Investigating the Mechanisms and Dynamic Response of Graded Aggregate Mud Pumping Based on the Hybrid DEM-FDM Method

Abstract

This study investigated the macro and meso mechanisms of void formation in graded aggregates within high-speed railway subgrades under train loads using a hybrid discrete element–finite difference method (DEM-FDM). First, a contact parameter inversion model based on a linear model (LM) was developed using extensive DEM simulations through angle of repose, drop, and inclined plate tests. The contact parameters for graded aggregates were further calibrated through physical and triaxial tests. Next, a refined hybrid DEM-FDM model was established to capture void formation behavior, characterized by the contact force chain ratio, and was validated against field measurements. Finally, simulations were conducted under different levels of void formation to explore the associated mechanisms based on dynamic response and meso-mechanical analysis. The results showed that the LM-based inversion model could accurately determine the contact parameters. The hybrid model’s predictions of dynamic displacement and acceleration under various train speeds fell within the range of the field data. When the fine particle loss ratio l_p was ≤3%, the dynamic displacement and acceleration remained below the standard limits of 0.22 mm and 10 m/s². As l_p increased, the contact between the roadbed and base weakened, and complete separation occurred at l_p ≥ 11%, preventing effective load transfer. These findings offer new insights into void formation in graded aggregates and support the safe operation of high-speed railways.

1. Introduction

The smoothness of track and subgrade structures is crucial to maintaining the daily safe operation of high-speed railways at 350 km/h [1,2]. It is notable that high-speed railway track structures typically adopt high-performance ballastless tracks. In particular, the CRTS III system consists of track slabs, self-compacting concrete, and a base layer, offering excellent performance and high construction efficiency. In contrast, subgrade structures are commonly constructed with coarse-grained soils [3,4]. Specifically, the roadbed beneath the track structure is constructed with graded aggregates, serving as the key area for transmitting train loads from the track to the subgrade [5,6]. Nevertheless, there are significant differences in continuity and stiffness between graded aggregates and ballastless tracks, leading to various issues (e.g., mud pumping and void formation) in graded aggregates under long-term train loads and environmental factors [7,8]. Hence, it is necessary to investigate the graded aggregate mud pumping and void formation mechanism at speeds of 350 km/h and above to maintain the daily safe operation of high-speed railways.

It is well known that mud pumping occurs when subgrade fillers are continuously expelled in slurry form under the coupled impact of water and train loads, commonly occurring in traditional railway ballast subgrades [9,10,11,12,13]. Excessive fine particle loss often results in void formation within the subgrade. In past years, indoor simulation systems for train load–moisture coupling have been developed by Ding et al. [14], Israr et al. [15], Zhang et al. [16], and Gao et al. [17], which have been used to investigate the mud pumping mechanism in ballast track subgrades. It has been indicated that an increase in dynamic pore water pressure under saturated conditions caused by train loads is a key factor in mud pumping. Moreover, mud pumping in ballastless track subgrades has emerged as a new problem in high-speed railways, particularly in areas with frequent rainfall [18]. Existing research indicates that in areas near mud pumping in ballastless tracks, numerous fine-graded aggregate particles accumulate on the sealing layer, leading to graded aggregate void formation beneath the base of the ballastless track, which further impacts the track structure smoothness [19,20,21]. It is worth noting that graded aggregates form slurry and generate a pore pressure gradient under the coupling of train loads and water, as reported by Bian et al. [22], which is then pumped to the sides of the base. A remediation method for graded aggregate void formation in ballastless tracks was proposed by Huang et al. [23] through onsite experiments, which effectively controlled abnormal subgrade vibrations. Nevertheless, existing research mainly focuses on the migration characteristics of fine particles under the coupling of train loads and water. However, there is a lack of research on the relationship between graded aggregate void formation, the subgrade dynamic response, ballastless track deformation, and train operation safety.

It is well known that numerical simulations, compared to model and onsite experiments, have the advantages of controllable variables and repeatability and could provide an efficient method to investigate the relationship between graded aggregate void formation and the subgrade dynamic response. Moreover, the discrete element method (DEM) has excellent capabilities in representing meso characteristics and could effectively quantify the meso-structural features within graded aggregates [24,25,26,27]. For example, the breakage characteristics of graded aggregates were investigated by Xiao et al. [5] and Xie et al. [6] using triaxial and vibratory compaction DEM models. However, it is very difficult to establish a full-scale train–track–subgrade DEM model due to computational efficiency limitations. To improve computational efficiency, a full-scale hybrid discrete element–finite difference method (DEM-FDM) for ballast track subgrade has been established by Tan et al. [28], Chen et al. [29], Xiao et al. [30,31], and Li et al. [32]. Model parameters are key factors influencing the accuracy of numerical simulation results. The macro parameters required for the FDM model (e.g., elastic modulus, Poisson’s ratio, and density) were directly obtained from existing studies [33,34], whereas the contact parameters required for the DEM model (e.g., stiffness, friction coefficient, and damping coefficient) had to be calibrated [35,36,37,38]. Hence, it was necessary to establish a refined hybrid DEM-FDM model for ballastless track subgrade by calibrating the graded aggregate contact parameters. This model was used to investigate the relationship between graded aggregate void formation and the subgrade dynamic response.

In summary, to address the graded aggregate void formation mechanism, this study aims to establish a refined hybrid DEM-FDM model for ballastless track subgrade to investigate the relationship between void formation and the subgrade dynamic response. Firstly, the contact parameters of graded aggregates in the DEM model are calibrated. Secondly, a refined hybrid DEM-FDM model for ballastless track subgrade, representing graded aggregate void formation, is established. Finally, the relationship between graded aggregate void formation and the subgrade dynamic response is investigated based on the macro- and meso-characteristic evolution of the graded aggregates. The research findings provide a theoretical basis for addressing a graded aggregate void formation disaster, contributing to maintaining the daily safe operation of high-speed railways.

2. Graded Aggregate Void Formation Simulation with a Hybrid DEM-FDM Model

2.1. Graded Aggregate Void Formation

It is well known that high-speed railways operating at a speed of 350 km/h in China have all been constructed with ballastless track structures compared to traditional rapid railways and freight railways [3,4]. It is inevitable that mud pumping and void formation occur between the base of the ballastless track and the roadbed under the coupled impact of high-frequency train vibrations and environmental factors (e.g., wind, rain, and corrosion) [18,19,20,21]. Figure 1 illustrates the process of roadbed mud pumping and void formation. In the normal stage, water is prevented from entering the roadbed by the sealing layer and sealant, which effectively enhance the subgrade service performance. Nevertheless, in the seepage stage, the sealant is damaged under the impact of train loads and environmental factors, leading to water entering the roadbed through the gaps between the base and the sealing layer [22]. It is worth noting that the roadbed is constructed with well-performing graded aggregates [5,6]. Hence, in the mud pumping stage, fine particles in the graded aggregates are continuously expelled under the coupled impact of the water and train loads, leading to graded aggregate void formation and, consequently, reducing the service performance of high-speed railway subgrade [23].

It is significant to investigate the graded aggregate void formation mechanism to improve the service performance of high-speed railway subgrade. For graded aggregate, a typical coarse-grained soil, meso-structure evolution (i.e., particles, contacts, and voids) is essential to its mechanical strength [5,6,35]. Compared to onsite and model experiments, the discrete element method (DEM) has become a mainstream method for investigating the coarse-grained soil meso structure due to its advantages of repeatability and cost-effectiveness [24,25,26,27]. Nevertheless, it is computationally difficult to establish a full-scale DEM model for high-speed railway subgrade. For non-research focus objects such as the track structure and subsoil, the finite difference method (FDM) can be used for simulation to reduce the model complexity [28,29,30,31,32]. Hence, a hybrid DEM-FDM model can be established to investigate the relationship between void formation and the subgrade dynamic response from macro and meso perspectives.

2.2. Hybrid DEM-FDM Model

2.2.1. Model Setup

To improve the computational efficiency of the hybrid DEM-FDM model, the graded aggregates beneath the base (measuring 2.5 m in length and 0.4 m in height) were simulated using DEM, while the remaining structures were simulated using FDM. It is worth noting that the model cross-sectional dimensions were designed based on the Code for the design of high-speed railways [39]. As shown in Figure 2a, the establishment of the high-speed railway subgrade hybrid DEM-FDM model mainly includes three steps:

• Step 1: Layered compaction

In the FDM, the model was directly established through solid elements, whereas specimen preparation was a complex process in the DEM. Hence, the FDM models for the subsoil, subgrade, and half of the roadbed were first established. Then, the DEM model for the roadbed was established through three layers of compaction. In each layer, the graded aggregates were compacted under cyclic loading until the displacement stabilized [28,40,41]. Ultimately, the track structure was established using the FDM, including the rails, sleepers, slab, self-compacting concrete, base, and sealing layer.

• Step 2: Gravity balancing

For the FDM model, gravity balancing is a key step before applying a load. Hence, the hybrid DEM-FDM model was calculated in the gravity field until the maximum unbalanced force ratio was less than 10⁻⁵ [28].

• Step 3: Application load

The vibration impact of trains on the subgrade can be simulated using an “M”-shaped wave based on research by Liu et al. [42]. Hence, the train load was represented using the three-level Fourier series proposed by Sun et al. [43,44], which was calculated by Equation (1). Moreover, as shown in Figure 2b,c, numerous train loads corresponding to different speeds v and axle loads A_l were represented by changing the parameters in Equation (1), which was used to investigate the impact of graded aggregate void formation on the subgrade dynamic performance under different v and A_l.

$$F(t)=\left\{\begin{array}{ll}{A}_l+{\displaystyle \sum _{n=1}^3\left[A_n\cos n2\pi ft+B_n\cos n2\pi ft\right]}& (0<t\le \frac{3T}{10})\\ 0& (\frac{3T}{10}<t\le \frac{7T}{10})\\ A_l+{\displaystyle \sum _{n=1}^3\left[A_n\cos n2\pi f(t+\frac{2T}{10})+B_n\cos n2\pi f(t+\frac{2T}{10})\right]}& (\frac{7T}{10}<t\le T)\end{array}\right.$$

(1)

where F(t) is the train load, A_l is the axle load, t is the loading time, A_n and B_n are the third-level Fourier coefficients [43], f is the loading frequency, and T is the cycle period.

2.2.2. Model Parameters

It is well known that the model parameters are key factors determining the simulation results for hybrid DEM-FDM models [45,46]. Firstly, in the FDM, an elastic model was used to accurately represent the physical–mechanical behavior of the track structure, subgrade, and subsoil. Elastic modulus E, Poisson’s ratio v, and density ρ are three macroscopic parameters required for the elastic model and were directly obtained through existing research and physical experiments [33,34]. Secondly, as shown in

Table 1. The effective contact parameters in LM.

Schematic Diagram	LM Contact Parameter	Symbol
	Effective modulus for particle to particle	Ebb
	Stiffness ratio for particle to particle	krbb
	Friction coefficient for particle to particle	fbb
	Damping coefficient for particle to particle	βbb
	Effective modulus for particle to base	Ebc
	Stiffness ratio for particle to base	krbc
	Friction coefficient for particle to base	fbc
	Damping coefficient for particle to base	βbc

, a linear model (LM) was used in the DEM to accurately represent the interactions between the graded aggregates; this is a commonly used contact model for coarse-grained soils [37,40,41,47,48]. It was observed that there were numerous contact parameters in the LM, such as normal stiffness k_n, tangential stiffness k_s, normal damping β_n, tangential damping β_s, and friction f_r, which were difficult to obtain directly. Moreover, stiffness was converted into effective modulus E_e and stiffness ratio k_r by Equations (2) and (3). Finally, in the hybrid part of the DEM and FDM, the interactions between the graded aggregates and other structures were also accurately represented using the LM. Hence, it was necessary to calibrate the contact parameters within the graded aggregates listed in Table 1 to improve the accuracy of the DEM-FDM hybrid model.

$$E_e=\left\{\begin{array}{ll}{\displaystyle \frac{\left(r_a+r_b\right)k_n}{\min {\left(r_a,r_b\right)}^2\mathsf{\pi }}}& \mathrm{Particle}\mathrm{to}\mathrm{particle}\\ \\ \hspace{1em}{\displaystyle \frac{k_n}{\mathsf{\pi }{r}_a^2}}& \mathrm{Particle}\mathrm{to}\mathrm{zone}\end{array}\right.$$

(2)

$$k_r={\displaystyle \frac{k_n}{k_s}}$$

(3)

where r_a and r_b are the radius of the A and B particles.

According to the research by Xiao et al. [5], Li et al. [35], Qi et al. [37,47], and Zhang et al. [40,41,48], the damping and friction coefficients in the LM are directly calibrated through the macro-dynamic responses of particles in physical tests. It is worth noting that the particle dynamic responses in the angle of repose tests, dropping tests, and inclined plate tests correspond to the angle of repose θ, rebound height h, and critical inclination angle of the plate α, respectively. Moreover, normal and tangential stiffness are indirectly calibrated through triaxial tests. The influence of the parameters and particle morphology on the results of the calibration tests are discussed in the research by Xiao et al. [5]. As shown in Figure 3, the LM contact parameter calibration mainly included the following three steps:

• Step 1: Parametric inversion model

Numerous DEM simulations through angle of repose tests, dropping tests, and inclined plate tests were conducted using contact parameters as variables. Based on the simulated dataset, the relationship between the contact parameters and the dynamic responses was established and was used to develop an inversion model for the contact parameters.

• Step 2: f and β calibration

The actual dynamic responses of graded aggregates were obtained through indoor tests for the angle of repose, dropping, and inclined plate. Furthermore, the dynamic responses were incorporated into the parameter inversion model established in Step 1 to calibrate the contact parameters f_bb, β_bb, β_bc, and f_bc.

• Step 3: E_e and k_r calibration

The DEM simulations were conducted using E_bb as a variable, and E_bb was calibrated by comparing the simulation results with the indoor test results. The remaining parameters, E_bc, k_rbc, and k_rbb, were calculated using the contact theory in Equations (4)–(6) [40,41]:

$$v_{bc}={\displaystyle \frac{v_c{E}_{b1}(1+v_c)+v{E}_c(1+v)}{E_{b1}(1+v_c)+E_c(1+v)}}$$

(4)

$$k_{rb}={\displaystyle \frac{2-v}{2(1-v)}},k_{rbc}={\displaystyle \frac{2-v_{bc}}{2(1-v_{bc})}}$$

(5)

$$E_{bw}={\displaystyle \frac{2E_{b1}{E}_c(2-v_{bc})(1+v_{bc})}{E_{b1}(2-v_c)(1+v_c)+E_c(2-v_c)(1+v)}}$$

(6)

where v_bc is the Poisson’s particle-to-concrete ratio, E_c is the effective modulus for concrete, set to 32.5 GPa [33,34], and v_c is the Poisson’s ratio for concrete, set to 0.167 [33,34].

As shown in Figure 4a, an angle-of-repose DEM model was established that allowed the graded aggregates to accumulate under gravity, which is consistent with the methods of Qi et al. [47] and Deal et al. [49]. Then, the particle positions on the accumulated surface were exported and fitted using a linear function. As shown in Equation (7), θ was calculated using the slope of the fitted line. Furthermore, as shown in Figure 4b, the relationship between θ and f_bb could be represented by an exponential function based on numerous DEM simulation data. Hence, when θ of the graded aggregates in the indoor tests was determined, f_bb could be inverted using Equation (8):

$$\theta ={\displaystyle \frac{180\times \arctan \left|k\right|}{\mathsf{\pi }}}$$

(7)

$$\theta =28.913-47.163e^{\left(f_{bb}/0.15\right)}$$

(8)

where k is the slope of the fitted line.

As shown in Figure 5a, a dropping DEM model was established that allowed the graded aggregates to free fall under gravity, which is consistent with the methods of Xiao et al. [5] and Li et al. [35]. Then, the first rebound height h for the particles under different β_bb and β_bc was exported. Furthermore, as shown in Figure 5b, the relationship between h, β_bb, and β_bc could be represented by an exponential function based on numerous DEM simulation data. Hence, when h of the graded aggregates in the indoor tests was determined, β_bb and β_bc could be inverted using Equation (9):

$$\left\{\begin{array}{l}h=0.033+0.459e^{\left({\beta }_{bb}/0.15\right)}\\ h=0.033+0.459e^{\left({\beta }_{bc}/0.15\right)}\end{array}\right.$$

(9)

As shown in Figure 6a, an inclined plate DEM model was established that allowed the graded aggregates to slide freely under gravity, which is consistent with the methods of Zhang et al. [48]. Then, f_bc for the particles sliding under different α were exported. Furthermore, as shown in Figure 6b, the relationship between α and f_bc could be represented by an exponential function based on numerous DEM simulation data. Hence, when α of the graded aggregates in the indoor tests was determined, f_bc could be inverted using Equation (10):

$$\alpha =-0.594+0.632e^{\left(f_{bc}/47.198\right)}$$

(10)

Additionally, as shown in Figure 7a, indoor triaxial tests were conducted to obtain the actual dynamic responses of the graded aggregates. Then, four DEM triaxial models consistent with the indoor tests were established, with parameter E_bb values of 1 × 10⁷ Pa, 5 × 10⁷ Pa, 1 × 10⁸ Pa, and 1 × 10⁹ Pa. As shown in Figure 7b, in both the indoor tests and DEM simulation results, deviatoric stress q exhibited softening characteristics. It was observed that the numerical model with E_bb set to 1 × 10⁸ Pa exhibited an excellent fit with the test results. Hence, the parameter E_bb was calibrated to 1 × 10⁸ Pa, while the remaining parameters were calculated by Equations (4)–(6):

2.2.3. Model Verification

As shown in

Table 2. Parameter calibration results for graded aggregate in the DEM model.

Parameter	Ebb	krbb	fbb	βbb	Ebc	krbc	fbc	βbc
Value	1 × 108 Pa	1.35	0.85	0.18	5 × 108 Pa	1.30	0.55	0.22

and

Table 3. Parameter values in the FDM model [33,34].

Parameter	Rail	Sleeper	Slab	Self-Compaction Concrete	Base	Roadbed	Subgrade	Subsoil
E (GPa)	206	36	36	32.5	32.5	0.23	0.20	0.16
v	0.3	0.167	0.167	0.167	0.167	0.3	0.35	0.40
ρ (kg/m3)	7830	2450	2450	2450	2450	2300	2100	2000

, the simulation parameters for the hybrid DEM-FDM model for the high-speed railway subgrade were determined based on the parameter calibration methods in Section 2.2.2 and existing research [33,34]. To ensure the reliability of the hybrid DEM-FDM model, it was necessary to verify the accuracy of the simulation parameters through onsite experiments.

As shown in Figure 8a, the roadbed dynamic response was obtained using acceleration sensors and displacement sensors installed on the surface of the roadbed. The sensitivity of the acceleration sensors was 500 pC/g, with a range of 150 g, and the sensitivity of the displacement sensors was 1 mV/μm, with a range of 3 mm. It was noted that the test train had an axle load of 14 t, a fixed axle spacing of 2.5 m, and a total length of 201.4 m. Moreover, to minimize the impact of the environment on the results during onsite experiments, multiple tests were conducted at different speeds. As shown in Figure 8b,c, 95% confidence bands were established for the measured roadbed dynamic displacement and acceleration. The simulation results from the hybrid DEM-FDM model fell within these bands, indicating good agreement with the field measurements. This suggested that the hybrid model accurately captured the dynamic response of the high-speed railway subgrade. It was therefore suitable for investigating the relationship between void formation and subgrade behavior.

2.2.4. Simulation for Graded Aggregate Void Formation

It is crucial to propose a scientifically sound simulation method to effectively investigate the mechanism within graded aggregate void formation. Existing research indicates that the skeletal structure in graded aggregate consists of coarse particles, with fine particles filling voids between coarse particles. As shown in Figure 9a, when the contact forces between the fine and coarse particles are weak, the fine particles are repeatedly displaced within the voids and are further extruded from the roadbed under the coupled impact of water and train loads. Hence, fine particles are removed based on the contact forces to simulate graded aggregate void formation. As shown in Equation (11), the fine particle contact force ratio f_p was set as the threshold for particle loss; when the fine particle contact force was less than f_p times the average contact force, the fine particles were lost. Furthermore, the extent of fine particle loss was quantified using the ratio of lost particle mass l_p, calculated using Equation (12). Based on the hybrid DEM-FDM model established for high-speed railway subgrade, the relationship between l_p and f_p is shown in Figure 9b:

$$f_p={\displaystyle \frac{{\displaystyle \sum _{i=1}^m{F}_i}}{{\displaystyle \sum _{j=1}^C{f}_j}/C}}\times 100\%$$

(11)

$$l_p={\displaystyle \frac{m_l}{m}}\times 100\%$$

(12)

where m is the contact number for fine particles, F_i is the contact force for fine particles, C is the total contact number, f_j is the contact force, l_p is the ratio of the lost fine particle mass to the total particle mass, m_l is the lost fine particle mass, and m is the total particle mass.

3. Results and Analysis

3.1. Dynamic Response

3.1.1. Roadbed Dynamic Displacement

To describe the deformation behavior of the roadbed under train loading, a quantitative analysis of dynamic deformation was conducted [50,51]. According to the technical regulations for dynamic acceptance for high-speed railway construction (TB10761-2013) [52], the dynamic displacement of ballastless track subgrades must be less than 0.22 mm. Hence, it was necessary to investigate the impact of void formation on the subgrade dynamic performance based on dynamic displacement evolution characteristics.

As shown in Figure 10a–e, there was a consistent trend in the impact of v and A_l on roadbed dynamic displacement when l_p was the same. When l_p ≤ 3%, the roadbed primarily exhibited elastic deformation, and the dynamic displacement evolution was consistent with the applied load, presenting an “M” shape. When l_p > 3%, the roadbed primarily exhibited plastic deformation, and the dynamic displacement gradually increased with the loading time. Moreover, as shown in Figure 10f, dynamic displacement under different v and A_l exhibited a trend of a slow increase followed by a rapid increase as l_p increased. It is worth noting that when l_p ≤ 3%, the dynamic displacements under different v and A_l were all less than the limit of 0.22 mm. When l_p > 3%, the arrangement within the graded aggregates was changed by the loss of fine particles. The graded aggregate underwent significant plastic deformation due to rearrangement, leading to void formation in the roadbed, consequently reducing the high-speed railway roadbed service performance.

3.1.2. Roadbed Acceleration

One essential mechanical indicator used to characterize subgrade performance under high-speed loading is the acceleration response of the roadbed [50,51]. According to the technical regulations for dynamic acceptance for high-speed railway construction (TB10761-2013) [52], the roadbed acceleration of ballastless track subgrades must be less than 10 m/s². Hence, it was necessary to investigate the impact of void formation on the subgrade dynamic performance based on the acceleration evolutionary characteristics.

As shown in Figure 11a–e, there was a consistent trend in the impact of v and A_l on roadbed acceleration when l_p was the same. It was clearly observed that the acceleration was zero as the second train bogie passed when l_p = 11%, indicating a complete void formation between the graded aggregate and the base. Furthermore, as shown in Figure 11f, roadbed acceleration increased slowly at first and then decreased rapidly with the increase in l_p, reaching its peak at l_p = 9%. It is worth noting that when l_p ≤ 3%, the acceleration under different v and A_l were all less than the limit of 10 m/s². Moreover, when l_p > 3%, the void formation level of the roadbed gradually increased, and the relative coordinated deformation between the subgrade surface and the base slab became more significant. Nevertheless, when l_p > 9%, there was a complete void formation between the graded aggregate and the base, resulting in the acceleration approaching zero.

3.1.3. Roadbed Dynamic Stress

The roadbed dynamic stress was also one of the key mechanical indicators for evaluating the high-speed railway subgrade service performance [50,51]. According to the technical regulations for dynamic acceptance for high-speed railway construction (TB10761-2013) [52], the roadbed dynamic stress must be less than 100 kPa. Hence, it was necessary to investigate the impact of void formation on the subgrade dynamic performance based on the dynamic stress evolution characteristics.

As shown in Figure 12a–e, there was a consistent trend in the impact of v and A_l on the roadbed dynamic stress when l_p was the same. When l_p ≤ 3%, there was an “M”-shaped distribution in the roadbed dynamic stress, whereas the dynamic stress gradually decreased with the increasing loading time when l_p > 3%. Furthermore, as shown in Figure 12f, the dynamic stress under different v and A_l increased rapidly and then slowly decreased with the increase in l_p, reaching its peak at l_p = 3%. It is notable that the roadbed dynamic stress under different void formation levels were all less than the limit of 100 kPa, making it difficult to evaluate the high-speed railway subgrade service performance based on dynamic stress.

3.2. Mesoscopic Analysis

3.2.1. Coordination Number

The coordination number Z [5,6,37,40], a key physical indicator for characterizing internal compactness within coarse-grained soils, typically refers to the number of contacts a particle has with its neighboring particles and is calculated using Equation (13). Hence, Equation (13) can be used to investigate the impact of graded aggregate void formation on the subgrade dynamic performance based on the evolution characteristics of the coordination number:

$$Z={\displaystyle \frac{2C-N_1}{N-N_1-N_0}}$$

(13)

where N is the total particle number, and N₀ and N₁ are the numbers of particles with zero and one contact, respectively.

As shown in Figure 13a–e, there was a consistent trend in the impact of v and A_l on the coordination number within the roadbed when l_p was the same. Moreover, the variation in the coordination number gradually decreased as l_p increased, and the coordination number remained stable when the second bogie passed, especially when l_p > 9%. This indicated that the graded aggregate gradually separated from the base when l_p > 9%, resulting in a gradual reduction in the train load transmitted from the track structure to the roadbed. Furthermore, as shown in Figure 13f, the coordination number under different v and A_l gradually decreased with the increase in l_p, indicating a reduction in the internal compactness within the roadbed and an increase in the void formation level.

3.2.2. Strong Force Chain

The strong force chain F is a key mechanical indicator for characterizing the internal contact interactions within coarse-grained soils, and it is closely related to the ability of coarse-grained soils to bear external loads [5,6,37,40]. As shown in Equation (14), the strong force chain is typically defined by two indicators: the average contact force F_a and the contact direction δ:

$$F=\left\{\begin{array}{l}{F}_s>F_a\\ \delta >45°\end{array}\right.$$

(14)

where F_s is the force chain, F_a is the average force chain, and δ is the angle between the force chain and the horizontal plane.

As shown in Figure 14a–e, there was a consistent trend in the impact of v and A_l on the strong force chain within the roadbed when l_p was the same. Moreover, the strong force chain exhibited an “M” shape consistent with the train load when l_p<11%, whereas it approached zero as the second bogie passed when l_p = 11%. This indicated that there was a complete void formation between the graded aggregates and the base when l_p = 11%, which prevented the train load from being transmitted to the roadbed and stabilized the internal contacts within the graded aggregates. Furthermore, as shown in Figure 14f, the strong force chain exhibited a trend of a slow increase followed by a rapid decrease under different v and A_l. When l_p ≤ 3% and the internal skeletal structure within the graded aggregates remained unchanged, the loss of fine particles enhanced the contact interactions within the coarse particles, leading to an increase in the strong force chain. Then, as l_p increased, the loss of fine particles led to a decrease in the contact interactions between the roadbed and the base, reducing the effect of train loads on the strong force chain. Especially when l_p = 11%, the strong force chain approached zero. In summary, the strong force chain was the most sensitive meso-structural indicator for characterizing the graded aggregate void formation level.

3.2.3. Fabric Anisotropy

Fabric anisotropy is a key characteristic distinguishing coarse-grained soil from other continuous materials. It forms a complex, multi-level, and interrelated system composed of factors such as the particle shape, arrangement, contact, and gradation. Hence, the contact fabric anisotropy of the graded aggregates was quantified to further reveal the void formation mechanism in the roadbed. The contact fabric anisotropy in the 2D model was calculated using Equation (15), as proposed by Rothenburg [53]:

$$E\left(\theta \right)={\displaystyle \frac{1}{2\mathsf{\pi }}}\left[1+a_n\cos 2\left(\theta -{\theta }_n\right)\right]$$

(15)

where a_n is a parameter defining the magnitude of anisotropy in the contact orientations, and θ_n defines the direction of anisotropy.

As shown in

Table 4. Normal contact force anisotropy (v = 400 km/h and A_l = 13.5 t).

	t = 0.02 s	t = 0.11 s	t = 0.18 s
lp = 0%
lp = 3%
lp = 5%
lp = 7%
lp = 9%
lp = 11%

, under a train speed of 400 km/h and an axle load of 13.5 t, the evolution of normal contact force anisotropy in the roadbed was analyzed at three key time points (t = 0.02 s, 0.11 s, and 0.18 s). When l_p = 0%, the principal direction of anisotropy followed a vertical–horizontal–vertical pattern, indicating a temporary redistribution of internal force chains under dynamic loading. In contrast, when l_p > 0%, the principal direction remained consistently vertical. Figure 14 further demonstrates the spatial evolution of strong force chains, showing that fine particle loss weakened the interlocking among the coarse particles. As l_p increased, the anisotropy magnitude gradually declined, and the principal axis deviated from the vertical. These changes reflected a loss of structural integrity within the graded aggregates, leading to particle rearrangement and the diminished ability of the roadbed surface to support vertical loads. Thus, the evolution of contact anisotropy effectively captured the meso-structural degradation induced by the void formation.

4. Conclusions

This study aims to investigate the macro and meso mechanisms of graded aggregate void formation in the high-speed railway subgrade structure under train-induced excitation using the hybrid DEM-FDM numerical simulation approach. Firstly, a refined hybrid DEM-FDM model for high-speed railway subgrade was established by calibrating the graded aggregate contact parameters. Secondly, an approach to represent void formation in the roadbed was proposed based on the contact force chain ratio. Finally, the impact mechanism of void formation on the high-speed railway subgrade dynamic performance was revealed based on the macro- and meso-characteristics evolution. The main conclusions are as follows:

(1)

In the hybrid DEM-FDM model, the linear model contact parameters for graded aggregates were calibrated as E_bb = 1 × 10⁸ Pa, k_rbb = 1.35, f_bb = 0.85, and β_bb = 0.18. Moreover, the contact parameters between the graded aggregates and the base were calibrated as E_bc = 5 × 10⁸ Pa, k_rbc = 1.30, f_bc = 0.55, and β_bc = 0.22.

(2)

The model accuracy was validated, as the roadbed dynamic displacement and acceleration at different train speeds in the hybrid DEM-FDM model were within the onsite measured range. Moreover, the roadbed void formation was accurately represented by setting the contact force chain ratio as the particle loss threshold.

(3)

As the fine particle mass loss ratio l_p increased, the physical indicator (i.e., dynamic displacement) continuously increased, while the mechanical indicators (i.e., acceleration and dynamic stress) first increased and then decreased, reaching their peak values at l_p = 9% and l_p = 3%, respectively. Under the conditions of 400 km/h and 13.5 t, the maximum acceleration and dynamic stress reached 28.22 m/s² and 58.57 kPa. It is noteworthy that when l_p > 3%, both the dynamic displacement and acceleration exceeded the standard limits of 0.22 mm and 10 m/s².

(4)

As l_p increased, the coordination number in the roadbed decreased from about 3.5 to 3.0. Strong force chains peaked at l_p = 3%, with a maximum contact force of 138 N under 400 km/h and 13.5 t. When l_p ≤ 3%, fine particle loss enhanced coarse particle contact. For 3% < l_p < 11%, the roadbed–base contact weakened. At l_p ≥ 11%, full separation occurred, and the train load could not be effectively transferred, with the contact force dropping to 20.7 N.

This study treated the dynamic response of the ballastless track roadbed as a key characteristic for analyzing void formation caused by mud pumping. Based on this analysis, the critical voiding threshold of the roadbed, corresponding to the condition where the dynamic characteristics satisfied the specification limits, was determined to be l_p = 3%. Hence, when the voiding level approached the critical threshold, roadbed inspection efforts were strengthened to promptly assess the service condition of the roadbed. When the voiding degree exceeded 3%, maintenance actions such as grouting were carried out in a timely manner to restore the integrity of the roadbed and prevent further deterioration.