1 Introduction

Ballasted track bed is one of the main forms of railway track structure in China [1,2,3]. However, with the increase of train running speed and bearing capacity, the deterioration of granular ballast bed is accelerated, and the interaction between train, track, and ballast bed is intensified [4,5,6,7]. The granular ballast bed is the weakest part in the track structure. In the long running process, the ballast bed will inevitably have a series of problems, such as settlement [8, 9], ballast particles breakage [10, 11], slurry, and mud, which seriously threaten the safety of train operation [12, 13]. This will not only increase the amount of track maintenance work, but also affect train operation safety and ride comfort. Therefore, it is necessary to control the deterioration of the track bed. As an effective measure to protect the railway structure, under sleeper pads (USPs) have many advantages and has been widely used in the railway industry in various countries [14,15,16].

In terms of experimental research, Kraśkiewicz [17] tested the stiffness performance of four kinds of USPs through fatigue test and assigned the best stiffness of the USPs used in track structure. Esmaeili et al. [18] studied the settlement and stiffness of ballast bed with USPs of different thickness through the ballast box tests and found that installing USPs under the sleeper can reduce the settlement and degradation of ballast bed. Ngamkhanong and Kaewunruen [19] conducted field tests to analyse the vibration behaviour of sleepers with USPs under impact load. The results show that the USPs can reduce the force between wheel and rail by about 10% under extreme impact load. Jayasuriya et al. [20] studied the deformation and degradation of ballast particles in ballast bed with USPs under cyclic load with different stiffness using indoor test devices. They found that the vertical and horizontal displacements of the ballast particles could be reduced by the action of the USPs and that the breakage of ballast particles could also be reduced; however, the contact area between the sleeper and the ballast particles was increased. Mottahed et al. [21] studied the vibration impact of sleepers when the train passes through the bridge deck with USPs, through numerical simulation and field tests. The results showed that the sleepers were able to reduce the vibration by 58% of the maximum value and the vertical displacement by 15% of the maximum value. Navaratnarajah et al. [22] studied the influence of the USPs on ballast bed under cyclic load through large-scale triaxial test. Tests showed that the use of the USPs reduced the amount of stress transferred from the sleeper to the ballast and reduced ballast deterioration. Ngo and Indraratna [23] conducted indoor tests and found that installing USPs between sleepers and ballast could reduce the vertical and horizontal deformation of ballast, increase the contact area between sleepers and ballasts, and reduce ballast breakage. Sol-sánchez et al. [24] used discarded tires to make USPs. Through fatigue testing, they studied the effect of the USPs on the stress distribution and lateral resistance of the ballast bed and found that the use of the USPs can increase the lateral resistance of the ballast bed. Kaewunruen et al. [25] conducted a situ test on the vibration reduction characteristics of the USPs and showed that the use of the USPs increased the vibration of the rails and sleepers and reduced the interaction between the ballast particles.

In terms of numerical simulation, Li and McDowell [26] built the USP model based on the discrete element method, discretized the continuous USPs, analysed the meso-mechanism of the ballast bed under the action of the USPs, and concluded that the use of the USPs can increase the contact number among ballasts and sleepers, so that the load transmitted by the superstructure can more evenly distributed in the ballast bed, and reduced the particle breakage under the sleeper. Zbiciak et al. [27] established the finite element model of the USPs and ballast bed and developed the vibration isolation theory of the USPs by analysing the dynamic characteristics of four kinds of USPs. Paixão et al. [28] used the finite element method to analyse the influence of the USPs on the dynamic characteristics of the transition zone. The results showed that the use of the USPs in the transition zone can control the vertical stiffness of the track, reduce the load transmitted from the sleeper to the ballast, and reduce the vibration of the ballast bed. Krishnamoorthy et al. [29] used the finite element software ABQUS to establish the finite element model of track structure and studied the influence of the USPs with different thickness on the mechanical characteristics of sleepers and ballast beds. The results showed that with the increase of the thickness of the USPs, the sleeper stress was reduced, and the application range of the lower thickness of wood sleepers was obtained. Li et al. [30] established a finite element model for the wheel set–rail–sleeper system; they studied the influence of the USPs on the vertical contact forces between wheel and rail and between sleeper and ballast and concluded that increasing the sleeper width and using the USPs with small stiffness can reduce the contact force between sleeper and ballast. Qu et al. [31] established a finite element structural model to study the vibration reduction effect of ballast bed with elastic elements. The results showed that the vibration between 5 and 30 Hz are amplified, and the vibration above 30 Hz has a good vibration reduction.

In summary, existing experimental studies have mainly focused on the macroscopic aspects of ballast settlement, ballast deterioration, and vibration reduction effects of the USPs, while neglecting the mesoscopic mechanism of the interaction between the ballast, sleeper, and ballast bed with USPs. In the aspect of numerical simulation, most of the above research is to establish the finite element model of ballast bed with USPs, treat the granular ballast bed as a continuum, or establish the 2D discrete element model of ballast bed with USPs while ignoring the dispersity and integrity of the ballast bed. In addition, the existing studies on ballast bed considering the role of USP did not use coupling simulation to increase the accuracy of the analysis, and the studies using coupling analysis to analyse the role of ballast bed did not consider the role of USP. Therefore, in order to overcome the shortcomings of the two present conditions, this article adopts the coupling analysis method to analyse the meso-dynamical behaviour of a high-speed railway ballast bed with USPs, and the differences between ballast track with USPs and without USPs are analysed. The physical model of the ballast particles was reconstructed using 3D scanning techniques, and a 3D multi-flexible dynamic model of the rail–sleeper–USPs–ballast bed system was constructed. Using coupled simulation with discrete element method (DEM) and multi-flexible-body dynamics (MFBD), we analyse the micromechanical dynamic behaviour of a common ballast bed with or without USPs under dynamic load of a train.

2 Modelling

2.1 Establishment of ballast particle model

Four typical representative ballast particles obtained from the ballast bed were selected to acquire point cloud images based on a 3D scanning technique, and the 3D scanning platform is shown in Fig. 1. The geometrically closed entity of the ballast is then obtained by reverse engineering technique. In EDEM, the cluster approach is used to populate the entities. The more spherical particles are populated, the more realistic the geometric description of the ballast particles becomes. In this paper, considering the balance between computational efficiency and model accuracy, 20–40 balls are used for modelling each ballast particle, and ballast breakage is not considered. Figure 2 shows the discrete element models of four typical ballast particles in EDEM, which will be used as template particles to generate ballast bed. Besides, Hertz–Mindlin (no slip) contact model is used to simulate the contact situation among particles, as well as between ballast particles and sleepers and boundaries.

Fig. 1
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Particle 3D scanning platform

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Fig. 2
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Discrete element models of four typical ballast particles

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2.2 Model establishment of ballast bed with USPs

The ballast bed of high-speed railway is composed of ballast particles of different shapes and gradations. According to industry standard TB/T 2140–2008 “Railway Gravel Ballast”, a ballast bed model with specified grading with USPs is established. The specified gradation is shown in Fig. 3. In EDEM, we mainly built the model of granular ballast bed. In addition, the sleepers and rails were built in RecurDyn, and USPs were also built at the bottom of each sleeper. According to industry standard TB/T 10621–2014 “Code for Design of High Speed Railway”, type III sleeper was selected to conduct research in this work. It should be noted that the model developed in this paper simplifies the fasteners, whose main function is to fix the rails and sleepers in place. In the simplified model, fixed constraints are used to fix the rails and sleepers; in order to meet the actual situation and ensure the transmission of force, vertical springs are established between sleepers and USPs. The rails and sleepers are only considered for vertical motion.

Fig. 3
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Model grading

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A three-dimensional model of ballast bed with USPs is established in EDEM, which includes seven sleepers. In the process of model generation, considering the accuracy of calculation and efficiency of the model, the time step is set to 24% Rayleigh time step, i.e., 3 × 10−6. Main parameters of this model are set as follows: The thickness of the ballast bed is 0.35 m, the length between the two ballast shoulders is 3.6 m, the slope is 1:1.75, and the diameter of the ballast particles is between 22.4 and 64 mm. Figure 4 shows the top view and the cross-sectional view of the model. In order to reduce the impact of boundary conditions on the simulation results, the simulation region includes only the middle three sleepers, as shown in Fig. 5.

Fig. 4
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Top view and cross-sectional view of the ballast bed

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Fig. 5
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Simulation area

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In order to achieve the coupling simulation between the multi-flexible body dynamics and discrete elements, the soft processing was performed on rails, sleepers, and USPs in RecurDyn. According to the existing research, the stiffness of the USPs was selected as 80 kN/mm, and the thickness was 11 mm [9]. In order to simplify the model and improve the efficiency of the coupling calculations, only the sleepers and the USPs were meshed in this study. In RecurDyn, the meshes of the sleeper and the USP are shown in Fig. 6. Each sleeper has a total of 1306 nodes and 1218 meshes, and each USP has a total of 1188 nodes and 520 meshes. Figure 7 shows the DEM–MFBD coupling model of the ballast bed with USPs.

Fig. 6
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Meshes in a sleepers and b USPs

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Fig. 7
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DEM–MFBD coupling model

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The material of the ballast is granite, and its main parameters include the elastic modulus, Poisson’s ratio, and density. The physical parameters of each track structure are listed in Table 1.

Table 1 Physical parameters of track structure
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It is necessary to accurately define the contact model and contact parameters when calculating discrete elements. In granular materials, the contact parameters include static friction coefficient, rolling friction coefficient, and impact recovery coefficient, which are only related to the properties of the material itself. In this paper, Hertz–Mindlin (no slip) contact model is selected to simulate the contact situation of granular ballast bed. The static friction coefficient, rolling friction coefficient, and impact recovery coefficient between the ballast and other structures should be calibrated.

Due to the diversity of the ballast, there are still large voids in the model when generating the discrete element model of the granular ballast bed. To ensure the accuracy of the calculated results, it is necessary to initially load the model before the coupled simulation. First, the three sleepers in the middle were loaded simultaneously with a cosine load of amplitude 5−10 kN and frequency 2 Hz for 50 cycles and initially compacted to simulate a running track line. After compacted, the calculated void ratio of the ballast bed is 0.332, which is within the range of internal void ratio of normal clean compacted ballast bed [32].

2.3 Calibration of friction coefficient

The static friction coefficient and impact recovery coefficient were calibrated using the test principles as shown in Figs. 8 and 9, respectively.

Fig. 8
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Principle of static friction coefficient test

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Fig. 9
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Principle of impact recovery coefficient test

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In the process of static friction coefficient test, the ballast particles were first placed on a plate and slowly rotated at a constant speed. During this process, the angle between the plate and the horizontal plane was recorded when the ballast started to move. However, the shape of the ballast was extremely irregular, in order to reduce the testing error, three test data of a typical ballast particle were recorded during the test, and its average value was taken. The static friction coefficient was

$$\mu = \tan \theta,$$
(1)

where μ is the coefficient of static friction, θ is the angle between the plate and the horizontal plane, and the final static friction coefficient of the ballast obtained is 0.861.

Impact recovery coefficient refers to the ability of granular material to recover its deformation after impact, which is mainly related to the properties of the material itself. The impact recovery coefficient is defined as

$$E = \frac{{v_{{2}} - v_{{1}} }}{{v_{{{10}}} - v_{{{20}}} }},$$
(2)

where E is the impact recovery coefficient, \(v_{{2}} - v_{{1}}\) is the velocity difference between the two objects after the impact, and \(v_{{{10}}} - v_{{{20}}}\) is the velocity difference before the impact.

The measurement procedure of the impact recovery coefficient was as follows: Place the ballast sample at the height H of the distance from the ground, and let it fall in the way of free fall, bounce back after collision with the bottom plate during the fall process, record the rebound height H of the ballast sample at this time with a camera, and obtain the speed of the ballast through the fee fall law. Then, the test was repeated ten times, and the average value was taken to be the measured impact recovery coefficient (0.6).

The rolling friction coefficient represents the difficulty of an object’s rolling motion. The inclined rolling test is usually used to obtain the rolling friction coefficient of the sample. In the experiment, the ballast sample was placed on a 60° inclined plate and rolled downwards along the bottom plate at an initial speed of 0 m/s at a fixed height of 30 mm. The distance of the ballast sample rolling on the plane was recorded. The experiment was repeated ten times, and the average value was calculated. Finally, the horizontal rolling distance was 34.532 mm.

The accuracy of the rolling friction coefficient was verified by the natural reposed angle \(\alpha\) as defined in Eq. (3), which refers to the angle between the surface of the material and the horizontal plane when the material was in the natural packing state. During the test, the ballast particles was stacked in a cylinder with a smooth inner wall, and then the cylinder was lifted at a constant speed. The ballast particles in the cylinder will naturally fall under the action of gravity The measurement method of this test are shown in Fig. 10.

Fig. 10
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Reposed angle test

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The reposed angle is

$$\alpha = {\text{arc}}\text{tan} \; \frac{{h_{{1}} }}{r},$$
(3)

where \(\alpha\) is the reposed angle, h1 is the distance from the highest point of the ballast to the floor in the natural packing state, and r is the radius of the ballast pile on the floor under the same circumstances.

In order to reduce the influence of the error, the test was repeated ten times, and the average value was taken. According to the formula, the average value of the reposed angle was finally measured as 37.5°.

It was necessary to obtain the reposed angle in the software simulation through the reposed angle simulation test and compared it with the reposed angle obtained in the laboratory test to verify the correctness of the rolling friction coefficient. In the discrete element software EDEM, a simulation model was established under the same experimental conditions. When verifying the rolling friction coefficient, the impact recovery coefficient and static friction coefficient among the ballast particles were found to be between 0.5 and 1 using the trial-and-error method, he rolling friction coefficient was between 0 and 0.3 using the same method, and the optimal rolling friction coefficient was finally determined to be 0.03 through the optimal arrangement and combination.

In addition, the friction coefficients between ballast particles and sleepers, as well as between ballast particles and subgrade were all obtained by trial-and-error method. The contact parameter values of ballast particles were finally obtained, as shown in Table 2.

Table 2 Discrete element contact parameters
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2.4 Coupling of multi-flexible body dynamics and discrete element model

The coupling of the DEM–MFBD method is achieved on the basis of energy conservation. The difficulty lies in the transfer of mechanical parameters between DEM and MFBD. The coupling principle is shown in Fig. 11. The load, velocity, and other parameters of the sleeper can be obtained by MFBD, while micro-mechanical parameters such as particle displacements and contact forces can be accurately obtained by the discrete element method. The bidirectional transfer of mechanical parameters between DEM and MFBD is achieved through the coupling wall. The specific coupling procedure is as follows: In a cyclic coupling step, MFBD first computes a time step and transmits the data of displacements to DEM through the coupling interface. Then, DEM calculates the interaction between particles and geometry, and then transmits the force and torque back to the MFBD [33]. Based on the data returned by the DEM, MFBD recalculates the geometric displacement information. The coupling principle is as follows.

Fig. 11
figure 11

DEM–MFBD coupling principle

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The fundamental kinetic equation of the particle is

$$\left\{ {\begin{array}{*{20}c} {m^{i} {\varvec{\ddot{r}}}^{i} = \sum\limits_{j = 1}^{{N_{i} }} {{\varvec{F}}^{ij} + } {\varvec{R}}^{i} \;\;\;\;\;\;\;\;\;\;} \\ {J^{i} {\varvec{\ddot{\theta }}}^{i} = \sum\limits_{j = 1}^{{N_{i} }} {{\varvec{q}}^{ij} \times {\varvec{F}}^{ij} + {\varvec{K}}_{{\text{e}}}^{i} } \;\;\;\;} \\ \end{array} } \right.i = 1,\;2, \cdots ,\;N,$$
(4)

where \({m}^{i}\) is the mass of particle i, \({J}^{i}\) is the moment of inertia of particle i, \({\varvec{\ddot{r}}}^{i}\) and \({\varvec{\ddot{\theta }}}^{i}\) are respectively the central position and angle vector of particle i, \({{\varvec{F}}}^{ij}\) is the force exerted on particle \(j\) by contacting particle\(i\), \({\varvec{q}}^{ij}\) is the moment from force \({\varvec{F}}^{ij}\) to the centroid of particle\(i\), \({\varvec{R}}^{i} \;\) and \({\varvec{K}}_{{\text{e}}}^{i}\) are respectively the external force and external moment exerted on particle\(i\), \({N}_{i}\) is the number of particles in contact with particle\(i\), and \(N\) is the total number of particles in the system.

The equilibrium equation of the continuum is

$${\varvec{M\ddot{u}}} = - \left( {{\varvec{C\dot{u}}} + {\varvec{Ku}}} \right) + {\varvec{F}}_{{\text{e}}},$$
(5)

where \(\ddot{{\varvec{u}}}\), \(\dot{{\varvec{u}}}\) and \({\varvec{u}}\) are the acceleration, velocity and displacement vectors of continuum nodes respectively; \({\varvec{M}}\), \({\varvec{C}}\) and \({\varvec{K}}\) are mass, damping and stiffness matrixes, respectively; \({{\varvec{F}}}_{\mathrm{e}}\) is the load of the node. Here, \({\varvec{C}}\) is a constant matrix whose elements take a value of 0.7 according to Refs. [34,35,36].

Since the contact forces on the coupling surface are usually not located at the nodes of the finite element meshes, type function interpolation is used to transfer the contact forces from the discrete element domain to the finite element domain. In the finite element analysis of the track mechanism, the 8-node hexahedral iso-parameter elements are used for mesh generation. The virtual work done by the contact force on the contact surface is

$${\updelta }{\varvec{W}} = {\updelta }{\varvec{U}}^{{\text{T}}} {\varvec{P}},$$
(6)

where W is the virtual work done by the contact force on the contact surface; \({\varvec{P}}\) is the contact force on the coupling contact surface; and \({\varvec{U}}\) is the displacement vector at the contact point, which is obtained by interpolating of type function and the node displacement, and the expression is

$${\varvec{U = N}}_{i}^{{8}} {\varvec{u}}_{i} \;\;\;\left( {i = {1,}\;{2,}...{,}\;{8}} \right),$$
(7)

where \({\varvec{N}}_{i}^{{8}}\) is the shape function of the 8-node isoparametric element at the contact point, and \({{\varvec{u}}}_{i}\) is displacement of the element node.

The virtual work done by the equivalent node force at the node is

$$\updelta {\varvec{W}}_{{\text{e}}} {\varvec{ = }}\updelta {\varvec{u}}_{i} {\varvec{F}}_{{\text{e}}} ,$$
(8)

where \({\varvec{W}}_{{\text{e}}}\) is virtual work done by the equivalent node force at the node; \({\varvec{F}}_{{\text{e}}}\) is equivalent nodal force.

According to the principle of virtual work, the equivalent nodal force can be expressed as

$${\varvec{F}}_{{\text{e}}} = \left[ {{\varvec{N}}_{i}^{{8}} } \right]^{{\text{T}}} {\varvec{P}}\;\;\;\;\;\;\left( {i = 1,\;2,...,\;8} \right),$$
(9)

where

$$N_{i} \left( {\xi ,\eta ,\zeta } \right) = \frac{1}{8}\left( {1 + \xi \xi_{i} } \right)\left( {1 + \eta \eta_{i} } \right)\left( {1 + \zeta \zeta_{i} } \right)\;\;\;\;\;\;\;\left( {i = 1,\;2,...,\;8} \right),$$
(10)

in which \({\xi }_{i}\), \({\eta }_{i}\), and \({\zeta }_{i} \, (i=1, 2,..., 8)\) are the coordinate value of node \(\left(\xi , \eta , \zeta \right)\) in the local coordinate system, which can be taken as 1 or − 1.

3 Model verification and dynamic load application

A test bench for high-speed railway ballast bed cyclic loading was self-developed at a 1:1 scale at the School of Mechanical and Electrical Engineering of Kunming University of Science and Technology, China. The test bench consists mainly of a moving loading system, a cyclic loading system, a hydraulic control system, and a ballast bed structure, as shown in Fig. 12, where rails, sleepers, and ballast bed of actual size are used. Considering the impact and vibration of the surrounding buildings during the test, in order to avoid the occurrence of accidents, an independent vibration isolation design was innovatively proposed when designing the cyclic loading test bench to minimise the impact of vibration.

Fig. 12
figure 12

Cyclic loading test bench

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This test bench can simulate the settlement, vibration, and other physical and mechanical properties of the ballast bed under different train speeds and axle loads. In addition, the test bench has three loading mechanisms, which can exert different forces through the hydraulic system.

Figure 13 shows the principle of the high-speed railway ballast bed cyclic loading test platform, including the rails, sleepers, and ballast bed. The test platform has an overall width of 7 m and a length of 12 m, and the width of the two ballast shoulders is 3600 mm. The platform contains 11 type III sleepers, with standard spacing of 600 mm. The thickness of the ballast layer below the sleeper is 350 mm, the side slope of the ballast bed is 1:1.75, and the shoulder height is 150 mm. In addition, at the bottom of the ballast layer, there are a gravel bottom ballast layer with a thickness of 200 mm and a clay layer with a thickness of 200 mm to simulate the subgrade of the real track bed.

Fig. 13
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Test principle

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Due to the technical limitations of independently designed loading devices, in order to be consistent with realistic conditions of train axle loads, three loading pressures (i.e. 15, 20, and 25 MPa) were chosen for the analysis. To be clear, 15, 20, and 25 MPa represent the oil pressures produced from the loading equipment, which correspond to the train axle loads 18.8, 25, and 31 t, respectively. The forces exerted on the three loading mechanisms are as follows:

$$\begin{gathered} F_{3} = \frac{1}{2}A_{0} \left[ {\sin \left( {2{\uppi }f} \right)t + 1} \right], \hfill \\ F_{4} = \frac{1}{2}A_{0} \left[ {\sin \left( {2{\uppi }ft - \frac{{\uppi }}{2}} \right) + 1} \right], \hfill \\ F_{5} = \frac{1}{2}A_{0} \left[ {\sin \left( {2{\uppi }ft - {\uppi }} \right) + 1} \right], \hfill \\ \end{gathered}$$
(11)

where \({A}_{0}\) is the load amplitude, \(f\) is the frequency, and \(t\) is time. Different phases are used to make the middle sleeper bear 50% of the load, and the sleepers on both sides bear 25% of the load amplitude.

Loading frequency is also a key parameter in the test. Taking China’s CRH3 series high-speed trains as an example, each car of a train is equipped with bogies at the beginning and end, the length of the carriage is 25 m, the wheelbase of the bogies is 2.5 m, and the distance between the two bogies is 17.5 m, as shown in Fig. 14. The frequency of cyclic loading can be calculated by Eq. (12).

$$f = \frac{1000}{{60 \times 60 \times l}} \cdot v,$$
(12)

where \(l\) is the distance between two bogies of the same carriage (m), and \(v\) is the train speed (km/h). In this article, the loading frequency is 2 Hz.

Fig. 14
figure 14

Structure diagram of CHR380 bogie (unit: m)

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Laboratory tests were conducted on the test bench as show in Fig. 12 to verify the correctness of the model. In the test, the acceleration of vibration of the middle sleeper and the settlement of ballast particles 15 mm below the middle sleeper were measured with a load amplitude of 20 MPa. In this case, the corresponding axle weight of the train is 25 t. Figures 15, 16 show the laboratory tested and simulated values of the sleeper acceleration, respectively, under the load of 20 MPa for the same time period. As can be seen from the figures, the peak values of the tested and simulated accelerations are different, but the trend of the curve distribution is essentially the same, within the errors.

Fig. 15
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Laboratory tested values of middle sleeper acceleration

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Fig. 16
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Simulated values of middle sleeper acceleration

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Figure 17 shows the laboratory tested values and simulated values of the displacement of ballast particles under the load of 20 MPa for the same time period, in which the ballast was 15 mm below the middle sleeper. As can be seen from the figure, the change trend of displacement curves is the same; the displacement curves present a trend of rapid increase at first, and then tends to be flat. However, due to the dispersity of the ballast particles, the arrangement of ballast particles in the ballast bed is not consistent. Despite the preload, there are still some differences between the laboratory model of the ballast bed, which makes the displacement curves of the two particles different, within the errors. Thus, the correctness of the DEM–MFBD coupling model can be verified, and hence can be used in subsequent work.

Fig. 17
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Laboratory tested and simulated values of ballast displacement

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4 Analysis of calculation results

4.1 Contact quantity among sleepers and ballast particles

The number of contacts between sleepers and ballast particles reflects the size of the contact region [37]. The larger the number of contacts between them, the larger the contact area, and the more uniform the load transfer from the sleepers to the ballast bed; conversely, the smaller the number of contacts, the smaller the contact area, and the more concentrated the load transferred from the sleeper to the ballast bed.

Figure 18 shows the contact quantity between sleeper and ballast particles for the two track structures (i.e. with and without USPs) in three loading conditions. At a load of 15 MPa, there is little difference in the number of contacts between the three sleepers with USPs and the ballast particles. At the applied load of 20 MPa, the number of contacts between sleepers No. 3 and No. 5 and ballast particles was less than that between sleeper No. 4 and ballast particles. At the applied load of 25 MPa, the number of contacts between sleeper No. 4 and ballast particles is the largest. At zero time, the number of contacts between the sleepers and ballast particles is small, but it increases rapidly as the load is applied. The reason is that the sleeper moves downwards under the load, further compressing the ballast bed, resulting in an increase in the number of contacts between the two. Besides, we can see that under the action of the USPs, the larger the load amplitude, the greater the number of contacts between the sleepers and ballast particles. Under the load transmitted from the sleeper, the elasticity USPs causes them to deform; this increases the contact area between ballast particle and sleepers, thus leading to a higher number of contacts between sleepers and ballast particle. As the load amplitude increases, the number of contacts between the sleepers and ballast particle increases further. At an applied load of 25 MPa, the number of contacts between the sleepers and ballast particle increases significantly. Table 3 shows the maximum and average values of the number of contacts between sleeper No. 4 and ballast particles for the three operating conditions with or without USP.

Fig. 18
figure 18

Contact quantity between sleepers and ballast particles under different loads with or without USPs: a 15 MPa with USPs; b 15 MPa without USPs; c 20 MPa with USPs; d 20 MPa without USPs; e 25 MPa with USPs; f 25 MPa without USPs

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Table 3 Number of contacts between sleeper No. 4 and ballast particles under different load amplitudes
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It can be seen from Table 3, for the three operating conditions, the number of contacts between the sleepers and ballast particles in the ballast bed with the elastic USPs is larger than that in the ordinary ballast bed. At a load of 15 MPa, the maximum and average number of contacts between the elastic sleepers and ballast particles is 1.64% and 2.1%, respectively, more than that in a normal ballast bed; at a load of 20 MPa, the maximum and average contact number of the ballast bed with the elastic sleeper is 1.94% and 2.32% more than that of the normal ballast bed, respectively; at a load of 25 MPa, the maximum and average number of contacts between the sleepers and the ballast particles is 9.42% and 13.97%, respectively, larger in the ballast bed with the elastic sleepers than that in the normal ballast bed. For the ordinary ballast bed, however, it is worth noting that compared with the two working conditions with loads of 15 and 20 MPa, the contact quantity between sleepers and ballast particles is less under the load of 25 MPa. When the ordinary ballast bed is subjected to a large load, the ballast particles are subjected to a greater force and move greatly, therefore resulting in a decrease in the contact quantity of sleepers and ballast particles.

4.2 Contact force of ballast particles

In order to analyse the spatial distribution law of the contact forces of the ballast particles under the influence of the cyclic load of the train. Figure 19 shows the vectorial distribution of the contact forces of the ballast particles in the cross section of the two track structures (i.e. ballast bed with or without USPs) for three loading conditions.

Fig. 19
figure 19

Distributions of ballast contact force vectors under different loads with or without USPs: a 15 MPa with USPs; b 15 MPa without USPs; c 20 MPa with USPs; d 20 MPa without USPs; e 25 MPa with USPs; f 25 MPa without USPs

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As can be seen from Fig. 19, the colour depth indicates the value of the contact forces, and the arrows indicate the direction of the contact forces of the ballast particles. The contact forces among the ballast particles appear mainly below the sleepers and spread to both sides in a fan-like pattern, indicating that the contact forces among the ballast particles are not evenly distributed, but mainly concentrated in the bearing region of the ballast bed. As the depth increases, the indirect contact forces of the ballast particles gradually decrease. The contact forces of the ballast particles appear at the shallow ballast bed below the sleepers and becomes relatively small at the bottom of the ballast bed. As the depth of the ballast particles increases, the contact forces of the ballast particle propagate to both sides, resulting in an enlargement of the contact area and a decrease in the amplitude of the contact forces, which disperse and absorb the train load.

In both track structures (i.e. with and without USPs), the distribution of the contact force vector among the ballast particles shows little change when the load amplitude is increased from 15 to 20 MPa, but it increases significantly when the load amplitude is increased to 25 MPa. As the load amplitude increases, the contact forces also increase. At load amplitudes of 15 and 20 MPa, the effect of the USPs on reducing the contact forces of the ballast particle is less pronounced. When the load amplitude is changed to 25 MPa, the contact forces among ballast particles increase significantly, and the effect of the USPs in reducing the contact forces among the ballast particles is also apparent. An increase in the load amplitude causes an increase in the indirect contact forces among the ballast particles, and the effect of installing the USPs is noticeable when the load amplitude is larger.

4.3 Vibration analysis of ballast at different depth

To further analyse the mesoscopic motion law of the granular ballast bed under the cyclic load of the train, five monitoring points below the sleepers were chosen to detect the motion of the ballast particles at different depths. The monitoring points selected in the discrete element model is shown in Fig. 20, where point 1 located 200 mm below sleeper No. 3, monitoring points 2, 3, and 4 located at 50, 200, and 350 mm below sleeper No.4, respectively, and monitoring point 5 located at 200 mm below sleeper No.5. Points 1, 3, and 5 are at the same depth.

Fig. 20
figure 20

Schematic diagram of monitoring points (unit: mm)

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The vibrational level of the ballast bed is important for assessing its stability. As the speed and bearing capacity of trains increases, the interactions between track and sleepers, between sleepers and ballast particles, and among the ballast particles increase. In order to explore the vibrational response law of the ballast particles at different depths of the granular ballast bed under the cyclic load of the train, the vibrational transport law of the granular ballast bed was explored by recording the vibrations of the ballast particles at each monitoring point. Figure 21 shows the acceleration time histories at monitoring points of the two kinds of ballast bed structures under three loading conditions.

Fig. 21
figure 21

Acceleration time histories of monitoring points in different working conditions: a 15 MPa with USPs; b 15 MPa without USPs; c 20 MPa with USPs; d 20 MPa without USPs; e 25 MPa with USPs; f 25 MPa without USPs

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As can be seen from Fig. 21, during the initial stage of loading, the velocity fluctuations are large because of the large gap among the ballast particles and the dislocation of the rearrangement of the ballast particle, so that only the time-history curve of the ballast acceleration from 0.5 to 5 s is recorded. In both track structures, the peak acceleration of the ballast at each monitoring point increases with the load amplitude. Under the same operating conditions, monitoring points 2, 3, and 4 are located at different depths under the same sleeper, and the peak acceleration of the ballast particle decrease with the increased depth. The USPs had a noticeable effect of reducing vibration in all the three loading conditions. On the one hand, it reduces the peak acceleration of ballast particle at most monitoring points. On the other hand, the change in ballast acceleration is very small. Due to the discreteness of the ballast particles, not all monitoring points satisfy the above rules.

4.4 Analysis of mesoscopic settlement characterristics of ballast bed

Under cyclic load, discrete ballast particles move and dislocate under stress, which further reduces the gap between the ballast particles and makes the ballast bed more compact, which is generally manifested as settlement of the ballast bed. The settling mechanism of granular ballast bed is very complex and is related to many factors. In order to visually analyse the settlement of the granular ballast bed, we take the vertical displacement of the sleepers as the settlement of the ballast bed. Figure 22 shows a comparison plot of the settlement of the ballast bed with USPs and the normal ballast bed at three load amplitudes.

Fig. 22
figure 22

Sleeper settlement in different working conditions: a 15 MPa with USPs; b 15 MPa without USPs; c 20 MPa with USPs; d 20 MPa without USPs; e 25 MPa with USPs; f 25 MPa without USPs

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As can be seen from Fig. 22, the settlement of the ballast bed with or without USPs can be roughly divided into two stages: The initial rapid settlement stage and the later slow settlement stage. In the early stage of rapid settlement, there are large gaps in the granular ballast bed and the number of contacts among ballast particles is small. When subjected to the train load, the ballast particles in the ballast bed squeeze and bite into each other. The movements and rearrangements of the ballast particles will reduce the internal space of the ballast bed and further compact the ballast bed, resulting in rapid settlement of the ballast bed. In the slow settlement stage, the internal clearance of the ballast bed is small, and there are fewer particle dislocations and rearrangements. Although there is no local and large-scale movement of ballast particles, the ballast bed also settles under the influence of the train load, but the settling rate becomes slower. This simulation does not consider ballast crushing, so the crushing settlement stage is not reflected in the curves. Due to the dispersion of ballast particles and the difference of load application, the settlement of each sleeper is different; however, the overall settling trend is the same. It can be seen from the figure that under the three working conditions, the settlement of each sleeper is reduced by the elastic sleeper in different ranges, but the settlement and reduced range of each sleeper is different.

It can be seen from Table 4 that the sleeper settlement gradually increases with the increase of the load. Especially when the load is 25 MPa, the sleeper settlement increases sharply. At a load of 20 MPa, the settling depth of sleepers 3, 4, and 5 is 4.55, 4.55, and 5.05 mm, respectively. When the load is increased to 25 MPa, the settling depth of sleepers 3, 4, and 5 is 8.66, 8.13, and 8.17 mm, respectively, showing a respective increase of 90.3%, 78.7%, and 61.8% in settlement compared with the results at 20 MPa. This indicates that load has a large impact on the settlement of the sleepers. Also, as can be seen from Table 4, the application of USPs can significantly reduce the settlement of sleepers. At a load of 15 Mpa, the settlement of sleepers with USPs is reduced by 6.3% to 27.2% compared with that without USPs. At a load of 20 Mpa, the settlement is reduced by 15.4% to 28%. At a load of 25 Mpa, the settlement is reduced by 15.6% to 27.5%.

Table 4 Sleeper settlement under different load amplitudes
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From the above analysis, the settlement of sleepers is reduced when USPs are applied. The settlement of different sleepers is different under the same working condition, and the reduction of the settlement of each sleeper with USP is also different, but the data show that the application of the USPs can reduce the sleeper settlement.

Taking the ballast particles at the monitoring points in Fig. 20 as the object, by recording the changes of the vertical displacement of them with the application of load, the settlement of ballast particles is characterised by the vertical displacement, and the movement rule of the ballast particles at different positions is analysed.

Figure 23 shows the displacement curves of the ballast particles at the monitoring points inside the ballast bed. We can see that monitoring point 2 is the closest to the sleeper under the three loads, regardless of whether it is the ballast bed with USPs or without USPs. The contact force at point 2 is greater than other points, resulting in larger displacement. Monitoring point 4 is the farthest from the bottom of the sleeper and it receives less force, where the displacement of ballast particle is small. Monitoring points 1, 3, and 5 are at the same depth, so their displacement is close. Due to the dispersion of the ballast particles, the vertical displacement of the ballast particles at each monitoring point shows strong stochasticity. The amplitude of displacement of ballast particles varies with monitoring points, but the trend is the same. In all the three operating conditions, the ballast displacement at the monitoring points is reduced to varying degrees by the action of the USPs. The main reason is that the USPs makes the load transmitted from the sleeper to the ballast bed more uniform, reducing the stress on the ballast particles, which in turn reduces the displacement of the ballast particles.

Fig. 23
figure 23

Ballast displacement curve at monitoring points: a 15 MPa with USPs; b 15 MPa without USPs; c 20 MPa with USPs; d 20 MPa without USPs; e 25 MPa with USPs; f 25 MPa without USPs

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As can be seen from Fig. 23, the settlement of the granular ballast bed occurs mainly in the region at the bottom of the sleepers, as the ballast particles have the largest range of motion and displacement at this location. By installing an elastic USP at the bottom of the sleeper, the contact area among the sleeper and the ballast particles is increased. As a result, the load transmitted from the sleeper to the ballast bed is more evenly distributed, and the settlement of sleeper is reduced. In this case, the stress on the ballast particles is also reduced, so does its vertical displacement.

5 Conclusions

  1. (1)

    In this paper, the geometrically irregular and refined discrete element models of ballast particles are established by 3D scanning, and the DEM–MFBD coupling model of multi-structure interaction in engineering scale is established. The correctness of the coupling model is verified by experiments.

  2. (2)

    As the amplitude of the load increases, the deformation of the elastic USPs increases, which increases the area of contact between the sleepers and the ballast particles, so the load transmitted from the sleepers to the ballast bed can be evenly distributed.

  3. (3)

    The contact force between ballast particles is mainly concentrated at the bottom of the sleeper and extends to both sides of the sleeper in a fan-shaped attenuation. Under the action of the elastic USPs, the contact forces between ballast particles are decreased, and the reduction effect is more obvious when the load amplitude is larger.

  4. (4)

    With the depth of ballast bed increasing, the peak acceleration of the ballast bed decreases gradually. After applying the USPs, the peak acceleration of the ballast at most monitoring points decreases, which makes the acceleration fluctuation of the ballast decrease.

  5. (5)

    The settlement of the granular ballast bed mainly occurs at the shallow position of the bottom of sleeper. The application of elastic USPs at the bottom of the sleeper can effectively reduce the stress on the ballast particles and reduce their range of motion.