Numerical calculation of crack driving force using the configurational force concept for elastic

Abstract

One of the main reasons for mechanical failures in railway structures is attributed to the propagation of defects and cracks. Exact prediction of crack growth in rails and wheels may lead to solutions to reduce damages, improve efficiency and facilitate maintenance. In this paper, the driving force of surface cracks on the rail under contact forces caused by wheel movement on the rail is investigated. The contact problem of the rails and wheels is simulated by a 2D model in finite element software and the elastic–plastic material model is considered for the rail. In order to calculate the crack driving force, the configurational force method is used to accurately predict the crack behavior and rail damage in plastic dominated zones. The parametric results of load, slip, size, crack orientation and the influence of interaction of cracks are studied for the surface cracks in rail and crack growth path and driving forces are estimated for inclined and vertical cracks. It is shown that the inclined cracks are more prone to propagation than vertical cracks under the same contact loading of rails and wheels.

Keywords

Contact forces rail damage configurational force elastic–plastic material model slip

Nomenclature

Configurational stress

Length increment in contour

Configurational force vector

Scalar configurational force

Deformation gradient

g _i ^I

Configurational force component in node I by integrating configurational force in element e

Identity tensor

Scalar J-integral

J-integral vector

N ^I

Shape function according to node I

N _.X ^I

Matrix gradient

Normal unit vector

Radius

Radius of the wheel

1st Piola-Kirchhoff stress tensor

$\dot{S}$

Slip ratio

Cauchy stress tensor

u _i

Component of the displacement vector

u ^I

Displacement vector in node I

u _{.

x

_j}

Displacement gradient

v _tip

Crack tip velocity vector

Linear velocity

V _e

The volume of element e

𝛤_r

Contour around the crack tip with radius of r

𝜃

Angle

𝜙

Free Helmholtz energy

𝜓_tip

Energy dissipation due to moving of crack tip

𝜔

Angular velocity

1. Introduction

Through the progression of transportation, efficient and high-speed railways have become one of the most important topics to investigate. For enhancing the railway industry, important effective factors in design, safety, and cost of tracks and machinery should be examined and reliable plans must be presented. For an optimum design, safe life of the rails and wheels should be assessed under different high loading conditions. Damage and failure of these parts are mostly caused by cracks. Cracks in rails and wheels initiate or propagate as a result of fatigue loading, high mechanical loading, contact pressure, inappropriate environment, thermal stresses, manufacturing, welding process’ defect and so on. These factors should be examined carefully.

Calculating crack driving force and predicting the crack progression in order to estimate the safe life of important components has been implemented widely in the railway industry. The stress intensity factor can demonstrate how a crack tends to grow and could be used as a fracture criterion for linear materials. The evaluation of elastic intensity factor and studying various effects along with predicting crack growth or initiation has been developed using experimental setups and numerical methods [11,21–23].

The concept of crack tip opening displacement (CTOD) can also be used as a fracture criterion. Thermal cracking and rolling fatigue crack in the wheel using a 3D model for elastic–plastic materials has been investigated [3], where the calculation of the crack driving force has been performed using crack tip displacement and crack tip opening method by considering three-point displacement around the cracks.

For elastic–plastic materials, the J-integral method could provide more accurate results for characterizing crack behavior, and it has been utilized by researchers in numerous occasions for evaluating cracks in the railway industry [14,20].

Recently, the configurational force method was used to compute the crack driving force. For example, crack growth in rail under fatigue loading condition has been assessed using the configurational force method [2]. The effects of friction coefficient, initial crack angle, and size in a 2D rail model subjected to a moving load were investigated. The propagation of cracks in surface layers of the rail caused by excessive plastic deformation has been evaluated using the concept of configurational forces [18]. The configurational force was calculated using a weight function formulation and crack growth regarding various assumed fracture thresholds was evaluated. Estimating crack driving force using configurational forces has been carried out for cracks in rails under rolling fatigue contact loading and the influence of friction coefficients between the rails and wheels, angle of pre-existing cracks, crack spacing, and the number of cracks in a rails containing a network crack have been examined [32].

The stress and strain state in a twin disk test and a real scale rig experiment has been studied and the results have been compared to achieve a scaling factor for crack patch size and depth of plastic deformation between two aforementioned tests [16]. Also, calculating crack driving force and predicting the angle of crack propagation have been investigated by a 2D model and the configurational force concept was used in this work. This paper focusses on the wear effects. The calculation of the crack driving force using configurational force has been restricted to one loading condition and a comparison between the different situations was lacking. In another research [15], the configurational force has been evaluated under sliding/rolling contact of the rail wheel and by changing effective parameters, and the influences of these changes on vertical crack growth behaviour have been studied. In [15], the results of the crack driving force using the configurational force concept and J-integral for elastic material have been compared for verification purposes. Propagation of shear crack in rails has been analysed with a 2D model in [5]. In that work, a mesh network of triangle elements have been utilized around the crack tip and the crack growth has been predicted by the configurational force concept during the tenth loading cycle for the inclined crack.

The configurational force concept is able to predict the crack driving force more accurately than other methods. Although some researchers in the railway industry implemented this method to analyse various types of railway cracks under different loading conditions, the effects of critical parameters on inclined rail cracks, which are extremely common in the railway industry and are also responsible for the failure of rails and structures, have not been precisely studied. Here, these types of cracks and mentioned issues are investigated.

In this study, a summary of the configurational force theory and its advantages over classic J-integral are presented. Crack driving force using the configurational force concept has been evaluated for rail cracks under the wheel and rail contact forces. A 2D simulation is carried out to achieve and improve previous studies on this topic. The elastic–plastic material model has been chosen to represent rail material around the crack and a parametric study has been performed. A finite element simulation and numerical post-processing calculation of configurational forces are explained in detail and the results of the crack driving force are presented and discussed. The configurational force for inclined cracks in rails and the influence of some noticeable factors such as crack orientation, slip and interactions of cracks have been assessed while wheels move over the rail. In addition, the crack’s tendency to propagate and the growth path has been estimated.

2. Governing equations

Industrial parts fail as a result of loading, time and environmental factors such as temperature and corrosion [31]. Researchers have tried to present better criteria for designing industrial structures, which would also take more influence parameters into account, and to obtain the results closer to the actual materials behavior.

Linear elastic fracture mechanic was invented as a solution to estimate pre-existing crack growth in materials with linear elastic manner. Since not all the cases are restricted to the linear elastic condition, J-integral was provided to characterize crack behaviour in materials that are not linear elastic. The J-integral theory is based on the assumption of the deformation theory of plasticity and using nonlinear elastic material instead of elastic–plastic behaviour. In other words, in this approach, elastic–plastic materials are treated as non-linear elastic [30]. In the loading situation, the behavior of these two mentioned cases is identical. But when unloading happens, the behaviors of nonlinear elastic and elastic–plastic materials are entirely different. Consequently, J-integral is able to provide accurate results in the loading state for elastic–plastic material but not for unloading. Non-proportional loading may occur while unloading happens either because the whole body is externally unloaded or because of the formation of voids or crack extension [1,13,24]. Therefore, if a considerable crack extension occurs or a cyclical loading affects the material, the deformation theory of plasticity and J-integral describes no physical meaning [13]. The main advantage of considering the deformation theory of plasticity is the capability to achieve a path-independent approach, but if irreversible plastic deformation occurs, no fracture parameter could be defined by utilizing the deformation theory of plasticity and J-integral [30]. Hence, researchers have attempted to present a theory that adapts to the non-proportional loading conditions. The configurational force for any consecutive parts without any presumptions has been obtained in [28,29]. Configurational forces are responsible for kinematic changes in references such as boundary growth or crack propagation due to crack tip displacement [30]. The explicit equation for bulk configurational stress could be calculated as [6,8]: $\begin{eqnarray}\displaystyle \boldsymbol{C}={\phi}\boldsymbol{I}-\boldsymbol{F}^{T}\boldsymbol{S} & & \displaystyle\end{eqnarray}$ (1) where 𝜙 is free Helmholtz energy and in this case, it is equal to strain energy density, I is the identity tensor, F ^T is the transposed deformation gradient matrix, and S is the 1st Piola-Kirchhoff stress tensor.

The divergence of configurational stress is non-zero in defects. The configurational force in defects can be calculated by: $\begin{eqnarray}\displaystyle \boldsymbol{f}=-{\nabla}\cdot \boldsymbol{C}=-{\nabla}\cdot ({\phi}\boldsymbol{I}-\boldsymbol{F}^{T}\boldsymbol{S}) & & \displaystyle\end{eqnarray}$ (2) where f is a vector that represents the magnitude and direction of the configurational force.

The configurational force in the crack tip, assuming that no thermal heat and inertia forces exist, could be found using [13]: $\begin{eqnarray}\displaystyle \boldsymbol{f}_{tip}=-\lim _{r\rightarrow 0}\int _{{\Gamma}_{r}}({\phi}\boldsymbol{I}-\boldsymbol{F}^{T}\boldsymbol{S})mdl & & \displaystyle\end{eqnarray}$ (3) where 𝛤_r is the arbitrary contour around the crack tip, m is its normal unit vector and dl is the increment of that contour. By evoking the second law of thermodynamic and deriving dissipation due to the propagation of crack tip and plastic deformation (𝜓_tip = (− f _tip) ⋅ v _tip ≥ 0), it has been observed that (− f _tip) can be considered as the new crack driving force in the crack tip [13], so: $\begin{eqnarray}\displaystyle \boldsymbol{J}_{tip}=-\boldsymbol{f}_{tip}. & & \displaystyle\end{eqnarray}$ (4) The scalar J-integral can be calculated by the product of J-integral vector and unit vector of the crack growth direction: $\begin{eqnarray}\displaystyle j_{tip}=e\cdot (-\boldsymbol{f}_{tip})=e\cdot \boldsymbol{J}_{tip}=e\cdot \lim _{r\rightarrow 0}\int _{{\Gamma}_{r}}({\phi}\boldsymbol{I}-\boldsymbol{F}^{T}\boldsymbol{S})mdl & & \displaystyle\end{eqnarray}$ (5) As it is presented, the crack driving force can be calculated using the configurational force concept using the previous equations. This new crack driving force is not restricted to the proportional loading condition and can be utilized in more general loading situations such as incremental plasticity and cyclic loadings. For more details about deriving the configurational force and its applications, see [13,28–30].

3. Simulation

In order to evaluate the behavior of a rail with surface cracks, a 2D model (approximately similar to the model used in [15]) using ABAQUS software has been developed and the stresses and displacements around the crack tip have been calculated. With a post-processing method using MATLAB software, the crack driving force was computed by applying the configurational force. In the following section, a summary of the finite element simulation and numerical implementations are outlined.

3.1. Finite element model

As is illustrated in Fig. 1, the model includes a rectangular part to represent the rail, a sector of the ring (in order to save computation time, a full ring has been avoided) represents the wheel tread and a rigid analytical part represents the wheel hub which is responsible for the movement and loading of the wheel. An initial crack located in the middle of the upper surface of the rail has been employed to examine the crack driving force. The dimensions used in this model are presented in Table 1 [15]. R260 steel was chosen as the rail material, and an elastic–plastic material with cyclic hardening [17] was employed to represent the rail material around the crack. For the wheel and the sides of the rail, elastic material was chosen. The properties of the utilized steel are listed in Table 2. C _i and 𝛾_i parameters describe the established kinematic hardening model [4].

Fig. 1.

The 2D finite element model.

Table 1

Geometric dimensions of the 2D model [15]

Characteristics	Length (mm)
Rail length	300
Rail height	40
The outer radius of the wheel	450
The inner radius of the wheel	410
The distance of the crack mouth to the rail sides	150

Table 2

Properties of R260 steel [17]

Young’s modulus	206 GPa
Poisson ratio	0.28
Initial yield stress	320 MPa
Back stresses of the rail material model
C _i (GPa)	𝛾_i (–)
350	5000
80	1000
15	150
10	50
8	20
0.19	0.01

A constant friction coefficient of 0.5 was applied to all the surfaces in contact and a penalty contact formula is applied to reflect the contact conditions. A vertical point load was utilized in order to indicate the track load. The wheel rotated clockwise and moved 70 mm across the rail throughout each of the disquisitions in this study. The bottom and side surfaces of the rail are considered motionless. Figure 2 shows a schematic view of the loading, displacement and boundary conditions. The thicknesses of the rail and the wheel were thought to have a similar value of 10 mm. Therefore, for meshing this model, the plane strain elements were selected. Near the crack tip, square elements with a dimension of 0.1 × 0.1 mm have been utilized, as can be observed in Fig. 3, for both vertical and inclined cracks. In the contact region for involved surfaces, smaller elements have been used and coarse elements have been utilized in other parts to reduce processing and calculation time. Figure 4 represents the finite element model and the corresponding mesh network.

Fig. 2.

Applied loading, movements and boundary conditions.

Fig. 3.

Mesh network around the vertical crack tip (left) and the inclined crack tip (right).

Fig. 4.

Mesh network of the rail and wheel model.

3.2. Numerical calculation

In the previous section the extracting configurational force’s relations have been discussed. In the following section, a summary of the numerical calculations related to this method using finite element results is explained. The configurational force could be calculated in nodes using [25]: $\begin{eqnarray}\displaystyle \boldsymbol{g}_{i}^{I}=\int _{V_{e}}\boldsymbol{N}^{I}\boldsymbol{f}dV=\int _{V_{e}}\boldsymbol{N}_{.X}^{I}\boldsymbol{C}_{ij}dV & & \displaystyle\end{eqnarray}$ (6) where $\boldsymbol{g}_{i}^{I}$ represents the configurational force component in node I by integrating the configurational force in element e. N ^I is the shape function according to node I, and $\boldsymbol{N}_{.X}^{I}$ is the matrix gradient. To obtain the total configurational force, all the $\boldsymbol{g}_{i}^{I}$ values should be summed from the elements surrounding node K. $\begin{eqnarray}\displaystyle \boldsymbol{g}_{i}^{K}=\mathop{\sum }_{e}\boldsymbol{g}_{i}^{I} & & \displaystyle\end{eqnarray}$ (7) When a displacement gradient matrix is so small compared to the identity matrix, using F = I + 𝛻u could cause errors in the calculations of configurational forces. Assuming that no bulk force exists (𝛻C ₀ = 𝛻C), using the following equation is recommended instead of Eq. (1) [15]: $\begin{eqnarray}\displaystyle \boldsymbol{C}_{\mathbf{0}}={\phi}\boldsymbol{I}-{\nabla}\boldsymbol{u}^{T}\boldsymbol{S} & & \displaystyle\end{eqnarray}$ (8) In this equation, 𝜙 can be calculated directly from the finite element result as the strain density and the displacement gradient can be calculated by: $\begin{eqnarray}\displaystyle \boldsymbol{u}_{.x_{j}}=\boldsymbol{N}_{.x_{j}}^{I}\boldsymbol{u}^{I} & & \displaystyle\end{eqnarray}$ (9) where u ^I, is the displacement vector in node I that can be obtained from the finite element results. By using the following equation, S is the 1st Piola-Kirchhoff stress tensor that would be calculated using the Cauchy stress tensor that can be derived from the finite element output: $\begin{eqnarray}\displaystyle \boldsymbol{S}=\det \boldsymbol{F}\boldsymbol{T}\boldsymbol{F}^{-T} & & \displaystyle\end{eqnarray}$ (10) where F is the deformation gradient and can be computed by using the displacement gradient.

Lastly, the configurational force can be calculated using the following equations [26]: $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{g}_{i}^{I}=\int _{V_{e}}\boldsymbol{N}_{.X}^{I}\boldsymbol{C}_{\mathbf{0}ij}dV\end{eqnarray}$ (11) $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \boldsymbol{g}_{i}^{I}=\left\{\begin{array}{@{}c@{}}g_{i1}^{I}\\[3.0pt] g_{i2}^{I}\\[3.0pt] g_{i3}^{I}\end{array}\right\}=\int _{V_{e}}\left\{\begin{array}{@{}c@{}}N._{X1}^{I}C_{0_{11}}+N._{X2}^{I}C_{0_{12}}+N._{X3}^{I}C_{0_{13}}\\[4.0pt] N._{X1}^{I}C_{0_{21}}+N._{X2}^{I}C_{0_{22}}+N._{X3}^{I}C_{0_{23}}\\[4.0pt] N._{X1}^{I}C_{0_{31}}+N._{X2}^{I}C_{0_{32}}+N._{X3}^{I}C_{0_{33}}\end{array}\right\}dV.\end{eqnarray}$ (12) For more information about computing configurational force in finite element discretization, refer to [27].

4. Results and discussion

In order to validate the results of the constructed model, the contact parameter obtained from the simulation is compared to the results of the Hertz contact theory in this section [12]. The mesh convergence has been evaluated by comparing the J-integral for a variety of element sizes around the crack tip. The configurational force has been computed for perpendicular and inclined cracks in different loading situations to achieve tendency of propagation of the crack and to estimate crack growth direction.

4.1. Hertz contact model verification

In Table 3, the maximum contact pressures and the contact patch sizes obtained from the performed finite element model and the Hertz contact theory are compared, considering the free rolling movement of the wheel and 50 kN of axial load. As can be seen, the difference between the modeling result and the Hertz contact theory for maximum contact pressure is about 2.51% and 0.77% for the contact patch, which indicates the acceptable accuracy of the considered model.

Table 3
Comparison of the contact parameters with the Hertz contact theory

Maximum contact pressure (MPa) Contact patch size (mm)

Finite element 644.9 10.205

Hertz contact theory 628.7 10.126

	Maximum contact pressure (MPa)	Contact patch size (mm)
Finite element	644.9	10.205
Hertz contact theory	628.7	10.126

4.2. Mesh convergence

In order to evaluate the effect of mesh size on the results, a 1 mm crack is considered at the same loading condition as mentioned above. The first J-integral contour for two principal directions of cracks employing three element sizes of 0.05, 0.1 and 0.2 mm is compared and presented in Fig. 5. The difference between the J-integrals in each figure is less than 13.15%, and it could be concluded that the results are almost meshed size independent. For the next computations, a mesh size of 0.1 mm around the crack was selected.

Fig. 5.

J-Integral in (a) the x-direction and (b) the y-direction for the three mentioned mesh sizes. Free rolling case and 50 kN axial load have been applied.

4.3. Crack driving force calculation

The crack driving force was calculated considering an axial load of 100 kN, by configurational force and J-integral methods for the free rolling state. The results of the crack driving force are shown in the following figures for x and y and the resultant directions. Figure 6(a) shows that the tendency of crack growth in the x-direction varies when the wheel passes the crack. The negative sign of the crack driving force states that crack tends to grow in opposite directions. When the wheel has not passed the crack mouth and is on the left side of the crack, the crack tends to propagate in the left direction and when the wheel has passed the crack mouth and the wheel is located on the right side of the crack, the crack’s trend of propagation changes and it tends to propagate in the right direction. Figure 6(b) shows that the crack tends to propagate downwards and the peak of the crack driving forces occurs where the wheel reaches the crack mouth.

Fig. 6.

Crack driving force in (a) the x-direction, (b) the y-direction, and (c) the resultant direction. Free rolling case and 100 kN axial load have been applied.

By observing Fig. 6, it could be concluded that the crack driving force calculated by the configurational force and J-integrals method are close and they follow the same trend. Therefore, the validity of the numerical computation of the crack driving force using the configurational force concept can be assured. The difference between the configurational forces and J-integral results is due to the formed plastic zone around the crack tip.

Unlike the configurational force concept, J-integral could only be used as the crack driving force when proportional loading conditions are completely fulfilled. However, crack growth could be considered as a non-proportional loading state due to the presence of unloading behind the crack tip. Even for a stationary crack when large deformation occurs the same situation exists because the axes of principal stresses rotate at a fixed point of material during the deformation [1,24,30]. The computations of J-integral considering the deformation theory of plasticity are based on assuming a stored energy density which includes the plastic work. However, the plastic work has already been dissipated and cannot influence crack driving force [30].

4.4. Crack orientation effect

Rail cracks initiate and propagate in different directions, and each of them could act distinctly under the same loading condition, so the evaluation of crack behavior under various orientations is imperative. In this section, 1 mm pre-existing cracks which originated from the middle of the upper surface of the rail with different primary angles to the surface have been examined. Figure 7 shows the crack position towards the rail surface and the principal directions of cracks. The free rolling condition with 50 kN of axle load is employed to obtain the crack driving force using the configurational force concept.

Fig. 7.

Schematic view of an oriented crack and its principal directions.

Figure 8(a) shows the driving force in direction I for cracks with various initial angles (10 to 30 degrees). The crack driving force reaches its peak value almost when the wheel is located above the crack mouth. As can be seen, for cracks with angles of 10 to 25 degrees, with increasing the angle, the crack driving force increases, and afterwards this trend changes and the crack driving force for the 30 degrees crack in direction I was reduced. The crack driving force in direction II is presented Fig. 8(b). Similar to direction I, the maximum value of the crack driving force occurs when the wheel has moved for approximately half of its route. In this situation, the crack driving force is reduced by increasing the initial angle of the crack.

Fig. 8.

Crack driving force in (a) direction I and (b) direction II. Free rolling case and 50 kN axial load have been applied.

According to Fig. 8, it can be observed that cracks will propagate in mixed-mode conditions. However, for cracks with a lower angle, the values of the crack driving force in direction II are higher than direction I. For example, considering a crack with an angle of 10 degrees, the maximum driving force in direction I is 125.41 N/m and is 214.79 N/m in direction II. Therefore, it can be concluded that cracks with a lower angle tend to propagate more in direction II and the shear mode is more dominant for propagation of these cracks. On the other hand, for cracks with higher initial angles, the driving force in direction I is higher than direction II. For instance, the maximum driving force in directions I and II are 198.57 and 115.42 N/m, respectively for a 30 degrees crack. It can be concluded that the crack tends to grow more in its normal direction.

4.5. Effect of crack size

The effect of crack size has been examined through the evaluation of the crack driving force for three crack sizes, considering an initial angle of 10 degrees, free rolling condition, and 50 kN axial force. The results are presented in Fig. 9. As reported, the trend of changes is similar, but the longer crack tends to grow more than the shorter one. For instance, the driving force of the 1.5 mm crack in directions I and II are 2.36 and 1.83 times the magnitude of the crack driving force for 0.5 mm crack, respectively. As can be observed, increasing the crack size can increase the crack driving force significantly. Therefore, the influence of the crack size is very significant and increasing the crack driving force with crack growth can instantly lead to rail failure.

Fig. 9.

Crack driving force in (a) direction I and (b) direction II of cracks with various sizes. Free rolling case and 50 kN axial load have been considered.

4.6. Load effect

In this section, the effect of axial load on the crack driving force has been examined for an inclined crack with an initial angle of 45 degrees in the free rolling case. Figure 10(a) shows that the trend of the crack driving force is similar for all axial loadings. The value of the crack driving force is negative until the wheel passes the crack mouth and its peak occurs when the contact center of the rail and wheel is above the crack. As the wheel passes about 60% of its route, the crack driving force’s sign changes and decreases until the end of the wheel movement. As is shown in Fig. 10(b), the crack driving force has a positive value until the wheel passes the crack mouth. This indicates that the crack tends to grow on the positive side of direction II. After that, the sign of the crack driving force changes and the growth tendency consequently also changes. As can be seen in these figures, increasing the axial load in the free rolling case leads to an increase in the value of crack driving forces. Rising stress at the crack tip and forming a plastic zone in a shorter time are the reasons that the crack driving force is affected in a shorter distance rolled by the wheel when the axial load increases. For instance, for an axial load of 100 kN, the crack driving force increases significantly at 39.6% of the rolling distance of the wheel, but for 200 kN of axial load this happens at 33.1% of the mentioned distance.

Fig. 10.

Crack driving force for the inclined crack with an angle of +45 degrees considering a free rolling case in (a) direction I and (b) direction II.

As can be seen, the first peak value is more significant than the second one. Considering the signs of driving forces in principal directions, it could be concluded that cracks tend to propagate towards the upper surface of the rail. If this process occurs in different rail layers, spalling in the rail tread surface could occur [9].

4.7. The opposite crack

The driving force for oppositely inclined cracks has been evaluated by using 45 degrees cracks. Figure 11(a) shows the conditions of the two examined cracks. The crack mouth is located in the middle of the upper surface of the rail and the size of each crack is 1 mm. Figure 11(b) shows the assumed direction of mode I and II of propagation of a crack with the initial angle of −45 degrees. The results of cracks with an angle of +45 degrees were presented in the previous section. In this section, the crack driving force values for the crack with an angle of −45 degrees in the same condition as the aforementioned crack is presented and compared.

Fig. 11.

(a) Schematic view of oppositely inclined cracks and (b) principal directions for cracks with an angle of −45 degrees.

As can be seen in Fig. 12, the trend of changes for all axial loads is similar, and the crack driving force increases when the axial load increases, similar to the previously obtained results for the +45 degrees crack. When the wheel arrives in the vicinity of the crack mouth, the crack driving force values reach their maximum. By comparing crack driving forces for these cracks, it can be observed that a crack with an angle of +45 degrees tends to grow more than a crack with an angle of −45 degrees in direction I and the reverse takes place for direction II.

Fig. 12.

Crack driving force for the inclined crack with an angle of −45 degrees considering a free rolling case in (a) direction I and (b) direction II.

4.8. Slip effect

Slip is a common phenomenon that can occur in the rail and wheel contact region, and affects the wear mechanism which can cause different damages ranging from slight oxidation and peeling to destructive fatigue wear or spalling [19,34], so examining the effect of slip on the crack driving force is crucial. In order to obtain the slip effect, the slip ratio parameter ( ${\dot{S}}=(\frac{{\omega}R}{V}-1)\times 100\%$ )) was used. In this formula, 𝜔 is the angular velocity, V is the linear velocity and R is the radius of the wheel.

Fig. 13.

Slip ratio effect on the crack driving force in (a) direction I and (b) direction II when 50 kN axial load is applied.

A slip ratio from 0 to 1% and 50 kN axial load was applied to a rail containing a 1 mm crack with the initial angle of 45 degrees to obtain the crack driving force, and the results are displayed in Fig. 13. For the 0% slip ratio that indicates a free rolling case, the result is considerably different from sliding/rolling cases and it can be concluded that even a small partial slip can significantly affect the crack driving force and consequently the crack’s tendency to propagate. In the slip zone, the tangential forces have their maximum value and are proportional to the normal force, but in the stick region, the tangential forces are related to micro slip velocities between rails and wheels in the contact area. In the free rolling case, of which almost the whole contact area is a stick zone, the tangential forces are at their lowest, which can explain the difference between free rolling and other situations.

Fig. 14.

Slip ratio effect on the crack driving force in (a) direction I and (b) direction II when 200 kN axial load is applied.

According to the results, generally rising the slip ratio leads to an increase in the driving force in both the opening and shear mode, but the changes of the crack driving force for a slip ratio higher than 0.7% are insignificant, and it can be concluded that a full slip condition has been achieved. The main reason for decreasing crack driving force in this case after reaching a full slip condition is the relationship between slip ratio and traction force that has been presented in [7,33]. The traction force is proportional to creepage until it reaches a critical value, and after the saturation point, the traction force is constant or could even be dropped off due to the decrease of the adhesion coefficient [10].

The influence of the slip ratio on the crack driving force has also been studied considering 200 kN of the normal force and the opening and shear mode are shown in Fig. 14. In this case, the slip ratio changed from 0 to 3% and the full slip situation took place almost at the slip ratio of 2.5%. Similar to previous calculations, the increase of the slip ratio led to an increase in the crack driving force and the crack in rolling/sliding conditions tends to propagate more than in a pure rolling situation.

4.9. Multiple cracks effect

In order to evaluate the influence of multiple cracks, a network of three parallel and equal cracks with an angle of 45 degrees was selected. The distance between two cracks is 2 mm and the middle crack is located in the middle of the upper surface of the rail, similar to all the previously obtained results. The crack driving force in directions I and II for these three cracks are presented in Fig. 15, respectively. As can be seen, the trend in all graphs is identical, and they are approximately similar to what has been calculated for a single crack with an angle of 45 degrees (as shown in Fig. 10). The maximum of the crack driving force occurs when the wheel is almost above the crack mouth and so the occurrence time of the maximum crack driving force is different for each crack. It seems that the left crack has been affected more than the other two cracks and the value of the crack driving force is slightly higher. Table 4 shows the extremum of the crack driving force for the network of cracks and this has been compared to the crack driving force for a single crack in directions I and II. It can be concluded that the existence of more cracks could lead to a decrease in the strength of the rail and the tendency of the crack to grow could be increased by 37.24% if other cracks exist.

Fig. 15.

Crack driving force in (a) direction I and (b) direction II for multiple cracks. Free rolling case and 50 kN axial load have been applied.

Table 4

Comparison of the crack driving force between single and multiple cracks

	Extremum of the crack driving force in direction I (N/mm)	Extremum of the crack driving force in direction II (N/mm)
Left crack	−217.326	39.285
Middle crack	−204.027	32.08
Right crack	−216.201	35.588
Single crack	−179.213	28.624

5. Conclusions

For the first time, a parametric study on inclined cracks using the configurational force method was performed in this study. It was found that the crack driving force for inclined cracks are excessively greater than perpendicular cracks, which means they tend to propagate more and faster. The effect of the crack size, slip ratio, initial orientation, and axial load has been implemented for these types of cracks, and the influence of the existing several cracks has been evaluated. The most important findings of this study are:

In the examined cases, by increasing the initial angle of the cracks, the driving force increases in the opening mode, and decreases in the shear mode.

The propagation mode of the inclined cracks is generally mixed-mode, however, for cracks with lower initial angles, the shear mode was more decisive and for bigger initial angels, the opening mode was more affected.

In the free rolling contact, increasing the crack size will significantly increase the crack driving force and the tendency of propagation. In the investigated case it was shown that if the crack size increases by three times of its initial size, the crack driving force could grow by 136%.

When the slip is considered, the configurational force curve is entirely different from the case when free rolling is assumed. The slip tends to increase the crack driving force considerably.

The examined inclined cracks tend to propagate toward the upper surface of the rail, which might cause spalling to the rail surface.

Evaluating a crack network reveals that more cracks can lead to an increase of the crack driving force and consequently a reduction in the safe life of the rails.

Footnotes

Conflict of interest

None to report.

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Numerical calculation of crack driving force using the configurational force concept for elastic–plastic rail cracks

Abstract

Keywords

Nomenclature

1. Introduction

2. Governing equations

3.1. Finite element model

4.1. Hertz contact model verification

Table 3 Comparison of the contact parameters with the Hertz contact theory Maximum contact pressure (MPa) Contact patch size (mm) Finite element 644.9 10.205 Hertz contact theory 628.7 10.126

Footnotes

Conflict of interest

References

Table 3
Comparison of the contact parameters with the Hertz contact theory

Maximum contact pressure (MPa) Contact patch size (mm)

Finite element 644.9 10.205

Hertz contact theory 628.7 10.126