Accepted manuscript

Effect of residual stress on contact fatigue of a carburized generator according to multiaxial fatigue criteria Wei Wang, Huaiju Liu, Caichao Zhu, Xuesong Du, Jinyuan Tang PII: DOI: References:

S0020-7403(18)31588-1 https://doi.org/10.1016/j.ijmecsci.2018.11.013 MS 4639

Appearing in:

International Journal of Mechanical Sciences

Received date: Revision date: Acceptance date:

16 May 2018, 10 October 2018, 13 November 2018

Diesen Artikel zitieren als: Wei Wang, Huaiju Liu, Caichao Zhu, Xuesong Du, Jinyuan Tang, Effect of Residual Stress on Contact Fatigue of a Carbureted Wind Turbine Gear with Multiaxial Fatigue Criteria, International Journal of Mechanical Sciences (2018), doi: https://doi.org/10.1016/j.ijmecsci.2018.11.013

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Rolling contact fatigue damage and life of hardened gears are calculated considering gradients of mechanical properties and residual stress within the hardening layer.

The influence of the initial residual stress on the fatigue life of rolling contacts can be represented by the Fatemi-Socie criterion and the result shows excellent agreement.

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with the existing reference.

The effects of the peak value of the initial residual stress in the rolling contact

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Fatigue damage is shown under different loading conditions.

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Influence of residual stress on contact fatigue of a carburized wind turbine gear with multiaxial fatigue criteria Wei Wang1, Huaiju Liu1*, Caichao Zhu1, Xuesong Du1, Jinyuan Tang2 1 State Key Laboratory of Mechanical Transmissions, Chongqing University, Chongqing, China, 400030 2 State Key Laboratory of High Performance Complex Manufacturing, South Central University, Changsha, Hunan, China, 410083

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*Corresponding author:[email protected]Abstract

Hardened gears are widely used in heavy-duty machinery such as wind turbines, ships, high-speed rails, etc. Variations in hardness and residual stress created by the carburizing and quenching processes have a significant impact on rolling contact fatigue (RCF) during repeated meshing.

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In this work, an attempt is made to investigate the effect of residual stress on rolling contact fatigue and to predict the rolling contact fatigue life of a carburized wind turbine gear based on multiaxial fatigue criteria. A finite element elastic-plastic contact model is developed that accounts for hardness gradients and initial residual stress. The initial distribution of the residual stresses is determined by experimental measurements. The stabilized stress-strain field is performed considering the shaking state under heavy load conditions. The Fatemi Society and the

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The Brown-Miller criteria are used to estimate the contact fatigue damage of the carburized gear. The numerical results show that the initial residual stress affects the contact fatigue failure

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it can be reflected in the Fatemi-Socie criterion while it is not included in the Brown-Miller criterion since the two parameters appearing in the criterion are not affected by the residual stress. In terms of the Fatemi-Socie criterion, under moderate loading conditions where an elastic response occurs, a

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Significant compressive residual stress would change the orientation of the crack initiation plane, whereas under extremely severe loading conditions where plasticity occurs, the influence of the initial residual stress on contact fatigue damage within the non-contact zone is limited. plastic near the surface. He

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The effect of initial residual stress on contact fatigue life presents a bilinear curve. As the magnitude of the residual tensile stress increases, the RCF life decreases linearly. As the

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If the magnitude of the compressive residual stress exceeds a certain value, the RCF life will no longer increase.

Keywords: carburetor gearbox; rolling contact fatigue life; initial residual stress; Fatemi Society Criterion; Brown-Miller criterion

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Nomenclature

C

Pure strain amplitude, %

γf

Shear Fatigue Ductility Coefficients Orthogonal Shear Strain, %

ah

The amplitude of normal deformation, %

e f

Axial fatigue ductility coefficients

εx

Axialdehnung x, %

j

Axial deformation y, %

θ σb σ f

The angle of the critical plane

to the max

Tensile strength, MPa. Resistance coefficients against axial fatigue. Maximum normal stress, MPa. Initial residual stress, MPa

X

Axialspannung x, MPa

j

Axial span and, MPa

per se

The initial elastic limit, MPa

b BM c

CC CRS

The orthogonal shear stress, MPa. The fatigue strength exponents. The Brown-Miller criterion

Exponents of fatigue ductility

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t-xy

Shear Fatigue Resistance Coefficient, MPa

The effective cementing depth, mm. Initial compressive residual stress, MPa

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t f

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tu

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γx y

Contact fatigue damage according to the Brown-Miller criteria

SFD

Contact fatigue damage according to Fatemi-Socie criteria

E FS G HV M

The equivalent modulus of elasticity, MPa

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DBM

You give Fate-Society-Criterium

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The shear modulus of elasticity, MPa

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N.f.

FCR RS

Vickers hardness, HV. Modulus of linear hardening, MPa. Number of cycles before cracking occurs at the site of the material. rolling contact fatigue. Initial residual stress, MPa

T0

Input torque for example gearbox, kNm

TRS x y

Initial residual tensile stress, MPa

z

The rolling direction. The depth direction. The tooth width direction

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1. Introduction With the increasing demand for higher power densities and load capacities of gear driven mechanical machines, contact fatigue failure problems of gears have become limiting factors affecting the reliability of these machines, such as: As wind turbines influence. Due to the large width of the teeth and the high flexural strength of the tooth root, another type of failure occurs, namely flexural fatigue of the tooth root

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In technical practice, there are only minor failures in gearboxes in wind turbines. Transmission failure caused by rolling contact fatigue can have many causes, such as: B. micropitting [1], pitting [2] and tooth flank fracture [3]. Figure 1 shows a typical contact fatigue failure in a 2 MW wind turbine gearbox. Case hardening processes such as carburizing [4] are often used in large format, heavy-duty gears. Rolling contact fatigue failures of carburized gears are still unavoidable, although compared to non-carburized gears, they have longer life due to longer life

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contact fatigue resistance. After the carburizing and quenching process, the change in carbon content creates a hardness gradient from the case to the core. At the same time, residual stresses are induced in the hardened shell as a result of the heat treatment. Predicting localized contact fatigue damage in carburized gears remains a difficult problem because the mechanical properties vary due to the hardness gradient and the significant residual strength.

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stress level.

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Figure 1: Contact fatigue failure in a 2MW wind turbine gearbox (a)-(b) and contact fatigue surface microtopography (c)

Many efforts have been made to study the contact fatigue behavior of gears. Evans et al. [5]

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simulated the effect of surface roughness on contact fatigue life of gears using a mixed lubrication model. Liu et al. [6, 7] studied the influence of lubrication and surface roughness on the stress distribution in the subsurface of gears. Seghdei et al. [8] used the continuous damage approach to study the evolution of fatigue damage during contact fatigue. Anisetti [9] studied the contact fatigue behavior of gears considering the effects of tribodynamics. Dong et al. [10] studied the influence of temperature on gears based on the mixed EHL theory. Liu et al. [11] applied the Dang-Van criterion to study the contact fatigue behavior of a pair of spur gears. Qiao [12] applied several multi-axial criteria to study RCF lifetime and failure probability. However, these investigations do not take into account residual stresses and fluctuations in the mechanical properties. The influence of residual stress on fatigue behavior has been extensively studied in recent years.

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Years. Fukumasu et al. [13] studied the influence of residual surface tension generated by shot peening on gear contact failure and found resistance to gear pitting with additional shot peening treatment. Maasi [14] studied the influence of residual stress on the contact fatigue behavior of lubricated contacts, taking into account the surface roughness. Pape [15] studied the influence of the initial state of residual stress on the fatigue life of the bearing contact using the Ioannides-Harris model. Batista et al. [16] developed a numerical model to predict the relaxation of residual stresses during the contact process of automotive gears. Hiroyuki [17] studied the effect of shear process induced residual stresses on fatigue life using an experimental method and a

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A higher fatigue life was observed for the compressive residual stress sample. As many experimental observations and simulation results show, the rolling contact fatigue behavior of carburized gears is very sensitive to fluctuations in the mechanical properties and the residual stress within the hardened layer. Joe et al. [18] investigated the fatigue behavior of case-hardening steels using a two-layer model and found good agreement with the experimental results. Narita et al. [19] simulated the fatigue behavior of a traction drive element and a good

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Agreement with the experimental results was found when the hardness gradient and the residual stress are taken into account. These studies confirm the need to consider mechanical property variations and residual stresses within the hardened layer when evaluating the fatigue behavior of carburized gears.

In this study, the influence of the initial residual stress on the rolling contact fatigue life of carburized gears is investigated. An elastic-plastic finite element contact model is developed that takes care of this

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Consideration of hardness gradients and initial residual stress. The initial distribution of the residual stresses is determined by experimental measurements. The Fatemi Society and Brown Miller Criteria

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are used to estimate the contact fatigue damage of the carburized gear. A set of loading conditions is applied to study the effect of residual stress, thus providing theoretical support for a deeper understanding of the role of residual stress in gear contact fatigue.

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2. Simulationsmethodik

The example gear pair is from a megawatt wind turbine gearbox. This intermediate

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In engineering practice, it has been found that the parallel gear stage, especially the small output gear, suffers severe premature failures due to rolling contact fatigue. The basic parameters of this pair of gears are shown in Table 1. The gear material is 18CrNiMo7-6 with case hardening, quenching and

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Tempering treatments, after which the gear is ground as the final manufacturing process. Table 1 Transmission parameters

number of teeth

Z1=121, Z2=24

tooth width of the gear

B=0,295m

Gear module normal

m0=0,011 m

pressure angle

α0=20°

Poissonzahl

v=0,3

modulus of elasticity

E=210000 MPa

Radius am Wegpunkt

R1=0,684 m, R2=0,136 m

rated input torque

T0=282,8 kNm

According to the theory of contact mechanics [20], the contact of the gear pair at any point in time of the meshing could be represented in a simplified way as if two bodies with different radii of curvature were in contact with each other.

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other. Planar stretching is assumed in the study, which means that the influence of the helix angle and the modification of the crown tooth is neglected. The schematic diagram of gear contact.

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The model is shown in Figure 2.

Figure 2 The schematic diagram of gear contact model 2.1 The characterization of residual stress and hardness gradient

The coordinate system used in the numerical model is defined as follows: The roll direction is

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represented as x-direction, the depth direction is represented in y-direction, while the width of the tooth is given in z-direction. The hardness gradient from box to core induced by cementing can be determined by the Vicker test [21] or empirical methods [22, 23], among others Thomas presented an empirical formula for representing the hardness curve. Hardness for case hardened gears, expressed as [23]:

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aa y 2 ba y ca HV( y ) ab y 2 bb y cb HV Kernel

550 - HVsuperficie; CHD2 - 2 yHV,max CHD

ba = -2 aa yHV,max ;

ca HVsurface ;

(1)

ab

H(CHD) 2 (CHD ycore )

;

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aa

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(0 y CHD) (CHD y ycore ) ( ycore y )

bb 2 aa ycore ; cb 550 ab CHD2 bb CHD ;

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Where HVsurface and HVcore are the surface hardness and the core hardness, respectively; yHV,max is the depth of maximum hardness, in this study it is set to 0. ycore is the depth with

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HV( y) HVcore . The effective cementing depth (CHD) is defined as the depth at 550 HV. The hardness of the sample gear is also measured experimentally with Vicker's indentation testing machine from the surface to the core. Figure 3 shows the Thomas fitted hardness curve as well as the experimental hardness data. As can be seen, the empirical result and the tested result agree well, with only minor deviations very close to the surface, which is probably due to a slight grinding burn in the area close to the surface. The maximum hardness value is at a depth of about 0.7 mm. Additionally, it should be noted that the surface hardness, core hardness, and effective hard belt depth used in the empirical Thomas method may be determined by actual engineering requirements. Therefore, the surface hardness, the core hardness, and the CHD are controlled to 670 HV, 420 HV, and 2.2 mm, respectively.

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according to the technical design requirements. The solid red line in Figure 3 shows the hardness gradient profile based on the Thomas method, from which it can be seen that at the depth of 3.6

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mm, the hardness gradually decreases to the core hardness value.

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Figure 3 The comparison between the measured hardness data and Thomas's empirical method. In addition to the hardness gradient, the carburizing process also involves the variation of other mechanical properties such as yield strength and tensile strength from the shell to the core. A large number of studies have shown that there are certain relationships between the hardness and other mechanical properties of case-hardening steels [24, 25]. In terms of contact fatigue life

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Prediction, fatigue parameters and yield strength expressed as functions of hardness are used in the work. According to references [26], the relationships between the

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Hardness, yield strength and tensile strength are expressed as: (2)

σb ( y) 1200 (1950 1200)/(670- 420)* (HV( y) - 420)

(3)

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σ ys (y) 800 (1300 800)/(670- 420)* (HV(y) - 420)

where σ ys ( y ) is the yield point and σ b ( y ) is the tensile strength, HV is the Vickers value

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Hardness. The yield strength distribution resulting from the cemented model is shown in Figure 4 (left). In this work, the Seeger relationships between fatigue parameters and tensile strength are used [27],

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which are expressed as:

σf 1,5σ b , εf 0,59ψ , b 0,087, c 0,58

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If σ b /E 0.003, ψ 1 ; if σ b /E 0.003, ψ 1.375 125(σ b /E) . where ε f and σ f are the fatigue resistance coefficients and the axial fatigue ductility coefficients, respectively. b and c are the fatigue strength and fatigue ductility exponents, respectively, and E is the modulus of elasticity. The distributions of the fatigue parameters ε f and σ f of the hardened gear are described in Figure 4 (right). The shear fatigue coefficient and the shear fatigue ductility can be further calculated as [28]: σ τ f f , γf 3 εf 3

(5)

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Figure 4 Yield strength profiles (left) and fatigue parameters (right)

It is worth noting that the cementing and quenching process would also introduce a large amount of residual stress. The final grinding process also contributes to the formation of the residual stress distribution. Residual stresses at different depth positions can be measured using the established method of electropolishing X-ray diffraction. In the meantime, some empirical methods [29, 30] for determining the carburization residual stress have also been made available.

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Components In this work, the residual stress is measured with an X-ray diffractometer (PROTO LXRD system) with Cr_K-alpha radiation. The point size is set to 1 mm. The two normal components of the residual stress along the x and z directions, σ r,x and σ r,z, are measured, and after electropolishing, the residual stress is then measured at a deeper position. The flat area after electropolishing must be large enough to allow an accurate measurement. Initial residual stress

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The toothed profile components are shown in Figure 5. The reason why σ r,y is not measured is that this component would be negligibly small [20]. Hertter's empirical formula for predicting residual stress based on hardness [30] can be expressed as follows:

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σ r ( y) 1,25 (HV( y) HVcore ) σ r ( y) 0,2857 (HV( y) HVcore ) 460

When(HV( y ) HVcore ) 300 When(HV( y ) HVcore ) 300

(6)

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where HVcore is the hardness of the core. HV(y) is the current local hardness at depth location y.

Figure 5 Different components of the initial residual stress in a tooth profile The results of the initial residual stress are shown in Figure 6. The results show that σ r,x and σ r,z at the same depth level are therefore close to each other, the residual stress is represented by σ r. The measured points are represented by pink hollow circles, based on which a fitted curve is formed (dotted blue line). The first measurement is at the tooth surface and the residual stress value is -132

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MPa The residual stress measured within the hardened layer represents the compressive property, and the maximum value of the compressive residual stress occurs at a depth of about 1.75 mm in the subsoil. This figure also shows the empirical Hertter curve [30]. The comparison shows that the residual stress distribution determined from the measurement and the empirical results agree. In this work, the empirical residual stress is described as a

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function of hardness.

Figure 6 Distribution of initial residual stress along depth It should be noted that the distribution of residual stress can vary significantly from one to another. Although the same hardness profile is guaranteed, the residual stress profile could vary

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change [31]. To investigate the influence of the initial residual stress, 13 sets of residual stress profiles with a maximum value in the range of -300 MPa to 300 MPa are created using Hertter's empirical method. As shown in Figure 7, VCRS and VTRS represent the maximum initial residual

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emphasize. In some technical processes, there may be a residual tensile stress within the hardening layer.

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such as those that grind burn cases [32, 33].

Figure 7 The distributions of the compressive residual stress (left) and the tensile residual stress (right) with different VRS

2.2 The elastic-plastic finite element model of the contact The stress and strain response is essential for further predicting the contact fatigue life. As part of ABAQUS, a finite element elastic-plastic contact model is being developed using the Python programming environment. The calculation area is fixed at 10 mm x 10 mm,

ACCEPTED MANUSCRIPT 0 mm and 10 mm. The upper rigid circle with the appropriate radius of curvature rolls from x=-4mm to x=4mm to ensure that the material points involved experience full load cycles. In this model, different mechanical properties are applied at different depths of the body. The thickness of each layer is 0.025mm, thin enough to capture the effect of mechanical property gradients and initial residual stress. Because material points on the same horizontal plane experience the same stress cycle, the 160 evenly spaced material points on the black dotted line (x 0 mm and y [0.4] mm) are most affected. The element

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The size in the critical zone (the purple part in Figure 8) is set to 0.025mm 0.025mm.

Figure 8 The finite element contact model in layers In this work a constitutive equation of kinematic hardening and the Mises creep criterion are presented.

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It is believed to represent the elastic-plastic response under cyclic loading conditions. As shown in Figure 9, the linear hardening modulus M is defined as 5% of the Young's modulus E.

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according to the tested tensile stress-strain curve. It should be noted that under extremely high loading conditions, a concussion condition can occur, which means that multiple loading cycles must be applied to achieve a stabilized stress and strain response. For a better understanding, the variation of the

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The equivalent plastic strain over each load cycle under the heaviest load condition is shown in Figure 10. The numerical results show that the stress-strain field stabilizes after 5 load cycles for all selected cases. In this work, 5 loading cycles were carried out to ensure stabilization

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Response and in the following sections the results are presented based on the stabilized mechanics

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answer.

Figure 9 The fabric model of the plastic-elastic material

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Figure 10 The variation of the equivalent plastic deformation in each cycle under the load T=1133 kNm 2.3 The fatigue life criterion

In order to better understand the complex state of stress when the transmission is running, the distributions of

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The stress components x y , x , y at a depth of 0.3 mm are shown in Figure 11. As shown in Figure 11, the stress state during the contact process is a typical multi-axial stress state. . The stress components do not vary in equal proportion to each other over time. Therefore, under such complex stress conditions, it is necessary to adopt an appropriate multi-axial fatigue criterion. In addition, the example equipment in the present study is from a medium parallel stage in the megawatt range.

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wind turbine gearbox. In fact, an elastic-plastic reaction can occur in heavy-duty gears throughout their lifetime due to random loading. Due to the high local stress in the contact area, plastic deformation is often observed when the gear is supported. therefore, the

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The fatigue process is dominated by the combined effect of strain and stress.

Figure 11 The stress history at a depth of 0.3mm To capture the state of multiaxial stresses during service, many multiaxial fatigue criteria have been proposed, among which the critical plane approach is widely accepted. The critical plane approach is based on the physical fact that damage always occurs in the plane that has the maximum value of the shear strain amplitude and that crack propagation occurs as a result of the critical plane.

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normal stresses or strains acting on it during load cycles. Since the critical plane is not known prior to the analysis, all candidate planes must be examined to identify the critical plane for each material point. The parameter denotes the angle between the direction of the normal of the candidate plane and the positive direction of the x-axis, as shown in Figure 12 (right). Since the plane strain condition is assumed, it is only necessary to examine the candidate planes within the x-y plane. The normal stress, normal stress and shear strain in a candidate plane are thus obtained from [12]: (7)

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σθ (t ) σ x cos2 θ σ y sen 2 θ 2 τ xy sen θ cosθ γθ (t ) γxy (cos2 θ sen 2 θ ) 2(εy εx )sen θ cosθ θ (t ) εx cos2 θ εz sen 2 θ γxy sen θ cosθ

(8) (9)

Fatemi and Socie have proposed a well-accepted multiaxial fatigue model using the maximum amplitude of shear strain and the maximum normal stress. The fatigue life in this criterion is usually

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expressed as [34]:

γmax σ τ [1 k max ] f (2 N f ) b γf (2 N f )c 2 σ ys G

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Here Δ γmax /2 is the amplitude of the shear stress in the critical plane and σ max is the maximum normal stress in this plane. τ f and γf are the shear fatigue strength and shear fatigue ductility coefficients, respectively. b and c are the resistance to fatigue and fatigue

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Ductility Index, or G is Young's modulus in shear and Nf is the number of cycles for cracking. The material constant k is chosen to be 1 [35].

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The fatigue damage of this criterion is then defined as follows[36]:

γmáx σ [1 k máx ] 2 σ ys

(11)

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Brown and Miller suggested that multiaxial fatigue behavior should be governed by the normal strain amplitude and plane shear strain amplitude components [37]. In her opinion,

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Cyclic shear loads contribute to crack formation and normal loads encourage their growth. The relationship between shear strain amplitude and normal strain amplitude is normal

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suggested as:

γmax σ S ε n C1 f (2 N f ) b C2 εf (2 N f ) c 2 E

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Where Δ γmax /2 γa , Δ εn /2 εa , C1 1.3 0.7S and C2 1.5 0.5S . The fatigue damage of the Brown-Miller criterion is defined as [36]:

γmax S εn (13) 2 The procedure for estimating the fatigue life of a material under cyclic loading is presented in DBM

Figure 12 (left). First, the stress-strain field in the shaking state is determined after 5 load cycles and the stress and strain curves are recorded for each point in the material. Next, for each angle , the stress and strain in the candidate plane are calculated and the damage is further calculated.

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depending on the selected fatigue criteria. Then the contact fatigue life for each point in the material is

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Rated maximum damage.

Figure 12 Flowchart for contact fatigue life prediction (left) and definition of critical plane angle (right) 3. Results and Discussion

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3.1 Comparison between the BM and FS criteria

Under the nominal load condition T0=282.8 kNm without considering the initial residual stress, the damage parameters are evaluated using both the BM criteria and the FS criteria. Figure 13 shows the damage parameters and DBM damage based on the BM criteria. As can be seen, the maximum shear strain amplitude γa is reached at the angles for each material point

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from θ 0 ( 180 ) and θ 90 . The maximum magnitude of γa occurs at a depth of about 0.3 mm. The maximum normal strain amplitude εa is reached at angle θ 90, which means that normal strain would accelerate the contact fatigue failure process along this direction.

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As a result, the maximum value of DBM damage occurs at an angle of θ 90. For comparison, the damage parameters γa, σ max and the DFS damage calculated with the FS criterion are shown in Figure 14. With this criterion, the maximum normal stress σ max reaches its value

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Minimum value (almost 0) at angle θ 90, while it reaches its maximum value at angle θ 0 (180). This indicates that the maximum normal stress accelerates the contact.

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Fatigue failure at angle θ 0 (180 ), while this damage parameter has no effect on damage at angle θ 90 . As a result, the maximum value of DFS damage appears at an angle of θ 0 (180), which is quite different from the Brown-Miller result.

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Criteria.

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Figure 13 The amplitude of the shear strain γa (left), the amplitude of the normal strain εa (middle) and the contact

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DBM fatigue damage distributions (right) based on BM criteria

Figure 14 The amplitude of the shear stress γa (left), the maximum normal stress σ max (middle) and

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DFS distributions for contact fatigue damage (right) based on FS criteria

The influence of the initial residual stress in the stress and deformation field is shown in Figures 15 and 16 under the load condition T0=282.8 kNm. The results show that the initial residual stress has no effect on the stress and strain amplitude as it affects only the mean and maximum values. In addition, the influence of the initial residual stress on the mean and maximum values of stress and strain varies with the change in angle. As seen in Figure 15, the first is

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Residual stress affects σ max significantly at angle around θ 0 (180), CRS decreases σ max while TRS increases σ max at around θ 0 (180) by a similar amount to the box without RS. Since in the BM criterion the fatigue life is determined by εa and

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γa, which are not affected by the initial residual stress, the predicted damage based on the BM

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The DBM criterion does not change when the residual stress varies.

Figure 15 The maximum normal stress σ max with CRS (left), without RS (middle) and with TRS (right) under the load case T0=282.8 kNm. For the FS criterion, on the other hand, it applies that the shear strain amplitude reaches its minimum at the angle of θ 0 ( 180 ) and θ 90 the state of the normal stress σ max ultimately determines the direction of failure. As shown in Figure 16, the maximum damage occurs at an angle of θ 0 (180) when the initial residual stress due to normal stress action is absent. if the initial

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from angle θ 0 ( 180 ) to angle θ 90 . Hence the initial residual stress

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influences the state of damage by its effect on the maximum normal stress.

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Figure 16 The DFS distributions with CRS (left), without RS (middle) and with TRS (right) under load T0=282.8 kNm. Figure 17 (left) shows the influence of the initial residual stress on damage according to the FS criterion. As can be seen, compared to the non-RS case, the damage of the TRS model increases by 13.9%, while the CRS model reduces damage by 1.1%. Figure 17 (right) shows the experimental results of the existing reference on the influence of residual stress on fatigue strength [38]. Self

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Although the result was obtained through a flexural fatigue test, it still clearly reflects the difference between the influence of residual tensile stress and residual compressive stress on fatigue life. May be

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In concrete terms, the TRS leads to a significant deterioration due to fatigue damage and the influence of the CRS on the damage is moderate. Similar results were also found in experiments on the influence of mean stress on rolling contact fatigue [39]. Based on this result, the left side of the Dang Van

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The diagram has been changed.

Figure 17 The effects of residual stress in DFS (left) and the experimental results of the influence of residual stress on fatigue strength according to Ref. [38] (right) 3.2 Effect of residual stress at different stress levels according to FS criterion In this section, the effect of the Residual stress at different stress levels depending on the studied

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SF criteria. Under extremely severe working conditions, as in the case of T0=848.4 kNm and in the case of T0=1131.2 kNm, plasticity occurs in the underground zone. Plastic deformation caused by loading would change the initial residual stress profile. Figure 18 shows the development of the residual stress under the two heavy load conditions. As can be seen, the residual stress changes significantly after cyclic loading compared to the initial residual stress, and the residual stress in the plastic zone tends to be uniform. Under the most severe loading condition, T0=1131.2 kNm, the residual stress profiles after cyclic loading are almost identical, suggesting this

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the trace of the initial residual stress disappears completely and the effect of plasticity appears.

Figure 18 The evolution of residual stress under heavy load conditions. At T0=848.4 kNm the influence of the initial residual stress on the maximum

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Normal stress is shown in Figure 19. As can be seen, plasticity occurs within a certain range of depths, as shown within the dashed lines. Once the initial residual tensile stress is in place, the span is

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The plasticity zone is largest compared to the case without residual stress and the case with compressive residual stress. Meanwhile, the presence of the initial compressive residual stress reduces the range of plasticity compared to the case of no residual stress. He

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The maximum normal stress in the plastic zone is almost identical in these three cases, while there is a clear difference in the maximum normal stress in the non-plastic zone near the surface. The maximum normal stress in the non-plastic zone near the surface is higher in the TRS case and restraining in the CRS case

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the increase in maximum normal stress in that zone.

Figure 19 The maximum normal stress σ max with CRS (left), without RS (middle) and with TRS (right) under heavy load condition T0=848.4 kNm

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Based on the FS criterion, the damage in each candidate plane at each material point is shown in Figure 20 under three cases of initial residual stress. For the CRS, the maximum damage depth is around 0.4mm, while for the TRS sling and non-RS sling, the maximum damage occurs at a depth of around 0.2mm. Based on the FS criterion, the CRS influences

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the depth of crack initiation by influencing the maximum normal stress.

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Figure 20 DFS with CRS (left), without RS (middle) and with TRS (right) under heavy load conditions T0=848.4 kNm

Based on the FS criterion, the influence of peak residual stress value on contact fatigue damage under different loading conditions is shown in Figure 21. The load range is T0=141.4 kNm–848.4 kNm, as can be seen in all selected cases. In some cases, the relationships between the damage and the peak residual stress follow a bilinear curve. The switching point threshold is

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It depends on the state of charge. As the state of charge increases, the switching point of the two segments of the bilinear curve shifts to the left. As the magnitude of the residual tensile stress increases, the

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Contact fatigue damage within the gear surface increases linearly. However, the influence of compressive residual stress on contact fatigue damage shows the same slope only in a limited range where the magnitude of compressive residual stress is relatively small. However,

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If the residual compressive stress exceeds a certain value, the contact fatigue damage remains unchanged with increasing magnitude. These results were also found in experiments

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studies. Terrin et al. [40] used a twin-disc dynamometer to evaluate the influence of residual compressive stress on the gear material 17NiCrMo6-4, which is similar to the material used in the present work. The experimental results show that, based on the residual stress present,

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The residual compressive stresses generated by shot peening cannot prevent contact fatigue damage.

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Figure 21 Effect of maximum value of residual stress in DFS under different load conditions

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3.3 Effect of Residual Stress on RCF Life Based on FS Criteria

Based on the FS criterion, the RCF lifetimes are shown in Figure 22. The minimum contact fatigue life with CRS under the rated load condition is 9.00×107, this value increases by 28% compared to the free result. from RS, while the minimum contact fatigue life with TRS

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decreases by 73% compared to the RS-free result.

Figure 22 Contact fatigue life under three residual stress conditions using the FS criteria

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The curves γ - N f of this cemented gear are obtained considering various states of the output gear.

residual stress. The TRS is 300 MPa and the CRS is -300 MPa. The selected load range is T0=141.4 kNm-1131 kNm. As shown in Figure 23, the curves with CRS and without RS are very close.

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If one compares the two curves with the TRS curve, however, a clear discrepancy can be seen.

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Figure 23 The simulated γ - N f curve based on the FS criterion

The influence of the residual stress state on the service life of the gear RCF can be observed in Figure 24 as a supplement to Figure 23. With increasing load, the contact fatigue life decreases exponentially, confirming the rule of Basquin's equation [41]. The charging area can be roughly divided into two levels. Stress level 1 represents the purely elastic response, while stress level 2 produces plasticity.

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At low loads, the effect of compressive residual stress on RCF life improvement is significantly smaller than the degrading effect of tensile residual stress on life. However, if an extremely high load occurs, the initial residual stress changes somewhat

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Plasticity, the influence of the initial residual stress on the contact fatigue life is mitigated.

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Figure 24 The effects of residual stress on contact fatigue life under different loading conditions 4. Conclusions In this study, the influence of initial residual stress on the rolling contact fatigue life of carburized gears is investigated using a numerical simulation method. Based on the Fatemi-Soice criterion, an innovative model is developed that takes into account the hardness gradient and the initial residual stress. A finite element elastic-plastic contact model of a carburized gear of a megawatt wind turbine gearbox is proposed. The conclusions are summarized below. 1. The initial residual stress does not affect the stress and strain amplitude, only the average value and the maximum value. The initial residual stress influences the

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The maximum normal stress and hence the effect of the initial residual stress on contact fatigue failure can be reflected by the Fatemi-Socie criterion. 2. According to the Fatemi-Socie criteria, the initial residual stress influences the damage state through its effect on the maximum normal stress. The residual tensile stress leads to a noticeable worsening of the contact fatigue damage, while the influence of the residual compressive stress on the damage is moderate. Under severe loading conditions where plasticity occurs, the initial residual stress profiles are greatly altered, with post-evolution residual stress still playing a significant role in the severity of fatigue damage.

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especially in the near-surface area where no plasticity occurs. 3. According to the Fatemi-Socie criterion, the relationship between the damage and the maximum value of the residual stress follows a bilinear curve. As the state of charge increases, the switching point of the two segments of the bilinear curve shifts to the left. If the residual compressive stress exceeds the switching point, the fatigue life of the rolling contact no longer increases with increasing size.

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5. Recognition

The work is supported by the National Natural Science Foundation of China (Grant Nos. 51775060, 51575060, 51535012) 6. References

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GRAPHICALLY ABSTRACT