 0 0 0 0 0 0

0 33.33 0 33.33 0 0 33.33 0 0 0 0 0 0 0 0

0 14.29 0 0 0 0 0 0 0 14.29 0 57.14 14.29 0 0

The critical plane does not always coincide with the plane where the fatigue damage parameter takes its maximum value. Therefore, if we define the maximum damage plane on which the fatigue damage parameter assumes its maximum value as the critical plane, the Eq. (3) becomes 
  Dg sn C 1K Z f ðNf Þ (4) D3n 2 2sy max The correlation of the fatigue life and the KBM parameter, the proposed parameter which takes into account the maximum shear strain as the critical plane (WYKgmax) and the proposed parameter which takes the maximum damage plane as the critical plane (WYKDmax), are shown in Fig. 4(a)–(c), respectively. As can be observed from Fig. 4(a), the phase difference between the axial stress and the torsional stress has a direct influence on the amplitude of the KBM parameter under the same Mises’ equivalent stress loading. The amplitude of the KBM parameter under the 908 out-of-phase loading is much larger than that under the in-phase loading under the same Mises’ equivalent stress loading. This will lead to a considerable difference in the predicted fatigue life between in-phase loading and out-of-phase loading situations. The proposed parameter, which takes the maximum shear strain plane as the critical plane, shows a good correlation with the fatigue life under in-phase and out-of-phase loading as shown in Fig. 4(b). However, compared with Fig. 4(c), the amplitudes of the proposed parameter shown in Fig. 4(b) is smaller: this means that the predicted fatigue life given by the proposed parameter, which takes the maximum shear strain plane as the critical plane, is greater than those given by the model which takes the maximum damage plane as the critical plane. The critical plane coincides with the plane where the fatigue damage parameter takes its maximum value under 908 out-of-phase loading, so the amplitude of

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Fig. 3. Correlation between the fatigue life and the parameters on the Dgmax plane: (a)Dgmax, (b)D3n, (c) sn.

two different parameters shown in Fig. 4(b) and (c), respectively, is the same. The above analysis can be considered as qualitative ones, and further quantitative study on the predictive capability of the three multiaxial fatigue parameters will be shown in the following sections. 3.3. Critical plane The critical plane concept is based on the physical observations that cracks initiate and grow on preferred planes. These planes on which cracks initiate and grow, and hence on which the dominant fatigue damage should occur, are called critical planes [32]. However, the critical plane does not always coincide with the plane where the fatigue damage parameter takes its maximum value. The stress and strain states on the edge of the hole were analyzed by elastic–plastic finite element analyses. The finite element code MSC/NASTRAN was used as the analysis solver. Here, tetrahedron elements were used. The finite element model of the specimen consists of 6330 elements and 11,216 nodes. The linear elastic, linear strain hardening plastic stress–strain curve was chosen to describe the material deformation behavior. Isotropic hardening and the Von Mises

yield criterion were used for elastic–plastic FEA. The distributions of fatigue cracks, and the maximum shear strain and the damage parameter are shown in Fig. 5. The column above each interval has a height equal to the number of relative frequency of crack nucleation positions falling in this interval as shown in Table 4. The solid lines show the relative magnitude and distribution of the maximum shear strain with orientation; the broken lines show the relative magnitude and distribution of the damage parameter, the quantity in square brackets of Eq. (4), with orientation. It is found that the distribution of the damage parameter is similar to that of the maximum shear strain. Furthermore, the distributions of these two parameters are consistent with that of the fatigue cracks detected for all loading cases considered. Therefore, it is rational to define both the plane where the fatigue damage parameter takes its maximum value and the maximum shear strain plane as the critical plane.

4. Experimental verifications For tension–tortion loading situations, the maximum shear strain amplitude and the normal strain amplitude on the plane that experiences the maximum shear strain are

Fig. 4. Correlation of fatigue life with three fatigue damage models: (a) KBM parameter, (b) WYKgmax, (c) WYKDmax

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Fig. 5. Frequency histograms showing the distribution of the fatigue crack nucleation positions (a) Proportional, (b) 458 out-of-phase, (c) 908 out-of-phase.

given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gmax Z ð1 C neff Þ2 32 C g2  1Kneff 3n Z 3 2

(5)




(6)

where neff is an effective Poisson’s ratio. It is defined by neff Z ðne 3e C np 3p Þ=3t

relationship is obtained: 
  Dg smax C 1K n D3n 2 2sy max s0 Z ð1 C ne Þ f ð2Nf Þb C 1:53f0 ð2Nf Þc E    sf0 sf0 b b 0 c C 1K ð2Nf Þ ð1Kne Þ ð2Nf Þ C 0:53f ð2Nf Þ 4sy E

(7)

(11)

where ne is the elastic Poisson’s ratio, np is the plastic Poisson’s ratio assuming it is equal to 0.5 for all the materials considered in this paper, 3e, 3p, 3t are elastic, plastic, total strain, respectively. For proportional loading conditions, the maximum values of these quantities can be obtained by setting 3Z3a and gZga. And for proportional loading conditions, the maximum shear stress and the maximum shear strain occur on the same plane. Therefore, the stress normal to the plane of the maximum shear strain is given by

Predictions using Eqs. (10) and (11) appear in Figs. 6 and 7.

sn Z

s 2

(8)

The widely used strain-life relationship is so-called Manson-Coffin’s equation. It is written as: D3 sf0 Z ð2Nf Þb C 3f0 ð2Nf Þc 2 E

(9)

Combining Eq. (3) by using the so-called MansonCoffin’s equation will result in:
 Dgmax s s0 C 1K n D3n Z ð1 C nÞ f ð2Nf Þb 2 2sy E   0 sf 0 c b (10) C 1:53f ð2Nf Þ C 1K ð2Nf Þ 4sy   s0 ! ð1KnÞ f ð2Nf Þb C 0:53f0 ð2Nf Þc E Combining Eq. (4) by using the so-called MansonCoffin’s equation, the following approximate life

4.1. Evaluation by the experimental data for LY12CZ aluminum alloy The experimental data listed in Table 3 for LY12CZ are used to evaluate the KBM parameter, the WYKgmax parameter and the WYKDmax parameter. The results’ comparisons are shown in Fig. 6. As it can be seen from Fig. 6, the WYKDmax parameter yields the most satisfactory result. Correlation of the WYK Dmax parameter is within a factor of 3 under in-phase loading, while the predictions given by the KBM parameter and the WYKg max parameter fall far away from the experimental data under in-phase loading as shown in Fig. 6(a). As it can be seen from Fig. 6(b), the correlation of the WYKDmax parameter is within a factor of 2, and the correlation of the KBM parameter is within a factor of 3. The predictions given by the WYKgmax parameter are all on the nonconservative side under our-of-phase loading, in which the predictions are in disagreement by a factor of 3. 4.2. Evaluation by the experimental data for 1045HR steel and 6061 aluminum alloy In order to examine the general fitness of the WYKDmax parameter, the experimental data found in literature for SAE-1045 steel reported in Kurath et al. [30], and 6061 aluminum alloy tested by Itoh et al. [31] are used to evaluated such a parameter. Furthermore, prediction by the

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Fig. 6. Predicted versus experimental lives for LY12CZ: (a) inphase loading, (b) out-of-phase loading.

KBM parameter is also included for comparison. All the specimens consist of thin-walled tubular specimens, and the corresponding strain paths used are in-phase and 908 out-ofphase loading for two materials. The mechanical and cyclic properties of SAE-1045 steel and 6061 aluminum alloy are listed in Tables 1 and 2, respectively. The assessment of the obtained results is shown in Fig. 7. As it can be seen from Fig. 7, the prediction of the WYK Dmax parameter is better than that of the KBM parameter. Correlation of the WYKDmax parameter is within a factor of 2 in life except for the data of 1045 steel under out-ofphase loading, and it shows to be more conservative than that of the KBM parameter.

5. Conclusions For LY12CZ aluminum alloy under the same Mises’ equivalent stress amplitude loading, the fatigue life under nonproportional loading is lower than that under

proportional loading, and the fatigue life under 908 out-ofphase loading has been observed to be the lowest possible. The stress and strain states on the edge of the hole were analyzed by elastic–plastic finite element analyses. The obtained distributions of fatigue cracks, the maximum shear strain and the results given by the proposed damage parameter are compared. It is found that the distribution of the proposed damage parameter is similar to that of the maximum shear strain. Furthermore, the distributions of these two parameters are consistent with that of the fatigue cracks for all the loading cases considered. Therefore, it is rational to define both the plane where the fatigue damage parameter is taken its maximum value and the maximum shear strain plane as the critical plane. A multiaxial fatigue parameter based on the critical plane concept is proposed. The relevant values of this parameter consist of the amplitude of the shear strain, the normal stain range and the amplitude of the normal stress range acting on the critical plane. The plane on which the damage parameter reaches its maximum value is taken as the critical plane.

Fig. 7. Predicted versus experimental lives for 1045 steel and 6061 aluminum alloy.

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The prediction given by the WYKDmax parameter is better than that of the KBM parameter for LY12CZ aluminum alloy, SAE-1045 steel and 6061 aluminum alloy under inphase and our-of-phase loading. Correlation of the WYK Dmax parameter is within a factor of 2 in life except for the data of 1045 steel under out-of-phase loading, and is more conservative than that of the KBM parameter. It has been shown that the new critical plane criterion gives good correlation of multiaxial fatigue lives for various proportional and nonproportional loading conditions for different material fatigue data.

Acknowledgements The authors gratefully acknowledge the financial support of the National Natural Science Foundation of China (10377007), the National Doctoral Foundation of China (Project No. 20020287022) and the Innovative and Excellent PhD Thesis Foundation of NUAA (Project No. 4003-019001).

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