Dynamic Response and Rock Damage of Different Shapes of Cavities under Blasting Loads

Abstract

In order to investigate the dynamic response and rock mass damage characteristics of cavities with different shapes under blasting loads, this paper, through a combination of model tests and numerical simulations, studies the stress distribution, strain, failure modes, and blasting fragment size distribution of cavities with different shapes subjected to blasting loads. The results show that under the action of blasting loads, the presence of cavities with different shapes significantly affects the blasting effects and rock mass damage. Spherical cavities exhibit excellent blast resistance, whereas rectangular and triangular cavities are prone to stress concentration at their tips, which in turn promotes rock mass damage and failure. Subsequent analysis of the blasting fragment sizes reveals that rectangular and triangular cavities yield more favorable blasting results than spherical cavities. The research findings provide important theoretical foundations and practical guidance for the design and construction of underground engineering blasting, contributing to enhancing engineering safety and promoting the sustainable development of the underground engineering industry.

1. Introduction

With the rapid pace of urbanization in our country, there has been a significant increase in the construction and development of underground engineering projects. Against this background, the construction of underground facilities such as tunnels, water-diversion tunnels, and hydroelectric power projects often necessitates passage through complex and variable geological environments [1,2]. However, in actual engineering practice, traditional blasting techniques frequently encounter issues such as over-excavation and inaccurate excavation pit formation [3,4,5]. Under such complex geological conditions, the presence of rock mass with different-shaped caves, is of paramount importance for the stability and safety of underground engineering projects [6]. The action of impact loads may lead to excessive excavation damage, inaccurate cutting, or overdevelopment of fractures, which can lead to a sudden water inflow accident [7,8]. Therefore, the stability of rock mass containing different-shaped cavities is not only directly related to the safety and efficiency of construction projects but also plays an indispensable and vital role in ensuring the safe operations of workers on the construction site [9]. Therefore, conducting in-depth research on the dynamic responses and rock mass damage characteristics of cavities with different shapes under the action of impact loads is of great theoretical significance and practical application value for guiding the design and construction of underground engineering projects and enhancing their safety.

Scholars both at home and abroad have conducted in-depth research on the dynamic response and damage characteristics of rock masses under impact loads [10,11,12,13]. Impact loads refer to time-varying loads such as explosions, vibrations, stress shocks from adjacent rockbursts, and earthquakes [14]. The research primarily focuses on the macroscopic response characteristics of rock masses under impact loads, such as stress, strain, and displacement, as well as the influence of impact vibrations on the surrounding environment [15,16,17,18].

Tao et al. [19] investigated the behavior of rock with preset circular holes under static and dynamic loads using both theoretical and laboratory testing methods. The results showed that when an underground cavity is subjected to both static and dynamic loads, the combined effect of static and dynamic stress concentration can induce primary and ultimate macroscopic failure of the rock. Li et al. [20] studied the dynamic response surrounding a circular tunnel subjected to concentrated blast stress waves through theoretical analysis and numerical simulation, thus revealing the reasons for the dynamic stress concentration around pre-set holes.

In recent years, with the development of numerical simulation technology, scholars have started to use numerical simulation methods to study the damage and failure mechanisms of rock masses under blasting loads. Li et al. [21] investigated the failure mode of tunnels subjected to blasting loads when there are cracks nearby, using indoor tests and LS-DYNA numerical simulation. The study found that the direction of the blasting load has a significant impact on the final failure mode of the tunnel in the fractured rock mass. Lin et al. [22] explored the interaction mechanism between circular holes and cracks induced by explosions through experiments and numerical simulation. The research results showed that pre-set circular holes do not simply promote or inhibit the propagation of blast-generated cracks but are determined by the relative position between the circular holes and the blastholes. Ma et al. [23] analyzed the foundation of blast-fill dams using discrete element software, with the results indicating that appropriate construction progress and flexible waterproofing materials are key characteristics of the waterproofing structure. Liu et al. [8] conducted research on potential water inflow disasters caused by blasting vibrations during the excavation of submarine tunnels, combining experiments with numerical simulation and establishing a model for the evolution of seepage characteristics during the surrounding rock damage process. Despite numerous studies that have explored the dynamic response and damage issues of rock masses under blasting loads, there are still the following issues and controversies: (1) the dynamic response laws of cavities with different shapes under blasting loads are not clearly defined; (2) existing research is mostly focused on macroscopic response characteristics, lacking studies on the micro-damage mechanisms of cavities with different shapes.

In response to the aforementioned issues and controversies, this paper aims to analyze the dynamic response patterns of cavities with different shapes under blasting loads, providing a theoretical basis for underground engineering construction. The study seeks to investigate the damage mechanisms and blast fragment size distributions of cavities with different shapes under blasting loads, explore the influence of cavity geological bodies on the size of blast fragments and rock mass damage characteristics, and reveal the mechanisms of rock mass damage and failure. The goal is to provide guidance for underground engineering design and construction to enhance project safety. Addressing the cavity structural conditions in mining geology, this study combines field tests with numerical simulations. Firstly, concrete model specimens with pre-set spherical, triangular, and rectangular holes (representing cavities with different shapes) were cast. A series of field experiments were designed and conducted to study the dynamic response and damage of cavities with different shapes under blasting loads, deeply investigating the relationship between cavity geological structures and blasting effects and providing important information on blast fragment size distributions for engineering practice and geological research. Subsequently, using the LS-DYNA finite element software, numerical simulation models of cavities with different shapes were established to simulate their dynamic responses and damage processes under blasting loads. By simulating the blasting processes of cavities with different shapes, the results can validate the field tests and further reveal how these geological structures affect rock fracturing and fragmentation and subsequently influence the block size distribution of blast products. This is of guiding significance for blasting construction safety in engineering projects. In the experiments, it is possible to predict the blasting effects under different geological structures, including block size distribution and degree of fragmentation, thereby providing predictions and references for actual projects. The experimental results can provide important data support for rock blasting and excavation in engineering projects, helping engineers make reasonable decisions and reduce uncertainty and risk.

This study contributes to enriching the theory of rock mass dynamic response and damage under blasting loads, providing new perspectives and methods for subsequent research. (1) It offers a theoretical basis for underground engineering design and construction, enhancing project safety; (2) it aids in optimizing blasting parameters, reducing the impact of blasting vibrations on the surrounding environment; and (3) it provides technical support for China’s underground engineering construction, fostering the sustainable development of the underground engineering industry. In summary, the research on the dynamic response and rock mass damage of caves with different shapes under blasting loads holds significant theoretical importance and practical application value.

2. Experimental Methodology

2.1. Specimen Making

Based on the experimental scheme for cavity structures, ten concrete casting molds were prepared. Before casting the samples, the molds were first checked to ensure their integrity. After the mold inspection, the molds were moistened with water to prevent the wooden boxes from absorbing moisture from the concrete, which could lead to an imbalance in the water-cement ratio and affect the strength and quality of the concrete, potentially causing cracking or detachment. Following the mold wetting, the concrete casting process commenced. The concrete was mixed using fine river sand, cement, and water in a ratio of 1:2:0.5 for cement, sand, and water, respectively. A small amount of clay was added to one batch, as shown in Figure 1. After thorough mixing, the concrete was placed into the molds. Blastholes were pre-made using PVC pipes with a diameter equal to the blasthole size. The strength of the PVC pipe is negligible compared to the strength of the concrete and the power of the explosive, and thus the pipes were left inside during the explosion experiment. For some models that required the placement of spheres, cubes, or other pre-made cavity shapes, foam objects were placed at the designated positions as the concrete was filled. After filling the mortar with concrete, a vibrating rod was used to expel air bubbles from the cement mortar, improving the compactness of the concrete. The vibration rod should be moved uniformly within the wooden box during compaction. Once the casting was completed, the samples were cured under standard conditions for 28 days. The finished concrete casting products are shown in Figure 1.

After the concrete samples were cured for 28 days, the process of demolding the samples began. Due to the adhesion between the concrete and the mold, tools such as pry bars and hammers were needed. For some molds with a nine-parallelepiped shape, the demolding was done in a sequence of first the edges and then the center. During the demolding process, iron tools such as hammers should not be used directly on the concrete model to avoid breaking the samples and ensure the accuracy of the experiment. After the samples were separated from the mold, they should be properly labeled to facilitate the identification of sample conditions and proportions during testing. As some concrete may enter the pre-made blastholes during the casting process, after demolding, it is necessary to check each blasthole to ensure that it is not obstructed. In cases of obstruction, a rod should be used to clear the hole. For more severe blockages, a drill should be used to clear the hole. When using a drill, the drilling direction must align with the direction of the blasthole, without deviation, to prevent damage to the sample.

2.2. Experimental Program

Due to the large volume of the samples and the relatively large amount of explosives used during the blast, the large samples were all placed in the blast pit, arranged in order as shown in Figure 2. During the explosion experiments, the blasts were conducted sequentially to ensure that each fragment came from the same sample. High-speed cameras and associated equipment were used in the experiment, with the high-speed cameras being used to record the destruction process of the samples and to measure key parameters during the blast, such as the propagation velocity of the blast waves and the fragmentation and separation process of the explosive material.

These parameters are crucial for evaluating the blasting effect and studying the blasting mechanism. The various experimental equipment was connected to set up the explosion test system, with the high-speed camera aimed at the sample to be blasted, adjusting the angle and focal length, and checking through the computer screen whether the adjustments were successful. Lighting was added to enhance the lighting effects for better photography, and it was necessary to turn off the lights promptly after each experiment to prevent damage to the light sources. The process of loading the explosive is shown in Figure 2. Considering the large volume of the samples, to achieve a better blasting effect, the single-hole loading quantity was kept at 20 g. An instantaneous electronic detonator was placed in each blasthole. The explosive was first loaded into the blasthole, followed by the digital electronic detonator, to ensure that the explosive would be initiated fully and effectively.

3. Experimental Results

3.1. Specimen Fracture and Break

Due to the frame rate of the high-speed camera being set at 20,000 Hz, the interval between each image is 50 μs. Key time-point images were selected for an overall analysis of the damage characteristics of the samples containing cavities under blasting loads. The damage and destruction process is shown in Figure 3. Generally speaking, the process of damage and failure of the rock mass under blasting loads is as follows: Firstly, the electronic detonator initiates the explosive charge, causing the PVC pipe in the blasthole to be propelled out by the blast wave, but no cracks were observed on the side of the sample. Subsequently, a vertical crack appeared at the position closest to the blasthole, and the crack extended further until it pierced through the entire sample, with a branching crack also forming at the middle of the side. As the damage progressed, the branching crack pierced through the sample, with the entire crack forming an “X” shape. At this point, the explosive gas began to be ejected, and the blocked sand and earth were continuously thrown out. New cracks continued to form at the top of the side and gradually expanded towards the left free surface. An additional branching crack also formed in the middle of the “X” crack, expanding towards the right. Finally, as the action of the explosive gas intensified further, the width of the cracks kept increasing, and no new cracks were produced.

Figure 4 shows the crack propagation process of the control group (i.e., with no structural components) under blasting loads. From the figure, it can be observed that with the significant displacement of the PVC pipe, obvious deformation was also seen on the upper surface of the sample. Subsequently, a noticeable crack propagation was observed, and the propagation speed was relatively fast. After the basic formation of cracks, the width of the cracks continued to increase. Throughout the explosion process, the emission of explosive gases from the blast lasted for a considerable period. When no new cracks were produced, the process of width increase began to see the efflux of explosive gases and debris. This indicates that the action time of the explosive gas was prolonged, and the energy of the explosive was fully utilized, which is conducive to the fragmentation of the sample.

The results of the aforementioned research indicate that under the intense shock wave generated by the explosion of the explosive in the rock mass, the shock and compression of the surrounding rock mass cause local pulverization. After passing through the crushed zone, the compressive stress wave continues to propagate outward, but its intensity has decreased to the point where it can no longer directly cause rock fracturing. When the compressive stress wave reaches the free surface, it reflects from the free surface as a tensile stress wave. Since the tensile strength of rock is significantly lower than its compressive strength, if the strength of the reflected tensile waves exceeds the dynamic tensile strength of the rock, the rock will begin to produce layer-by-layer tensile fracture damage from the free surface [24]. This mode of fracture is also known as “falling in pieces”. As the reflected wave propagates towards the explosion source, the falling in pieces continues until the rock within the explosion range is completely pulled apart. High-speed camera images clearly show that the free surface cracks are primarily generated in the area closest to the blasthole, as the stress waves arrive there first, producing a tensile effect and concentrating stress, which in turn leads to crack formation. Subsequently, under the action of the explosive gases, the cracks continue to propagate, and their widths keep increasing. At this point, it is clearly visible that explosion gases and debris are erupting from the blasthole.

3.2. Blasting Block Size and Effectiveness

The presence of cavities within concrete structures affects the propagation of stress waves, and the impact of these cavities varies depending on their location. For blasting experiments conducted on structures with cavities of different shapes, a corresponding size analysis is carried out. The analysis focuses on the blasting block sizes X₅₀ and X₁₀₀ at distances varying from the upper free surface. X₅₀ reflects the average size of the fragments, while X₁₀₀ represents the maximum size of the fragments, which effectively describes the proportion of large pieces generated by the explosion. Additionally, due to limitations in the experimental conditions, the spherical cavity model only yielded block size data at distances of 8 cm and 12 cm from the upper free surface. The block size distribution is shown in Figure 5.

A large amount of measured data and literature indicate that the five-parameter Swebrec modified distribution function can well describe the distribution law of rock blasting fragments [25]. Therefore, this article, after obtaining the sizes of each rock block through ImagJ (https://imagej.net/ij/) image processing, uses the Swebrec modified distribution function to obtain the size distribution curves for each group of rock blasting tests. The basic expression of this function is as follows:

$$P(x)=1/\left\{1+A\left[\ln (x_{\max }/x)/\ln (x_{\max }/x_{50})\right]B+(1-a){\left[\left({\displaystyle \frac{x_{\max }}{x}}-1\right)/\left({\displaystyle \frac{x_{\max }}{x_{50}}}-1\right)\right]}^C\right\}$$

(1)

where x_max is the maximum size at P(x_max) = 100%, which can be obtained by directly measuring the size of the largest fragment; x₅₀ is the median size at P(x₅₀) = 50%; A is the grading coefficient, with a range of values from 0 to 1; and B, C are the fitting parameters, with a range of values around 2.

The distribution patterns of blasting fragment sizes under different shapes and distances from the upper free surface are illustrated in Figure 6a–c showing good fitting effects with R² values greater than 0.99. Figure 6a reveals that when the spherical cavity is 8 cm away from the upper free surface, the X₅₀ value (average fragment size) is 10.64 cm, and the X₁₀₀ value (maximum fragment size) is 35.97 cm. When the cavity is 12 cm away from the upper free surface, the X₅₀ value is 6.84 cm, and the X₁₀₀ value is 19.25 cm. The fragment size analysis indicates that a spherical cavity closer to the free surface yields a better blasting effect, resulting in smaller overall fragment sizes and a reduced proportion of large pieces.

Similarly, for concrete structures with parallelepiped cavities at different positions, different blasting effects were observed. Figure 6b shows the blast fragments from different positions of a cube. In this case, the cavity is located in three different positions. The best blasting effect was achieved when the cavity was 12 cm away from the upper free surface, with X₅₀ = 5.98 cm and X₁₀₀ = 12.73 cm, resulting in no obvious large pieces, as shown in Figure 6b. When the cavity was 4 cm or 8 cm away from the upper free surface, the blasting effect was poorer, with values of X₅₀ and X₁₀₀ being relatively close to each other.

In summary, there is little difference in the average fragment size among the three samples. However, for the maximum fragment size, the sample with the spherical cavity 12 cm away from the upper free surface had significantly smaller values compared to the other two groups. Figure 6c displays the fitting curves of the blasting fragment sizes for concrete structures with conical cavities under explosive shock loads. For the three different positions of the conical structures, the best blasting effect was obtained when the cavity was 8 cm away from the upper free surface, with the smallest average fragment size, X₅₀ = 6.99 cm and X₁₀₀ = 14.59 cm. The worst effect was achieved when the cavity was 4 cm away from the upper free surface, with X₅₀ = 10.84 cm and X₁₀₀ = 25.54 cm. The fitting curve was relatively flat when the cavity was 12 cm away from the upper free surface, indicating a large discrepancy in blasting fragment sizes, with X₅₀ = 7.93 cm and X₁₀₀ = 22.62 cm. Overall, the conical cavities exhibited consistent blasting fragment size distribution patterns with other shapes, meaning that the best blasting effect is achieved when the conical cavity structure is located in the middle of the sample, while the worst effect is observed when the cavity is in the middle upper position. The shape of the cavity affects the propagation of explosion stress waves, and for spherical, parallelepiped, and conical cavities, the average fragment size after blasting for spherical cavities was significantly larger than for the other two shapes. The overall average sizes of the parallelepiped and conical cavities were relatively close, suggesting that these two shapes have similar effects on the blasting fragment sizes.

4. Numerical Simulation Study on the Response and Fracture Characteristics of Cavity Blasting

In this section, corresponding numerical simulation analyses are carried out for different blasting conditions in similar model tests, and the blasting model, intrinsic relationship and parameter selection of numerical simulation are calibrated by comparing numerical simulation and similar model test results to verify the accuracy of numerical simulation. On this basis, a two-dimensional mechanical response and three-dimensional dynamic damage numerical simulation are carried out to further obtain the influence of different cross-section shapes on the stress redistribution and damage process around the cavity, so as to provide a theoretical basis and practical guidance for the on-site engineering blasting and the stability of the cavity. The software selected for numerical simulation is the dynamic finite element program ANSYS/LS-DYNA R7.0.

4.1. Parameter Calibration and Cavity Stress Response

In order to obtain the material parameters, samples were taken on-site in the model test to make standard tests for mechanical property tests, and atriaxial testing machine was used to carry out uniaxial compression, Brazilian splitting and straight shear tests, and the mechanical property test data are shown in

Table 1. Experimental test data for similar materials.

Sample	Density ρ/(kg∙m−3)	Static Compressive Strength σs/MPa	Dynamic Compressive Strength σd/MPa	Tensile Strength σt/MPa	Modulus of Elasticitye/GPa	Poisson’s Ratio μ	Cohesion c/MPa	Angle of Internal Friction φ/°
1	2069.76	4.83	8.13	0.47	2.68	0.23	0.53	20.43
2	2103.71	4.66	8.48	0.45	2.59	0.24	0.51	20.29
3	1882.84	4.46	8.58	0.49	2.53	0.22	0.55	21.95
Average value	2018.77	4.65	8.4	0.47	2.60	0.23	0.53	20.89

. In order to study the mechanical response characteristics of the cavity, the rock material is set as an elastic material, the keyword *MAT_ELASTIC is used, and the parameter settings are consistent with the mean values in Table 1. The foam material is used to simulate the cavity—ANSYS/LS-DYNA software simulation foam has a variety of materials available. In this paper, the MAT_SOIL_AND_FOAM material is used and the specific parameters determined by reference [26] are shown in

Table 2. Parameter settings for simulated cavity materials.

Density ρ/(kg∙m−3)	Tensile Strength σt/MPa	Volume Modulus/MPa	a0	a1	a2
400	0.64	3	0.0034	703.3	0.3 × 109
vcr	ref	lcid	eps1	eps2	eps3
0	0	0	0	−0.104	−0.161
eps4	eps5	eps6	eps7	eps8	eps9
−0.192	−0.224	−0.246	−0.271	−0.283	−0.290
eps10	p1	p2	p3	p4	p5
−0.4	2 × 105	2 × 105	4 × 105	6 × 105	1.2 × 106
p6	p7	p8	p9	p10
2 × 106	4 × 106	6 × 106	8 × 105	41 × 105

.

The two-dimensional elastic numerical modeling is shown in Figure 7, which is consistent with the top-view cross-section size of the similar model test, and the model is a rectangular structure with the size of 60 cm × 30 cm, and the total number of model units is about 130,000 units. The explosive size is 2 cm × 2 cm, and the equivalent blasting curve is used to apply the load and generate the blasting stress wave propagating to the cavity. The shapes of the cavities are circular, parallelepiped, and conical, and the diameters and side lengths are 4 cm and 12 cm, respectively, so as to realize the numerical simulation of the process of blasting stress wave incident on cavities with different cross-sectional shapes. The needle-shaped wave is used to represent the blast stress wave, in which the waveform rises rapidly and falls slowly, with the energy concentrated primarily at the front, effectively approximating the stress wave generated by blasting. Additionally, the pressure is applied along the contour of the blasthole.

The stress variation and effective strain cloud diagrams of different cross-sectional shapes of the cavities under blasting loads are shown in Figure 8, Figure 9 and Figure 10. From Figure 9a, it can be seen that the blasting load produces a significant stress concentration at the bottom of the top of the cavity when it propagates to the circular cavity. As the stress wave propagates to the right free surface, a conical region of small effective stresses appears around the cavity due to the presence of the upper and lower free surfaces. This means that the circular cavity absorbed a large amount of energy during the incidence of the blast stress wave and played a good anti-detonation role. Subsequently, due to the free boundaries on the left and right sides, the reflected wave and the incident wave acted together on the cavity, resulting in the cavity being subjected to the stresses on both sides and an X-shaped high-stress distribution appeared around the circular cavity, which means that the cavity is prone to butterfly damage in the second half of the propagation of the blast stress wave, which is in line with the conclusions of the existing studies [28]. The effective strain cloud in Figure 9b is also consistent with the above analysis. The circular cavity first shows significant compressive strains on the wave-facing side, and with the propagation of the stress wave, compressive strains also appear on the back-blast side of the circular cavity. Finally, under the combined effect of the incident wave and the reflected wave, significant compressive strains appear in the whole cavity interior, but the strains on the periphery of the cavity are larger, which is due to the X-shape of the high-stress distribution.

Figure 10 shows the process of changes in effective stress and strain within a parallelepiped cavity subjected to blast impact. Due to the large effective impact area of the parallelepiped cavity, significant stress concentration first appeared at the two corners on the left. Subsequently, the stress wave produced significant reflection and scattering between the blasthole and the cavity, resulting in a conical region of high effective stress between the blasthole and the cavity. As the stress wave propagated to the free surfaces on both sides, the incident wave and the reflected wave simultaneously acted on the parallelepiped cavity, causing sustained dynamic stress concentration on the side facing the blast, while stress concentration also appeared at the two corners on the right side of the parallelepiped cavity. From Figure 10b, it can be observed that when the blast stress wave reached the parallelepiped cavity, a significant compression strain rapidly appeared on the side facing the blast, and as the stress wave propagated, the entire cavity exhibited compression strain, although the strain on the side facing the blast was much greater than that on the upper and lower sides and the side opposite the blast, with the strain on the opposite side being the smallest. This is because the incident wave, reflected wave, and scattered waves directly acted on the side facing the blast, where the majority of the stress wave energy was consumed, while the side opposite the blast was affected the least. Finally, when the incident wave and the reflected wave produced by the right side’s free surface acted on the cavity together, a significant compression strain also appeared on the opposite side, but it was still smaller than that on the side facing the blast. Additionally, the strain near the four sides of the parallelepiped was much greater than the strain within the cavity itself. In summary, during the plastic stage, the parallelepiped cavity is prone to develop cracks first at the four corners of the parallelepiped, which then gradually propagate to the four sides, thus initiating structural failure.

As shown in Figure 10a, the propagation of stress waves to the conical cavity resulted in significant scattering. The stress waves were bifurcated by the corners facing the blast, propagating along the edges of the conical cavity to the right. Between the blasthole and the conical cavity, a larger conical region appeared with high effective stress, and the corners facing the blast exhibited significant dynamic stress concentration. Subsequently, the reflected waves generated by the free surface interacted with the incident waves simultaneously on the conical cavity, causing significant stress concentration at the three corners of the conical cavity, with the effective stress at the corner facing the blast being greater than that at the other two corners. From Figure 10b, it can be observed that when the blast stress wave reached the cavity, a significant compression strain appeared at the corner facing the blast. As the stress wave propagated, both edges of the side facing the blast also showed a considerable compression strain. Finally, when both the incident wave and the reflected wave acted on the conical cavity simultaneously, a region of significant compression strain appeared at the two corners opposite the blast, but no such strain was observed near the edges on the opposite side. Therefore, during the plastic stage, the conical cavity is prone to damage and failure at the three corners and the two sides adjacent to the blast, with most of the energy being consumed by the rock mass of the cavity on the side facing the blast, thereby protecting the structure of the cavity on the opposite side.

The comprehensive analysis reveals that different cross-sectional shapes have a significant impact on the propagation of blast stress waves and the characteristics of stress and strain changes around the cavity. The circular cavity, due to its curvature, experiences only transient stress concentration at the top and bottom, and the effective strain throughout the process is far greater than that of the parallelepiped or conical cavities. This indicates that the circular cavity absorbs a large amount of blast energy for structural deformation of the cavity and subsequent cavity damage, thus reducing the energy used for the destruction of the surrounding rock mass. Therefore, the circular cavity serves as a better blast-resistant effect on the surrounding rock mass. Furthermore, both the parallelepiped and conical cavities exhibit significant stress concentration at the corners, with the effective strain of the conical cavity being less than that of the parallelepiped cavity due to the larger effective impact area of the parallelepiped cavity. However, because of the tip effect of the conical, the effective stress at the corner facing the wave is much greater than that of the parallelepiped.

4.2. Three-Dimensional Numerical Modeling and Damage Characteristics of Cavities

Based on the stress distribution characteristics of cavities under blasting, a three-dimensional elastic-plastic model was constructed to further study the destructive characteristics of cavities under blasting. Some scholars have employed the MAT_PLASTIC_KINEMATIC model for simulating the dynamic damage of rock due to its computational efficiency and alignment with the dynamic rock damage criteria [29,30]. The rock material is primarily defined using the material keyword MAT_PLASTIC_KINEMATIC (kinematic plasticity model material). This material’s constitutive model can reflect the elastic and plastic failure processes of soils, rocks, concrete, and other media under the action of explosive blasting. According to the mechanical testing results of cement mortar specimens from similar model tests, the specific parameters used are listed in

Table 3. Material parameter settings.

Density kg/m3	Modulus of Elasticity/GPa	Poisson’s Ratio	Yield Strength/MPa	Tensile Strength/MPa	Failure Strain
2000	2.60	0.23	4.65	0.47	0.06

.

The keyword MAT_ADD_EROSION can define various parameters of material failure, such as compression stress, tensile stress, equivalent stress, strain, and failure time. For the MAT_PLASTIC_KINEMATIC material, therefore, it is only necessary to use this keyword to define the material’s tensile strength [31]. In this simulation, any model that defines the tensile strength parameter with MAT_ADD_EROSION is set to −0.47 MPa (the tensile stress value for MAT_ADD_EROSION is labeled as a negative value). Additionally, the MAT_HIGH_EXPLOSIVE_BURN material is used to define high-energy explosives and void material. This paper uses No. 2 rock emulsion explosives with the same explosive performance as the similar model test, with a density of 1050 kg/m³ and a detonation velocity of 4200 m/s. Besides material parameters, in ANSYS/LS-DYNA, it is also necessary to set the *EOS_JWL equation of state parameters [32].

The EOS_JWL equation of state is as follows:

$$P=A[1-{\displaystyle \frac{\omega }{R_{1}V}}]e^{-R_{1}V}+B[1-{\displaystyle \frac{\omega }{R_{2}V}}]e^{-R_{2}V}+{\displaystyle \frac{\omega E}{V}}$$

(2)

where A, B, R₁, R₂, ω are parameters related to explosives determined by experiments, V₀ is relative specific volume, E₀ is initial specific energy. The specific parameters of the equation of state of the explosive are shown in

Table 4. Parameters of the explosive equation of state.

A	B	R1	R2	OMEG	E0	V0
2.144 × 1011	0.182 × 109	4.2	0.9	0.15	4.192 × 109	1.0

.

The three-dimensional numerical modeling employs the Lagrange (Lagrange) method with shared nodes, which requires not only the definition of explosive and rock materials but also the additional definition of void material. The ALE12 algorithm for *SECTION_SOLID is used to calculate the void material, which provides the necessary volume for the expansion of the explosive, thereby preventing negative volume errors during the simulation. The void material is co-nodally coupled with the rock mass without requiring additional special boundaries, ensuring that the transmission of the blasting stress wave remains unaffected. The void material (void) is defined using the same material parameters as the explosive material. The three-dimensional numerical model is shown in Figure 11. Consistent with similar model tests, the model is rectangular in structure with dimensions of 60 cm × 40 cm × 30 cm, and the total number of elements is approximately 640,000. The blue part in the figure represents the explosive elements, with a size of 4 cm × 2 cm × 2 cm, which is close to the 20 g charge mass in the similar model test. The red part represents the void material elements, which are shaped as cubes with an edge length of 20 cm. The yellow part represents the filled area; in this figure, an individual fracture model is used as an example, with a size consistent with the corresponding similar model test. The green part represents the rock elements, with a size of 60 cm × 40 cm × 30 cm, and the void material elements partially overlap with the rock elements in the modeling. Generally, when simulating the blasting of rock mass, a non-reflecting boundary is applied at the model’s boundary using the keyword BOUNDARY_NON_REFLECTING to simulate different numbers of free surfaces. The 3D model is not set with a reflection-free boundary to consistency with the experimental specimen.

Due to the presence of free surfaces, there is a limiting and nature-altering effect on the energy of the blast stress waves. This causes the stress wave energy to be confined within the expected range of blasting action through reflection, and it can alter the nature of the stress wave energy, converting the original compressive wave energy into tensile and shear wave energy. As a result, compared to the case without free surfaces, the rock mass is subjected to a longer duration of stress wave destructive action, leading to tensile and shear damage to the rock mass.

The damage development of the rock mass during a simulated typical rock blasting process is shown in Figure 12 and is specifically manifested as follows:

(1) At 0.1 ms, under the impact load of the explosive, the rock mass elements surrounding the explosive reach their failure strength and undergo massive destruction, forming a blast cavity. At 0.3 ms, the blast cavity has basically expanded to its limit, indicating that the action of the explosive impact load has essentially ended. The rock mass outside the cavity is subjected to hoop stress from the direct blast stress waves, forming radial cracks that radiate outward. At 0.7 ms, the radially spreading cracks have further developed. As the blast stress wave reaches the edge of the rock mass, due to the reflection of the stress wave at the free surface, the rock mass undergoes multi-layer fracturing under the action of the reflected wave, forming multiple cracks parallel to the free surface.

(2) At 0.9 ms, the radiating radial cracks and the parallel free surface layer cracks further expand. The reflected waves from adjacent free surfaces begin to converge, and under the superimposed stress field, the rock mass exhibits angular cracks, with the angular crack cracks beginning to develop. At 4.5 ms, in addition to the already fully developed radiating radial cracks, parallel free surface layer cracks, and angular cracks, the rock mass shows a wider variety of cracks under the action of stress waves, including edge cracks at 45° to the free surface. These various cracks are mostly interconnected, and the rock mass is overall in a state of destruction.

(3) Compared to the 4.5 ms moment, at 10 ms, the cracks have only slightly developed, and their expansion has essentially halted, indicating that the destructive effect of the rock mass blast stress waves has ceased. At this point, the cracks are primarily expanding under the quasi-static pressure of the explosive gases, and at the same time, the broken rock mass is being propelled outward by the action of the explosive gases, with the rock mass edges showing noticeable bulging. When comparing the 20 ms with the 10 ms moment, there is almost no further development of rock mass cracks, and in some cases, the edges of certain cracks have even exhibited a closure phenomenon. The main cracks have further extended, and the bulging of the rock mass at the free surface is more pronounced.

The damage situations of different cross-sectional cavity numerical models under blasting are shown in Figure 13. From Figure 13, it can be seen that the shape of the blast cavity within the rock mass model has not changed significantly with the variation in the shape of the cavity cross-section. This indicates that these conditions are set at a considerable distance, and they have little effect on the compressive and shearing destructive actions in the blasting crushing zone. At the five free surfaces near the explosive, under the action of blasting, there have been obvious tensile failure cracks, and these cracks exhibit variations in shape under different conditions.

To analyze the effects of blasting further statistically under different conditions, the information files from the simulations can be used to count the unit failure in the rock mass model. This allows us to understand the development of rock mass damage and the final damage effects during the rock blasting simulation process.

Table 5. Number and proportion of failed units for different condition models.

Numerical Model Type	Complete Model	Parallelepiped Cavity Model	Conical Cavity Model	Spherical Cavity Model
Number of failed units	70,800	70,771	70,644	62,857
Number of rock units	639,566	625,742	633,580	638,661
Percentage of failed units	11.07%	11.31%	11.15%	9.84%

shows the number of failed units and the ratio of failed units to the total number of units for all models up to 20 ms, reflecting the impact of different conditions on the rock mass damage effects. The proportion of failed units in the parallelepiped cavity is slightly greater than that of the conical cavity and much greater than that of the circular cavity. Additionally, the proportion of failed units in the intact model is less than that of the parallelepiped and conical cavities but greater than that of the circular cavity. This means that the circular cavity absorbs more blast energy, hinders rock mass damage, and reduces the degree of rock mass destruction, while the parallelepiped and conical cavities promote rock mass damage. This is because parallelepiped and conical shapes have significant dynamic stress concentration at their corners, which induces damage and fractures in the cavities. From Figure 13, it can be observed that the cracks produced by the parallelepiped and conical cavities connect with the blasting crushing zone area, and the main cracks in the parallelepiped and conical cavities occur on the blast-facing side, which is consistent with the results of the two-dimensional numerical model. However, due to the larger effective incoming area of the parallelepiped compared to the conical, the area between the blastholes and the cavities experiences significant reflection and scattering of stress waves, thereby intensifying the damage to the rock mass in that area. Moreover, the average block sizes of the spherical, conical, and parallelepiped cavities in the experimental results are consistent with the numerical simulation results. This indicates that the spherical cavity hinders the blast stress wave from damaging the rock mass, while the conical and parallelepiped cavities promote the blasting damage to the rock mass.

5. Conclusions

This study has conducted an in-depth exploration of the dynamic response and rock mass damage issues under blasting loads for different shapes of cavities. Through field tests and numerical simulations, the mechanisms of rock mass damage and destruction under blasting loads were revealed, and the following conclusions were drawn:

Firstly, the dynamic response laws of different shaped cavities under blasting loads were analyzed, and the damage mechanisms and blast block size distributions under blasting loads were studied. The influence of cavity geological bodies on the size of blasting fragments and the characteristics of rock mass damage were revealed. A combination of field tests and numerical simulations was used to verify the experimental results and further reveal the influence of different geological structures on the blasting effect. The results show that the presence of spherical, conical, and parallelepiped cavities has a significant impact on the blasting effect and rock mass damage under blasting loads. The spherical cavity, with its good anti-blast performance, can absorb a large amount of blast energy and reduce the degree of rock mass destruction, while parallelepiped and conical cavities are prone to stress concentration at the corners, promoting rock mass damage and destruction. Additionally, the study of blast block size distribution indicates that the blasting effect of the parallelepiped and conical cavities is better than that of spherical cavities. The research results of this study contribute to the enrichment of the theory of rock mass dynamic response and damage under blasting loads, providing new ideas and methods for subsequent research. These research findings serve as an important theoretical basis and practical guidance for the design and construction of underground engineering blasting, helping to improve engineering safety and promote the sustainable development of the underground engineering industry.