- Email: [email protected]
A 2D peridynamic model for fatigue crack initiation of railheads
International Journal of Fatigue 135 (2020) 105536
Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
A 2D peridynamic model for fatigue crack initiation of railheads Xiaochuan Ma a b
a,b,⁎
a
a
b
a
b
, Jinhui Xu , Linya Liu , Ping Wang , Qingsong Feng , Jingmang Xu
T
Engineering Research Center of Railway Environment Vibration and Noise, Ministry of Education, East China Jiaotong University, Nanchang 330013, China MOE Key Laboratory of High-speed Railway Engineering, Southwest Jiaotong University, Chengdu 610031, China
ARTICLE INFO
ABSTRACT
Keywords: Fatigue Peridynamic theory Railhead cracks Initiation Wheel-rail contact
Due to the use of PDEs (Partial Differential Equations), classical continuum mechanics is difficult to solve the problems of discontinuous deformation. Using integral equations, peridynamic theory can overcome this difficulty. A 2D method for predicting fatigue crack initiation of railheads was presented based on the peridynamic theory. First, the reasonable values of the model parameters were analyzed. Second, the accuracy of the method was verified through comparisons with the results of classical continuum mechanics and existing conclusions in the literature. Finally, the effects of wheel-rail rolling-sliding status, friction coefficient, and wheel load on fatigue crack initiation were analyzed and discussed.
1. Introduction When the wheel rolls on the rail, tremendous local stress is generated on the interface between the wheel and the rail, which is one of the main causes for the rail’s wear and fatigue [1]. In recent years, with the application of an optimized profile, anti-wear steels and lubrication, rail wear is gradually suppressed [2]; therefore, fatigue has overtaken wear [3] and gradually become the main damage that reduces the service life of rails. Cracks, squats, spalling, brittle fractures, etc. are all manifestations of rail fatigue damage. While fatigue damage can affect the stability and safety of vehicles operation, and even leads to derailment of vehicles [4]; the tremendous expenses required in repairing rail fatigue damage also significantly increases the cost of railway transportation [5]. In fact, the source of most fatigue damages is crack initiation. After crack initiation, the fatigue layer on the surface of railheads can be removed by grinding to stop the further development of cracks, which improves the service life of the rail [6]; however, when the crack enters the expansion period, the removal of a large amount of metal through grinding is adverse to the control of the maintenance cost and service life of the rails. Therefore, the optimum period for rail grinding is the period between the crack initiation and the initial propagation. As a result, research on the mechanism of crack initiation is extremely important and can provide a theoretical basis for the development of rail maintenance programs and intervals. For homogeneous steels without inclusions and defects, the re-
searchers proposed two predominant views to explain how rail fatigue cracks are initiated. The first view is the ratchet fatigue failure theory proposed by Kapoor [7]. He believes that the stress-strain state of the steel is in the ratchet region and irreversible plastic deformation occurs under load. Crack initiation occurs when the cumulative amount of plastic deformation reaches the toughness limit of the material. With this view, Franklin et al. established the “BRICK” model [1,8] to determine the crack initiation and wear of the steel by calculating the cumulative shear strain of each BRICK unit. Pun et al. [9] proposed to evaluate the ratcheting performance of the steel using the crack initiation life, and analyzed the influence of the wheel-rail contact interface parameters on the ratchet performance of three different steels. The second view was put forward by Ringsberg [5], Jiang, and Sehitoglu [10] et al., who believe that the stress-strain response of steels mainly occurs in the plastic stabilization stage; therefore, the low cycle fatigue criteria need to be applied. Under this view, the researchers used low-cycle fatigue criteria such as Dang Van [11], Fatemi-Socie [12,13], SWT [14], and Jiang-Sehitoglu [15] to evaluate the life cycle of crack initiation, and the direction of the crack was evaluated using the critical plane method [16]. In the above analysis method, the classical continuum mechanics theory is used to solve the stress-strain response of the rail steel, and the external criterion is used to judge the location and life of the crack initiation. Because of the use of PDEs to build mathematical frameworks, classical continuum mechanics can fail at discontinuities of the material (such as cracks and steel boundaries),
⁎ Corresponding author at: Engineering Research Center of Railway Environment Vibration and Noise, Ministry of Education, East China Jiaotong University, Nanchang 330013, China. E-mail address: [email protected] (X. Ma).
https://doi.org/10.1016/j.ijfatigue.2020.105536 Received 3 November 2019; Received in revised form 1 January 2020; Accepted 7 February 2020 Available online 08 February 2020 0142-1123/ © 2020 Elsevier Ltd. All rights reserved.
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
Nomenclature Notation R x, x′ δ X, X′ U, U′ ρ b ξ η F c h κ w s s0 ψ λ N ε smax, smin A1, m1
critical bond broken numbers nc l, h length/height of 2D rail peridynamic model height of railhead h1 x1, x2, x3 longitudinal/lateral/vertical direction of the railway line Δ particle spacing external load application field Rf Rc fictitious boundary region ur line support stiffness P wheel load I inertia moment of the rail cross section pn normal wheel-rail contact stress d virtual thickness of the 2D peridynamic model a, b half-axis length of the wheel-rail contact patch in the longitudinal/lateral Direction of the railway line b0 transverse equivalent length of the contact patch ν Poisson's ratio E elastic modulus A, B coefficients related to the radius of curvature of the wheelrail profiles m, n, θ parameters related to the elliptic integral R1, R3 radii of curvature of the wheel and rail in the Ox1x3 plane, respectively R2, R4 radii of curvature of the wheel and the rail in the Ox1x2 plane, respectively Δt time interval D fictitious diagonal density matrix f force density matrix which composed of internal and external forces c damping coefficient fx relative longitudinal creepages of the wheel-rail μ wheel-rail friction coefficient
specification a continuous medium object particles horizon in peridynamic theory position vectors of the particles before the object is deformed displacement vectors of the particles after the object is deformed mass density of the particle external force density of the particle relative position vector of particles before the object is deformed relative displacement vector of particles after the object is deformed pairwise response function between the particles bond constant in the peridynamic theory thickness of the 2D structures bulk modulus of the material a scalar function to indicate the damage of the bond bond stretch critical value of the bond stretch local damage of a particle remaining life of the bond number of wheel load cycles bond stretch change under a single cyclic loading maximum/minimum values of bond stretch during a single cyclic loading fitting parameters related to the cyclic strain curve of the rail steel
making it difficult to simulate the process of steel change from its previously sound condition to the occurrence of macroscopic cracks. Furthermore, using different external criteria will result in large differences in the results of the predictions made [13]. To overcome the difficulties of classical continuum mechanics in solving discontinuous problems, Dr. Silling of Sandia National Laboratories proposed the peridynamic theory based on a series of new concepts, such as bond, the pairwise response and horizon et al. [17]. The theory successfully avoids the failure of the mathematical framework at discontinuities by using the integral method to construct the equation of motion, and thus has significant advantages in solving the fracture problem. After two decades of development, peridynamic theory has been widely used in the study of composite fractures [18], rock-based steel failure [19], corrosion damage [20] and crack propagation model [21]. Based on the peridynamic theory, Freimanis et al. [22] studied the generation and development of rail squats. However, they failed to consider the superposition of rail bending and wheel-rail local contact deformations and the moving effects of wheel load in the study, which have a great impact on rail fatigue behavior [2,23]. Although the peridynamic theory was proposed to solve the discontinuity problem, it still falls within the scope of continuum mechanics and thus, may also be used to solve the continuity problem. Based on the peridynamic fatigue theory and considering the supporting effect of sleepers on the rail and the moving effect of the wheel load, this study proposed a 2D prediction model for the fatigue crack initiation of railheads. First, the parameters of the analysis model are reasonably valued. Second, the correctness of the model is verified through comparisons with the classical continuum mechanics theory and the calculation results of existing references. Finally, the influence of wheel-rail rolling-sliding status, wheel-rail friction coefficient and wheel load on the location and life of fatigue crack initiation of railheads are analyzed.
2. Peridynamic fatigue theory 2.1. Bond-based peridynamic theory The peridynamic theory is a continuum model based on non-local ideas [17]. As shown in Fig. 1, during the deformation of the continuous medium object R, there is an interaction between any particle x and the particle x′ in a limited range around it. Before the object is deformed, the coordinates of the particle x′ must satisfy the requirement of ||X′-X|| < δ, where δ is the horizon, X and X′ represent the position vectors of the two particles before the object is deformed, respectively, U and U′ represent the displacement vectors of the two particles after the deformation process of the object, respectively. According to Newton's second law of motion, the equation motion of particle x at any time can be expressed as:
(x ) U¨ (x , t ) =
F ( , , t )dH + b (x , t ) H
(1)
Fig. 1. Schematic diagram of particle interactions in peridynamic theory. 2
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
particle ψ is defined as:
(x , t ) = 1
(
wdVx )/( H
where ρ is the mass density of the particle x, and b is the external force density that the particle x is subjected to. ξ is the relative position vector of the particle x and the particle x′ before the object is deformed, and η is the relative displacement vector of the two particles during the deformation process of the object. The calculation formulas are:
X (t ) U (t )
(0) = 1 d / dN = A1 m1 = |smax smin|
(2)
In Eq. (1), F is the pairwise response function between the particles, which acts as the material constitutive relation in the classical continuum mechanics theory. In the bond-based peridynamic theory, it is assumed that the particle interactive force has a linear relationship with the bond stretch, and its expression is:
F = c(
( ) (3)
In the equation, c is the bond constant in the peridynamic theory. For isotropic steels, the bond constant can be determined by considering an infinite homogeneous body under isotropic expansion [24]. Eq. (3) gives a bond constant expression for 2D structures, where h is the thickness of the 2D structures and κ is the bulk modulus of the steel. 2.2. Fatigue damage theory In the theory of peridynamic fatigue damage [25], a scalar function w is usually used to indicate the damage of the bond. When the bond stretch exceeds the critical value, the bond between the two particles will be broken permanently, and the value of the function w changes from 1 to 0. Therefore, when considering the damage to the bond, the equation motion of the particle can be rewritten as:
(x ) U¨ (x , t ) = =
3. Prediction model for fatigue crack initiation
wF ( , , t )dH + b (x , t )
A 2D rail peridynamic model with length l and height h is constructed (Fig. 3). Under the Cartesian plane coordinate system Ox1x3 (x1 and x3 represent the longitudinal and vertical directions of the railway line, respectively), the rail domain R is dispersed into uniformly distributed particles. For evenly distributed particles, the particle spacing is Δ, and the horizon δ = 3.015Δ [24]. Select a layer of particles on the top surface of the rail as the external load application field Rf for applying the wheel load. The moving range of the wheel load is x1 ∈ [-l1, l1], and the value of l1 in this paper is 40 mm [27]. On the left and right
H
w= s=
1, 0,
s s0 s > s0
| + | | | | |
(6)
where λ represents the remaining lifespan of the bond, N is the number of wheel load cycles, and ε is the change in bond stretch under a single cyclic loading. smax and smin are the maximum and minimum values of bond stretch during a single cyclic loading, respectively. A1 and m1 are the fitting parameters related to the cyclic strain curve of the rail steel, and their values in this paper are 426.00 and 2.77 [22], respectively. If the bond is broken under a single cyclic loading, the corresponding ε can be regarded as the critical value of the bond stretch. According to formula (6), the critical value of the bond stretch of the rail steel in this paper is 0.112. To improve efficiency, the method of “critical bond broken numbers” can be used in the calculation process [26]. First, the stretch values of each bond are obtained after the wheel load moves from -l1 to l1, and the numbers of load cycles required to break each bond are calculated by formula (6). Then, the nc bonds with the smallest number of load cycles are broken in the peridynamic model, and the damage of other unbroken bonds can be reflected in the remaining lifespan. Finally, the next moving wheel load is applied, and the above loop is executed again. In this process, nc refers to the critical bond broken numbers, and the ideal case is nc = 1 when the calculation efficiency is not needed to be considered.
| + | || + )· | + | ||
c = 12 / h 3, 2D Structures
(5)
As shown in Fig. 2, when the local damage of the particle reaches 0.5, it is considered to be the initiation location of the macroscopic crack of the steel [24], which marks the end of the crack initiation phase. Following this, the crack near this particle will enter the propagation period. In general, the bond stretch in brittle materials can easily exceed the critical value. However, during deformation of ductile materials (such as the rail), it is difficult for the bond stretch to exceed the critical value directly, and cracks can only be generated under cyclic loads. Therefore, the concept of remaining lifespan is given to each bond, and the relationship between its initial value and the number of load cycles is:
Fig. 2. Schematic diagram of the relationship between particle local damage and macroscopic crack.
= X (t ) = U (t )
dVx ) H
(4)
where s represents bond stretch and s0 represents the critical value of the bond stretch. In addition, in order to describe the degree of damage of any particle, according to the function w, the local damage of a
Fig. 3. Schematic diagram of 2D rail peridynamic model. 3
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
Fig 4. Schematic diagram of normal and tangential contact between wheel and rail.
corrected to keep the bending deformation consistent. According to the formula (6) in the literature [28], the equivalent calculation formula of line support stiffness is:
P2D P3D
ur ,2D = ur ,3D · 3
4
I3D I2D
(7)
where ur represents the line support stiffness, P is the wheel load, I is the inertia moment of the rail cross section. The subscripts 2D and 3D represent the two-dimensional and three-dimensional rail structures, respectively. For the 2D rail structure of this study, the formula for calculating the wheel load and the inertia moment of the rail cross section is:
P2D = 2·
j i=0
pn
(
2i + 1 · 2
)·
·d
I2D = d ·h3/12
(8)
where pn is the normal wheel-rail contact stress, d is the virtual thickness of the 2D peridynamic model, j is the largest positive integer which satisfying (i + 1/2)Δ ≤ a, and a is the half-axis length of the wheel-rail contact patch in the longitudinal direction of the railway line. 3.1. Load and boundary conditions 3.1.1. Load conditions As shown in Fig. 4, the Hertzian contact theory is used to simulate the interaction between the wheel and the rail, and the normal and tangential contact forces are applied to the particles in the corresponding Rf. In the figure, b represents the half-axis length of the wheelrail contact patch in the lateral direction of the railway line, and x2 represents the lateral direction of the railway line. According to Carter's 2D rolling contact theory [29], the normal contact stress at any position within the wheel-rail contact patch can be expressed as: Fig. 5. Flow chart of the computer program.
pn (x ) =
2P ab0
1
x 2 /a2
b0 = 4b /3
borders along the rail, a fictitious boundary region Rc with length δ is constructed for applying the boundary conditions of the model. In order to study the fatigue crack initiation of railheads, this paper sets the railhead region as the failure zone, and the bonds in this area are allowed to be broken while the bonds in the no failure zone are not allowed to be broken. According to the size of the Chinese high-speed railway CHN60 rail, the height of railhead h1 is 48.5 mm. The supporting effect of the sleeper on the rail can be simplified to a continuously supported model [28]. The 2D rail structure and the actual 3D rail structure have large differences in bending deformation. Therefore, the line support stiffness in the peridynamic model must be
a = m· 3
2) P 3(1 2E (A + B )
b = n· 3
2) P 3(1 2E (A + B )
(9)
where b0 is the transverse equivalent length of the contact patch, ν is the Poisson's ratio of the wheel-rail material, E is the elastic modulus of the wheel-rail steel, A and B are the coefficients related to the radius of curvature of the wheel-rail profiles, m, n and θ are the parameters related to the elliptic integral which can be obtained by the table provided in the appendix. The values of A, B and θ can be calculated as: 4
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
Fig. 6. Effect of model sizes on displacement of particles (x1 = 0). (a) Effect of rail length on the results. (b) Effect of particle spacing on the results.
Fig. 7. Effect of the critical bond broken numbers on calculation accuracy and efficiency. (a) Effect of nc on the crack initiation life of the railhead. (b) Effect of nc on the calculation time required.
region on both sides of the constrained rail is 0, which is the boundary condition in the peridynamic model. 3.2. Numerical methods and processes In the peridynamic theory, the equation concerning the motion of a particle is in a dynamic form. Therefore, in solving static or quasi-static problems, ADR (Adaptive Dynamic Relaxation) method is usually used to solve the steady state of the transient response [31]. This method recalculates the damping coefficient at each time interval, which greatly improves the efficiency of numerical calculation. In this study, the central-difference explicit integration is used to calculate the particle velocity of the next time interval:
Fig. 8. Schematic diagram of finite element calculation model for rail deformation.
A+B=
1 2
B
1 2
A=
( (
1 R1
+
1 R1
cos
1 R2 1 R2
=
+
1 R3
+
1 R3
B A A+B
+
1 R4 1 R4
) )
(2
(10)
U
where at the position of wheel-rail contact point, R1 and R3 are the radii of curvature of the wheel and rail in the Ox1x3 plane, respectively, and R2 and R4 are the radii of curvature of the wheel and the rail in the Ox1x2 plane, respectively. When the wheel is in a pure sliding state, the tangential contact stress can be obtained by Coulomb's law of friction; when the wheel is in a frictionless state, the tangential contact stress is 0; when the wheel is in an adhesion-sliding state [30], the tangential contact stress can be solved through the Carter 2D rolling contact theory.
n + 1/2
=
c n t ) U n 1/2 + 2 tD 1f n (2 + c n t ) tD 1f 0 , 2
, n>0
n=0
(11) th
In this formula, n indicates the n iteration, Δt is the time interval, D is the fictitious diagonal density matrix, f is the force density matrix which composed of internal and external forces, and c is the damping coefficient. For the specific calculation method, refer to the literature [31]. The main calculation process of the peridynamic model for predicting fatigue crack initiation of railheads is as follows: Step 1: Enter all calculation parameters; Step 2: One quasi-static analysis will be done by moving the wheel load from x1 = −l1 to x1 = l1. According to the remaining lifespan of the bond and the formula (6), the numbers of wheel load cycles required to break each bond are calculated separately. Then, the nc bonds
3.1.2. Boundary conditions The actual rail structure can be considered to be of infinite length. Therefore, the displacement of the particle in the fictitious boundary 5
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
Fig. 9. Displacement variation of rail nodes/particles. (a) Calculation results of the classical continuum mechanical model; (b) Calculation results of the peridynamic model; (c) Calculation error of the peridynamic model according to the classical continuum mechanical model.
Fig. 10. Damage image of the rail, N = 3.89 × 106 cycles of wheel loading.
which having the fewest number of load cycles will be broken, and the numbers of wheel load cycles required to break the nc bonds are added to the total fatigue life; Step 3: The remaining life of other unbroken bonds will be reduced and updated according to formula (6); Step 4: The local damage of particles will be calculated by formula (5). If the maximum local damage of the particle is greater than or equal to 0.5, then stop the calculation and output the result; otherwise, skip to step 2 for a loop calculation. According to the above calculation process, this study uses the Fortran language to compile the calculation program, which can be used to predict the location and life of fatigue crack initiation of railheads. The flow chart of the computer program is shown in Fig. 5.
4. Analysis of the value selection of the model parameters 4.1. Model sizes analysis The larger the length of the rail model, the more it is able to eliminate the boundary effect on the calculation result; the smaller the spacing of the particles, the more accurate is the numerical solution. However, both the increase in the length of the model and the decrease in the spacing of the particles reduce the computational efficiency of the model. In this study, by analyzing the influence of rail length and particle spacing on the displacement of the particle, the reasonable model sizes are determined. The pure sliding state of the wheel was calculated as an example. The wheel and rail profiles are LMA and CHN60, respectively, which 6
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
Fig. 11. Damage images of the rail with different wheel-rail rolling-sliding status. (a) Frictionless state of the wheel, N = 6.58 × 106; (b) Adhesion-sliding state of the wheel, N = 5.75 × 106; (c) Pure sliding states of the wheel, N = 3.89 × 106.
than 3 mm, the calculation result of the particle displacement tends to be stable. Therefore, the reasonable model sizes are l = 0.9 m and Δ = 3 mm. 4.2. Critical bond broken numbers analysis Theoretically, the prediction results obtained when nc = 1 is the most accurate, but the calculation time required will be long, which seriously affects the calculation efficiency of the model. This study uses a wheel in pure sliding state as an example. Through the cyclic wheel loading, the effect of nc on the crack initiation life of railhead is shown in Fig. 7(a), and the calculation time required with different nc is shown in Fig. 7(b). It can be seen from Fig. 7 that when the critical bond broken numbers are quite large, the calculation results of the crack initiation life of railhead are scattered. When nc does not exceed 24, the prediction results are relatively stable. In addition, an increase of nc will increases the time required for the calculation. Therefore, for the overall effect on the calculation accuracy and efficiency, nc = 24 is a reasonable value for this model.
Fig.12. The variation of the maximum local damage of the particle vs. the number of wheel load cycles with different rolling-sliding states of wheel.
5. Model verification
are often used in China's high-speed railway; the wheel load is 60.8 kN (half the static axle weight of the CRH2 vehicle); the wheel-rail friction coefficient is 0.3; the elastic modulus and Poisson's ratio of the rail steel are 2.1 × 105 MPa and 1/3, respectively. When the center of the wheel load moves to the position of x1 = 0, the influence of the rail length on the displacement of particles (x1 = 0) is shown in Fig. 6(a), and the effect of particle spacing on the displacement of particles (x1 = 0) is shown in Fig. 6(b). The displacement of particles at x1 = 0 gradually increases as the location changes from bottom to the top of the rail. As shown in Fig. 6, when the rail length is greater than 0.9 m and the particle spacing is less
The correctness of the peridynamic model established in this study is verified by comparing the results of quasi-static deformation and crack initiation prediction of railheads with the classical continuum mechanics model and the conclusions from the references. 5.1. Quasi-static deformation of rail Based on the classical continuum mechanics theory, the finite element analysis model of the rail deformation is established. As shown in Fig. 8, the parameters in the model are consistent with the peridynamic 7
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
Fig. 13. Damage images of the rail under different wheel-rail friction coefficients. (a) μ = 0.1, N = 6.28 × 106; (b) μ = 0.2, N = 5.23 × 106; (c) μ = 0.3, N = 3.89 × 106; (d) μ = 0.4, N = 2.19 × 106; (c) μ = 0.5, N = 1.00 × 106.
model. In the finite element analysis model, the PLANE42 unit is used to simulate a 2D rail with a certain amount of thickness, while the COMBIN14 unit is used to simulate the supporting effect of the sleeper on the rail. The wheel load is applied to the node on the surface of the rail. In addition, the freedoms in x1 and x3 directions of the nodes on both sides of the rail are restricted. Supposing that there is no damage to the rail (that is, when solving the continuous problem), when the wheel load moves to the location of x1 = 0 in a pure sliding state, the displacement variation pattern of the rail nodes or the particle is calculated as shown in Fig. 9(a) and (b), where (a) is the calculation results of the classical continuum mechanical model, and (b) is the calculation results of the peridynamic
model. The variation pattern of the rail displacement obtained by the two models is very consistent. In order to further evaluate the degree of agreement between the two models, the displacements of the nodes or particles on the surface of the rail are extracted separately. Based on the classical continuum mechanical model, the calculation error of the peridynamic model is shown in Fig. 9(c). As shown in Fig. 9(c), the variation of the surface nodes/particles displacement of the rail obtained by the two models is very much consistent, and the calculation error of the peridynamic model is within 5%. The displacement of rail node/particle at the center of the wheelrail contact is the largest. The maximum displacements of the rail node/ particle obtained by the classical continuum mechanicals model and 8
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
respectively. During the transition from frictionless to pure sliding state, the fatigue crack initiation life of railhead gradually decreases, indicating that the existence of the tangential wheel-rail contact stress will significantly reduce the fatigue crack initiation life of railhead. 6.2. Wheel-rail friction coefficient
Based on the prediction model of railhead fatigue crack proposed in this study, the damage image of the rail is obtained when the maximum local damage of the particle reaches 0.5 (Fig. 10). As shown in Fig. 10, the location of the fatigue crack initiation of railhead is about 3 mm below the surface of the rail, which is consistent with the conclusion that “the crack initiation point is 2.3 mm below the rail surface” described in the literature [2]. In addition, the fatigue crack initiation life of railhead is 3.89 × 106 cycles of wheel loading, and the prediction result in the literature [2] is 1.30 × 106 cyclic load. The main reason for the difference in prediction results is that the static wheel load is used in this study, and the dynamic impact load is used in the literature [2]. Therefore, the predicted life of fatigue crack initiation in this paper is larger than the results in the literature [2]. From this point of view, the prediction results of this paper are also reasonable.
With the change in conditions such as climate and lubrication, the wheel-rail friction coefficient changes greatly. For example, in the case of complete dryness, the friction coefficient can be as high as 0.5; under wet conditions, the friction coefficient of is about 0.3; under oil-lubricated conditions, the friction coefficient can be reduced to 0.1. Based on the pure sliding state of the wheel, the damage images of the rail under different wheel-rail friction coefficients are simulated and shown in Fig. 13, where μ represents the wheel-rail friction coefficient. As shown in Fig. 13, when the friction coefficient is 0.1, 0.2 and 0.3, the initiation location of fatigue crack is 3 mm below the rail surface, and when the friction coefficient is increased to 0.4 and 0.5, the initiation location of fatigue crack is on the rail surface. It can be seen that with the increase in wheel-rail friction coefficient, the initiation location of fatigue crack gradually shifts from the inside of the rail to the surface of the rail. Under different wheel-rail friction coefficients, the variation pattern of the maximum local damage of the particle vs. the number of load cycles is shown in Fig. 14. It can be seen from Fig. 14 that the number of load cycles required for the fatigue crack initiation of railhead is reduced from 6.28 × 106 to 1.00 × 106 times during the process of increasing the friction coefficient from 0.1 to 0.5, indicating that the increase of the friction coefficient will significantly reduce the fatigue crack initiation life of railhead. Based on the above conclusions, by reducing the wheel-rail friction coefficient, the initiation life of the fatigue crack can be effectively improved. However, when the friction coefficient is small, the initiation location of the fatigue crack is found inside the rail. At this time, the fatigue crack will further grow because it is difficult to maintain by grinding and other measures even if it can be detected. In contrast, if the friction coefficient is increased, although the life of the fatigue crack initiation is reduced, the crack initiation position will be located on the rail surface. At this time, the growth of fatigue cracks can be eliminated or suppressed by natural wear or manual grinding [3]. From this point of view, improving the friction coefficient is also an effective way to improve the overall service life of the rail. Of course, the value of the friction coefficient should not be too large, because the actual situation that the rail cannot go through excessive wear needs to be considered.
6. Numerical simulations, results and discussions
6.3. Wheel load
6.1. Wheel-rail rolling-sliding status
Due to the irregularity of the track, when the wheel rolls through the rail, dynamic loads are bound to occur between the wheel and rail, which is quite different from the static load. In order to eliminate the influence of tangential contact stress, the wheel is set in a frictionless state, and the damage images of the rail under different wheel loads are calculated and shown in Fig. 15. As shown in Fig. 15, with the frictionless state of wheel, the initiation location of fatigue crack is always located about 6 mm below the surface of the rail regardless of the change of the wheel loads, indicating that the effect of the normal contact stress on the initiation location of fatigue crack is negligible. According to the analysis and conclusions in sections 6.1 and 6.2, the authors believe that the ratio of tangential stress to normal stress determines the initiation location of fatigue crack, that is, the sliding and adhesive friction coefficient are significant factors which affecting the crack initiation location of railhead. The variation pattern of the maximum local damage of the particle vs. the number of load cycles under different wheel loads is shown in Fig. 16. It can be seen from Fig. 16 that when the wheel load is increased
Fig. 14. The variation of the maximum local damage of the particle vs. the number of load cycles under different wheel-rail friction coefficient.
peridynamic model are 0.238 mm and 0.231 mm, respectively, and the calculation error of peridynamic model is about 3%. The correctness of the peridynamic model in solving the quasi-static deformation problem of rail is verified by the above results. 5.2. Crack initiation of railheads
Affected by the changes of track conditions, the wheel and rail will be relatively rolling and sliding. The damage images of the rail in different wheel rolling-sliding states are shown in Fig. 11. In this figure, (a), (b), and (c) represent the frictionless, adhesion-sliding and pure sliding states of the wheel. The relative longitudinal creepages of the wheel-rail in adhesion-sliding state is fx = −1.0 × 10−3. It can be seen from Fig. 11 that during the transition from frictionless to pure sliding state, the location of the fatigue crack initiation of railhead is transferred from 6 mm below the rail surface to 3 mm below the rail surface, indicating that the existence of tangential contact stress will make the location of the crack initiation closer to the rail surface. In the different rolling-sliding states of wheel, the variation pattern of the maximum local damage of the particle vs. the number of wheel load cycles is shown in Fig. 12. As shown in Fig. 12, when the wheel is in frictionless, adhesionsliding and pure sliding states, the corresponding crack initiation life of railhead is 6.58 × 106, 5.75 × 106 and 3.89 × 106 load cycles, 9
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
Fig. 15. Damage images of the rail under different wheel load conditions. (a) P = 60.8 kN, N = 6.58 × 106; (b) P = 91.2 kN, N = 4.71 × 106; (c) P = 121.6 kN, N = 3.35 × 106; (d) P = 152.0 kN, N = 3.11 × 106; (e) P = 182.4 kN, N = 2.95 × 106.
from 60.8 kN to 182.4 kN, the number of load cycles required for the fatigue crack initiation of railhead is reduced from 6.58 × 106 to 2.95 × 106 times, indicating an increase of the wheel load, will significantly reduce the fatigue crack initiation life of railhead.
calculation accuracy and efficiency. It is reasonable to take the rail length of 0.9 m, the particle spacing of 3 mm, and the critical bond broken number of 24. Second, when calculating the deformation of a non-damaged rail (solving continuous problem), the correctness of the model was verified by comparing with the results of the classical continuum mechanics; when predicting the fatigue crack initiation of railhead (solving discontinuity problem), the correctness of the model was verified by comparing with the conclusions of the existing literature. Finally, the effects of wheel-rail rolling-sliding states, wheel-rail friction coefficients and wheel loads on the fatigue crack initiation of railhead were explored. Based on the results and discussions in Section 6, the authors argue
7. Conclusions Taking into account the supporting effect of the sleepers on the rail and the moving effect of the wheel load, a 2D prediction model for the fatigue crack initiation of railheads is constructed based on the bondbased peridynamic theory and its fatigue damage theory. First, the reasonable values of the model parameters were obtained according to 10
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
racy and applicability of this approach. After comparison, the results of this paper match well with the general findings and work in the field, supporting the credibility of the new approach. In addition, the main purpose of this study is to introduce a new method to solve railhead fatigue problems. Some simple parametric studies are only for verification and demonstration purposes. In future work, this method can be used to analyze more complex problems for new results. In order to improve and develop the peridynamic model of this article, a lot of research works are needed in the future. (1) There will be some deviations because the conditions of the peridynamic model cannot be completely consistent with the numerical experiments in the literature. Therefore, the authors plan to design a twin-disc testing in future work. This individual test can be used to extend and verify the work in this paper. (2) The railhead has a finite thickness, therefore the 2D model fails to consider the effect of transversal action on fatigue crack initiation. In the future work, the authors will propose a 3D peridynamic model to predict fatigue crack initiation of railhead. (3) In this study, the supporting effect of sleepers on the rail is simplified as a continuously bearing model. In fact, the sleeper is a concentrated support with a certain width to the rail. The question of how to consider this factor in the 2D model, and analyze the effect of supporting parameters of the sleeper on the fatigue crack initiation of railheads is a subject to be studied in the future work.
Fig.16. The variation pattern of the maximum local damage of the particle vs. the number of load cycles with different wheel loads.
that (1) The fatigue life is reduced as the wheel-rail friction coefficient increases. However, when the friction coefficient is small, the crack is initiated inside the rail, which significantly increases the difficulty of maintenance. Therefore, appropriately increasing the wheel-rail friction coefficient is beneficial to eliminate or inhibit the further growth of fatigue cracks and increase the overall service life of the rail. (2) The sliding and adhesive friction coefficient is one of the main factors affecting railhead crack initiation location. In addition, in a state of frictionless wheel-rail contact, the initiation location remains unchanged, regardless of the wheel load, which demonstrates that the effect of normal contact stress on initiation location is negligible. Although the results shown in the paper have been known for a long time in the field, the authors consider them to be meaningful. The use of peridynamic theory to analyze fatigue crack initiation in railheads is a completely new approach, therefore it is necessary to verify the accu-
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors acknowledge the support of the National Natural Science Foundation of China (51968025, 51808221), Natural Science Foundation of Jiangxi Province (20192BAB216035, 20181BAB216029) and the open research fund of MOE Key Laboratory of High-speed Railway Engineering, Southwest Jiaotong University (2018).
Appendix Table A1. Table A1 The values for θ, m and n. θ/°
m
n
θ/°
m
n
θ/°
m
n
0.5 1.0 1.5 2.0 3.0 4.0 6.0 8.0 10
61.40 36.89 27.48 22.26 16.65 13.31 9.790 7.860 6.604
0.102 0.131 0.152 0.169 0.196 0.219 0.255 0.285 0.311
18 20 25 30 35 40 45 50 55
4.156 3.813 3.152 2.731 2.397 2.136 1.926 1.754 1.611
0.394 0.412 0.456 0.493 0.530 0.567 0.604 0.641 0.678
60 65 70 75 80 85 90 95 100
1.486 1.378 1.284 1.202 1.128 1.061 1.000 0.944 0.893
0.717 0.759 0.802 0.846 0.893 0.944 1.000 1.061 1.128
[5] Ringsberg JW, Loo-Morrey M, Josefson BL, Kapoor A, Beynon JH. Prediction of fatigue crack initiation for rolling contact fatigue. Int J Fatigue 2000;22(3):205–15. [6] Zhi S, Li J, Zarembski AM. Predictive modelling of the rail grinding process using a distributed cutting grain approach. Proc Instit Mech Eng Part F – J Rail Rapid Transit 2015;230(6):1540–60. [7] Kapoor A. A re-evaluation of the life to rupture of ductile metals by cyclic plastic strain. Fatigue Fract Eng Mater Struct 1994;17(2):201–19. [8] Franklin FJ, Chung T, Kapoor A. Ratcheting and fatigue-led wear in rail-wheel contact. Fatigue Fract Eng Mater Struct 2003;26(10):949–55. [9] Pun CL, Kan QH, Mutton PJ, Kang GZ, Yan WY. An efficient computational approach to evaluate the ratcheting performance of rail steels under cyclic rolling
References [1] Franklin FJ, Widiyarta I, Kapoor A. Computer simulation of wear and rolling contact fatigue. Wear 2001;251(1–12):949–55. [2] El-sayed HM, Lotfy M, Zohny HNE, Riad HS. Prediction of fatigue crack initiation life in railheads using finite element analysis. Ain Shams Eng J 2018;9(4):2329–42. [3] Zhou Y, Han YB, Mu DS, Zhang CC, Huang XW. Prediction of the coexistence of rail head check initiation and wear growth. Int J Fatigue 2018;112:289–300. [4] Zerbst U, Lundén R, Edel KO, Smith RA. Introduction to the damage tolerance behaviour of railway rails - a review. Eng Fract Mech 2009;76(17):2563–601.
11
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al. contact in service. Int J Mech Sci 2015;101:214–26. [10] Jiang YY, Sehitoglu H. A model for rolling contact failure. Wear 1999;224(1):38–49. [11] Reis T, Lima ED, Bertelli F, dos Santos AA. Progression of plastic strain on heavyhaul railway rail under random pure rolling and its influence on crack initiation. Adv Eng Softw 2018;124:10–21. [12] Kiani M, Fry GT. Fatigue analysis of railway wheel using a multiaxial strain-based critical-plane index. Fatigue Fract Eng Mater Struct 2018;41(2):412–24. [13] Reis L, Li B, de Freitas M. A multiaxial fatigue approach to Rolling Contact Fatigue in railways. Int J Fatigue 2014;67:191–202. [14] Pun CL, Kan Q, Mutton PJ, Kang G, Yan W. A single parameter to evaluate stress state in rail head for rolling contact fatigue analysis. Fatigue Fract Eng Mater Struct 2014;37(8):909–19. [15] Andersson R, Ahlström J, El Kabo, Larsson F, Ekberg A. Numerical investigation of crack initiation in rails and wheels affected by martensite spots. Int J Fatigue 2018;114:238–51. [16] Wang JX, Xu YD, Lian SL, Wang LY. Probabilistic prediction model for initiation of RCF cracks in heavy-haul railway. Int J Fatigue 2011;33:212–6. [17] Silling SA. Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 2000;48(1):175–209. [18] Zhou W, Liu D, Liu N. Analyzing dynamic fracture process in fiber-reinforced composite materials with a peridynamic model. Eng Fract Mech 2017;178:60–76. [19] Ai DH, Zhao YC, Wang QF, Li CW. Experimental and numerical investigation of crack propagation and dynamic properties of rock in SHPB indirect tension test. Int J Impact Eng 2019;126:135–46. [20] Jafarzadeha S, Chenb Z, Zhaoa J, Bobarua F. Pitting, lacy covers, and pit merger in
stainless steel: 3D peridynamic models. Corros Sci 2019;150:17–31. [21] Jung J, Seok J. Mixed-mode fatigue crack growth analysis using peridynamic approach. Int J Fatigue 2017;103:591–603. [22] Freimanis A, Kaewunruen S. Peridynamic analysis of rail squats. Appl SciencesBasel 2018;8(11):1–18. [23] Beheshti A, Khonsari MM. On the prediction of fatigue crack initiation in rolling/ sliding contacts with provision for loading sequence effect. Tribol Int 2011;44:1620–8. [24] Silling SA, Askari E. A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 2005;83(17):1526–35. [25] Silling SA, Askari A. Peridynamic model for fatigue cracking Tech. rep. no. SAND2014-18590. Albuquerque (NM, United States): Sandia National Laboratories (SNL-NM); 2014. [26] Zhao SL, Yu Y, Xu W. Ordinary state-based peridynamics method for fatigue multicrack propagation. J Harbin Instit Technol 2019;51(4):19–25. (in Chinses). [27] Reisa T, de Abreu Limaa E, Bertelli F, dos Santos Juniora AA. Progression of plastic strain on heavy-haul railway rail under random pure rolling and its influence on crack initiation. Adv Eng Softw 2018;124:10–21. [28] Ma XC, Wang P, Xu JM, Chen R. Effect of the vertical relative motion of stock/ switch rails on wheel–rail contact mechanics in switch panel of railway turnout. Adv Mech Eng 2018;10(7):1–13. [29] Carter FW. On the action of locomotive driving wheel. Proc R Soc Lond 1926:151–7. [30] Sichani M, Enblom R, Berg M. An alternative to FASTSIM for tangential solution of the wheel-rail contact. Veh Syst Dyn 2016;54(6):748–64. [31] Kilic B, Madenci E. An adaptive dynamic relaxation method for quasi-static simulations using the peridynamic theory. Theor Appl Fract Mech 2010;53(3):194–204.
12
Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
A 2D peridynamic model for fatigue crack initiation of railheads Xiaochuan Ma a b
a,b,⁎
a
a
b
a
b
, Jinhui Xu , Linya Liu , Ping Wang , Qingsong Feng , Jingmang Xu
T
Engineering Research Center of Railway Environment Vibration and Noise, Ministry of Education, East China Jiaotong University, Nanchang 330013, China MOE Key Laboratory of High-speed Railway Engineering, Southwest Jiaotong University, Chengdu 610031, China
ARTICLE INFO
ABSTRACT
Keywords: Fatigue Peridynamic theory Railhead cracks Initiation Wheel-rail contact
Due to the use of PDEs (Partial Differential Equations), classical continuum mechanics is difficult to solve the problems of discontinuous deformation. Using integral equations, peridynamic theory can overcome this difficulty. A 2D method for predicting fatigue crack initiation of railheads was presented based on the peridynamic theory. First, the reasonable values of the model parameters were analyzed. Second, the accuracy of the method was verified through comparisons with the results of classical continuum mechanics and existing conclusions in the literature. Finally, the effects of wheel-rail rolling-sliding status, friction coefficient, and wheel load on fatigue crack initiation were analyzed and discussed.
1. Introduction When the wheel rolls on the rail, tremendous local stress is generated on the interface between the wheel and the rail, which is one of the main causes for the rail’s wear and fatigue [1]. In recent years, with the application of an optimized profile, anti-wear steels and lubrication, rail wear is gradually suppressed [2]; therefore, fatigue has overtaken wear [3] and gradually become the main damage that reduces the service life of rails. Cracks, squats, spalling, brittle fractures, etc. are all manifestations of rail fatigue damage. While fatigue damage can affect the stability and safety of vehicles operation, and even leads to derailment of vehicles [4]; the tremendous expenses required in repairing rail fatigue damage also significantly increases the cost of railway transportation [5]. In fact, the source of most fatigue damages is crack initiation. After crack initiation, the fatigue layer on the surface of railheads can be removed by grinding to stop the further development of cracks, which improves the service life of the rail [6]; however, when the crack enters the expansion period, the removal of a large amount of metal through grinding is adverse to the control of the maintenance cost and service life of the rails. Therefore, the optimum period for rail grinding is the period between the crack initiation and the initial propagation. As a result, research on the mechanism of crack initiation is extremely important and can provide a theoretical basis for the development of rail maintenance programs and intervals. For homogeneous steels without inclusions and defects, the re-
searchers proposed two predominant views to explain how rail fatigue cracks are initiated. The first view is the ratchet fatigue failure theory proposed by Kapoor [7]. He believes that the stress-strain state of the steel is in the ratchet region and irreversible plastic deformation occurs under load. Crack initiation occurs when the cumulative amount of plastic deformation reaches the toughness limit of the material. With this view, Franklin et al. established the “BRICK” model [1,8] to determine the crack initiation and wear of the steel by calculating the cumulative shear strain of each BRICK unit. Pun et al. [9] proposed to evaluate the ratcheting performance of the steel using the crack initiation life, and analyzed the influence of the wheel-rail contact interface parameters on the ratchet performance of three different steels. The second view was put forward by Ringsberg [5], Jiang, and Sehitoglu [10] et al., who believe that the stress-strain response of steels mainly occurs in the plastic stabilization stage; therefore, the low cycle fatigue criteria need to be applied. Under this view, the researchers used low-cycle fatigue criteria such as Dang Van [11], Fatemi-Socie [12,13], SWT [14], and Jiang-Sehitoglu [15] to evaluate the life cycle of crack initiation, and the direction of the crack was evaluated using the critical plane method [16]. In the above analysis method, the classical continuum mechanics theory is used to solve the stress-strain response of the rail steel, and the external criterion is used to judge the location and life of the crack initiation. Because of the use of PDEs to build mathematical frameworks, classical continuum mechanics can fail at discontinuities of the material (such as cracks and steel boundaries),
⁎ Corresponding author at: Engineering Research Center of Railway Environment Vibration and Noise, Ministry of Education, East China Jiaotong University, Nanchang 330013, China. E-mail address: [email protected] (X. Ma).
https://doi.org/10.1016/j.ijfatigue.2020.105536 Received 3 November 2019; Received in revised form 1 January 2020; Accepted 7 February 2020 Available online 08 February 2020 0142-1123/ © 2020 Elsevier Ltd. All rights reserved.
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
Nomenclature Notation R x, x′ δ X, X′ U, U′ ρ b ξ η F c h κ w s s0 ψ λ N ε smax, smin A1, m1
critical bond broken numbers nc l, h length/height of 2D rail peridynamic model height of railhead h1 x1, x2, x3 longitudinal/lateral/vertical direction of the railway line Δ particle spacing external load application field Rf Rc fictitious boundary region ur line support stiffness P wheel load I inertia moment of the rail cross section pn normal wheel-rail contact stress d virtual thickness of the 2D peridynamic model a, b half-axis length of the wheel-rail contact patch in the longitudinal/lateral Direction of the railway line b0 transverse equivalent length of the contact patch ν Poisson's ratio E elastic modulus A, B coefficients related to the radius of curvature of the wheelrail profiles m, n, θ parameters related to the elliptic integral R1, R3 radii of curvature of the wheel and rail in the Ox1x3 plane, respectively R2, R4 radii of curvature of the wheel and the rail in the Ox1x2 plane, respectively Δt time interval D fictitious diagonal density matrix f force density matrix which composed of internal and external forces c damping coefficient fx relative longitudinal creepages of the wheel-rail μ wheel-rail friction coefficient
specification a continuous medium object particles horizon in peridynamic theory position vectors of the particles before the object is deformed displacement vectors of the particles after the object is deformed mass density of the particle external force density of the particle relative position vector of particles before the object is deformed relative displacement vector of particles after the object is deformed pairwise response function between the particles bond constant in the peridynamic theory thickness of the 2D structures bulk modulus of the material a scalar function to indicate the damage of the bond bond stretch critical value of the bond stretch local damage of a particle remaining life of the bond number of wheel load cycles bond stretch change under a single cyclic loading maximum/minimum values of bond stretch during a single cyclic loading fitting parameters related to the cyclic strain curve of the rail steel
making it difficult to simulate the process of steel change from its previously sound condition to the occurrence of macroscopic cracks. Furthermore, using different external criteria will result in large differences in the results of the predictions made [13]. To overcome the difficulties of classical continuum mechanics in solving discontinuous problems, Dr. Silling of Sandia National Laboratories proposed the peridynamic theory based on a series of new concepts, such as bond, the pairwise response and horizon et al. [17]. The theory successfully avoids the failure of the mathematical framework at discontinuities by using the integral method to construct the equation of motion, and thus has significant advantages in solving the fracture problem. After two decades of development, peridynamic theory has been widely used in the study of composite fractures [18], rock-based steel failure [19], corrosion damage [20] and crack propagation model [21]. Based on the peridynamic theory, Freimanis et al. [22] studied the generation and development of rail squats. However, they failed to consider the superposition of rail bending and wheel-rail local contact deformations and the moving effects of wheel load in the study, which have a great impact on rail fatigue behavior [2,23]. Although the peridynamic theory was proposed to solve the discontinuity problem, it still falls within the scope of continuum mechanics and thus, may also be used to solve the continuity problem. Based on the peridynamic fatigue theory and considering the supporting effect of sleepers on the rail and the moving effect of the wheel load, this study proposed a 2D prediction model for the fatigue crack initiation of railheads. First, the parameters of the analysis model are reasonably valued. Second, the correctness of the model is verified through comparisons with the classical continuum mechanics theory and the calculation results of existing references. Finally, the influence of wheel-rail rolling-sliding status, wheel-rail friction coefficient and wheel load on the location and life of fatigue crack initiation of railheads are analyzed.
2. Peridynamic fatigue theory 2.1. Bond-based peridynamic theory The peridynamic theory is a continuum model based on non-local ideas [17]. As shown in Fig. 1, during the deformation of the continuous medium object R, there is an interaction between any particle x and the particle x′ in a limited range around it. Before the object is deformed, the coordinates of the particle x′ must satisfy the requirement of ||X′-X|| < δ, where δ is the horizon, X and X′ represent the position vectors of the two particles before the object is deformed, respectively, U and U′ represent the displacement vectors of the two particles after the deformation process of the object, respectively. According to Newton's second law of motion, the equation motion of particle x at any time can be expressed as:
(x ) U¨ (x , t ) =
F ( , , t )dH + b (x , t ) H
(1)
Fig. 1. Schematic diagram of particle interactions in peridynamic theory. 2
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
particle ψ is defined as:
(x , t ) = 1
(
wdVx )/( H
where ρ is the mass density of the particle x, and b is the external force density that the particle x is subjected to. ξ is the relative position vector of the particle x and the particle x′ before the object is deformed, and η is the relative displacement vector of the two particles during the deformation process of the object. The calculation formulas are:
X (t ) U (t )
(0) = 1 d / dN = A1 m1 = |smax smin|
(2)
In Eq. (1), F is the pairwise response function between the particles, which acts as the material constitutive relation in the classical continuum mechanics theory. In the bond-based peridynamic theory, it is assumed that the particle interactive force has a linear relationship with the bond stretch, and its expression is:
F = c(
( ) (3)
In the equation, c is the bond constant in the peridynamic theory. For isotropic steels, the bond constant can be determined by considering an infinite homogeneous body under isotropic expansion [24]. Eq. (3) gives a bond constant expression for 2D structures, where h is the thickness of the 2D structures and κ is the bulk modulus of the steel. 2.2. Fatigue damage theory In the theory of peridynamic fatigue damage [25], a scalar function w is usually used to indicate the damage of the bond. When the bond stretch exceeds the critical value, the bond between the two particles will be broken permanently, and the value of the function w changes from 1 to 0. Therefore, when considering the damage to the bond, the equation motion of the particle can be rewritten as:
(x ) U¨ (x , t ) = =
3. Prediction model for fatigue crack initiation
wF ( , , t )dH + b (x , t )
A 2D rail peridynamic model with length l and height h is constructed (Fig. 3). Under the Cartesian plane coordinate system Ox1x3 (x1 and x3 represent the longitudinal and vertical directions of the railway line, respectively), the rail domain R is dispersed into uniformly distributed particles. For evenly distributed particles, the particle spacing is Δ, and the horizon δ = 3.015Δ [24]. Select a layer of particles on the top surface of the rail as the external load application field Rf for applying the wheel load. The moving range of the wheel load is x1 ∈ [-l1, l1], and the value of l1 in this paper is 40 mm [27]. On the left and right
H
w= s=
1, 0,
s s0 s > s0
| + | | | | |
(6)
where λ represents the remaining lifespan of the bond, N is the number of wheel load cycles, and ε is the change in bond stretch under a single cyclic loading. smax and smin are the maximum and minimum values of bond stretch during a single cyclic loading, respectively. A1 and m1 are the fitting parameters related to the cyclic strain curve of the rail steel, and their values in this paper are 426.00 and 2.77 [22], respectively. If the bond is broken under a single cyclic loading, the corresponding ε can be regarded as the critical value of the bond stretch. According to formula (6), the critical value of the bond stretch of the rail steel in this paper is 0.112. To improve efficiency, the method of “critical bond broken numbers” can be used in the calculation process [26]. First, the stretch values of each bond are obtained after the wheel load moves from -l1 to l1, and the numbers of load cycles required to break each bond are calculated by formula (6). Then, the nc bonds with the smallest number of load cycles are broken in the peridynamic model, and the damage of other unbroken bonds can be reflected in the remaining lifespan. Finally, the next moving wheel load is applied, and the above loop is executed again. In this process, nc refers to the critical bond broken numbers, and the ideal case is nc = 1 when the calculation efficiency is not needed to be considered.
| + | || + )· | + | ||
c = 12 / h 3, 2D Structures
(5)
As shown in Fig. 2, when the local damage of the particle reaches 0.5, it is considered to be the initiation location of the macroscopic crack of the steel [24], which marks the end of the crack initiation phase. Following this, the crack near this particle will enter the propagation period. In general, the bond stretch in brittle materials can easily exceed the critical value. However, during deformation of ductile materials (such as the rail), it is difficult for the bond stretch to exceed the critical value directly, and cracks can only be generated under cyclic loads. Therefore, the concept of remaining lifespan is given to each bond, and the relationship between its initial value and the number of load cycles is:
Fig. 2. Schematic diagram of the relationship between particle local damage and macroscopic crack.
= X (t ) = U (t )
dVx ) H
(4)
where s represents bond stretch and s0 represents the critical value of the bond stretch. In addition, in order to describe the degree of damage of any particle, according to the function w, the local damage of a
Fig. 3. Schematic diagram of 2D rail peridynamic model. 3
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
Fig 4. Schematic diagram of normal and tangential contact between wheel and rail.
corrected to keep the bending deformation consistent. According to the formula (6) in the literature [28], the equivalent calculation formula of line support stiffness is:
P2D P3D
ur ,2D = ur ,3D · 3
4
I3D I2D
(7)
where ur represents the line support stiffness, P is the wheel load, I is the inertia moment of the rail cross section. The subscripts 2D and 3D represent the two-dimensional and three-dimensional rail structures, respectively. For the 2D rail structure of this study, the formula for calculating the wheel load and the inertia moment of the rail cross section is:
P2D = 2·
j i=0
pn
(
2i + 1 · 2
)·
·d
I2D = d ·h3/12
(8)
where pn is the normal wheel-rail contact stress, d is the virtual thickness of the 2D peridynamic model, j is the largest positive integer which satisfying (i + 1/2)Δ ≤ a, and a is the half-axis length of the wheel-rail contact patch in the longitudinal direction of the railway line. 3.1. Load and boundary conditions 3.1.1. Load conditions As shown in Fig. 4, the Hertzian contact theory is used to simulate the interaction between the wheel and the rail, and the normal and tangential contact forces are applied to the particles in the corresponding Rf. In the figure, b represents the half-axis length of the wheelrail contact patch in the lateral direction of the railway line, and x2 represents the lateral direction of the railway line. According to Carter's 2D rolling contact theory [29], the normal contact stress at any position within the wheel-rail contact patch can be expressed as: Fig. 5. Flow chart of the computer program.
pn (x ) =
2P ab0
1
x 2 /a2
b0 = 4b /3
borders along the rail, a fictitious boundary region Rc with length δ is constructed for applying the boundary conditions of the model. In order to study the fatigue crack initiation of railheads, this paper sets the railhead region as the failure zone, and the bonds in this area are allowed to be broken while the bonds in the no failure zone are not allowed to be broken. According to the size of the Chinese high-speed railway CHN60 rail, the height of railhead h1 is 48.5 mm. The supporting effect of the sleeper on the rail can be simplified to a continuously supported model [28]. The 2D rail structure and the actual 3D rail structure have large differences in bending deformation. Therefore, the line support stiffness in the peridynamic model must be
a = m· 3
2) P 3(1 2E (A + B )
b = n· 3
2) P 3(1 2E (A + B )
(9)
where b0 is the transverse equivalent length of the contact patch, ν is the Poisson's ratio of the wheel-rail material, E is the elastic modulus of the wheel-rail steel, A and B are the coefficients related to the radius of curvature of the wheel-rail profiles, m, n and θ are the parameters related to the elliptic integral which can be obtained by the table provided in the appendix. The values of A, B and θ can be calculated as: 4
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
Fig. 6. Effect of model sizes on displacement of particles (x1 = 0). (a) Effect of rail length on the results. (b) Effect of particle spacing on the results.
Fig. 7. Effect of the critical bond broken numbers on calculation accuracy and efficiency. (a) Effect of nc on the crack initiation life of the railhead. (b) Effect of nc on the calculation time required.
region on both sides of the constrained rail is 0, which is the boundary condition in the peridynamic model. 3.2. Numerical methods and processes In the peridynamic theory, the equation concerning the motion of a particle is in a dynamic form. Therefore, in solving static or quasi-static problems, ADR (Adaptive Dynamic Relaxation) method is usually used to solve the steady state of the transient response [31]. This method recalculates the damping coefficient at each time interval, which greatly improves the efficiency of numerical calculation. In this study, the central-difference explicit integration is used to calculate the particle velocity of the next time interval:
Fig. 8. Schematic diagram of finite element calculation model for rail deformation.
A+B=
1 2
B
1 2
A=
( (
1 R1
+
1 R1
cos
1 R2 1 R2
=
+
1 R3
+
1 R3
B A A+B
+
1 R4 1 R4
) )
(2
(10)
U
where at the position of wheel-rail contact point, R1 and R3 are the radii of curvature of the wheel and rail in the Ox1x3 plane, respectively, and R2 and R4 are the radii of curvature of the wheel and the rail in the Ox1x2 plane, respectively. When the wheel is in a pure sliding state, the tangential contact stress can be obtained by Coulomb's law of friction; when the wheel is in a frictionless state, the tangential contact stress is 0; when the wheel is in an adhesion-sliding state [30], the tangential contact stress can be solved through the Carter 2D rolling contact theory.
n + 1/2
=
c n t ) U n 1/2 + 2 tD 1f n (2 + c n t ) tD 1f 0 , 2
, n>0
n=0
(11) th
In this formula, n indicates the n iteration, Δt is the time interval, D is the fictitious diagonal density matrix, f is the force density matrix which composed of internal and external forces, and c is the damping coefficient. For the specific calculation method, refer to the literature [31]. The main calculation process of the peridynamic model for predicting fatigue crack initiation of railheads is as follows: Step 1: Enter all calculation parameters; Step 2: One quasi-static analysis will be done by moving the wheel load from x1 = −l1 to x1 = l1. According to the remaining lifespan of the bond and the formula (6), the numbers of wheel load cycles required to break each bond are calculated separately. Then, the nc bonds
3.1.2. Boundary conditions The actual rail structure can be considered to be of infinite length. Therefore, the displacement of the particle in the fictitious boundary 5
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
Fig. 9. Displacement variation of rail nodes/particles. (a) Calculation results of the classical continuum mechanical model; (b) Calculation results of the peridynamic model; (c) Calculation error of the peridynamic model according to the classical continuum mechanical model.
Fig. 10. Damage image of the rail, N = 3.89 × 106 cycles of wheel loading.
which having the fewest number of load cycles will be broken, and the numbers of wheel load cycles required to break the nc bonds are added to the total fatigue life; Step 3: The remaining life of other unbroken bonds will be reduced and updated according to formula (6); Step 4: The local damage of particles will be calculated by formula (5). If the maximum local damage of the particle is greater than or equal to 0.5, then stop the calculation and output the result; otherwise, skip to step 2 for a loop calculation. According to the above calculation process, this study uses the Fortran language to compile the calculation program, which can be used to predict the location and life of fatigue crack initiation of railheads. The flow chart of the computer program is shown in Fig. 5.
4. Analysis of the value selection of the model parameters 4.1. Model sizes analysis The larger the length of the rail model, the more it is able to eliminate the boundary effect on the calculation result; the smaller the spacing of the particles, the more accurate is the numerical solution. However, both the increase in the length of the model and the decrease in the spacing of the particles reduce the computational efficiency of the model. In this study, by analyzing the influence of rail length and particle spacing on the displacement of the particle, the reasonable model sizes are determined. The pure sliding state of the wheel was calculated as an example. The wheel and rail profiles are LMA and CHN60, respectively, which 6
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
Fig. 11. Damage images of the rail with different wheel-rail rolling-sliding status. (a) Frictionless state of the wheel, N = 6.58 × 106; (b) Adhesion-sliding state of the wheel, N = 5.75 × 106; (c) Pure sliding states of the wheel, N = 3.89 × 106.
than 3 mm, the calculation result of the particle displacement tends to be stable. Therefore, the reasonable model sizes are l = 0.9 m and Δ = 3 mm. 4.2. Critical bond broken numbers analysis Theoretically, the prediction results obtained when nc = 1 is the most accurate, but the calculation time required will be long, which seriously affects the calculation efficiency of the model. This study uses a wheel in pure sliding state as an example. Through the cyclic wheel loading, the effect of nc on the crack initiation life of railhead is shown in Fig. 7(a), and the calculation time required with different nc is shown in Fig. 7(b). It can be seen from Fig. 7 that when the critical bond broken numbers are quite large, the calculation results of the crack initiation life of railhead are scattered. When nc does not exceed 24, the prediction results are relatively stable. In addition, an increase of nc will increases the time required for the calculation. Therefore, for the overall effect on the calculation accuracy and efficiency, nc = 24 is a reasonable value for this model.
Fig.12. The variation of the maximum local damage of the particle vs. the number of wheel load cycles with different rolling-sliding states of wheel.
5. Model verification
are often used in China's high-speed railway; the wheel load is 60.8 kN (half the static axle weight of the CRH2 vehicle); the wheel-rail friction coefficient is 0.3; the elastic modulus and Poisson's ratio of the rail steel are 2.1 × 105 MPa and 1/3, respectively. When the center of the wheel load moves to the position of x1 = 0, the influence of the rail length on the displacement of particles (x1 = 0) is shown in Fig. 6(a), and the effect of particle spacing on the displacement of particles (x1 = 0) is shown in Fig. 6(b). The displacement of particles at x1 = 0 gradually increases as the location changes from bottom to the top of the rail. As shown in Fig. 6, when the rail length is greater than 0.9 m and the particle spacing is less
The correctness of the peridynamic model established in this study is verified by comparing the results of quasi-static deformation and crack initiation prediction of railheads with the classical continuum mechanics model and the conclusions from the references. 5.1. Quasi-static deformation of rail Based on the classical continuum mechanics theory, the finite element analysis model of the rail deformation is established. As shown in Fig. 8, the parameters in the model are consistent with the peridynamic 7
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
Fig. 13. Damage images of the rail under different wheel-rail friction coefficients. (a) μ = 0.1, N = 6.28 × 106; (b) μ = 0.2, N = 5.23 × 106; (c) μ = 0.3, N = 3.89 × 106; (d) μ = 0.4, N = 2.19 × 106; (c) μ = 0.5, N = 1.00 × 106.
model. In the finite element analysis model, the PLANE42 unit is used to simulate a 2D rail with a certain amount of thickness, while the COMBIN14 unit is used to simulate the supporting effect of the sleeper on the rail. The wheel load is applied to the node on the surface of the rail. In addition, the freedoms in x1 and x3 directions of the nodes on both sides of the rail are restricted. Supposing that there is no damage to the rail (that is, when solving the continuous problem), when the wheel load moves to the location of x1 = 0 in a pure sliding state, the displacement variation pattern of the rail nodes or the particle is calculated as shown in Fig. 9(a) and (b), where (a) is the calculation results of the classical continuum mechanical model, and (b) is the calculation results of the peridynamic
model. The variation pattern of the rail displacement obtained by the two models is very consistent. In order to further evaluate the degree of agreement between the two models, the displacements of the nodes or particles on the surface of the rail are extracted separately. Based on the classical continuum mechanical model, the calculation error of the peridynamic model is shown in Fig. 9(c). As shown in Fig. 9(c), the variation of the surface nodes/particles displacement of the rail obtained by the two models is very much consistent, and the calculation error of the peridynamic model is within 5%. The displacement of rail node/particle at the center of the wheelrail contact is the largest. The maximum displacements of the rail node/ particle obtained by the classical continuum mechanicals model and 8
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
respectively. During the transition from frictionless to pure sliding state, the fatigue crack initiation life of railhead gradually decreases, indicating that the existence of the tangential wheel-rail contact stress will significantly reduce the fatigue crack initiation life of railhead. 6.2. Wheel-rail friction coefficient
Based on the prediction model of railhead fatigue crack proposed in this study, the damage image of the rail is obtained when the maximum local damage of the particle reaches 0.5 (Fig. 10). As shown in Fig. 10, the location of the fatigue crack initiation of railhead is about 3 mm below the surface of the rail, which is consistent with the conclusion that “the crack initiation point is 2.3 mm below the rail surface” described in the literature [2]. In addition, the fatigue crack initiation life of railhead is 3.89 × 106 cycles of wheel loading, and the prediction result in the literature [2] is 1.30 × 106 cyclic load. The main reason for the difference in prediction results is that the static wheel load is used in this study, and the dynamic impact load is used in the literature [2]. Therefore, the predicted life of fatigue crack initiation in this paper is larger than the results in the literature [2]. From this point of view, the prediction results of this paper are also reasonable.
With the change in conditions such as climate and lubrication, the wheel-rail friction coefficient changes greatly. For example, in the case of complete dryness, the friction coefficient can be as high as 0.5; under wet conditions, the friction coefficient of is about 0.3; under oil-lubricated conditions, the friction coefficient can be reduced to 0.1. Based on the pure sliding state of the wheel, the damage images of the rail under different wheel-rail friction coefficients are simulated and shown in Fig. 13, where μ represents the wheel-rail friction coefficient. As shown in Fig. 13, when the friction coefficient is 0.1, 0.2 and 0.3, the initiation location of fatigue crack is 3 mm below the rail surface, and when the friction coefficient is increased to 0.4 and 0.5, the initiation location of fatigue crack is on the rail surface. It can be seen that with the increase in wheel-rail friction coefficient, the initiation location of fatigue crack gradually shifts from the inside of the rail to the surface of the rail. Under different wheel-rail friction coefficients, the variation pattern of the maximum local damage of the particle vs. the number of load cycles is shown in Fig. 14. It can be seen from Fig. 14 that the number of load cycles required for the fatigue crack initiation of railhead is reduced from 6.28 × 106 to 1.00 × 106 times during the process of increasing the friction coefficient from 0.1 to 0.5, indicating that the increase of the friction coefficient will significantly reduce the fatigue crack initiation life of railhead. Based on the above conclusions, by reducing the wheel-rail friction coefficient, the initiation life of the fatigue crack can be effectively improved. However, when the friction coefficient is small, the initiation location of the fatigue crack is found inside the rail. At this time, the fatigue crack will further grow because it is difficult to maintain by grinding and other measures even if it can be detected. In contrast, if the friction coefficient is increased, although the life of the fatigue crack initiation is reduced, the crack initiation position will be located on the rail surface. At this time, the growth of fatigue cracks can be eliminated or suppressed by natural wear or manual grinding [3]. From this point of view, improving the friction coefficient is also an effective way to improve the overall service life of the rail. Of course, the value of the friction coefficient should not be too large, because the actual situation that the rail cannot go through excessive wear needs to be considered.
6. Numerical simulations, results and discussions
6.3. Wheel load
6.1. Wheel-rail rolling-sliding status
Due to the irregularity of the track, when the wheel rolls through the rail, dynamic loads are bound to occur between the wheel and rail, which is quite different from the static load. In order to eliminate the influence of tangential contact stress, the wheel is set in a frictionless state, and the damage images of the rail under different wheel loads are calculated and shown in Fig. 15. As shown in Fig. 15, with the frictionless state of wheel, the initiation location of fatigue crack is always located about 6 mm below the surface of the rail regardless of the change of the wheel loads, indicating that the effect of the normal contact stress on the initiation location of fatigue crack is negligible. According to the analysis and conclusions in sections 6.1 and 6.2, the authors believe that the ratio of tangential stress to normal stress determines the initiation location of fatigue crack, that is, the sliding and adhesive friction coefficient are significant factors which affecting the crack initiation location of railhead. The variation pattern of the maximum local damage of the particle vs. the number of load cycles under different wheel loads is shown in Fig. 16. It can be seen from Fig. 16 that when the wheel load is increased
Fig. 14. The variation of the maximum local damage of the particle vs. the number of load cycles under different wheel-rail friction coefficient.
peridynamic model are 0.238 mm and 0.231 mm, respectively, and the calculation error of peridynamic model is about 3%. The correctness of the peridynamic model in solving the quasi-static deformation problem of rail is verified by the above results. 5.2. Crack initiation of railheads
Affected by the changes of track conditions, the wheel and rail will be relatively rolling and sliding. The damage images of the rail in different wheel rolling-sliding states are shown in Fig. 11. In this figure, (a), (b), and (c) represent the frictionless, adhesion-sliding and pure sliding states of the wheel. The relative longitudinal creepages of the wheel-rail in adhesion-sliding state is fx = −1.0 × 10−3. It can be seen from Fig. 11 that during the transition from frictionless to pure sliding state, the location of the fatigue crack initiation of railhead is transferred from 6 mm below the rail surface to 3 mm below the rail surface, indicating that the existence of tangential contact stress will make the location of the crack initiation closer to the rail surface. In the different rolling-sliding states of wheel, the variation pattern of the maximum local damage of the particle vs. the number of wheel load cycles is shown in Fig. 12. As shown in Fig. 12, when the wheel is in frictionless, adhesionsliding and pure sliding states, the corresponding crack initiation life of railhead is 6.58 × 106, 5.75 × 106 and 3.89 × 106 load cycles, 9
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
Fig. 15. Damage images of the rail under different wheel load conditions. (a) P = 60.8 kN, N = 6.58 × 106; (b) P = 91.2 kN, N = 4.71 × 106; (c) P = 121.6 kN, N = 3.35 × 106; (d) P = 152.0 kN, N = 3.11 × 106; (e) P = 182.4 kN, N = 2.95 × 106.
from 60.8 kN to 182.4 kN, the number of load cycles required for the fatigue crack initiation of railhead is reduced from 6.58 × 106 to 2.95 × 106 times, indicating an increase of the wheel load, will significantly reduce the fatigue crack initiation life of railhead.
calculation accuracy and efficiency. It is reasonable to take the rail length of 0.9 m, the particle spacing of 3 mm, and the critical bond broken number of 24. Second, when calculating the deformation of a non-damaged rail (solving continuous problem), the correctness of the model was verified by comparing with the results of the classical continuum mechanics; when predicting the fatigue crack initiation of railhead (solving discontinuity problem), the correctness of the model was verified by comparing with the conclusions of the existing literature. Finally, the effects of wheel-rail rolling-sliding states, wheel-rail friction coefficients and wheel loads on the fatigue crack initiation of railhead were explored. Based on the results and discussions in Section 6, the authors argue
7. Conclusions Taking into account the supporting effect of the sleepers on the rail and the moving effect of the wheel load, a 2D prediction model for the fatigue crack initiation of railheads is constructed based on the bondbased peridynamic theory and its fatigue damage theory. First, the reasonable values of the model parameters were obtained according to 10
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al.
racy and applicability of this approach. After comparison, the results of this paper match well with the general findings and work in the field, supporting the credibility of the new approach. In addition, the main purpose of this study is to introduce a new method to solve railhead fatigue problems. Some simple parametric studies are only for verification and demonstration purposes. In future work, this method can be used to analyze more complex problems for new results. In order to improve and develop the peridynamic model of this article, a lot of research works are needed in the future. (1) There will be some deviations because the conditions of the peridynamic model cannot be completely consistent with the numerical experiments in the literature. Therefore, the authors plan to design a twin-disc testing in future work. This individual test can be used to extend and verify the work in this paper. (2) The railhead has a finite thickness, therefore the 2D model fails to consider the effect of transversal action on fatigue crack initiation. In the future work, the authors will propose a 3D peridynamic model to predict fatigue crack initiation of railhead. (3) In this study, the supporting effect of sleepers on the rail is simplified as a continuously bearing model. In fact, the sleeper is a concentrated support with a certain width to the rail. The question of how to consider this factor in the 2D model, and analyze the effect of supporting parameters of the sleeper on the fatigue crack initiation of railheads is a subject to be studied in the future work.
Fig.16. The variation pattern of the maximum local damage of the particle vs. the number of load cycles with different wheel loads.
that (1) The fatigue life is reduced as the wheel-rail friction coefficient increases. However, when the friction coefficient is small, the crack is initiated inside the rail, which significantly increases the difficulty of maintenance. Therefore, appropriately increasing the wheel-rail friction coefficient is beneficial to eliminate or inhibit the further growth of fatigue cracks and increase the overall service life of the rail. (2) The sliding and adhesive friction coefficient is one of the main factors affecting railhead crack initiation location. In addition, in a state of frictionless wheel-rail contact, the initiation location remains unchanged, regardless of the wheel load, which demonstrates that the effect of normal contact stress on initiation location is negligible. Although the results shown in the paper have been known for a long time in the field, the authors consider them to be meaningful. The use of peridynamic theory to analyze fatigue crack initiation in railheads is a completely new approach, therefore it is necessary to verify the accu-
Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors acknowledge the support of the National Natural Science Foundation of China (51968025, 51808221), Natural Science Foundation of Jiangxi Province (20192BAB216035, 20181BAB216029) and the open research fund of MOE Key Laboratory of High-speed Railway Engineering, Southwest Jiaotong University (2018).
Appendix Table A1. Table A1 The values for θ, m and n. θ/°
m
n
θ/°
m
n
θ/°
m
n
0.5 1.0 1.5 2.0 3.0 4.0 6.0 8.0 10
61.40 36.89 27.48 22.26 16.65 13.31 9.790 7.860 6.604
0.102 0.131 0.152 0.169 0.196 0.219 0.255 0.285 0.311
18 20 25 30 35 40 45 50 55
4.156 3.813 3.152 2.731 2.397 2.136 1.926 1.754 1.611
0.394 0.412 0.456 0.493 0.530 0.567 0.604 0.641 0.678
60 65 70 75 80 85 90 95 100
1.486 1.378 1.284 1.202 1.128 1.061 1.000 0.944 0.893
0.717 0.759 0.802 0.846 0.893 0.944 1.000 1.061 1.128
[5] Ringsberg JW, Loo-Morrey M, Josefson BL, Kapoor A, Beynon JH. Prediction of fatigue crack initiation for rolling contact fatigue. Int J Fatigue 2000;22(3):205–15. [6] Zhi S, Li J, Zarembski AM. Predictive modelling of the rail grinding process using a distributed cutting grain approach. Proc Instit Mech Eng Part F – J Rail Rapid Transit 2015;230(6):1540–60. [7] Kapoor A. A re-evaluation of the life to rupture of ductile metals by cyclic plastic strain. Fatigue Fract Eng Mater Struct 1994;17(2):201–19. [8] Franklin FJ, Chung T, Kapoor A. Ratcheting and fatigue-led wear in rail-wheel contact. Fatigue Fract Eng Mater Struct 2003;26(10):949–55. [9] Pun CL, Kan QH, Mutton PJ, Kang GZ, Yan WY. An efficient computational approach to evaluate the ratcheting performance of rail steels under cyclic rolling
References [1] Franklin FJ, Widiyarta I, Kapoor A. Computer simulation of wear and rolling contact fatigue. Wear 2001;251(1–12):949–55. [2] El-sayed HM, Lotfy M, Zohny HNE, Riad HS. Prediction of fatigue crack initiation life in railheads using finite element analysis. Ain Shams Eng J 2018;9(4):2329–42. [3] Zhou Y, Han YB, Mu DS, Zhang CC, Huang XW. Prediction of the coexistence of rail head check initiation and wear growth. Int J Fatigue 2018;112:289–300. [4] Zerbst U, Lundén R, Edel KO, Smith RA. Introduction to the damage tolerance behaviour of railway rails - a review. Eng Fract Mech 2009;76(17):2563–601.
11
International Journal of Fatigue 135 (2020) 105536
X. Ma, et al. contact in service. Int J Mech Sci 2015;101:214–26. [10] Jiang YY, Sehitoglu H. A model for rolling contact failure. Wear 1999;224(1):38–49. [11] Reis T, Lima ED, Bertelli F, dos Santos AA. Progression of plastic strain on heavyhaul railway rail under random pure rolling and its influence on crack initiation. Adv Eng Softw 2018;124:10–21. [12] Kiani M, Fry GT. Fatigue analysis of railway wheel using a multiaxial strain-based critical-plane index. Fatigue Fract Eng Mater Struct 2018;41(2):412–24. [13] Reis L, Li B, de Freitas M. A multiaxial fatigue approach to Rolling Contact Fatigue in railways. Int J Fatigue 2014;67:191–202. [14] Pun CL, Kan Q, Mutton PJ, Kang G, Yan W. A single parameter to evaluate stress state in rail head for rolling contact fatigue analysis. Fatigue Fract Eng Mater Struct 2014;37(8):909–19. [15] Andersson R, Ahlström J, El Kabo, Larsson F, Ekberg A. Numerical investigation of crack initiation in rails and wheels affected by martensite spots. Int J Fatigue 2018;114:238–51. [16] Wang JX, Xu YD, Lian SL, Wang LY. Probabilistic prediction model for initiation of RCF cracks in heavy-haul railway. Int J Fatigue 2011;33:212–6. [17] Silling SA. Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids 2000;48(1):175–209. [18] Zhou W, Liu D, Liu N. Analyzing dynamic fracture process in fiber-reinforced composite materials with a peridynamic model. Eng Fract Mech 2017;178:60–76. [19] Ai DH, Zhao YC, Wang QF, Li CW. Experimental and numerical investigation of crack propagation and dynamic properties of rock in SHPB indirect tension test. Int J Impact Eng 2019;126:135–46. [20] Jafarzadeha S, Chenb Z, Zhaoa J, Bobarua F. Pitting, lacy covers, and pit merger in
stainless steel: 3D peridynamic models. Corros Sci 2019;150:17–31. [21] Jung J, Seok J. Mixed-mode fatigue crack growth analysis using peridynamic approach. Int J Fatigue 2017;103:591–603. [22] Freimanis A, Kaewunruen S. Peridynamic analysis of rail squats. Appl SciencesBasel 2018;8(11):1–18. [23] Beheshti A, Khonsari MM. On the prediction of fatigue crack initiation in rolling/ sliding contacts with provision for loading sequence effect. Tribol Int 2011;44:1620–8. [24] Silling SA, Askari E. A meshfree method based on the peridynamic model of solid mechanics. Comput Struct 2005;83(17):1526–35. [25] Silling SA, Askari A. Peridynamic model for fatigue cracking Tech. rep. no. SAND2014-18590. Albuquerque (NM, United States): Sandia National Laboratories (SNL-NM); 2014. [26] Zhao SL, Yu Y, Xu W. Ordinary state-based peridynamics method for fatigue multicrack propagation. J Harbin Instit Technol 2019;51(4):19–25. (in Chinses). [27] Reisa T, de Abreu Limaa E, Bertelli F, dos Santos Juniora AA. Progression of plastic strain on heavy-haul railway rail under random pure rolling and its influence on crack initiation. Adv Eng Softw 2018;124:10–21. [28] Ma XC, Wang P, Xu JM, Chen R. Effect of the vertical relative motion of stock/ switch rails on wheel–rail contact mechanics in switch panel of railway turnout. Adv Mech Eng 2018;10(7):1–13. [29] Carter FW. On the action of locomotive driving wheel. Proc R Soc Lond 1926:151–7. [30] Sichani M, Enblom R, Berg M. An alternative to FASTSIM for tangential solution of the wheel-rail contact. Veh Syst Dyn 2016;54(6):748–64. [31] Kilic B, Madenci E. An adaptive dynamic relaxation method for quasi-static simulations using the peridynamic theory. Theor Appl Fract Mech 2010;53(3):194–204.
12