A novel energy-based equivalent damage parameter for multiaxial fatigue life prediction

Accepted Manuscript A novel energy-based equivalent damage parameter for multiaxial fatigue life prediction Haipeng Zhu, Hao Wu, Yingya Lu, Zheng Zhong PII: DOI: Reference:

S0142-1123(18)30719-9 https://doi.org/10.1016/j.ijfatigue.2018.11.025 JIJF 4913

To appear in:

International Journal of Fatigue

Please cite this article as: Zhu, H., Wu, H., Lu, Y., Zhong, Z., A novel energy-based equivalent damage parameter for multiaxial fatigue life prediction, International Journal of Fatigue (2018), doi: https://doi.org/10.1016/j.ijfatigue. 2018.11.025

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A novel energy-based equivalent damage parameter for multiaxial fatigue life prediction Haipeng Zhu1, Hao Wu1*, Yingya Lu1, Zheng Zhong1, 2*, 1

School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China 2

School of Science, Harbin Institute of Technology, Shenzhen 518055, China * Corresponding authors: [email protected], [email protected]

Abstract: Energy-based damage parameters are widely used, due to their availability, in the prediction of multiaxial fatigue life, usually by integrating the product of stress and strain range into a single scalar quantity. However, in most engineering applications, stress and the strain histories cannot be known simultaneously. Although incremental plasticity methods can be used to estimate the response from measured or specifically designed loads, these processes are too complicated to be employed. In this paper, a novel energy-based Equivalent Damage Parameter (EDP) based on uniaxial fatigue data is proposed to predict the fatigue life under multiaxial fatigue loadings. In this way, the augmentation of uniaxial tensile elastoplastic work can be estimated thanks to the non-proportional (NP) hardening factor FNP and the energy-based material constant αw. Moreover, the contributions to total elastoplastic work from different loading components can be separated and quantified by using the Moment Of Inertia (MOI) method and weighting factor ξ, introduced as a parameter. The efficiency of the proposed parameter is validated by reasonable correlations with the experimental fatigue data of 316L steel tubular specimens subjected to various proportional or NP loadings. Keywords: Multiaxial fatigue; Equivalent damage parameter; Fatigue life prediction; Moment Of Inertia; Loading path Nomenclature: o o I XX , IYY = Moment of inertia with respect to the perimeter centroid of the loading path

D = convex enclosures (smallest circle) diameter of the loading path dext, dint = external and internal diameters of the test specimen dp = infinitesimal length of the loading path = EDP = Equivalent Damage Parameter FNP = non-proportional factor G = shear modulus 1

Hc = Ramberg–Osgood uniaxial cyclic hardening coefficient hc = Ramberg-Osgood’s uniaxial cyclic hardening exponent non-proportional IXX, IYY , IXY = Moment of inertia with respect to the origin k = material-dependent parameter MOI = Moment Of Inertia Nf = fatigue life NP = non-proportional P = applied axial load p = perimeter of the loading path PC = perimeter centroid s, e = deviatoric stress and strain Sy = monotonic yield stress T = applied torque Xc, Yc = coordinates of the perimeter centroid αNP = material parameter related to additional stress hardening αw = energy-based material constant γxy, γxz, γyz = shear strains ΔW0 = uniaxial elastoplastic work ΔW0NP = augmentation of uniaxial elastoplastic work ΔWel , ΔWpl = axial elastic and plastic work ΔWel’, ΔWpl’ = shear elastic and plastic work ΔWτ = pure shear elastoplastic work Δγ = shear strain range Δε = strain range ΔεI = maximum principal strain range ΔεNP = non-proportional strain range

Δσ = stress range Δτ = shear stress range εx, εy, εz = normal strains 2

ξ = weighting parameter σ⊥ = normal stress perpendicular to candidate planes σc, εc, b, c = material parameters of Coffin-Manson’s equation τc, γc, bγ, cγ = material parameters of shear version of Coffin-Manson’s equation σeq, σ0, σOP = von Mises equivalent stress amplitudes for a measured loading path, a pure tensile path and a circular path σh, εh = hydrostatic stress and strain σx, σy, σz = normal stresses τxy, τxz, τyz = shear stresses υ = Poisson's ratio ψ= Symbol of EDP

Subscripts ⊥ = perpendicular to candidate planes a = amplitude e = elastic eq = equivalent max = maximum min = minimum p = plastic 1. Introduction: 316L austenitic stainless steels are widely used in various industrial applications due to their excellent mechanical properties and superior resistance to corrosion [1, 2]. Structural components such as the core of a fusion reactor, made of 316L austenitic stainless steels, are subjected not only to severe internal pressure but also to various external multiaxial fatigue loadings during periodic operation. A comprehensive understanding of the mechanical properties of 316L steels is helpful to guarantee their designed fatigue life. A common way to assess environmental and mechanical damages in order for designing the survivability of engineering components is to modify traditional S-N curves by introducing different correction factors. Therefore, many modified S-N relations have been proposed to predict fatigue life under multiaxial loading. However, due to the lack of exact multiaxial stress/strain relations, this type of empirical approaches provides poor estimation for mechanical design and structural integrity evaluations [3, 4]. In fact, the traditional S-N or ε-N curves, measured under proportional loading, cannot be directly used in multiaxial fatigue estimation since the variation of principal directions under NP loading may influence the crack initiation direction 3

and then the associated fatigue life. In addition, austenitic stainless steels display complex transient elastoplastic loops due to strain-controlled hardening or stress-controlled softening under NP multiaxial loading, for which S-N based method cannot give precise multiaxial fatigue assessment [5]. Therefore, to quantify multiaxial damage accumulation including both stress and strain information, several energy-related parameters have been proposed, such as Garud’s[6] or Morrow’s[7], which accumulates to a single scalar quantity based on the elastoplastic work per cycle or more specifically, the area within each closed hysteresis loop in a multiaxial load history path. Such a parameter in terms of the equivalent elastoplastic work can be used to estimate the multiaxial fatigue crack initiation life when it reaches a critical value which is material-dependent. Smith, Watson and Topper [8] proposed a parameter to correlate both elastic and plastic energy densities per cycle with fatigue life, which can be regarded as an energy parameter as well and is called SWT parameter later, but it is not applicable for large compressive mean stress conditions. Taking into account of the compressive effect on fatigue life, a deviatoric version of SWT parameter has been proposed by Kujawski [9]. Ince has also presented a mean stress correction model based on the distortional strain energy by modifying the elastic strain energy term using maximum deviatoric stress [10]. Moreover, in order to consider the contribution from all strain and stress components, two generalized fatigue damage parameters were proposed by Ince and Glinka [11]. Another modified version for general multiaxial fatigue life prediction has been proposed by Zhu et al. [12] by introducing the maximum shear strain plane as the critical plane. In most engineering applications, to predict transient response induced by multiaxial NP loading, incremental plasticity methods are usually employed which correlate infinitesimal increments of stress components with associated infinitesimal strain variations, or inversely, as presented for stress and strain estimations of notched components [13,14]. Nevertheless, the procedure is very complex so that the quantification of the loading path dependent effects, such as NP hardening, easily introduces accumulated errors which will decrease the accuracy of prediction [15-16]. Moreover, excessive computational effort makes it hard to be applicable in practical engineering problems. From a more fundamental point of view, the mentioned hardening effects are caused by the rotation of the principal stresses and the NP variation of their magnitudes, which are originally induced by complex loading paths. Therefore, it would be a good alternative to consider multiaxial loading path effects from several empirical models such as MOI’s[17], Itoh’s[18] and Bishop’s[19]. Although all multiaxial fatigue life predictions based on typical damage models assume that both stress and strain histories are known, in most real cases, one of them is unknown beforehand and should be derived from another. Under linear elastic conditions, Hooke’s law is enough to correlate multiaxial stresses and strains. However, for a general elastoplastic NP loading history, incremental elastoplasticity models must be adopted to correlate stress and strain increments. Moreover, the consistency condition should be introduced to guarantee the stress state during any plastic straining process always remains on the yield surface, significantly increasing the computer burden. To simplify the solution process and to improve its effectiveness and practicability under multiaxial cyclic loading, a novel energy-based equivalent damage parameter (EDP) is presented in this paper. The initiation fatigue crack life 4

under different multiaxial loadings is estimated through the path-dependent damage parameter, by introducing the MOI concept. And the damage from different loading directions can be distinguished and then integrated to a unified damage parameter without any incremented calculation. A series of strain-controlled multiaxial fatigue tests is also performed to validate the capability of the proposed method. 2. Material and experimental setup The material tested is 316L austenitic stainless steel. The chemical compositions of the material other than the balanced Fe are given in Table1. Straight bars are machined to tubular specimens with external diameter dext = 16mm, internal diameter dint = 13mm and gauge length l0 = 30 mm for multiaxial fatigue testing. The minimum 1.5mm Since the average grain size of 316L steel is found to be 20±10 μm [21], the number of grains is estimated approximately from 50 to 150 through the wall thickness so that the averaged continuum response of the material can be ensured.

C 0.024

Si 0.46

Ni 12.53

Mn 1.59

Cr 17.51

Mo 2.55

S 0.001

P 0.039

N 0.0859

The normal engineering stress σxeng caused by the applied axial load P is simply calculated through the following equation, regardless of elastic or plastic conditions,



where dext and dint specimen.



2  d2  (1)  xeng  4P /   dext int   are the external and internal diameters of the critical section of the test 5

The engineering shear stress τxyeng caused by the applied torque T under elastic conditions can be expressed as:





eng 4  d4  (2)  xy  16  T  dext /   dext int   On the other hand, the extremely high torsional loadings can produce torsional plastic collapse, resulting in a constant shear stress along the entire wall, which gives:





eng 3  d3   xy  12  T /   dext int



(3)

In this paper, an empirical uniformly-distributed shear stress expression recommended by ASTM E2207-08 under general elastoplastic conditions has been used to adjust the shear stress equation.







eng 2  d2  d   xy  16  T /   dext int ext  dint  

(4)

eng  xy  tan1(  xy  x  ln( 1   xeng ) )   eng eng  1  y  ln( 1   y ) and  xz  tan (  xz )    tan1(  eng ) eng yz yz  z  ln( 1   z ) 

(5)

Fig. 2. Tension-torsion testing machine and extensometer mounted on a tubular specimen.

6

 x   xeng     eng y  y    eng z  z  eng  xy   xy  eng  xz   xz   yz   eng yz 

eng  ( 1   eng y )  ( 1   z )  ( 1   xeng )  ( 1   zeng )

( 1   xeng )  ( 1   eng  y )  eng  eng ( 1   eng y )  ( 1   z )   yx   yx  eng  eng ( 1   eng y )  ( 1   z )   zx   zx  eng ( 1   xeng )  ( 1   zeng )   zy   zy

( 1   xeng )  ( 1   zeng )

(6)

( 1   xeng )  ( 1   eng  y )  ( 1   xeng )  ( 1   eng  y ) 

Fig. 3. Cyclic axial response with strain range under a uniaxial tension-compression test

Hc = 74 MPa and exponent hc = 0.123 (used in Eq. (11) and (12)), with Poisson ratio ν = 0.3, Young’s modulus E = 193 GPa, and shear modulus G = E/(2 + 2ν) ≈ 74 GPa. Other properties of 316L steel are adopted from the literature [22]: σc=745.4MPa, εc= 0.161, b= -0.092 and c= -0.419. Strain-controlled multiaxial fatigue tests under constant-amplitude loading are carried out under different strain paths described in the x × xy/3 diagram (see Fig. 4).

7

Fig. 4. Applied periodic εx × γxy/√3 strain paths on tension-torsion tubular specimens Table 2. The experimental results for 316L stainless steel test loading εa γa/√3 Nf test number path (%) (%) (cycles) number 1 II 0 0.6 27149 2 3 III 0.6 0.6 772 4 5 V 0.6 0.6 976 6 7 VII 0.6 0.6 842 8 9 VIII 0.6 0.2 906 10 11 IV 0.5 0.5 2404 12 13 VIII 0.1 0.8 11303 14 15 IV 0.4 0.4 4874 16 17 V 0.4 0.4 7212 18 * Unbalanced loading path with εmin = 0 and εmax = 1.0%.

loading path I IV VI VIII IX I III VI IV*

εa (%) 0.6 0.6 0.6 0.6 0.6 0.4 0.4 0.4 0.5

γa/√3 (%) 0 0.6 0.6 0.4 0.6 0 0.4 0.4 0.5

Nf (cycles) 1115 837 1535 1086 1133 9050 3117 7944 750

8

a 

a  σc, τc, εc, γc

c

 2N  E

  c  2N f



 2N  G

  c  2N f



b

f

c

b

f

c

c

b, c, bγ, cγ

for uniaxial axial loading

(7)

for pure torsional loading

(8)

parameters. ε

 a ,eq 

c

 2N  E

b

f

  c  2N f



c

(9)

ε

strain-controlled to

HNP = Hc ∙ (1+αNP ∙FNP) (10) αNP additional hardening coefficient (typically 0 ≤αNP ≤ 1) and FNP is the NP factor which depends solely on the shape of the load history path (typically 0 ≤ FNP ≤ 1).

 a ,eq

 a ,eq

     a ,eq  E  H NP 

1 / hc

(11)

9

Analogous to the stress or strain-based EDP, the energy-based EDP, such as the plastic work per cycle, is usually required in low cycle fatigue by calculating the area encompassed by each closed hysteresis loop. Moreover, the addition of elastic work per cycle can extend such energy-based parameter to be applicable for high cycle fatigue, avoiding an estimated infinite life with zero plastic work under pure elastic cases. Accordingly, the accumulated energy-based parameter can be quantified as the total work in all applied loading direction, i.e., the sum of the axial elastic work ΔWel, the shear elastic work ΔWel’, the axial plastic workΔWpl and the shear plastic work ΔWpl’ as shown in Eq.(12). ' Wel

Wel

W 'pl

W pl

2 2    max    max   1  hc  1  hc           p              p  1  hc  1  hc   2E   2G   

  2  2N f  c  2E 



2b

   2  2N 2b f     c   2G  

   4  c  b   c c  2N f b  c    c c  2N f  cb  plastic strength





b  c



(12)

elastic strength

The left side of the equation can be regarded as a composite measure of the amount of fatigue damage per cycle for proportional tension-torsion histories with zero mean tension and torsion loads. The right side of the equation is the elastic and plastic strength of a material, through ε -N and γ -N curve incorporated by the elastic and plastic parts, using:

  p b c    E  e   c  2N f  ,   c  2N f   max  2 2 2     b c   p  G  e   c  2N f  ,   c  2N f   max  2 2 2  b  1  hc c  b  hc  b / c   c 1  h cb  

(13)

Note that ξ is a weighting factor used to consider that shear work is not as damaging as tensile work. For tensile-sensitive materials, ξ tends to 0, and for shear-sensitive materials, ξ increases, usually less than 1. As special cases of Eq.(12), Garud’s and Morrow’s models remove all elastic work and elastic strength terms and assume ξ = 0 and 0.5, respectively. Note that the original expressions of ε-N and γ-N curves assume zero mean stresses and strains, and they are only suitable for these fully-reversed tests under zero mean tension and torsion loadings before and after the NP hardening process. Therefore, some energy densities per cycle including the mean or maximum stress/strain have been proposed. For most metallic materials that fail due to a single dominant crack, the mean stress/strain effects are reflected by the mean or maximum normal stresses perpendicular to the free face, so-called critical plane. Therein the SWT damage parameter [8] considers the maximum positive stress and the strain amplitude on the critical plane which correlates both elastic and plastic energy densities per cycle with fatigue life. Therefore, the SWT method can predict, as expected, an initiation fatigue life in

10

the presence of higher peak stresses σ⊥max induced by mean stress and NP hardening. The energy-based EDP symbolized by ψ is expressed as:

   max 

  2

(14)

Similar to the SWT parameter, the Chu-Conle-Bonnen (CCB) parameter [24] includes not only normal but also shear work, such that:

        max     max    2 2 max 

(15)

where σ⊥max and τmax are the maximum normal and shear stresses on the critical plane; Δε⊥ and Δγ⊥ are normal strain and shear strain range on the critical plane. The ξ in CCB’s parameter is assumed to be 1 and it might lead to the overestimation of fatigue life for tensile-sensitive material. The critical plane is also adopted in Fatemi-Socie (FS)’s [25] parameter, as shown in Eq.(16). Plenty of other similar available parameters have also been presented, such as by Glinka [26, 27], Liu [28], Walker [29], etc.



 2

    1  k max  Sy  

(16)

where k is a material-dependent parameter, Δγ is shear strain range, σm is mean stress and Sy is monotonic yield stress. The basic idea of the critical plane method is to identify where the crack is expected to initiate with the maximized damage accumulation for all counted candidate plane projection. Even though the critical plane method is probably the most accepted approach to estimate multiaxial fatigue crack initiation lives for most metallic materials that fail due to a single dominant crack, some assumptions may not be true for some materials: 1) the direction of the dominant microcrack is assumed to be invariant during the early crack propagation; 2) the contributions to fatigue damage on the critical plane from other possible planes are neglected. Moreover, the critical plane method may be inappropriate for materials that fail due to cavitation or multiple cracks. Nevertheless, the prediction of multiaxial fatigue crack initiation life is more important than that of crack direction from the engineering point of view. As a result, the approach proposed in the paper is focused on the energy-based EDP that can be integrated along any multiaxial loading path without the need to project histories onto any candidate planes and the fatigue crack initiation life can be predicted when the scalar accumulated elastoplastic work reaches a critical value. In general, multiaxial fatigue predictions are quite challenging for NP loading histories, particularly when the elastoplastic (EP) stresses and strains are induced, so a good multiaxial EDP should involve the following important features: 1) Stress-strain relationships: Uniaxial Ramberg-Osgood equation cannot be directly employed in NP multiaxial fatigue problems. Traditional energy-based damage parameters usually include both stress and strain components so that a material constitutive model is required. 11

2) NP strain-hardening effect: The peak of normal stress σ⊥max induced by NP hardening can be quantified by the NP factor FNP. 3) Mean stress effect: Since the mean stress σm always remains zero in fully-reversed loading even after NP hardening process, the mean stress effect can be reflected by adding an additional term such as in Glinka’s parameter or by σ⊥max which exists e.g. in SWT’s parameters. 4) Path dependence: This effect can be modeled by FNP, based on integrals of the multiaxial load history path such as in MOI’s estimation. 5) Availability for both high and low cycle fatigue: As mentioned previously, the energy-based damage parameters are appropriate for low cycle fatigue problems, and it can be extended to high cycle fatigue problems by introducing elastic energy density (or work). 6) Simple and effective application to practical engineering problems: Most energy-based damage parameters use both stress and strain components to quantify fatigue damage accumulation, and they usually need the calculation or estimation of response from measured or specified designing loads based on the yield function, the plastic flow rule and the hardening rule, which is not easy to implement in practice. Hereafter a novel energy-based EDP based on MOI concept will be presented. The proposed parameter uses only the uniaxial fatigue data and the MOI along the history path that can incorporate the important features stated above, such as the combination of response of designing loads, NP hardening, mean stress effects and path-dependence. 4. Proposed energy-based EDP Itoh’s empirical model [18] provides reasonable prediction of FNP which reveals phenomenologically the relation with strain ranges ΔεNP in multiaxial fatigue calculations as shown in:

 NP  1   NP FNP    I

(17)

where ΔεI is the maximum principal strain range under NP straining. Obviously, for tests having the same strain range, NP tests have a higher stress range than that of proportional tests, by a factor as high as (1 + 1×1) =2. When NP hardening is significant, fatigue life will be much shorter than that from proportional histories with the same strain range, since the NP hardening increases the corresponding stress range. Analogously, the NP hardening increases the corresponding work, and the augmentation of the uniaxial elastoplastic work ΔW0NP is proposed as:

W0 NP  1   w FNP   W0

(18)

where αw is an energy-based material constant, αw = 0.38 for 316L stainless steel [30], which is independent of the loading path and calibrates the influence of material sensitivity to the non-proportionality of the loads. ΔW0 is the uniaxial elastoplastic work per cycle with the same equivalent designing loading level; for example, for the diamond path (Case V in Fig. 4) with strain amplitude 0.6% × 0.6% in the x × xy/3 diagram, ΔW0 is the area encompassed by the uniaxial loading and unloading curves of 0.6% tensile strain amplitude in Fig.3.

12

Moreover, to compensate for the flaws of FNP in directional estimates, the MOI concept is introduced in the parameter. MOI method is presented firstly by Meggiolaro and Castro [31] to calculate the equivalent stress (or strain) range and FNP in general NP load histories, assumed to be a homogeneous wire with unit mass, as illustrated in Fig.5. It defines (Xc, Yc) as o o the perimeter centroid (PC) of a 2D path. ( I XX , IYY ) and (IXX, IYY) are the MOI with respect to

the origin and the PC in 2D loading path.

Fig.5. History path, assumed as a homogeneous wire with unit mass [31]. Obviously, the history path segments stretching farther away from the PC contribute more to the value of MOI, which can be evaluated for each cycle through integrating along the contour of the history path, as:

p



O dp, I XX 

1 p



Y 2 dp, IYYO

1 p



O X 2 dp, I XY 

1 p



XYdp

(19)

where dp is the length of an infinitesimal arc of the path and p is the perimeter of the path. Note that the MOI spacing from the PC is obtained based on the parallel axis theorem, given by: O I XX  I XX  Yc 2 O IYY  IYY  X c2

(20)

O I XY  I XY  X cYc

In this study, the MOI in all 18 different loading paths are calculated. Being different from FNP which cannot separate the directionality of loading path under proportional or uniaxial loading, the F1 term based on MOI concept introduced in the damage parameter can reflect the contributions to total elastoplastic work from different loading components, expressed in the following equations:

 12I XX 12IYY F1     2 D D2 

  

(21)

where D is a convex enclosure (smallest circle) diameter of the loading path. The factor ξ = ΔWτ/ΔW0 is kept to distinguish the damage weights caused by tensile and torsional effects, where ΔWτ is the pure shear elastoplastic work with the same loading level. Besides, to avoid possible argument on the estimation of the value of FNP, the definition of FNP is used in the calculation, which can be measured by test results as [32]: 13

FNP 

 eq /  0  1  OP /  0  1

(22)

where σeq, σ0 and σOP are von Mises equivalent stress amplitudes for the measured loading path, pure tensile path and circular path under the similar strain level, respectively. Correspondingly, the proposed energy-based EDP is proposed as:



    

12I XX 12IYY  2 D D2

    1   w FNP   W0 

(23)

The parameter includes the following special features: 1) F1 includes the weighting factor ξ which reflects the ratio of damage caused by shear work to that by tensile work with the same effective loading amplitude. For the tensile-sensitive material, e.g. 316L stainless steel, the damage caused by the torsional loading is much smaller than that by tensile loading because the tensile stress tends to open fatigue microcracks, significantly reducing the fatigue lives, contrary to the torsional damage as no crack opening is contributed. Hence, the weighting factor ξ therein is obtained as 0.15 through fitting experimental data, different from those proposed by Morrow (ξ=0) and by Garud (ξ=0.5). 2) Ixx and Iyy quantifying the distance away from X and Y axes present the contribution of torsional and tensile damage to the total damage. The work generated by the combined tension-torsion loading is related to the uniaxial tensile work through the proposed equivalent parameter in Eq (18). The NP hardening effect has been revealed not only through the material-dependent αw, but also through load path dependent FNP. Note that for pure shear test, the vanishing of Iyy leads to Weq    W0 ; for uniaxial tension-compression test, Weq  W0 with Ixx = 0; and for 45°proportional loading, i.e. the case IX of Fig.4, Weq  1     W0 with I XX 

1 1 2 D and IYY  D 2 , 12 12

with the combination of tensile and torsional work under proportional loadings. 3) As Ixx and Iyy of its corresponding homogeneous wire can be interpreted as descriptors of the applied loading path, the path-dependent F1 combined with Ixx and Iyy can associate the directional damages induced by actual shape of the loading path with the equivalent damage parameter. 4) The proposed parameter based on the uniaxial elastoplastic fatigue work ΔW0 can significantly decrease experimental or computational cost for multiaxial constitutive modeling. In addition, for low cycle fatigue problems, the plastic work is dominant, while for high cycle fatigue problems, the elastic work has been taken into account. Note also that the NP hardening effect is transient until it converges to a steady-state with a sufficiently large accumulated plastic strain [33]. The transient NP hardening effect can be reflected by the proposed parameter, as the MOI-based F1 can be integrated along any multiaxial load history path. 5) It is also worthy of noting that the proposed parameter can also give a much more reasonable reflection of the mean stress effect without introducing additional terms, 14

because the mean component can be easily estimated through the application of the parallel axis theorem from original point to PC of hypothesized homogeneous wire. Furthermore, to improve the computational efficiency and reduce the possible influence of hydrostatic stress σh on the plastic strain estimation for pressure-insensitive materials, e.g., 316L stainless steel, the computation is performed in the deviatoric space as recommended in the literature [34]. The stresses and strains (σ, ε) in a general six-dimensional (6D) space can be transformed to deviatoric stresses and strains (s,e) in 5D deviatoric space with the following expressions:

sx   x   h   2 x   y   z  / 3

ex   x   h   2 x   y   z  / 3

s y   y   h   2 y   x   z  / 3 ey   y   h   2 y   x   z  / 3

(24)

sz   z   h   2 z   x   y  / 3 ez   z   h   2 z   x   y  / 3





where the hydrostatic stress and strain can be expressed as  h   x   y   z / 3 and

 h   x   y   z  / 3 . Indeed, the calculation can be further simplified in 2D sub-deviatoric space when σy = σz = τxz = τyz = 0, with γxz = γyz = 0 and εy = εz = -νεx, resulting in:

s1   x , s2  0, s3   xy 3, s4  0, s5  0





e1   x 1  , e2  0, e3 

 xy 2

3, s4  s5  0

(25)

(26)

whereν is an effective Poisson ratio that combines elastic and plastic effects. 5. Results and discussion Finally, the fatigue crack initiation lives under all 18 strain-controlled loading paths including the unbalanced circle path are employed to assess the prediction capabilities of the proposed fatigue damage parameter. The proposed parameter versus the fatigue crack initiation life are plotted in Fig.6(a). The predicted and observed fatigue lives in number of blocks, where each block consists of full load period, are compared in Fig. 6(b). For these quasi-constant loading paths, cycle detection is simple as the peaks and valleys of each stress or strain component in general match with the peaks and valleys of the other components. Every time a full load cycle with extreme points is counted, the MOI method is used to calculate the area-equivalent elastoplastic work to estimate the damage contribution of one block. Note however that for variable amplitude NP multiaxial loading, the multiaxial counting methods, e.g., the modified Wang-Brown method, are needed to identify the individual cycles.

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(a) (b) Fig.6. Fatigue life prediction based on the energy-based EDP: (a) EDP vs. Nf correlation; (b) comparison of predicted and experimental lives. As shown in Fig. 6, the parameter correlated with experimental data is favorable as all points fall within a factor of three scatter band of the test lives. The conclusion from these comparisons is that all the predicted fatigue lives correlate well with the experimental data under tension, torsion and simultaneous tension-torsion with different proportional and NP strain-controlled loading paths, illustrating a robust prediction capability of the proposed parameter. It is also noted that the prediction of certain loading path (circular, square and rhombic paths) which expresses high NP hardening effects tends to be a little bit conservative. The reason for such phenomenon might be the weighting factor ξ which tends to be non-constant with the increase of FNP. Basically speaking, the fatigue crack is induced by localized cyclic dislocation movements which associate with shear stress (or strain) range. With the propagation of the shear-controlled microcrack, the mixed tensile/shear or tensile-controlled crack is formed, whose dominated driving force is the normal stress σ⊥ or maximum stress σmax. However, to measure the transition of these two phases in practice is not easy because the material-dependent threshold is normally much smaller than the reliably detectable value from macroscopic measurements. As a result, crack initiation models usually assume the presence of both phases, depending on the percentage of the initiation life spent at each phase. For instance, Fatemi-Socie’s damage parameter combines the shear strain range Δγ and the peak normal stress σ⊥max in the initiation of shear cracks, and CCB’s parameter includes not only normal stress and strain but also shear stress and strain. When the propagation phase of the microcracks is dominant over its initiation, SWT’s parameter is particularly useful. The proposed energy-based damage parameter involves shear and normal components, and the former relate to the movement of dislocations while the latter affect the propagation of the dominant microcrack. The actual 2D stress or strain loading path is finally calculated as a function of the moment of inertia of such path with respect to its centroid to quantify the damage contributions from shear or normal direction, through Ixx and Iyy in Eq.(23).

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Besides, the path dependence of the strain histories can be accounted through not only FNP but also the MOI of the plastic strain path about y-axis and x-axis, i.e., Ixx and Iyy, which could be integrated along any multiaxial load history path. According to the comparative analysis with some existing models, it can be discovered that the proposed model has several advantages: (i) the complex incremental plasticity algorithms are not required in multiaxial fatigue life prediction even for NP histories, correspondingly a set of differential equations need not to be integrated in such cases; (ii) the damages from different loading direction can be distinguished which makes it possible to be used in general six-dimensional space; (iii) the introduction of the MOI into the damage parameter makes it possible to use the traditional properties of MOI to get hints about the behavior of any arbitrarily shaped multiaxial load histories without losing information. Note that the proposed energy-based EDP is to try to find an accumulated elastoplastic work, derived from different loading components with different weighting factors, as most energy-based parameters do. However, such linear relation might not describe well the materials which behave in fatigue and further searches need to be explored for more complex-shaped load history paths. It is worthwhile to note as well that, being different from laboratory pre-designed fatigue paths, the realistic service load histories make multiaxial fatigue damage calculations more challenging and laborious because the peaks and valleys of the stresses and the corresponding strains do not usually coincide under general random multiaxial histories. In this case, the multiaxial rainflow counting algorithm e.g. modified Wang–Brown method [35] can be used to obtain different equivalent ranges necessary for damage calculation. Moreover, in practical applications the measured signals will always be contaminated by unavoidable noise, to reduce the calculation burden, the multiaxial racetrack amplitude filter is also recommended to synchronously filter complex loading histories with a user-defined filtering amplitude while preserving the load path shape information, which is an important feature for multiaxial fatigue analyses [36]. 6. Conclusions In this work, a novel energy-based EDP is proposed and the following essential conclusions can be drawn: (1) The proposed parameter is applicable to predict the fatigue life under NP multiaxial histories, without having to deal with complex multiaxial constitutive modeling. (2) Based on the MOI concept, path-dependent damages from different loading paths can be distinguished. And an integrated uniaxial equivalent elastoplastic work has been used to estimate the fatigue life generated by proportional or NP loadings, successfully generalizing the ε-N curve to deal with multiaxial histories. (3) Non-proportional tension-torsion experiments under different strain-controlled loading paths validate the prediction capability of the proposed parameter with 316L steel tubular specimens. The results show that the proposed parameter is simple, effective, and easy to be implemented in engineering practice. Acknowledgments

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This work was supported by Shenzhen Municipal Government through the Fundamental Research Project (grant number JCYJ20170307151049286) and the National Natural Science Foundation of China (Nos. 11572227 and 11772106).

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[18] Itoh T, Chen X, Nakagawa T, Sakane M. A Simple Model for Stable Cyclic Stress-Strain Relationship of Type 304 Stainless Steel Under Nonproportional Loading. Trans, ASME, J. Engng Mater Tech. 2000; 122:1-9. [19] Bishop JE. Characterizing the non-proportional and out-of-phase extent of tensor paths. Fatigue Fract Engng Mater Struct 2000; 23:1019-32. [20] ASTM E2207-08. Standard practice for strain-controlled axial-torsional fatigue testing with thin-walled tubular specimens; 2013. [21] Mazánová V., Škorík V., Kruml T., Polák J. Cyclic response and early damage evolution in multiaxial cyclic loading of 316L austenitic steel. Int J Fatigue 2017;100: 466-76. [22] Wong YK, Hu XZ, Norton MP. Plastically elastically dominant fatigue interaction in 316L stainless steel and 6061-T6 aluminium alloy. Fatigue Fract Engng Mater Struct 2010;25: 201-13. [23] Socie, DF, Marquis, GB. Multiaxial Fatigue, SAE 1999. [24] Chu CC, Conle FA, Bonnen JF. Multiaxial stress-strain modeling and fatigue life prediction of sae axle shafts. ASTM STP 1191; 1993: 37-54. [25] Fatemi A, Socie DF. A critical plane approach to multiaxial damage including out-of-phase loading. Fatigue Fract Engng Mater Struct 1988;11(3): 149-65. [26] Glinka G, Wang G, Plumtree A. Mean stress effects in multiaxial fatigue. Fatigue Fract Engng Mater Struct 1995;18(7-8): 755-64. [27] Glinka G, Shen G, Plumtree A. A multiaxial fatigue strain energy density parameter related to the critical fracture plane. Fatigue Fract Engng Mater Struct 1995;18(1) : 37-46. [28] Liu KC. A method based on virtual strain-energy parameters for multiaxial fa-tigue life prediction. ASTM STP 1191; 1993: 67-84. [29] Walker K. The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 7075-T6 aluminum. ASTM STP 462; 1970: 1-14. [30] 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 Engng Mater Struct 2018; 41(6): 1402-11. [31] Meggiolaro MA, Castro JTP, Wu H. An improved multiaxial rainflow algorithm for non-proportional stress or strain histories - Part I: Enclosing surface methods Int J Fatigue 2012; 42: 217-26. [32] Meggiolaro MA, Castro JTP. Prediction of non-proportionality factors of multiaxial histories using the Moment Of Inertia method. Int J Fatigue 2014; 61:151-9. [33] Meggiolaro MA, Wu H, Castro JTP. Non-proportional hardening models for predicting mean and peak stress evolution in multiaxial fatigue using Tanaka’s incremental plasticity concepts Int J Fatigue 2016; 82:146-57. [34] Meggiolaro MA, Castro JTP, Wu H. A general class of non-linear kinematic models to predict mean stress relaxation and multiaxial ratcheting in fatigue problems – Part I: Ilyushin spaces Int J Fatigue 2016;82: 158-66. [35] Meggiolaro MA, Castro JTP. An improved multiaxial rainflow algorithm for non-proportional stress or strain histories – Part II: The Modified Wang-Brown method Int J Fatigue 2012;42: 194-206. [36] Wu H, Meggiolaro MA, Castro JTP. Validation of the multiaxial racetrack amplitude filter. Int J Fatigue 2016;87: 167-79. 19

Highlights 1. A novel energy-based equivalent damage parameter based on uniaxial fatigue data is proposed. 2. The Moment Of Inertia method and a weighting factor are introduced in the parameter. 3. Incremental plasticity algorithms are avoided in the prediction of non-proportional multiaxial fatigue life. 4. A series of multiaxial fatigue tests validate the prediction capability of the proposed parameter.

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