1 INTRODUCTION
In the 21st century, there has been a sustained program of exploration and development of underground space and resources, resulting in the extraction of a significant volume of underground resources from various depths (Guo, Ding, Wei, Yang, Yang, 2024). The characterization of deep rock masses is defined by the concept of “three highs and one disturbance” with the geological stress factor remaining constant despite changes in the surrounding geological environment (Zhu et al., 2012). Additionally, deep-buried tunnel engineering is subjected to various dynamic loads of different intensities and frequencies related to different spectral bands (Guo, Ding, Wei, Yang, 2024). When local stress and dynamic loads act simultaneously, the deep underground rock mass may experience fractures, rock bursts, or even collapse. Therefore, it is crucial to study the fracture behavior of fissured tunnel rock under this coupled action.
To date, some progress has been made in the study of the structural fracture characteristics of deep rock masses (He et al., 2018; Li et al., 2021; Zhang et al., 2021). For instance, He et al. (2018) studied the crack propagation process and dynamic strain behavior of granite materials under horizontal pressure. Their findings revealed that as the in situ stress K increases from 0 to 1, stress concentration occurs around the pore, and the crack propagation direction is mainly controlled by the circumferential tensile stress and biaxial precompression ratio. Furthermore, stress concentration phenomena were observed at the top of the pore and the upper part of the sample. In a further development of this field of study, Lü et al. (2024) proposed an extraction method that optimizes the main frequency of acoustic emission from rocks, enhances the extraction of high-frequency acoustic emission signals, resolves the issue of inaccurate extraction via traditional methods, and accurately predicts precursor signals of rock failure. Zhou et al. (2024) reviewed the in situ fracture toughness of rocks under environmental conditions over the past 20 years. These researchers found that there is a synergistic effect between high temperature and soil stress that enhances the fracture toughness of rocks. However, there is also a competitive relationship where high soil stress increases fracture toughness, while high temperature decreases it. Wang et al. (2024) analyzed the stress-deformation characteristics of the rock surrounding a tunnel, the deformation characteristics of tunnel movement, and the dynamic instability process of the tunnel through a blast test. The dynamic instability process of the tunnel was categorized into six stages: initial calmness, tunnel vibration, particle ejection, atomization, local block ejection, and restoration of calmness. The influence of geological stress on rock cracks during blasting was analysed by Yi et al. (2018) and Li et al. (2021), who also explained the crack extension mechanism. In the context of rock mass ejection in deep tunnels, Li et al. (2024) utilized five distinct materials: cement mortar, polyurea, modified epoxy resin, aramid-fabric modified epoxy resin (AME), and carbon-fiber fabric-modified epoxy resin (CME). These materials were applied to the rock surface to investigate its ductile and brittle characteristics. These scholars revealed that the inherent failure characteristics of the rock remained unchanged despite the application of these materials. However, such treatment can enhance the compressive strength and deformation stiffness. Xie et al. (2022) studied the rock mechanical parameters (strength, failure modes, crack characteristics, and brittle ductility) before and after microwave irradiation. In addition, a new model was developed by comparing it with the linear Mogi criterion and spatial mobilization plane (SMP) criterion. Nevertheless, the majority of studies have concentrated on intact rock masses, with few investigations focusing on fractured tunnel rock masses.
In recent years, notable advancements have been made in the study of the fracture characteristics of cracked rock. For instance, Zhu et al. (2008, 2014) employed numerical methods to study the damage behavior of the surrounding rock of a tunnel under explosive loads and reported that the main shattered area near the blast hole is induced by tensile stress, whereas circumferential cracks are induced by reflected tensile waves. Kyaw et al. (2023) and Phyo Myat et al. (2020) proposed an effective method to calculate the stress intensity factors (SIFs) of inclined surface cracks in a semi-infinite body under arbitrary stress fields. This method utilizes a single noninclined crack model and computes the SIFs for different crack angles. In a creep test conducted by Mei et al. (2023) on cement mortar specimens with adjacent cracks, tensile shear failure was identified as the primary mode of failure. The macroscopic fracture surface exhibited a Z-shaped configuration, which facilitated the formation of microcracks. The interaction between cracks was characterized by an initial inhibition of wing crack propagation in the rock bridge area, followed by subsequent promotion of crack propagation during later stages. By employing numerical analysis software, Fu et al. (2022) investigated the mechanical properties of brittle materials under the influence of various factors, including the position of holes, the presence of double cracks (Wang et al., 2023), the use of a tunnel boring machine (TBM) disc-cutter machine (Yang et al., 2024), and the type of filling material (Zhou et al., 2022). Peng et al. (2024) conducted an in-depth investigation into the fracture behavior of ISSC-reinforced plates, with a particular focus on examining the effects of the aspect ratio, spacing ratio, crack angle, relative depth, stiffness coefficient, and loading method on their fracture behaviors. It was discovered that the reinforcement exerts a markedly more pronounced inhibitory influence on extensive surfaces. Additionally, the constraint effect is more pronounced at surface points than at the deepest point. Chen et al. (2021) investigated the mechanical properties (shear forces and crack propagation trends) of sandstone samples with different joint angles via the acoustic emission (AE) technique. The results show that the fracture propagation trend of the samples shifted from shear fracture (top to bottom) to tensile fracture (center to center), with an intensification observed at angled joints. The maximum level of this intensification was observed at an angle of 60°. Xu et al. (2023) assumed that the crack was propagated as a finite-length crack, thus enabling finite element analysis. The theoretical solution for the propagation of a finite-length crack is obtained on the basis of the energy release rate. Xue et al. (2024) conducted a three-point bending test to investigate the fracture behavior and characteristics of mode I cracks in granite materials, as well as the deformation field, strain field, and acoustic emission characteristics. In a study of rock engineering on the Tibetan Plateau, Zhang et al. (2022) studied the fatigue deformation characteristics (fatigue fracture and micromorphological evolution), energy evolution laws, and freeze–thaw (F–T) fatigue-coupled damage of sandstone materials via digital image correlation (DIC) technology and scanning electron microscopy (SEM) techniques. The aforementioned research has been conducted on rock materials with cracks. However, most studies have concentrated on fractured rock in the absence of confining pressure, with relatively few investigations conducted into the fracture characteristics of fractured rock masses under the influence of geological stress.
In summary, investigations of fracture characteristics in deep underground tunnels, especially fractured deep tunnels, are still in their infancy and are currently unable to meet the safety requirements of deep rock and soil engineering (Li et al., 2019; Xie et al., 2022). In light of the above, in this study, three methodologies are employed: (1) CFT technology is employed in conjunction with large-scale physical experiments and numerical simulations to investigate the dynamic fracture response of the tunnel model in the absence of in situ stress, with the objective of validating the reliability of the numerical model. (2) The AUTODYN numerical simulation software is employed to simulate the dynamic fracture characteristics of the tunnel model under in situ stresses ranging from 0 to 2.5 MPa. (3) A calculation of the dynamic stress concentration factor (DSCF) of the tunnel is conducted on the basis of stress wave theory and fracture mechanics via numerical software. This evaluation assesses the stability of the fracture tunnel model under various in situ stress loads, thereby establishing a theoretical foundation for evaluating the stability of deep-buried tunnels under dynamic perturbation loads.
2 EXPERIMENTAL DESIGN
2.1 Impact sample
The focus of this engineering investigation is on the Meishan Iron Mine in Jiangsu Province, China. An outside crack around a tunnel (OCT) was employed in the test. The rock slab used in the test measures 325 mm × 200 mm × 30 mm, with a tunnel-shaped cavity of 50 mm × 50 mm × 30 mm (Figure 1). To meet practical requirements, a minimum crack width of 1 mm was specified. The purpose of this study is to analyze the fracture characteristics and distribution of a cracked tunnel model under static and dynamic loading.
The diagram of impact OCT sample. (mm).
2.2 Impact test apparatus
Dynamic experiments were conducted using a DID tester, which previous studies have demonstrated to be more suitable for studying dynamic fracture experiments. The objective was to investigate the fracture characteristics of the OCT model under varying in situ loading conditions. The acquisition system included an impact hammer, an incident rod, a transmission rod, a sensor, and a data acquisition system. The incident and transmission rods were made of aluminum alloy and measured 300 mm in width, 40 mm in thickness, and 3000 and 2000 mm in length, respectively. A strain gauge was installed between the impact and transmission rods to capture the stress wave effect during the dynamic impact process. The loading hammer had a range of 0–10 m, and the impact velocity was measured using an infrared velocimeter (Figure 2).
Physics experiment test system.
The crack propagation velocity and CBT are significant parameters in fracture mechanics and have been the focus of researchers investigating the dynamic fracture behavior of rocks. In this study, the CBT was evaluated via CFT, which is capable of calculating the crack propagation velocity and fracture toughness (Ying et al., 2019, 2022; Zhou et al., 2018).
2.3 Test results
The stress inside a specimen can be determined by attaching strain gauges to the device base using one-dimensional theory. The results obtained from the strain gauges are presented in Figure
3, and the dynamic calculation results are as follows (Yao & Xia,
2019):
(1)
where
,
, and
are strain gauge signals received from the incident, reflected and transmitted points, respectively;
,
are stress received from the incident and transmitted points of specimen, respectively;
and
denote the elastic moduli of the two bars.
Strain time-history and crack fracture test (CFT) test result.
The rate of crack propagation in fractured rock under dynamic loading represents a crucial mechanical parameter. This study uses a crack parameter extensometer to measure this rate. The testing methodology of the crack parameter extensometer has been previously described in the literature. The results of the experiment are presented in Figure 4a,b displaying the variation in the crack extension velocity and propagation distance over time. The graphical representation indicates that the crack extension velocity observed in these results is not constant; instead, the crack propagation speed generally ranges from 300 to 650 m/s (gray area). This finding aligns with results reported in the literature (Wang et al., 2021; Ying et al., 2019).
Strain time-history result. (a) Crack fracture test (CFT) test result and (b) experimental crack growth rate results.
2.4 Numerical analysis without in situ stresses
Numerical simulation is commonly used to validate the validity of experimental results and analyze stress propagation details that cannot be easily observed during experiments. In previous studies (Zhu et al.,
2008,
2014), descriptions of constitutive equations have been provided, and the advantages of using AUTODYN for analyzing dynamic problems have been highlighted. The pressure or deformation of rock is insignificant during impact processes and has little influence on thermodynamic entropy. The failure behavior of sandstone was described via the linear state equation (EOS), which can be expressed as follows (Xu & Shen,
2003):
(2)
where
represents the pressure of the sandstone,
is the bulk modulus,
and
mean the reference density and actual density of the rock, respectively.
Rock, as a brittle material, can withstand compressive stress but not tensile stress. Its failure mode is mainly tensile stress exceeding the allowable stress of the material. To obtain the failure mode under impact loading, this study uses principal stress to determine the failure mode. Failure of the component occurs when the principal stress exceeds the dynamic strength (tensile or shear) of sandstone. The stress criterion is as follows:
(3)
where
σ
1 represents the principal tensile stress, [
σ] denotes the material tensile strength,
τ
12 indicates the principal shear stress, and [
τ
r] manifests the material shear strength.
The selection of the mesh in numerical calculations is crucial, and in recent years, there have been extensive studies and interesting progress in the subdivision of mesh units (Areias & Rabczuk, 2017; Areias et al., 2016). In our study, a triangular unstructured mesh is used in AUTODYN to minimize the influence on crack propagation behavior. The total number of units is 74 710, as shown in Figure 6. To reproduce the results of the physical tests, we apply a stress‒time curve obtained from the experimental results to the incident plate, with a peak value of 90.05 kN, as shown in Figure 5. Regarding boundary settings, the bottom concrete base is set as a nonreflective boundary.
Dynamic load curves.
2.5 Numerical analysis results without in situ stress
The application of numerical modeling allows for the effective replication of the failure behavior of rock masses under dynamic loads. To simulate the initiation and propagation time of cracks under dynamic loading, monitoring points were positioned in a strategic manner along the anticipated crack extension route, as depicted in Figure 6. The monitoring points, designated as “ti,” were utilized to document the time at which the cracks were initiated at each respective location. Importantly, the component does not undergo instantaneous complete failure. Instead, the crack tip is influenced by dynamic stress waves, reaching a state of complete failure in approximately 6.67 µs. Consequently, the crack tip CBT can be calculated as 203.64 μs + 6.67 μs. This value differs from the experimental result by 191.84 µs, resulting in an error of approximately 9.6%.
Numerical model grid diagram.
To determine the crack propagation velocity, the space between adjacent monitoring points in the numerical model was set to 2 mm, which was the same as the distance between adjacent wires in the CFT. The crack propagation velocities obtained from both the numerical and experimental data are shown in Figure 7. During the simulation process, the crack extended to 40 mm after being subjected to dynamic stress waves for 428.05 µs. At this point, the crack propagation rate reached its maximum value of 729.99 m/s, accompanied by crack arrest. The time for the fissure to extend to the vault of the tunnel model was 214.01 µs, with an average crack extension velocity of 709.72 m/s. Both the test and numerical results reveal that crack propagation initially decreases rapidly, then increases, and later decreases again. During the transmission of stress waves, in addition to reflection at the boundaries of the tunnel, stress is also reflected through the bottom of the specimen. Furthermore, one is compressive stress, and the other is tensile stress, with the tensile stress accumulating first, followed by the compressive stress, which is analyzed in detail in Section 3. A comparison of the numerical and test results reveals that the numerical results generally have higher velocities than the experimental results. This is because the numerical results do not consider the presence of internal voids and assume a straight-line crack propagation path, whereas, in the experiment, there are internal voids in the material, which accelerate energy dissipation. Moreover, the crack extension route is not a straight line, as shown in Figure 8 (the black line represents the numerical results, and the pink line represents the experimental results). The maximum deviation is 6.9 mm, which indicates that the calculated experimental results are in error. The nonlinear crack propagation path observed in the experiment may be attributed to experimental errors or material heterogeneity. In conclusion, the AUTODYN program can effectively express the dynamic fracture characteristics of a fractured model under dynamic loads. This method will continue to be used in future studies on the propagation patterns of external cracks under different stress conditions.
Comparison of numerical and experimental crack growth rates under nonground stress conditions.
The crack path between numerical result and test result under nonground stress conditions. (mm).
3 NUMERICAL ANALYSES OF DIFFERENT IN SITU STRESSES
Owing to the presence of initial stress, the stress state of deep rock masses differs from that of shallow rock masses. To investigate the fracture behavior of tunnels under various initial stresses, a numerical model was established based on unconfined loading conditions. The model was proportionally scaled down according to the actual engineering scenario, and low in situ stresses and dynamic shock wave values were set. The numerical model was developed using the combined static (static implicit) and dynamic (dynamic explicit) modules in Workbench. In the static module, initial boundary conditions and in situ stresses were applied horizontally, whereas displacement constraints were set vertically to replicate the stress state of deep rock masses. The strain, displacement, and stress at different positions of the OCT were calculated and imported into the AUTODYN software's initial condition section in.rst file format. To obtain a stable stress field, the same stress boundary conditions as those used in the static analysis were used in AUTODYN settings. According to the in situ stress equation (Cai et al., 2000), the maximum principal stress in the horizontal direction at a depth of 120 m perpendicular to the tunnel axis was 6.27 MPa, and the horizontal and vertical principal stress were 2.85 and 3.29 MPa, respectively. The in situ stress K was set numerically, and the value ranged from 0 to 2.5 MPa, with an increment of 0.5 MPa. The crack propagation characteristics of the tunnel under low in situ stresses were used as a guide for determining crack characteristics under high initial stresses. The introduction of in situ stresses in the static analysis resulted in changes in the stress field around the tunnel, which was manifested as different initial stress distribution patterns in the OCT model, as shown in Figure 9. A detailed analysis can be found in Section 3.3.
Initial stress law of crack tip with vary in situ stresses. (a) 0 MPa, (b) 0.5 MPa, (c) 1.0 MPa, (d) 1.5 MPa, (e) 2.0 MPa and (f) 2.5 MPa.
Figure 10 illustrates the change in the first principal stress at the crack tip over time with various in situ stresses. In this study, t = 0 µs was defined as the moment when the loading wave touched the top of the sample. The time curve displayed four peak values, resulting in three distinct superposition effects of stress waves (highlighted by the red dashed box in the figure). The first superposition occurred at approximately 240 µs, the second occurred at approximately 540 µs, and the third occurred at approximately 710 µs. Observations revealed that the second and third wave peaks were higher than the first wave peaks, which can be attributed to the longer wavelength of the loading wave. Before the complete passage of the incident wave through the crack tip, it had already reflected back from the free surface and underwent multiple superpositions with the incident wave. Experimental testing, which utilized the longitudinal wave velocity of the green sandstone material, determined that the three superimposed waves occurred at 223.3, 519.69, and 751.21 µs, with error values of 7.40%, 3.91%, and 5.49%, respectively. When the superposed stress waves exceeded the rock tensile strength, the crack tip element failed and could no longer withstand tensile stress. The time curve of the maximum principal stress of the monitoring unit dropped to 0 MPa, and cracks began to propagate, similar to the case of K = 0 MPa. When the K exceeded 2 MPa, the incident wave was unable to initiate crack propagation at the crack tip. Since superimposed waves can influence the stress intensity factor at the crack tip, affecting both crack initiation rates and crack extension rates, a comprehensive analysis was conducted in subsequent chapters. The numerical calculation results demonstrated the effective simulation of fracture characteristics in the OCT model under various K values using the numerical model.
Max. stress-time curves of the crack tip under varying
K.
3.1 Crack break time and crack propagation velocity
Figure 11 depicts the correlation between the crack break time (CBT) of the crack tunnel and various initial stresses. When K is within the range from 0 to 1.5 MPa, the CBT of the crack follows an exponential growth pattern. Specifically, under an in-situ stress condition of K = 0 MPa, the crack initiates at t = 203.06 µs. As the K increases to 0.5 MPa, the CBT is delayed to t = 219.54 µs. Similarly, for in situ stresses of 1.0 and 1.5 MPa, the CBTs are 338.69 and 1071.0 µs, respectively. However, when K exceeds 2.0 MPa, the prefabricated crack within the crack tunnel model is no longer initiated, as indicated by the gray area in the figure. Once a crack breaks, it continues to propagate forward as the input of stress waves. Consequently, the propagation speed of the crack becomes a crucial parameter in fracture mechanics. Figure 12 presents the average extension speed of the crack as it intersects with the tunnel under the K = 0 MPa in situ stress condition, which is 393.26 m/s. With an in situ stress of K = 0.5 MPa, the average extension speed of the prefabricated crack towards the tunnel is 235.67 m/s. As the K increases, the crack average extension speed of the crack decreases rapidly. When K = 1.0 MPa, the average propagation speed of the crack decreases to 55.48 m/s, and the crack exhibits a prolonged stoppage behavior at 5 mm. When K = 1.5 MPa, the crack only extends 5 mm, yet its average propagation speed reaches 568.99 m/s. This result indicates that under higher in-situ stress conditions, the crack experiences an extremely fast release speed shortly after initiation. Further explanations regarding the average extension speed of the crack are provided in Section 3.2.
Crack break time (CBT) with varying
K.
Crack propagation length with varying
K.
3.2 Discussion of the crack propagation velocity and crack extension length
Figure 12 illustrates the correlation between crack propagation rate and length under varying ground stress conditions. The gray area in the figure indicates the noninitiation of pre-existing cracks. Once initiated, crack propagation persists due to stress wave effects, with different ground stresses exerting distinct impacts on the propagation rate. Under ground stress of K = 0–1.0 MPa, the average crack propagation rate markedly decreases. This finding is attributed to the ground stress causing the closure of cracks originally initiated by expansion forces, thereby impeding crack propagation. Consequently, there is a rapid decrease in the crack propagation velocity, and at K = 1.0 MPa, the crack ceases to propagate after expanding by 5 mm. After a delay of 432.5 µs, the crack recommences propagation and connects with the tunnel, resulting in an overall substantial decrease in the average propagation rate. However, under ground stress of K = 1.5 MPa, the CBT is 1071.0 µs, which is a result of multiple reflections and superimpositions of stress waves, leading to a substantial energy accumulation at the crack tip during initiation. Consequently, the crack initiates and propagates rapidly. Additionally, as the crack only expands by only 4 mm and does not propagate further, the limited sample size for calculating the average velocity results in a significant degree of randomness in the calculated average propagation rate. Nonetheless, the crack propagation velocity still falls within the range of 300–650 m/s, which is consistent with sandstone's typical range. When K exceeds 1.5 MPa, pre-existing cracks are no longer initiated under the influence of ground stress. Considering both the tunnel and the crack, the hazardous area in the model extends from the crack zone to the tunnel area, thus weakening the influence of cracks around the tunnel on the overall structural stability under high ground stress.
When affected by initial stress, the crack propagation length becomes shorter, as depicted in Figure 13. The impact of ground stress causes the crack propagation path to become more tortuous because of increased internal friction in the material. Furthermore, since the adopted constitutive model is based on softening damage, the failure of grid cells does not occur instantaneously but requires a certain amount of time. Therefore, the grid can withstand stress within a short period, resulting in slight movement of the overall structure and potential deviation in the crack propagation path. The crack propagation length is significantly affected by the ground stress. Figure 14 shows the relationship between the crack propagation length and dynamic load under varying ground stresses. For clarity, cracks formed between the crack and the tunnel rock bridge are denoted as Crack 1, whereas those between the tunnel and the transmission end are labeled as Crack 2. The figure clearly shows that, for Crack 1, when the ground stress exceeds 1.0 MPa, the crack length is impacted. Furthermore, when the ground stress exceeds 1.5 MPa, Crack 1 is no longer initiated, and the dynamic load induces the formation of Crack 2 in the tunnel arch. For Crack 2, when the ground stress exceeds 0.5 MPa, the propagation length of Crack 2 is affected. With increasing ground stress, the propagation length of Crack 2 diminishes rapidly. This is attributed to the hindrance of the lateral expansion of the model by ground stress, resulting in closed crack states and rendering crack formation more challenging.
Crack propagation results with varying
K. (a) 0 MPa, (b) 0.5 MPa, (c) 1.0 MPa, (d) 1.5 MPa, (e) 2.0 MPa and (f) 2.5 MPa.
Crack propagation lengths at two places with varying
K.
3.3 Dynamic stress distribution of the OCT
Under dynamic loading, the stress field distribution of a fractured tunnel model is influenced by the diffusion of stress waves. In Figure 9, the stress distribution characteristics of the model specimen under different in situ stresses at time t = 0 µs are depicted. When in situ stresses were applied, the distribution pattern of the stress exhibited symmetry along the specimen, with compressive stresses focused on the crack tip and the rock bridge between the crack and tunnel. Tensile stresses were primarily distributed around the crack surface and the tunnel sidewall, with the most significant tensile stress occurring around the crack. On increasing the K, the stress values at various positions within the OCT model also increased accordingly, whereas the overall stress distribution remained relatively stable.
As time progresses, the dynamic stress transmits from the top of the specimen to the bottom, resulting in changes in the internal stress distribution of the sample. Figure 15 shows the distribution pattern of stress waves around the pre-crack and tunnel, comparing K = 0.5 MPa and K = 2.0 MPa. When t = 0 µs, there is compressive stress at the crack tip, and as K increases, the compressive stress at the crack tip also increases. Moreover, the tensile stress on the crack surface increases with increasing K, as shown in Figure 15a. When t = 213.6 µs and K = 0.5 MPa, the pre-existing crack starts to break, and concentrated tensile stress occurs in the roof and floor of the tunnel, which is more dangerous. However, the compressive stress appears at the crack tip, which leads to the pre-crack not cracking when K = 2.0 MPa. The tunnel undergoes overall lateral compression, as presented in Figure 15b. As the stress wave continues to input, when t = 427.8 µs and K = 0.5 MPa, crack 1 is already connected with the tunnel, and there is concentrated tensile stress in the roof area of the tunnel where new cracks are most likely to occur. However, when K = 2.0 MPa, the pre-existing crack still does not break, but the compressed stress decreases, and the tunnel corner experiences compressive stress, with little change in stress on the sidewalls of the tunnel. The tensile stress in the roof arch of the tunnel decreases, as displayed in Figure 15c. Finally, when t = 1071.0 µs, for K = 0.5 MPa, Crack 2 has already connected with the bottom of the specimen, and the specimen is split in two, whereas for K = 2.0 MPa, the stress state at the pre-existing crack changes with tensile stress, but the value is small. This means that the SIF at the crack tip cannot meet the dynamic initiation toughness (DIT) of the rock. The roof and floor of the tunnel are in a tensile state, and Crack 2 extends from the roof of the tunnel to the bottom of the sample, but the extent of the crack is short, as revealed in Figure 15d.
The stress time distribution of outside crack around a tunnel (OCT) model. (a)
t = 0 μs, (b)
t = 213.6 μs, (c)
t = 427.8 μs and (d)
t = 1071.0 μs.
Therefore, when the material has a low compressive strength, or when the peak load is high, the tunnel's shoulder, arch foot, and crack tip are susceptible to compression damage, which can impact engineering stability and safety. In conclusion, high ground stress can partially impede the propagation of cracks or internal cracks in tunnels. Nevertheless, high ground stress, coupled with dynamic loading, is also a significant cause of rock burst incidents in rock masses. Hence, in deep surrounding rock masses, prioritizing the prevention and control of rock bursts or slabbing disasters is crucial.
3.4 Dynamic initiation toughness (DIT)
When studying the dynamic fracture behavior, it is essential to consider parameters not only the CBT, crack length, and crack propagation velocity but also the DIT. To determine the SIFs at the crack tip, there are two primary methods: the displacement extrapolation method and the
J-integral method. In this study, the displacement extrapolation method is employed to calculate the dynamic SIFs at the crack tip. Specifically, AUTODYN software is utilized to compute and extract the open displacement at the crack tip, which is then used in the following equation to calculate the SIFs (Li,
2012):
(4)
where
and
are the SIFs before and after normalization;
represents elastic modulus;
stands for the space from monitored point to crack tip;
signifies the crack opening displacement;
Poisson's ratio of the material;
represents the initial stress;
represents the length of crack.
The determination of DIT is dependent on the CBT. The results of DIT under various K values are presented in Figure 16. The crack does not break when the K exceeds 2.0 MPa. Hence, the maximum value of SIFs was selected for analysis in this study. The figure shows that the in situ stress significantly impacts DIT, and the results are all greater than 1.552 MPa·m1/2. When 0 MPa < K < 1.5 MPa, the DIT of the material is small and has little influence. Crack 1 propagates through the tunnel. However, when K > 1.0 MPa, DIT rapidly increases, making crack break and propagation highly challenging. For instance, at K = 1.5 MPa, the Crack 1 only initiates and expands by only 3 mm. Nevertheless, when the K exceeds 1.5 MPa, the Crack 1 no longer breaks.
Dynamic initiation toughness (DIT) under different lateral pressure.
3.5 The DSCF of the tunnel
The presence of tunnels and cracks leads to various reflections and transmissions of stress waves during extension, resulting in a complex stress field surrounding the tunnel. The stress concentration at the surrounding rock and crack tips is notable. Researchers have extensively focused on forecasting the failure characteristics of fractured tunnels under dynamic perturbation loads by investigating tunnel stresses (radial and tangential). To facilitate analysis, a local coordinate origin is established in a polar coordinate system, with the center of the tunnel as the reference point. The angle increases clockwise, where
θ = 0° is the direction of the vault, as shown in Figure
17a. The radial and tangential stress formulas can be computed as follows (Herakovich,
2016):
(5)
(6)
where
and
are radial stress and circumferential stress, respectively;
,
, and
are the
x directions stress,
y directions stress and shear stress, respectively, of the monitoring points.
Variation rule of the dynamic stress concentration factor (DSCF) around the tunnel. (a) Stress distribution around the tunnel in polar coordinates, (b) DSCF under different in situ stress.
As a typical brittle material, the fracture characteristics of rock are highly correlated with the distribution of the circumferential stress. The DSCF is employed to study the failure characteristics of fractured tunnels subjected to dynamic loads. Here, the DSCF refers to the tunnel tangential stress divided by the amplitude of the incident stress wave without an outside crack specimen in the same position. The magnitude of the negative DSCF value indicates the level of tensile stress concentration, with smaller values indicating higher levels of stress concentration. Figure 17b shows the DSCF of the tunnel under various K values. Despite the nonuniform grid in the numerical computation model, the overall model maintains approximate symmetry, as it is a symmetrical structure. Therefore, only the 0°–180° range of the tunnels is considered to explain the overall distribution in this article.
Overall, under different stress conditions, the magnitudes of the DSCF exhibit a similar trend and are approximately symmetric as the angle varies; therefore, only half of the change law needs to be considered. Within the 0°–180° range, there are two decreases and two increases in the DSCF. The first decrease occurs at the roof of the tunnel. As the angle increases, the DSCF gradually increases and reaches its maximum value at 45°. Subsequently, as θ continues to increase, the DSCF becomes stable and begins increasing again at approximately 120°. When the angle reaches 130°, the DSCF reaches the maximum value. Finally, within the 130°–180° range, the DSCF decreases as the tensile stress increases. Under all ground stress conditions, the maximum DSCF is observed at 180°, which means that the cracks mostly appear at the floor of the tunnel. When K = 0 MPa, the maximum tensile stress occurs at 0°, 135°, 180°, and 225°. The DSCF gradually increases from 0° and peaks at approximately 40°. Then, it decreases, and it turns at 60°. The DSCF starts to decrease at approximately 120°. A comparison of the effects of the presence and absence of ground stress clearly reveals that ground stress significantly influences the DSCF of a tunnel. Therefore, the DSCF pattern for deep rock engineering cannot be accurately explained without considering in situ stress.
