Damage Behavior Study of Specimens with Double-Prefabricated Cracks under Dynamic–Static Coupling Loads

Abstract

The surrounding rock of the deep-buried chamber contains high-ground stress and initial cracks. Under a dynamic load, cracks will develop and expand, leading to the fracture and collapse of the confining pressure. Therefore, it is essential to study the failure process of fissured surrounding rock under the joint action of static stress and a dynamic load. In this paper, samples with cracks are used to simulate the defective rock mass. Similar modeling tests and numerical simulation studies were carried out to reveal the damage process of cracked deep rock mass under dynamic disturbance and investigate the impact threshold of rock mass damage under a certain level of hydrostatic pressure. The model test investigated the damage behavior for specimens with double-prefabrication cracks under pressure from a dynamic–static coupling load. The influence of the mechanisms of the angle of a crack, the initial static pressure, and impact capacity on specimen damage was analyzed. It was perceived that, with an increase in the angle of the crack, the omen of specimen damage is less obvious, and the specimen is subjected to sudden damage. On this basis, the damage process of the specimen containing prefabricated cracks under combined dynamic and static loads is realized through numerical simulation, and tests verify the accuracy of the results. The analysis allowed us to come up with a variation rule for the single-disturbance energy threshold for specimens with a prefabricated crack angle and the initial static load level of the specimen containing double-prefabrication cracks. The study lays the foundation for the future analysis of any deep rock mass failure process under dynamic disturbance and the protection of a deeply buried chamber.

1. Introduction

Tunnels are often needed during the construction of highways, railways, and subways. In construction, the disturbance of rock mass under a high static load occurs occasionally. Mechanical excavation, blasting excavation, and other dynamic disturbances will make the cracks in the surrounding rock continue to develop and expand until the rock mass is unstable, leading to severe engineering disasters.

During the formation and development of a rock body, the interior of the rock body is not entirely homogeneous, and “structural surfaces” such as fissures, joints, and faults are widely present [1]. These “structural surfaces” (fissures, joints, and faults) have an important influence on the deformation behavior and strength characteristics of rock bodies [2,3,4,5]. In addition, underground engineering rock bodies may be subjected to dynamic and static loads. On the one hand, the self-gravitational stress of the rock mass increases with an increase in the burial depth, and the rock mass is in a state of static prestress [6]. On the other hand, blasting, earthquakes, and other perturbations will subject the rock mass to dynamic perturbations [7]. Therefore, studying the mechanical behavior of crack-containing specimens under pressure from both dynamic and static loads is significant in analyzing the failure mechanism of deeply buried caverns. It can provide a reference for site construction to help improve the tunnel’s safety factor and speed up the safety construction of the tunnel.

Existing studies have adequately analyzed the damage behavior of fracture-containing rocks under separate static and dynamic pressure loads. Under the action of separate static loads, numerous scholars have investigated the mechanical properties and fracture mechanisms of specimens containing single cracks [8,9], double cracks [10,11,12,13], or multiple cracks [4,14,15,16] by conducting uniaxial, biaxial, and triaxial tests [17,18,19]. It was found that parameters such as crack length, the inclination of cracks, the spacing between cracks, and rock bridge length have an essential influence on the deformation damage characteristics of the specimens [20,21,22]. In separate dynamic loading studies, some scholars analyzed the mechanical properties and crack extension inclination of crack-containing specimens using the SHPB system [23,24,25,26]. In addition, there has also been some research about the cracking state and dynamic toughness of rocks under dynamic loading, and these studies proposed development guidelines for tip, closed, and secondary cracks [27,28].

With the gradual deepening of the research, it was found that there is a big difference between the mechanical behavior of rock specimens under dynamic–static coupled pressure and that under a pure static loading pressure [29]. Moreover, the magnitude of axial pressure will have a specific effect on the rock’s strength and dynamic fracture toughness [30]. Based on this, many scholars have carried out research on the damage process of crack-containing specimens under the action of dynamic–static coupled loading. Liu et al. [31] analyzed the crack extension characteristics within different rock materials under explosive loading and detailed the whole extension process of cracks. Feng et al. [32] investigated the mechanical properties of specimens containing non-parallel cracks and their damage characteristics under the action of different dynamic–static coupled loads. They analyzed the effect of strain rate on the damage characteristics of the specimens. Li et al. analyzed the damage process, mechanical response, and energy evolution characteristics of defective rocks under static–dynamic coupling based on the improved split Hopkinson pressure bar (SHPB) system to further understand the damage mechanism of the specimens [33,34,35]. In addition, there has been research and analysis of the mechanical properties and crack patterns of rock specimens subjected to triaxial dynamic–static coupling loading, and a series of valuable conclusions have been obtained [36].

Previous research has extensively investigated the mechanical behavior and damage process of fracture-containing rocks under separate static and dynamic loads. Scholars have analyzed the crack extension characteristics, mechanical properties, and damage characteristics of rock specimens under different loading conditions. For example, Yang et al. studied the fracture coalescence behavior of red sandstone containing two unparallel fissures under a uniaxial compression [1]. Li et al. analyzed the damage process and energy evolution characteristics of defective rocks under a static–dynamic coupling [29]. Z.T. Bieniawski [37] predicted the fracture initiation stress based on Griffith’s theory and studied crack arrays and fracture propagation in rock. CD Martin and NA Chandler [38] conducted uniaxial and triaxial sample tests of Lac du Bonnet granite and found that the crack initiation stress remains relatively constant and is independent of the damage accumulated in the specimen.

The crack damage stress mainly depends on the cumulative damage amount. For the determination of rock cracking stress and crack damage stress, there are mainly three types of research methods: the direct observation method based on observation equipment such as CT and SEM [39], the acoustic emission method [40], and the determination method based on the crack volume strain [41].

However, as for the actual deeply buried chamber environment, when the top of the deeply buried chamber is subjected to dynamic loads such as blast impacts, dynamic stress concentrations will be generated at the side walls, and the left and right sides of the chamber will be the first to suffer damage [42]. It is important to investigate the damage patterns and damage thresholds of the perimeter rock on both sides of the chamber under dynamic and static coupling loads in order to lay a foundation for the protection of the chamber.

However, the existing studies focus mainly on the expansion mode of cracks and the damage mode of specimens and less on the safety threshold of rock damage. Therefore, this study builds on and extends previous research by focusing on the damage law of specimens with double-parallel cracks under dynamic–static coupling loads. The aims of this study are to use cement mortar as a similar material, adopt a model test and numerical simulation to study the dynamic expansion mode of fracture-containing specimens under dynamic and static coupled loading, analyze the damage mechanism of deeply buried chambers under impact disturbance, and determine the damage threshold of surrounding rock. This study contributes to the existing body of knowledge by providing insights into the behavior of specimens with double-parallel cracks under dynamic–static coupling loads, which can inform the design and construction of structures in practical applications.

2. Experimental Design

2.1. Specimen Fabrication

2.1.1. Mold Preparation and Prefabrication of Cracks

In order to produce cement mortar specimens containing cracks, a square mold was made by splicing acrylic plates (Figure 1a), and the space inside the mold was 150 mm × 60 mm × 30 mm. A combined structure is shown in Figure 1b. The insert size is length × width × thickness = 30 mm × 20 mm × 2 mm. The combined structure is inserted into a predetermined position during the specimen fabrication to produce cracks as required. The specimen designed in this paper contains two prefabricated cracks that run through the thickness direction of the specimen. The positional relationship between the two prefabricated cracks is parallel.

The prefabricated cracks were distributed symmetrically along the center of the specimen. The lower apex of the upper crack was 2 cm from the right edge of the specimen and 6.5 cm from the upper edge of the specimen; the upper apex of the lower crack was 2 cm from the left edge of the specimen and 6.5 cm from the lower edge. The length of the cracks was 20 mm, and the width was 2 mm. The crack inclination angles in the prefabricated specimens containing the cracks were 30°, 45°, and 60°, respectively (Figure 2).

2.1.2. Raw Material Selection and Specimen Preparation

In the process of structural simulation experiments in the laboratory, cement mortar has the characteristics of good uniformity, noticeable cracking effect, high brittleness, and easy molding, which can better simulate the macroscopic performance of the crack-containing deep surrounding rock under the action of upland pressure, quasi-static loading, and impact loading. In recent years, many scholars have studied artificial rock and similar materials, in which the Young’s modulus, $E$, bulk weight, $\gamma$, internal friction angle, $\varphi$, and cohesion, $c$, of cement mortar are relatively stable and can basically simulate medium-strength rocks.

In addition, cement mortar was chosen because it is more difficult to take rock specimens containing cracks from their original site. It is also not easy to ensure the consistency of the specimens. At the same time, in order to eliminate the influence of the discrete type of rock specimens on the test results, this paper adopts mortar as a consistent material and pours the specimens containing prefabricated cracks.

The specific preparation process is as follows:

• The mass ratio of various materials is sand:Portland cement:calcium sulfate white powder:water = 3:1.2:0.8:0.6 (Figure 3). The density is 1997 kg/m³. After carrying out a mechanical property test, the strength of the cement mortar selected in this paper was 25.8 MPa, and the Young’s modulus was 3.54 GPa. Among them, the same batch and the same storage conditions are used for cement and white powder to eliminate the dispersion differences caused by batch changes and different storage conditions. Fine sand from the Yangtze River was screened by a 1.25 mm aperture sieve and dried for reserve. The purpose of using calcium sulfate white powder is to fill the pores, make the internal particles of the specimen uniform, and reduce the problem of significant differences in test results caused by local defects of the specimen due to large particle sizes.
• Lubricating oil is applied evenly inside the acrylic mold after splicing, holes are cut in the bottom of the mold, and it is placed on the flat floor for use. The purpose of applying lubricating oil and opening holes at the bottom is to facilitate the release of the specimen and to evenly apply lubricating oil for the production of precracked acrylic sheets, which is convenient for the subsequent removal and insertion of concrete after the initial setting.
• After preparing the materials, 11.4 kg of sand, 4.5 kg of cement, and 3 kg of calcium sulfate white powder are taken out and weighed. Three materials are poured into the JLS-16 mortar mixer (Figure 4) and stirred for 3 min to ensure the three materials are thoroughly mixed.
• Then, 2 L of tap water is slowly poured into the thoroughly mixed ingredients and stirred well until smooth. Then, the mixture is loaded into the mold made of an acrylic plate, thoroughly vibrated, eliminating the internal cavity, making the material dense, and then using the tool to smooth the surface of the specimen.
• The acrylic sheet is vertically inserted into the mortar specimen along the predetermined position. At the same time, it is constantly vibrating to prevent a cavity forming inside the specimen. Figure 5 shows the insertion method.
• After the initial setting of the specimen, the acrylic sheet is pulled out to form a prefabricated crack. The specimen is demoulded after 24 h of curing, and then the integrity of the specimen is checked, and unqualified specimens are removed. After 14 days of maintenance, it is taken out for a loading test. The finished specimens are shown in Figure 6.
• After the specimens are cured, a number of cube specimens with a size of 100 mm are poured with the same material and ratio, maintained under the same conditions, and dismantled after 14 days. Then, the different strengths of the specimens are tested using a uniaxial compression testing machine. The strength test results of the crack-containing specimens are shown in

  Table 1. Comparison of the static compressive strength of specimens (at a loading rate of 0.2 mm/min).

  Prefabricated Crack Angle (°)	Static Compressive Strength (MPa)
  30	17.35
  45	16.24
  60	14.05

  .

2.2. Test Device and Test Method

The impact and static load testing of mortar specimens containing preformed cracks was accomplished on a rock creep impact tester.

The arrangement of the test setup and specimens is shown in Figure 7, in which the test setup consists of a constant pressurization system, a power perturbation application system, and a measurement and monitoring system, and the test is carried out indoors at a constant temperature, so the temperature is not artificially controlled. A computer controls the experimental operation and data. Before the experiment, the bottom floor is leveled with a level ruler to prevent test errors caused by an uneven force during the test. At the same time, a high-speed camera is set up right in front of the testing machine. When the specimen is loaded, the crack development, failure mode, and penetration mode of the specimen are photographed simultaneously.

The static load is applied through the creep meter; the rate of load application is controlled by displacement; the application rate is 0.2 mm/min; the endpoint of load application is controlled by force; and the value of the endpoint force can be adjusted according to the needs of the test.

Before the experiment, to verify the stability of the test machine’s hydrostatic loading, a stability test of the test machine was carried out, and the test results proved that the creep instrument could meet the test stability requirements. After the static load was applied to a stabilized value, the impact test was carried out using a drop hammer. The impact hammer slides down the steel tube from top to bottom. In order to reduce friction and to stabilize the falling direction of the hammer, the steel tube is prefabricated with slide rails. The drop hammer slides down and impacts the pallet, transferring the impact load to the specimen below. The impact energy can be varied by controlling both the falling height and the mass of the hammer. To control the test variables, the mass of the impact drop hammer used in this test was 1.06 kg.

Before the test, a dynamic pressure sensor was pre-embedded above the block to test the impact loading capacity of the drop hammer (Figure 8). In order to minimize the friction effect in the drop hammer impact test device, the size of the block bearing the impact surface was designed to be 100 mm × 100 mm, and the thickness of each block was 50 mm. A steel plate was placed above the specimen, and Vaseline lubricant was uniformly applied between the steel plate and the specimen to reduce the friction coefficient of the contact surface. The impact disturbance of the falling hammer was transmitted to the specimen and the sensor after passing through the steel plate. The axial static load was applied to the specimen by lifting the cylinder, and the static load of the specimen was controlled to be 2 MPa. After the static pressure was stabilized, the falling hammer was released from a certain height to impact the specimen, and the dynamic disturbance was applied. The pressure waveforms, measured by the dynamic pressure sensor below the steel plate and set at different impact energies, are shown in Figure 9. The free fall height of the heavy object was 4 cm and 5 cm, and the corresponding impact energy was 0.42 J and 0.53 J. When the impact energy was 0.42 J, the peak pressure was 1.6 MPa; when the impact energy was 0.53 J, the peak pressure was 1.9 MPa. The peak pressure increased with the increase in impact energy, which shows that the equipment has sufficient dynamic and static combination loading capacity.

2.3. Test Program

The combined dynamic and static loading test means that the impact load is applied after stabilizing hydrostatic loading. During the test, an impact load is applied to the same specimen continuously and cyclically until the specimen is destroyed. In other words, the specimen is not destroyed under one impact; the impact disturbance continues to be applied until the specimen is destroyed, and the damage pattern of the specimen and the number of impacts are recorded. Therefore, in this paper, the damage pattern of the specimen and the number of impact perturbations during the damage of the specimen were analyzed by varying the crack angle, the level of static stress, and the magnitude of the impact energy. The specific experimental content was:

• The specimens were controlled to be at different levels of static force. An axial force was applied to the specimen, and then we waited for the pressure to stabilize for a while. We observed that the creep meter static load readings did not change significantly, then applied an impact energy of 0.53 J to the impact disturbance. We noticed that the axial force values between multiple groups of specimens were different, and we investigated which relates to the destruction of a specimen. If the specimen was not destroyed by a single impact, then the initial impact was followed by more until the specimen was destroyed. The time interval between the two impacts should be 60 s.
• Different angles for controlling prefabricated cracks were studied. A static pressure of 12.22 MPa was applied to two groups of specimens with prefabricated cracks of 30° and 45°, respectively. When the pressure was stabilized, and no significant change in the static load reading of the creep meter was observed, a shock perturbation with a shock energy of 0.53 J was applied. If the specimen was not destroyed by a single impact, then the initial impact was followed by more until the specimen was destroyed. The time interval between the two impacts should be 60 s.
• Different controlling shock perturbation energies were studied. A static load of 8.89 MPa was applied to the specimen with a prefabricated 45° crack under this condition. When the pressure was stabilized for a while and no significant change in the static load reading of the creep meter was observed, a shock perturbation was applied. The corresponding impact energies were 0.42 J and 0.53 J. If the specimen was not destroyed by a single impact, then the initial impact was followed by more until the specimen was destroyed. The time interval between the impacts should be 60 s.

3. Test Results and Analysis

3.1. Specimen Damage Process analysis

In order to study the damage behavior of the specimen under combined dynamic and static loading, a static load of 11.11 MPa was applied to the specimen containing 30° prefabricated cracks, at which time the static compressive strength was 65% of the compressive strength of the specimen. When the creep meter pressure sensor showed no apparent change, an impact disturbance with an impact energy of 0.53 J was applied, and the damage process of the specimen is shown in Figure 10.

Under a static loading pressure, no nascent cracks appeared in the specimen, indicating that the cracking stress of the specimen had not been reached at this time. Under the first impact disturbance, however, nascent wing cracks appeared, indicating that the stress level in the specimen under the effect of the impact disturbance exceeded the initiation stress. The angle between the nascent and prefabricated cracks was about 70°, which meets the maximum tensile stress criterion in fracture mechanics. The specimen in the impact process showed a small number of examples of small debris collapses, and debris collapsed on the block surface, producing small pits. After the second impact disturbance, the cracks continued to expand, and the rock bridges on either side of the prefabricated cracks tended to connect but were not yet fully connected. There was a slight increase in the amount of debris chipping out and a slight increase in the crack width during this process as compared to the first impact. After the third impact disturbance, the cracks continued to expand, the rock bridges on both sides of the prefabricated cracks were wholly connected, and the cracks began to expand to the top and bottom of the specimen. There was a slight increase in the amount of debris chipping out and a slight increase in the crack width during this process compared to the results after the second impact. However, the expansion of the rock bridge was more pronounced between the two impacts, and the cracks expanding to the top and bottom of the specimen increased in crack width only when they were close to the prefabricated cracks.

From the fourth impact perturbation to the sixth impact perturbation, the rock bridges on both sides of the prefabricated cracks continued to expand and become more expansive. The cracks extending to the specimen’s top and bottom also gradually expanded. A debris collapse accompanied each impact process, but the specimen did not show overall damage at this time. After the fifth impact, a nascent crack appeared on the specimen, expanding to the left along the prefabricated crack below. The main reason for this is that the release of energy is realized through crack expansion, specimen deformation, and debris chipping out. Therefore, when the input energy reaches a certain level but the specimen has not yet reached the destructive load, cracks will appear, triggering the dissipation and transfer of energy.

After the seventh impact disturbance, the specimen showed overall damage formed by lateral crack extension and by the cracks extending to the upper and lower sides of the specimen. These were accompanied by the appearance of other nascent cracks. It can be seen that the seventh impact perturbation reached the critical value of specimen damage, and the impact perturbation promoted crack initiation and extension. When the final specimen was destroyed, it was an instantaneous destruction under the impact, and there was no apparent premonition in the early stage. Moreover, when the specimen was destroyed, this destruction was accompanied by a small amount of debris flying out.

By analyzing the damage process of the specimens containing prefabricated cracks under dynamic–static combined loading, the following laws can be obtained:

• The moment of the application of the impact disturbance is accompanied by the appearance of debris, and the kinetic energy released by the flying debris allows some of the energy to be released and transferred;
• The damage of the specimen under the impact disturbance load is instantaneous; the specimen is suddenly destabilized, and there are a large number of pieces of debris combined with the certain speed of an outward avalanche;
• If the energy value of the disturbance is small and lower than the damage load of the specimen, the specimen can withstand multiple impact loads without damage.

3.2. Analysis of Factors Affecting Specimen Damage Patterns

3.2.1. Initial Static Level

As shown in Figure 11a,b, the specimens containing 45° prefabricated cracks are in a state of damage when they are damaged under combined dynamic and static loads. The control two sets of specimens were subjected to static pressures of 8.89 MPa and 12.22 MPa, corresponding to 55% and 75% of the specimen strength, respectively. The figure shows that with an increase in the initial static force level, the specimens were damaged more violently, the cracks more thoroughly developed, and the size of the damaged block became smaller. The number of block ejections and the ejection velocity shown in the video also increased with the initial stress. At an axial force of 8.89 MPa, the specimen lost its load-bearing capacity after five impacts, but at an axial force of 12.22 MPa, one impact was sufficient to result in the destruction. It can be seen that with an increase in the axial force, the ability of the specimen to resist the dynamic load is gradually weakened, with a greater degree of weakening.

To verify the above law, specimens containing 30° prefabricated cracks were controlled at different initial stress levels, corresponding to static pressures of 11.11 MPa and 12.22 MPa. The static load application method, dynamic disturbance application method, and control are the same as those of the specimens containing the 45° prefabricated cracks.

As shown in Figure 11c,d, the specimens containing 30° prefabricated cracks are in a state of damage when placed under combined dynamic and static loads. With an increase in the static load, the cracks in the specimen developed more entirely and were more violent in the state of destruction. When the static pressure was 11.11 MPa, the specimen lost its load-bearing capacity after seven impacts, and when the static pressure was 12.22 MPa, it was after five impacts that the specimen was destroyed. With an increase in the axial force, the ability of the specimen to resist the dynamic load gradually decreases.

With an increase in the initial static force level, the initially stored elastic energy inside the specimen increased, so that the damage state was more violent when damage occurred. Moreover, the excess elastic energy was converted into consumable energy for crack expansion and kinetic energy for debris chipping out. These findings lead us to the conclusion that the crack development is more complete when the specimen is destroyed at a high stress level, which leads to a smaller size of the destroyed block when the specimen is destroyed. The adjustment and equilibrium of the initial stress is completed earlier when the initial stress level is higher, which contributes more to the extension of any crack. The increase in the kinetic energy of the debris disintegration is manifested by an increase in block ejections and also by their ejection velocity.

3.2.2. Prefabricated Crack Angle

A comparison of Figure 11b,d shows the effect of different crack angles on the damage mode of the specimen subjected to the same initial static pressure and impact energy.

When the prefabricated crack angle was 30°, the specimen was damaged under a static load of 12.22 MPa after undergoing five impact disturbances of 0.53 J. The specimen was then subjected to a static load of 12.22 MPa. The specific damage process is such that the double-rock bridges in the middle of the two prefabricated cracks are connected, extending along one side of the rock bridge to both the upper and lower sides of the specimen. At final destruction, the specimen exhibited tensile damage. During the process of impact disturbance, there was a continuous eruption of debris, and the closer it approached the failure of the specimen, the larger the size and the faster the velocity of the debris. The cracks in the specimen are more fully developed, so there were no large chunks falling out when the final damage occurred.

When the prefabricated crack angle was 45°, a single impact disturbance was able to induce the sudden destruction of the specimen under a static load of 12.22 MPa. The specific damage process is such that one side of the rock bridge extends downward from the upper prefabricated crack, and the overall damage of the specimen occurs before it reaches the lower crack. The damage to the specimen was caused by the cracks extending to the side, and during the damage, the fragments on the side crumbled outwards and at a faster rate.

With an increase in the crack angle, the ability of the specimen to withstand the impact under the same static force is weakened, the specimen’s ability to resist the impact load is reduced, and overall instability damage is more likely to occur under an impact disturbance. On the other hand, when the crack angle is small under the impact, the crack gradually develops, expands, and releases part of its energy and there is a precursor to the damage of the specimen. In a scenario like this, the probability of overall damage to the specimen suddenly occurring is low, similar to that of the ductile damage under the impact. On the other hand, specimens containing large angular cracks are more likely to undergo brittle damage under an impact, which occurs suddenly and without apparent warning.

3.2.3. Impact Disturbance Energy

An initial hydrostatic pressure of 8.89 MPa was applied to the specimens with prefabricated 45° cracks. The impact perturbation was applied when it was noticed that there was no significant fluctuation in the pressure sensor reading on the creep meter. The control of the falling height of the hammer was 4 cm and 5 cm, respectively, with corresponding impact energies of 0.42 J and 0.53 J. If the specimen was not destroyed under the action of one impact, it was impacted again after an interval of 60 s until the specimen was destroyed.

The effect of the impact energy on the damage pattern of the specimen is shown in Figure 12. When the axial pressure was 8.89 MPa, it was at a low level. The result of this is that the destructive process of the specimen was not drastic.

When there was an impact energy of 0.42 J, due to the fact that the axial force and impact energy were small, the dynamic and static combination of load failed to reach the stress of crack expansion, so the crack never occurred, and even after ten impacts, the specimen was not damaged; just the end of the bear impact disturbances appeared as a flake off.

When the impact energy was 0.53 J, under multiple impact perturbations, the crack continued to expand until after the fifth impact, when the indication of the stress sensor began to decline, and the specimen lost its load-bearing capacity. During the impact process, debris was avalanched outwards, but it was not significant in size, only fine particles.

As the energy of a single impact increases, the smaller the number of impact disturbances that can cause damage to the specimen. In addition, when the energy of a single impact disturbance is reduced, the specimen can withstand multiple low-level impacts. When the sum of the energy values of the low-level impacts is higher than that of the higher-level impacts of the damage at the time of its occurrence, the specimen has not yet been damaged. As the impact energy decreases, the percentage of energy used for specimen destruction (utilization of energy) also decreases.

3.3. Energy Process Analysis

An essential feature of physical processes is the transformation and transfer of energy. The deformation and destruction of rock mass is a process of energy absorption and transformation. Moreover, specimen destruction is an irreversible energy dissipation process. Analyzing the evolution of energy in the specimen provides insight into the nature of specimen damage, leading to a better understanding of the phenomena in the damage process.

During the test, the work done by the external force on the specimen, in addition to causing the deformation of the specimen, also contributes to the fact that a part of the specimen causes damage, which is the dissipated energy. Neglecting the heat exchange between the system and the outside world during the test, we may see that according to the first law of thermodynamics, the energy input from the outside world, $U_{+}$, will be converted into the elastic and dissipative energy of the specimen during this physical process, i.e.:

$$U_{+}=U_d+U_e$$

(1)

where $U_d$ is the dissipated energy and $U_e$ is the elastic energy.

Under static and dynamic combination loading, the external force that works on the specimen has two parts, respectively: quasi-static load work and impact disturbance work. The work done by these two parts are $U_c$ and $U_f$, respectively. When the sum of the work done by the two parts exceeds the total energy, U, required for uniaxial compression damage, the specimen undergoes destabilizing damage. Thus, both types of the energy are critical for specimen damage.

$$U_c+U_f\ge U$$

(2)

$U_c$ is obtained from the stress–strain relationship of the specimen before the impact disturbance.

$$U_c=Al{\displaystyle \int \sigma }d\epsilon =Al{\displaystyle \sum _{i=1}^n\frac{1}{2}}({\sigma }_{i+1}+{\sigma }_i)({\epsilon }_{i+1}-{\epsilon }_i)$$

(3)

where A is the loaded area of the specimen and l is the height of the specimen. The stress–strain relationship can be obtained from load–displacement curves during the hydrostatic phase of the creep instrument. The higher the static load level, the more work is done by the quasi-static load, and the more quickly the specimen containing prefabricated cracks will be damaged.

$U_f$ can be expressed in the following equation:

$$U_f={\displaystyle \sum _{i=1}^n{U}_{fi}{\eta }_i}$$

(4)

where ${\eta }_i$ is the effective utilization rate of single-impact disturbance energy, indicating the ratio of the energy used for specimen damage in the impact process to the single-impact energy.

When the energy of multiple shock disturbances is the same, Equation (4) can be reduced to the following form:

$$U_f=U_{fi}\overline{\eta }N$$

(5)

where $\overline{\eta }$ is defined as the average impact energy utilization. From the previous section, when the impact energy was 0.53 J, the specimen was damaged under five impacts, and at this time, the total impact energy was 2.65 J. When the impact energy was 0.42 J, the specimen remained undamaged after 10 impacts, when the total impact energy was 4.2 J. The comparison reveals that the average shock energy utilization increases as the shock perturbation energy increases.

Therefore, the energy of a high-energy shock perturbation is more likely to be used for specimen damage than for a low-level shock. Both experimental and theoretical derivations show that if no damage occurs under a single impact, the specimen can withstand multiple impacts with corresponding energy values without damage. Therefore, it is crucial to find the critical threshold of the specimen’s single-disturbance energy. Finding the critical perturbation energy threshold under the corresponding working conditions can help determine the level of hydrostatic pressure and impact perturbation during the construction process. At the same time, it can be used to prejudge an engineering situation and guide construction planning.

4. Determination of Critical Threshold for Single Perturbation

4.1. Modeling and Load Application

4.1.1. Modeling Ideas

This paper focuses on simulating the damage law of the specimen under impact disturbance and static load, which is primarily a dynamic problem. To address this, LS-DYNA R13 is chosen as the simulation software due to its better applicability in handling dynamic problems. Based on existing research, the model parameters of the RHT model were determined through preliminary experiments and calibration [43]. The parameters obtained from the preliminary tests were imported into the software, and the impact block unit and specimen unit were established (Figure 13). Through the sensitivity analysis of the grid, the computational efficiency and accuracy can be taken into account when using the 1mm cube grid. The number of grids is 405,000. The specific process involves applying the disturbance load to the specimen through the downward impact of the impact block in order to simulate a dynamic disturbance load under actual working conditions.

The mortar specimen is a three-dimensional solid unit, and the RHT model is adopted. This model can describe the mechanical behavior of the material in the shock compression stage well and is suitable for analyzing the concrete structure models aftfected by various loads such as an impulse load. The main parameters [43] for the mortar specimen, including its material properties, dimensions, and crack characteristics, are provided in

Table 2. Main parameters of the RHT intrinsic model.

Parameters	Retrieve a Value	Parameters	Retrieve a Value
Material density	1997 (kg/m3)	Failure surface parameters, N	0.61
Young’s modulus of cement mortar	3.54 (GPa)	Relative shear strength, FS	0.18
Parameters of the equation of state, B0	1.22	Relative tensile strength, FT	0.1
Parameters of the equation of state, B1	1.22	Rhodes Angle Parameter, Q0	0.865
Parameters of the equation of state, T1	100	Rhodes Angle Parameter, B	0.0105
Parameters of the equation of state, T2	0	Failure surface arameters, A	1.4

. The size of the specimen is 150 mm × 60 mm × 30 mm. The specimen is prefabricated with two cracks, both with a length of 20 mm and a width of 2 mm. The two cracks pass through the specimen along the thickness direction, and the specific location of the cracks is the same as described in the previous section. According to the test conditions, three angles of cracks were prefabricated, and the prefabricated crack angles were 30°, 45°, and 60°. The material simulation cracking phenomenon is realized through the epsf parameter definition in the RHT. The impact block, representing the impact hammer, is modeled by an elastic–plastic principal structure, *MAT_PLASTIC_KINEMTIC, and the main parameters are shown in

Table 3. Main parameters of the PLASTIC_KINEMTIC intrinsic model.

Parameters	Retrieve a Value
Material density	7852 (kg/m3)
Young’s modulus of cement mortar, E	206 (GPa)
Poission’s ratio, PR	0.3

. The mass of the block is 1.06 kg, which is the same as the mass of the actual impact hammer. In order to ensure that no shear damage occurs during the impact, the size of the contact surface between the impact block and the specimen is the same as that of the upper surface of the specimen, which is 60 mm × 30 mm. The contact between the impact speed and the specimen is defined by *CONTACT_AUTOMATIC_SURFACE_TO_SURFACE.

These modeling choices and parameters are crucial for accurately simulating the specimen’s behavior under an impact disturbance and static load, providing valuable insights into the damage law and response of the specimen under dynamic–static coupling loads.

4.1.2. Load Application

The static load is applied through the LOAD_SEGMENT method in LS-DYNA, and in order to improve the computational efficiency, both the load curve and the damping curve are defined so that both act simultaneously. This approach eliminates the vibration effect caused by the external load, resulting in more accurate static loading. At the end of the static loading, the specimen is impacted using a restart at a pre-set impact block speed. The impact energy can be varied by controlling and varying the impact velocity of the impact block.

The first step during the numerical simulation tests is determining the static load level. The impact block impacts the specimen once the static load is applied and stabilized. This impact aims to assess whether the specimen is damaged under a single impact. If the specimen is destroyed, the impact energy is reduced in subsequent tests. On the other hand, if the specimen remains undamaged, the impact energy is increased in subsequent tests. The change in the value of the impact block velocity between two impact tests is set to 0.1 m/s.

The minimum impact energy at which the specimen breaks during a single impact is defined as the single-perturbation critical threshold. This threshold represents the minimum energy required to cause failure in the specimen under a single impact. By determining this critical threshold, the specimen’s impact resistance and damage behavior can be evaluated.

4.2. Numerical Simulation Validation

In order to verify the accuracy of the numerical simulation, a comparative study with physical experiments was carried out in terms of static loading and combined dynamic and static loading, respectively.

4.2.1. Static Loading Verification

Using cement mortar as a similar material, the specimens with prefabricated cracks were poured. After a mechanical property test, specimens with different material strengths were obtained for testing. An axial force was applied to the specimens at each of the three angles until the specimens were damaged. This was carried out in order to obtain the compressive strengths of the specimens at different material strengths. The specific working conditions and results are shown in

Table 4. Compressive strength of specimens with different material strength and crack angle.

Serial Number	Crack Angle (°)	Material Strength (MPa)	Compressive Strength (MPa)	Serial Number	Crack Angle (°)	Material Strength (MPa)	Compressive Strength (MPa)
1	30	20	14.5	7	45	35	20.5
2	30	26	17	8	45	40	23
3	30	35	21	9	60	20	12
4	30	40	25	10	60	26	14
5	45	20	14	11	60	35	16.5
6	45	26	16	12	60	40	18

, and four material strength parameters are listed for each prefabricated cracking angle. As shown in Figure 14, the trend of the compressive strength of the three prefabricated cracked specimens is plotted against the material strength. The compressive strength of the specimen gradually increases with an increase in material strength. The interrelationship between the specimen compressive strengths and the angle of cracks within them was examined. It was found that, all with the same material strength, the larger the crack angle, the lower the compressive strength of the specimens. The regularity is consistent with the pre-tests, proving the reliability of the simulation results.

By comparing the compressive strengths of the specimens, it was found that the compressive strength of the specimens was more consistent with that of the specimens under actual working conditions when the material strength was 26 MPa (

Table 5. Comparison of compressive strength of test and simulated specimens.

Crack Angle (°)	Data Sources	Compressive Strength (MPa)
30	test	17.35
	simulation	17
45	test	16.24
	simulation	16
60	test	14.05
	simulation	14

).

Figure 15, shows a comparison between the damage mode diagram of the specimen calculated by numerical simulation and the damage state of the specimen in the test case. The numerical simulation results show that a single crack controls the damage to the specimen containing 30° cracks. The other crack was compacted during the loading process, and the damage pattern of the specimen is consistent with the test results. The damage to the specimens containing 45° cracks was a connecting rock bridge on one side between the double cracks, which then expanded to both sides, showing tensile damage. Newborn cracks appeared on both sides of the upper crack and developed in an upward direction. The damage pattern of the specimen is consistent with the test results. The damage to the specimen containing 60° cracks is such that rock bridge connectivity occurs between the double cracks and extends to both sides, and the specimen, as a whole, shows tensile damage. It is consistent with the test results on the right side.

A comparison of the damage patterns of specimens containing cracks at the three angles shows that as the crack angle increases, connecting cracks are more likely to appear between the prefabricated cracks in the specimens, and the specimens are more likely to be damaged. The comparison shows that the simulation results are consistent with the test results, which illustrates the validity of this numerical simulation and shows that the tangible approach used in this paper to simulate hydrostatic loading is feasible. So, the material’s compressive strength was selected as 26 MPa and corresponding dynamic and static combination validation studies were carried out.

4.2.2. Comparative Validation of Dynamic and Static Combination Loading

The above analysis proves that the static loading method is feasible. In order to verify the feasibility of the dynamic–static combination loading method selected for the numerical simulation, the test results and simulation results of the prefabricated specimen with a crack angle of 45°were selected, and the comparison results of the two are shown in Figure 16.

According to the calculated results of the numerical simulation, the compressive strength of the specimen containing 45° prefabricated cracks is 16 MPa when the material strength is 26 MPa. The axial static load is maintained at 80% of the compressive strength of the specimen (12.8 MPa); meanwhile, a dynamic impact disturbance is carried out. According to the calculated results of the numerical simulation, the critical threshold of a single disturbance of the specimen under such conditions is 0.26 J. That is, in this state, when the impact energy is 0.26 J, the specimen is just destroyed, and the state at the time of destruction is shown in Figure 16.

During the test, an impact disturbance with an impact energy of 0.53 J was applied after applying a prestressing force of 12.2 MPa to the specimen containing a 45° prefabricated crack. The result of this was that the specimen was damaged, and the state at the time of damage is shown in Figure 16. Through the comparison of the simulation and test results, it can be found that the damage to the specimens is always caused by the impact disturbance, and the damage patterns are basically the same. When the specimens were damaged, this was caused by the rock bridge connection between the double-prefabrication cracks, which then expanded to both sides. During the test, the impact energy applied to the specimen was higher, so the damage process of the specimen was more violent. During the test, after the application of an impact perturbation, side cracks suddenly appeared and began to extend. The specimen crack extension process is also accompanied by block ejection, and the strain-type rock burst is basically the same. The main reason for this is that when the impact disturbance energy is too large, the combined force of the elastic energy stored inside the block and the absorbed disturbance energy needs to be released through the destruction of the block specimen and the ejection of the block.

The accuracy of the numerical simulation is verified through a comparative study of both static loading and combined dynamic and static loading. LS-DYNA simulation software and the selected load application method can effectively simulate and extend the test process.

4.3. Single-Impact Energy Threshold for Specimens Containing Prefabricated Cracks

The preceding description investigated the damage process of specimens containing prefabricated cracks under a certain level of static load and the corresponding laws when the specimens were subjected to multiple impact disturbances. However, due to the limitations of the experimental procedure, it is challenging to obtain the energy threshold for a single shock perturbation in different initial hydrostatic states. Therefore, to extend the experiments’ limitations, corresponding numerical simulation studies were carried out. Through numerical simulations, the effects of different prefabricated crack angles (30°, 45°, and 60°) and different static pressure levels (50%, 60%, 70%, 80%, and 90%) on the critical energy thresholds of a single perturbation in the specimen were investigated with the expectation of obtaining the corresponding laws to be used in guiding the construction of the project and provide a deeper understanding of the behavior of specimens under dynamic–static coupling loads.

4.3.1. Containing 30° Prefabricated Crack Specimens

The single-perturbation energy thresholds of the specimens containing 30° cracks under an impact perturbation are shown in

Table 6. Single-disturbance energy thresholds for specimens containing 30° cracks.

Serial Number	Initial Static Pressure Level	Single-Disturbance Energy Threshold (J)
1	50%	1.44
2	55%	1.2
3	60%	0.966
4	65%	0.85
5	70%	0.76
6	75%	0.55
7	80%	0.43
8	85%	0.23
9	90%	0.084

. As can be seen from the table, the single-disturbance critical energy threshold gradually decreases as the hydrostatic pressure level increases. When the static load is at a high level, a small amount of disturbance energy can cause damage to the specimen. Therefore, it is necessary to pay attention to the level of static force under the actual working conditions during a construction process in order to avoid the sudden destruction of a rock body under high stress.

The damage pattern of the specimen containing a 30° crack under combined dynamic and static loading is shown in Figure 17. From the figure, it can be seen that with an increase in the hydrostatic pressure level, the damage to the specimen is more intense, and the extension area of the crack becomes large. After the appearance of a double-rock bridge between the two prefabricated cracks on the specimen, it gradually extends to the two sides, which finally leads to damage to the specimen. The overall damage mode of the specimens was tensile damage. Some of the specimens were damaged with inclined cracks underneath, which were mainly caused by crack expansion and shear stresses present in the specimens. However, when the hydrostatic pressure level is 90%, the damage to the specimen in the simulation results shows that the prefabricated crack below the specimen suddenly extends to the two sides and destroys the specimen very quickly. Before other cracks in the specimen have had time to develop and expand, a penetrating crack has already formed. As shown in Figure 17, damage was produced inside the specimen. (The unit of the value in Figure 17 is J).

4.3.2. Comparative Analysis

From the simulation results, it can be seen that the presence of prefabricated cracks weakens the specimen’s ability to resist damage. It can also be seen that a change in the crack angle also leads to a change in the specimen’s ability to resist damage. Figure 18 shows the variation in the result of a single-perturbation energy threshold with the initial hydrostatic pressure level for specimens with different crack angles in the numerical simulation. The single-disturbance energy threshold of the specimen decreases with an increase in the prefabricated crack angle for the same initial hydrostatic pressure level. The difference is among the three decreases in crack angles with increasing hydrostatic pressure levels. It can be seen that the ability of the specimen to resist damage decreases as the prefabricated crack angle increases, and the change in resistance is more uniform as the prefabricated crack angle changes.

The change in the prefabricated crack angle has an essential effect on the specimen’s damage process and damage mode. The damage process of the specimen with 30° prefabricated cracks is the formation of double-rock bridges in the middle of the prefabricated cracks and then an expansion to both sides of the specimen. On the other hand, the damage processes for the specimen with 45° prefabricated cracks and the specimen with 60° prefabricated cracks are both the formation of single-rock bridges in the middle of the prefabricated cracks, then an expansion to both sides of the specimen, and then their final destruction. The specimens containing 60° prefabricated cracks underwent slip-type damage along the prefabricated cracks at higher stress levels. The main reason for this is that when the prefabricated crack angle is large, the time taken for cracks to sprout and develop in the specimen is short, and some of the cracks are damaged before the specimen has time to expand. It can be seen that the crack angle has an essential influence on the damage process and damage mode of the surrounding rock of roadways under the action of dynamic and static combined loading. In the actual construction of these roadways, cracks with smaller angles in the direction of the principal stress (the crack inclination angle is larger) must be controlled.

The initial hydrostatic pressure level has an equally important effect on the specimen’s damage mode and damage process. As shown in Figure 18, for specimens containing different prefabricated crack angles, the single-perturbation energy threshold decreases with increasing hydrostatic pressure level, and the two are roughly linearly correlated. When the prestressing level is high enough, minor disturbances can induce damage to the specimen, and the damage process is more violent, with debris crashing out. In specimens containing 60° prefabricated cracks, with an initial static pressure increase, the specimen’s damage mode transfers from tensile damage to shear damage, with an intermediate transition process of tensile–shear composite damage. With an increase in the initial static pressure, the shock perturbation is more likely to cause damage to the specimen, and the damage time is shorter and occurs in a more violent process. As a result, damage to the specimen occurs before the crack has time to expand fully.

In summary, the observed trends in single-perturbation energy thresholds can be attributed to prefabricated cracks and the influence of the initial hydrostatic pressure level. A change in the prefabricated crack angle affects the specimen’s ability to resist damage. At the same time, an increase in the hydrostatic pressure level makes the specimen more susceptible to damage, even with more minor disturbances.

The initial static load has an essential influence on the damage process and damage mode of a tunnel enclosure under combined dynamic and static loads, and the principal stress needs to be monitored and controlled in an actual construction. Suppose excessive principal stresses occur in the surrounding rock of the roadway. Timely measures should be taken to reduce the stress level, including drilling holes in advance to relieve pressure and supporting the rock as soon as possible.

4.3.3. Combined Dynamic and Static Load Analysis

Table 7. Peak loads for dynamic damage to specimens containing 45° prefabricated cracks.

Static Stress Level (%)	Static Stress (MPa)	Critical Shock Energy (J)	Peak Dynamic Load (MPa)	Peak Total Load (MPa)
90	14.4	0.047	2.7	17.1
85	13.6	0.13	2.9	16.5
80	12.8	0.26	3.8	16.6
75	12	0.39	4.5	16.5
70	11.2	0.51	7.3	18.5
65	10.4	0.61	7.6	18
60	9.6	0.76	7.8	17.4
55	8.8	0.98	8.1	16.9
50	8	1.19	8.6	16.6

presents the peak load on the upper surface of the specimen at the time of damage under combined dynamic and static loads. It includes specimens with 45° prefabricated cracks at various initial static stress levels. From the data in the table, we can see that when the impact energy is at the single-disturbance threshold, comparing the stress value of the dynamic–static combined load obtained from the simulation with the stress value of the specimen damage under quasi-static loading, it can also be seen that the total peak load of the dynamic and static load exceeds the compressive strength of the specimen under quasi-static loading by 16 MPa. The main reason for this is that under dynamic loading, the cracks in the specimen do not have time to expand, and the peak strength of the specimen is increased. However, at the same time, the excess energy input by the dynamic perturbation is released during the specimen destruction process. The pathways for releasing energy during specimen damage include debris avalanches and rapid crack expansion. However, a crack expansion under an impact disturbance is incomplete, and the excess energy is released as kinetic energy from a debris avalanche. Therefore, a high-stress tunnel enclosure under a dynamic disturbance is highly susceptible to dynamic hazards such as rock bursts. Therefore, in cases where there is a high-stress roadway perimeter rock and where a dynamic disturbance exists, care should be taken to guard against rock explosions to avoid casualties and property damage.

5. Conclusions

In order to reveal the damage mechanism of cracked rocks under the coupling of stress and dynamic disturbances, a simulation test of the damage process of double-cracked specimens under combined static and dynamic loads was carried out using a rock creep impact testing machine. The threshold value of a single perturbation that leads to the damage of the specimens under different static stresses was analyzed through numerical simulation. We obtained some meaningful conclusions:

• The damage process of specimens containing double cracks under dynamic–static coupled loading was simulated through tests. The influence mechanisms of the crack angle, initial static pressure, and impact energy on the damage of the specimens were analyzed.
• It was obtained from the tests that when the crack angle increases, the intensity of damage to the specimen increases, and the pre-damage sign decreases. Increasing the crack angle will change the damage characteristics from “ductile” to “brittle”.
• The damage process of specimens containing prefabricated cracks under combined dynamic and static loading was simulated using LS-DYNA. The credibility of the numerical simulation was verified by comparing the specimen’s peak value and damage pattern under quasi-static loading and dynamic–static combined loading in the test process. This also confirms the scientific validity of the load application method.
• Through simulation, the variation rule of the single-disturbance energy threshold of specimens containing prefabricated cracks with a prefabricated crack angle and initial static load level was obtained. The influence of the prefabricated crack angle and initial static load level on the damage law and damage mode of the specimens under a dynamic–static combined load was obtained.

This study takes rock mass with double-prefabricated cracks as the research object. It demonstrates the influence of two parameters, crack angle and initial static load level, on the safety threshold, which helps analyze the mechanical mechanism of rock mass failure. The results have practical and theoretical significance for improving people’s understanding of rock mass engineering and geological disasters and can help with taking corresponding preventive measures. However, there are still some shortcomings and improvements needed in this paper, and further research in this direction could be carried out on the following points:

• Cement mortar is used as a substitute material for rock. Although the two have many similar properties, the mechanical parameters of the material used in the test, including the Young’s modulus, E, of the material and Poisson’s ratio, μ, are not discussed in the natural rock. Therefore, the difference between the material and natural rock is still unclear. Subsequently, the required materials can be cut directly from the intact natural rocks to study the corresponding crack propagation laws of natural rocks.
• This paper studied the law of the influence of the double-prefabricated crack angle on the failure of specimens. However, in a natural fractured rock mass, the fracture relationship is complex, and the crack spacing and length of cracks will affect the performance of the rock mass under a load. Switching the focus to other changing parameters, a relatively complete theoretical system of prefabricated double-crack growth laws is obtained.
• The specimens in the test are all made of customized molds of exact specifications, and the size effect between the specimen size and the crack length is limited. Subsequently, specimens of different sizes can be made to obtain more applicable rock failure laws.
• According to the current test results, the number of experimental groups is small, and quantitative conclusions cannot be fully drawn. Only a qualitative analysis of the test law can be used. In the later stage, the intermediate group can be considered to enrich the test, and more accurate quantitative conclusions can be obtained to provide a more accurate basis for construction decisions.