A modified energy-based model for low-cycle fatigue life prediction under multiaxial irregular loading

International Journal of Fatigue 128 (2019) 105187

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A modified energy-based model for low-cycle fatigue life prediction under multiaxial irregular loading Yingya Lua, Hao Wua, , Zheng Zhongb, ⁎

a b

T

⁎

School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China School of Science, Harbin Institute of Technology, Shenzhen 518055, China

ARTICLE INFO

ABSTRACT

Keywords: Damage accumulation Multiaxial fatigue Life prediction Effective energy parameter Irregular loading path

This work provides a damage accumulation algorithm to calculate low cycle fatigue life under multiaxial irregular loading path, using an intuitively constructed energy-based parameter which combines both cyclic plastic work and non-proportional (NP) hardening effect. The cumulative fatigue damage is calculated according to actual shape of the loading path, which is different from the traditional damage accumulation models that consider block sequence effect. The effectiveness of this proposed damage accumulation algorithm has been validated by experimental results of 316L stainless steel under nine regular loading paths and nine irregular multiaxial loading paths.

1. Introduction Engineering components are generally subjected to multiaxial irregular or random loading and the multiaxial fatigue life prediction for multiaxial irregular loading is quite challenging. For damage computation of a multiaxial variable loading, a general idea is to divide the multiaxial variable amplitude histories into approximately irregular, regular or even constant amplitude loading paths and then to count the cycles. In this procedure, damage caused by every divided cycle can be calculated by means of the fatigue criteria for constant amplitude loading. So a multiaxial fatigue analysis process consists of three parts: (a) identify and count every cycle; (b) calculate the damage of every cycle by the fatigue criteria; (c) accumulate the fatigue damage for different cycles. Bannantine and Socie counted cycles on different candidate planes and determined the critical plane which has largest fatigue damage [1]. Wang and Brown proposed a cyclic counting method based on an equivalent strain and indicated that hysteresis strain hardening could provide a rational basis for multiaxial cycle counting [2,3]. Moreover, many damage accumulation theories have been proposed, such as linear cumulative damage theory [4], nonlinear cumulative damage theory [5,6], damage curve method [7], two-stage linear damage theory [8], continuum damage mechanics method [9,10] and so on. Among these cumulative damage theories, Miner linear cumulative theory is widely adopted due to its convenience and relative accuracy.

⁎

In order to avoid the complex determination of multiaxial elastoplastic constitutive relation, different fatigue criteria have been proposed based on plastic work. Morrow [11] postulated that accumulation of plastic work is the main reason of material fatigue failure. For proportional tension-torsion loading histories, Garud [12] presented an exponential model associating with plastic work, and multiaxial fatigue life can be estimated based on uniaxial fatigue data through introducing a weighting factor to reflect the shear work contribution in total fatigue damage. Besides, Ellyin [13–15] and his collaborators employed total strain energy density per cycle as a fatigue parameter, which can be calculated from von Mises equivalent stress and strain amplitude, to introduce mean stress effects into fatigue life prediction. Liu [16] generalized the uniaxial energy-based model into a virtual strain energy model which can estimate the multiaxial elastic and plastic work on a specific plane. Chu [17] also proposed a similar energy parameter by using the maximum stress ranges instead of the stress ranges to include the effect of mean stresses. Zhu et al. [18] proposed an energy-based equivalent damage parameter based on uniaxial fatigue data and the non-proportional hardening factor to predict the fatigue life under multiaxial fatigue loadings. In Zhu’s model, the contributions to total elastoplastic work from different loading components can be quantified by using the moment of inertia method. Although the energy-based models are usually more convenient than critical plane methods and more accurate than equivalent strain methods, the traditional energy-based models could not consider variable amplitude loading paths. As Socie [19] indicated, it is difficult to

Corresponding authors. E-mail addresses: [email protected] (H. Wu), [email protected] (Z. Zhong).

https://doi.org/10.1016/j.ijfatigue.2019.105187 Received 10 March 2019; Received in revised form 3 July 2019; Accepted 3 July 2019 Available online 06 July 2019 0142-1123/ © 2019 Elsevier Ltd. All rights reserved.

International Journal of Fatigue 128 (2019) 105187

Y. Lu, et al.

Nomenclature

A, r D Di Dblock E FNP i FNP K n Nf T Ti v W0 W0i Wc

Weff i Weff Weq Wf W fi

material constants for Garud’s Model cumulative damage cumulative damage of ith loading path in a cycle cumulative damage during a block elastic modulus non-proportional hardening factor non-proportional hardening factor for ith regular shape cyclic strength coefficient cyclic strain hardening exponent fatigue life period of one cycle the ratio in a period Poisson’s ratio uniaxial or proportional plastic work uniaxial or proportional plastic work for ith regular shape plastic work per cycle

w

p eq

p

eq

p

i

eq u

y

define cycles for energy approaches because they are scalar quantities. However, variable amplitude cycles could be identified by rain flow counting method, which usually involves irregular cycles. And an irregular loading path means a complex geometrical shape in loading diagram. Therefore, an energy parameter applicable for irregular loading paths is very necessary and meaningful. The objective of this paper is to extend the authors’ previous work [20] to an irregular loading path condition based on Miner linear cumulative theory. The emphasis will be placed on the feasibility of the proposed energy parameter for an irregular path, rather than a counting cycle method.

effective energy parameter effective plastic work under ith loading path in a cycle equivalent plastic work per cycle fatigue toughness fatigue toughness under ith loading path in a cycle non-proportional plastic work amplification coefficient tensile stress range plastic strain range tensile stress range tensile stress range plastic strain angle of maximum principal strain direction central angle of ith arc Pi equivalent stress amplitude ultimate tensile stress yield stress

2. Multiaxial fatigue test Fatigue tests under 18 loading paths with von Mises equivalent strain amplitudes 0.4–0.85% have been performed on tubular 316L stainless steel specimens in MTS809 axial/torsional test system. These loading paths are divided into 2 groups: 9 regular loading paths illustrated in Figs. 1 and 9, irregular loading paths illustrated in Fig. 2. An irregular loading path can contains either a single cycle such as Path V6-V9 or several cycles which can be simply counted such as Path V1V5. Indeed, paths V1-V5 are designed to examine the combined loading block; Path V8-V9 are designed to examine the irregular loading path;

C1

C2

C3

C4

C5

C6

C7

C8

C9

Fig. 1. 9 typical constant amplitude loading paths. 2

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Y. Lu, et al.

V1

V2

V3

V4

V5

V6

V7

V8

V9

Fig. 2. 9 variable amplitude loading paths.

and Path V6-V7 are for the both conditions. The dimension of testing specimens with wall thickness t = 1.5 mm is presented in Fig. 3. The chemical composition of specimen material is 0.016 C; 0.710 Si; 11.250 Ni; 18.050 Cr; 1.422 Mo; 0.018 S; 0.050 P; 0.013 N (mass fraction, %) and balanced by Fe. Yield stress of the material at room temperature is 265 MPa, and ultimate tensile stress is 585 MPa. Relevant material constants acquired from uniaxial test have been detailed in Table 1. All the fatigue tests were conducted under strain-controlled condition at room temperature and terminated when the main crack propagation length reached over 10 mm (excluded the test under pure shear condition with 0.4% strain amplitude which has not appeared a main crack after more than 200 000 cycle). The test results are described in Table 2.

Table 1 Mechanical properties of 316L stainless steel. E (GPa) 193

y

(MPa)

265

u

(MPa)

585

0.3

K (MPa)

n

A

r

916.7

0.1567

34,476

−1.8312

3. Proposed damage accumulation algorithm 3.1. Effective energy model for constant amplitude loading paths Traditional plastic work models postulated that dissipated hysteresis energy is the essence of fatigue failure. Based on this postulation,

Fig. 3. Specimen for multiaxial fatigue tests of 316L stainless steel (mm). 3

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Table 2 Fatigue test results and the computed results. Specimen number

path

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36

C1 C1 C2 C2 C3 C3 C4 C4 C4 C5 C5 C5 C6 C7 C7 C8 C9 C9 V1 V1 V2 V2 V3 V3 V4 V4 V5 V5 V6 V6 V7 V7 V8 V8 V9 V9

eq/2

0.4 0.6 0.4 0.6 0.4 0.6 0.4 0.5 0.6 0.6 0.6 0.8 0.6 0.4 0.6 0.75 0.4 0.6 0.4 0.6 0.4 0.6 0.4 0.6 0.57 0.85 0.4 0.6 0.4 0.6 0.47 0.71 0.45 0.68 0.56 0.84

(%)

FNP

Test Life (Block)

Cycles of a block

Predicted Life by Eqs. (2) and (3) (Block)

Predicted Life by Eq.(17) (Block)

0 0 0 0 0.43 0.43 1 1 1 0.42 0.72 0.14 0 0.81 0.81 0.54 0.77 0.77 – – – – – – – – – – 0.89 0.89 0.61 0.61 0.71 0.71 0.74 0.74

9050 1115 > 200,000 27,149 7944 1535 4874 2404 837 906 1086 11,303 1133 7212 976 842 3117 772 3137 454 1451 348 1766 358 917 288 1942 294 5983 983 6097 936 3278 923 2488 751

1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 3 3 4 4 1 1 1 1 1 1 1 1

7937 2194 372,548 42,628 4780 1321 4401 2112 1216 1870 1541 17,690 2194 4961 1372 1294 4215 1165 – – – – – – – – – – – – – – – – – –

7937 2194 372,548 42,628 4780 1321 4401 2112 1216 1870 1541 17,690 2194 4961 1372 1294 1683 532 2831 782 1257 383 2332 645 978 291 1567 433 4664 1289 3267 962 4747 1311 3329 964

the empirical relationships of fatigue life and plastic work per cycle have been presented. For uniaxial or proportional tension-torsion loading histories, Garud [12] proposed an energy based model, in which fatigue life is related to plastic work per cycle, as follows:

Nf = AWcr

W0 =

where A and r are material constants. To consider path dependence of multiaxial fatigue life, the authors [20] proposed an effective energy parameter Weff to modify Garud’s model, expressed as follows: (2)

Weff = (1 +

(3)

w FNP ) W0

where w is a material constant reflecting material sensitivity to a NP path, W0 denotes uniaxial plastic work, and FNP accounts for the path dependency of NP cyclic hardening. Besides, Weff can be regarded as effective plastic work which combines uniaxial plastic work and NP hardening effect. The uniaxial cyclic stress-strain relationship can be described by the Osgood-Ramberg equation:

Weq =

=

2E

+

2K

(5)

eq

p eq

1 n 1+n

(6)

where eqp , eq and Weq are von Mises equivalent plastic strain amplitude, von Mises equivalent stress amplitude and equivalent uniaxial plastic work, respectively. The effective energy parameter Weff is accordingly updated as:

1/ n

2

1 n 1+n

where p , and n are the uniaxial plastic strain amplitude, the uniaxial stress amplitude and the cyclic strain hardening exponent, respectively. For a uniaxial case, the proposed model degenerates to Garud’s model. In this model, FNP can be obtained by measuring experimentally or estimating theoretically by MOI method [21,22], Itoh’s model [23,24], Bishop’s model [25] or Tanaka’s model [26] for arbitrary constant amplitude loading paths, such as the paths shown in Fig. 1. Moreover, this model is very convenient for engineering applications since it avoids estimation of multiaxial constitutive relations. However, it is difficult using this model to deal with irregular loading paths because of the ambiguity to define the uniaxial strain amplitude corresponding to an irregular path, such as the path V9 with several reversals in a strain-time diagram (see Fig. 4). Hence, in order to extend the application of this model to multiaxial irregular loading paths, W0 in Eq. (3) is replaced by:

(1)

Nf = A (Weff )r

p

(4)

where E , K and n are the elastic modulus, the cyclic strength coefficient and the cyclic strain hardening exponent, respectively. The uniaxial plastic work can be obtained by integrating the Osgood-Ramberg equation as:

Weff = (1 +

w FNP ) Weq

(7)

Eq. (6) means replacing an irregular loading path with a regular loading path with equivalent perspective of energy. For constant amplitude 4

International Journal of Fatigue 128 (2019) 105187

Y. Lu, et al.

1+r Wf = Nf · Weff = AWeff = A [(1 +

1+r w FNP ) Weq ]

(9)

Fatigue toughness denotes the limit of energy absorption of a material under user-defined loading conditions. To deal with irregular path cases, fatigue damage of a cycle is defined as: i Di = Weff /W fi

(10)

i and W fi are the damage, the effective plastic work and where Di , Weff the fatigue toughness under ith loading path in a cycle, respectively. Fatigue failure under ith loading path occurs when i D = Nf ·Di = Nf ·Weff / W if = 1

(11)

For a combined loading path consisting of m different shapes of loading paths, the Palmgren-Miner’s cumulative damage of a block is calculated as: m

D = Nf · Fig. 4. The variation of strain with time under Path V9.

i=1

r w FNP ) Weq ]

W fi

(12)

Fatigue failure occurs when D = 1, and fatigue life is obtained as:

loading path, Eq. (6) degenerates to Eq. (5), and Eq. (7) degenerates to , eqp = p . Eq. (3), since eq = Besides, it is proposed to estimate W0 by von Mises strain amplitude than by tensile or shear strain amplitude. Fig. 5 shows three regular strain paths: circular path, diamond path and square path. They have the same tensile strain amplitude 0.6% and the same shear strain amplitude 0.6% × √3, but von Mises strain amplitude of the square path is larger than that of the other two. The detailed fatigue test data are shown in Table 3. The tensile plastic work W and the shear plastic work W are calculated based on the measured hysteresis loop. The NP factor is calculated by Itoh’s method. Substituting Eq. (7) into Eq. (1), we obtain:

Nf = A [(1 +

i Weff

m

Nf = 1/ i=1

i Weff

W if

(13)

3.3. Effective energy-based model for irregular loading paths Traditional energy-based models deal with regular loading paths such as square path, circular path, cross-shaped path, instead of irregular loading paths, such as Paths V6-V9 in Fig. 2.The plots of normal strain vs shear strain, normal stress vs shear stress, normal strain vs strain and shear strain vs shear stress are shown in Fig. 7 for Path V9. However, periodic loading path in engineering application is usually irregular. Any irregular path can be approximately divided into several regular path shapes, such as Path V6-V7 shown in Fig. 2. The path V6 can be divided into half a diamond and half a circle; the path V7 can be divided half a diamond, a quarter of a circle and a quarter of a square. If a general irregular path can be divided into k regular shapes which are respectively contained Ti (i = 1, 2, , k; 0 T 1) subcycle, the corresponding fatigue damage during a block can be estimated as:

(8)

In this model, the loading amplitude effect is reflected by the equivalent uniaxial work and the path effect is reflected by the NP factor. The relation between the effective energy parameter and the fatigue life is based on the Garud’s exponential model. Note that equivalent uniaxial plastic work should be computed by the shear form of the Osgood-Ramberg equation, if the shear strain is dominant. The strain amplitude ratio = is a criterion for determining the dominant strain between the tensile strain and the shear E strain. When > 0 = G· = 2(1 + v )/ , the shear strain is dominant, otherwise the tensile strain is dominant. Herein v is the Poisson’s ratio, and is a weight factor introduced by Garud [12]. For 316L stainless steel, = 0.47 , and 0 = 6.4 .

k

Dblock =

Ti i=1

i Weff

W if

(14)

The accumulated damage of Nf blocks is calculated as: k

D = Nf ·

Ti i=1

i Weff

W fi

(15)

3.2. Effective energy model for combined loading blocks Several typical combined non-proportional loading blocks are shown in Fig. 2. A combined loading block is a simple variable amplitude loading path since it can be explicitly divided into several constant amplitude paths. For example, a block of circle-diamond combined loading path shown in Fig. 6 consists of a cycle of circle loading path and a cycle of diamond loading path. It can start from any intersection and then follow different track such as A-E-B-F-C-G-D-H-A along circle then A-J-B-K-C-L-D-M-A along diamond, or A-J-B-K-C-L-D-M-A along diamond then A-E-B-F-C-G-D-H-A along circle. On the contrary, traditional energy-based models can deal with the multiaxial fatigue problem for a single regular loading path instead of a combined loading path. Effective plastic work represents the absorbed energy by a material under cyclic loading condition. According to Eq. (2), fatigue toughness of a material under cyclic loading of constant amplitude can be defined as:

Fig. 5. Three regular strain paths: circular path, diamond path and square path. 5

International Journal of Fatigue 128 (2019) 105187

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Table 3 the detailed comparison among circle path, diamond path and square path. Loading path

/2 (%)

C4 C4 C7 C7 C9 C9

0.4 0.6 0.4 0.6 0.4 0.6

/2 3 (%) 0.4 0.6 0.4 0.6 0.4 0.6

Von Mises strain amplitude (%)

αw

FNP

Wσ (MJ/m3)

Wτ (MJ/m3)

Test Life (cycles)

0.4 0.6 0.4 0.6 0.57 0.85

0.38 0.38 0.38 0.38 0.38 0.38

1 1 0.81 0.81 0.77 0.77

2.29 5.53 2.24 5.18 3.22 7.28

0.93 3.33 0.37 2.15 1.91 5.42

4874 837 7212 976 3117 772

k

Nf = A/

[Ti ·(1 + i=1

Fig. 6. Circle-diamond combined loading path.

Fatigue failure occurs when D = 1, and fatigue life is obtained as: k

Ti i=1

i Weff

W if

(17)

where are the NP factor and the equivalent plastic work for ith regular shape, respectively. Every subcycle Ti is divided from the same one cycle. Obviously, a geometric segmentation algorithm is necessary to divide an irregular path into several approximate regular parts. When the divided segments are smallish, any of them can be approximatively regarded as an arc of a circular path. And then every Ti can be determined as the ratio of the central angle corresponding to the arc to 2 * Pi rather than the ratio of the length of the arc to the perimeter of the path, illustrated by Fig. 8. Defining Ti using central angle is more reasonable than that using arc length since NP hardening effect is caused by the rotation of the principal strain axes during cyclic loading [19,27]. Another key point is how to choose the reference point so as to compute the central angle. When the origin is outside a closed strain path, the origin will be inappropriate to act as the reference point, since in this case the sum of the central angles of the divided segments with respect to the origin is less than 2π, see Fig. 9(a). The perimeter i FNP

Nf = 1/

i i r r w FNP ) ·(Weq ) ]

(16)

That is

i and Weq

Fig. 7. The plot of normal strain vs shear strain, normal stress vs shear stress, normal strain vs normal stress and shear strain vs shear stress under Path V9. 6

International Journal of Fatigue 128 (2019) 105187

Y. Lu, et al.

illustrated in Fig. 12, the path ABCEDFA consists of half circle ABC and half diamond CEDFA. If the reference point is located in the incenter, the rotation angle of A-B-C and C-E-D-F-A with respect to CC are respectively 180 degree, which will lead to the same damage during the two stages. However, the path CEDFA leads to a more severe damage than the path ABC actually, see the test results in Table 2. If the reference point is located in the perimeter centroid, the rotation angle of C-E-D-F-A will be greater than 180 degree and that of A-B-C opposite, which is consistent with the test results qualitatively. From above analysis, the reference point cannot be arbitrarily chosen and the perimeter centroid is a reasonable choice whose corresponding central angle better reflects the rotation of the principal strain axes during cyclic loading. In fact, the popular critical plane methods require a specific reference plane and their damage is also relative to their respective reference planes, which leads to different critical plane model [1–3,28–30]. When Itoh computed the non-proportional factor, he defined the orientation where the maximum of the principal strain is obtained as the reference orientation and then calculated the angle of the principal strain axis [23,24]. In Itoh’s method, the reference orientation is unique. It is obvious that a reference point or line is necessary to consider the non-proportional hardening effect and the computed damage is dependent on the choice of the reference point. Since every little segment is regarded as an arc of a circular path whose NP factor is 1, Eq. (17) can be simplified as:

Fig. 8. The definition of Ti for an irregular path.

centroid (PC) is a more reasonable choice as the reference point. If the path is analogous to a homogeneous wire with unit mass, the perimeter centroid is equal to the center of gravity and can be obtained by the Moment Of Inertia method [21], see Fig. 9(b). The perimeter centroid is more reasonable than the area centroid (AC). As illustrated in Fig. 10(a), an area element AE within the path contributes no effect on the rotation of the principal strain axes, while a line element of the path LE does. In addition, if the reference point is located in the area centroid, the path P1 and the path P2 have nearly the same area centroid and then their damage per cycle will be nearly the same, which is obviously unreasonable. If the reference point is located in the perimeter centroid, the long arm part in the path P2 will have effect on the site of the reference point and then every line element in the long arm part will lead to a bigger and significant central angle, see Fig. 10(b). The perimeter centroid is more reasonable than the circumcenter (CC). As illustrated in Fig. 11, the path ABCEDFA consists of half circle ABC and half diamond CEDFA. If the reference point is located in the circumcenter, the rotation angle of A-B-C and C-E-D-F-A with respect to CC are respectively 180 degree, which will lead to the same damage during the two stages. However, the path ABC leads to a more severe damage than the path CEDFA actually, see the test results in Table 2. If the reference point is located in the perimeter centroid, the rotation angle of A-B-C will be greater than 180 degree and that of C-E-D-F-A opposite, which is consistent with the test results qualitatively. The perimeter centroid is more reasonable than the incenter (IC). As

k

Nf = A (1 +

w)

i [Ti ·(Weq ) r]

r/ i=1

(18)

The above geometric segmentation algorithm implies

Ti =

i

2

where

k

, i=1 i

k

Ti = i=1

i

2

=1

(19)

is the central angle of ith arc.

4. Results and discussion To compare and discuss the differences between regular and irregular case, it is necessary to recalculate the regular path (PathC1-C9 in Fig. 1) according to Eqs. (2)–(7) instead of Eqs. (2) and (3) which are employed by the authors’ previous model. The Minimum Ball (MB) method [31] is introduced to obtain the equivalent strain amplitude of single cycle loading path. In the two-dimensional case, the MB method seeks for the circle with minimum radius that contains the single cycle path in / 3 diagram and defines the diameter of the circle as the equivalent strain amplitude of the path. For the combined pathsV1-V5, one block contains 2, 3 or 4 cycles so

Fig. 9. The difference between the origin and the perimeter centroid for computing central angle. 7

International Journal of Fatigue 128 (2019) 105187

Y. Lu, et al.

Fig. 10. The difference between locating in the area centroid and the perimeter centroid.

Fig. 11. The difference between locating in the circumcenter and the perimeter centroid.

Fig. 13. The equivalent strain amplitude and the Minimum Ball method.

Fig. 12. The difference between locating in the incentre and the perimeter centroid.

that we should distinguish the different cycles and then compute equivalent strain amplitude of certain specific cycle, which can be used to predict the fatigue life. For the combined path V6-V7, a similar procedure can be carried out but the difference is that less than one cycle of every typical path is contained in these paths. As a result, how to define the ratios of the contained loading cycles for an irregular path needs to been discussed.

Fig. 14. Comparison of the predicted values by the proposed model and observed fatigue life.

8

International Journal of Fatigue 128 (2019) 105187

Y. Lu, et al.

The ratio of every segment of curve to its corresponding regular path cycle can be determined by its central angle relative to 2Pi rather than its length relative to the perimeter of the regular path. Such as Path V6, it is obvious that the path contains half a circle and half a diamond, which means that the arc is corresponding to the central angle Pi. And the polygonal line is corresponding to the other central angle Pi as well. Although both of their corresponding central angles are Pi, their lengths are not equal. If the ratio of the cycle is calculated by the length of curve, the accumulated damage will not be accurate. For the irregular path V8-V9, the key point is how to divide the irregular path into several regular curves so that they can be treated as regular loading paths. Fig. 4 shows the variation of strain under Path V9. Eq. (18) can be employed to estimate the fatigue life of an irregular loading path. The irregular path can be dealt with by the geometric segmentation algorithm introduced in Section 3.3. In this procedure, Miner linear cumulative damage theory is implicitly adopted, otherwise the interplay between the segment of curve corresponding to high strain level and low strain level must be considered. Itoh’s method is used to calculate the NP factor since it considers just the shape of the path. Note that the value of FNP for the same path can be a little different for different material, since Itoh’s method is influenced by the Poisson’s ratio. The process of obtaining the equivalent strain amplitude by Minimum Ball method is illustrated in Fig. 13. In particular, Path V6-V7 can be respectively treated as not only a combined loading path but also an irregular loading path, which implies that a combined loading path and an irregular loading path can be equivalent when Miner linear cumulative damage theory does work. All the computed results for 18 paths are shown in Table 2 and the error rates by the new model are shown in Fig. 14. For most of regular paths, Eq. (18) gives the similar results with the authors’ previous model (see Table 2). As revealed in Fig. 14, all the results fall with the factor of two error lines. Compared to the previous model, the modified model does not introduce new parameter, since it is not necessary to discuss again the difference between the tensile work and the shear work. Another consideration is that it is difficult to obtain the uniaxial plastic work using the previous model for irregular loading path because of the frequent reversal of strain loading and the difficulty in defining the strain amplitude as well, unless an accurate constitutive relation is provided. Although the current model does not discuss the tensile work and the shear work respectively, it provides an accurate estimation in the effective plastic work. For example, the Path C9, the tensile and shear plastic work are 7.28 MJ/m3 and 5.42 MJ/m3 respectively.In the previous model, the effective plastic work is calculated by the uniaxial plastic work and then magnified by the non-proportional effect. Since the uniaxial plastic work based on tensile or shear is underestimated, the effective plastic work of this path is underestimated as 6.71 MJ/m3. In the current model, the plastic work is calculated by the equivalent strain from the Minimum Ball method as 9.76 MJ/m3. If Garud’s suggestion that the shear work is not as damaging as the tensile work is accepted, the effective plastic work will be calculated as 9.83(7.28 + 0.47 * 5.42 = 9.83) MJ/m3 according to the test measured data.

accurate in a wider range of application. (4) The reasonability of the algorithm was described based on comparison with the previous model. The possible error rate was described based on the influence of material constants and path parameters. (5) A geometric segmentation algorithm of the loading path considering nonlinear cumulative damage theory is the key point to generalize the model for more general loading paths. Acknowledgements This work was supported by the Shenzhen Municipal Government through the Fundamental Research Project (Grant No. JCYJ20170307151049286) and the National Natural Science Foundation of China (Nos. 11572227 and 11772106). References [1] Bannantine JA. A variable amplitude multiaxial fatigue life prediction method; 1989. [2] Wang CH, Brown MW. Life prediction techniques for variable amplitude multiaxial fatigue—Part 1: theories. J Eng Mater Technol 1996;118:367–70. [3] Wang CH, Brown MW. Life prediction techniques for variable amplitude multiaxial fatigue—Part 2: Comparison with experimental results; 1996. [4] Miner MA. Cumulative damage in fatigue. J Appl Mech 1945;68:339–41. [5] Richart FE, Newmark NM. A hypothesis for the determination of cumulative damage in fatigue. Aust Fam Phys 2015;41:523–7. [6] Marco SM, Starkey WL. A concept of fatigue damage; 1954. [7] Manson SS, Nachtigall AJ, Ensign CR, Freche JC. Further investigation of a relation for cumulative fatigue damage in bending. J Eng Indust 1965;87:25. [8] Corten HT, Dolan TJ. Cumulative fatigue damage; 1956. [9] Dattoma V, Giancane S, Nobile R, Panella FW. Fatigue life prediction under variable loading based on a new non-linear continuum damage mechanics model. Int J Fatigue 2006;28:89–95. [10] Giancane S, Nobile R, Panella FW, Dattoma V. Fatigue life prediction of notched components based on a new nonlinear continuum damage mechanics model. Proc Eng 2010;2:1317–25. [11] Morrow JD. Cyclic plastic strain energy and fatigue of metals. Astm Stp Astm 1965. [12] Garud YS. A new approach to the evaluation of fatigue under multiaxial loadings. J Eng Mater Technol Trans Asme 1981;103:118–25. [13] Ellyin F, Golos K. Multiaxial fatigue damage criterion. J Eng Mater Technol 1988;110:63. [14] Ellyin F, Kujawski D. Multiaxial fatigue criterion including mean-stress effect, ASTM; 1993. p. 171–74. [15] Ellyin F, Xia Z. A general fatigue theory and its application to out-of-phase cyclic loading. J Eng Mater Technol 1993;115:411–6. [16] Liu KC. A Method based on virtual strain-energy parameters for multiaxial fatigue life prediction; 1993. p. 67–84. [17] Chu CC, Conle FA, Bonnen JJF. Multiaxial stress-strain modeling and fatigue life prediction of SAE axle shafts; 1993. [18] Zhu H, Wu H, Lu Y, Zhong Z. A novel energy-based equivalent damage parameter for multiaxial fatigue life prediction. Int J Fatigue 2019;121:1–8. [19] Socie DF, Marquis GB. Multiaxial fatigue. Warrendale, PA: Society of Automotive Engineers; 2000. [20] Lu Y, Wu H, Zhong Z. A simple energy-based model for nonproportional low-cycle multiaxial fatigue life prediction under constant-amplitude loading. Fatigue Fract Eng Mater Struct 2018;41. [21] Meggiolaro MA, de Castro JTP. An improved multiaxial rainflow algorithm for nonproportional stress or strain histories – Part I: Enclosing surface methods. Int J Fatigue 2012;42:217–26. [22] Meggiolaro MA, de Castro JTP. Prediction of non-proportionality factors of multiaxial histories using the Moment of Inertia method. Int J Fatigue 2014;61:151–9. [23] Itoh T, Sakane M, Ohnami M, Socie DF. Nonproportional low cycle fatigue criterion for type 304 stainless steel. J Eng Mater Technol 1995;117:285–92. [24] Itoh T, Sakane M, Ohsuga K. Multiaxial low cycle fatigue life under non-proportional loading. Int J Press Vessels Pip 2013;110:50–6. [25] Bishop JE. Characterizing the non-proportional and out-of-phase extent of tensor paths. Fatigue Fract Eng Mater Struct 2010;23:1019–103. [26] Tanaka E. A nonproportionality parameter and a cyclic viscoplastic constitutive model taking into account amplitude dependences and memory effects of isotropic hardening. Eur J Mech A Solids 1994;13:155–73. [27] Kanazawa K, Miller KJ, Brown MW. Cyclic deformation of 1% Cr-Mo-V steel under out-of-phase loads. Fatigue Fract Eng Mater Struct 1979;2:217–28. [28] Smith RN, Watson P, Topper TH. A stress-strain function for the fatigue of metals. J Mater 1970;5:767–78. [29] Fatemi A, Socie DF. A Critical plane approach to multiaxial fatigue damage including out of plane loading. Fatigue Fract Eng Mater Struct 1988;11:149–65. [30] Kandil FA, Brown MW, Miller KJ, Biaxial low-cycle fatigue failure of 316 stainless steel at elevated temperatures: mechanical behaviour and nuclear applications of stainless steel at elevated temperatures; 1982. [31] Van KD, Papadopoulos IV. High-cycle metal fatigue. Vienna: Springer; 1999.

5. Concluding remarks (1) Multiaxial low cycle fatigue tests under nine regular loading paths and nine combined or irregular loading paths were conducted on tubular 316L stainless steel specimens. (2) A modified empirical model based on the effective energy parameter was presented. The modified model is applicable for regular and irregular loading paths based on Miner linear cumulative damage theory. It calculates the fatigue life based on equivalent uniaxial plastic work and the complicated critical plane analysis or the incremental plasticity procedure can be avoided. (3) A detailed comparison was made with the authors’ previous model. It can be concluded that the current model is more convenient and 9