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Effect of the residual stress on contact fatigue of a wind turbine carburized gear with multiaxial fatigue criteria
Accepted Manuscript
Effect of the residual stress on contact fatigue of a wind turbine carburized gear with multiaxial fatigue criteria Wei Wang , Huaiju Liu , Caichao Zhu , Xuesong Du , Jinyuan Tang PII: DOI: Reference:
S0020-7403(18)31588-1 https://doi.org/10.1016/j.ijmecsci.2018.11.013 MS 4639
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
16 May 2018 10 October 2018 13 November 2018
Please cite this article as: Wei Wang , Huaiju Liu , Caichao Zhu , Xuesong Du , Jinyuan Tang , Effect of the residual stress on contact fatigue of a wind turbine carburized 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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Highlight
The rolling contact fatigue damage and life of the carburized gear are calculated by considering gradients of mechanical properties and residual stress within the hardening layer.
The influence of the initial residual stress on rolling contact fatigue life can be reflected by the Fatemi-Socie criterion and the result have an excellent agreement
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with the existing reference.
The effects of the peak value of the initial residual stress on the rolling contact
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fatigue damage are illustrated under a series of load circumstances.
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Effect of the residual stress on contact fatigue of a wind turbine carburized 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, Central South University, Changsha, Hunan, China, 410083
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*Corresponding author: [email protected] Abstract
Carburized gears are extensively applied in heavy duty machines such as wind turbines, ships, high-speed rails, etc. The variations of hardness and residual stress introduced by carburizing and quenching processes remarkable affect the rolling contact fatigue (RCF) during repeated gear
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meshing. The present work attempts to study the effect of the residual stress on rolling contact fatigue and predict the rolling contact fatigue life of a wind turbine carburized gear based on multiaxial fatigue criteria. A finite element elastic-plastic contact model is developed which takes the gradients of hardness and initial residual stress into account. Initial residual stress distribution is obtained through experimental measurements. The stabilized stress strain field is carried out by considering the shakedown state under the heavy load conditions. The Fatemi-Socie and the
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Brown-Miller criterions are used to estimate the contact fatigue damage of the carburized gear. Numerical results reveal that the effect of the initial residual stress on the contact fatigue failure
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can be reflected by the Fatemi-Socie criterion whereas it is not reflected by the Brown-Miller criterion since the two parameters appeared in the criterion is not influenced by residual stress. In terms of the Fatemi-Socie criterion, under modest load conditions where elastic response occur, a
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considerable compressive residual stress would alter the plane orientations of crack initiation, while under extremely heavy load conditions where plasticity occur, the influence of the initial residual stress on contact fatigue damage is limited within the near-surface non-plastic zone. The
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effect of the initial residual stress on contact fatigue life represents a bilinear curve. As the magnitude of the tensile residual stress increases, the RCF life decreases linearly. As the
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magnitude of the compress residual stress exceeds a certain value, the RCF life does not increase anymore.
Keywords: Carburized gear; Rolling contact fatigue life; Initial residual stress; Fatemi-Socie criterion; Brown-Miller criterion
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Nomenclature
γa
The amplitude of sheer strain, %
γf
The shear fatigue ductility coefficients The orthogonal shear strain, %
εa
The amplitude of normal strain, %
ε f
The axial fatigue ductility coefficients
εx
The x-axial strain, %
εy
The y-axial strain, %
θ σb σ f
The angle of critical plane
σ max
The tensile strength, MPa The axial fatigue strength coefficients The maximum normal stress, MPa The initial residual stress, MPa
σx
The x-axial stress, MPa
σy
The y-axial stress, MPa
σ ys
The initial yield strength, MPa
b BM c
CHD CRS
The orthogonal shear stress, MPa The fatigue strength exponents The Brown-miller criterion
The fatigue ductility exponents
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τ xy
The shear fatigue strength coefficients, MPa
The effective case hardening depth, mm Compressive initial residual stress, MPa
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τ f
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σr
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γx y
The contact fatigue damage under Brown-miller criterion
DFS
The contact fatigue damage under Fatemi-Socie criterion
E FS G HV M
The equivalent elasticity modulus, MPa
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DBM
The Fatemi-Socie criterion
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The shear elastic modulus, MPa
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Nf
RCF RS
The Vickers hardness, HV The linear hardening modulus, MPa The number of cycles to crack initiation for material point The Rolling contact fatigue Initial residual stress, MPa
T0
Input torque for sample gear, kNm
TRS x y
Tensile initial residual stress, MPa
z
The rolling direction The depth direction The tooth width direction
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1. Introduction With the growing demand on higher power densities and loading capacities of gear-driven mechanical machines, contact fatigue failure problems of gears have become limiting factors influencing the reliabilities of those machines such as wind turbines. Due to the large teeth width and high tooth root bending strength, the other failure type, namely tooth root bending fatigue
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failure, occurs in a low proportion in the wind turbine gear during engineering practice. The gear failure induced by rolling contact fatigue takes many modes, such as micropitting [1], pitting [2] and tooth flank fracture [3]. Figure 1 shows a typical contact fatigue failure on a 2 MW wind turbine gearbox. Case hardening processes, such as the carburizing [4], are widely used in large-scale heavy duty gears. The rolling contact fatigue failures of carburized gears still can not be avoided, even though they have longer lives compared with un-carburized gears due to higher
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contact fatigue resistance. The hardness gradient is generated from the case to the core after carburizing and quenching process by changing the carbon content, at the meantime, the residual stress is also induced within the hardened layer due to the heat treatment. The prediction of the local material contact fatigue damage of carburized gears is still a difficult problem due to the variations of mechanical properties introduced by hardness gradient and the considerable residual
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stress level.
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Figure 1 the contact fatigue failure on a 2 MW wind turbine gearbox (a)-(b) and microtopography of contact fatigue surface (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 gear contact fatigue life by using a mix-lubrication model. Liu et al. [6, 7] investigated the effect of the lubrication and the surface roughness on gear subsurface stress distribution. Seghdei et al. [8] used the continuous damage approach to study the fatigue damage evolution during the contact fatigue. Anisetti [9] studied the gear contact fatigue behavior by considering the impact of tribo-dynamic. Dong et al. [10] investigated 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 spur gear pair. Qiao [12] applied various multiaxial criteria to investigate the RCF life and failure probability. However, those research do not take the residual stress and the variations of mechanical properties into account. The influences of the residual stress on fatigue behavior have been widely investigated in recent
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years. Fukumasu et al. [13] studied the influence of the surface residual stress produced by shot peening on gear contact failure and a pitting resistance was found of the gear with an additional shot-peeting treat. Maasi [14] investigated the effect of the residual stress on contact fatigue behavior in lubricated contact by considering the surface roughness. Pape [15] studied the influence of the initial residual stress state on bearing contact fatigue life by using Ioannides-Harris model. Batista et al. [16] developed a numerical model to predict residual stress relaxation during the contact process of automotive gears. Hiroyuki [17] studied the effect of residual stresses induced by cutting process on fatigue life through experimental method and a
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higher fatigue life was found for the specimen with compressive residual stress. As revealed by many experimental observations and simulation results, the rolling contact fatigue behavior of carburized gears is quite sensitive to the variations of mechanical properties and the residual stress within the hardened layer. Jo et al. [18] studied the fatigue behavior of carburized steels by using a two-layer model and found a good correspondence to the experimental results. Narita et al. [19] simulated the fatigue behavior of a traction drive element and a good
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correspondence to the experimental results was found for the case where the hardness gradient and the residual stress are considered. Those research confirm the necessity of the consideration of the mechanical properties variations and the residual stress within the hardened layer when the fatigue behaviors of carburized gears are evaluated.
In this study, the influence of the initial residual stress on rolling contact fatigue life of carburized gears is investigated. A finite element elastic-plastic contact model is developed which takes the
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gradients of hardness and initial residual stress into account. Initial residual stress distribution is obtained through experimental measurements. The Fatemi-Socie and the Brown-Miller criterions
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are used to estimate the contact fatigue damage of the carburized gear. A series of load circumstances are applied to investigated the effect of residual stress, which provides a theoretical support for a deeper understanding on the role of the residual stress in gear contact fatigue.
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2. Simulation methodology
The sample gear pair comes from a megawatt level wind turbine gearbox. This intermediate
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parallel gear stage, especially the small output gear, is found to suffer from severe premature rolling contact fatigue failure in engineer practice. The basic parameters of this gear pair is listed in table 1. The material of the gear is 18CrNiMo7-6 with the carburizing, quenching and
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tempering treatments, after which the gear is ground as the final manufacturing process. Table 1 Gear parameters
Teeth number
Z1=121, Z2=24
Gear tooth width
B=0.295 m
Gear normal module
m0=0.011 m
Pressure angle
α0=20°
Poisson’s ratio
v=0.3
Young’s modulus
E=210000 MPa
Radius at the pitch point
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 meshing moment could be simplified as two bodies with different radius of curvature contacting with each
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other. The plane strain assumption is assumed in the study which means the influence of the helix angle and crown tooth modification are neglected. The schematic diagram of the gear contact
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model is shown in figure 2.
Figure 2 The schematic diagram of the gear contact model 2.1 The characterization of the residual stress and the hardness gradient
The coordinate system used in the numerical model is defined as follows: the rolling direction is
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represented as the x direction, the depth direction is represented the y direction, while the tooth width is denoted by the z direction. The hardness gradient from case to the core induced by carburizing can be obtained through the Vicker’s test [21] or empirical methods [22, 23], among which, Thomas presented an empirical formula to depict the hardness curve for case carburized gears, expressed as [23]:
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aa y 2 ba y ca HV( y ) ab y 2 bb y cb HV core
550 - HVsurface ; 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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Where
(0 y CHD) (CHD y ycore ) ( ycore y )
bb 2 aa ycore ; cb 550 ab CHD2 bb CHD ;
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Where HVsurface , HVcore are the surface hardness and the core hardness, respectively; yHV,max is the depth of the maximum hardness, in this study, it is set to be 0. ycore is the depth with
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HV( y) HVcore . The effective case hardening depth (CHD) defined as the depth at 550 HV. The hardness of the sample gear is also experimentally measured by the Vicker's indentation test machine from the surface to the core. Figure 3 shows the Thomas fitted hardness curve as well as the hardness experimental data. As can be seen, the empirical result and the tested result corresponds well with each other, only with some slight variations very close to the surface, which is probably due to the slight grinding burn at near-surface area. The maximum value of the hardness occurs at the depth around 0.7 mm. In addition, it should be noticed that the surface hardness, the core hardness, the effective case hardening depth used in the empirical Thomas method could be determined by the actual engineering requirements. Thus, the surface hardness, the core hardness and the CHD are controlled as 670 HV, 420 HV and 2.2 mm, respectively,
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according to the engineering design requirement. The red solid 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 the empirical Thomas method Besides the hardness gradient, the case-carburizing process also introduce the variation of other mechanical properties such as the yield strength and the tensile strength from case to core. A large amount of studies have demonstrated that certain relations exist between the hardness and other mechanical properties for carburized steels [24, 25]. With regard to the contact fatigue life
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prediction, the material fatigue parameters and the yield strength, expressed as functions of the hardness, are utilized in the work. According to the references [26], the relations between the
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hardness and the yield strength and the 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 strength, and σ b ( y ) is the tensile strength, HV is the Vickers
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hardness. The resulting yield strength distribution of carburized model is shown in figure 4 (left). The Seeger relations between fatigue parameters and the tensile strength [27] are used in this work,
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which are expressed as:
σf 1.5σ b , εf 0.59ψ , b 0.087, c 0.58
(4)
When σ b /E 0.003, ψ 1 ; when σ b /E 0.003, ψ 1.375 125(σ b /E) . where ε f and σ f are the axial fatigue ductility and axial fatigue strength coefficients, respectively. b and c are the fatigue strength and the fatigue ductility exponents, respectively, and E is the Young’s modulus. The distributions of fatigue parameters ε f and σ f of the carburized 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 The profiles of the yield strength (left) and fatigue parameters (right)
It is worth to note that the carburizing 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. The residual stress at different depth position can be measured through the well-established X-ray diffraction method with electro-polishing, meanwhile, some empirical methods [29, 30] have also been provided for the determination of the residual stress of carburized
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components. In the present work, the residual stress is measured using an X-ray diffractometer (PROTO LXRD system) with Cr_K-Alpha radiation. Spot size is set to be 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 then after electro-polishing, the residual stress at a deeper position is measured. The flat area after electro-polishing should be large enough for the accurate measurement. The initial residual stress
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components with a gear tooth profile are shown in figure 5. The reason that σ r,y is not measured because this component would be negligibly small [20]. The Hertter empirical formula for the prediction of the residual stress based on the hardness [30] can be expressed as:
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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 the depth position y .
Figure 5 Various initial residual stress components with a gear tooth profile The initial residual stress results are depicted in figure 6. Results show that σ r,x and σ r,z at the same depth level are close to each other, hence the residual stress is represented by σ r . Measured points are represented by the pink hollow circles, based on which a fitted curve (dotted blue line) is formed. The first measurement is made on the tooth surface and the residual stress value is -132
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MPa The measured residual stress within the hardening layer represents the compressive characteristic, and the maximum value of compressive residual stress occurs at the subsurface with the depth of around 1.75 mm. In this figure the Hertter empirical curve [30] is also depicted. By comparison, the residual stress distribution obtained from measurement and the empirical results are found to agree with each other. In this work the empirical residual stress is described as a
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function of the hardness.
Figure 6 Distribution of the initial residual stress along the depth It should be highly noted that the residual stress distribution might vary significantly from one to another. Even though the same hardness profile is guaranteed, the residual stress profile might
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change [31]. In order to investigate the influence of initial residual stress, 13 sets of residual stress profiles are constructed with the peak value varies from -300 MPa to 300 MPa using the Hertter empirical method. As shown in figure 7, VCRS and VTRS represent the peak value of initial residual
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stress. Tensile residual stress may exists within the hardening layer in some engineering practices
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such as those grinding burn cases [32, 33].
Figure 7 The compressive residual stress (left) and the tensile residual stress (right) distributions with different VRS
2.2 The finite element elastic-plastic contact model The stress and strain response is essential for the further prediction of contact fatigue life. A finite element elastic-plastic contact model is developed in the framework of ABAQUS with the aid of the Python programming environment. The calculation domain is set as 10 mm x 10 mm ,
ACCEPTED MANUSCRIPT 0 mm y 10 mm . The upper rigid circle with the equivalent radius of curvature rolls from x=-4 mm to x=4 mm in order to make sure that interested material points experience complete loading cycles. In this model different mechanical properties are applied at different depth of the body. The thickness of each layer is 0.025 mm which is fine enough to capture the effect of the gradients of mechanical properties and the initial residual stress. Because the material points at the same horizontal level experience the same stress cycle, the 160 equally spaced material points located on the black dotted line ( x 0 mm and y [0,4]mm ) are thus mostly concerned. The element
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size at the critical zone (the purple part in figure 8) is controlled as 0.025mm 0.025mm .
Figure 8 The layered finite element contact model In this work a kinematic hardening constitutive equation and the von Mises yield criterion are
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considered 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 is worth to note that, under extremely heavy load conditions, shakedown state may occur which means several loading cycles should be applied to get the stabilized stress and strain response. For better understanding, the variation of the
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equivalent plastic strain over each load cycle under heaviest load condition is shown in figure 10. Numerical results reveal that the stress strain field is found stabilized after 5 loading cycles for all the selected cases. In this work 5 loading cycles were performed to guarantee the stabilized
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response and in the following sections the results are presented based on the stabilized mechanical
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response.
Figure 9 The elastic-plastic material constitutive model
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Figure 10 The variation of the equivalent plastic strain over each cycle under the load T=1133 kNm 2.3 The fatigue life criteria
To better understand the complex stress condition when the gears are running, the distributions of
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the stress components x y , x , y at the depth of 0:3 mm are shown in figure 11. As shown in figure 11, the stress condition during contact process is a typical multiaxial stress states. The stress components do not vary with time in the same proportion to each other. Thus, under such a complex stress condition, a proper multiaxial fatigue criterion needs to be adopted. In addition, the sample gear in the present study comes from an intermediate parallel stage of a megawatt level
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wind turbine gearbox. In fact, for heavy duty gears, during the whole service lifetime, elastic-plastic response might occur due to the random loading, due to the high localized stress at the contact region, plastic deformation is often observed in the gear rolling process. Thus the
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fatigue process is dominated by the combined effect of both the strain and the stress.
Figure 11 The stress history at the depth 0.3 mm In order to capture the multiaxial stress state during operation, many multiaxial fatigue criteria have been proposed among which the critical plane approach is widely accepted. The critical plane approach relies on the physical fact stating that the damage will always occur on the plane which has the maximum value of shear strain amplitude, and the crack will propagate as a result of the
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normal stress or strain acting on it during the load cycles. Since the critical plane is not known before the analysis, all the candidate planes have to be examined in order to identify the critical plane for each material point. The parameter denotes the angle between the direction of the normal line of candidate plane and the positive direction of x-axis, as shown in figure 12 (right). Since the plane strain condition is assumed, only candidate planes within the x-y plane need to be examined. The normal stress, the normal and shear strain on a candidate plane is thus obtained by [12]: (7)
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σθ (t ) σ x cos2 θ σ y sin 2 θ 2 τ xy sin θ cosθ γθ (t ) γxy (cos2 θ sin 2 θ ) 2(εy εx )sin θ cosθ εθ (t ) εx cos2 θ εz sin 2 θ γxy sin θ cosθ
(8) (9)
Fatemi and Socie proposed a well-accepted multi-axial fatigue model using the maximum shear strain amplitude and the maximum of 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
(10)
Where Δ γmax /2 is the amplitude of the shear strain on the critical plane, and σ max is the maximum normal stress to that plane. τ f and γf are the shear fatigue strength and the shear fatigue ductility coefficients, respectively. b and c are the fatigue strength and the fatigue
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ductility exponents, respectively. G is the shear elastic modulus, and N f is the number of cycles to crack initiation. The material constant k is chosen as 1 [35].
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The fatigue damage of this criterion is then defined as below[36]:
γmax σ [1 k max ] 2 σ ys
(11)
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Brown and Miller suggested that the multiaxial fatigue behavior should be governed by the normal strain amplitude and the shear strain amplitude components of the plane [37]. In their opinion,
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cyclic shear strains will help to nucleate cracks and the normal strain will assist in their growth. The relationship between the shear strain amplitude and the normal strain amplitude is usually
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proposed as:
γmax σ S ε n C1 f (2 N f ) b C2 εf (2 N f ) c 2 E
(12)
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 to estimate the fatigue life of a material subjected to cyclic loading is presented in DBM
figure 12 (left). First, the shakedown state stress-strain field is obtained after 5 loading cycles, and the stress and strain histories for each material point are recorded. Next, for each angle , the stress and strain on the candidate plane are calculated and the damage is further calculated
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depending on the selected fatigue criterion. Then, the contact fatigue life for each material point is
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evaluated by using the maximum damage.
Figure 12 The flow chart of the contact fatigue life prediction (left) and the definition of the critical plane angle (right) 3. Results and Discussion
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3.1 Comparison between the BM and FS criterion
Under the nominal load condition T0=282.8 kNm without the consideration of the initial residual stress, the damage parameters are evaluated based on both the BM criterion and the FS criterion. Figure 13 shows the damage parameters and the damage DBM based on the BM criterion. As can be seen, for each material point, the maximum shear strain amplitude γa is achieved at the angles
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of θ 0 ( 180 ) and θ 90 . The maximum magnitude of γa occurs at the depth of around 0.3 mm. The maximum normal strain amplitude ε a is achieved at the angle of θ 90 , which means the normal strain would accelerate the contact fatigue failure process along this direction.
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As a consequence, the maximum value of the damage DBM appears at the angle of θ 90 . For comparison, the damage parameters γa , σ max and the damage DFS calculated with the FS criterion are shown in figure 14. With this criterion, the maximum normal stress σ max achieves its
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minimum value (nearly 0) at the angle of θ 90 , while it achieves its maximum value at the angle of θ 0 ( 180 ). This indicates that the maximum normal stress accelerates the contact
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fatigue failure at the angle of θ 0 ( 180 ) while this damage parameter does not affect the damage at the angle of θ 90 . As a consequence, the maximum value of the damage DFS appears at the angle of θ 0 ( 180 ), significantly different with the result of the Brown-Miller
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criterion.
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Figure 13 The shear strain amplitude γa (left), normal strain amplitude ε a (middle) and contact
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fatigue damage DBM (right) distributions based on the BM criterion
Figure 14 The shear strain amplitude γa (left), the maximum normal stress σ max (middle) and
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contact fatigue damage DFS (right) distributions based on the FS criterion
The influence of initial residual stress on the stress strain field is depicted in figures 15 and figure 16 under the load condition T0=282.8 kNm. Results reveal that the initial residual stress does not affect the amplitude of the stress and strain, because it only affects the mean value and the maximum value. In addition, the influence of the initial residual stress on the mean and maximum value of the stress and strain varies as the angle changes. As can be seen in figure 15, the initial
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residual stress significantly affect σ max at the angle of around θ 0 ( 180 ), the CRS decreases the σ max while the TRS increases the σ max at around θ 0 ( 180 ) by a similar value compared with the RS-free case. Since in the BM criterion the fatigue life is governed by εa and
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γa , which are both unaffected by the initial residual stress, the predicted damage based on the BM
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criterion DBM is unchanged as 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 Meanwhile for the FS criterion, since the shear strain amplitude achieves its minimum at the angle of θ 0 ( 180 ) and θ 90 , the state of the normal stress σ max will finally determine the failure direction. As shown in figure 16, when the initial residual stress is absent, due to the action of the normal stress, the maximum damage occurs at the angle of θ 0 ( 180 ). If the initial
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from the angle of θ 0 ( 180 ) to the angle of θ 90 . Thus the initial residual stress
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influences the damage state through 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 based on the FS criterion. As can be seen, compared with the RS-free case, the damage of the TRS model increases 13.9%, while the CRS decreases the 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]. Even
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though the result is obtained by bending fatigue test, it still clearly reflects the difference between the influence of the tensile residual stress and the compressive residual stress on fatigue life. To be
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specific, the TRS leads to a remarkable deterioration for the fatigue damage, and the influence of the CRS on damage is moderate. Similar results were also found in experiments on the effect of mean stress on rolling contact fatigue [39]. Based on this result, the left side of the Dang Van
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diagram was modified.
Figure 17 The effects of the residual stress on DFS (left) and the experimental results of the influence of residual stress on fatigue strength according to the Ref. [38] (right) 3.2 Effect of the residual stress at different load levels based on the FS criterion In this section the effect of the residual stress at different load levels is investigated based on the
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FS criterion. Under extremely heavy duty conditions such as the case of T0=848.4 kNm and the case of T0=1131.2 kNm, plasticity occur at the subsurface zone. The load-induced plastic strain would change the profile of the initial residual stress. Figure 18 shows the evolutions of the residual stress under the two heavy load conditions. As can be seen, compared with the initial residual stress, the residual stress after cyclic loading is significantly changed, moreover, the residual stress in plastic zone incline to unanimous. Under the heaviest load condition T0=1131.2 kNm, the profiles of the residual stress after cyclic loading are almost identical, which indicates
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the trace of the initial residual stress disappear completely and the effect of plasticity takes over.
Figure 18 The evolutions of the residual stress under heavy load conditions Under the case of 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 depth, as shown within the dotted lines. Once the tensile initial residual stress is present, the range
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of the plasticity zone is the largest compared with the case without the residual stress and the case with compressive residual stress. Meanwhile the existence of the compressive initial residual stress shrinks the plasticity region compared with the case without the residual stress. The
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maximum normal stress of plastic zone is almost identical in such three cases, while a remarkable maximum normal stress difference is found at near-surface non-plastic zone The maximum normal stress at the near-surface non-plastic zone is largest in the case of TRS, and the CRS case inhibits
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the increasing of the maximum normal stress in this zone.
Figure 19 The maximum normal stress σ max with CRS (left), without RS (middle) and with TRS (right) under a heavy load condition T0=848.4 kNm
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Based upon the FS criterion, the damage on each candidate plane at each material point is depicted in figure 20 under three cases of the initial residual stress. When the CRS is considered, the depth with the maximum damage is around 0.4 mm, while the maximum damages of the TRS case and the RS-free case occur at the depth around 0.2 mm. Based on the FS criterion, the CRS influences
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the depth of crack initiation by affecting the maximum normal stress.
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Figure 20 DFS with CRS (left), without RS (middle) and with TRS (right) under a heavy load condition T0=848.4 kNm
Based on the FS criterion, the influence of the peak value of the residual stress on contact fatigue damage under various loading conditions is shown in figure 21. The load range is T0=141.4 kNm-848.4 kNm, as can be seen, in all the selected cases, the relationships between the damage and the residual stress peak value follow bilinear curve. The threshold of the switching point is
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dependent with the load level. As the load level increases, the switching point of the two segments of the bilinear curve shifts to the left. As the magnitude of the tensile residual stress increases, the
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contact fatigue damage within the gear subsurface increases linearly. But the influence of the compressive residual stress on the contact fatigue damage represents the same slope only within a limited range where the magnitude of the compressive residual stress is relatively small. However,
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if the compressive residual stress exceeds a certain value, as its magnitude further increases, the contact fatigue damage keeps unchanged. These results have also been found in experimental
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studies. Terrin et al. [40] employed a two‐ disc test rig to assess the influence of the compressive residual stress on gear material 17NiCrMo6‐ 4, which is similar to the material used in the present work. The experimental results reveal that, on the basis of the existing residual stress, additional
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compressive residual stresses introduced by shot peening are ineffective in preventing contact fatigue damages.
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Figure 21 Effect of residual stress peak value on DFS under different load conditions
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3.3 Effect of residual stress on RCF life based on the FS criterion
Based on the FS criterion, the RCF lives are shown in figure 22. The minimum contact fatigue life with CRS under the nominal load condition is 9.00×107, the value of which increases 28% compared with the RS-free result, meanwhile, the minimum contact fatigue life with TRS
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decreases 73% compared with the RS-free result.
Figure 22 Contact fatigue lives under three residual stress states with the FS criterion
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γ - N f curves of this carburized gear are obtained by considering different states of the initial
residual stress. The TRS is 300 MPa and the CRS is -300 MPa. The chosen 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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meanwhile a significant discrepancy can be found if the two curves are compared with the TRS curve.
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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 gear RCF life can be observed through figure 24, as a supplement of figure 23. As the load increases, the contact fatigue life decreases exponentially, validates the rule of the Basquin’s equation [41]. The load region can be roughly divided into two stages. Load Stage 1 represents the purely elastic response while in Load Stage 2 plasticity occurs.
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When the load is modest, the effect of the compressive residual stress on the improvement of RCF life is significantly less than the degradation effect of tensile residual stress on life. However, if an extreme heavy load is applied, since the initial residual stress is somewhat changed due to
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plasticity, the influence of the initial residual stress on contact fatigue life is weakened.
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Figure 24 The effects of the residual stress on contact fatigue life under different load conditions 4. Conclusions In this study, the influence of the initial residual stress on rolling contact fatigue life of carburized gears is investigated by using a numerical simulation methodology. An innovative model based on Fatemi-Soice criterion is developed which takes the hardness gradient and the initial residual stress in consideration. A finite element elastic-plastic contact model of a carburized gear from a megawatt wind turbine gearbox is proposed. Conclusions are summarized as follows. 1. The initial residual stress does not affect the amplitude of the stress and strain, and it only affects the mean value and the maximum value. The initial residual stress influences the
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maximum normal stress and hence the effect of initial residual stress on contact fatigue failure can be reflected by the Fatemi-Socie criterion. 2. Based on the Fatemi-Socie criterion, the initial residual stress influences the damage state through its effect on the maximum normal stress. The tensile residual stress leads to a remarkable deterioration for the contact fatigue damage, while the influence of the compressive residual stress on damage is moderate. Under heavy load conditions where plasticity occur, the profiles of the initial residual stress are strongly modified, the residual stress after evolution still plays a considerable role on the severity of fatigue damage,
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especially at the near-surface area where no plasticity occurs. 3. Based on the Fatemi-Socie criterion, the relationship between the damage and the residual stress peak value follows a bilinear curve. As the load level increases, the switching point of the two segments of the bilinear curve shifts to the left. If the compressive residual stress exceeds the switching point, as its magnitude further increases, the rolling contact fatigue life does not increase anymore.
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5. Acknowledgement
The work is supported by the National Natural Science Foundation of China (Grant Nos. 51775060, 51575060, 51535012) 6. References
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GRAPHICAL ABSTRACT
Effect of the residual stress on contact fatigue of a wind turbine carburized gear with multiaxial fatigue criteria Wei Wang , Huaiju Liu , Caichao Zhu , Xuesong Du , Jinyuan Tang PII: DOI: Reference:
S0020-7403(18)31588-1 https://doi.org/10.1016/j.ijmecsci.2018.11.013 MS 4639
To appear in:
International Journal of Mechanical Sciences
Received date: Revised date: Accepted date:
16 May 2018 10 October 2018 13 November 2018
Please cite this article as: Wei Wang , Huaiju Liu , Caichao Zhu , Xuesong Du , Jinyuan Tang , Effect of the residual stress on contact fatigue of a wind turbine carburized 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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Highlight
The rolling contact fatigue damage and life of the carburized gear are calculated by considering gradients of mechanical properties and residual stress within the hardening layer.
The influence of the initial residual stress on rolling contact fatigue life can be reflected by the Fatemi-Socie criterion and the result have an excellent agreement
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with the existing reference.
The effects of the peak value of the initial residual stress on the rolling contact
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fatigue damage are illustrated under a series of load circumstances.
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Effect of the residual stress on contact fatigue of a wind turbine carburized 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, Central South University, Changsha, Hunan, China, 410083
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*Corresponding author: [email protected] Abstract
Carburized gears are extensively applied in heavy duty machines such as wind turbines, ships, high-speed rails, etc. The variations of hardness and residual stress introduced by carburizing and quenching processes remarkable affect the rolling contact fatigue (RCF) during repeated gear
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meshing. The present work attempts to study the effect of the residual stress on rolling contact fatigue and predict the rolling contact fatigue life of a wind turbine carburized gear based on multiaxial fatigue criteria. A finite element elastic-plastic contact model is developed which takes the gradients of hardness and initial residual stress into account. Initial residual stress distribution is obtained through experimental measurements. The stabilized stress strain field is carried out by considering the shakedown state under the heavy load conditions. The Fatemi-Socie and the
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Brown-Miller criterions are used to estimate the contact fatigue damage of the carburized gear. Numerical results reveal that the effect of the initial residual stress on the contact fatigue failure
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can be reflected by the Fatemi-Socie criterion whereas it is not reflected by the Brown-Miller criterion since the two parameters appeared in the criterion is not influenced by residual stress. In terms of the Fatemi-Socie criterion, under modest load conditions where elastic response occur, a
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considerable compressive residual stress would alter the plane orientations of crack initiation, while under extremely heavy load conditions where plasticity occur, the influence of the initial residual stress on contact fatigue damage is limited within the near-surface non-plastic zone. The
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effect of the initial residual stress on contact fatigue life represents a bilinear curve. As the magnitude of the tensile residual stress increases, the RCF life decreases linearly. As the
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magnitude of the compress residual stress exceeds a certain value, the RCF life does not increase anymore.
Keywords: Carburized gear; Rolling contact fatigue life; Initial residual stress; Fatemi-Socie criterion; Brown-Miller criterion
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Nomenclature
γa
The amplitude of sheer strain, %
γf
The shear fatigue ductility coefficients The orthogonal shear strain, %
εa
The amplitude of normal strain, %
ε f
The axial fatigue ductility coefficients
εx
The x-axial strain, %
εy
The y-axial strain, %
θ σb σ f
The angle of critical plane
σ max
The tensile strength, MPa The axial fatigue strength coefficients The maximum normal stress, MPa The initial residual stress, MPa
σx
The x-axial stress, MPa
σy
The y-axial stress, MPa
σ ys
The initial yield strength, MPa
b BM c
CHD CRS
The orthogonal shear stress, MPa The fatigue strength exponents The Brown-miller criterion
The fatigue ductility exponents
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τ xy
The shear fatigue strength coefficients, MPa
The effective case hardening depth, mm Compressive initial residual stress, MPa
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τ f
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σr
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γx y
The contact fatigue damage under Brown-miller criterion
DFS
The contact fatigue damage under Fatemi-Socie criterion
E FS G HV M
The equivalent elasticity modulus, MPa
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DBM
The Fatemi-Socie criterion
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The shear elastic modulus, MPa
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Nf
RCF RS
The Vickers hardness, HV The linear hardening modulus, MPa The number of cycles to crack initiation for material point The Rolling contact fatigue Initial residual stress, MPa
T0
Input torque for sample gear, kNm
TRS x y
Tensile initial residual stress, MPa
z
The rolling direction The depth direction The tooth width direction
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1. Introduction With the growing demand on higher power densities and loading capacities of gear-driven mechanical machines, contact fatigue failure problems of gears have become limiting factors influencing the reliabilities of those machines such as wind turbines. Due to the large teeth width and high tooth root bending strength, the other failure type, namely tooth root bending fatigue
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failure, occurs in a low proportion in the wind turbine gear during engineering practice. The gear failure induced by rolling contact fatigue takes many modes, such as micropitting [1], pitting [2] and tooth flank fracture [3]. Figure 1 shows a typical contact fatigue failure on a 2 MW wind turbine gearbox. Case hardening processes, such as the carburizing [4], are widely used in large-scale heavy duty gears. The rolling contact fatigue failures of carburized gears still can not be avoided, even though they have longer lives compared with un-carburized gears due to higher
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contact fatigue resistance. The hardness gradient is generated from the case to the core after carburizing and quenching process by changing the carbon content, at the meantime, the residual stress is also induced within the hardened layer due to the heat treatment. The prediction of the local material contact fatigue damage of carburized gears is still a difficult problem due to the variations of mechanical properties introduced by hardness gradient and the considerable residual
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stress level.
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Figure 1 the contact fatigue failure on a 2 MW wind turbine gearbox (a)-(b) and microtopography of contact fatigue surface (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 gear contact fatigue life by using a mix-lubrication model. Liu et al. [6, 7] investigated the effect of the lubrication and the surface roughness on gear subsurface stress distribution. Seghdei et al. [8] used the continuous damage approach to study the fatigue damage evolution during the contact fatigue. Anisetti [9] studied the gear contact fatigue behavior by considering the impact of tribo-dynamic. Dong et al. [10] investigated 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 spur gear pair. Qiao [12] applied various multiaxial criteria to investigate the RCF life and failure probability. However, those research do not take the residual stress and the variations of mechanical properties into account. The influences of the residual stress on fatigue behavior have been widely investigated in recent
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years. Fukumasu et al. [13] studied the influence of the surface residual stress produced by shot peening on gear contact failure and a pitting resistance was found of the gear with an additional shot-peeting treat. Maasi [14] investigated the effect of the residual stress on contact fatigue behavior in lubricated contact by considering the surface roughness. Pape [15] studied the influence of the initial residual stress state on bearing contact fatigue life by using Ioannides-Harris model. Batista et al. [16] developed a numerical model to predict residual stress relaxation during the contact process of automotive gears. Hiroyuki [17] studied the effect of residual stresses induced by cutting process on fatigue life through experimental method and a
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higher fatigue life was found for the specimen with compressive residual stress. As revealed by many experimental observations and simulation results, the rolling contact fatigue behavior of carburized gears is quite sensitive to the variations of mechanical properties and the residual stress within the hardened layer. Jo et al. [18] studied the fatigue behavior of carburized steels by using a two-layer model and found a good correspondence to the experimental results. Narita et al. [19] simulated the fatigue behavior of a traction drive element and a good
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correspondence to the experimental results was found for the case where the hardness gradient and the residual stress are considered. Those research confirm the necessity of the consideration of the mechanical properties variations and the residual stress within the hardened layer when the fatigue behaviors of carburized gears are evaluated.
In this study, the influence of the initial residual stress on rolling contact fatigue life of carburized gears is investigated. A finite element elastic-plastic contact model is developed which takes the
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gradients of hardness and initial residual stress into account. Initial residual stress distribution is obtained through experimental measurements. The Fatemi-Socie and the Brown-Miller criterions
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are used to estimate the contact fatigue damage of the carburized gear. A series of load circumstances are applied to investigated the effect of residual stress, which provides a theoretical support for a deeper understanding on the role of the residual stress in gear contact fatigue.
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2. Simulation methodology
The sample gear pair comes from a megawatt level wind turbine gearbox. This intermediate
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parallel gear stage, especially the small output gear, is found to suffer from severe premature rolling contact fatigue failure in engineer practice. The basic parameters of this gear pair is listed in table 1. The material of the gear is 18CrNiMo7-6 with the carburizing, quenching and
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tempering treatments, after which the gear is ground as the final manufacturing process. Table 1 Gear parameters
Teeth number
Z1=121, Z2=24
Gear tooth width
B=0.295 m
Gear normal module
m0=0.011 m
Pressure angle
α0=20°
Poisson’s ratio
v=0.3
Young’s modulus
E=210000 MPa
Radius at the pitch point
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 meshing moment could be simplified as two bodies with different radius of curvature contacting with each
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other. The plane strain assumption is assumed in the study which means the influence of the helix angle and crown tooth modification are neglected. The schematic diagram of the gear contact
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model is shown in figure 2.
Figure 2 The schematic diagram of the gear contact model 2.1 The characterization of the residual stress and the hardness gradient
The coordinate system used in the numerical model is defined as follows: the rolling direction is
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represented as the x direction, the depth direction is represented the y direction, while the tooth width is denoted by the z direction. The hardness gradient from case to the core induced by carburizing can be obtained through the Vicker’s test [21] or empirical methods [22, 23], among which, Thomas presented an empirical formula to depict the hardness curve for case carburized gears, expressed as [23]:
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aa y 2 ba y ca HV( y ) ab y 2 bb y cb HV core
550 - HVsurface ; 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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Where
(0 y CHD) (CHD y ycore ) ( ycore y )
bb 2 aa ycore ; cb 550 ab CHD2 bb CHD ;
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Where HVsurface , HVcore are the surface hardness and the core hardness, respectively; yHV,max is the depth of the maximum hardness, in this study, it is set to be 0. ycore is the depth with
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HV( y) HVcore . The effective case hardening depth (CHD) defined as the depth at 550 HV. The hardness of the sample gear is also experimentally measured by the Vicker's indentation test machine from the surface to the core. Figure 3 shows the Thomas fitted hardness curve as well as the hardness experimental data. As can be seen, the empirical result and the tested result corresponds well with each other, only with some slight variations very close to the surface, which is probably due to the slight grinding burn at near-surface area. The maximum value of the hardness occurs at the depth around 0.7 mm. In addition, it should be noticed that the surface hardness, the core hardness, the effective case hardening depth used in the empirical Thomas method could be determined by the actual engineering requirements. Thus, the surface hardness, the core hardness and the CHD are controlled as 670 HV, 420 HV and 2.2 mm, respectively,
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according to the engineering design requirement. The red solid 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 the empirical Thomas method Besides the hardness gradient, the case-carburizing process also introduce the variation of other mechanical properties such as the yield strength and the tensile strength from case to core. A large amount of studies have demonstrated that certain relations exist between the hardness and other mechanical properties for carburized steels [24, 25]. With regard to the contact fatigue life
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prediction, the material fatigue parameters and the yield strength, expressed as functions of the hardness, are utilized in the work. According to the references [26], the relations between the
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hardness and the yield strength and the 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 strength, and σ b ( y ) is the tensile strength, HV is the Vickers
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hardness. The resulting yield strength distribution of carburized model is shown in figure 4 (left). The Seeger relations between fatigue parameters and the tensile strength [27] are used in this work,
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which are expressed as:
σf 1.5σ b , εf 0.59ψ , b 0.087, c 0.58
(4)
When σ b /E 0.003, ψ 1 ; when σ b /E 0.003, ψ 1.375 125(σ b /E) . where ε f and σ f are the axial fatigue ductility and axial fatigue strength coefficients, respectively. b and c are the fatigue strength and the fatigue ductility exponents, respectively, and E is the Young’s modulus. The distributions of fatigue parameters ε f and σ f of the carburized 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 The profiles of the yield strength (left) and fatigue parameters (right)
It is worth to note that the carburizing 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. The residual stress at different depth position can be measured through the well-established X-ray diffraction method with electro-polishing, meanwhile, some empirical methods [29, 30] have also been provided for the determination of the residual stress of carburized
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components. In the present work, the residual stress is measured using an X-ray diffractometer (PROTO LXRD system) with Cr_K-Alpha radiation. Spot size is set to be 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 then after electro-polishing, the residual stress at a deeper position is measured. The flat area after electro-polishing should be large enough for the accurate measurement. The initial residual stress
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components with a gear tooth profile are shown in figure 5. The reason that σ r,y is not measured because this component would be negligibly small [20]. The Hertter empirical formula for the prediction of the residual stress based on the hardness [30] can be expressed as:
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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 the depth position y .
Figure 5 Various initial residual stress components with a gear tooth profile The initial residual stress results are depicted in figure 6. Results show that σ r,x and σ r,z at the same depth level are close to each other, hence the residual stress is represented by σ r . Measured points are represented by the pink hollow circles, based on which a fitted curve (dotted blue line) is formed. The first measurement is made on the tooth surface and the residual stress value is -132
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MPa The measured residual stress within the hardening layer represents the compressive characteristic, and the maximum value of compressive residual stress occurs at the subsurface with the depth of around 1.75 mm. In this figure the Hertter empirical curve [30] is also depicted. By comparison, the residual stress distribution obtained from measurement and the empirical results are found to agree with each other. In this work the empirical residual stress is described as a
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function of the hardness.
Figure 6 Distribution of the initial residual stress along the depth It should be highly noted that the residual stress distribution might vary significantly from one to another. Even though the same hardness profile is guaranteed, the residual stress profile might
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change [31]. In order to investigate the influence of initial residual stress, 13 sets of residual stress profiles are constructed with the peak value varies from -300 MPa to 300 MPa using the Hertter empirical method. As shown in figure 7, VCRS and VTRS represent the peak value of initial residual
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stress. Tensile residual stress may exists within the hardening layer in some engineering practices
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such as those grinding burn cases [32, 33].
Figure 7 The compressive residual stress (left) and the tensile residual stress (right) distributions with different VRS
2.2 The finite element elastic-plastic contact model The stress and strain response is essential for the further prediction of contact fatigue life. A finite element elastic-plastic contact model is developed in the framework of ABAQUS with the aid of the Python programming environment. The calculation domain is set as 10 mm x 10 mm ,
ACCEPTED MANUSCRIPT 0 mm y 10 mm . The upper rigid circle with the equivalent radius of curvature rolls from x=-4 mm to x=4 mm in order to make sure that interested material points experience complete loading cycles. In this model different mechanical properties are applied at different depth of the body. The thickness of each layer is 0.025 mm which is fine enough to capture the effect of the gradients of mechanical properties and the initial residual stress. Because the material points at the same horizontal level experience the same stress cycle, the 160 equally spaced material points located on the black dotted line ( x 0 mm and y [0,4]mm ) are thus mostly concerned. The element
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size at the critical zone (the purple part in figure 8) is controlled as 0.025mm 0.025mm .
Figure 8 The layered finite element contact model In this work a kinematic hardening constitutive equation and the von Mises yield criterion are
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considered 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 is worth to note that, under extremely heavy load conditions, shakedown state may occur which means several loading cycles should be applied to get the stabilized stress and strain response. For better understanding, the variation of the
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equivalent plastic strain over each load cycle under heaviest load condition is shown in figure 10. Numerical results reveal that the stress strain field is found stabilized after 5 loading cycles for all the selected cases. In this work 5 loading cycles were performed to guarantee the stabilized
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response and in the following sections the results are presented based on the stabilized mechanical
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response.
Figure 9 The elastic-plastic material constitutive model
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Figure 10 The variation of the equivalent plastic strain over each cycle under the load T=1133 kNm 2.3 The fatigue life criteria
To better understand the complex stress condition when the gears are running, the distributions of
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the stress components x y , x , y at the depth of 0:3 mm are shown in figure 11. As shown in figure 11, the stress condition during contact process is a typical multiaxial stress states. The stress components do not vary with time in the same proportion to each other. Thus, under such a complex stress condition, a proper multiaxial fatigue criterion needs to be adopted. In addition, the sample gear in the present study comes from an intermediate parallel stage of a megawatt level
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wind turbine gearbox. In fact, for heavy duty gears, during the whole service lifetime, elastic-plastic response might occur due to the random loading, due to the high localized stress at the contact region, plastic deformation is often observed in the gear rolling process. Thus the
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fatigue process is dominated by the combined effect of both the strain and the stress.
Figure 11 The stress history at the depth 0.3 mm In order to capture the multiaxial stress state during operation, many multiaxial fatigue criteria have been proposed among which the critical plane approach is widely accepted. The critical plane approach relies on the physical fact stating that the damage will always occur on the plane which has the maximum value of shear strain amplitude, and the crack will propagate as a result of the
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normal stress or strain acting on it during the load cycles. Since the critical plane is not known before the analysis, all the candidate planes have to be examined in order to identify the critical plane for each material point. The parameter denotes the angle between the direction of the normal line of candidate plane and the positive direction of x-axis, as shown in figure 12 (right). Since the plane strain condition is assumed, only candidate planes within the x-y plane need to be examined. The normal stress, the normal and shear strain on a candidate plane is thus obtained by [12]: (7)
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σθ (t ) σ x cos2 θ σ y sin 2 θ 2 τ xy sin θ cosθ γθ (t ) γxy (cos2 θ sin 2 θ ) 2(εy εx )sin θ cosθ εθ (t ) εx cos2 θ εz sin 2 θ γxy sin θ cosθ
(8) (9)
Fatemi and Socie proposed a well-accepted multi-axial fatigue model using the maximum shear strain amplitude and the maximum of 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
(10)
Where Δ γmax /2 is the amplitude of the shear strain on the critical plane, and σ max is the maximum normal stress to that plane. τ f and γf are the shear fatigue strength and the shear fatigue ductility coefficients, respectively. b and c are the fatigue strength and the fatigue
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ductility exponents, respectively. G is the shear elastic modulus, and N f is the number of cycles to crack initiation. The material constant k is chosen as 1 [35].
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The fatigue damage of this criterion is then defined as below[36]:
γmax σ [1 k max ] 2 σ ys
(11)
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DFS
Brown and Miller suggested that the multiaxial fatigue behavior should be governed by the normal strain amplitude and the shear strain amplitude components of the plane [37]. In their opinion,
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cyclic shear strains will help to nucleate cracks and the normal strain will assist in their growth. The relationship between the shear strain amplitude and the normal strain amplitude is usually
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proposed as:
γmax σ S ε n C1 f (2 N f ) b C2 εf (2 N f ) c 2 E
(12)
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 to estimate the fatigue life of a material subjected to cyclic loading is presented in DBM
figure 12 (left). First, the shakedown state stress-strain field is obtained after 5 loading cycles, and the stress and strain histories for each material point are recorded. Next, for each angle , the stress and strain on the candidate plane are calculated and the damage is further calculated
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depending on the selected fatigue criterion. Then, the contact fatigue life for each material point is
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evaluated by using the maximum damage.
Figure 12 The flow chart of the contact fatigue life prediction (left) and the definition of the critical plane angle (right) 3. Results and Discussion
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3.1 Comparison between the BM and FS criterion
Under the nominal load condition T0=282.8 kNm without the consideration of the initial residual stress, the damage parameters are evaluated based on both the BM criterion and the FS criterion. Figure 13 shows the damage parameters and the damage DBM based on the BM criterion. As can be seen, for each material point, the maximum shear strain amplitude γa is achieved at the angles
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of θ 0 ( 180 ) and θ 90 . The maximum magnitude of γa occurs at the depth of around 0.3 mm. The maximum normal strain amplitude ε a is achieved at the angle of θ 90 , which means the normal strain would accelerate the contact fatigue failure process along this direction.
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As a consequence, the maximum value of the damage DBM appears at the angle of θ 90 . For comparison, the damage parameters γa , σ max and the damage DFS calculated with the FS criterion are shown in figure 14. With this criterion, the maximum normal stress σ max achieves its
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minimum value (nearly 0) at the angle of θ 90 , while it achieves its maximum value at the angle of θ 0 ( 180 ). This indicates that the maximum normal stress accelerates the contact
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fatigue failure at the angle of θ 0 ( 180 ) while this damage parameter does not affect the damage at the angle of θ 90 . As a consequence, the maximum value of the damage DFS appears at the angle of θ 0 ( 180 ), significantly different with the result of the Brown-Miller
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criterion.
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Figure 13 The shear strain amplitude γa (left), normal strain amplitude ε a (middle) and contact
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fatigue damage DBM (right) distributions based on the BM criterion
Figure 14 The shear strain amplitude γa (left), the maximum normal stress σ max (middle) and
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contact fatigue damage DFS (right) distributions based on the FS criterion
The influence of initial residual stress on the stress strain field is depicted in figures 15 and figure 16 under the load condition T0=282.8 kNm. Results reveal that the initial residual stress does not affect the amplitude of the stress and strain, because it only affects the mean value and the maximum value. In addition, the influence of the initial residual stress on the mean and maximum value of the stress and strain varies as the angle changes. As can be seen in figure 15, the initial
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residual stress significantly affect σ max at the angle of around θ 0 ( 180 ), the CRS decreases the σ max while the TRS increases the σ max at around θ 0 ( 180 ) by a similar value compared with the RS-free case. Since in the BM criterion the fatigue life is governed by εa and
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γa , which are both unaffected by the initial residual stress, the predicted damage based on the BM
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criterion DBM is unchanged as 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 Meanwhile for the FS criterion, since the shear strain amplitude achieves its minimum at the angle of θ 0 ( 180 ) and θ 90 , the state of the normal stress σ max will finally determine the failure direction. As shown in figure 16, when the initial residual stress is absent, due to the action of the normal stress, the maximum damage occurs at the angle of θ 0 ( 180 ). If the initial
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from the angle of θ 0 ( 180 ) to the angle of θ 90 . Thus the initial residual stress
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influences the damage state through 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 based on the FS criterion. As can be seen, compared with the RS-free case, the damage of the TRS model increases 13.9%, while the CRS decreases the 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]. Even
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though the result is obtained by bending fatigue test, it still clearly reflects the difference between the influence of the tensile residual stress and the compressive residual stress on fatigue life. To be
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specific, the TRS leads to a remarkable deterioration for the fatigue damage, and the influence of the CRS on damage is moderate. Similar results were also found in experiments on the effect of mean stress on rolling contact fatigue [39]. Based on this result, the left side of the Dang Van
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diagram was modified.
Figure 17 The effects of the residual stress on DFS (left) and the experimental results of the influence of residual stress on fatigue strength according to the Ref. [38] (right) 3.2 Effect of the residual stress at different load levels based on the FS criterion In this section the effect of the residual stress at different load levels is investigated based on the
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FS criterion. Under extremely heavy duty conditions such as the case of T0=848.4 kNm and the case of T0=1131.2 kNm, plasticity occur at the subsurface zone. The load-induced plastic strain would change the profile of the initial residual stress. Figure 18 shows the evolutions of the residual stress under the two heavy load conditions. As can be seen, compared with the initial residual stress, the residual stress after cyclic loading is significantly changed, moreover, the residual stress in plastic zone incline to unanimous. Under the heaviest load condition T0=1131.2 kNm, the profiles of the residual stress after cyclic loading are almost identical, which indicates
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the trace of the initial residual stress disappear completely and the effect of plasticity takes over.
Figure 18 The evolutions of the residual stress under heavy load conditions Under the case of 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 depth, as shown within the dotted lines. Once the tensile initial residual stress is present, the range
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of the plasticity zone is the largest compared with the case without the residual stress and the case with compressive residual stress. Meanwhile the existence of the compressive initial residual stress shrinks the plasticity region compared with the case without the residual stress. The
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maximum normal stress of plastic zone is almost identical in such three cases, while a remarkable maximum normal stress difference is found at near-surface non-plastic zone The maximum normal stress at the near-surface non-plastic zone is largest in the case of TRS, and the CRS case inhibits
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the increasing of the maximum normal stress in this zone.
Figure 19 The maximum normal stress σ max with CRS (left), without RS (middle) and with TRS (right) under a heavy load condition T0=848.4 kNm
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Based upon the FS criterion, the damage on each candidate plane at each material point is depicted in figure 20 under three cases of the initial residual stress. When the CRS is considered, the depth with the maximum damage is around 0.4 mm, while the maximum damages of the TRS case and the RS-free case occur at the depth around 0.2 mm. Based on the FS criterion, the CRS influences
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the depth of crack initiation by affecting the maximum normal stress.
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Figure 20 DFS with CRS (left), without RS (middle) and with TRS (right) under a heavy load condition T0=848.4 kNm
Based on the FS criterion, the influence of the peak value of the residual stress on contact fatigue damage under various loading conditions is shown in figure 21. The load range is T0=141.4 kNm-848.4 kNm, as can be seen, in all the selected cases, the relationships between the damage and the residual stress peak value follow bilinear curve. The threshold of the switching point is
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dependent with the load level. As the load level increases, the switching point of the two segments of the bilinear curve shifts to the left. As the magnitude of the tensile residual stress increases, the
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contact fatigue damage within the gear subsurface increases linearly. But the influence of the compressive residual stress on the contact fatigue damage represents the same slope only within a limited range where the magnitude of the compressive residual stress is relatively small. However,
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if the compressive residual stress exceeds a certain value, as its magnitude further increases, the contact fatigue damage keeps unchanged. These results have also been found in experimental
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studies. Terrin et al. [40] employed a two‐ disc test rig to assess the influence of the compressive residual stress on gear material 17NiCrMo6‐ 4, which is similar to the material used in the present work. The experimental results reveal that, on the basis of the existing residual stress, additional
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compressive residual stresses introduced by shot peening are ineffective in preventing contact fatigue damages.
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Figure 21 Effect of residual stress peak value on DFS under different load conditions
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3.3 Effect of residual stress on RCF life based on the FS criterion
Based on the FS criterion, the RCF lives are shown in figure 22. The minimum contact fatigue life with CRS under the nominal load condition is 9.00×107, the value of which increases 28% compared with the RS-free result, meanwhile, the minimum contact fatigue life with TRS
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decreases 73% compared with the RS-free result.
Figure 22 Contact fatigue lives under three residual stress states with the FS criterion
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γ - N f curves of this carburized gear are obtained by considering different states of the initial
residual stress. The TRS is 300 MPa and the CRS is -300 MPa. The chosen 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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meanwhile a significant discrepancy can be found if the two curves are compared with the TRS curve.
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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 gear RCF life can be observed through figure 24, as a supplement of figure 23. As the load increases, the contact fatigue life decreases exponentially, validates the rule of the Basquin’s equation [41]. The load region can be roughly divided into two stages. Load Stage 1 represents the purely elastic response while in Load Stage 2 plasticity occurs.
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When the load is modest, the effect of the compressive residual stress on the improvement of RCF life is significantly less than the degradation effect of tensile residual stress on life. However, if an extreme heavy load is applied, since the initial residual stress is somewhat changed due to
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plasticity, the influence of the initial residual stress on contact fatigue life is weakened.
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Figure 24 The effects of the residual stress on contact fatigue life under different load conditions 4. Conclusions In this study, the influence of the initial residual stress on rolling contact fatigue life of carburized gears is investigated by using a numerical simulation methodology. An innovative model based on Fatemi-Soice criterion is developed which takes the hardness gradient and the initial residual stress in consideration. A finite element elastic-plastic contact model of a carburized gear from a megawatt wind turbine gearbox is proposed. Conclusions are summarized as follows. 1. The initial residual stress does not affect the amplitude of the stress and strain, and it only affects the mean value and the maximum value. The initial residual stress influences the
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maximum normal stress and hence the effect of initial residual stress on contact fatigue failure can be reflected by the Fatemi-Socie criterion. 2. Based on the Fatemi-Socie criterion, the initial residual stress influences the damage state through its effect on the maximum normal stress. The tensile residual stress leads to a remarkable deterioration for the contact fatigue damage, while the influence of the compressive residual stress on damage is moderate. Under heavy load conditions where plasticity occur, the profiles of the initial residual stress are strongly modified, the residual stress after evolution still plays a considerable role on the severity of fatigue damage,
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especially at the near-surface area where no plasticity occurs. 3. Based on the Fatemi-Socie criterion, the relationship between the damage and the residual stress peak value follows a bilinear curve. As the load level increases, the switching point of the two segments of the bilinear curve shifts to the left. If the compressive residual stress exceeds the switching point, as its magnitude further increases, the rolling contact fatigue life does not increase anymore.
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5. Acknowledgement
The work is supported by the National Natural Science Foundation of China (Grant Nos. 51775060, 51575060, 51535012) 6. References
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GRAPHICAL ABSTRACT