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The effect of wear on short crack propagation under fretting conditions
International Journal of Mechanical Sciences 157–158 (2019) 552–560
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International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
The effect of wear on short crack propagation under fretting conditions Xin Liu a, Jinxiang Liu a,∗, Zhengxing Zuo a, Huayang Zhang b a b
School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China School of Aeronautical Engineering, Zhengzhou University of Aeronautics, Zhengzhou 450046, China
a r t i c l e
i n f o
Keywords: Short crack Crack propagation rate Wear Fretting
a b s t r a c t The effect of wear on the crack propagation behavior for short cracks under fretting conditions is investigated by using numerical method. The numerical simulation effectively combines the remeshing technology with the adaptive meshing technology, and the evolution of contact force and crack length caused by wear can be considered in this study. The critical distortional elastic energy criterion is used to determine the threshold condition of short crack propagation, and the Archard equation is employed for wear simulation. The results show that wear reduces the short crack propagation rate. Furthermore, the effect of wear on the short crack propagation rate is more obvious for the shorter initial crack length.
1. Introduction Fretting fatigue caused by micron-level oscillatory motion between two contacting surfaces occurs in a wide range of industry fields, such as aviation, aerospace, shipbuilding industry, etc [1]. The fretting fatigue could be affected by many factors, such as fatigue bulk stress, contact shear stress, contact pressure, slip amplitude, contact geometry, wear, etc. Among these factors, the wear has a highly complex effect on fretting fatigue, especially on the initial stage of fretting fatigue crack propagation. In the initial stage of fretting fatigue crack propagation, the fretting crack is a short crack and the crack length is in the same order of magnitude with the removed material depth due to wear. On the one hand, the wear can change the geometry of the contact surface, which finally results in the change of contact force distribution. On the other hand, the wear directly reduces the length of the existing crack, and then influences the crack propagation rate. In extreme cases, the wear can even totally remove the existing short cracks [2]. In addition, the microstructure evolution caused by wear can causes a change in the coefficient of friction [3], but the friction coefficient will be stable in a relatively short period [4], so this study does not consider the effect of microstructure evolution. In summary, understanding the effect of wear on the short crack propagation is essential for the fatigue design under fretting conditions. In order to analyze the effect of wear on the short crack propagation under fretting conditions, the change of crack length and contact surface profile caused by wear is a major concern. However, without suspending the fretting experiment, it is difficult to accurately measure the change of crack length and wear profile in micron-scale. However,
∗
suspending the experiment will change the contact state and further affect the subsequent crack propagation behavior and wear behavior. The numerical method provides an alternative approach to investigate the effect of wear on the short crack propagation for fretting fatigue. Among the numerical methods, the finite element method is widely used to analyze fretting wear behavior and fretting fatigue crack propagation behavior. By means of finite element method, Pereira et al. proposed a multiscale procedure to study roughness effect on fretting wear [5]. Yue et al. used finite element method to analyze the stress singularity in fretting wear [6]. According to FE results, Pereira et al. found that the extended maximum tangential stress criterion is more suitable for predicting the crack path than the classical maximum tangential stress criterion [7]. By combining the extended finite element method with fatigue crack growth criteria, Martìnez et al. predicted the fretting fatigue crack trajectory of a railway axle [8]. In addition, there are many successful applications of finite element methods in the study of fretting wear and fretting fatigue [9–14]. Thus, a proper combination of wear simulation and crack propagation simulation through finite element method can provide a better understanding of the effect of wear on the short crack propagation for fretting fatigue. In this paper, the behavior of short crack propagation is implemented by the FE-based remeshing technology, and the wear behavior is implemented by the FE-based adaptive meshing technology. Considering the number of cycles that need to be simulated, a cycle jump technique is adopted to improve the computational efficiency. Finally, the effect of wear on the short crack propagation under fretting conditions is investigated for different initial crack lengths.
Corresponding author. E-mail address: [email protected] (J. Liu).
https://doi.org/10.1016/j.ijmecsci.2019.05.001 Received 2 February 2019; Received in revised form 6 April 2019; Accepted 1 May 2019 Available online 6 May 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.
X. Liu, J. Liu and Z. Zuo et al.
International Journal of Mechanical Sciences 157–158 (2019) 552–560
Table 1 Analytic formulation of coefficients in Eq. (3).
2. Methodology 2.1. Theory background
KIY
TY 7−16𝜈+16𝜈 2 19−16𝜈+16𝜈 2
𝐾 √ IY 2 2𝜋𝜉
√
f1 7−16𝜈+16𝜈 2 1−𝜈+𝜈 2
32(1−10𝜈+10𝜈 2 ) √ 15𝜋 (1−𝜈+𝜈 2 )(7−16𝜈+16𝜈 2 )
The variables in this criterion are KI , KII and T. The coefficients in this criterion are KIY , KIIY , TY and f1 , and the analytic formulation of the coefficients is listed in Table 1 [22]. The Poisson’s ratio 𝜈 and the mode I threshold stress intensity factor Δ𝐾Ith(𝑅=−1) are the material parameters. Refer to the relationship between the stress intensity factor range ΔK and fatigue crack propagation rate dl/dN from Pairs’ law, the relationship between f and dl/dN is established. Then, the short crack propagation rate could be estimated by the equation [22]: d𝑙∕d𝑁 = 𝛼(𝑓max )𝛾 .
(5)
Where fmax is the maximum value of index f in a loading cycle, 𝛼 and 𝛾 are fatigue coefficients. When describing the stress field around the crack tip region, the application premise of the elastic material assumption is that the plastic zone radius ry at the crack tip is at least less than a quarter of the crack length l under cyclic loading conditions. For plane strain problem, the plastic zone radius ry can be expressed as [26]: 2𝑟y ≅
(1)
0 −𝜋
1 3π
(
𝐾 𝜎y
)2 .
(6)
Where 𝜎 y is yield stress, K represents the stress intensity factor. A promising approach which is called interaction integral is provided for the calculation of T-stress and stress intensity factors KI , KII [27,28]. The interaction integral method separates and obtains the KI , KII and T-stress from the real field by establishing the auxiliary field. In order to extract the stress intensity factor KI and KII , the auxiliary stress fields 𝝈 aux , strain fields 𝜺aux and displacement fields uaux can be selected as the asymptotic fields around the tip of a semi-infinite crack in an homogeneous body, as shown in Fig. 1(a), and then the interaction integral M is given by [7]: [ ( ) ( )] 1 𝑎𝑢𝑥 𝑎𝑢𝑥 𝑎𝑢𝑥 𝑎𝑢𝑥 𝑎𝑢𝑥 𝑎𝑢𝑥 𝑀 = lim 𝜀𝑗𝑘 𝛿1𝑖 − 𝜎𝑗𝑘 𝑢𝑗,1 + 𝜎𝑗𝑘 (7) 𝜎𝑗𝑘 𝜀𝑗𝑘 + 𝜎𝑗𝑘 𝑛𝑖 dΓ Γ→0 ∫Γ 2
Where 𝜎′ and 𝜀′ represent the deviatoric parts of the stress and strain tensors, 𝜉 is a length scale parameter and it can be identified by experiments, r and 𝜃 denote the polar coordinates centered at the crack tip. The stress tensors 𝜎 for the crack tip region can be well described by a asymptotic expansion equation for an isotropic elastic material, and then the expression for 𝜎 is as follows [24]: 𝜎𝑖𝑗 (𝑟, 𝜃) = 𝐾I 𝑟−1∕2 𝑓𝑖𝑗(1) (𝜃) + 𝐾II 𝑟−1∕2 𝑓𝑖𝑗(2) (𝜃) + 𝑇 𝑓𝑖𝑗(3) (𝜃).
√
Fig. 1. (a) Crack tip in an infinite homogeneous medium and (b) a point force applied at the crack-tip in the direction parallel to the crack.
𝜋
1 𝑈= tr (𝜎 ′ ⋅ 𝜀′ )𝑟d𝜃d𝑟. ∫ ∫ 2
𝐾IY
Δ𝐾Ith(𝑅=−1) ∕2
According to the suggestion of Chapetti [15], the crack initiation is in the microstructural short crack regime and short crack propagation is in the physical short crack regime. These two regimes are relatively independent, and this paper focuses on the latter regime. In addition, there is a critical crack length lc to divide long cracks and short cracks [16]. Unlike the long cracks, the short cracks can propagate below the threshold of stress intensity factor [17]. Therefore, in the numerical simulation of short crack propagation, a suitable criterion is needed to evaluate whether the stress state around the crack tip satisfies the threshold condition of short crack propagation. Furthermore, under fretting conditions, the variation of the contact force and the crack length caused by wear significantly affects the short crack propagation rate. Therefore, a wear model suitable for fretting conditions also needs to be adopted. At present, the physical short crack propagation model [18], the short crack propagation model with equivalent strain parameter [19] and the critical distortional elastic energy criterion [20], etc. have been successfully applied in the study of short crack propagation. Compared with other criteria, the critical distortional elastic energy criterion can more precisely describe the stress state of the crack tip which is crucial in the accurate prediction of short crack propagation. In addition, the critical distortional elastic energy criterion is mature in the study of short crack propagation under fretting conditions [20–22]. Therefore, this study chooses the critical distortional elastic energy criterion to analyze the effect of wear on short crack propagation under fretting conditions. The criterion is based on the assumption that fatigue cracks propagate because free surfaces are created at the crack tip when it experiences plastic deformation. The plastic deformation occurs when the distortional elastic energy exceeds the critical value. This criterion can predict the propagation threshold of short cracks under the multiaxial loading conditions like fretting conditions. Fretting problems is usually simplified as a plane strain problem, therefore, the distortional elastic energy U over a domain within a distance 𝜉 to the crack front can be expressed as [23]: 𝜉
KIIY
(2)
Where KI is mode I stress intensity factor, KII is mode II stress intensity factor, T stands for a stress acting parallel to cracked plane, 𝑓𝑖𝑗(𝑛) represents the normalized polynomial. For short cracks, the stress intensity factor K and the T-stress have sufficient accuracy to describe the stress field around the crack tip [25]. Substituting Eq. (2) into Eq. (1), and the U can be expressed in terms of KI , KII , and T-stress. In this study, the yield threshold KIY is used to determine the critical distortional elastic energy Uc . With the above mentioned assumption, the short cracks propagate when the U satisfies Eq. (3). ( ) 𝑈 𝐾I , 𝐾II , 𝑇 > 𝑈c (𝐾IY , 0, 0). (3)
Where Γ stands for the integral loop around the crack tip, 𝛿 1i is the Kronecker symbol, ni represents the outward unit normal on the contour. For multi-axis loading conditions, M-integral can be related to the stress intensity factor as follows: 𝑀=
) 2 ( 𝐾 𝐾 aux + 𝐾II 𝐾IIaux 𝐸′ I I
(8)
Let 𝐾Iaux = 1, 𝐾IIaux = 0 and 𝐾Iaux = 0, 𝐾IIaux = 1, the real field stress intensity factor KI and KII can be expressed as [7]:
A few mathematical simplifications of Eq. (3) allow expressing the short crack propagation criterion f as follows: ( ) ( ) ( )2 𝐾I 2 𝐾II 2 𝐾 𝑇 𝑇 𝑓 = + + + 𝑓1 I − 1 > 0. (4) 𝐾IY 𝐾IIY 𝑇Y 𝐾IY 𝑇Y
𝐾I =
𝐸′ 𝐸′ 𝑀1 , 𝐾II = 𝑀 . 2 2 2
(9)
Where 𝐸 ′ = 𝐸∕(1 − 𝜈 2 ) for plane strain condition, and E is Young’s modulus. 553
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International Journal of Mechanical Sciences 157–158 (2019) 552–560
In order to extract the T-stress, the auxiliary field due to a point force F in the x direction is adopted, as shown in Fig. 1(b). Thus, the interaction integral M can be expressed as [29]: [ ( ) ( )] 1 𝑎𝑢𝑥 𝑎𝑢𝑥 𝑎𝑢𝑥 𝑀 = lim (10) 𝜎𝑗𝑘 𝜀𝑎𝑢𝑥 𝑗𝑘 + 𝜎𝑗𝑘 𝜀𝑗𝑘 𝛿1𝑖 − 𝜎𝑖𝑗 𝑢𝑗,1 + 𝜎𝑖𝑗 𝑢𝑗,1 𝑛𝑖 dΓ Γ→0 ∫Γ 2 For isotropic materials, the relationship between M-integral and the Tstress is obtained as [29]: 𝑇 =
𝐸′ 𝑀. 𝐹
(11)
In this study, the evolution of contact surface profile caused by fretting wear is described by using the Archard equation [30] which is often applied in fretting conditions, and it can be expressed as: 𝑉 =
𝐾W 𝑃 𝑆 . 𝐻
(12)
Where V represents total wear volume, KW stands for the wear coefficient, P is the normal load, S denotes the sliding distance and H is the material hardness. In general, the FE-based wear simulation and the FE-based crack propagation simulation are independent. Therefore, based on the above theories, the present work is need to present a FE-based calculation methodology, which considers the effect of wear in the simulation of short crack propagation. 2.2. Calculation procedure Under the fretting loading, there are usually multiple crack initiation sites on the surface. Some of crack initiation sites merge into one larger crack, which will continue to propagate into material and forms the main crack [31,32]. The others will stop propagating into material, and may be gradually removed by wear. These multiple initiation cracks would have a slight influence on the early propagation stage of main crack. For further propagation of main crack, the influence of multiple initiation cracks is deemed negligible. Meanwhile, considering the complexity of simulation of multiple initiation cracks, this research only studies the main crack. Fig. 2 shows the flow chart of numerical procedure for fretting fatigue short crack propagation under the influence of wear used in this study. Firstly, the fretting fatigue FE model containing a crack with initial length li is established, and then the initial conditions are specified. Although the initiation angle of cracks under fretting conditions is generally inclined, we also found some researches show that the initial crack is straight under fretting conditions for the material titanium alloy [33,34]. For this reason and also for simplification, a straight crack at the contact surface is presented to study the short crack propagation behavior, and some scholars have also used the same method to study fretting fatigue crack propagation [35,36]. Secondly, during a loading cycle, the stress intensity factors KI , KII and T-stress at the crack tip are calculated by using the interaction integral method, and then the f is obtained by Eq. (3). Combined with the maximum value fmax of f in a loading cycle, the crack propagation increment dl can be obtained by Eq. (5). At the same time, the contact pressure p(x) and the relative slip amplitude 𝛿(x) along the contact surface is obtained by the finite element method. In order to simulate the evolution of contact surface profile caused by wear, a option is to calculate the local wear. Then, the Archard equation is modified by Eq. (13) [37]: Δℎ(𝑥, 𝑡) = 𝑘𝑝(𝑥)𝛿(𝑥).
Fig. 2. Flow chart of numerical procedure for fretting crack propagation with considering the effect of wear.
wear depth of ΔN cycles, and the nodal wear depth increment Δh(x) for one loading cycle is calculated by the Eq. (14). Δℎ(𝑥) = Δ𝑁𝑘𝑝(𝑥)𝛿(𝑥).
(14)
The remeshing technology [40,41] is used to update the coordinates of crack tip, and then the crack length can increase along the propagation direction. The spatial position of the contact nodes is adjusted according to Δh(x), and then the mesh quality of the FE model is improved by the adaptive meshing technology [42,43]. Adjustment of the contact node will reduce the actual crack length. If the actual crack length l is larger than critical crack length lc , the crack exceeds the category of short cracks, and the calculation is terminated. In addition, if fmax < 0, the calculation is also terminated. Otherwise, repeating the above whole procedure. Thus, the behavior of short crack propagation under the effect of fretting wear is simulated. 3. Finite element modelling The scheme of the fretting assembly used in the study is shown in Fig. 3. The material for the pad and specimen is Ti-6Al-4V alloy, and the material parameters are listed in Table 2 [22,30,44]. The sinusoidal fatigue bulk stress 𝜎 B is applied on the right side of specimen, and the constant pressure Pc is applied on the top of the pad. Due to symmetry condition, only one pad and half of the specimen are taken in the twodimensional FE model. The FE model of fretting fatigue is given in Fig. 4. The pad radius R, the specimen width w and the specimen thickness b are 40 mm, 10 mm and 2.58 mm, respectively. In order to analyze the influence of initial crack length, two kinds of crack lengths, namely 6 μm and 10 μm, are adopted in the study. The average grain size of the Ti-6Al-4V alloy is about 6 μm [45], so the two crack lengths selected are all in the physical short crack regime. Considering that the crack usually initiates near the edge of the contact
(13)
Where Δh(x), p(x) and 𝛿(x) are the incremental wear depth, contact pressure and relative slip at point x, respectively, the k takes the place of KW /H as the wear coefficient. It is inefficient to model each cycle, and then a method called the cycle jumping technique is adopted in this study [38,39]. The wear rate is assumed to be constant over a given cycle jump size ΔN. Thus, one loading cycle can simulate the cumulative 554
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International Journal of Mechanical Sciences 157–158 (2019) 552–560
Fig. 3. The scheme of the fretting assembly. Table 2 The material parameters of Ti-6Al-4V.
Fig. 5. Comparison between the theoretical solution and the corresponding unworn FE results of contact pressure distribution.
Material properties
Value
Young’s modulus E Poisson’s ratio 𝜈 Yield stress 𝜎 y The threshold value of mode I SIF (Δ𝐾Ith(𝑅=−1) ) Length scale parameter 𝜉 Fatigue coefficient 𝛼 Fatigue coefficient 𝛾 Critical crack length lc Wear coefficient k
1.26 × 105 MPa 0.32 830 MPa 4.2 MPa · m1/2 1 μm 9 × 10−10 1 20 μm 2.75 × 10−8 MPa−1
without considering wear behavior, according to Hertzian solution, the distribution of contact pressure p(x) along the contact surface can be expressed as [6,47]: √ 𝑥2 𝑝 (𝑥 ) = 𝑝 0 1 − (15) 𝑎2 Where a represents the half-width of the contact area, p0 stands for the maximum contact pressure, and they are given by: √ 4𝑃 𝑅 𝑎= (16) 𝜋𝐸 ∗ √ 𝑝0 =
𝑃 𝐸∗ 𝜋𝑅
(17)
Where P is the applied normal load, and E∗ is the composite modulus of the two contacting bodies. For plane strain conditions, the latter is given by: ( )−1 1 − 𝜈12 1 − 𝜈22 ∗ 𝐸 = + (18) 𝐸1 𝐸2 Where E1 , E2 are the Young’s moduli and 𝜈 1 , 𝜈 2 are the Poisson’s ratios of the pad and the specimen, respectively. R is the relative curvature given by: ( )−1 1 1 𝑅= + (19) 𝑅1 𝑅2
Fig. 4. A 2D fretting fatigue FE model with a crack.
Where R1 , R2 are the radii of the contacting surfaces of the pad and the specimen, respectively. Since the contact surface of the specimen is a plane, the R2 is equal to ∞. Fig. 5 shows a comparison between the theoretical solutions and the corresponding unworn FE results of contact pressure distribution. For theoretical solutions, 𝑝0 = 230.651 MPa and 𝑎 = 0.276 mm. For FE results, 𝑝0 = 232.099 MPa and 𝑎 = 0.280 mm. It can be seen that the FM results are in good agreement with the theoretical solutions.
zone under fretting conditions [30], the initial crack is placed near the edge of the contact area and explicitly introduced within the FE mesh, as shown in the partial enlarged view of the Fig. 3. Since the r1/2 singularity could improve the calculation accuracy, the crack tip is modeled with a rosette of collapsed quadrilateral elements. The pad and the specimen adopt four-node plane strain elements, and a truss element is used to simulate springs. Considering the computational efficiency, the region away from the contact area is divided by coarse mesh, but in order to capture high stress gradient change in the contact area, the mesh of the contact area is divided by fine mesh. The friction behavior between pad and specimen is described by the Coulomb’s law with a friction coefficient of 0.8, which represents the friction behavior of a dry Ti-6Al-4V/Ti-6Al-4V contact [46]. The FE model is developed by using commercial FE code ABAQUS. The constant pressure Pc , the sinusoidal fatigue bulk stress 𝜎 B and the stress ratio are set to be 10 MPa, 700 MPa and −1, respectively, to ensure that the fretting is in gross slip state. In order to improve the calculation efficiency under the premise of ensuring the calculation accuracy, a compromise is reached by using the cycle jump size ΔN of 200. In order to verify the finite element model, the theoretical solution of the Hertzian contact pressure distribution is employed to compare the numerical solution obtained by finite element calculation. In the case
4. Results and discussion 4.1. Contact variables The removal of surface material caused by fretting wear can reduce crack length, and then affect short crack propagation rate. Fig. 6 shows the wear profile on the contact surface at different loading cycles under gross slip conditions. In the figure, the zero of the abscissa represents crack initiation location, the solid line represents the wear profile on the left side of the crack, and the solid line with ring represents the wear profile on the right side. It can be seen that the wear profile is bilaterally symmetrical, and the middle position of the wear profile is the deepest, which is in good agreement with the shape of wear profile measured by experiment [48]. With the increase of the number of loading cycles, 555
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International Journal of Mechanical Sciences 157–158 (2019) 552–560
Fig. 8. The frictional tangential forces on the left and right sides of the crack initiation location.
Fig. 6. Distribution of wear profile on the contact surface at different loading cycles.
The integral of contact shear stress along the contact surface is the frictional tangential force. The frictional tangential force plays an important role in the short crack propagation behavior. This is because that if the frictional tangential forces on both sides of the crack have the same value and direction, the crack can not open. Therefore, the frictional tangential force even can act as a barrier to short crack propagation. Fig. 8 shows the evolution of frictional tangential force on the both sides of the crack with the increase of cycle number. It can be seen that the frictional tangential force on the right side increases with the increase of the number of loading cycles, while that on the left decreases. The whole frictional tangential force is constant under gross slip conditions, and therefore the difference of frictional tangential force between left side and right side of the crack decreases with the increase of cycle number. The results show that the difference decreases from 73.88 N to 56.31 N in the first 6000 cycles, and the decrease is 27.4%. However, if the fretting wear behavior is not considered, the frictional tangential force difference between left side and right side of the crack will not change with the increase of cycle number. Therefore, it can be concluded that fretting wear reduces the contact force which drives the short crack propagation.
Fig. 7. Distribution of contact pressure on the contact surface at different loading cycles.
the depth and width of the wear profile increase. Moreover, at the crack initiation location, the wear depth increment in the early wear stage is slightly larger than that in the later wear stage. It indicates that the reduction of crack length due to wear is most obvious in the early stage of fretting. In addition, it can be seen from the local magnification of the image, the wear profile curve is discontinuous due to the existence of the crack, and the wear profile near the crack has a slight depression. Under the fretting conditions, in addition to fatigue loading, the specimen is also subjected to the contact pressure and the contact shear stress. The distributions of contact pressure on the contact surface at the tensile stage of cyclic loading at different loading cycles are shown in Fig. 7. For the cylinder-plate contact, the contact profile directly determines the distribution of contact pressure under the same loading conditions. The contact area is relatively small during the initial stage of fretting, so the contact pressure has a larger amplitude. However, with the increase of cycle number, the contact area gradually increases, which leads to the decrease of contact pressure. In addition, it can be seen from the local magnification of the image, the contact pressure at the crack initiation location is much higher than that at the other locations. The reason is that when fatigue loading is tensile, a tiny angle is formed between the specimen surface near the crack initiation location and the pad surface. To some extent, the tiny angle forms a local line-surface contact. As a result, the contact area near the crack location will withstand higher contact pressure. Higher contact pressure results in larger wear rate, which is why the wear profile near the crack has a slight depression. In addition, since the contact state between specimen and pad is in the gross slip domain, the distribution of contact shear stress is simply proportional to the contact pressure (via COF).
4.2. Variables around the crack tip Under fretting loading, the stress state near the crack tip is affected by fatigue loading, contact stress and crack length. Fretting wear can affect the stress state by changing contact stress and crack length. In this study, the KI , KII and T-stress are used to describe the stress state around the crack tip, and the fmax is used to describe the propagation rate of short cracks. In order to analyze the effect of initial crack lengths, two kinds of initial crack lengths, i.e. the shorter initial crack length 6 μm and the longer initial crack length 10 μm, are adopted in the calculation. From the calculation results of Eq. (6), it can be seen that the plastic zone radii ry at the crack tip for 6 μm and 10 μm initial crack lengths are 1.46 μm and 2.36 μm, respectively, and they are all less than a quarter of the crack length. Therefore, the elastic material assumption can be adopted in this research. The relationship among number of cycles, crack length and the variables in Eq. (3), such as KI , KII , T-stress and f, are shown in Fig. 9. It can be seen that the KI without considering wear increases with the increase of the crack length for both initial crack lengths, as shown in Fig. 9(a). Under the condition without considering wear, the fatigue loading and contact stress do not change with the increase of cycle number. Therefore, the crack length plays an important role in the KI . The evolution of KI with considering wear is obviously different from that without considering wear. For the 6 μm initial crack, the calculation is terminated after 6000 cycles because there is not enough energy to drive the crack propagation. Due to the effect of fretting wear, although the crack continues to propagate before 6000 cycles, the actual crack length changes 556
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International Journal of Mechanical Sciences 157–158 (2019) 552–560
Fig. 9. The relationship among number of cycles, crack length and (a) KI , (b) KII , (c) T, (d) f (left - 6 μm case, right - 10 μm case).
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Fig. 10. The evolution of |KI /KIY |, |KII /KIIY |, |T/TY | and f during the first loading cycle.
Fig. 11. The value of |KI /KIY |/|T/TY | versus the crack length.
little. Moreover, although the crack length changes little, the KI with considering wear decreases with the increase of cycle number. For the 10 μm initial crack, although the crack length increases with the increase of cycle number, the KI with considering wear decreases before 6000 cycles. It indicates that the crack length is not the only factor affecting the KI . From the conclusion of Section 4.1, the reduction of the frictional tangential force difference caused by wear can restrain the crack propagation, and the frictional tangential force difference decreases obviously before 6000 cycles. Therefore, the KI with considering wear does not always increase with the increase of crack length before 6000 cycles. Fig. 9(b) shows the evolution of KII with the variation of the crack length and cycle number. It can be seen that the value of KII is reduced due to the influence of wear. Although under the influence of fretting wear, both KI and KII decrease, and the KI decreases greatly. However, KI is still much larger than KII . Therefore, in the studied cases, the mode I crack propagation is dominant, and the effect of wear on the direction and path of crack propagation is small. Fig. 9(c) shows the evolution of T-stress with the variation of the crack length and cycle number. Without considering wear, the T-stress decreases with the increase of the crack length for both initial crack lengths. However, the decrease of T-stress with considering wear is obviously larger than that without considering wear before 6000 cycles. For the 6 μm initial crack, although the crack length changes little, the T-stress with considering wear decreases sharply with the increase of cycle number. For the 10 μm initial crack, the T-stress with considering wear decreases sharply with the increase of crack length before 6000 cycles. The f can be obtained by Eq. (3). The calculation results show that the f1 in Eq. (3) is negative, and other variables and coefficients are positive. Therefore, the f can be expressed as: | 𝐾 || 𝑇 | | 𝐾 |2 | 𝐾 |2 | 𝑇 |2 𝑓 = || I || + || II || + || || + ||𝑓1 |||| I |||| || − 1. | 𝐾IY || 𝑇Y | | 𝐾IY | | 𝐾IIY | | 𝑇Y |
Fig. 12. Short crack propagation rate versus loading cycles.
4.3. Rate of short crack propagation The short crack propagation rate dl/dN versus loading cycles with different initial crack lengths are shown in Fig. 12. As can be seen from the figure, without considering the effect of wear, the short crack propagation rate dl/dN increases with the increase of cycle number. However, with considering the effect of wear, the short crack propagation rate does not always increase with the increase of the cycle number. For the initial crack length 10 μm, the short crack propagation rate has a downward trend in the first 6000 cycles, and then increases in the following cycles. For the initial crack length 6 μm, the short crack propagation rate dl/dN decreases with the increase of cycle number. When 6000 fretting cycles is reached, the short crack propagation rate dl/dN is equal to 0, and the crack stops propagating. In summary, the short crack propagation rate dl/dN is reduced by fretting wear. The short crack propagation rate is actually the movement rate of the crack tip. However, the actual crack length increment is not equal to the moving distance of the crack tip, and the reduction length of the crack caused by fretting wear needs to be considered. Fig. 13 shows the evolution of the contact surface profile and crack path versus loading cycles with different initial crack lengths. The patterning represents the crack tip, the solid line represents the contact surface profile. For the initial crack length 6 μm, the crack tip is almost static after 4000 cycles, but still about 0.6 μm crack length has been worn away, as shown in Fig. 13(a). It means that although the crack no longer propagates, the actual crack length will gradually decrease or even be eliminated due to the wear. For the initial crack length 10 μm, the contact surface profile gradually moves away from the initial position due to the material removal, and the crack tip moves along the depth direction with the increase of the number of loading cycles, as shown in Fig. 13(b). When the loading cycle number reaches 14000, the actual crack length exceeds 20 μm, and the specimen fails. However, it
(20)
The time histories of |KI /KIY |, |KII /KIIY |, |T/TY | and f for the case with an initial crack length of 10 μm during the first loading cycle are presented in Fig. 10. The results show that |KII /KIIY | is far less than |KI /KIY | and |T/TY | when the f is the largest, and the case with an initial crack length of 6 μm has the same result. Therefore, the value of fmax is mainly determined by |KI /KIY | and |T/TY |. As can be seen from Fig. 11, the ratios of |KI /KIY | to |T/TY | increase with the increase of crack length. When the crack length propagates from 6 μm to 20 μm, the ratio increases from 2.12 to 4.31. It means that the weight of KI in index fmax increases with the increase of crack length. As shown in Fig. 9(d), the variation trend of fmax is very similar with that of the KI because KI has the greatest weight in fmax . 558
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International Journal of Mechanical Sciences 157–158 (2019) 552–560
Fig. 13. Predicted evolution of the contact surface profile and the crack path with an initial crack length of (a) 6 μm and (b) 10 μm.
diction of short crack propagation life with considering wear is 50% higher than that without considering wear. Based on the present computations, it can be concluded that the fretting wear can significantly affect the short crack propagation rate, and this effect cannot be ignored. Under gross slip conditions, the life prediction of short crack propagation without considering wear will overestimate the risk of components.
Acknowledgments This study is based upon work supported by the National Natural Science Foundation of China (Grant No. 51675045) and Aeronautical Science Foundation of China (Grant No. 2016ZE55011). Fig. 14. Short crack propagation lives with and without considering fretting wear.
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can be seen that the crack tip is located at the depth of 23.5 μm from the initial contact surface. Finally, the predicted fretting fatigue crack propagation life with and without considering wear versus different initial crack length are summarized in Fig. 14. For the shorter initial crack length 6 μm, the predicted crack initiation life with considering wear is infinite, and that without considering wear is only 14121. For the initial crack length 10 μm, the predicted crack initiation life with considering wear is about 50% higher than that without considering wear. The results clearly demonstrate that the predicted crack initiation lives with considering wear is longer than that without considering wear for all the initial crack lengths, and the wear has the larger effect on the crack propagation life of the specimen with shorter initial crack length. In summary, the prediction of short crack propagation life without considering wear will overestimate the risk of components. 5. Conclusions The application of the calculation methodology in this study provides a detailed understanding of the effect of wear on the short crack propagation under fretting conditions, and the conclusions are as follows. Under fretting loading, the short crack propagation rate does not always increase with the increase of the cycle number. For the shorter initial crack length, the short crack propagation rate decreases with the increase of cycle number until the crack stops propagation. For the longer initial crack length, the short crack propagation rate has a downward trend in the first 6000 cycles, and then increases in the following cycles. Retardation of short crack propagation is more remarkable for the shorter initial crack length under the effect of fretting wear. In this study, for the shorter initial crack length 6 μm, the prediction of short crack propagation life with considering wear is infinite, while that without considering wear is 14121. For the initial crack length 10 μm, the pre559
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[21] Brugier F, Pommier S, de Moura PR, Mary C, Soria D. A novel approach to predict the growth rate of short cracks under multiaxial loadings. In: International conference on multiaxial fatigue & fracture; 2015. [22] Bellecave J, Pommier S, Nadot Y. T-stress based short crack growth model for fretting fatigue. Tribol T 2014;6:23–34. [23] RDM P, Pommier S, Mary C. A novel methodology to predict the endurance domain for a material and its evolution using a generalized fracture mechanics framework. Int J Fatigue 2012;42:183–93. [24] Williams ML. On the stress distribution at the base of a stationary crack. J Appl Mech-T ASME 1957;24:109–14. [25] Zerbst U, Vormwald M, Pippan R. About the fatigue crack propagation threshold of metals as a design criterion-a review. Eng Fract Mech 2016;153:190–243. [26] Stephens RI, Fatemi A, Stephens RR, Fuchs HO. Metal fatigue in engineering. 2nd. New York: Wiley-interscience; 2000. p. 134–5. [27] Muthu N, Falzon BG, Maiti SK, Khoddam S. Modified crack closure integral technique for extraction of SIFs in meshfree methods. Finite Elem Anal Des 2014;78:25–39. [28] Gupta M, Alderliesten RC, Benedictus R. A review of T-stress and its effects in fracture mechanics. Eng Fract Mech 2015;134:218–41. [29] Paulino GH, Kim JH. A new approach to compute T-stress in functionally graded materials by means of the interaction integral method. Eng Fract Mech 2004;71:1907–50. [30] Madge JJ, Leen SB, Mccoll IR, Shipway PH. Contact-evolution based prediction of fretting fatigue life: effect of slip amplitude. Wear 2007;262:1159–70. [31] Garcia D, Grandt A. Fractographic characteristics associated with fretting fatigue cracks in turbomachinery alloys. In: 44th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics, and materials conference; 2003. [32] Antoniou RA, Radtke TC. Mechanisms of fretting-fatigue of titanium alloys. Mater Sci Eng A 1997;237:229–40. [33] Garcia DB, Grandt AF, Bartha BB, Golden PJ. Threshold fatigue measurements and fractographic examination of fretting induced cracks in Ti-17. Eng Fail Anal 2007;14:529–40. [34] Garcia DB, Grandt AF. Fractographic investigation of fretting fatigue cracks in Ti-6Al-4V. Eng Fail Anal 2005;12:537–48.
[35] Kubota M, Kataoka S, Takazaki D. A quantitative approach to evaluate fretting fatigue limit using a pre-cracked specimen. Tribol Int 2017;108:48–56. [36] Giner E, Navarro C, Sabsabi M. Fretting fatigue life prediction using the extended finite element method. Int J Mech Sci 2011;53:217–25. [37] Madge JJ, Leen SB, Shipway PH. A combined wear and crack nucleation-propagation methodology for fretting fatigue prediction. Int J Fatigue 2008;30:1509–28. [38] Arnaud P, Fouvry S, Garcin S. Fretting wear rate impact on Ti-6Al-4V fretting crack risk: experimental and numerical comparison between cylinder/plane and punch/plane contact geometries. Tribol Int 2017;108:32–47. [39] Labergere C, Saanouni K, Sun ZD. Prediction of low cycle fatigue life using cycles jumping integration scheme. Appl Mech Mater 2015;784:308–16. [40] Noraphaiphipaksa N, Manonukul A, Kanchanomai C. Fretting-contact-induced crack opening/closure behaviour in fretting fatigue. Int J Fatigue 2016;88:185–96. [41] Ghosh A, Paulson N, Sadeghi F. A fracture mechanics approach to simulate sub-surface initiated fretting wear. Int J Solids Sturct 2015;58:335–52. [42] Basseville S, Cailletaud G. An evaluation of the competition between wear and crack initiation in fretting conditions for Ti-6Al-4V alloy. Wear 2015;328–329:443–55. [43] Shen F, Hu W, Meng Q. A damage mechanics approach to fretting fatigue life prediction with consideration of elastic-plastic damage model and wear. Tribol Int 2015;82:176–90. [44] Wallace JM, Neu RW. Fretting fatigue crack nucleation in Ti-6Al-4V. Fatigue Fract Eng M 2010;26:199–214. [45] Chong Y, Bhattacharjee T, Shibata A. Investigation of the grain size effect on mechanical properties of Ti-6Al-4V alloy with equiaxed and bimodal microstructures. In: IOP conference series: materials science and engineering; 2017. [46] Jin O, Mall S. Effects of slip on fretting behavior: experiments and analyses. Wear 2004;256:671–84. [47] Mccoll IR, Ding J, Leen SB. Finite element simulation and experimental validation of fretting wear. Wear 2004;256:1114–27. [48] Paulin C, Fouvry S, Meunier C. Finite element modelling of fretting wear surface evolution: application to a Ti-6Al-4V contact. Wear 2008;264:26–36.
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Contents lists available at ScienceDirect
International Journal of Mechanical Sciences journal homepage: www.elsevier.com/locate/ijmecsci
The effect of wear on short crack propagation under fretting conditions Xin Liu a, Jinxiang Liu a,∗, Zhengxing Zuo a, Huayang Zhang b a b
School of Mechanical Engineering, Beijing Institute of Technology, Beijing 100081, China School of Aeronautical Engineering, Zhengzhou University of Aeronautics, Zhengzhou 450046, China
a r t i c l e
i n f o
Keywords: Short crack Crack propagation rate Wear Fretting
a b s t r a c t The effect of wear on the crack propagation behavior for short cracks under fretting conditions is investigated by using numerical method. The numerical simulation effectively combines the remeshing technology with the adaptive meshing technology, and the evolution of contact force and crack length caused by wear can be considered in this study. The critical distortional elastic energy criterion is used to determine the threshold condition of short crack propagation, and the Archard equation is employed for wear simulation. The results show that wear reduces the short crack propagation rate. Furthermore, the effect of wear on the short crack propagation rate is more obvious for the shorter initial crack length.
1. Introduction Fretting fatigue caused by micron-level oscillatory motion between two contacting surfaces occurs in a wide range of industry fields, such as aviation, aerospace, shipbuilding industry, etc [1]. The fretting fatigue could be affected by many factors, such as fatigue bulk stress, contact shear stress, contact pressure, slip amplitude, contact geometry, wear, etc. Among these factors, the wear has a highly complex effect on fretting fatigue, especially on the initial stage of fretting fatigue crack propagation. In the initial stage of fretting fatigue crack propagation, the fretting crack is a short crack and the crack length is in the same order of magnitude with the removed material depth due to wear. On the one hand, the wear can change the geometry of the contact surface, which finally results in the change of contact force distribution. On the other hand, the wear directly reduces the length of the existing crack, and then influences the crack propagation rate. In extreme cases, the wear can even totally remove the existing short cracks [2]. In addition, the microstructure evolution caused by wear can causes a change in the coefficient of friction [3], but the friction coefficient will be stable in a relatively short period [4], so this study does not consider the effect of microstructure evolution. In summary, understanding the effect of wear on the short crack propagation is essential for the fatigue design under fretting conditions. In order to analyze the effect of wear on the short crack propagation under fretting conditions, the change of crack length and contact surface profile caused by wear is a major concern. However, without suspending the fretting experiment, it is difficult to accurately measure the change of crack length and wear profile in micron-scale. However,
∗
suspending the experiment will change the contact state and further affect the subsequent crack propagation behavior and wear behavior. The numerical method provides an alternative approach to investigate the effect of wear on the short crack propagation for fretting fatigue. Among the numerical methods, the finite element method is widely used to analyze fretting wear behavior and fretting fatigue crack propagation behavior. By means of finite element method, Pereira et al. proposed a multiscale procedure to study roughness effect on fretting wear [5]. Yue et al. used finite element method to analyze the stress singularity in fretting wear [6]. According to FE results, Pereira et al. found that the extended maximum tangential stress criterion is more suitable for predicting the crack path than the classical maximum tangential stress criterion [7]. By combining the extended finite element method with fatigue crack growth criteria, Martìnez et al. predicted the fretting fatigue crack trajectory of a railway axle [8]. In addition, there are many successful applications of finite element methods in the study of fretting wear and fretting fatigue [9–14]. Thus, a proper combination of wear simulation and crack propagation simulation through finite element method can provide a better understanding of the effect of wear on the short crack propagation for fretting fatigue. In this paper, the behavior of short crack propagation is implemented by the FE-based remeshing technology, and the wear behavior is implemented by the FE-based adaptive meshing technology. Considering the number of cycles that need to be simulated, a cycle jump technique is adopted to improve the computational efficiency. Finally, the effect of wear on the short crack propagation under fretting conditions is investigated for different initial crack lengths.
Corresponding author. E-mail address: [email protected] (J. Liu).
https://doi.org/10.1016/j.ijmecsci.2019.05.001 Received 2 February 2019; Received in revised form 6 April 2019; Accepted 1 May 2019 Available online 6 May 2019 0020-7403/© 2019 Elsevier Ltd. All rights reserved.
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International Journal of Mechanical Sciences 157–158 (2019) 552–560
Table 1 Analytic formulation of coefficients in Eq. (3).
2. Methodology 2.1. Theory background
KIY
TY 7−16𝜈+16𝜈 2 19−16𝜈+16𝜈 2
𝐾 √ IY 2 2𝜋𝜉
√
f1 7−16𝜈+16𝜈 2 1−𝜈+𝜈 2
32(1−10𝜈+10𝜈 2 ) √ 15𝜋 (1−𝜈+𝜈 2 )(7−16𝜈+16𝜈 2 )
The variables in this criterion are KI , KII and T. The coefficients in this criterion are KIY , KIIY , TY and f1 , and the analytic formulation of the coefficients is listed in Table 1 [22]. The Poisson’s ratio 𝜈 and the mode I threshold stress intensity factor Δ𝐾Ith(𝑅=−1) are the material parameters. Refer to the relationship between the stress intensity factor range ΔK and fatigue crack propagation rate dl/dN from Pairs’ law, the relationship between f and dl/dN is established. Then, the short crack propagation rate could be estimated by the equation [22]: d𝑙∕d𝑁 = 𝛼(𝑓max )𝛾 .
(5)
Where fmax is the maximum value of index f in a loading cycle, 𝛼 and 𝛾 are fatigue coefficients. When describing the stress field around the crack tip region, the application premise of the elastic material assumption is that the plastic zone radius ry at the crack tip is at least less than a quarter of the crack length l under cyclic loading conditions. For plane strain problem, the plastic zone radius ry can be expressed as [26]: 2𝑟y ≅
(1)
0 −𝜋
1 3π
(
𝐾 𝜎y
)2 .
(6)
Where 𝜎 y is yield stress, K represents the stress intensity factor. A promising approach which is called interaction integral is provided for the calculation of T-stress and stress intensity factors KI , KII [27,28]. The interaction integral method separates and obtains the KI , KII and T-stress from the real field by establishing the auxiliary field. In order to extract the stress intensity factor KI and KII , the auxiliary stress fields 𝝈 aux , strain fields 𝜺aux and displacement fields uaux can be selected as the asymptotic fields around the tip of a semi-infinite crack in an homogeneous body, as shown in Fig. 1(a), and then the interaction integral M is given by [7]: [ ( ) ( )] 1 𝑎𝑢𝑥 𝑎𝑢𝑥 𝑎𝑢𝑥 𝑎𝑢𝑥 𝑎𝑢𝑥 𝑎𝑢𝑥 𝑀 = lim 𝜀𝑗𝑘 𝛿1𝑖 − 𝜎𝑗𝑘 𝑢𝑗,1 + 𝜎𝑗𝑘 (7) 𝜎𝑗𝑘 𝜀𝑗𝑘 + 𝜎𝑗𝑘 𝑛𝑖 dΓ Γ→0 ∫Γ 2
Where 𝜎′ and 𝜀′ represent the deviatoric parts of the stress and strain tensors, 𝜉 is a length scale parameter and it can be identified by experiments, r and 𝜃 denote the polar coordinates centered at the crack tip. The stress tensors 𝜎 for the crack tip region can be well described by a asymptotic expansion equation for an isotropic elastic material, and then the expression for 𝜎 is as follows [24]: 𝜎𝑖𝑗 (𝑟, 𝜃) = 𝐾I 𝑟−1∕2 𝑓𝑖𝑗(1) (𝜃) + 𝐾II 𝑟−1∕2 𝑓𝑖𝑗(2) (𝜃) + 𝑇 𝑓𝑖𝑗(3) (𝜃).
√
Fig. 1. (a) Crack tip in an infinite homogeneous medium and (b) a point force applied at the crack-tip in the direction parallel to the crack.
𝜋
1 𝑈= tr (𝜎 ′ ⋅ 𝜀′ )𝑟d𝜃d𝑟. ∫ ∫ 2
𝐾IY
Δ𝐾Ith(𝑅=−1) ∕2
According to the suggestion of Chapetti [15], the crack initiation is in the microstructural short crack regime and short crack propagation is in the physical short crack regime. These two regimes are relatively independent, and this paper focuses on the latter regime. In addition, there is a critical crack length lc to divide long cracks and short cracks [16]. Unlike the long cracks, the short cracks can propagate below the threshold of stress intensity factor [17]. Therefore, in the numerical simulation of short crack propagation, a suitable criterion is needed to evaluate whether the stress state around the crack tip satisfies the threshold condition of short crack propagation. Furthermore, under fretting conditions, the variation of the contact force and the crack length caused by wear significantly affects the short crack propagation rate. Therefore, a wear model suitable for fretting conditions also needs to be adopted. At present, the physical short crack propagation model [18], the short crack propagation model with equivalent strain parameter [19] and the critical distortional elastic energy criterion [20], etc. have been successfully applied in the study of short crack propagation. Compared with other criteria, the critical distortional elastic energy criterion can more precisely describe the stress state of the crack tip which is crucial in the accurate prediction of short crack propagation. In addition, the critical distortional elastic energy criterion is mature in the study of short crack propagation under fretting conditions [20–22]. Therefore, this study chooses the critical distortional elastic energy criterion to analyze the effect of wear on short crack propagation under fretting conditions. The criterion is based on the assumption that fatigue cracks propagate because free surfaces are created at the crack tip when it experiences plastic deformation. The plastic deformation occurs when the distortional elastic energy exceeds the critical value. This criterion can predict the propagation threshold of short cracks under the multiaxial loading conditions like fretting conditions. Fretting problems is usually simplified as a plane strain problem, therefore, the distortional elastic energy U over a domain within a distance 𝜉 to the crack front can be expressed as [23]: 𝜉
KIIY
(2)
Where KI is mode I stress intensity factor, KII is mode II stress intensity factor, T stands for a stress acting parallel to cracked plane, 𝑓𝑖𝑗(𝑛) represents the normalized polynomial. For short cracks, the stress intensity factor K and the T-stress have sufficient accuracy to describe the stress field around the crack tip [25]. Substituting Eq. (2) into Eq. (1), and the U can be expressed in terms of KI , KII , and T-stress. In this study, the yield threshold KIY is used to determine the critical distortional elastic energy Uc . With the above mentioned assumption, the short cracks propagate when the U satisfies Eq. (3). ( ) 𝑈 𝐾I , 𝐾II , 𝑇 > 𝑈c (𝐾IY , 0, 0). (3)
Where Γ stands for the integral loop around the crack tip, 𝛿 1i is the Kronecker symbol, ni represents the outward unit normal on the contour. For multi-axis loading conditions, M-integral can be related to the stress intensity factor as follows: 𝑀=
) 2 ( 𝐾 𝐾 aux + 𝐾II 𝐾IIaux 𝐸′ I I
(8)
Let 𝐾Iaux = 1, 𝐾IIaux = 0 and 𝐾Iaux = 0, 𝐾IIaux = 1, the real field stress intensity factor KI and KII can be expressed as [7]:
A few mathematical simplifications of Eq. (3) allow expressing the short crack propagation criterion f as follows: ( ) ( ) ( )2 𝐾I 2 𝐾II 2 𝐾 𝑇 𝑇 𝑓 = + + + 𝑓1 I − 1 > 0. (4) 𝐾IY 𝐾IIY 𝑇Y 𝐾IY 𝑇Y
𝐾I =
𝐸′ 𝐸′ 𝑀1 , 𝐾II = 𝑀 . 2 2 2
(9)
Where 𝐸 ′ = 𝐸∕(1 − 𝜈 2 ) for plane strain condition, and E is Young’s modulus. 553
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International Journal of Mechanical Sciences 157–158 (2019) 552–560
In order to extract the T-stress, the auxiliary field due to a point force F in the x direction is adopted, as shown in Fig. 1(b). Thus, the interaction integral M can be expressed as [29]: [ ( ) ( )] 1 𝑎𝑢𝑥 𝑎𝑢𝑥 𝑎𝑢𝑥 𝑀 = lim (10) 𝜎𝑗𝑘 𝜀𝑎𝑢𝑥 𝑗𝑘 + 𝜎𝑗𝑘 𝜀𝑗𝑘 𝛿1𝑖 − 𝜎𝑖𝑗 𝑢𝑗,1 + 𝜎𝑖𝑗 𝑢𝑗,1 𝑛𝑖 dΓ Γ→0 ∫Γ 2 For isotropic materials, the relationship between M-integral and the Tstress is obtained as [29]: 𝑇 =
𝐸′ 𝑀. 𝐹
(11)
In this study, the evolution of contact surface profile caused by fretting wear is described by using the Archard equation [30] which is often applied in fretting conditions, and it can be expressed as: 𝑉 =
𝐾W 𝑃 𝑆 . 𝐻
(12)
Where V represents total wear volume, KW stands for the wear coefficient, P is the normal load, S denotes the sliding distance and H is the material hardness. In general, the FE-based wear simulation and the FE-based crack propagation simulation are independent. Therefore, based on the above theories, the present work is need to present a FE-based calculation methodology, which considers the effect of wear in the simulation of short crack propagation. 2.2. Calculation procedure Under the fretting loading, there are usually multiple crack initiation sites on the surface. Some of crack initiation sites merge into one larger crack, which will continue to propagate into material and forms the main crack [31,32]. The others will stop propagating into material, and may be gradually removed by wear. These multiple initiation cracks would have a slight influence on the early propagation stage of main crack. For further propagation of main crack, the influence of multiple initiation cracks is deemed negligible. Meanwhile, considering the complexity of simulation of multiple initiation cracks, this research only studies the main crack. Fig. 2 shows the flow chart of numerical procedure for fretting fatigue short crack propagation under the influence of wear used in this study. Firstly, the fretting fatigue FE model containing a crack with initial length li is established, and then the initial conditions are specified. Although the initiation angle of cracks under fretting conditions is generally inclined, we also found some researches show that the initial crack is straight under fretting conditions for the material titanium alloy [33,34]. For this reason and also for simplification, a straight crack at the contact surface is presented to study the short crack propagation behavior, and some scholars have also used the same method to study fretting fatigue crack propagation [35,36]. Secondly, during a loading cycle, the stress intensity factors KI , KII and T-stress at the crack tip are calculated by using the interaction integral method, and then the f is obtained by Eq. (3). Combined with the maximum value fmax of f in a loading cycle, the crack propagation increment dl can be obtained by Eq. (5). At the same time, the contact pressure p(x) and the relative slip amplitude 𝛿(x) along the contact surface is obtained by the finite element method. In order to simulate the evolution of contact surface profile caused by wear, a option is to calculate the local wear. Then, the Archard equation is modified by Eq. (13) [37]: Δℎ(𝑥, 𝑡) = 𝑘𝑝(𝑥)𝛿(𝑥).
Fig. 2. Flow chart of numerical procedure for fretting crack propagation with considering the effect of wear.
wear depth of ΔN cycles, and the nodal wear depth increment Δh(x) for one loading cycle is calculated by the Eq. (14). Δℎ(𝑥) = Δ𝑁𝑘𝑝(𝑥)𝛿(𝑥).
(14)
The remeshing technology [40,41] is used to update the coordinates of crack tip, and then the crack length can increase along the propagation direction. The spatial position of the contact nodes is adjusted according to Δh(x), and then the mesh quality of the FE model is improved by the adaptive meshing technology [42,43]. Adjustment of the contact node will reduce the actual crack length. If the actual crack length l is larger than critical crack length lc , the crack exceeds the category of short cracks, and the calculation is terminated. In addition, if fmax < 0, the calculation is also terminated. Otherwise, repeating the above whole procedure. Thus, the behavior of short crack propagation under the effect of fretting wear is simulated. 3. Finite element modelling The scheme of the fretting assembly used in the study is shown in Fig. 3. The material for the pad and specimen is Ti-6Al-4V alloy, and the material parameters are listed in Table 2 [22,30,44]. The sinusoidal fatigue bulk stress 𝜎 B is applied on the right side of specimen, and the constant pressure Pc is applied on the top of the pad. Due to symmetry condition, only one pad and half of the specimen are taken in the twodimensional FE model. The FE model of fretting fatigue is given in Fig. 4. The pad radius R, the specimen width w and the specimen thickness b are 40 mm, 10 mm and 2.58 mm, respectively. In order to analyze the influence of initial crack length, two kinds of crack lengths, namely 6 μm and 10 μm, are adopted in the study. The average grain size of the Ti-6Al-4V alloy is about 6 μm [45], so the two crack lengths selected are all in the physical short crack regime. Considering that the crack usually initiates near the edge of the contact
(13)
Where Δh(x), p(x) and 𝛿(x) are the incremental wear depth, contact pressure and relative slip at point x, respectively, the k takes the place of KW /H as the wear coefficient. It is inefficient to model each cycle, and then a method called the cycle jumping technique is adopted in this study [38,39]. The wear rate is assumed to be constant over a given cycle jump size ΔN. Thus, one loading cycle can simulate the cumulative 554
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Fig. 3. The scheme of the fretting assembly. Table 2 The material parameters of Ti-6Al-4V.
Fig. 5. Comparison between the theoretical solution and the corresponding unworn FE results of contact pressure distribution.
Material properties
Value
Young’s modulus E Poisson’s ratio 𝜈 Yield stress 𝜎 y The threshold value of mode I SIF (Δ𝐾Ith(𝑅=−1) ) Length scale parameter 𝜉 Fatigue coefficient 𝛼 Fatigue coefficient 𝛾 Critical crack length lc Wear coefficient k
1.26 × 105 MPa 0.32 830 MPa 4.2 MPa · m1/2 1 μm 9 × 10−10 1 20 μm 2.75 × 10−8 MPa−1
without considering wear behavior, according to Hertzian solution, the distribution of contact pressure p(x) along the contact surface can be expressed as [6,47]: √ 𝑥2 𝑝 (𝑥 ) = 𝑝 0 1 − (15) 𝑎2 Where a represents the half-width of the contact area, p0 stands for the maximum contact pressure, and they are given by: √ 4𝑃 𝑅 𝑎= (16) 𝜋𝐸 ∗ √ 𝑝0 =
𝑃 𝐸∗ 𝜋𝑅
(17)
Where P is the applied normal load, and E∗ is the composite modulus of the two contacting bodies. For plane strain conditions, the latter is given by: ( )−1 1 − 𝜈12 1 − 𝜈22 ∗ 𝐸 = + (18) 𝐸1 𝐸2 Where E1 , E2 are the Young’s moduli and 𝜈 1 , 𝜈 2 are the Poisson’s ratios of the pad and the specimen, respectively. R is the relative curvature given by: ( )−1 1 1 𝑅= + (19) 𝑅1 𝑅2
Fig. 4. A 2D fretting fatigue FE model with a crack.
Where R1 , R2 are the radii of the contacting surfaces of the pad and the specimen, respectively. Since the contact surface of the specimen is a plane, the R2 is equal to ∞. Fig. 5 shows a comparison between the theoretical solutions and the corresponding unworn FE results of contact pressure distribution. For theoretical solutions, 𝑝0 = 230.651 MPa and 𝑎 = 0.276 mm. For FE results, 𝑝0 = 232.099 MPa and 𝑎 = 0.280 mm. It can be seen that the FM results are in good agreement with the theoretical solutions.
zone under fretting conditions [30], the initial crack is placed near the edge of the contact area and explicitly introduced within the FE mesh, as shown in the partial enlarged view of the Fig. 3. Since the r1/2 singularity could improve the calculation accuracy, the crack tip is modeled with a rosette of collapsed quadrilateral elements. The pad and the specimen adopt four-node plane strain elements, and a truss element is used to simulate springs. Considering the computational efficiency, the region away from the contact area is divided by coarse mesh, but in order to capture high stress gradient change in the contact area, the mesh of the contact area is divided by fine mesh. The friction behavior between pad and specimen is described by the Coulomb’s law with a friction coefficient of 0.8, which represents the friction behavior of a dry Ti-6Al-4V/Ti-6Al-4V contact [46]. The FE model is developed by using commercial FE code ABAQUS. The constant pressure Pc , the sinusoidal fatigue bulk stress 𝜎 B and the stress ratio are set to be 10 MPa, 700 MPa and −1, respectively, to ensure that the fretting is in gross slip state. In order to improve the calculation efficiency under the premise of ensuring the calculation accuracy, a compromise is reached by using the cycle jump size ΔN of 200. In order to verify the finite element model, the theoretical solution of the Hertzian contact pressure distribution is employed to compare the numerical solution obtained by finite element calculation. In the case
4. Results and discussion 4.1. Contact variables The removal of surface material caused by fretting wear can reduce crack length, and then affect short crack propagation rate. Fig. 6 shows the wear profile on the contact surface at different loading cycles under gross slip conditions. In the figure, the zero of the abscissa represents crack initiation location, the solid line represents the wear profile on the left side of the crack, and the solid line with ring represents the wear profile on the right side. It can be seen that the wear profile is bilaterally symmetrical, and the middle position of the wear profile is the deepest, which is in good agreement with the shape of wear profile measured by experiment [48]. With the increase of the number of loading cycles, 555
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Fig. 8. The frictional tangential forces on the left and right sides of the crack initiation location.
Fig. 6. Distribution of wear profile on the contact surface at different loading cycles.
The integral of contact shear stress along the contact surface is the frictional tangential force. The frictional tangential force plays an important role in the short crack propagation behavior. This is because that if the frictional tangential forces on both sides of the crack have the same value and direction, the crack can not open. Therefore, the frictional tangential force even can act as a barrier to short crack propagation. Fig. 8 shows the evolution of frictional tangential force on the both sides of the crack with the increase of cycle number. It can be seen that the frictional tangential force on the right side increases with the increase of the number of loading cycles, while that on the left decreases. The whole frictional tangential force is constant under gross slip conditions, and therefore the difference of frictional tangential force between left side and right side of the crack decreases with the increase of cycle number. The results show that the difference decreases from 73.88 N to 56.31 N in the first 6000 cycles, and the decrease is 27.4%. However, if the fretting wear behavior is not considered, the frictional tangential force difference between left side and right side of the crack will not change with the increase of cycle number. Therefore, it can be concluded that fretting wear reduces the contact force which drives the short crack propagation.
Fig. 7. Distribution of contact pressure on the contact surface at different loading cycles.
the depth and width of the wear profile increase. Moreover, at the crack initiation location, the wear depth increment in the early wear stage is slightly larger than that in the later wear stage. It indicates that the reduction of crack length due to wear is most obvious in the early stage of fretting. In addition, it can be seen from the local magnification of the image, the wear profile curve is discontinuous due to the existence of the crack, and the wear profile near the crack has a slight depression. Under the fretting conditions, in addition to fatigue loading, the specimen is also subjected to the contact pressure and the contact shear stress. The distributions of contact pressure on the contact surface at the tensile stage of cyclic loading at different loading cycles are shown in Fig. 7. For the cylinder-plate contact, the contact profile directly determines the distribution of contact pressure under the same loading conditions. The contact area is relatively small during the initial stage of fretting, so the contact pressure has a larger amplitude. However, with the increase of cycle number, the contact area gradually increases, which leads to the decrease of contact pressure. In addition, it can be seen from the local magnification of the image, the contact pressure at the crack initiation location is much higher than that at the other locations. The reason is that when fatigue loading is tensile, a tiny angle is formed between the specimen surface near the crack initiation location and the pad surface. To some extent, the tiny angle forms a local line-surface contact. As a result, the contact area near the crack location will withstand higher contact pressure. Higher contact pressure results in larger wear rate, which is why the wear profile near the crack has a slight depression. In addition, since the contact state between specimen and pad is in the gross slip domain, the distribution of contact shear stress is simply proportional to the contact pressure (via COF).
4.2. Variables around the crack tip Under fretting loading, the stress state near the crack tip is affected by fatigue loading, contact stress and crack length. Fretting wear can affect the stress state by changing contact stress and crack length. In this study, the KI , KII and T-stress are used to describe the stress state around the crack tip, and the fmax is used to describe the propagation rate of short cracks. In order to analyze the effect of initial crack lengths, two kinds of initial crack lengths, i.e. the shorter initial crack length 6 μm and the longer initial crack length 10 μm, are adopted in the calculation. From the calculation results of Eq. (6), it can be seen that the plastic zone radii ry at the crack tip for 6 μm and 10 μm initial crack lengths are 1.46 μm and 2.36 μm, respectively, and they are all less than a quarter of the crack length. Therefore, the elastic material assumption can be adopted in this research. The relationship among number of cycles, crack length and the variables in Eq. (3), such as KI , KII , T-stress and f, are shown in Fig. 9. It can be seen that the KI without considering wear increases with the increase of the crack length for both initial crack lengths, as shown in Fig. 9(a). Under the condition without considering wear, the fatigue loading and contact stress do not change with the increase of cycle number. Therefore, the crack length plays an important role in the KI . The evolution of KI with considering wear is obviously different from that without considering wear. For the 6 μm initial crack, the calculation is terminated after 6000 cycles because there is not enough energy to drive the crack propagation. Due to the effect of fretting wear, although the crack continues to propagate before 6000 cycles, the actual crack length changes 556
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Fig. 9. The relationship among number of cycles, crack length and (a) KI , (b) KII , (c) T, (d) f (left - 6 μm case, right - 10 μm case).
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Fig. 10. The evolution of |KI /KIY |, |KII /KIIY |, |T/TY | and f during the first loading cycle.
Fig. 11. The value of |KI /KIY |/|T/TY | versus the crack length.
little. Moreover, although the crack length changes little, the KI with considering wear decreases with the increase of cycle number. For the 10 μm initial crack, although the crack length increases with the increase of cycle number, the KI with considering wear decreases before 6000 cycles. It indicates that the crack length is not the only factor affecting the KI . From the conclusion of Section 4.1, the reduction of the frictional tangential force difference caused by wear can restrain the crack propagation, and the frictional tangential force difference decreases obviously before 6000 cycles. Therefore, the KI with considering wear does not always increase with the increase of crack length before 6000 cycles. Fig. 9(b) shows the evolution of KII with the variation of the crack length and cycle number. It can be seen that the value of KII is reduced due to the influence of wear. Although under the influence of fretting wear, both KI and KII decrease, and the KI decreases greatly. However, KI is still much larger than KII . Therefore, in the studied cases, the mode I crack propagation is dominant, and the effect of wear on the direction and path of crack propagation is small. Fig. 9(c) shows the evolution of T-stress with the variation of the crack length and cycle number. Without considering wear, the T-stress decreases with the increase of the crack length for both initial crack lengths. However, the decrease of T-stress with considering wear is obviously larger than that without considering wear before 6000 cycles. For the 6 μm initial crack, although the crack length changes little, the T-stress with considering wear decreases sharply with the increase of cycle number. For the 10 μm initial crack, the T-stress with considering wear decreases sharply with the increase of crack length before 6000 cycles. The f can be obtained by Eq. (3). The calculation results show that the f1 in Eq. (3) is negative, and other variables and coefficients are positive. Therefore, the f can be expressed as: | 𝐾 || 𝑇 | | 𝐾 |2 | 𝐾 |2 | 𝑇 |2 𝑓 = || I || + || II || + || || + ||𝑓1 |||| I |||| || − 1. | 𝐾IY || 𝑇Y | | 𝐾IY | | 𝐾IIY | | 𝑇Y |
Fig. 12. Short crack propagation rate versus loading cycles.
4.3. Rate of short crack propagation The short crack propagation rate dl/dN versus loading cycles with different initial crack lengths are shown in Fig. 12. As can be seen from the figure, without considering the effect of wear, the short crack propagation rate dl/dN increases with the increase of cycle number. However, with considering the effect of wear, the short crack propagation rate does not always increase with the increase of the cycle number. For the initial crack length 10 μm, the short crack propagation rate has a downward trend in the first 6000 cycles, and then increases in the following cycles. For the initial crack length 6 μm, the short crack propagation rate dl/dN decreases with the increase of cycle number. When 6000 fretting cycles is reached, the short crack propagation rate dl/dN is equal to 0, and the crack stops propagating. In summary, the short crack propagation rate dl/dN is reduced by fretting wear. The short crack propagation rate is actually the movement rate of the crack tip. However, the actual crack length increment is not equal to the moving distance of the crack tip, and the reduction length of the crack caused by fretting wear needs to be considered. Fig. 13 shows the evolution of the contact surface profile and crack path versus loading cycles with different initial crack lengths. The patterning represents the crack tip, the solid line represents the contact surface profile. For the initial crack length 6 μm, the crack tip is almost static after 4000 cycles, but still about 0.6 μm crack length has been worn away, as shown in Fig. 13(a). It means that although the crack no longer propagates, the actual crack length will gradually decrease or even be eliminated due to the wear. For the initial crack length 10 μm, the contact surface profile gradually moves away from the initial position due to the material removal, and the crack tip moves along the depth direction with the increase of the number of loading cycles, as shown in Fig. 13(b). When the loading cycle number reaches 14000, the actual crack length exceeds 20 μm, and the specimen fails. However, it
(20)
The time histories of |KI /KIY |, |KII /KIIY |, |T/TY | and f for the case with an initial crack length of 10 μm during the first loading cycle are presented in Fig. 10. The results show that |KII /KIIY | is far less than |KI /KIY | and |T/TY | when the f is the largest, and the case with an initial crack length of 6 μm has the same result. Therefore, the value of fmax is mainly determined by |KI /KIY | and |T/TY |. As can be seen from Fig. 11, the ratios of |KI /KIY | to |T/TY | increase with the increase of crack length. When the crack length propagates from 6 μm to 20 μm, the ratio increases from 2.12 to 4.31. It means that the weight of KI in index fmax increases with the increase of crack length. As shown in Fig. 9(d), the variation trend of fmax is very similar with that of the KI because KI has the greatest weight in fmax . 558
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Fig. 13. Predicted evolution of the contact surface profile and the crack path with an initial crack length of (a) 6 μm and (b) 10 μm.
diction of short crack propagation life with considering wear is 50% higher than that without considering wear. Based on the present computations, it can be concluded that the fretting wear can significantly affect the short crack propagation rate, and this effect cannot be ignored. Under gross slip conditions, the life prediction of short crack propagation without considering wear will overestimate the risk of components.
Acknowledgments This study is based upon work supported by the National Natural Science Foundation of China (Grant No. 51675045) and Aeronautical Science Foundation of China (Grant No. 2016ZE55011). Fig. 14. Short crack propagation lives with and without considering fretting wear.
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