Research on the Stability of Lining Structures Under Different Fault Moments Based on FDM-DEM

Abstract

Currently, research on employing finite difference method and discrete element method (FDM-DEM) coupling to assess the stability of tunnel lining structures is limited. This study utilized the FDM-DEM coupling approach, with the F2 fault of the East Tianshan Tunnel as a case study, to develop a numerical model in conjunction with PFC3D 6.0 and FLAC3D 6.0 software. We conducted a comprehensive analysis of the displacement deformation and crack progression of the tunnel lining structure under varying dislocation momentum conditions, unveiling the underlying mechanisms. The findings indicated that as the dislocation increased, the extent of damage to the vault intensified, and the particle contact force within the tunnel lining shifted from compression to tension, significantly contributing to the crack formation. Fault dislocation influenced the gradual expansion of cracks from the vault to the spandrel and arch waist, with the crack width increasing alongside the rising dislocation momentum. In particular, under substantial dislocation momentum, the overall stability of the tunnel lining was markedly diminished. The safety factor at the tunnel section declined progressively as the dislocation momentum escalated, with values of 2.53, 2.49, 2.43, 2.39, and 2.32 corresponding to dislocation momenta of 0.01 m, 0.05 m, 0.1 m, 0.15 m, and 0.2 m, respectively. This research offers valuable insights and a reference framework for investigating the stability of tunnel lining structures in proximity to fault dislocations, pinpointing potential failure points, and bolstering the structural integrity of tunnels.

1. Introduction

To achieve the goals of the ‘14th Five-Year Plan’ [1], China has accelerated the development of infrastructure and transportation construction in the western region. Consequently, an increasing number of infrastructure projects, including highway tunnels, railway tunnels, and water diversion tunnels, have been accomplished in Xinjiang. However, the presence of 23 seismic fault zones in China poses numerous challenges to the construction, development, and operation of tunnel engineering [2]. These zones are characterized by a high density of active faults, and for underground structures, fault dislocation often coincides with significant stratum deformation (earthquakes), which can have severe repercussions and result in irreversible damage to engineering structures [3,4]. For instance, the 2008 Wenchuan earthquake inflicted considerable damage on the Longxi Tunnel [5], rendering it non-functional for transportation purposes. Therefore, investigating the stress and deformation of a tunnel’s internal structure under fault dislocation is crucial for tunnel excavation support, seismic reinforcement, and operation and maintenance tasks [6,7,8].

Consequently, numerous scholars, both domestically and internationally, have conducted extensive research on how fault dislocation affects tunnel structures [9,10,11]. For instance, a team led by Zhiqiang Zhang et al. [12] carried out a large-scale physical model test on a subway tunnel crossing a reverse fault shear dislocation. This test was based on the Urumqi Metro Line 1 tunnel crossing the Wancaogou reverse fault, comparing and analyzing the mechanical properties of conventional and hinged tunnel structures. Pingliang Chen et al. [3] utilized dislocation load as a unidirectional harmonic velocity pulse to explore the damage mechanism of the Daliang tunnel. Yang Zhang et al. [13] employed a self-designed large-scale active fault dislocation simulation loading system to conduct a 60° inclination method fault rod slip dislocation model test, determining the crack range, failure characteristics, and failure modes of the tunnel lining structure. Xu Zhao et al. [14] verified the seismic effects of tunnel grouting with inactive faults and probed into the seismic mitigation mechanism behind this. Gaoming Yan et al. [15] proposed a method to estimate the response of multi-flexible joint tunnels under fault motion, obtaining an explicit solution for tunnels affected by fault movement. They studied various factors in detail, such as the fault displacement, lining stiffness, surrounding rock properties, and joint stiffness of the fault zone. Tianqiang Wang et al. [16] discussed the deformation and failure mechanism of fault-fractured tunnels, revealing the mechanical behavior of tunnels under fault dislocation action. Many researchers [17,18,19,20] have used FDM-DEM coupling to analyze and study tunnel excavation and backfill grouting from the perspectives of solid elements and particles.

Despite the multifaceted research conducted by the aforementioned scholars on the mechanical response of tunnels at fault zones, their primary approach has been through physical model testing. Such tests are often time-consuming, costly, and require numerous iterations for verification. Current numerical simulations are predominantly performed using the finite element method [21,22]. Although some researchers have employed discrete elements to investigate the mechanical response of chamber excavations at fault zones [23], these studies typically involve large numbers of particles and focus solely on analyzing the mechanical changes during rock layer excavation. They have not delved deeply into the impact of fault dislocation on the stability of tunnel support structures. Therefore, by examining the research methodologies and outcomes of previous fault studies and leveraging the context of the East Tianshan Tunnel Project in Xinjiang, this paper employed the FDM-DEM coupling method to explore the affected area of the tunnel support structure and the internal mechanical alterations of the lining structure under reverse fault dislocation. This investigation enabled a straightforward analysis of the tunnel’s overall stability under varying fault dislocation momenta through particle displacement and crack variations, allowing for a more intuitive and precise identification of unstable areas within the tunnel lining. Such insights provide enhanced data support for construction activities.

2. Project Profile

The East Tianshan Tunnel is situated in the Hami region of Xinjiang and serves as a crucial transportation link between Hami and Balikun, significantly boosting the external traffic connectivity of the Hami area. The tunnel’s entrance is located on the downstream side of the Songshutang Ski Resort, north of the Tianshan Mountains, while its exit is positioned on the right side of the Gourd Ditch. Spanning the Tianshan Mountains, the tunnel has a total length of 11,865 m and an average elevation exceeding 2000 m, classifying it as a high-altitude tunnel. According to the “China Fault Distribution Map” and advanced geological forecasting reports, the tunnel’s excavation intersects with three major fault zones. Among these, the F2 fault trends NW292°∼NW313° with a dip angle of approximately 75°, and the surrounding rock is classified as grade V, exhibiting distinct thrust fracture characteristics. The fault zone consists of materials such as mylonite, schist, and rock blocks, which are relatively well-cemented. The fault plane is curved and rough, with an intersection width of about 270 m between the tunnel and the F2 fault fracture zone. The lithology of the fault primarily comprises heterogeneous interbedded tuff and sandstone with a thin-layer structure, resulting in a fragmented rock mass and extensive fracturing. The stability of the surrounding rock is poor, making hole formation challenging. The location of the East Tianshan Tunnel is illustrated in Figure 1.

In this context, the section of the tunnel between ZK11 + 720 and ZK11 + 880 intersects with the fault zone, rendering the lining structure susceptible to cracking and failure due to geological activities, ultimately compromising its structural stability. Based on Tianshan Project data, the average burial depth of this tunnel section at the F2 fault junction is 620 m. The tunnel’s profile is designed in a horseshoe shape, with an excavation cavity measuring 1612 cm in width and 1129 cm in height. Prompt initial supports were implemented following excavation, and an inverted arch was installed once full-section excavation was complete, sealing the tunnel into a continuous loop. The initial support was reinforced by injecting 4 m long ϕ25 hollow grouting bolts and spraying a 20 cm thick layer of C25 plain concrete. After completing the initial support, a 60 cm thick layer of C50 concrete was applied to finalize the secondary lining. The tunnel structure is illustrated in Figure 2.

3. Research Program

3.1. Modeling, Research Programs

During the modeling process, a continuous mesh was utilized in FLAC3D to construct the surrounding rock and the primary lining structure, while the secondary lining structure was created using discrete particles in PFC3D. The coupling analysis was facilitated by a coupling wall, which converted the contact forces and moments exerted by the particles into equivalent force systems acting on the wall. These forces were then transferred to the nodes of the shell or regional elements. Simultaneously, the macroscopic stress and additional information from the solid elements in FLAC3D were conveyed to the PFC3D particles, ensuring synchronization with the coupling time. Information was updated and exchanged at each time step to enable a dynamic response analysis. This coupled analysis approach addressed the limitations of traditional finite element and discrete element methods, allowing for a comprehensive analysis of the simulation results from various perspectives, aspects, and levels. The assumptions made during the modeling process included the following:

3.2. Geometric Size of the Model

Because the model is difficult to be consistent with the actual project, the following assumptions are made for the numerical model:

(1)

The upper and lower walls of the model fault were assumed to be isotropic and to exhibit a continuous linear elastic behavior;

(2)

It was presumed that the fault movement occurs at a constant rate;

(3)

The boundary conditions for the lower wall of the fault were considered to be entirely fixed.

Regarding the assumption of material properties, the Mohr–Coulomb criterion was primarily employed, which may deviate somewhat from actual engineering conditions, potentially leading to minor errors within an acceptable range of results. Concerning the fault movement, given the relatively slow motion of tectonic plates, it could be reasonably assumed to progress at a constant speed over a certain time frame. As for the boundary conditions of the fault’s lower wall, although a positional shift may occur as the upper wall moves upward, this displacement is considered negligible compared to the overall extrusion deformation. Therefore, fixed boundary conditions were applied to the lower wall of the fault. Figure 3 illustrates the general modeling process and the protocol for studying the fault dislocation. Once the coupling had been established, fault dislocation in the tunnel was simulated by varying the fault dislocation momentum. Subsequently, the impact of dislocation momentum on the fault was assessed through an analysis of the particle and solid element displacements and cracks, enabling a further examination of the safety factors.

In this paper, the degree of damage to the tunnel structure under varying dislocation momenta of reverse fault types was examined, based on the classification of survey faults. Taking into account the unique geological characteristics of the fault activity, this study considered the impact range of the fault dislocation momentum on the surrounding rock. In line with the research focus, the interface was designated to simulate the fault sliding surface, and its physical and mechanical parameters were defined to facilitate analysis of the fault sliding behavior. A FDM-DEM model was constructed to simulate the extent and severity of reverse fault influence on the tunnel structure following excavation support, under different dislocation momenta. Due to computational limitations, simulations were not feasible for large models with a high number of discrete elements. Therefore, after careful consideration, a section extending 3 m on each side of the intersection between the tunnel axis and the fault (totaling 6 m) was chosen for investigation. To mitigate boundary effects, a three-dimensional model with dimensions of 50 m in length, 50 m in height, and 6 m in width (as depicted in Figure 4) was constructed, offering a more realistic and precise representation than a two-dimensional model.

Once the natural stress field reached equilibrium, the model’s displacement and velocity were reset. By adjusting the boundary conditions, various displacements were applied to the fault to simulate the dislocation effect. To investigate the impact of different dislocation momenta, five levels of fault dislocation were established: 0.01 m, 0.05 m, 0.1 m, 0.15 m, and 0.2 m. Monitoring points were placed at intervals of 0.1 m along the bottom of the vault, spandrel, arch waist, arch foot, and arch bottom to record the principal stress, shear force, and displacement induced by varying dislocation momenta. Additionally, monitoring points on the fault plane were set up to determine the influence range of the fault plane under different dislocation momenta. As illustrated in Figure 5, monitoring points were spaced every 0.5 m along each axis, totaling 11 monitoring points.

4. FDM-DEM Model Construction

4.1. Construction of Macro and Micro Parameters

Through field investigation and sampling, according to the relevant building material specifications in China, combined with an indoor force mechanics test, the physical and mechanical parameters of the rock strata shown in

Table 1. Physical and mechanical parameters of rock layer.

Name	Volumetric Weight/(kN·m−3)	Elastic Modulus/GPa	Poisson Ratio	Angle of Internal Friction/(°)	Cohesion/MPa
sandstone	25	8.5	0.48	30	5

, and the physical and mechanical parameters of lining structure shown in

Table 2. Structural physical and mechanical parameters.

Name	Volumetric Weight/(kN·m−3)	Elastic Modulus/GPa	Poisson Ratio	Compressive Yield Stress/MPa	Tensile Yield Stress/MPa
primary lining	24	28	0.13	16.7	1.78
secondary lining	25	30	0.15	20.1	2.01
anchor bar	38.5	200	0.3

were determined. The parameters of the fault contact surface were determined by the weak surface method. The internal friction angle was half of the internal friction angle of the original sandstone, and the remaining parameters were 1/20 of the corresponding mechanical parameters of the sandstone.

Traditional mechanical experiments treat the study sample as a homogeneous medium and measure macroscopic mechanical parameters, whereas discrete element studies adopt a microscopic perspective, focusing on particle-to-particle and particle-to-wall interactions. By altering various contact models, the mesoscopic mechanical behaviors of different materials can be investigated. PFC offers a range of contact models, and the classical parallel bonding model was employed in this study. This model features two contact interfaces: the first interface is force-transmitting only, while the second, a viscous interface, can transmit both force and moment. The parallel bonding model consists primarily of two sets of parameters: linear group parameters (such as the effective model and stiffness ratio) and parallel bonding group parameters (including the bonding effective model, etc.). Through PFC3D simulations of macroscopic mechanical tests (including uniaxial compression and Brazil splitting experiments), the same results as those from macroscopic experiments were achieved after repeated adjustments using a trial-and-error approach. Figure 6 and Figure 7 show a comparison between the simulated data from the uniaxial compression test and the Brazilian splitting test with the experimental data.

According to the following formula, the correctness of the simulation results was verified. According to the calculation results, the microscopic parameters of the parallel bond model were as shown in

Table 3. Secondary lining microscopic parameter table.

Mesoscopic Parameters	Taking Values	Mesoscopic Parameters	Taking Values
Rmin (cm)	5	Pb_emod (Gpa)	3.65
Rmax (cm)	7	Pb_kratio	1.97
porosity	0.03	Pb_ten (Mpa)	1.05
density (kg/m3)	25	Pb_coh (Mpa)	50
fric	0.6	Pb_fa	40
emod (Gpa)	3.19	kratio	1.58

.

$$\begin{array}{c}\hfill E={\displaystyle \frac{{\sigma }_i}{{\epsilon }_i}}\\ \hfill \mu =\left|{\displaystyle \frac{{\epsilon }_{\mathrm{d}}}{{\epsilon }_L}}\right|\\ \hfill {\sigma }_{\mathrm{t}}={\displaystyle \frac{2{\mathrm{p}}_t}{\pi DL}}\end{array}$$

(1)

4.2. Construction of Coupling Model

PFC3D and FLAC3D were interconnected, with FLAC3D handling the analysis and computation of macroscopic continuous mechanics, while PFC3D focused on the discrete domain mechanics. Data exchange between the two simulations was facilitated through the SocketO/I interface. In this research, the coupling wall (SocketO/I interface) was created using the built-in wall-zone command flow within the software. The finite element secondary lining was substituted with a discrete element secondary lining through a command flow that was randomly generated within the geometry of the balls, and the corresponding parameters were assigned. The model was embedded to a depth of 600 m, based on the soil stress formula, and the initial stress above the tunnel was preliminarily calculated using the thickness of the soil layers, with an appropriate vertical stress being applied to the top of the model. Regarding the boundary conditions, horizontal displacements of the left and right boundaries (X), horizontal displacements of the front and rear boundaries (Y), and vertical displacement of the bottom boundary (Z) were restricted. The rock strata were assigned a Mohr–Coulomb constitutive model, while the initial lining used an elastic-constitutive model, and cable structural elements were employed for the anchors. Considering only the gravitational field, the coupled model’s displacement is depicted in Figure 8. In conjunction with references [24], the displacement fields of the finite elements and discrete elements in the Figure 8 model are consistent, and the stratification of displacements within discrete regions is evident. The findings indicated that the macro/micro model had been constructed correctly, and the internal force transmission under the gravitational field functioned normally.

5. Results Analysis and Discussion

5.1. Displacement Characteristics

Figure 9 displays the displacement contours of the tunnel lining structure under varying fault dislocation momenta. The subfigures (a–e) in Figure 9 correspond to dislocation momenta of 0.01 m, 0.05 m, 0.1 m, 0.15 m, and 0.2 m, respectively. The figure illustrates that, as the fault dislocation increased, the displacement and deformation of each section of the tunnel progressively intensified. Notably, the vault experienced the most significant displacement changes. Specifically, at a dislocation momentum of 0.01 m, as shown in Figure 9a, although displacements occurred in various parts of the tunnel, the changes were relatively minor and mainly concentrated near the tunnel’s top. However, when the dislocation momentum increased to 0.05 m, as depicted in Figure 9b, the displacement deformation became more pronounced, and signs of instability began to emerge in the tunnel lining. Due to the characteristics of the discrete element method (DEM), the displacement at the tunnel’s apex was particularly noticeable. At a dislocation momentum of 0.1 m, as illustrated in Figure 9c, the deformation of the tunnel roof further intensified, and local detachment even occurred. This detachment phenomenon vividly demonstrated the severe impact of fault dislocation on the stability of the tunnel lining structure. As the dislocation momentum reached 0.15 m and 0.2 m (as shown in Figure 9d,e), the unstable areas of the tunnel lining not only expanded but also began to extend downward. This indicates that with increasing fault dislocation, the overall deformation of the tunnel became more pronounced, and the unstable area was no longer confined to the top but gradually spread to other sections of the tunnel. Observing the shifting clouds, it is evident that the unstable area of the tunnel lining consistently appeared at the top of the tunnel under different dislocation momenta and was close to the fault dislocation position. The findings reveal that fault dislocation had the most substantial impact on the tunnel’s apex, particularly when the dislocation momentum reached 0.15–0.2 m, and the displacement changes in each part of the tunnel became quite evident.

To observe the displacement changes of the tunnel along the depth direction more intuitively, we selected the vault monitoring points (T0–T10) that exhibited significant displacement variations (as shown in Figure 10) for data analysis and comparison. The monitoring data in the figure revealed that the maximum displacement consistently occurred near the fault dislocation surface. Among these monitoring points, the displacement value diminished progressively as the distance from the fault dislocation surface increased. This trend indicates that the influence of fault dislocation was distinctly localized, primarily concentrated around the fault dislocation surface and its adjacent areas. From the viewpoint of tunnel depth, the closer to the fault dislocation surface, the larger the displacement, which also demonstrates that the impact of fault dislocation on the tunnel structure exhibited notable spatial distribution characteristics.

5.2. Force Chain and Crack Analysis

Figure 11 and Figure 12 depict the contact force chain contours of the tunnel lining structure under varying fault dislocation momenta. These diagrams clearly illustrate the fluctuations in contact forces within the tunnel lining structure and the progression of cracks under different dislocation momentum conditions. Given the close proximity between the fault dislocation surface and the tunnel front, ranging from 1 to 2 m, Figure 11a–e reveals that when the dislocation momentum was 0.01 m, the fracture zone at the tunnel front was relatively small, and the crushing effect was not pronounced. However, as the dislocation momentum increased to 0.05 m, the crushing effect became more significant, and the affected area not only extended into the depth of the tunnel but also gradually spread to the spandrel region. This trend further intensified as the dislocation momentum rose to 0.1 m, 0.15 m, and 0.2 m, with a substantial expansion of the fracture zone and an increase in the severity of fragmentation.

Figure 11 and Figure 12 provide a detailed illustration of the evolution of the contact forces among the tunnel lining particles under varying dislocation momenta. As the fault dislocation progressed, several changes occurred in the particle contact forces within the tunnel lining. Some contact forces transitioned from compression to tension, leading to the emergence of cracks. Particularly at higher dislocation momenta, as depicted in Figure 12d,e, the number of cracks in the tunnel lining significantly increased, the crack widths gradually widened, the failure of the entire lining structure became more pronounced, and the loosening area continued to expand. Analyzing the evolution of contact forces allows for a better understanding of the dynamic, progressive failure characteristics of tunnel lining structures. For instance, when the dislocation momentum is low (e.g., 0.01 m), the contact forces within the lining structure are primarily compressive, with few or no cracks present. However, as dislocation increases, a portion of the contact forces shift from compression to tension, and cracks begin to emerge and progressively grow. Once the dislocation momentum reaches a certain threshold (such as over 0.1 m), not only do the cracks multiply, but their widths also significantly increase, and the lining structure sustains severe damage.

Figure 12 offers a clearer visualization of the crack evolution around the tunnel lining. In Figure 12a, due to the minimal dislocation momentum (0.01 m), the tunnel lining exhibited almost no cracks, and the structure remained predominantly intact. However, as shown in Figure 12b, when the dislocation momentum rose to 0.05 m, a few cracks began to emerge in the tunnel’s spandrel-to-vault region. With the continued increase in dislocation momentum, Figure 12b–e more distinctly demonstrate the progression of cracks from the vault to the spandrel and arch waist areas, accompanied by an increase in crack width. Figure 13a–h illustrates the variation in cracks at different times under a dislocation momentum of 0.1m, which was monitored every 12 h. It is evident from the figure that as time progressed, cracks started to appear and their width gradually increased, until reaching a stable state, Figure 13h.

5.3. Analysis of Safety Coefficient

The “Code for the Design of Highway Tunnels” stipulates that when the safety factor of a concrete lining is governed by compressive strength, the calculation formula for the safety factor is as follows:

$$KN\le \varphi \alpha R_{\alpha }bh$$

(2)

In the formula, K is the safety factor; $\varphi$ is the longitudinal bending coefficient of the component, which can be taken as 1 for the tunnel lining; $\alpha$ is the eccentric influence coefficient of axial force, which is 1 in this paper; $R_{\alpha }$ is the ultimate compressive strength of concrete or masonry; b is the width of the cross-section, take 1; h is the thickness of the cross-section, take 0.5.

Figure 14 illustrates the stress variation along the depth direction of the vault under different dislocation momenta of the tunnel. The fault was situated near monitoring point 3, where stress was more concentrated, making it more susceptible to damage. Therefore, when calculating the safety factor, several points near the fault were selected for computation. The chosen points are listed in

Table 4. The strength of the axle was calculated.

Dislocation	Axial Force (N)
	Monitoring Point 1	Monitoring Point 2	Monitoring Point 3	Monitoring Point 4	Monitoring Point 5
0.01 m	57,242.64	81,777.26	7,903,525.25	131,219.87	57,439.52
0.05 m	77,850.17	138,367.82	8,030,244.14	193,893.48	79,787.62
0.10 m	88,205.86	299,059.51	8,240,239.66	369,925.09	148,322.33
0.15 m	95,000.06	366,353.41	8,358,759.49	464,925.09	321,333.07
0.20 m	98,659.10	549,124.89	8,616,360.41	580,916.30	410,469.27

, and the corresponding safety factor calculations are presented in

Table 5. Factor of safety.

Dislocation	Factor of Safety K
	Monitoring Point 1	Monitoring Point 2	Monitoring Point 3	Monitoring Point 4	Monitoring Point 5
0.01 m	349.39	244.57	2.53	152.42	348.19
0.05 m	256.90	144.54	2.49	103.15	250.67
0.10 m	226.74	66.88	2.43	54.06	134.84
0.15 m	210.53	54.59	2.39	43.02	62.24
0.20 m	202.72	36.42	2.32	34.43	48.72

. To provide a clearer representation of the safety factor changes, the data were plotted as shown in Figure 14.

As depicted in Figure 15, the safety factor at each monitoring point of the tunnel decreased with increasing dislocation momentum. Notably, monitoring point 3, located at the fault position, clearly showed a significant reduction in the safety factor of the lining near the fault. Specifically, when the dislocation momentum reached 0.2 m, the minimum safety factor for the tunnel’s fault dislocation was 2.32. As the distance from the fault increased, the safety factor at each point of the tunnel significantly increased under the different dislocation momenta. Figure 16 displays a profile of the plastic zone along the central axis of the tunnel, corresponding to dislocation momenta of 0.01 m, 0.05 m, 0.1 m, 0.15 m, and 0.2 m (Figure 16a–e). These diagrams reveal that the plastic region underwent changes to some extent as the dislocation momentum increased, especially at the top of the tunnel, where the plastic zone’s alteration was more pronounced, and its area significantly expanded. This change aligned with the trend observed in the safety factor, reinforcing each other.

Compared to previous single finite element analyses, incorporating discrete element particles allows for a more detailed mesoscopic analysis, from displacement changes to observing force chain cracks and plastic zones, which are more intuitive. This approach can be applied in engineering practice, and optimizing a small part of the structure can significantly enhance its stability, which holds great significance for practical engineering applications.

6. Conclusions

This paper investigated the stability of a lining structure under fault dislocation using FDM-DEM coupling. It analyzed the displacement variations of the lining under different dislocation momenta (0.01 m, 0.05 m, 0.1 m, 0.15 m, and 0.2 m) and arrived at the following main conclusions:

(1)

As the fault dislocation increased, the displacement around the tunnel significantly intensified, leading to a gradual expansion of the fracture zone at the tunnel front and spreading both in the depth direction and towards the spandrel area. Especially under conditions of significant dislocation momentum (e.g., 0.15 m and 0.2 m), the scope of the fracture zone dramatically increased, signifying that the tunnel structure endured substantial damage and deterioration;

(2)

As the fault dislocation progressed, the particle contact force within the tunnel lining gradually transitioned from compression to tension. This shift in mechanical state was the primary cause of the crack formation. Under conditions of significant misalignment momentum (e.g., 0.15 m and 0.2 m), the tensile effect of the contact force started to intensify, leading to the appearance and expansion of a large number of cracks in the lining structure, with a significant increase in both the quantity and width of cracks;

(3)

The evolution of cracks exhibited characteristics of dynamic, gradual failure: when the dislocation momentum was low (e.g., 0.01 m), the lining structure remained nearly intact. However, as the dislocation momentum gradually increased (especially at 0.15 m and 0.2 m), cracks began to propagate from the vault towards the spandrel and arch waist areas, with a significant increase in crack width. This demonstrates that under conditions of significant dislocation momentum, the overall stability of the tunnel lining greatly diminished. This approach transcends the limitations of traditional finite element analysis by employing discrete elements to monitor the development of tunnel cracks, enabling the identification of weak points in the tunnel lining’s failure, which is beneficial for prioritizing protection during the construction process.

(4)

The safety factor at the faulted section of the tunnel was significantly diminished, reaching its lowest value of 2.32 when the dislocation momentum was 0.2 m. Changes in the plastic zone, particularly at the top of the tunnel, also indicated that as the dislocation momentum increased, the stability of the tunnel lining was progressively compromised, with the failure effect at the vault becoming more pronounced. This demonstrates that the stress distribution within a tunnel lining varies considerably under different dislocation momenta and should be flexibly considered during the design of the tunnel lining structure. These findings underscore the critical importance of monitoring and managing misalignment momentum to ensure the safety of tunnel structures.

Compared to the traditional finite element method (FEM), FDM-DEM coupling offers significant advantages in simulating dynamic failure and crack initiation in engineering structures. The traditional FEM relies on the continuum medium assumption, which encounters limitations when addressing nonlinear phenomena such as large strains, significant displacements, and material crack propagation. The FDM-DEM coupling method integrates the numerical stability of the finite difference method (FDM) with the precise simulation capabilities of the discrete element method (DEM) to model particle interactions. This approach can more accurately replicate the fracture, cracking, and large deformation behaviors of materials under stress. In future research, we aim to address challenges related to geometric discontinuities arising from the contact between spheres and planes or other shapes, the stress concentrations caused by these discontinuities, and discrepancies between particle shapes and their real-world counterparts.