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Rail. Eng. Science (2020) 28(4):408–423 https://doi.org/10.1007/s40534-020-00227-6 Mechanical characteristic variation of ballastless track in high- speed railway: effect of train–track interaction and environment loads 1 1 1 1 • • • Shengyang Zhu Jun Luo Mingze Wang Chengbiao Cai Received: 23 September 2020 / Revised: 4 November 2020 / Accepted: 6 November 2020 / Published online: 30 November 2020 The Author(s) 2020. This article is published with open access at Springerlink.com Abstract Due to the fact that ballastless tracks in high- stress and damage; the interface damage could be induced speed railways are not only subjected to repeated train– by the wheel–rail longitudinal vibrations at a high vehicle track dynamic interaction loads, but also suffer from running speed owing to the dynamic amplification effect complex environmental loads, the fundamental under- caused by short wave irregularities; the vehicle dynamic standing of mechanical performance of ballastless tracks load could produce considerable water pressure that pre- under sophisticated service conditions is an increasingly sents nonlinear spatial–temporal characteristics at the track demanding and challenging issue in high-speed railway interface, which would lead to the interface failure under a networks. This work aims to reveal the effect of train–track certain condition due to the coupled dynamic effect of interaction and environment loads on the mechanical vehicle load and water pressure. characteristic variation of ballastless tracks in high-speed railways, particularly focusing on the typical interface Keywords Ballastless track High-speed railway damage evolution between track layers. To this end, a finite Mechanical characteristic Interface damage Train–track element model of a double-block ballastless track involv- interaction Temperature gradient Dynamic water ing the cohesive zone model for the track interface is first pressure Cohesive zone model established to analyze the mechanical properties of the track interface under the loading–unloading processes of the negative temperature gradient load (TGL) followed by the same cycle of the positive TGL. Subsequently, the effect of wheel–rail longitudinal interactions on the non- 1 Introduction linear dynamic characteristics of the track interface is investigated by using a vehicle-slab track vertical-longi- With the advantages of strong stability, high smoothness tudinal coupled dynamics model. Finally, the influence of and less maintenance, ballastless tracks for high-speed dynamic water pressure induced by vehicle dynamic load railways overcome the disadvantages of ballasted track, on the mechanical characteristics and damage evolution of and have become the prior selection to enabling a rapid the track interface is elucidated using a fluid–solid coupling development of high-speed railways worldwide [1, 2]. method. Results show that the loading history of the pos- However, it is still quite difficult for ballastless tracks to itive and negative TGLs has a great impact on the non- avoid some typical damage and faults during its service linear development and distribution of the track interface time, attributed to the fact that they are exposed to the complicated loads of high-speed train dynamic load and environmental loads such as temperature change and rain & Shengyang Zhu water erosion. Meanwhile, due to the diversity of their syzhu@swjtu.edu.cn constituent materials and the complexity of inter-layer connection, the service performance of ballastless tracks is Train and Track Research Institute, State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu found to be deteriorating in practical engineering 610031, China 123 Mechanical characteristic variation of ballastless track in high-speed railway: effect of… 409 applications, with main defects of the interface damage or tracks on this high-speed railway line [3]. This occurrence cracks between track layers, as shown in Fig. 1. Usually, of the interface gaps affected the safety operation of high- the temperature load is the main cause of the initiation of speed trains, leading to speed limit treatment and wide- the interface damage which will be then accelerated by spread trains delay. Therefore, the effect of train–track train dynamic loads. Subsequently, the rainwater will be interaction and environment loads on the mechanical inevitably trapped at the interface gap to produce dynamic characteristics of ballastless track interface is very water pressure under train dynamic load, leading to the sophisticated, which is one of main concerns to improve aggravation of the interface damage. the design theory and maintenance technology of ballast- The interface damage could be quite harmful to the track less tracks, and to ensure the operation safety of high-speed and train systems through complicated mechanisms. railways. Firstly, the emergence and development of the interface To capture the mechanical characteristic variation of damage will lead to the gradual destruction of the integrity ballastless tracks under environment loads such as the of the track system; especially after the penetration of rain temperature load, Zhong et al. [4] established a three-di- water, it will have a splitting and scouring effect on the mensional finite element model of slab track to study the track interface under train dynamic loads, which will deformation and interface stress of a slab track under daily intensify the expansion of the interface gap to the depth, changing temperature. Zhang et al. [5] tried to improve the and cause the ballastless track to gradually lose its bearing mechanical properties of cement asphalt (CA) mortar and capacity, thus affecting the stability and durability of the concrete composite specimens in high-speed railway by track structure. Secondly, the change of interface bonding modifying the interfacial bonding relationship, and dis- state will result in the alteration of load transfer mechanism cussed the effects of temperature on the mechanical prop- of ballastless tracks. The interface bonding area will con- erties. Peng et al. [6] carried out pull-off tests to study the tinue to decrease due to the emergence of the interface interfacial bonding strength between CA mortar and con- damage, and the vehicle-track interaction load will be more crete slab, and found that temperature cycles decrease the concentrated in the middle of the track, causing the sub- interfacial bonding possibly due to the weakened grade to bear unfavorable concentrated loads. Thirdly, the mechanical interlocking forces at the interface. Wang et al. interface damage weakens the stiffness, strength and sta- [7] evaluated the mechanical performance of the CA bility of the track system, leading to an increased dynamic mortar of the slab track interlayer under extreme temper- displacement of rails under spatial train dynamic loads, atures, by conducting the flexural strength and fatigue tests which could even exceed the standard limit and finally pose and a transient thermal simulation. Yu et al. [8] presented a threat to the running safety of high-speed trains. For an in situ experiment to evaluate the influence of seasonal temperature variations on the curling behavior of concrete example, a field investigation of a high-speed railway found that the maximum amount of the interface gap can track slab, and created a thermomechanical coupled finite reach up to more than 10 mm, and under the condition of element model to compare with the experimental data. continuous abnormal high temperature in summer, 141 Chen et al. [9] studied the effect of acid rain environment interface gaps appeared in some sections of ballastless on the degradation of CA mortar at track interface in practical engineering; they found that one of the main reasons for the material degradation is the loss of asphalt and asphaltenes under the coupled effects of loads and rain. Cai et al. [10] investigated the arching mechanism of slab joints under high temperature conditions considering the buckling instability of track structure, and the damage process of joint concrete and its effects on the arching instability were analyzed with the help of the field inves- tigation. Li et al. [11] established a three-dimensional finite element model of the CRTS II track incorporating a cohesive zone model to simulate the nonlinear behavior of the track interface under the temperature gradient load (TGL). Zhu et al. were among the first that introduced the cohesive zone model to investigate the interface damage of slab tracks under temperature and vehicle dynamic loads [12], and further they obtained the damage constitutive model of the concrete interface of double-clock ballastless Fig. 1 Example of interface damage of a ballastless track in China’s track based on experimental tests, with which the interface high-speed railways Rail. Eng. Science (2020) 28(4):408–423 410 S. Zhu et al. damage evolution of the track under monotonic and cyclic railway, a cohesive zone model considering mixed-mode TGL was investigated to elucidate the practical application damage is employed to investigate the nonlinear mechan- of the proposed model [13]. However, few studies can be ical characteristics and damage evolution of the track found that focus on the simulation analysis of the track interface subject to complicated environment loads and interface damage under the alternative action of negative vehicle-track interaction load. Firstly, a finite element and positive TGLs. model of a double-block ballastless track involving the Concerning on the mechanical property of the concrete cohesive zone model is established to analyze the effect of interface of ballastless tracks subject to train dynamic loading–unloading processes of the negative and positive loads, much work has been done. Ren et al. [14] built a slab TGLs on the mechanical behavior of the track interface. track model based on the damage mechanics to study the Then, a vehicle-slab track vertical-longitudinal coupled effect of debonding on the concrete damage distribution dynamics model is adopted to capture the effect of wheel– and mechanical responses of track slabs, where the rail rail longitudinal interactions on the nonlinear dynamic supporting forces obtained from in situ tests were adopted characteristics of the track interface. Finally, the coupled as the train loads. Zhao et al. [15] carried out a prototype dynamic effect of vehicle load and water pressure on the fatigue test for a slab track considering the coupled effects mechanical characteristics and damage evolution of the of wheel load, temperature change, and water erosion, in track interface is revealed using a fluid–solid coupling which the repeated wheel loads were applied through a method. Some interesting and useful conclusions are drawn loading vehicle equipped with eccentric vibrator wheels, from the comprehensive numerical analysis, which may the TGL was generated by heating or cooling the slab track provide theoretical support for the maintenance strategy system, and the water erosion effect was simulated in the establishment and safe operation management of ballast- test. Zhang et al. [16] established a viscoelastic finite ele- less tracks in high-speed railways. ment model of a slab track incorporating viscoelastic parameters and a cohesive zone; the model was verified by experimental data and used to analyze the initiation 2 Constitutive model of the track interface mechanism of debonding under coupling actions of tem- perature and dynamic vehicle loadings. Xiao et al. [17] For simulating the nonlinear mechanical behavior of the proposed a method for analyzing inter-layer defects in slab track interface, the cohesive zone model could be an tracks under train and temperature loads based on fatigue optimal selection as it is widely used in the interface analysis and the extended finite element method. Cao et al. damage analysis of composite structures with different [18] studied the interface damage mechanism of a slab kinds of constitutive relationship forms such as the bilinear, track under the coupling effect of vehicle dynamic load and rectangular, trapezoidal, polynomial, and exponential the interface water pressure, and investigated the effects of cohesive zone models [20–24], in which the bilinear load characters, water viscosity, and crack shape on the cohesive zone model has been successfully employed in coupled hydro-mechanical fracture of the slab track. Zhu evaluating the interface damage development of ballastless et al. [19] developed a coupled dynamics model of a tracks due to its appropriate modeling for nonlinear dam- vehicle and the slab track involving nonlinear spring- age behavior of the track interface. In this paper, the damper elements for simulation of the interface damage. bilinear cohesive zone model considering mixed-mode By considering the random nature of the interface damage damage is employed for investigating the nonlinear parameters, the probability analysis was performed using mechanical characteristics and damage evolution of the the response surface method and Monte Carlo simulations, track interface subject to complicated environment loads and finally the damage assessment criterion and the cor- and vehicle-track interaction load, as shown in Fig. 2. responding safety threshold were suggested on the basis of It is shown in Fig. 2 that under external loads, the stress the concept of reliability for the long-term dynamic per- in the cohesive zone of the crack tip increases linearly with formance of slab tracks. However, most of the existing an increase in the displacement, and when the stress researches only considered the effect of vertical vehicle reaches its peak value, it will undergo a softening process dynamic load on the mechanical behavior of the track because of the interface damage initiation. As the dis- interface, while the coupling effect of the longitudinal and placement continues to increase, the stress decreases lin- vertical dynamic loads on the interface damage was rarely early, resulting in a decrease in the load-bearing capacity ascertained because of the complexity of train and track and a gradual expansion of the interface crack. When the interaction. stress is reduced to zero, the interface crack grows com- In this work, to reveal the mechanical characteristic pletely leading to the failure of the interface. variation and damage evolution at the track interface The nonlinear constitutive relationship of the bilinear between track layers of ballastless tracks in high-speed cohesive zone model can be expressed as Rail. Eng. Science (2020) 28(4):408–423 Mechanical characteristic variation of ballastless track in high-speed railway: effect of… 411 i to the area enclosed by the stress-displacement curve under 0 0 t ðs s Þ i 0 i s the current stress state. In this regard, the energy-based t ¼ i ¼ n, s, t ; ð1Þ power law, where the mixed mode damage is determined > s s > 0 i 0 t ðs [ s Þ i i f 0 by the combination of the critical fracture energies (normal s s i i and shear) according to the power law, is employed in this where subscript i ¼ n, s, t stands for the normal and two work, and thus the track interface failure at a material point shear directions, respectively; t represents the interface under mixed-mode conditions is governed by stress under mixed loading modes of tension and shear, a a a G G G n s t respectively, and s the corresponding interface displace- þ þ ¼ 1; ð6Þ C C C G G G n s t ment; t represents the peak value of the interface stress under a pure mode of tension or shear, and s the corre- where G , G , and G refer to the work done by the stress n s t sponding interface displacement; when the cohesive ele- and its corresponding relative displacement in the normal C C C ment at the interface completely fails, the interface and the two shear directions, respectively; G , G , and G n s t displacement attains its maximum value s . refer to the critical fracture energy in the normal and the The interface damage is assumed to be initiated when two shear directions; and the power a is set to 2 in the the following equation satisfies analysis. When the material point fails completely, the mixed-mode critical fracture energy is equal to the sum of 2 2 2 hi t t t n s t þ þ ¼ 1: ð2Þ each uniaxial fracture energy. It is worth pointing out that 0 0 t t t n s t under mixed-mode loading the interface failure could occur even if the fracture energy, such as the G , is smaller than To quantitatively describe the damage degree at the n the critical fracture energy G in the uniaxial state. In this interface, a scalar damage variable D that monotonously case, the normal stress-displacement curve no longer pre- increases from 0 to 1 is introduced in the numerical analysis. The damage scalar proposed by Camanho and sents a bilinear relationship when shear displacements appear in the cohesive zone, but it is enveloped in the Davila [25] is used to describe the nonlinear softening process, expressed as bilinear curve under a pure normal loading mode, as shown in Fig. 2. Moreover, with the increase in the proportion of f max 0 s s s m m m D ¼ ; ð3Þ shear fracture energy with respect to the total fracture max f 0 s s s m m m energy, the normal stress-displacement relationship is fur- max ther and further from the bilinear relationship while closer where s refers to the maximum value of the effective 0 to the abscissa axis. It can be also known from Eq. (6) that displacement attained during the loading history, s is the under mixed-mode loading the fracture energy in one effective displacement at the damage initiation, and s direction must be smaller than its corresponding critical denotes the effective displacement at the complete failure fracture energy. of a material point, which can be obtained as [25] qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 s ¼ hi s þs þ s : ð4Þ m n s t After the interface damage initiation, the stress components for the bilinear cohesive zone model can be calculated as t ¼ðÞ 1 D k s s [ 0 n n n n t ¼ðÞ 1 D k s ; ð5Þ s s s t ¼ðÞ 1 D k s t t t where k , k and k are the interface stiffness in the normal n s t and two shear directions, respectively. The energy-based damage evolution law is a rule that each uniaxial damage is combined according to a certain relationship by which the proportion of the normal and shear damages in the cohesive zone are determined quan- titatively. The energy dissipated with the development of damage, namely the fracture energy, can be used to prop- Fig. 2 Bilinear cohesive zone model considering mixed-mode erly define the interface damage evolution, which is equal damage Rail. Eng. Science (2020) 28(4):408–423 412 S. Zhu et al. supporting layer are not involved in the analysis. The 3 Effect of temperature loads on mechanical thermodynamic parameters of the track system can be property of track interface referred to in Ref.[12]. For the mechanical analysis, the element C3D8R is adopted for modeling the track slab and It is known that the temperature difference between upper and lower surfaces of ballastless tracks could be formed supporting layer, and the element COH3D8 with zero thickness is used to simulate the cohesive interface. The under the action of solar radiation and heat convection due to their poor heat conduction performance, which will ‘‘Tie’’ constraint is used between the cohesive layer and its adjacent layer, and the contact relationship is applied generate the TGL along the thickness direction of the track. A higher surface temperature of the track slab will form a between the track slab and the supporting layer. Symmetric constraints are applied to both ends of the model. Through positive TGL, resulting in an arching deformation of the numerical tests, the mesh size of the track slab and sup- track slab, whereas a lower surface temperature of the track porting layer is determined to be 0.1 m 9 0.1 m 9 0.1 m, slab produces the negative TGL, leading to an unwarping and it is 0.05 m 9 0.05 m for the cohesive layer, which deformation of slab edges. When the bonding strength of ballastless track interlayers is less than the stress caused by could eliminate the influence of the mesh size effect. The calculation length of the track is 10 m, and the other com- the temperature-induced deformation, cracks would be initiated at the track interface and expand along the inter- mon size and material parameters can be found in Ref.[13]. Elastic support constraints are applied to the bottom of the face under train dynamic loads and finally form the track slab void, which will significantly affect the service life of supporting layer to simulate the elastic subgrade. The sub- grade stiffness has a significant effect on the track defor- ballastless tracks. mation, which is selected as 310 MPa/m according to a field Ballastless tracks will experience the repeated processes test in a high-speed railway. The parameters for the cohe- of loading and unloading of the negative TGL, and fol- sive zone model are listed in Table 1, which were deter- lowed by the same cycle of the positive TGL. However, mined by conducting splitting loading tests on composite most previous studies just focused on the effect of the specimen. In the current simulation, the maximum nega- negative temperature gradient on the interface damage evolution, ignoring its unloading process and the following tive/positive TGL are selected according to the code for design of high-speed railway [26], and the temperature loading of the positive TGL. gradient loading process is shown in Fig. 4. 3.1 Finite element model of double-block ballastless 3.2 Stress and damage distribution at track interface In this section, the double-block ballastless track widely used in Chinese high-speed railways is taken as the research Figures 5 and 6 show the interface stress and damage subject. To analyze the mechanical characteristics of the distributions along the transverse direction of the track track interface under a cyclic variation of TGLs, a finite element model of the track involving the cohesive zone under cyclic temperature loads, respectively, where three loading states including the maximum negative tempera- model for the track interface is established using Abaqus software, as shown in Fig. 3. The model is mainly com- ture gradient (TGL = -76.9 C/m), the complete unload- ing (TGL = 0 C/m), and the maximum positive posed of the track slab, the cohesive layer, and the sup- porting layer, while the rails and fasteners are omitted due temperature gradient (TGL = 90 C/m) are plotted toge- ther. Since the interface stress, damage and displacement to their negligible influence on the track slab deformation. For the heat conduction analysis, the track slab is modeled do not change along the longitudinal direction of the track, their contour results are not shown here for brevity. with the element DC3D8, and the cohesive layer and Fig. 3 Finite element model of double-block ballastless track involving the cohesive zone model for the track interface Rail. Eng. Science (2020) 28(4):408–423 Mechanical characteristic variation of ballastless track in high-speed railway: effect of… 413 Table 1 Parameters for the cohesive zone model 0 0 f C 2 Parameters t (MPa) k (Pa/m) G (J/m ) d (mm) i d (mm) i i i i Normal direction (i = n) 1.43 0.0010 1.4 9 10 0.0075 5.4 The first shear direction (i = s) 0.36 0.0026 1.4 9 10 0.0110 2.0 The second shear direction (i = t) 0.36 0.0026 1.4 9 10 0.0110 2.0 Fig. 6 Interface damage distribution along the lateral direction of the track under the cyclic temperature load Fig. 4 Loading process of TGL formed a considerable crack length at the interface, the It can be seen that with the unloading of the negative undamaged area in the middle of the track suffers a large TGL, the interface stress decreases gradually and it reduces interface stress under the arching deformation of the track, to zero when the negative TGL is completely unloaded, but which leads to the propagation of the interface crack under the interface damage continues to increase. Subsequently, the positive TGL. According to the simulation calculation, with a gradual increase in the positive TGL, the track slab when there is no damage at the interface, the same positive presents an arching deformation, and the normal and shear TGL just causes the interface damage to emerge within contact stresses are generated due to the contact between around 0.05 m from the slab edge without forming any the track slab edges and the supporting layer. As the pre- cracks. This indicates that the ability to resist interface vious loading–unloading process of negative TGL has Fig. 5 Interface stress along the lateral direction of the track under the cyclic temperature load: a normal stress and b lateral shear stress Rail. Eng. Science (2020) 28(4):408–423 414 S. Zhu et al. applications in order to prevent the rapid development of the track interface damage. 3.3 Nonlinear mechanical characteristics of the track interface The stress, displacement and damage laws at different positions of the track interface (as shown in Fig. 3) are discussed here during loading and unloading of one cycle TGL. Figure 8 shows the nonlinear variation of the interface normal stress and displacement with the changing of the TGLs, and the interface damage evolution is shown in Fig. 9. In the loading stage of the negative TGL, the point A is subjected to tensile stress until the interface cracking occurs, and its normal displacement gradually increases with the increase in unwarping deformation of the track slab, while it starts to close slightly after the unloading of the negative TGL. With the increase in the positive TGL, the point A keeps in contact with the supporting layer and generates the contact stress. For the mechanical variation at the point B, it bears the compressive stress under the negative TGL, but it suddenly changes into the tensile stress within a small change of TGL and generates the interface damage during the unloading process. The dam- age process is quite fast which reflects the characteristic of brittle material fracture, whereas the displacement of the point B increases gradually under the loading of the posi- Fig. 7 Interface gap variation between the track slab and supporting tive TGL due to the arching deformation of the track slab. layer under the cyclic TGL: a height of interface gap and b depth of The point C is located in the middle of the track interface, interface gap and its damage occurs under the positive TGL. Its normal displacement is always zero until the damage initiation, damage and cracking under the positive TGL is greatly and it starts to increase due to the arching deformation of reduced after the interface separation. the track slab, which has a similar growth trend with the As shown in Fig. 7a, with the variation of the TGLs, the point B. height of the interface gap firstly decreases and then The variation characteristics of transverse shear stress increases, while the depth of the interface gap increases and displacement under the cyclic TGL are shown in substantially. Also, the shape of the track interface gap Fig. 10. It is clearly shown that the shear stress closer to the under the negative TGL is quite different from that under middle of the track interface is found to be larger than other the positive TGL, which presents, respectively, a warping locations. Note that the shear displacement undergoes a curve on both sides of the track slab and a ‘‘void’’ form reverse phenomenon when the TGL shifts from negative under the slab. Figure 7b shows the depth variation of the load to positive load. This is because the track slab pro- interface gap with the changing of TGLs. When the neg- duces warping deformations under the negative TGL, ative TGL is unloaded from the maximum value, the whereas the slab generates arching deformation under the interface gap does not close immediately, but expands positive TGL. rapidly until the negative TGL drops to -69 C/m. Sub- sequently, the depth of the interface gap keeps unchanged until the positive TGL reaches 64 C/m, and it continues to 4 Effect of vehicle-track interaction expand and stops at 72 C/m. Clearly, the depth variations on the interface mechanical characteristics of the interface gap under different TGLs exhibit strong nonlinear characteristics, and the TGL of track slabs at Speed adjustment of the motor cars of high-speed train is these ‘‘steep slope’’ sections should be avoided in practical usually achieved by traction or braking operations. Espe- cially in mountain high-speed railways such as Sichuan– Rail. Eng. Science (2020) 28(4):408–423 Mechanical characteristic variation of ballastless track in high-speed railway: effect of… 415 Fig. 8 Nonlinear mechanical variation of the track interface with the changing of the TGLs: a normal stress, and b normal displacement 4.1 Vehicle-track vertical-longitudinal coupled 1.0 dynamics model considering interface Loading Unloading Reloading interactions 0.8 0.6 In this section, the CRTS II slab track in high-speed rail- 0 /m ć 90 /m ć way is taken as an example to elucidate the effect of lon- 0.4 Point A gitudinal and vertical vehicle-track interactions on the Point B -76.9 /m ć 0.2 Point C interface mechanical characteristics. Figure 11 illustrates a vehicle-slab track vertical-longitudinal coupled dynamic 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 model which is developed using the framework of vehicle- Analysis time step (s) track coupled dynamics theory [27, 28], in which rail- rubber pad and track slab-CA mortar longitudinal interface Fig. 9 Damage evolution of the track interface with the changing of interactions are further considered by introducing longitu- the TGLs dinal degrees of freedom of the vehicle and slab track subsystems [29]. Such a large-scale dynamics model Tibet railway that have complex and large gradient sec- involving many nonlinear factors and time-varying tions, the wheel–rail longitudinal interactions may become parameters is solved by a new fast explicit integration more intense leading to longitudinal impacts on the track method proposed by Zhai [30] with a time step size of structures. By introducing system longitudinal vibrations –4 1 9 10 s. and implementing the interface interactions into the clas- The longitudinal resistance of the fasteners can be sical vehicle-track system, the effect of vehicle-track essentially regarded as a kind of friction force generated interaction on the interface mechanical characteristics is from the interface between the rail and rubber pad, which presented in this section. Fig. 10 Variation characteristics of transverse shear stress and displacement under the cyclic TGL: (a) transverse shear stress and b transverse shear displacement Rail. Eng. Science (2020) 28(4):408–423 Damage variable 416 S. Zhu et al. Fig. 11 Vehicle–track vertical-longitudinal coupled dynamics system can be well characterized by the Dahl friction model [31], d d t d tf t max j t0 Dt ¼ ; ð8Þ based on which the following explicit expressions for the d tðÞ d d t max j tf t0 longitudinal resistance are presented: F ðÞ x ¼ F for x ¼ x > Lf Lfs s > r ðÞ x x 0 s F ðÞ x ¼ F ðÞ F F exp for x [ x or increasing x; Lf Lm Lm Lfs s F ð7Þ Lm r ðÞ x x 0 s : F ðÞ x ¼ F þðÞ F þ F exp for x\ x or decreasing x Lf Lm Lm Lfs s Lm where F is the fastener longitudinal resistance, (x , F ) where d refers to the maximum effective displacement Lf s Lfs t max attained during the loading history; d denotes the is defined as a reference state and needs to be updated t0 during the motion, and F denotes the ultimate resistance effective displacement at damage initiation; and d Lm tf denotes the effective displacement at complete failure. force of the fasteners. Considering the fact that the interface damage in the Considering the irreversibility of damage, the nonlinear longitudinal cohesive force can be calculated by normal mode is negligible under the vertical vehicle dynamic load [12], only the longitudinal shear mode of the < k d t d t \d t t j t max j t0 cohesive zone model at the track interface was considered F d t ¼ k 1 Dt d t d d t \d ; t t j t j t j t0 t max j tf in the simulation. The interfacial bond-slip behavior 0 d t d t max j tf between the track slab and CA mortar is simulated as a ð9Þ series of nonlinear cohesive springs, whose constitutive law follows the bilinear discrete cohesive zone model. The where k is the initial stiffness of the cohesive spring damage variable D is adopted to describe the softening without damage, defined by k = F d ; and F is the t tm/ t0 tm process and track the extent of damage accumulated at the interfacial shear strength. interface. For the jth (j =1, 2, …) time step t , it can be expressed as Rail. Eng. Science (2020) 28(4):408–423 Mechanical characteristic variation of ballastless track in high-speed railway: effect of… 417 4.2 Nonlinear dynamic characteristics of the track (a) interface In the simulation, a motor car accelerating from 0 to 250 km/h under the combined excitation of track random irregularities and traction torques is investigated. Figure 12 shows the traction characteristic curve of a typical high- speed train in China, which is the total traction force of this train with respect to running speeds. The track irregularity characterized by wavelengths between 0.1–120 m is 0 50 100 150 200 250 assumed to move backward at the vehicle running speed to Running speed (km/h) simulate the vehicle traveling along the track. Without loss (b) of generality, the shear strength of the cohesive element is set as 0.015 MPa, the damage initiation displacement in sliding direction is selected as 0.02 mm, and the failure displacement in sliding direction is set as 2.0 mm in the current simulation. The wheel–rail interaction forces are portrayed in Fig. 13, from which one can observe that the maximum amplitude of wheel–rail vertical force increases from 54.2 to 83.3 kN during the vehicle acceleration process due to the combined effect of running speed and track irregular- 0 50 100 150 200 250 ities. Regarding the longitudinal counterpart, it increases Running speed (km/h) rapidly once the driving torque acts on the wheel axle and Fig. 13 Variations of wheel–rail interaction forces with increasing reaches the peak value of 11.01 kN at 2.35 km/h. As the speed: a vertical force and b longitudinal force running speed continues to increase, the amplitude of wheel–rail longitudinal force decreases gradually, which is the maximum value after the running speed exceeds basically coincident with the variation trend of the traction 232 km/h. characteristic curve as depicted in Fig. 12. Therefore, it can be concluded that the coupling of Figure 14 illustrates the evolution of damage at the track longitudinal and vertical vehicle dynamic loads under the slab-CA mortar interface during the speed-up process. It combined effect of track random irregularities and driving should be noted that the coordinate origin in Fig. 14a, b is torques could result in interface damage at the track fixed at the track slab beneath the centroid of the car body. interface layers, which should be seriously addressed in It can be observed that the interface damage occurs in the high-speed railway engineering practices, particularly in vicinity of the four wheel loading positions. The maximum complex mountain areas. damage variable is found to be 0.243 and the corresponding evolution process is displayed in Fig. 14c. As can be seen, the interface damage initiates at 188 km/h, undergoes rapid 5 Effect of dynamic water pressure on interface propagation at around 226 km/h, and eventually maintains mechanical property The engineering practice has shown that the dynamic water pressure at track interface induced by vehicle dynamic loads may play an important role in the expansion of the interface cracks between ballastless track layers. The crack growth rate at the track interface is faster in the areas with abundant rainfall or poor drainage. This is due to the fact that the interface damage will be accelerated by the dynamic water pressure induced by vehicle dynamic load once the rain water penetrates into the interface gap of ballastless tracks. The coupling effect between the inter- face water and the track slab is mainly reflected in two aspects. On the one hand, the movement of the track slab Fig. 12 Traction characteristic curve of a high-speed train Rail. Eng. Science (2020) 28(4):408–423 Wheel-rail vertical force (kN) Wheel-rail longitudinal force (kN) 418 S. Zhu et al. Fig. 14 Damage at the track interface: a three-dimensional view of time histories and distribution, b two-dimensional view of time histories and distribution, and c evolution of the maximum damage variable under vehicle dynamic loads will squeeze the interface momentum conservation law, and the energy conservation water and thus generate dynamic water pressure; on the law. On this basis, a calculation model of the dynamic other hand, the dynamic water pressure reacts to the track water pressure of the track interface is established as shown slab, resulting in the normal interface stress and damage in Fig. 15, where the upper, lower and end surfaces of the development. In this section, the mechanical characteristics interface gap are set as the fluid–solid coupling interface. and damage behavior at the interface of the double-block The turbulence influence is considered due to the high- ballastless track subject to the vehicle load-induced frequency vehicle load and the complicated flow of inter- dynamic water pressure is investigated by fluid–solid face water. The RNG (renormalization group) k-e turbu- coupling method. lence model is employed in the simulation as it can consider the effects of the separation flow and vortex flow 5.1 Modeling of dynamic water pressure and predict the near-wall flow accurately. The time history of vertical rail supporting forces obtained by the vehicle- To solve the fluid–solid coupling problem, the separation track coupled dynamics model is applied to the top of the method needs to obtain the responses in the solid region track slab, and the fixed constraint is applied to the bottom and the fluid region, respectively, as well as the data of the support layer. The element size of the track model is exchange of the fluid–solid coupling interface. For general smaller than 5 mm, and the total number of elements is compressible Newtonian fluid, the laws of physical con- about 76,000. The pressure at the opening end of the servation to follow include the mass conservation law, the interface water is set as 0, and the dynamic viscosity of Rail. Eng. Science (2020) 28(4):408–423 Mechanical characteristic variation of ballastless track in high-speed railway: effect of… 419 Fig. 15 Calculation model of the dynamic water pressure of the track interface induced by vehicle dynamics load –3 water is selected as 1.01 9 10 Pas. The same integration As can be seen, when the first wheelset is close to the time step should be used for the ballastless track model and monitoring section, the dynamic water pressure presents a the interface water model, and we found that the time step small fluctuation with an amplitude of ± 10 kPa. And it –5 of 2 9 10 s could enable a satisfied accuracy of simu- rises rapidly to the maximum value when the first wheelset lation results. moves at the monitoring section. Subsequently, the dynamic water pressure decreases sharply and forms a 5.2 Spatial and temporal characteristics of dynamic negative water pressure, which finally reduces to a mini- water pressure at the interface mum value when the mid-point of bogie reaches the monitoring section. As the second wheelset is approaching Take the interface gap with a height of 2 mm and a depth the monitoring section rapidly, the dynamic water pressure of 0.7 m as an example; the time-varying characteristics of increases significantly, but it can not reach the pressure the dynamic water pressure due to the vehicle passing at a value as the first wheelset passing through. The dynamic speed of 350 km/h is obtained, as shown in Fig. 16. water pressure exhibits the maximum positive value when the third wheelset moves at the section, and the maximum negative value is generated when the mid-point of the second bogie arrives at the section. After the high-speed vehicle passing, the dynamic water pressure continues to oscillate and gradually decreases. The maximum positive and negative values of the dynamic water pressure at the interface crack tip are obtained when the high-speed vehicle passing through the track with different depths of the interface gap, as shown in Fig. 17. It can be clearly seen that the dynamic water pressure is negligible when the depth of the interface gap is less than 0.3 m. The maximum positive and negative pressures rise slightly when the depth increases from 0.3 to 0.8 m, and Fig. 16 Time-varying characteristics of dynamic water pressure due they increase rapidly once the depth exceeds 0.8 m. Note to the vehicle passing at a speed of 350 km/h Rail. Eng. Science (2020) 28(4):408–423 420 S. Zhu et al. interface is dominated by positive pressures. A pressure difference is formed between the upper and lower surfaces of the track slab, leading to an upward dynamic force acted on the slab. When the fourth wheelset passes through the section, mainly negative dynamic water pressure is pro- duced by the vehicle dynamic load. The pressure is around 0 kPa in the range of 0.43–0.64 m from the gap opening point. The track bed plate bears the resultant force of the atmospheric pressure and the dynamic water force. Note that the variation time of the dynamic water pressure is about 0.014 s, and its significant change in a short time indicates that the track slab is subject to a high-frequency force due to the dynamic water pressure, and the force change its direction promptly. Fig. 17 Maximum positive and negative values of the dynamic water 5.3 Nonlinear mechanical properties of the track pressure at the interface gap with different depth interface that the maximum negative pressure will exceed the satu- Figure 19 exhibits the variation of the interface normal rated water vapor pressure at 20 C when the separation stress under the vehicle dynamic load. With the fast depth reaches 0.9 m. At this point, the liquid water will approaching of high-speed vehicles to the monitoring quickly vaporize into steam, causing unfavorable impact section, the dynamic water pressure at the track interface and denudation to the concrete materials at the ballastless increases rapidly, and it reaches the maximum negative track interface. value when the middle point of the first bogie passes, while Figure 18 shows the distribution characteristic of the the interface normal stress does not have an obvious dynamic water pressure along the depth of the track change. When the second wheelset passes through the interface gap, together with the contour of the pressure. section, the interface normal stress attains its maximum Here the dynamic water pressure induced by the second negative value, and then quickly returns to the vicinity of bogie passage is taken as an example considering that the zero with large amplitude fluctuations. The interface stress dynamic water pressure is found to have the maximum variation due to the second bogie passing has the same positive value when the third wheelset moves at the mon- characteristic as that induced by the first bogie, and it itoring section. oscillates with a certain amplitude after the fourth wheelset As can be seen, when the third wheelset reaches the passing. It is worth pointing out that this maximum nega- monitoring section, the dynamic water pressure is negative tive value of the interface stress is generated by the fourth within the range of 0.05–0.1 m away from the gap opening wheelset passing, and its maximum positive value occurs at point, while it increases with the gap depth increasing the time t after the vehicle passes. It can be concluded that gradually. Overall, the dynamic water pressure at the the maximum water pressure will be induced when the Fig. 18 Spatial distribution characteristic of the dynamic water pressure at the track interface gap: a the third wheelset passing and b the fourth wheelset passing Rail. Eng. Science (2020) 28(4):408–423 Mechanical characteristic variation of ballastless track in high-speed railway: effect of… 421 bonding area at the interface will be reduced, namely the contact area of vehicle dynamic load transferring to the supporting layer will decrease gradually, leading to a more concentrated interface pressure. That is, the negative interface stress will correspondingly increase. Similarly, the maximum stress of the track interface is obtained under different heights of the interface gap, in order to reveal the influence of the gap height on the interface mechanical behavior. It can be clearly seen from Fig. 21 that the maximum positive and negative values of interface normal stress decrease gradually with the inter- face gap height increasing. This is mainly attributed to the Fig. 19 Variation of interface normal stress under vehicle dynamic fact that the dynamic water pressure decreases as the gap load height increases, which leads to a reduced effect of the dynamic water pressure on the track slab, and the maxi- vehicle passes directly above the interface; however, the mum values of the interface stress are gradually closer to maximum positive value of interface normal stress appears those without interface water. after the vehicle passes due to the oscillation of the It is also found that the interface failure at material dynamic water pressure at the interface gap. Therefore, it points would occur due to the coupled dynamic effect of should be noted that the interface normal stress could be vehicle load and water pressure under the condition that the caused by the coupled effect of the vehicle dynamic loads interface gap depth reaches 0.9 m, the gap height is 2 mm, and dynamic water pressure, which will lead to the impact and the vehicle speed is 350 km/h. Figure 22 shows the failure or fatigue failure of ballastless track interface, as time varying characteristic of the interface damage when will be discussed below. the vehicle is passing through the section. Note that here To illustrate the influence of the interface gap depth, the the mechanical parameters of the cohesive zone model are maximum normal stress of the track interface is calculated the same as those in Sect. 3. under different depths of the interface gap, while the other As can be seen, there are obviously two damage parameters keep unchanged. development stage with instantaneous jumps. The damage As shown in Fig. 20, the maximum positive and nega- level jumps from 0 to 0.72 at time step t after the first tive values of the interface normal stress increase gradually bogie passing. Subsequently, the damage level remains at with the increase in the interface gap depth. This is due to 0.72 until it jumps to 1 at the time step t after the second the fact that the dynamic water pressure increases with an bogie passing, and the complete failure at the current increase in the gap depth, and its effect on the upward material point occurs as the damage level increases to 1. It movement of the track slab will be also gradually is worth pointing out that the time of interface material enhanced, resulting in an increase in the positive interface failure is consistent with the time when the interface stress stress. After the interface gap is further propagated, the reaches the maximum value after the second bogie passes Fig. 20 Influence of interface gap depth Fig. 21 Influence of interface gap height Rail. Eng. Science (2020) 28(4):408–423 422 S. Zhu et al. (2) The shapes of the track interface gap are quite different under the negative and positive TGLs, and the stress, displacement and damage laws at different positions of the track interface show different nonlinear variation patterns. The depth variation of the interface gap under different TGLs exhibit strong nonlinear characteristics. The TGL of track slabs that causes rapid damage development should be avoided in practical application in order to prevent the early damage of the track interface. (3) The coupling of longitudinal and vertical dynamic loads of a vehicle under the combined effect of track Fig. 22 Interface damage evolution when the vehicle is passing random irregularities and driving torques could result through the section in interface damage between track slab and CA mortar in the vicinity of the four wheel loading (see Fig. 19). The interface material failure happens in a positions. For evolution of the maximum damage very short time, which indicates the brittle fracture char- variable obtained in this paper, it initiates at acteristics of the track interface. 188 km/h, undergoes rapid propagation at around 226 km/h, and eventually maintains the maximum value after the running speed exceeds 232 km/h. 6 Conclusions (4) The dynamic water pressure presents different spatial distribution characteristic along the interface gap In this work, the effect of train–track interaction and when different wheelsets passing through, and its environmental loads on the mechanical characteristic significant change in a short time indicates a high variation at the interface between track layers of ballastless frequency force acting on the track slab. Under the tracks in high-speed railway has been investigated and calculated conditions, the maximum water pressures discussed systematically. A bilinear cohesive zone model rise gently when the interface gap depth increases considering mixed-mode damage has been employed in a from 0.3 to 0.8 m, whereas they increase rapidly finite element model of a double-block ballastless track to when the depth exceeds 0.8 m. study the effect of loading–unloading processes of the (5) The dynamic water pressure exhibits peak values negative and positive TGLs on the mechanical behavior of when the vehicle passes directly above the track the track interface. Further, the effect of wheel–rail lon- interface. However, the maximum positive value of gitudinal interactions on the nonlinear dynamic character- interface normal stress appears after the vehicle istics of the track interface has been illustrated by using a passage due to the oscillation of the dynamic water vehicle-slab track vertical-longitudinal coupled dynamics pressure at the interface gap, which could lead to model. Finally, the coupled dynamic effect of vehicle load impact failure or fatigue failure at ballastless track and water pressure on the mechanical characteristics and interface. The maximum interface stress gradually damage evolution of the track interface has been revealed increases with the increase in the interface gap depth, using a fluid–solid coupling method. The main findings can while it gradually decreases with the increase in the be summarized as follows: interface gap height. (1) The loading history of the loading–unloading process of the positive and negative TGLs has a significant Acknowledgements This work was supported by the National Nat- influence on the mechanical variation of the track ural Science Foundation of China (Nos. 51708457, 11790283, and 51978587), the Fund from State Key Laboratory of Traction Power interface. With the unloading of the negative TGL, (2019TPL-T16), the Young Elite Scientists Sponsorship Program by the interface stress decreases gradually but the CAST (2018QNRC001), and the 111 Project (Grant No. B16041), interface damage continues to increase. The followed which is gratefully acknowledged by the authors. loading of the positive TGL will lead to the further Open Access This article is licensed under a Creative Commons propagation of the interface crack, and the ability to Attribution 4.0 International License, which permits use, sharing, resist interface damage under the positive TGL will adaptation, distribution and reproduction in any medium or format, as be greatly reduced after the interface separation long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate induced by the former loading–unloading process of if changes were made. 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Railway Engineering Science – Springer Journals
Published: Nov 30, 2020
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