Open AccessArticle

Dynamic Response Analysis of Ballastless Tracks Considering the Temperature-Dependent Viscoelasticity of Cement-Emulsified Asphalt Mortar Based on a Vehicle–Track–Subgrade Coupled Model

School of Rail Transportation, Soochow University, Suzhou 215131, China

Authors to whom correspondence should be addressed.

Lubricants 2025, 13(2), 58; https://doi.org/10.3390/lubricants13020058

Submission received: 5 December 2024 / Revised: 27 January 2025 / Accepted: 28 January 2025 / Published: 30 January 2025

(This article belongs to the Special Issue Recent Advances in Lubricated Tribological Contacts)

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Browse Figures

Figure 1
CRTS I ballastless track. "> Figure 2
The FM model. "> Figure 3
Zener model. "> Figure 4
Fitting curves of the creep compliance. "> Figure 5
Finite element model for the compressive creep experiment with CA mortar. "> Figure 6
Simulation results of the creep experiment. "> Figure 7
Vertical vehicle–track–subgrade system model [<a href="#B30-lubricants-13-00058" class="html-bibr">30</a>]. "> Figure 8
Vertical track irregularity. "> Figure 9
The VTS model. "> Figure 10
Track subsystem simulation results. (a) Rail displacement; (b) slab displacement; (c) fastener force; (d) CA mortar stress. "> Figure 11
Comparison of the simulation results with the relevant literature. (a) Slab displacement and rail displacement; (b) fastener force and CA mortar stress. "> Figure 12
Comparison of time history curves of dynamic responses of the track subsystem. (a) Rail displacement; (b) slab displacement; (c) CA mortar displacement; (d) compressive displacement of CA mortar; (e) CA mortar stress; (f) base displacement. "> Figure 12 Cont.
Comparison of time history curves of dynamic responses of the track subsystem. (a) Rail displacement; (b) slab displacement; (c) CA mortar displacement; (d) compressive displacement of CA mortar; (e) CA mortar stress; (f) base displacement. "> Figure 13
Comparison of the maximum simulation outcomes. (a) Rail and slab displacement; (b) CA mortar and concrete base displacement; (c) compressive displacement of CA mortar; (d) CA mortar stress. "> Figure 14
Comparison of the maximum dynamic response results of the vehicle subsystem. (a) Vertical acceleration of the vehicle body; (b) wheel–rail force. "> Figure 15
Growth rate of dynamic responses at different temperatures. "> Figure 16
Comparison of the maximum dynamic response results of the track subsystem. (a) Slab displacement; (b) CA mortar displacement; (c) compression displacement of CA mortar; (d) CA mortar stress. "> Figure 17
Comparison of the maximum dynamic response results of the track subsystem. (a) Rail displacement; (b) CA mortar displacement; (c) CA mortar acceleration; (d) CA mortar stress. "> Figure 18
Position diagram for the CA mortar layer. "> Figure 19
Comparison of the maximum dynamic response results of CA mortar. (a) CA mortar displacement; (b) CA mortar stress. "> Figure 20
CA mortar layer compressive stress cloud diagram. ">

Versions Notes

Abstract

This study aims to explore the dynamic response of ballastless tracks under various temperatures of the cement-emulsified asphalt (CA) mortar layer and other environmental factors. CA mortar is the key material in the ballastless track structure, exhibiting notably temperature-dependent viscoelastic properties. It can be damaged or even fail due to the continuous loads from trains. However, the dynamic behaviors of ballastless tracks considering the temperature-dependent viscoelasticity of CA mortar have been insufficiently studied. This paper captures the temperature-dependent viscoelastic characteristics of CA mortar by employing the fractional Maxwell model and applying it to finite element simulations through a Prony series. A vehicle–track–subgrade (VTS) coupled CRTS I ballastless track model, encompassing Hertz nonlinear contact and track irregularity, is established. The model is constrained symmetrically on both of the longitudinal sides, and the bottom is fixed on the infinite element boundary, which can reduce the effects of reflected waves. After the simulation outcomes in this study are validated, variations in the dynamic responses under different environmental factors are analyzed, offering a theoretical foundation for maintaining the ballastless tracks. The results show that the responses in the track subsystem will undergo significant changes as the temperature rises; a notable effect is caused by the increase in speed and fastener stiffness on the entire system; the CA mortar layer experiences the maximum stress at its edge, which makes it highly susceptible to damage in this area. The original contribution of this work is the establishment of a temperature-dependent vehicle–track–subgrade coupled model that incorporates the viscoelasticity of the CA mortar, enabling the investigation of dynamic responses in ballastless tracks.

Keywords:

ballastless tracks; dynamic response; CA mortar; temperature; fractional viscoelasticity

1. Introduction

High-speed railways have swiftly advanced worldwide for their exceptional efficiency and excellent safety, as evidenced by notable developments in countries such as China, Germany, and Japan [1]. Ballastless tracks, serving as the carrier for high-speed trains, have emerged as the premier choice in contemporary railway construction [2] for high smoothness and low maintenance demands. In China, CRTS I ballastless tracks, which are shown in Figure 1, have surpassed 2700 km in total operating mileage, and the cement-emulsified asphalt (CA) mortar layer is a crucial structure throughout the construction and operation period, performing functions such as leveling and vibration reduction. Its mechanical performance significantly affects the operation safety of trains [3]. A CRTS I ballastless track will undergo damage or even failure under dynamic loads from the train, primarily manifested in the large-scale cracking and peeling of CA mortar layers, which constitutes a serious risk to train operation [4]. Therefore, evaluating how CA mortar’s mechanical characteristics effect the dynamic behaviors of ballastless tracks is of great importance.

Given that CA mortar is mainly constituted by emulsified asphalt, the mechanical behaviors of the CA mortar layer demonstrate typical viscoelastic characteristics. This time-dependent viscoelasticity has made it difficult to study the dynamic responses of ballastless tracks [5,6]. Studies have indicated a significant relationship between temperature and CA mortar viscoelastic properties. Fu et al. [7] conducted relaxation tests on CA mortar and established a stress relaxation function with temperature-sensitive factors. Wang et al. [8] carried out compressive tests on CA mortar and derived the stress–strain relationship at different temperatures. Zhu et al. [9] found that CA mortar’s creep behavior was aggravated when the temperature rose. Yuan et al. [10] examined the dynamic physical reactions of CA mortar and obtained viscoelastic curves in relation to temperature. You et al. [11] discovered that the elastic modulus of asphalt materials decreases as the temperature rises. Additionally, Xu et al. [12] developed a molecular chain network model to analyze how temperature variation influences the mechanical behavior of viscoelastic materials from a micro-level perspective. The above research mainly focused on physical experiments with CA mortar and connected different mechanical behaviors with the temperature, while the alterations in dynamic responses of the track system caused by this require further investigation. As the structure most prone to damage, the CA mortar layer gains considerable attention in dynamics studies of the track system. For instance, Dai [13] constructed a model of ballastless tracks in ABAQUS and carried out comparative analyses of the displacement and stress in slab when CA mortar was regarded as a viscoelastic or elastic material. Ren et al. [14] investigated the impact of the cracking area in the CA mortar layer on track structures with a vertical finite element model. However, this study regarded CA mortar as an elastic material. In Ref. [15], Ren simulated the deformation process of CA mortar and established a track structure model with the help of the time-hardening rate viscoelastic model in ABAQUS. Subsequently, they set different Young’s moduli for CA mortar to analyze the pattern of displacement variations. Deng et al. [16] wrote a subroutine to simulate the failure evolution process of CA mortar and then studied how the cracking area influences the fatigue accumulation of the CA mortar layer via a full-scale model in ABAQUS. Chen et al. [17] investigated the distortion between the layers of ballastless tracks and predicted the fatigue life of the ballastless tracks. Nevertheless, these studies generally use point loads to simulate the wheel–rail forces, neglecting track irregularity and wheel–rail contact, which leads to errors in load transfer. In analyzing the wheel–rail forces, Hertz nonlinear theory is typically employed for calculations [18]. Based on this, researchers have established corresponding models for different adhesion conditions between the wheel and rail to obtain the response of creeping forces [19,20]. Additionally, the aforementioned studies did not consider CA mortar’s temperature-dependent viscoelastic properties. This research aims to investigate how temperature variation in CA mortar affects the dynamic responses of ballastless tracks. In Section 2, the fractional Maxwell model is introduced to capture the creep curves, and we obtain the viscoelastic parameters required in ABAQUS to realize the creep progress of CA mortar using the Prony series. Then, a coupled vehicle–track–subgrade model is built. We employ Hertz nonlinear contact theory to calculate the vertical wheel–rail force. Additionally, the tangential friction coefficient is introduced to determine the lateral force. After comparing and verifying the simulation data with actual values, discrepancies in the dynamic responses under different environmental factors are further explored in Section 3. Section 4 presents the conclusions of this study.

2. Materials and Methods

Previous studies have often described the viscoelastic behavior of materials using the Zener model, Burgers model, or more complex classical models [21,22]. It was reported that fractional derivative models can describe the mechanical response of viscoelastic materials more accurately with fewer parameters [23]. The viscoelastic curves of CA mortar are captured by the fractional derivative model in this section.

2.1. The Fractional Maxwell (FM) Model

We can see from Figure 2 that by substituting the Newtonian dashpot in the Maxwell model with an Abel dashpot, a fractional Maxwell (FM) model can be obtained, and the constitutive relation is described by [24]

\{\begin{matrix} ε = ε_{e} + ε_{v e} \\ ε_{e} = σ / E \\ σ = η D^{α} ε_{v e} \end{matrix}

(1)

where ε_e and ε_ve are the strains of the elastic and viscoelastic bodies, E represents the elastic modulus, η represents the viscosity coefficient, and D^α denotes the fractional derivative, which is defined in the Riemann–Liouville form [25]:

D^{α} f (t) = {[Γ (1 - α)]}^{- 1} d / d t \int_{0}^{t} f (τ) \cdot {(t - τ)}^{- α} d τ

(2)

Researchers have established a bridge in the Laplace domain between different viscoelastic behaviors with the help of the transfer function T(s); this can be achieved using the ratio of stress to strain [26], which has been employed in viscoelastic models [27]. Table 1 presents the transfer functions of basic mechanical components.

According to the definition, the transfer function T(s) of the FM model is achieved:

T (s) = E η s^{α} / (η s^{α} + E)

(3)

In addition, the creep compliance J(t) is achieved from Equation (6) [26]:

J (t) = L^{- 1} [1 / s / T (s)]

(4)

For the FM model, the creep compliance is obtained by

J (t) = 1 / E + 1 / [η Γ (1 + α)]

(5)

2.2. Viscoelastic Description of CA Mortar Based on Creep Experimental Data

We adopt the results of the creep test on CA mortar with the temperature ranging from −20 to 40 °C in Ref. [28] and then employ the FM and Zener model to capture the creep curves. The Zener model consists of 2 springs and a Newtonian dashpot, as shown in Figure 3, and the creep compliance includes both elastic and viscoelastic components:

J (t) = 1 / E_{1} + 1 / [E_{2} (1 - e^{- E_{2} t / λ})]

(6)

where E₁ and E₂ represent the elastic modulus of the springs.

The fitting results are presented in Figure 4, and Table 2 lists the fitting parameters of the two models. It can be found that under the same condition of using three parameters, the FM model describes the creep behavior much better than the Zener model, with a goodness of fit no less than 0.95. Moreover, compared to other models with more parameters, such as the Burgers model, the advantage of having fewer parameters is that it avoids overfitting and is less likely to produce fitting results that lack physical meaning, making it more beneficial for researchers to analyze the relationship between parameters and environmental factors [24].

In ABAQUS, the viscoelastic properties of materials are expressed by the generalized Maxwell model, and we can simulate the creep progress of viscoelastic materials by its relaxation modulus, the Prony series [28]:

G (t) = G_{0} - G_{0} \sum_{i = 1}^{k} g_{i} [1 - \exp (- t / τ_{i})]

(7)

where G₀ represents the instantaneous modulus, k is the number of the Maxwell model, g_i is the relative modulus, and τ_i is the relaxation time for the i-th Maxwell model.

For the purpose of describing the viscoelasticity of CA mortar in ABAQUS, the FM model is converted to the Prony series in this section. By performing the Laplace transform in Equation (9), the transfer function with parameters of the Pony series is obtained:

T (s) = G_{0} (1 - \sum_{i = 1}^{k} g_{i} {(1 + τ_{i} s)}^{- 1})

(8)

According to Equation (2), we can obtain the parameters in Equation (7) by numerical inverse Laplace transform [29], and the parameters are optimized by a fitting program. The parameters are presented in Table 3, and k is set to 7, which is accurate enough to capture the creep curves.

To confirm the accuracy of the fitting parameters, referring to the specimen size in Ref. [28], a φ110 × 110 mm finite element model is established to simulate the uniaxial compression creep experiment in ABAQUS, as presented in Figure 5. The model adopts C3D8 elements with a total number of 13,328. The stress applied at the top of the model is 0.1 MPa, and three displacement degrees of freedom are fixed at the other end. The results of the simulation are presented below.

Figure 6 shows that despite the presence of diverse errors in the simulation of compressive creep under 4 temperatures, the accuracy is high enough to provide a favorable foundation for constructing a model of ballastless tracks, and the maximum error is 1.97%.

2.3. Establishing a Vehicle–Track–Subgrade (VTS) Finite Element Model

An accurate model that is congruent with reality is the basis for conducting dynamics studies. We constructed a VTS coupled model in ABAQUS. In this study, the vehicle subsystem is regarded as comprising rigid bodies to enhance the computational efficiency, and they are linked by the connection sections in ABAQUS. Stiffness and damping values are assigned to the connection sections to imitate the suspension system.

Figure 7 presents the vertical schematic diagram of this model, and the vertical motion equation in the vehicle subsystem is obtained by D’Alembert’s principle [18].

Heaving motion of vehicle body:

M_{v} {\ddot{Z}}_{v} + C_{v b} (2 {\dot{Z}}_{v} - {\dot{Z}}_{b 1} - {\dot{Z}}_{b 2}) + K_{v b} (2 Z_{v} - Z_{b 1} - Z_{b 2}) = - M_{v} g

(9)

Nodding motion of vehicle body:

J_{v} {\ddot{β}}_{v} + C_{v b} l_{b} (2 {\dot{β}}_{v} + {\dot{Z}}_{b 1} - {\dot{Z}}_{b 2}) + K_{v b} l_{b} (2 β_{v} + Z_{b 1} - Z_{b 2}) = 0

(10)

Heaving motion of front bogie:

\begin{array}{l} M_{b} {\ddot{Z}}_{b 1} + C_{b w} (2 {\dot{Z}}_{b 1} - {\dot{Z}}_{w 1} - {\dot{Z}}_{w 2}) + C_{v b} (2 {\dot{Z}}_{b 1} - {\dot{Z}}_{v} - l_{v} {\dot{β}}_{v}) \\ + K_{b w} (2 Z_{b 1} - Z_{w 1} - Z_{w 2}) + K_{v b} (2 Z_{b 1} - Z_{v} - Z_{v} β_{v}) = - M_{b} g \end{array}

(11)

Nodding motion of front bogie:

J_{b} {\ddot{β}}_{b 1} + l_{b} C_{b w} (2 l_{b} {\dot{β}}_{b 1} - {\dot{Z}}_{w 1} + {\dot{Z}}_{w 2}) + l_{b} K_{b w} (2 l_{b} β_{b 1} - Z_{b 1} + Z_{b 2}) = 0

(12)

Heaving motion of rear bogie:

\begin{array}{l} M_{b} {\ddot{Z}}_{b 2} + C_{b w} (2 {\dot{Z}}_{b 2} - {\dot{Z}}_{w 3} - {\dot{Z}}_{w 4}) + C_{v b} (2 {\dot{Z}}_{b 2} - {\dot{Z}}_{c} - l_{c} {\dot{β}}_{c}) \\ + K_{b w} (2 Z_{b 2} - Z_{w 3} - Z_{w 4}) + K_{v b} (2 Z_{b 2} - Z_{c} - Z_{c} β_{c}) = - M_{b} g \end{array}

(13)

Nodding motion of rear bogie:

J_{b} {\ddot{β}}_{b 2} + l_{b} C_{b w} (2 l_{b} {\dot{β}}_{b 2} - {\dot{Z}}_{w 3} + {\dot{Z}}_{w 4}) + l_{b} K_{b w} (2 l_{b} β_{b 2} - Z_{b 3} + Z_{b 4}) = 0

(14)

Heaving motion s of four wheelsets:

M_{w 1} {\ddot{Z}}_{w 1} + C_{b w} ({\dot{Z}}_{w 1} - {\dot{Z}}_{b 1} + l_{b} β_{b 1}) + K_{b w} (Z_{w 1} - Z_{b 1} + l_{b} β_{b 1}) = - M_{w 1} g + F_{r w, 1}

(15)

M_{w 2} {\ddot{Z}}_{w 2} + C_{b w} ({\dot{Z}}_{w 2} - {\dot{Z}}_{b 1} - l_{b} β_{b 1}) + K_{b w} (Z_{w 2} - Z_{b 1} - l_{b} β_{b 1}) = - M_{w 2} g + F_{r w, 2}

(16)

M_{w 3} {\ddot{Z}}_{w 3} + C_{b w} ({\dot{Z}}_{w 3} - {\dot{Z}}_{b 2} + l_{b} β_{b 2}) + K_{b w} (Z_{w 3} - Z_{b 2} + l_{b} β_{b 2}) = - M_{w 3} g + F_{r w, 3}

(17)

M_{w 4} {\ddot{Z}}_{w 4} + C_{b w} ({\dot{Z}}_{w 4} - {\dot{Z}}_{b 2} - l_{b} β_{b 2}) + K_{b w} (Z_{w 4} - Z_{b 2} - l_{b} β_{b 2}) = - M_{w 4} g + F_{r w, 4}

(18)

The meanings of the symbols in the formula above are shown in Table 4.

To reduce the boundary effect, this model has 21 track slabs, and the total length is 106 m. C3D8R hexahedral solid elements with eight nodes are employed to model the track and subgrade subsystem, with a total number of 100,440. The fastener system is replaced with connection sections, simulating the vibration reduction effect by defining stiffness and damping coefficients on connection sections. In actual engineering projects, the relative displacements between the various structures of the ballastless tracks are very small, so the layers are bound by tie connections to lower computational expenses. Since both the base and the subgrade are longitudinally continuous structures, symmetric constraints are applied at both ends of these structures to ensure that the model does not experience rigid body displacement [13,30]. The bottom is fixed at the infinite element boundary, which helps to reduce the effects of reflected waves, especially when the train is moving at high speeds [31].

The vehicle connects to the rail via the interaction between them, and the Hertz nonlinear theory provides a formula for calculating the vertical force [18]:

q (t) = {[Δ Z (t) / G]}^{3 / 2}

(19)

where G represents the wheel–rail contact constant, and ∆Z(t) is the total compression amount in both of them.

For high-speed trains, worn-profile tread wheels are used, and G is calculated as follows:

G = 3.86 R^{- 0.155} \times 10^{- 8}

(20)

where R represents the wheel radius.

The lateral friction coefficient µ determines the tangential wheel–rail force F(t), and it is set to 0.3.

F (t) = μ \times q (t)

(21)

Track irregularities will cause the train to vibrate irregularly, presenting a risk to the operational safety of the train. This paper employs the spectral density function for the American sixth-level track to simulate the vertical track irregularity; the formula is given by

S_{v} (ϕ) = j A_{v} ϕ_{c}^{2} / [ϕ^{2} (ϕ^{2} + ϕ_{c}^{2})]

(22)

where j is a constant coefficient that is taken as 0.25, A_v represents the coefficient of roughness (cm²·rad·m⁻¹), and ϕ and ϕ_c are the spatial frequency and the cut-off frequency (cm²·rad·m⁻¹), respectively, as shown in Table 5.

The data for vertical track irregularity along the advancing direction are obtained after numerical transformation [32] and are shown in Figure 8.

We can modify the coordinates of the rail elements to generate the track irregularity in ABAQUS. The constructed finite element model is shown in Figure 9.

2.4. Model Verification

In this model, the vehicle adopted is a power-distributed train developed in China. Table 6 lists the parameters of the vehicle and fastener [33].

The track–subgrade subsystem is simulated by C3D8R solid elements, and its physical property parameters are derived from actual values [34,35], as presented in Table 7. Additionally, the viscoelastic parameters listed in Table 3 are assigned to the CA mortar.

We utilize a computer with an Intel(R) Core(TM) i7-12700 CPU to perform calculations for this model following explicit dynamic steps. The simulation results in the vertical direction for the 4th fastener at the 11th slab at a speed of 300 km/h and temperature of 20 °C are presented in Figure 10.

Table 8 lists the calculated results (taking the maximum value) from this paper and compares them with relevant literature [36] and actual measurement data [37].

From the data presented, we can see that the results in this paper are similar to measured values from the actual lines, while the rail displacement is slightly larger than the measured value. This is because the simulation environment cannot fully reproduce the measured conditions, and both have certain discrepancies, leading to unavoidable errors in various parameters, but the errors are within an acceptable range. Furthermore, from the comparison in Figure 11, it can be found that the results of this paper are significantly better than those in the literature [36]; thus, it can be concluded that it is feasible to use this model for dynamics analysis.

3. Results and Discussion

3.1. Effect of Temperature on the Dynamic Responses

Applying the parameters in Table 3 to the CA mortar, Figure 12 and Figure 13 show the comparison of the track subsystem dynamic responses in the vertical direction at 300 km/h when the temperature ranges from −20 °C to 40 °C.

Figure 12a,f indicates that changes in temperature have a relatively small influence on the rail and base displacement. This may be because the rail is distant from the CA mortar layer and bears the moving load directly, while the base is continuous and has a mass that is much larger than that of the above two structures. Therefore, the vertical displacements of these two structures are less affected.

It can be observed from Figure 12b–e and Figure 13 that there are obvious differences in the dynamic responses of the CA mortar itself and in the slab displacement due to temperature variations. As the temperature rises, the elastic modulus decreases, leading to an increase in strain when the train passes, which is consistent with Ref. [11]. From −20 °C to 40 °C, the maximum displacement in the CA mortar layer rises from 0.316 mm to 0.352 mm, with a growth rate of approximately 11%; the maximum compression displacement rises from 0.020 mm to 0.055 mm, and growth rate reaches 175%. Since the two structures are adjacent to each other, the displacement in the slab rises from 0.320 mm to 0.357 mm, with a growth rate of 12%, which mirrors that of the CA mortar layer. We can see that the variation laws presented by them are identical: after undergoing an approximately linear increment ranging from −20 °C to 20 °C, the stress rises rapidly; the CA mortar stress diminishes when the temperature ascends, and the stress decreases from 54.878 kN to 41.199 kN, with a reduction rate of 25%. As observed in Ref. [38], a significant reduction in the stress growth rate of CA mortar occurs as temperature rises when subjected to pressure. Moreover, it can be seen that when the train is not passing, the stress in CA mortar is independent of temperature and approaches zero. Therefore, it can be deduced that higher temperatures result in less stress on the CA mortar when the train passes (in agreement with the calculated results in Figure 12e).

The results show that as the train moves over the track, the dynamic responses of track structures are notably influenced by the mechanical characteristics of CA mortar at various temperatures. Therefore, it is necessary to consider temperature as a factor in the mechanical analysis.

3.2. Effect of Speed on Dynamic Responses

Increasing train speed will intensify the wheel–rail interaction, leading to heightened vibrations throughout the entire system. Figure 14 presents the variations in vehicle body acceleration and wheel–rail force under four speeds. It can be observed from the results that the temperature exerts an extremely limited influence on these two variables. When the speed rises from 200 km/h to 350 km/h, the maximum vertical acceleration of the vehicle body at four temperatures rises linearly, with an increment of approximately 60%; the wheel–rail force growth pattern is analogous, and the growth rate is approximately 5%. The increase in wheel–rail force is not significant. This is because the wheelsets are regarded as rigid bodies in this simulation, thereby neglecting their compressive deformation, leading to slightly smaller calculated values using Hertz nonlinear theory.

Figure 15 and Figure 16 show a comparison of some dynamic responses of the track subsystem under different vehicle speeds. The results indicate that, when compared to the vehicle subsystem, temperature variation has a more remarkable impact on the track subsystem; additionally, the effect of vehicle speed is quite significant. When the speed rises from 200 to 350 km/h, the vertical displacements of the track slab and the CA mortar at four temperatures increase significantly by about 24%; the compression displacement and vertical stress of the CA mortar enhance at an accelerating rate with the increase in speed, attaining peak increments of approximately 16% and 18%, respectively.

3.3. Effect of Fastener Stiffness on Dynamic Responses

The fastener system undergoes aging after long-term exposure to train loads, resulting in an increase in its stiffness. Therefore, examining the impact of the vertical stiffness of fasteners on the track structure is very crucial, especially under high-speed conditions. We set the fastener stiffness values to 40 kN/mm, 50 kN/mm, and 60 kN/mm, and the speed was set to 350 km/h. According to the simulation data presented in Figure 17, we can observe that as the stiffness increases, both the rail and CA mortar layer displacements show a decreasing trend. The displacement reduction for the rail reaches 8%, while the change in displacement for the CA mortar is relatively small. Notably, the vertical acceleration and stress of the CA mortar exhibit opposite trends compared to the displacement: the maximum acceleration increases from 5.5 g to 5.8 g, an increase of 6%; the maximum stress rises from 48.4 kPa to 57.1 kPa, an increase of 18%. We can infer that the aging of the fastener system exacerbates the vibrations in track structures, which may reduce the fatigue life of ballastless tracks.

3.4. Effect of Position on Dynamic Responses

This paper analyzes the simulation results of the CA mortar layer at various positions, which are presented in Figure 18 and Figure 19, when the temperature is set to 20 °C. It is not difficult to see that the vertical displacement exhibits a tendency of first decreasing and then increasing from the center to the end, ultimately reaching a maximum value. The vertical stress shows a steady rise. Starting from the third fastener, its growth rate increases sharply. The vertical stress rises by about 72.4 kN in total. Therefore, we can infer that the CA mortar layer is vulnerable to be damaged at its end, thereby triggering catastrophes such as cracks and block shedding.

Throughout the entire dynamic simulation process, stress concentration frequently occurs in the part of the CA mortar layer near the lug boss, as shown in Figure 20. Therefore, we speculate that the displacement restriction of the lug boss is one of the reasons for the increased stress at the edges of the CA mortar layer. Creating a larger contact area between the lug boss and the CA mortar layer, such as by increasing the size of the lug boss or, alternatively, changing its shape to an ellipse, is likely to reduce the magnitude of the CA mortar edge stress.

4. Conclusions

This study establishes a vehicle–track–subgrade coupled model of a CRTS I ballastless track, taking the CA mortar’s temperature-dependent viscoelastic characteristics into account, and investigates the dynamic responses under different environmental factors. First, the creep curves of CA mortar under different temperatures are fitted by the fractional Maxwell (FM) model; subsequently, the creep parameters are transformed into an applicable Prony series in ABAQUS through the transfer function. Based on the Hertz nonlinear theory, a CRTS I finite element model is built in ABAQUS, and then we further analyze the influence of different environmental factors on the dynamic responses of the whole system. The main conclusions are outlined below:

The fractional Maxwell (FM) model accurately captures the creep curves of CA mortar, with a fitting degree of excellence above 0.95; moreover, the generalized Maxwell model parameters obtained from the transfer function can be conveniently and accurately implemented for the creep simulation of viscoelastic materials in finite element software.
The dynamic responses of the track are significantly impacted by the temperature-dependent viscoelastic characteristics of CA mortar, while the vehicle subsystem is slightly affected. All structural layers deform more as the temperature increases. The changes in the slab and the CA mortar layer are particularly significant. The CA mortar layer stress decreases as the temperature increases. Therefore, it is necessary to consider temperature changes when conducting dynamic analysis.
The increase in vehicle speed and fastener stiffness exacerbates the dynamic responses of the CRTS I ballastless track. The vertical acceleration of the vehicle body rises rapidly, which may affect the passenger’s travel experience. The vertical stress in the CA mortar layer rises in a linear manner. The entire system will thus face more intense vibrations. The compression of the CA mortar layer and the growth rate of the vertical stress also rise continuously. Due to the impact of the train load, the CA mortar edge of experiences the highest stress and is most likely to be damaged.

In the near future, we will conduct experiments related to the damage of CA mortar layers, such as viscoelastic responses after freeze–thaw cycles. We will use this model to analyze the impact of damage on the entire system. Finally, we will compare the calculated results with relevant high-speed railway standards to provide maintenance recommendations.

Author Contributions

Conceptualization, Y.C., B.W. and L.Y.; methodology, Y.C.; software, Y.C.; writing—original draft, Y.C.; supervision, X.S.; writing—reviewing and editing, L.Y. and X.S. resources, B.W. and X.S. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China, grant number 12302265; the Natural Science Foundation of Jiangsu Province, grant number BK20230469; and the Basic Science (Natural Science) Research Projects in Higher Education Institutions in Jiangsu Province, grant number 23KJB130008.

Data Availability Statement

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

Conflicts of Interest

The authors declare no conflicts of interest.

Abbreviations

ε	total strain of the viscoelastic model
ε_e	strain of the elastic body
ε_ve	strain of the viscoelastic body
σ	total stress of the viscoelastic model
E, η, α	parameters of the fractional Maxwell model
E₁, E₂, λ	parameters of the Zener model
T(s)	transfer function in the LC domain, where s is the complex variable
J(t)	creep compliance
G(t)	relaxation modulus
G₀, k, g_i, τ_i	parameters of the generalized Maxwell model
T	temperature
M_v, M_b, M_w	mass of the vehicle, bogie, and wheelset
J_v, J_b	pitch rotational inertia of the vehicle and bogie
Z_v, Z_b, Z_w	vertical displacement of the vehicle, bogie, and wheelset
β_v, β_b	pitch nodal displacement of the vehicle and bogie
l_v, l_b	half of the length of the vehicle and bogie
K_vb, K_bw	stiffness of the primary suspension and the secondary suspension
C_vb, C_bw	damping of the primary suspension and the secondary suspension
F_rw,k	wheel–rail force of the k-th wheelset
q(t)	vertical wheel–rail force
G	wheel–rail contact constant
∆Z(t)	compression amount between the wheel and rail
R	radius of the wheel
F(t)	tangential wheel–rail force
S_v(ϕ)	American power spectral density of track irregularity
j, A_v, ϕ, ϕ_c	parameters of the American power spectral density of track irregularity

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Figure 1. CRTS I ballastless track.

Figure 2. The FM model.

Figure 3. Zener model.

Figure 4. Fitting curves of the creep compliance.

Figure 5. Finite element model for the compressive creep experiment with CA mortar.

Figure 6. Simulation results of the creep experiment.

Figure 7. Vertical vehicle–track–subgrade system model [30].

Figure 8. Vertical track irregularity.

Figure 9. The VTS model.

Figure 10. Track subsystem simulation results. (a) Rail displacement; (b) slab displacement; (c) fastener force; (d) CA mortar stress.

Figure 11. Comparison of the simulation results with the relevant literature. (a) Slab displacement and rail displacement; (b) fastener force and CA mortar stress.

Figure 12. Comparison of time history curves of dynamic responses of the track subsystem. (a) Rail displacement; (b) slab displacement; (c) CA mortar displacement; (d) compressive displacement of CA mortar; (e) CA mortar stress; (f) base displacement.

Figure 13. Comparison of the maximum simulation outcomes. (a) Rail and slab displacement; (b) CA mortar and concrete base displacement; (c) compressive displacement of CA mortar; (d) CA mortar stress.

Figure 14. Comparison of the maximum dynamic response results of the vehicle subsystem. (a) Vertical acceleration of the vehicle body; (b) wheel–rail force.

Figure 15. Growth rate of dynamic responses at different temperatures.

Figure 16. Comparison of the maximum dynamic response results of the track subsystem. (a) Slab displacement; (b) CA mortar displacement; (c) compression displacement of CA mortar; (d) CA mortar stress.

Figure 17. Comparison of the maximum dynamic response results of the track subsystem. (a) Rail displacement; (b) CA mortar displacement; (c) CA mortar acceleration; (d) CA mortar stress.

Figure 18. Position diagram for the CA mortar layer.

Figure 19. Comparison of the maximum dynamic response results of CA mortar. (a) CA mortar displacement; (b) CA mortar stress.

Figure 20. CA mortar layer compressive stress cloud diagram.

Table 1. Transfer functions of basic mechanical components.

Component	Transfer Function T(s)
Hooke’s spring	E
Newtonian dashpot	λs
Abel dashpot	ηs^α

where λ is the viscosity coefficient of the Newtonian dashpot, and s is the complex number in the Laplace domain.

Table 2. Fitting parameters of the creep compliance.

T/°C	FM Model				Zener Model
T/°C	E/Mpa	η/Mpa·s^α	α	R²	E₁/Mpa	E₂/Mpa	λ/Mpa·s	R²
−20	46.477	268.708	0.253	0.977	34.191	45.280	245.191	0.970
0	27.892	477.358	0.320	0.995	23.453	42.088	399.271	0.977
20	23.132	146.751	0.219	0.992	17.201	36.594	264.657	0.962
40	10.984	140.754	0.190	0.957	10.077	41.962	103.763	0.907

Table 3. Fitting parameters of the Prony series.

No.	τ_i/s	g_i
		T/°C
		−20	0	20	40
1	1 × 10⁻²	6.160 × 10⁻²	1.662 × 10⁻²	6.395 × 10⁻²	4.072 × 10⁻²
2	1 × 10⁻¹	4.530 × 10⁻²	1.721 × 10⁻²	3.912 × 10⁻²	1.942 × 10⁻²
3	1 × 10⁰	7.066 × 10⁻²	3.402 × 10⁻²	5.758 × 10⁻²	2.814 × 10⁻²
4	1 × 10¹	1.060 × 10⁻¹	6.541 × 10⁻²	8.264 × 10⁻²	4.082 × 10⁻²
5	1 × 10²	1.381 × 10⁻¹	1.184 × 10⁻¹	1.078 × 10⁻¹	5.697 × 10⁻²
6	1 × 10³	1.600 × 10⁻¹	1.800 × 10⁻¹	1.314 × 10⁻¹	7.752 × 10⁻²
7	1 × 10⁴	1.210 × 10⁻¹	1.991 × 10⁻¹	1.223 × 10⁻¹	9.925 × 10⁻²
G₀/MPa		46.563	27.931	23.156	11.036

Table 4. Symbols in vehicle subsystem equations and their meanings.

Symbol	Component	Meaning
M_v, J_v	Vehicle	Mass and pitch rotational inertia
Z_v, β_v		Vertical displacement and pitch nodal displacement
l_v		Half of the length
M_b, J_b	Bogie	Mass and pitch rotational inertia
Z_b, β_b		Vertical displacement and pitch nodal displacement
l_b		Half of the length
M_w	Wheelset	Mass
K_vb, C_vb	Primary suspension	Stiffness and damping
K_bw, C_bw	Secondary suspension	Stiffness and damping
F_rw,k	The k-th wheelset	Wheel–rail force

Table 5. USA vertical track irregularity parameters.

Track Grade	A_v/(cm²·m/rad)	ϕ_c/(rad/m)
1	1.2107	0.8245
2	1.0181	0.8245
3	0.6816	0.8245
4	0.5376	0.8245
5	0.2059	0.8245
6	0.0339	0.8245

Table 6. Vehicle and fastener parameters.

Parameter	Value
Vehicle body mass/t	42.400
Bogie mass/t	3.400
Wheelset mass/t	2.200
Nodding moment of inertia of the vehicle body/(kg·m²)	2.74 × 10⁶
Nodding moment of inertia of the bogie/(kg·m²)	7.2 × 10³
Primary suspension stiffness/(N/m)	1.04 × 10⁷
Primary suspension damping/(N·s/m)	5 × 10⁴
Secondary suspension stiffness/(N/m)	4 × 10⁵
Secondary suspension damping/(N·s/m)	5 × 10⁴
Fastener stiffness/(N/m)	5 × 10⁷

Table 7. Track and subgrade parameters.

Structural	Density/(kg/m³)	Elastic Modulus/MPa	Poisson’s Ratio
Rail	7850	2.06 × 10⁵	0.3
Slab	2500	3.65 × 10⁴	0.2
Lug boss	2440	3.4 × 10⁴	0.2
Base	2500	3.6 × 10⁴	0.2
Subgrade surface	2250	2.5 × 10²	0.27
Subgrade bottom	2130	1.6 × 10²	0.32
Subgrade body	2000	1.2 × 10²	0.35
CA mortar	1800	——	0.45

Table 8. Comparison of the simulation results with relevant literatures.

Parameter	Result in This Paper	Results in the Literature [36]	Measured Value [36]	Measured Value [37]
Rail dis/mm	1.392	1.537	0.3–0.88	0.66–1.32
Slab dis/mm	0.331	0.367	0.27–0.39	0.16–0.37
CA mortar stress/KPa	46.583	125.501	25.63–38.48	≤45
Fastener force/kN	52.413	60.450	14.4–65.8	24.4–46.3

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Chen, Y.; Wu, B.; Yao, L.; Su, X. Dynamic Response Analysis of Ballastless Tracks Considering the Temperature-Dependent Viscoelasticity of Cement-Emulsified Asphalt Mortar Based on a Vehicle–Track–Subgrade Coupled Model. Lubricants 2025, 13, 58. https://doi.org/10.3390/lubricants13020058

AMA Style

Chen Y, Wu B, Yao L, Su X. Dynamic Response Analysis of Ballastless Tracks Considering the Temperature-Dependent Viscoelasticity of Cement-Emulsified Asphalt Mortar Based on a Vehicle–Track–Subgrade Coupled Model. Lubricants. 2025; 13(2):58. https://doi.org/10.3390/lubricants13020058

Chicago/Turabian Style

Chen, Yunqing, Bing Wu, Linquan Yao, and Xianglong Su. 2025. "Dynamic Response Analysis of Ballastless Tracks Considering the Temperature-Dependent Viscoelasticity of Cement-Emulsified Asphalt Mortar Based on a Vehicle–Track–Subgrade Coupled Model" Lubricants 13, no. 2: 58. https://doi.org/10.3390/lubricants13020058

APA Style

Chen, Y., Wu, B., Yao, L., & Su, X. (2025). Dynamic Response Analysis of Ballastless Tracks Considering the Temperature-Dependent Viscoelasticity of Cement-Emulsified Asphalt Mortar Based on a Vehicle–Track–Subgrade Coupled Model. Lubricants, 13(2), 58. https://doi.org/10.3390/lubricants13020058

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