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Jun 11, 2025

Evolution of crushing process of coarse-grained soil filler under the trains load | Scientific Reports

Scientific Reports volume 15, Article number: 35 (2025) Cite this article 870 Accesses Metrics details Research on the evolutionary behavior of the particle breakage processes in coarse-grained soil

Scientific Reports volume 15, Article number: 35 (2025) Cite this article

870 Accesses

Metrics details

Research on the evolutionary behavior of the particle breakage processes in coarse-grained soil under the action of train load is of practical significance for subgrade construction and maintenance. However, existing studies have not addressed the prediction of particle size distribution evolution. In this paper, the MTS loading system is used to simulate the dynamic train load effect on coarse-grained soil fillers. The study analyzes the influence of dynamic stress amplitude, loading frequency, and vibration times on both the macro-characteristics and micro-characteristics of particle breakage. The characteristics of particle fragmentation in coarse soil filler under high-speed train load are elucidated. Furthermore, a predictive model for the evolution of particle size distribution curves in relation to particle content and relative particle size is established using the ZHU continuous grading curve equation. This model captures the evolution process of particle breakage characteristics in coarse-grained soil fillers subjected to high-speed train loads. The applicability of this model has been verified. Based on the grading prediction model, an integral expression for the breakage rate index is derived, and the evolution characteristics of particle breakage in coarse-grained soil fillers under the action of train load are analyzed. The results indicate that during filling, the particle breakage mode of coarse-grained soil fillers during filling is primarily characterized by fracture and fragmentation; conversely, under dynamic cyclic loading conditions, it is predominantly characterized by fracture and grinding. The breakage rate aligns with the measured results, suggesting that the breakage rate index established in this study can effectively describe the evolution process of particle breakage in railway subgrade coarse-grained soil. After the reaching one million loading cycles, both deformation and particle breakage degree in coarse-grained soil fillers tend to stabilize. Under the action of dynamic stress amplitudes ranging from 10 to 200 kPa and loading frequencies between 2 and 12 Hz, the particle breakage index stabilizes below 1.1%. These research findings contribute to a deeper understanding of the evolutionary processes affecting engineering characteristics of railway subgrade coarse-grained soils and provide a theoretical as well as experimental foundation for railway subgrade construction and maintenance.

Coarse-grained soil is defined as a soil-rock mixture characterized by particle sizes ranging from 0.075 to 60 mm, with a mass ratio of particles exceeding 50%. Due to its favorable engineering properties, coarse-grained soil is extensively utilized as a filler in high-speed railway subgrade beds. Under the action of train dynamic load, coarse particles in the coarse-grained soil filler break down into smaller particles, resulting in changes in the particle size distribution of the soil mass, rearrangement, and distribution of particles, affecting the engineering properties of the subgrade. Therefore, research on the grading evolution of coarse-grained soil filler under train load is of great engineering significance for subgrade construction and maintenance.

In recent years, scholars have primarily focused their research on the evolution of soil gradation from two directions: quantitative indicators of soil particle breakage and the evolution process of particle breakage. Regarding the research on quantitative indicators of particle breakage, Marsal, Lee, Hardin, Einav, Xiao, and Han Huaqiang1,2,3,4,5,6 have used Bg, Bm, Br, B*r, Br50 and Dm to measure the size of particle breakage and analyzed the influence of stress–strain parameters on the breakage indicators. Although the breakage indicators can reflect the overall evolution of soil gradation, they cannot reflect the breakage and intermediate evolution processes of particles with different particle sizes in detail. Therefore, researchers attempted to establish a grading equation to quantify the grading composition of granular materials by mathematical formulas. Fuller et al.7 proposed a parabolic grading equation, and Talbot et al8. proposed a fractal grading equation, which has been widely applied in the physical and mechanical properties of grading and coarse-grained materials. Subsequently, Swamee et al.9 proposed a three-parameter grading equation that can reflect the inverse S-shaped curve. Zhu et al.10 proposed a two-parameter grading equation describing continuous grading soil, and Guo et al.11 verified its applicability for coarse-grained materials in earth-rock dam engineering. Wu et al.12 constructed a single-parameter Wu-continuous grading equation based on the Zhu-continuous grading equation.

In addition, researchers have conducted a large amount of research on the evolution characteristics of particle breakage in coarse-grained soils using particle breakage indicators and grading equations, considering factors such as particle size distribution and load parameters. Cai et al.13 studied the particle breakage law of rockfill materials under different gradations, densities, confining pressures, and established a relationship between fractal dimension, gradation and confining pressure. Einav4 synthesized previous research and established a modified Hardin breakage index based on Hardin’s breakage theory. Mcdowell14 and Lobo-Guerrero15 found that the survival probability of particles is consistent with the Weibull probability distribution between loads. Ozkan et al.16 started the perspective of stochastic processes and found that the process of particle breakage conforms to Markov processes, and established a Markov model for describing the evolution of gradation. Tong et al.17,18 further proposed a Markov chain model for describing the evolution of soil breakage. Chen et al.19 introduced the ratio of survival probability to breakage probability and used the log-odds regression method to establish a log-odds function expression for soil gradation curves.

In summary, most existing studies use the final grading curve to represent the change in particle grading, failing to predict the intermediate process of particle grading. However, under the action of high-speed train loads, the particle breakage of coarse-grained soil fillers is a gradual process. Therefore, it is necessary to reveal the intermediate process of soil particle grading evolution to grasp the intermediate process of subgrade engineering characteristics.By combing the related research results of soil particle breakage index and the strength, deformation and permeability of subgrade fill in the review, the author tries to use soil particle breakage index to reflect the change of engineering properties of subgrade fill of high-speed railway. For example, under the action of train load, the coarse-grained soil filler particles are broken, which leads to the gradation change, and then affects the dynamic deformation characteristics of the roadbed.

This article constructs a coarse-grained soil filler unit model and uses a high-performance hydraulic servo loading system (MTS) to simulate the dynamic load effect of trains, analyzes the influence of dynamic stress amplitude, loading frequency, and other factors on the particle breakage of coarse-grained soil fillers, and defines the particle breakage index in the particle content and relative particle size coordinate system using the Zhu-continuous grading equation. The relationship between the breakage index and grading parameters is derived, and a mathematical model for the evolution of particle grading in subgrade coarse-grained soil fillers is finally established to explore the evolution law of the breakage process of high-speed railway subgrade coarse-grained soil. Combined with the laboratory test and the research results of this paper, the grading evolution process of railway roadbed under long-term train load can be predicted and analyzed, and then the service status of high-speed railway roadbed can be analyzed, which plays an important reference for the selection and filling of roadbed fill, and provides a reference for the design and maintenance of roadbed, which has very important engineering practical significance.

To simulate the plane strain state of railway subgrades, test samples are designed as square cylinders with a side length of 20 cm and a height of 40 cm. To mitigate the influence of particle size, the maximum allowable particle size allowed for these test samples is set at 4 cm. The test samples are contained within a custom-made model box (as illustrated in Fig. 1). This model box primarily consists of a base, a straight track, a loading rod, a loading plate, a lateral steel plates, a vertical loading plate, etc. Figure 2 shows a dynamic cyclic loading test system for coarse-grained soil fillers using MTS and a model box.

Three-dimensional diagram of model box.

Test equipment.

To simulate the actual stress conditions experienced by subgrade soil, steel plates are used to constrain the deformation on both sides of the sample. A loading rod equipped with a spring and an adjustable loading plate simulates constraints imposed by adjacent soil along the direction of the line. The stiffness coefficient k of the spring is taken as 20 N/mm. The actuator from the MTS loading system is positioned above the sample to apply dynamic loads that mimic those generated by high-speed trains (as shown in Fig. 2).. The maximum range of the actuator used in the MTS loading system is 50 kN, the sensor accuracy is 0.5% of the indicated value, the maximum loading frequency is 30 Hz, and the maximum stroke is 15 cm, which can achieve high-frequency and high-vibration loading.

To obtain a coarse-grained soil filler that complies with the requirements outlined in the Code for Design on Subgrade of Railway, clay and siltstone gravel with particle sizes ranging from 2 to 40 mm are mixed. Among them, the basic physical properties of clay are shown in Table 1, Particle quantity (%) in each grain group is shown in Table 2, and the specific gravity of siltstone gravel Gs = 2.72 and density is 2.43 g/cm3.

Figure 3 shows the particle size distribution curve of the sample, which exhibits a non-uniformity coefficient Cu = 44.7, a curvature coefficient Cc = 2.03, and a good grading. The maximum particle size of the sample is 40 mm. Through compaction tests, the maximum dry density of the sample is obtained as ρdmax = 2.17 g/cm3, the optimal moisture content is ρwopt = 6%, and the saturated moisture content is wsat = 14.1%.

Particle size distribution curve of the sample.

According to the requirements for coarse-grained soil fillers in the Code for Design on Subgrade of Railway, the compaction degree is taken as 0.95. To ensure the degree of compaction and uniformity of the sample, a jack is used to compact it into 3 layers, with a height of about 13.3 cm per layer. After sample preparation, the particle size distribution curve of the sample is shown in Fig. 3. The non-uniformity coefficient Cu of the sample is 54.63, and the curvature coefficient Cc is 1.37.

The actual frequency of load-bearing for the subgrade is lower than that of the train load20,21,22,23,24. Under the influence of high-speed trains traveling at speeds ranging from 200 to 350 km/h, the primary frequency of load-bearing for the subgrade falls within a range of 2 to 16 Hz. When performing dynamic tests on the surface layer, bottom layer, and filler of the subgrade body of the subgrade bed, the maximum loading frequency can be taken as 3, 2, and 1 times the frequency v/L corresponding to the length L of the train compartment, respectively. The applied dynamic load adopts a full-pressure periodic sine function, which can be expressed as:

where σmax is the vertical dynamic stress amplitude of each structural layer of the subgrade, σ0 is the vertical static pressure value of each structural layer of the subgrade, and f is the loading frequency.

To simulate the dynamic load effects and stress environment exerted by trains on the bottom layer of the subgrade bed, the applied vertical static pressure is taken as σ0 = 25 kPa. Since the main frequency of the dynamic load on the bottom layer of the subgrade bed is between 2-8 Hz, the loading frequency for this test is set at 2, 4, 6, and 8 Hz, corresponding to train speeds of 89.7, 180.7, 271.6, and 362.5 km/h, respectively. The loading waveform is shown in Fig. 4.

Waveform Curve of Dynamic Loading.

During the testing process, the influence of dynamic stress amplitude, loading frequency, and number of loading cycles was mainly considered. A total of 19 samples were taken, in which samples 1–16 mainly investigated the influence of frequency and dynamic stress amplitude on the gradation of coarse-grained soil, with a loading cycle of 50,000 times. Samples 17 to 19 mainly consider the influence of loading times on the gradation of coarse-grained soil, with loading times of 20,000, 50,000, and 100,000 times, respectively. The literature points out that 99% of the dynamic stress on the surface of railway subgrades does not exceed 110.5 kPa25, and the literature gives the measured values of dynamic stress on the surface of multiple railway subgrades ranging from 9.5 to 100 kPa26. To fully consider the dynamic load effect of high-speed train loads on coarse-grained soil fillers, the dynamic stress amplitude range for the test loading is taken to be 25–200 kPa. The test adopts the stress control method, applying vertical static pressure first, and then applying dynamic load after the deformation stabilizes. Table 3 lists the test protocols and loading parameters for each sample.

In the actual situation, due to the influence of track irregularity and other factors, the load borne by the roadbed is not completely periodic, so there may be some limitations in using the periodic sine function to simulate the dynamic load of the train on the roadbed. However, the existing research has found that because the ballastless track has a good diffusion effect, the frequency characteristics of the bogie axle are not obvious, and the action frequency of the load borne by the roadbed is greatly reduced. The load borne by the roadbed has four main frequencies, and the first main frequency is basically consistent with the frequency corresponding to the vehicle length L. Therefore, according to the loading process and spectrum characteristics of each structural layer of the subgrade, the maximum loading frequency of the subgrade surface packing is taken as 3 times of the corresponding frequency v/L of the length of the carriage, that is, w = 3v/L. The maximum loading frequency is 2 times of the corresponding frequency v/L, that is, w = 2v/L. The maximum loading frequency v/L corresponding to the length of the carriage is taken as the maximum loading frequency, that is, w = v/L, when the subgrade body filler is carried out dynamic test.

To obtain the particle size distribution change characteristics of coarse-grained soil fillers under a load of high-speed trains, the entire sample was sieved with a total weight of approximately 33 kg per sample to avoid errors caused by sampling. First, a 0.075 mm sieve was used to manually wash and remove the clay particles, then allow them to air dry and bake naturally. Afterward, a vibrating screen was used to sieve them. According to the screening test and different vibration times, the content changes of each particle group of the sample are shown in Fig. 5.

Histogram of load applied and particle content change under different particle size.

As shown in Fig. 5, under the load test conditions, the particles with a particle size of 25–40 mm in the coarse-grained soil filler decreased by 0.4–2%, the particles with a particle size of 20–25 mm increased by 0.2–1.4%, and the content of particles below 0.075 mm increased by 0.1–1.2%. The changes in the content of particles in other particle groups were relatively small. In addition, the changes in particle size distribution after particle breakage are consistent for different dynamic stress amplitudes, vibration frequencies, and loading times, indicating that particle size distribution has a significant impact on the particle breakage mode of coarse-grained soils. Existing research has shown that the particle breakage of granular materials mainly occurs in three forms: cracking, crushing, and grinding. Due to the decrease in the content of the 25–40 mm particle size group, the content of the next particle size group increases, indicating that there is a form of fracture in the breakage of coarse-grained soil fillers under dynamic cyclic loading. The increase in the content of particles below 0.075 mm indicates that there is a grinding phenomenon between particles in the coarse-grained soil filler under dynamic cyclic loading.

As can be seen from the column chart of particle content changes after different vibration times, after applying dynamic loads of different vibration times to coarse-grained soil fillers, the increasing or decreasing particle groups are the same. Overall, the larger the number of loading times, the greater the change in the content of each particle group.

To analyze the evolution of coarse-grained soil particle breakage, it is essential to address the challenge of accurately representing grading. If a precise mathematical description of grading can be established, achieving a quantitative expression for particle breakage will be become more feasible27.To date, an accurate quantitative characterization of gradation continues to depend on the grading curve.

Zhu et al.10 statistically analyzed the shape of soil gradation curves in a large number of practical engineering applications and proposed the Zhu-continuous gradation equation with universal representation ability, as shown in Eq. (2):

where P is the passing mass percentage of particles with a particle size d, dmax is the maximum particle size, and m and b are characteristic parameters, where m determines the inclination of the curve and b determines the shape of the grading curve.

To facilitate analysis, normalization is performed by defining a relative particle size variable x and establishing a relative coordinate system.

Equation (2) can be transformed into:

where p′ is the mass percentage of particles with a relative particle size x passing through.

The analysis of the coarse-grained soil after sample preparation using Eq. (4) is shown in Fig. 6.

Zhu-continuously graded equation fitting sample gradation.

As shown in Fig. 6, the actual grading curve is ZHU-coincident with the curve drawn by the equation fitted from Eq. (4), and the correlation coefficient R2 is greater than 0.98, indicating that the continuous gradation equation has good applicability to coarse soil fillers in this study.

Due to the large range of gradation options for subgrade coarse-grained soil, and the fact that grading changes develop from initial gradations to more accurately describe the process of gradation change in subgrade coarse-grained soil fill under train load, Eq. (4) is expressed in incremental form:

where m0 and b0 are the characteristic parameters of the compacted sample after filling, b0 and b are the increments in characteristic parameters caused by dynamic load.

Thus, a predictive model for the intermediate process of particle size distribution in coarse-grained soil fillers has been established. The value of p′ consistently falls within the range (0, 1), which corresponds to the cumulative mass percentage of particles smaller than a specified size on the grading curve. The value of relative particle size x is also between (0, 1), corresponding to the particle size in the grading curve. The parameters of the mathematical expression for the grading curve can reflect changes in the particle size distribution of soil, allowing for predicting the intermediate stages of grain breakage evolution by using a mathematical formula.

The fitting parameters for the dynamic loading of the grading curve are shown in Table 4, by using Eq. (5) to fit. As shown in Table 4, the correlation coefficient R2 is greater than 0.98, indicating that the grading Eq. (5) also has good applicability to coarse-grained soils after dynamic testing loading.

The prediction model of particle size distribution evolution proposed in this paper has wider applicability to all kinds of continuously graded soil materials. At the same time, the prediction model of particle size distribution evolution and stress–strain evolution can be used to reflect the mathematical model of the evolution of coarse grain soil gradation with stress–strain, which will be very important for the design and maintenance of railway roadbed.

To visually analyze the influence of dynamic stress amplitude and loading frequency on the gradation of coarse-grained soil fillers, the relationship curves between the characteristic parameters ∆m and ∆b of gradation and the load parameters σdmax and f are plotted, as shown in Fig. 7.

Variation curve of parameters ∆m and ∆b.

As shown in Fig. 7, both σdmax and f have a significant impact on the characteristic parameters ∆m and ∆b. ∆m and ∆b exhibit a trend of rapid increase followed by a slow increase with σdmax, which can be approximated by hyperbolic functions. ∆m and ∆b increase with f, which can be represented by a power function. Therefore, a binary nonlinear regression analysis was conducted based on Fig. 8, and a relationship function between parameters ∆m and σdmax, f was constructed:

where a, c, e, g, h, and k are fitting parameters.

Relationship curve between ∆m, ∆b and N.

By fitting the data in Fig. 8 using Eqs. (6) and (7), the corresponding values for c, e, g, and h were obtained. The results are shown in Eqs. (8) and (9)

The relationship between characteristic parameters ∆m and ∆b and the number of loading cycles N was investigated. Taking 50,000 loading cycles as a benchmark, the normalized values of ∆m and ∆b after 20,000 and 100,000 loading cycles were calculated, as shown in Fig. 8.

It is not difficult to find that ∆m and ∆b increase with the number of loading times. Therefore, the relationship between ∆m, ∆b and loading times is described by a hyperbolic function:

Due to the similarity of changes in the particle size distribution of coarse-grained soil under different loads, the normalized expressions for characteristic parameters ∆m and ∆b to loading times N are substituted into the equations for characteristic parameters ∆m and ∆b. An expression for characteristic parameters ∆m and ∆b that takes into account dynamic stress amplitude a, loading frequency f, and loading times N was established:

Substituting Eqs. (12) and (13) into Eq. (5), the grading evolution equation of coarse-grained soil filler under train load was obtained:

To assess the accuracy of the prediction model presented in this article, the characteristic parameters of coarse-grained soil fillers were calculated using Eqs. (12) and (13), and these results were compared with experimental values, as illustrated in Fig. 9.

Comparison of parameter calculation values and experimental values.

As depicted in Fig. 10, both test values and predicted values predominantly align along the line y = x. Considering the characteristic parameters ∆m and ∆b of dynamic stress amplitude σdmax, loading frequency f, and number of loading N, it is reasonable to express them as a function that can reflect the changes in the grading characteristics of coarse-grained soil subgrade fill under train load to some extent.

Comparison of theoretical and experimental results for the grading curve.

To further validate the rationality of the evolution equation for coarse-grained soil filler gradation, Eq. (14) was used to calculate and obtain the gradation curves of coarse-grained soil fillers under different frequencies and dynamic stress loading amplitudes (Fig. 10).

As shown in Fig. 10, the grading curve calculated by Eq. (10) is consistent with that obtained from experiments, indicating that the expression for the evolution of grading curves established has strong adaptability.

Among the various indicators used to describe particle breakage, Bg and Br are the most commonly employed, however, both exhibit significant limitations. Bg and Br provide a way to define the fragmentation index, i.e., using ratios of the same dimension as the fragmentation rate indicator can quantify the degree of soil particle fragmentation by relative fragmentation rate. Therefore, in the coordinate system of particle content and relative particle size, a crushing rate index B′r is defined as shown in Fig. 11. Its expression is:

where B′r is the relative crushing rate of soil mass and its value ranges from 0 to 1; B′p is the crushing potential, which is the area enclosed by the initial grading curve S0 and the relative particle size coordinate; B′t is the amount of crushing, which is the difference between the area enclosed by the initial grading curve S0 and the relative particle size coordinate and the area enclosed by the post-test grading curve S1 and the relative particle size coordinate.

Schematic map of B′r.

The calculation result of Eq. (15) ranges from 0 to infinity, which can reflect the overall changes in the grading. The larger the value, the greater the change in the grading curve and the more obvious the particle breakage.

The area of the aforementioned region can be determined by integrating the grading equation. Given that the minimum value of particle size cannot be 0, a lower limit dk for the value of d needs to be set during the integration process. Existing research has shown that the engineering properties of coarse-grained soils mainly depend on the content of coarse particles and the properties of fine materials28. Therefore, this article refers to the limit value of the integration curve proposed by “Hardin” and takes dk = 0.074 mm. When the minimum particle size in the grading curve is greater than 0.075 mm, dk is directly taken as the minimum particle size. The area S enclosed by the grading curve, x = d/dmax = 1, and x = 0.074/dmax can be expressed as:

where dx is the differential form of the screening pass rate corresponding to the relative particle size of p′.

Since Eq. (16) cannot obtain an exact numerical solution, the adaptive “Lobatto” numerical integration method can be used to solve.

It should be pointed out that in Eq. (16), the lower limit of integration is 0.074/dmax. Obviously, for other types of soil, the lower integral limit can be set according to the need.

The grading parameters were determined by the initial grading m0 and b0. Assuming the characteristic parameters after particle breakage as ∆m and ∆b, the breakage rate index B′r was obtained by substituting Eq. (16) into Eq. (15):

In order to verify the adaptability and correctness of the particle breakage model, the experimental data of rock fill in Jia et al.29 's study and rockfill in Gao et al.30 's study on particle breakage in large-scale triaxial test were selected for verification in this paper. Formula (4) was used to obtain the initial m0 and b0 values of soil sample grading parameters before shearing, and then formula (5) was used to obtain the values of soil sample grading Δm and Δb after shearing. The results are shown in Tables 5 and 6. It can be seen that the correlation coefficient obtained by using the particle fragmentation evolution model established in this paper is greater than 0.99. In order to further verify the reasonableness of the crushing rate index B′r obtained by the grading equation in this paper, the crushing rate index B′r was calculated using Eq. (17), and the results are shown in Table 6.

The existing measured data indicate that the maximum dynamic stress generated by a train with an axle load ranging from 19.6 to 22.5 tons on the road surface is 185 kPa. Furthermore, when the train speed varies between 200 and 350 km/h, the main frequency of the load experienced by both the subgrade bed layer and the subgrade body is between 2 and 16 Hz, and the peak frequency at the bottom of the subgrade bed layer is below 13 Hz31,32. Therefore, this article analyzes the impact of train loads with a dynamic stress amplitude not exceeding 200 kPa and an action frequency not exceeding 12 Hz on coarse-grained soil subgrades. The 16 calculation conditions in Table 7 were analyzed by Eq. (17), which are shown in Fig. 12.

Evolution process of particle breakage in coarse-grained soil filler.

The relationship expression between dynamic cumulative strain and the number of actions for medium-coarse soil fillers through research was obtained33 and the dynamic cumulative strain of coarse soil fillers under different calculation conditions was calculated (Fig. 12).

It can be observed from Fig. 12 that, under the same frequency load, a larger the dynamic stress amplitude correlates with an increased particle breakage index and greater cumulative strain. Under the same dynamic stress amplitude, the larger the loading frequency, the larger the particle breakage index, and the larger the cumulative strain. Before 1 million loadings, the rate of increase in particle breakage index is relatively large and the development speed of cumulative strain is fast. After exceeding 1 million loadings, the growth rate of the particle breakage index becomes significantly smaller and gradually stabilizes, while the cumulative strain also gradually stabilizes. The analysis shows that the dynamic cumulative strain of coarse-grained soil fillers is significantly correlated with particle breakage. Under the load with a dynamic amplitude of 10–200 kPa and frequency of 2–12 Hz, the final cumulative strain value of coarse-grained soil filler is between 0.36 and 1.9%, which is similar to that obtained from full-scale model tests on subgrade under the ballastless track in reference34. It can be inferred that the breakage rate index of coarse-grained soil subgrade will eventually stabilize within 1.1% under train loading.

In order to further analyze the impact of train load on the crushing of coarse-grained soil fill material, it is necessary to discuss the relationship between the dynamic stress amplitude and the axle weight and running speed of the train. Currently, the design guideline for railway subgrade in China has proposed a formula for calculating the design dynamic stress amplitude of subgrade35.

where F is the static axle weight of the vehicle; v is the train speed (km/h); c is the impact coefficient; When v is 200–250 km/h, α = 0.004; When v is between 300 and 350 km/h, α = 0.003.

Based on the measured data of dynamic stress attenuation, Chen Yunmin et al. proposed a formula to describe the depth attenuation of dynamic stress in roadbed35:

where ϕ(z) is the attenuation coefficient of dynamic stress of roadbed at depth z, z is the depth of roadbed soil, and a′ , b′ are fitting coefficients. For the ballast track subgrade, the mean value of a′ is 0.64, and the mean value of b′ is 0.86. For ballastless track subgrade, the mean value of a′ is 2.12, and the mean value of b′ is 1.18.

Therefore, the variation law of dynamic stress along depth of railway roadbed can be expressed as:

Meanwhile, the author believes that the distribution law of the main frequency f in different positions of the subgrade along the depth can be expressed36:

where ψ(z) is the frequency attenuation coefficient of the load borne by the roadbed at depth z, f0 = v/l is the reference frequency of the train load, v is the train running speed (m/s), l is the disturbance wavelength (m), that is, the train wheelbase.

Taking CRH380A EMU as an example, axle load F = 140kN, carriage length L is 25 m. Calculate when the train is running on a ballasted track. The final conditions of the crushing of filler particles at the surface of the foundation bed (0 m), the bottom surface of the foundation bed (− 0.4 m) and the bottom surface of the foundation bed (− 2.7 m) are shown in Table 8, which can be used as a reference for the evaluation of the service status of the subgrade.

It can be clearly seen from the table that the higher the train loading speed, the higher the fragmentation rate. The lower the subgrade depth, the lower the particle fragmentation rate. By using the particle fragmentation method proposed in this paper to establish a database of particle fragmentation for coarse aggregate filler, the correspondence between particle fragmentation and train loading can be clearly seen, which can be applied in real engineering.

In future studies, we will consider optimizing the test device and adding different particle sizes for the test, so as to more comprehensively evaluate the impact of particle size on the crushing process." At the same time, more advanced analysis and calculation technology, such as three-dimensional discrete element analysis, is used to simulate the actual stress state more accurately, and obtain the evolution characteristics of the coarse-grained soil filler particle crushing process more in line with the engineering practice.

This paper employs the MTS loading system to simulate the dynamic loading of coarse-grained soil filler induced by train movement. The study analyzes the effects of dynamic stress amplitude, loading frequency, and vibration frequency on both macro and micro characteristics of particle breakage. The findings reveal the particle breakage behavior of coarse-grained soil filler under high-speed train loading.

Utilizing on the ZHU-continuous grading curve equation, a predictive model for the intermediate process of coarse soil filler particle gradation has been established within a coordinate system defined by particle content and relative particle size. The applicability of this model has also been validated, providing a quantitative framework to investigate the relationship between train load and particle gradation.

A new crushing rate index is defined within the coordinate system of particle content and relative particle size. Utilizing the grading prediction model, an integral expression for the crushing rate index has been derived. The evolution characteristics of coarse-grained soil particle crushing under train load are analyzed through the lens of the particle crushing rate index. The results indicate that the crushing rate index correlates well with measured data, suggesting that the index established in this study can effectively describe the evolution process of coarse-grained soil particle crushing in railway subgrades.

Under cyclic train loading conditions, the primary modes of particle breakage for coarse-grained soil fillers in subgrades are identified as fracture and grinding, with supplementary contributions from crushing. Notably, under varying dynamic stress amplitudes and loading frequencies, there exists a similarity and continuity in the particle breakage behavior of coarse-grained soil fillers.

A significant correlation is observed between dynamic cumulative deformation and particle breakage in coarse-grained soil fillers. As the number of load applications increases, both the degree of particle breakage within these fillings and cumulative deformation within the subgrade rise correspondingly. Initially, this growth occurs rapidly but gradually diminishes over time. After approximately 1 million loading cycles, both the deformation and particle breakage tend to stabilize. When subjected to dynamic stress amplitudes ranging from 10 to 200 kPa at loading frequencies between 2 and 12 Hz, it is found that the particle breakage index stabilizes below 1.1%.

The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.

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The authors appreciate the constructive comments from the anonymous reviewers. This work is supported by Hunan Provincial Natural Science Foundation of China (2023JJ50104); Scientific Research Fund of Hunan Provincial Education Department (22B0853; 24A0659); Hunan Institute of Technology Doctoral Initiation Fund Project (HQ21014, HQ21029).

College of Civil and Construction Engineering, Hunan Institute of Technology, Hengyang, 421002, China

Qiyun Wang, Jiajun Zeng & Qingqing Shen

College of Civil Engineering, Fujian University of Technology, Fuzhou, 350118, China

Huaming Lin

School of Railway Engineering, Hunan Technical College of Railway High-Speed, Hengyang, 421002, China

Yao Long

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Qiyun Wang: Conceptualization, Formal analysis, Investigation, Data Curation, Writing-Original Draft, Funding acquisition; Jiajun Zeng: Formal analysis, Data curation, Investigation, Writing-Review & Editing, Funding acquisition; Qingqing Shen: Investigation, Writing-Review & Editing; Huaming Lin: Data curation, Validation; Yao Long: Data curation, Validation.

Correspondence to Jiajun Zeng.

The authors declare no competing interests.

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Wang, Q., Zeng, J., Shen, Q. et al. Evolution of crushing process of coarse-grained soil filler under the trains load. Sci Rep 15, 35 (2025). https://doi.org/10.1038/s41598-024-83472-7

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Received: 08 September 2024

Accepted: 16 December 2024

Published: 02 January 2025

DOI: https://doi.org/10.1038/s41598-024-83472-7

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