1 Introduction

The ballasted track is one of the most basic track structures of railways [1]. Due to its low cost, easy maintenance, and significant advantages in adaptability and flexibility, it is widely used worldwide [2, 3]. The ballast bed is an integral part of the track structure, which has the functions of maintaining the stability of the track and mitigating the impact and vibration of the wheel rail [4]. However, it comprises ballast particle materials of varying sizes and gradations, resulting in internal voids [5]. Under the influence of train operations and external environmental factors, sand particles continuously permeate these voids, altering the granular composition and particle size distribution of the ballast bed. It leads to typical maintenance issues such as poor drainage and ballast clogging, thereby reducing the service performance of the ballast bed and increasing maintenance costs. In severe cases, it may even threaten the stability and safety of train operations [6, 7]. Therefore, in-depth research into the influence of wind-blown sand environments on the service quality of the granular ballast bed is essential for the scientific maintenance and repair of sandy railways, ensuring the safe and stable operation of trains. Figure 1 shows typical sandy railway lines in some countries around the world.

Fig. 1
figure 1

Typical sandy lines in some countries: a China railway; b Germany railway [8]; c Italy railway [9]; d Iran railway [10]; e Jordan railway [11]; f Austria railway [12]

Full size image

The sandy ballast bed is a discontinuous, non-uniform, and complex mixed multi-phase medium composed of ballast and sand particles of different sizes, exhibiting significant multi-scale effects. The internal stress and deformation mechanisms are complex. The penetration of sand particles into the ballast bed has caused a series of line service status issues, attracting the attention of many researchers worldwide.

Regarding experimental research, Kian et al. [13] conducted field tests on railways in sandy areas. The results show that sand penetration alters the stress mode of the sleepers. Moreover, as the sand content of the ballast bed increases, the pressure distribution under the concrete sleeper gradually becomes more uniform. Carrascal et al. [14] carried out indoor simulation tests on high-speed railway fasteners in sandy areas, showing that the presence of sand has little effect on individual fasteners. Kumara and Hayano [15] analyzed the relationship between sand content and ballast bed settlement based on a 1:5 scale indoor test of the ballasted track. Their results demonstrate that when the sand content of the ballast bed exceeds 30%, the cumulative deformation duration of the ballast bed in the early stage of loading is prolonged, and the ballast bed settlement amplitude becomes more extensive in the later stage. Sadeghi et al. [16] performed large-scale indoor direct shear tests and discovered that geogrids can enhance the shear strength of sandy railway ballast by 25%, and explored the optimal placement position and size of the geogrids. Huang and Qian [17] conducted large-scale triaxial tests, finding that incorporating a single layer of geogrid can effectively mitigate the settlement and deformation rates of fouled railway ballast under conditions of equivalent moisture content. Ebrahimi et al. [18] examined the plastic deformation of railway ballast under fouling conditions using a large-scale cyclic triaxial apparatus, investigating how fouling material affects the plastic deformation of railway ballast. Danesh et al. [19] introduced three different quantities of fouling material to clean ballast based on the fouling index. They conducted direct shear tests on clean ballast, sand-fouled ballast, and clay-fouled ballast and indicated that shear strength diminishes as the percentage of fouling material content increases. Indraratna et al. [20] executed a series of direct shear tests. They proposed a conceptual normalized shear strength model to predict the reduction in peak shear stress and peak shear angle under a given normal stress. Hussaini et al. [21] performed indoor experiments to evaluate the deformation and deterioration behavior of fouled ballast beds with geogrids and elastomers. Their results indicate that both geogrids and elastomers reduce the deformation of the ballast bed. Singh et al. [22] conducted field tests with the installation of geotextiles, finding that placing geotextiles reduces track fouling and extends the maintenance cycle. Li et al. [23] carried out a field evaluation on the impact of geotextiles on railway ballast degradation and deterioration, showing that geogrid reinforcement can significantly reduce the strain rate. Tyfour [24] highlighted that sand impedes the movement of ballast particles and increases the stiffness and damping of the ballast bed. Rahman et al. [25] and Parsons et al. [26] et al. validated the use of resistivity to evaluate the filling of ballast gaps through numerous experiments. Ichi et al. [27] studied the application of hyperspectral imaging (HSI) to characterize fouled ballast and explored how the presence of moisture and contaminants in the ballast affects its spectral characteristics. This provides a new approach to understanding and monitoring the condition of railway ballast.

Regarding numerical simulation, Ngo et al. [28] developed a direct shear simulation model of a fouled ballast bed and indicated that dirty media can reduce the interlocking capacity of the ballast, leading to a decrease in shear strength. Ngamkhanong et al. [29] discovered that dirt can reduce the lateral stability of the ballast bed by 50% and increase the risk of track buckling. Xiao et al. [30] established a 2D DEM model of a sandy ballast bed based on the dimensional analysis method. Their results show that sand penetration reduces the vibration level of the ballast bed under dynamic train load and mitigates the deformation of the ballast bed. Esmaeili et al. [31] applied the finite element theory to study the impact of sand penetration on environmental vibration along the railway. Their results indicate that as the sand content increases, the environmental vibration along the railway becomes more pronounced. Kian et al. [32] employed a finite element model to investigate the impact of railway dust pollution on the load action mode of prestressed concrete sleepers and proposed a new design and calculation method for sandy areas. Zakeri et al. [33] established a 2D finite element model of a sand-containing railway track under plane strain conditions. Their findings demonstrate that sand penetration increases the vertical vibration acceleration of the track structure. Charoenwong et al. [34] introduced a new numerical modeling approach combined with a novel ballast settlement model to simulate the impact of ballast fouling material content and moisture content on the evolution of track geometry. Their results indicate that higher levels of ballast fouling lead to rapid deterioration of track geometry, especially in the presence of moisture. Liang et al. [35] validated the feasibility of using infrared thermography for detecting the degree of fouling in ballast beds. Their findings suggest that moisture exacerbates the difference in heat transfer rates between clean and fouled ballast beds, with the fouled ballast bed exhibiting a higher heat capacity than the clean ballast bed. Li et al. [36] developed a numerical simulation model based on random irregular polygons and collision detection algorithms using the finite-difference time-domain algorithm for simulation. The degree of fouling in the ballast bed was identified using Hilbert transform energy, S-transform, and energy integral curve. Gong and Qian [37] statistically analyzed the RGB (red, green, blue) color distributions of fouled ballast images and found a strong linear correlation between the fouling index and variance, which can be used to quantify the degree of ballast fouling. Koohmishi and Guo [38] utilized large-scale flume experiments to measure the water level height on the cross section of the clay-fouled ballast bed and developed machine-learning models to predict the water level height. Their results indicate that the rainfall rate and fouling index are the most influential variables affecting the water height in the clay-fouled ballast layer of the railway track.

Based on the above, it is evident that while existing research has explored the mechanical properties of sandy ballast beds, there remain some gaps. For instance, most experimental studies are based on scaled-down laboratory experiments with limited field tests, which may not effectively capture the internal working mechanisms and action principles of the multi-scale overall ballast bed granular system during wind-blown sand penetration. Additionally, there is a scarcity of research on the evolution of ballast bed quality under the effect of wind-blown sand penetration, particularly in microscopic analysis. Furthermore, the interaction between particles of different sizes has not been fully considered. In terms of research methods, most existing studies on sandy railways have relied on the finite element method, which lacks a comprehensive multi-scale analysis from the microscopic to the macroscopic level and cannot accurately reflect the microscopic contact characteristics between 3D ballast and sand particles.

Building on this, the present study adopts the actual shapes of track ballast particles and considers the complex contact relationships between ballast and sand particles to establish 3D multi-scale analysis models of sandy ballast beds with different sand contents. These models can precisely simulate the movement of the granular ballast bed structure and provide detailed information at the particle microscopic scale. By combining field experiments with model simulations, this study systematically analyzes the influence of wind-blown sand content on the mechanical quality state of ballast beds from both macroscopic and microscopic perspectives. It reveals the changes in particle contact characteristics, resistance properties, and stiffness features of the ballast bed with different sand contents. This research provides a theoretical basis and technical guidance for the rational design and scientific maintenance of ballasted tracks in sandy railways.

2 Establishment and validation of numerical simulation model

2.1 Establishment of sandy ballast bed model

The sandy ballast bed is a unique track structure formed by the combined action of sand particles, external environmental factors, and dynamic train loads. To precisely simulate and analyze the formation mechanism of the sandy ballast bed and its impact on the evolution of the mechanical quality state of ballasted tracks, the 3D numerical simulation method is particularly crucial. Initially, a clean ballast bed model (with 0% sand content) is established. Based on this, generate sand particles, spread them in layers, compact them layer by layer, and stabilize the model through the action of gravity, thereby obtaining a sandy ballast bed model.

2.1.1 Clean ballast bed model

Railway ballast particles are distinct in shape, with precise edges and a significant multi-scale effect [39, 40]. The geometric characteristics of ballast particles notably impact the mechanical properties of the ballast bed, such as stiffness and cumulative deformation [41, 42]. The restoration and reconstruction of the actual contour of ballast particles are crucial for achieving an accurate simulation of the ballast bed [43].

Randomly select three samples of ballast particles in each of four shapes: spherical, ellipsoidal, polyhedral, and flattened spheroidal, from sections severely affected by wind-blown sand. Initially, laser scanning is used to reproduce the actual contours of the ballast particles. Subsequently, a method of filling spheres is employed to simulate the ballast particles.

Researches [44, 45] has demonstrated that accurately representing the interlocking and wedging action between real particles can be achieved by filling eight or more spheres to simulate a single ballast. Therefore, to balance computational time and accuracy, larger sphere units are filled in the central area of the ballast profile template, while relatively smaller sphere units are filled in the boundary range of the template. As illustrated in Figs. 2, 14–17 sphere units are utilized to simulate ballast particles.

Fig. 2
figure 2

Modeling of irregular ballast particles

Full size image

According to the Chinese railway industry standard "Railway Ballast" (TB/T 2140-2008) [46], and consider the actual ballast gradation distribution in the sandy railway, ballast particles are generated by the gradation of grade I for existing lines, as presented in Table 1.

Table 1 Ballast particles gradation
Full size table

The ballast particles are generated using the “rainfall method” to form a pile under their weight, with a ballast bed thickness of 350 mm. The sleepers are new type II concrete sleepers, 2500 mm long with a spacing of 600 mm. To enhance the accuracy and reliability of the numerical simulation results, a ballasted track model containing three sleepers (with a longitudinal length of 2100 mm) is established, and the middle sleeper is chosen for simulation analysis. This approach effectively minimizes the impact of boundary effects and ensures reasonable simulation computational efficiency. Once the ballast particles are generated, walls are used to shape the ballast bed's slopes and shoulders, with a top width of 3500 mm for the ballast bed, a slope of 1:1.75 for the side slopes, and a shoulder height of 150 mm. The ballast bed is compacted using a layered compaction method, with a compaction density of 1750 kg/m3. Finally, a geometric model of new type II concrete sleepers is established and imported into the DEM module, completing the establishment of the numerical simulation model for a clean ballast bed, as illustrated in Fig. 3.

Fig. 3
figure 3

DEM model of a clean ballast bed

Full size image

2.1.2 Sand gradation acquisition

The gradation of sand particles is one crucial physical property, as particles of different sizes affect the ballast bed's mechanical properties in various ways, such as compaction, drainage performance, and shear strength [47]. By analyzing the sand particle gradation, a more accurate model can be established to analyze the mechanical behavior of the ballast bed under sand penetration, thereby enabling a better assessment of its mechanical performance.

A sieving test is conducted to determine the size distribution of the sand particles from the Ganqimaodu–Wanshuiquan railway in China. By sieving a certain amount of sand particles through screens with different aperture sizes and based on the size of the holes, the sieved particles are divided into different grades. The percentage of the total mass represented by each particle size grade is calculated to obtain a sand gradation curve, which reflects the composition and distribution of the sand particles. The sieving process and results are presented in Fig. 4.

Fig. 4
figure 4

Screening of sand particles

Full size image

The sand particle gradation curve obtained is shown in Fig. 5. In sandy ballast beds, the significant difference in particle size between ballast and sand particles leads to a pronounced multi-scale particle effect, which cannot be overlooked. Notably, in the established 3D DEM model, numerous suspended sand particles appear in the voids of the ballast, making it challenging for the model to achieve a balanced state. This also significantly reduces computational efficiency. Kozicki et al. [48] and Ahmadi and Sizkow [49] conducted research and concluded that appropriately enlarging the size of sand particles has a relatively small impact on the results of the DEM model, and the use of the particle scaling simulation method is considered reasonably effective. From the literature review, it is known that the particle size was enlarged by three times in DEM simulations [50, 51]. Therefore, this study also adopts the method of enlarging the particle size by three times for simulation, resulting in modeling particle sizes ranging from 0.225 to 3 mm.

Fig. 5
figure 5

Sand particle gradation curve

Full size image

2.1.3 Ballast bed models with different sand contents

To investigate the impact of wind-blown sand content on the evolution of the mechanical quality state of ballast beds, the generation of sand particles continues using the rainfall method based on the clean ballast bed model described in Sect. 2.1.1. The actual structure of sand particles is irregular, but simulating them according to their actual shape would significantly increase the computational workload. Therefore, to simplify the calculation and improve the computational efficiency of the model, this study simulates the sand particles using single spherical particles, with the gradation of sand represented by the particle gradation curve shown in Fig. 5. The percentage void contamination (PVC) index, proposed by Feldman and Nissen [52], is used to evaluate the sand content in the ballast bed by calculating the proportion of sand particles in the total porosity of the ballast bed. This method defines sand particles with a particle size of up to 9.5 mm, suitable for calculating fine particle materials with a diameter of 9.5 mm or smaller. However, despite the theoretical clarity of this assessment standard, its practical application is limited due to the operational difficulties of volume-based calculations. Considering this limitation, this study proposes a new sand content assessment indicator to enhance its practicality in field operations. The formula for calculating it is shown in Eq. (1).

$$\alpha = \frac{{m_\text{s} }}{{m_\text{b} }} \times 100\%,$$
(1)

where \(\alpha\) is the sand content, \(m_\text{s}\) is the sand mass, and \(m_\text{b}\) is the ballast mass.

The actual modeling process involves the following steps:

  1. 1)

    Calculation of sand particle mass: When establishing a clean ballast bed, five base plates are reserved at the bottom of the sleeper in the depth direction, at depths of 70, 140, 210, 280, and 350 mm, respectively. Based on the total mass of ballast particles in the clean ballast bed model described in Sect. 2.1.1 and a sand content of 5%, the required mass of sand particles is calculated.

  2. 2)

    Model generation with a sand content of 5%: A layered sand spreading method is employed, involving a total of five sand spreading operations. Each operation distributes an amount equivalent to 1/5 of the total mass. The rainfall method is utilized to generate the required mass of sand particles. The sand spreading operation for different ballast bed layers is completed through the manipulation of the base plates. Subsequently, the model is stabilized under the action of gravity and compacted layer by layer. Finally, a sandy ballast bed model with a sand content of 5% is obtained, as illustrated in Fig. 6a. In this model, dark gray particles represent ballast particles, yellow particles represent sand particles, and green objects represent sleepers.

  3. 3)

    Increasing the sand content to 10%: Based on step 2, the layered sand spreading method is continued, with the rainfall method being used to add an additional 5% sand content, completing five more sand spreading operations. After stabilization and compaction, a sandy ballast bed model with a sand content of 10% is formed, as illustrated in Fig. 6b.

  4. 4)

    Further increasing the sand content: Based on the previous step, the sand content is sequentially increased by spreading sand in layers and compacting each layer. Finally, ballast bed models with sand contents of 15% and 20% are obtained, as illustrated in Fig. 6c and d.

Fig. 6
figure 6

Simulation model of sandy ballast bed with different sand contents: a 5%; b 10%; c 15%; d 20%

Full size image

It is important to note that since the models with higher sand content are constructed based on the models from the previous steps in the modeling process, the differences between the various models can be attributed to solely differences in sand content.

2.2 Contact model and parameters

This study utilizes the DEM to explore the impact of wind-blown sand content on the evolution of the mechanical quality state of ballast beds. Since both sand and ballast particles are non-cohesive materials in a dry state, the Hertz–Mindlin (no-slip) model is employed for contact force update calculations to improve model computational efficiency and ensure accuracy [53, 54]. In the DEM model, it is necessary to calculate contact forces between sand particles, between ballast particles, between ballast and sand particles, between sand particles and sleepers, and between ballast particles and sleepers. These contacts can all be approximated as contacts between spheres. Therefore, in calculating contact forces, the main considerations are the overlap and the nature of contact between spheres, as illustrated in Fig. 7.

Fig. 7
figure 7

Particle contact model

Full size image

The contact force between particles is divided into normal contact force \({\varvec{F}}_\text{n}\) and tangential contact force \({\varvec{F}}_\text{t}\). The normal contact force is calculated based on the Hertz contact theory [55]. The tangential contact force is computed based on the study of Mindlin and Deresiewicz [56]. Rolling friction is obtained by contact-independent directional constant torque modeling [57]. The specific calculation formulas are derived from Eqs. (2) and (3):

$${\varvec{F}}_\text{n} = {\varvec{F}}_\text{n}^\text{e} + {\varvec{F}}_\text{n}^\text{d},$$
(2)
$${\varvec{F}}_\text{t} {\varvec{ = }}{\varvec{F}}_\text{t}^\text{e} {\mathbf{ + }}{\varvec{F}}_\text{t}^\text{d},$$
(3)

where \({\varvec{F}}_\text{n}^\text{e}\) is the normal elastic force, \({\varvec{F}}_\text{n}^\text{d}\) is the normal damping force, \({\varvec{F}}_\text{t}^\text{e}\) is the tangential elastic force, and \({\varvec{F}}_\text{t}^\text{d}\) is the tangential damping force. Damping describes the energy dissipation during the contact process between particles and is typically divided into normal damping and tangential damping [58]. Normal damping simulates the energy loss during the collision process between particles, preventing excessive rebound. Tangential damping simulates the energy dissipation when particles slide on the contact surface, affecting the frictional behavior.

The normal elastic force \({\varvec{F}}_\text{n}^\text{e}\) and damping force \({\varvec{F}}_\text{n}^\text{d}\) can be obtained by

$${\varvec{F}}_\text{n}^\text{e} = \frac{4}{3}{{E}}^{*} \left( {{{R}}^{*} } \right)^{1/2} {{\varvec{\delta}}}_\text{n}^{3/2},$$
(4)
$${\varvec{F}}_\text{n}^\text{d} = - 2\sqrt{\frac{5}{3}} \beta \left( {{{E}}^{*} m^{*} } \right)^{1/2} ({{R}}^{*} {{\varvec{\delta}}}_\text{n} )^{1/4} {\varvec{v}}_\text{n}^\text{rel},$$
(5)

where \(E^{*}\), \(R^{*}\), and \(m^{*}\), respectively, represent the equivalent elastic modulus, equivalent radius, and equivalent mass; \(\beta\) is the damping ratio; \({{\varvec{\delta}}}_\text{n}\) is normal overlap; \({\varvec{v}}_\text{n}^\text{rel}\) is the normal relative velocity.

The tangential elastic force \({\varvec{F}}_\text{t}^\text{e}\) and damping force \({\varvec{F}}_\text{t}^\text{d}\) can be calculated by.

$${\varvec{F}}_\text{t}^\text{e} = - 8{{G}}^{*} \left( {{{R}}^{*} {{\varvec{\delta}}}_\text{n} } \right)^{1/2} {{\varvec{\delta}}}_{\tau },$$
(6)
$${\varvec{F}}_\text{t}^\text{d} = - 4\sqrt{\frac{5}{3}} \beta \left( {{{G}}^{*} m^{*} } \right)^{1/2} \left( {{{R}}^{*} {{\varvec{\delta}}}_\text{n} } \right)^{1/4} {\varvec{v}}_\text{t}^\text{rel},$$
(7)

where \({\varvec{v}}_{\text{t}}^{\text{rel}}\) is the tangential relative velocity, \({{\varvec{\delta}}}_{\tau }\) is tangential overlap, and \(G^{*}\) is the equivalent shear modulus.

The displacement of particles during motion can be obtained using Newton’s second law. The equation of motion for particle a is shown in Eq. (8).

$$\left. \begin{gathered} m_{a} {\varvec{a}}_{a} = \sum {\varvec{F}} \hfill \\ {{I}}_{a} {{\varvec{\alpha}}}_{a} = \sum {\varvec{M}} \hfill \\ \end{gathered} \right\},$$
(8)

where \(m_{a}\) is the mass of particle a, \({\varvec{a}}_{a}\) is the acceleration, \(I_{a}\) is the moment of inertia, \({{\varvec{\alpha}}}_{a}\) is the angular acceleration, \(\sum {\varvec{F}}\) is the external force acting on the particle at its center of mass, and \(\sum {\varvec{M}}\) is the external torque acting on the particle at its center of mass.

The parameters involved in the model primarily include the shear modulus, density, Poisson’s ratio, as well as the static friction coefficient and rolling friction coefficient. Among these, the shear modulus, density, and Poisson’s ratio are basic physical properties of the material, and these parameters are mainly selected based on existing literature [4, 30, 42]. The static friction coefficient and rolling friction coefficient are contact parameters, initially selected based on existing literature [59,60,61,62]. Through the method of controlling variables and combining the results of the lateral resistance and support stiffness tests of the ballast bed in Sect. 2.3, the simulation model’s calculation parameters are proposed, as illustrated in Table 2.

Table 2 Model parameters
Full size table

2.3 Model reliability test

2.3.1 Field experiments

Field experiments have been conducted on both severely sandy and clean sections of the Ganqimaodu–Wanshuiquan railway in China. The test railway line is classified as a grade I line, with ballast material consisting of basalt crushed stone. The track system comprises 60 kg/m steel rails, new type II concrete sleepers, type II elastic fastenings, and a ballast bed. For all track structures, the dimensional parameters are consistent with the DEM simulation model described in Sect. 2.1. The trains operating on the site are C70E, with a fixed axle distance of 1830 mm and an axle load of 23 t, traveling at a speed of 45 km/h through the sections.

The on-site ballast bed lateral resistance test is depicted in Fig. 8. Before the test, the fasteners and pads of the test sleeper are removed, and the hydraulic jack, pressure transducer reaction frame, and support device are installed horizontally on one side of the sleeper. During the field ballast bed lateral resistance test, multiple load levels are applied to the sleeper by manually controlling the hydraulic jack. As the sleeper slowly moves under the action of horizontal thrust, the readings of the high-precision dial indicator and the lateral resistance measured by the pressure sensor are recorded with the integrated measurement & control (IMC) dynamic data acquisition device. For each load level applied, it is necessary to wait for the sleeper displacement value to stabilize before applying the next load level. The test is considered to have reached the resistance limit, and loading can be stopped when the resistance remains roughly constant while the sleeper displacement continues to increase. The test results are then compiled.

Fig. 8
figure 8

Field lateral resistance test: a sandy section; b clean section; c detailed installation of equipment in the sandy section; d detailed installation of equipment in the clean section

Full size image

Additionally, field tests have been conducted to measure the support stiffness of the ballast bed, as depicted in Fig. 9. When conducting the ballast bed support stiffness test, it is also necessary to remove the constraints between the sleeper and the rail. The fasteners of the test sleeper should be dismantled, and the rubber pad should be pulled out. A reaction frame is installed between the rail and the test sleeper, using the rail as a reaction support point to apply a vertical load with a jack, and a pressure sensor is used to monitor the vertical force on the sleeper. Displacement sensors are installed on both sides of the rail at equal distances from the test sleeper position to collect the vertical displacement of the sleeper. To improve the accuracy of the test results and reduce randomness, five sleepers have been tested in both the sandy section and the clean section.

Fig. 9
figure 9

Field test for ballast bed support stiffness measurement: a detailed installation of equipment in the sandy section; b detailed installation of equipment in the clean section

Full size image

2.3.2 Model validation

The field track structure of the wind-blown sand-affected section is shown in Fig. 10. As seen in Fig. 10, the wind-blown sand has almost completely covered the ballast bed. Based on the ballast bed density data provided on-site, the sand content in the ballast bed at that time is calculated to be about 20%.

Fig. 10
figure 10

Field photograph of track structure in a section severely affected by wind-blown sand

Full size image

In the simulation of the lateral resistance of the ballast bed, the sleeper's movement speed is set to 0.001 m/s, and the calculation time step is set to 3 × 10–6 s. Based on the loading condition at the test site, it is observed that when the lateral displacement of the sleeper reaches 3 mm, the lateral resistance of the ballast bed becomes relatively stable. Therefore, the maximum lateral displacement of the sleeper during simulation is set to 3 mm, and both the lateral displacement of the sleeper and the lateral contact force of the sleeper are monitored. To verify the rationality of the parameter values in the model, the lateral resistance–displacement curves of the ballast bed at 0% and 20% sand content are plotted, as illustrated in Fig. 11. Based on existing research [63], the resistance value at a sleeper lateral displacement of 2 mm is taken as the evaluation value for the lateral resistance of the ballast bed.

Fig. 11
figure 11

Comparison between test and simulation results

Full size image

From Fig. 11, it can be observed that as the sleeper displacement increases, the experimental and simulated results of the lateral resistance for both types of ballast beds show a generally consistent trend. For the ballast bed with a 20% sand content, the experimental result of the lateral resistance is 11.77 kN, while the simulated result is 11.38 kN, differing by only 3.31%. For the clean ballast bed, the experimental result is 8.60 kN, and the simulation result is 8.79 kN, differing by only 2.21%. It can be seen that the experimental and simulation results for both types of ballast beds are relatively close.

To further verify the reliability of the model, the field test results and simulation results of the ballast bed support stiffness are compared and analyzed, as illustrated in Fig. 12. The secant stiffness from the load–displacement curve during vertical loading is taken as the ballast bed support stiffness K, which can be calculated by the following Eq. (9).

$$K = \frac{{{{P}}_{35} - {{P}}_{7.5} }}{{{{S}}_{35} - {{S}}_{7.5} }},$$
(9)

where \(P_{35}\) and \(P_{7.5}\) are the loads borne by the sleeper, taken as 35 and 7.5 kN, and \(S_{35}\) and \(S_{7.5}\) are the sleeper settlement under the loads of 35 and 7.5 kN, respectively.

Fig. 12
figure 12

Load–displacement curve verification: a sandy section; b clean section

Full size image

As depicted in Fig. 12, the load–displacement curve obtained from the numerical simulation closely resembles the field test results, with a high degree of data agreement. Numerically, the average value of the support stiffness test results for the ballast bed with a sand content of 20% is 286.54 kN/mm, while the simulation result is 283.11 kN/mm, differing by only 1.20%. Furthermore, the average value of the test results for the clean ballast bed is 206.86 kN/mm, while the simulation result is 205.98 kN/mm, differing by only 0.43%. Consequently, it can be observed that the test and simulation results for both types of ballast beds are closely aligned.

In summary, from the perspectives of both lateral resistance and support stiffness of the ballast bed, the reliability of the model established in this study has been effectively verified.

3 Results analysis

To investigate the impact of wind-blown sand content on the evolution of the mechanical quality state of the ballast bed, the ballast bed models with different sand contents established in Sect. 2 are utilized. Computational analyses are conducted for various conditions with sand contents of 0%, 5%, 10%, 15%, and 20%, aiming to elucidate the influence of different sand contents on the ballast bed from both macroscopic and microscopic perspectives.

3.1 Particle contact characteristics

Under the train load, sleepers in ballasted tracks distribute the load in different proportions at various positions. For clean ballast beds, it is generally considered that the train load is borne by the five sleepers directly beneath the wheel, with the load distribution ratio sequentially being 0.1:0.2:0.4:0.2:0.1 [64]. To simulate the train load accurately, the Gaussian function is used to fit the train load curve, resulting in a fitted curve for the load function of a single bogie, as illustrated in Eq. (10) and Fig. 13.

$$P(v,t) = \sum\limits_{i = 1}^{m} {0.27{{P}}_{0} }\, {\text{exp}}{{ \left[- \frac{{(vt - x_{i} )^{2} }}{0.78}\right]}},$$
(10)

where \(P_{0}\) is the static wheel load of the train, v is the train running speed, t is the time, and xi is the wheel position coordinates.

Fig. 13
figure 13

Train load

Full size image

The sandy ballast bed is a complex particle system composed of multi-element mixed materials that interact with each other, and its mechanical properties depend on the microscopic contact force between the particle systems [65]. Different sand contents significantly impact the contact force transmission within the ballast bed. To study the influence of the wind-blown sand content on the microscopic contact force of the ballast bed, the average contact force change curve of the ballast particles beneath the sleeper under different sand contents is drawn, as illustrated in Fig. 14.

Fig. 14
figure 14

Variation curve of the average contact force of the ballast particles under different sand contents

Full size image

From Fig. 14, it is evident that the average contact force between the ballast particles under the sleeper generally follows the same trend as the numerical changes in train load across different sand contents. The occurrence of peak and trough values under average contact force is roughly the same for different sand contents. As the sand content increases, there is a gradual decrease in the average contact force between ballast particles. This indicates that the increase in sand particles somewhat weakens the direct contact between ballast particles, leading to a reduction in average contact force. Additionally, the presence of sand reduces the interlocking effect between ballast particles, diminishing their mutual resistance. At lower sand contents (less than 10%), the lubricating effect of the sand is more pronounced, reducing friction between particles and thereby significantly lowering the contact force between ballast particles.

The decay rate of the peak value of the average contact force between ballast particles is further studied at various depths beneath the sleeper under different sand contents. The results are presented in Table 3.

Table 3 Decay rate of the peak value of the average contact force under different sand contents
Full size table

According to Table 3, as the sand content increases from 0 to 20%, there is a noticeable decrease in the average contact force between ballast particles at both 0.05 and 0.35 m depths beneath the sleeper. It is worth noting that although the average contact force at both depths decreases, the rate of decrease at 0.35 m is relatively smaller. Additionally, the rate of attenuation of the contact force from 0.05 to 0.35 m depth also decreases with increase in sand content. This indicates that sand intrusion alters the distribution and magnitude of internal stresses within the ballast bed. As sand content increases, sand particles form a sort of micro-support between ballast particles, reducing local particle movement and rotation, thereby slowing the attenuation of contact force. Furthermore, at higher sand contents, the difference in contact forces between different layers of ballast bed is smaller, suggesting more uniform force transmission and dispersion, but the overall contact force is lower. Therefore, appropriate control of the sand content is crucial for maintaining the mechanical stability of the ballast bed.

The coordination number between particles is a crucial indicator of contact characteristics between particles and plays a vital role in evaluating the overall stability of the ballast bed [66]. The curve depicting the variation of the coordination number among ballast particles under different sand contents with train load changes is illustrated in Fig. 15.

Fig. 15
figure 15

Variation of average coordination number between ballast particles under different sand contents

Full size image

Figure 15 reveals that the coordination number between ballast particles generally shows a downward trend with increase in sand content. In the clean state, i.e., at 0% sand content, the coordination number fluctuates significantly over time and increases notably with the magnitude of the train load. As the sand content increases, sand particles fill the gaps between ballast particles, leading to a reduction in contact points between ballast particles, resulting in a less tight connection. Due to the presence of sand particles, the original interlocking effect between ballast particles is replaced by the contact among ballast–sand–ballast, thus causing a reduction in the coordination number between ballast particles. Particularly at 20% sand content, the coordination number is the lowest, suggesting the weakest interlocking effect between ballast particles under this condition. Moreover, with the fluctuation over time, the variation in the coordination number is not significant, indicating that a high content of sand has formed a specific stable and compact structure, causing ballast consolidation and thus reducing its adaptability to dynamic train loads. Additionally, as the sand content increases, the average coordination number between ballast particles gradually decreases, but the number of contact points per individual ballast particle increases, inevitably leading to a decrease in the average contact force. This also explains the relationship between sand content and ballast particle contact force observed in Fig. 14. Therefore, controlling the sand content at an appropriate level is crucial for optimizing the mechanical performance of the ballast bed and maintaining its stability.

3.2 Ballast bed macroscopic characteristics

3.2.1 Ballast bed resistance

The longitudinal and lateral resistance of the ballast bed are important indicators representing the quality state of the ballast bed and of significance in ensuring the stability of seamless tracks in ballasted railways [67]. To study the influence of wind-blown sand content on the mechanical quality state of ballast beds, curves depicting the variations in lateral and longitudinal resistance of the ballast bed under different sand content levels are plotted, as illustrated in Fig. 16.

Fig. 16
figure 16

Variation rules of the resistance of the ballast bed with different sand contents: a lateral resistance; b longitudinal resistance

Full size image

From Fig. 16a, it is evident that the resistance values of the ballast bed under different sand contents exhibit an increasing trend with displacement. At the 2 mm displacement position, the lateral resistance values of the ballast bed under 0%, 10%, and 20% sand content conditions are 8.79, 7.92, and 11.38 kN, respectively. Compared to the clean state, the lateral resistance of the ballast bed with a 10% sand content decreases by 9.90%, while with a 20% sand content, it increases by 29.47%. This indicates that the lateral resistance of the ballast bed responds nonlinearly to changes in the sand content, generally showing a trend of first decreasing and then increasing. This is because when the sand content is between 0 and 10%, the lubricating effect between sand particles is dominant, and the increased contact between ballast and sand particles leads to a gradual decrease in lateral resistance. However, when the sand content exceeds 10%, as sand particles continue to penetrate, the number of sand particles between ballast particles increases. The sand particles can form a kind of “cementing” effect between ballast particles, causing the connection between ballast particles to become a connection among ballast-sand-ballast. This makes the entire ballast bed structure more compact and stable, leading to an increase in the lateral resistance of the ballast bed.

From Fig. 16b, it can be observed that the longitudinal resistance of the ballast bed shows a trend of initially decreasing and then increasing with the increase in the sand content. The maximum longitudinal resistance value occurs at a 20% sand content, with a value of 17.06 kN. In comparison, the minimum longitudinal resistance value occurs at a 10% sand content, with a value of 13.65 kN. This result indicates that the impact of sand content on the longitudinal resistance of the ballast bed is complex and nonlinear. At lower sand contents, the sand particles reduce the contact and friction with ballast particles, thereby lowering the longitudinal resistance. However, at higher sand contents, the increased sand particles lead to more complex interactions between materials, with close ballast–sand–ballast contact, thereby enhancing the longitudinal resistance.

The distribution characteristics of micromechanical contact forces between particles inside the ballast bed are key factors reflecting the macroscopic mechanical quality state and stability of the ballast bed. Figure 17 depicts the distribution cloud map of contact forces between particles inside the ballast bed under different sand contents when the lateral displacement of the sleeper is 2 mm.

Fig. 17
figure 17

Front view of lateral contact force distribution of the ballast bed (sleeper moves 2 mm to the right)

Full size image

From Fig. 17, it can be seen that the areas with higher contact forces are mainly distributed in the ballast shoulder region in terms of the lateral resistance of the ballast bed. Due to the rightward movement of the sleeper, the ballast resistance forms a certain angle with the horizontal line, showing strong and clearly defined contact forces. This indicates that the interlocking contact action between particles in the ballast bed is effective, significantly resisting lateral loads. Under different sand contents, the distribution range of red particles (indicating stronger contact forces) first decreases and then increases with the increase in the sand content. This contact relationship reflects the variation trend of the lateral resistance of the ballast bed shown in Fig. 16a.

Figure 18 shows the distribution cloud map of the contact force between particles inside the ballast bed under different sand contents when the longitudinal displacement of the sleeper is 2 mm.

Fig. 18
figure 18

Top view of longitudinal contact force distribution of the ballast bed (sleeper moves down 2 mm)

Full size image

From Fig. 18, it can be observed that in terms of the longitudinal resistance of the ballast bed, the areas alongside the sleeper within the length range of the sleeper play a dominant role. There is a precise concentration of contact forces along the longitudinal movement direction of the ballast bed, which first becomes less dense and then more compact with the increase in sand content. When the sand content is less than 10%, the ballast particles primarily bear the longitudinal load. This indicates that within this range of the sand content, the contact and interaction between ballast particles are dominant. The interactions among these particles provide sufficient friction and interlocking forces to maintain the longitudinal stability of the ballast bed. However, as the sand content exceeds 10%, the influence of sand particles in the ballast bed becomes more pronounced, and these sand particles start to play a more significant role in bearing the longitudinal load. Therefore, the sand content in the ballast bed is one of the key factors affecting its mechanical behavior, especially in terms of the mechanism of longitudinal load transfer.

As one of the critical parameters in maintaining the stability of railway lines, the ballast bed resistance is essentially a comprehensive manifestation of the resistance from different sources within the ballast bed [68]. The frictional force between the ballast particles and the sleepers is the primary source of ballast bed resistance. The magnitude of this resistance is closely related to the frictional resistance provided by the ballast shoulder, the side of the sleeper, and the ballast under the sleeper. To further study the impact of sand intrusion on the proportion of longitudinal and lateral resistance at different positions in the ballast bed, the proportion of resistance shared by different parts under different sand contents is statistically analyzed, as illustrated in Fig. 19.

Fig. 19
figure 19

Resistance sharing ratio of the ballast bed at different positions: a lateral resistance; b longitudinal resistance

Full size image

Figure 19a indicates that with the increase in the sand content, the proportion of lateral resistance shared by the area under the sleeper significantly decreases, the share in the area between the sleeper rises to some extent, and the share in the area of the ballast shoulder increases. Regarding the distribution, at the 0% sand content, the area under the sleeper dominates, while at the 15% sand content, the load-sharing is predominantly at the ballast shoulder. This suggests that sand intrusion primarily alters the distribution of resistance between the bottom of the sleeper and the ballast shoulder, with less impact on the side of the sleeper. Due to the larger exposed surface area of the ballast shoulder and the more significant accumulation of sand in this area, more sand particles participate in the formation of the load-bearing chain of the ballast bed, thereby being able to share a greater lateral load.

As can be seen from Fig. 19b, under different sand content conditions, the order of the proportion of longitudinal resistance shared in the ballast bed is always the highest on the side of the sleeper, followed by that at the bottom of the sleeper, and the lowest at the ballast shoulder. With the continuous increase in sand content, the proportion shared by the sleeper side area continues to rise, the proportion in the sleeper bottom area keeps declining, and the change in the proportion at the ballast shoulder is insignificant. This indicates that the variation in sand content affects not only the overall mechanical performance of the ballast bed but also the change in the mechanical behavior and interactions of different parts within the ballast bed.

3.2.2 Ballast bed stiffness

Adequate support stiffness is essential to ensure the stability, comfort, and safety of train operations [69]. By applying a linearly increasing load to the rail seat groove position on one side of the sleeper, gradually from 0 to 40 kN, with a load change frequency of 1 kN/s, and synchronously collecting the vertical displacement at the load application point, the load–displacement curves of the ballast bed support stiffness with varying degrees of wind-blown sand penetration are obtained, as illustrated in Fig. 20.

Fig. 20
figure 20

Load–displacement curves of different sand contents

Full size image

Figure 20 shows that as the sand content increases from 0 to 20%, the compressive load borne by the ballast bed under the same displacement conditions significantly increases. This indicates that an increase in sand content notably enhances the load-bearing capacity and stiffness of the ballast bed.

The curve illustrates the ballast bed support stiffness variation with different sand contents in Fig. 21.

Fig. 21
figure 21

Variation of ballast bed support stiffness value

Full size image

From Fig. 21, it is evident that the support stiffness of the ballast bed gradually increases with the increase in the sand content, reflecting that sand intrusion fills the gaps between ballast particles, enhancing the compactness and interconnection of the particle materials in the ballast bed. Numerically, as the sand content increases from 0 to 20%, the support stiffness of the ballast bed rises from 205.98 to 283.11 kN/mm, with an increase of 37.45%. This demonstrates that sand significantly contributes to enhancing the overall structural rigidity of the ballast bed. Moreover, the rate of stiffness increase is not linear but accelerates in a nonlinear manner with the increase in sand content. This indicates that at lower sand contents (less than 10%), the contribution of sand to stiffness is relatively limited. However, once the sand content exceeds a critical point of 10%, the contribution of sand to stiffness significantly increases, leading to an accelerated growth rate in the overall support stiffness of the ballast bed.

4 Conclusions

For railways constructed in desert regions, the intrusion of sand particles makes the macroscopic and microscopic force characteristics of the ballast bed extremely complex. To study the impact of wind-blown sand content on the evolution of the mechanical quality state of the ballast beds, this research considers the complex contact relationship between ballast and sand particles. A new assessment indicator for sand content is proposed. Based on the DEM, many multi-scale models of sandy ballast beds with varying sand content are established and validated through field experiments. The relationship between sand contents and the macroscopic and microscopic mechanical properties of the ballast bed is analyzed. The main conclusions drawn are as follows.

(1) The penetration of wind-blown sand alters the distribution of contact forces between ballast particles, leading to a decrease in average contact force and coordination number with increased sand content. At high sand contents (exceeding 10%), the difference in contact forces between particles in different layers of the ballast bed is reduced, indicating a more uniform distribution of forces.

(2) The longitudinal and lateral resistance of the ballast bed is essential in its quality state and ensures the stability of seamless tracks in ballasted railways. The analysis reveals that a 10% sand content is a critical point for the change in the resistance characteristics. When the sand content is less than 10%, the longitudinal and lateral resistance gradually decreases with the increase in sand content. However, when the sand content exceeds 10%, the values begin to increase again. Compared to the clean state, the lateral resistance with a 10% sand content decreases by 9.90%, while at 20%, it increases by 29.47%.

(3) The sand content significantly influences the resistance sharing ratio. As the sand content increases, the proportion of lateral resistance shared by the area under the sleeper significantly decreases, the proportion shared by the sleeper side increases to some extent, and the fluctuation in the proportion shared by the ballast shoulder increases. When the sand content is 15%, the sharing ratio is dominated by the ballast shoulder. The order of the proportion of longitudinal resistance shared in the ballast bed always remains the same, with the highest on the sleeper side, followed by the area under the sleeper, and the lowest at the ballast shoulder.

(4) The stiffness of the ballast bed increases with the increase in sand content. The support stiffness of the ballast bed increases from 205.98 to 283.11 kN/mm as the sand content increases from 0 to 20%, with an increase of 37.45%. When the sand content exceeds 10%, the contribution of sand particles to stiffness significantly increases, leading to an accelerated growth rate of the overall support stiffness of the ballast bed. This affects the good drainage performance of the ballast bed structure and is highly detrimental to the long-term service performance of the ballast bed.

In conclusion, it is recommended that maintenance and repair measures be implemented promptly when the sand content in the ballast bed reaches or exceeds 10% to maintain the performance of the ballast bed and extend its service life.