Numerical Analysis of the Stability of Underground Granite Chamber Under the Combined Effect of Penetration and Explosion

Abstract

In recent years, the majority of countries have focused on the development of earth-penetrating weapons and the construction of deep underground fortifications. It is therefore necessary to assess the damage to underground structures under the attacks of earth-penetrating weapons. In this paper, fluid–solid coupling and restarting methods are used to simulate the damage processes of a granite chamber subjected to the combined action of penetration and explosion with the commercial software of LS-DYNA R11. The applicability of the penetration model and the blasting model is verified by the previous penetration and blasting tests. The verified models are used to simulate the complete process of the underground granite chamber attacked by the earth-penetrating weapons. Based on peak particle velocity (PPV) damage criteria, the numerical results show that the hypervelocity impact of the earth penetrating weapons only causes local damage to the granite rock mass, and more serious damage is caused by the subsequent explosion. During the subsequent explosion, part of the detonation products and energy can escape along the penetration trajectory with the blast loading, resulting in the attenuation of blast stress waves. Subsequently, the relationship between the overlay thickness and the vibration attenuation in granite is also studied, which provides a fast method to determine the threshold damage level for an underground chamber to collapse under the attacks of earth-penetrating weapons.

1. Introduction

Following the Second World War, earth penetrating weapons (EPWs) and underground fortifications have evolved as adversarial counterparts, continuously advancing under the mutual threat posed by each other [1]. Nations are enhancing the penetration and explosion synergistic lethality in bunker busters while simultaneously harnessing natural rock formations to construct strategic strongholds deep beneath the surface—buried tens to hundreds of meters underground [2]. Since the Gulf War in 1991, investigations into the projectile–target interaction processes have emerged as a research frontier for civil engineers and ballisticians in realistic military scenarios. For ballisticians, it is a lethality demonstration of the bunker-busting bomb. While to civil engineers, the dynamic response of the target provides a basis for the stability assessment and anti-vibration design of underground structures in rock mass [3,4,5]. Generally, three ways are adopted to address the problems, namely empirical equations [6,7,8], cavity expansion analysis [9,10,11], and numerical simulations [12,13,14,15].

In practice, the projectile–target interaction is mainly studied through the discussions of penetration depth and target damage. In the past, when computing power was modest and not easily accessible, the penetration depth was usually evaluated by using empirical equations and cavity expansion theory combined with field tests. For example, to predict the maximum depth of penetration, Forrestal et al. [16] developed an empirical formula by conducting a series of tests of ogive-nosed projectiles penetrating concrete targets at normal incidence. This empirical equation predicts the penetration depth as a function of a dimensionless empirical constant S, which depends upon the unconfined compressive strength of concrete targets and is determined from the penetration depth versus striking velocity data. Then, taken as a basis, Frew et al. [17] extended it for limestone targets, and very encouraging predictions were produced through comparing the penetration results in experiments. However, although penetration depth is the most direct reflection of penetration capability, these and most similar empirical formulae were usually obtained by different researchers based on their collected experimental data, and the field tests are not only time- and cost-consuming, but also short of insights into the phenomenon and mechanism of the penetration process. In a theoretical investigation for penetration depth, Bishop et al. [18] proposed the quasi-static theoretical models for expansion of cylindrical and spherical cavities to calculate the pressure acting on the conical heads when they slowly punched into metal targets. Furthermore, Chadwick [19] and Hopkins [20] developed the dynamic cavity-expansion models of metal targets, and by combining these models and carrying out large amounts of experiments, Young [21] established a number of classical semi-empirical equations to determine the effects of different conditions on penetration depth, such as projectile nose shape, projectile mass and area, and impact velocity, etc. These theoretical models and semi-empirical equations, even though present some explanation for the interaction mechanism between projectile and target, they were always derived with several critical assumptions and simplifications, and some vague definitions of influencing factors often caused uncertainties in applications and some parameters closely related to penetration depth were even purely determined by fitting of the test data [22,23,24,25].

As for the target damage, it is generally divided into the penetration-induced damage and the subsequent explosion damage to the target. Like the penetration depth, considerable experimental efforts have been undertaken to directly observe and assess these two kinds of damages, but initially, the most accessible literature only concentrates on the damage induced by penetration. For instance, considering the situation of normal impact, the process of an explosively formed projectile penetrating a concrete target was recorded by Hu et al. using a high-speed camera [26], and the impact damage to the target was demonstrated through the physical phenomena of front spalling, rear scabbing crater, and penetration borehole. By employing the same method of high-speed image capturing, Chen et al. [27] investigated the penetration trajectories when projectiles impact targets with oblique angles and yaws, and defined the damage modes of the target as perforation, embedment, and ricochet. In the previous literature, not many experimental works on the explosion damage immediately following penetration were available and only a few papers [28,29,30,31] analyzed the explosion damage through pre-burying explosive at penetration depths in targets, rather than the continuous process of penetration and explosion, although the subsequent explosion damage is crucial for estimating the stability of underground structures. Since the initial damage induced by penetration has a significant influence on the damage evolution of the target compared to the purely internal explosion, it is more practical to consider the combined effect of penetration and explosion. In recent years, some researchers have gradually begun to conduct experimental studies on the composite damage effects of an explosion after penetration [32,33,34]. However, these studies are based on simplified models using small projectiles, targets, or explosives, which limit their practical application in civil defense and military engineering. As for tests of penetration and explosion with full-size EPWs, due to prohibitive costs, existing investigations are very limited.

With the proliferation of faster and cost-effective computing power, hydrocodes have gradually emerged as reasonable alternatives in predicting and revealing the complex interaction of a projectile with a target, while properly conducted experiments remain the bedrock of a logical design strategy. Building upon the laboratory-scale penetration experiments, Tham [3] simulated the penetration process in AUTODYN-2D using three available computational algorithms, namely Lagrange, Euler–Lagrange coupling, and smooth particle hydrodynamics SPH–Lagrange coupling, to predict the maximum penetration depth and radial stress-time response of the concrete target. Propelled by relentless advances in numerical algorithms and models, the numerical results are consistent with the prediction by using an empirical equation and yield good agreement with that obtained from the experiment, which increases the confidence of civil engineers when using a numerical approach in complementing full-process analysis subjected to penetration and explosion. With this in mind, some attempts have been made to numerically display the integral process of a projectile penetration and following explosion by means of building projectile and charge models and developing different algorithms. For accurately describing the combined effects of penetration and explosion, Sun et al. [32] used numerical simulation methods to calculate the depth and the radius of the final crater in plain concrete targets. Then, a numerical investigation about the composite damage actions under penetration and explosion of EPWs was conducted by Shu et al. to compare the destruction levels and dynamic responses of the dam subjected to an internal blasting with and without consideration of the initial penetration damage [33]. On the design of protective structures anti full-scale EPWs, Cheng et al. [34] modelled a full-size 370 mm-caliber EPW and ultra-high-performance concrete (UHPC) targets to assess the anti-strike resistance quantitatively. However, the current studies are focused on conventional concrete or UHPC targets, while the research on rock targets is limited. Moreover, one may note that the numerical simulations above mainly emphasized the damage and failure in targets [35,36], while propagation of blast-induced stress waves in rock mass, which is critical for the design of underground protective structures, is ignored. In order to qualitatively and quantitatively analyze the damage of underground structures inside rock formations subjected to blast waves generated by EPWs, employing the peak particle velocity criterion, which was utilized by Wu et al. [37] to predict rock damage during internal explosion, could be a plausible scheme.

In the paper, the process of the GBU-28 bunker buster carrying BLU-113 warhead impacting into underground granite chambers at zero angle of attack was simulated by the commercial software of LS-SYNA R11 to study the chamber response and evaluate the granite damage with PPV criterion. The mechanical parameters of granite in simulations were the same as those in sub-scale experiments. The EPW and chamber were modelled by Hypermesh 14.0, and model parameters were validated through the consistent results of sub-scale experiments and simulations. Furthermore, the effects of initial penetration damage and protective layer thickness were analyzed and discussed. The above studies provide a method to calculate the minimum thickness of the protective layer and serve as a reference for the structural design of civil defense engineering built in hard rock. More significantly, this study enhances the defensive capabilities of underground facilities against the bunker busters, thereby increasing conflict costs and reducing military adventurism, aligning with the United Nations Sustainable Development Goals.

2. Numerical Model and Calibration

2.1. Penetration and Explosion Tests

To perform an accurate numerical simulation for evaluating the combined effect of penetration and explosion on the stability of an underground structure, the validity of the numerical model should first be calibrated by comparing its numerical results with the experimental data before conducting a specific numerical investigation. In a large number of previous experimental studies, two penetrating and blasting tests carried out in granite are chosen for the current model examination.

2.1.1. Penetration Test

An experimental study on the penetration effect of oval-shaped long-bar high-strength steel projectiles on granite was carried out by Li et al. [38] using a two-stage light gas gun (Army Engineering University of PLA, Nanjing, China). The body of a 3.0 caliber-radius-head steel rod projectile is shown in Figure 1a. The projectile length L and the rod diameter of the projectile are 54 mm and 10.8 mm, respectively. The initial mass and density of the projectile made of high-strength alloy steel 30CrMnSiNi2A are 32.45 g and 7850 kg/m³. The granite rock used in the experiment was taken from Wulian County, Shandong Province, China, and the main physical and mechanical parameters of the granite rock measured before the experiment are listed in

Table 1. Main physical and mechanical parameters of granite rock.

Rock Type	Density (kg/m3)	Uniaxial Compressive Strength (MPa)	Young’s Modulus (GPa)	P-Wave Velocity (m/s)	Shear Modulus (GPa)	Poisson’s Ratio
Wulian granite [38]	2670	150	54.6	4200	21	0.21
Barre granite [39,40]	2660	155.7~164.8	35.1~48.2	4040~4530	21.9	0.15~0.23

. For a target sample shown in Figure 1b, a granite block with dimensions of 300 mm × 300 mm × 800 mm was placed in the middle of a cylindrical steel sleeve whose wall thickness and outer diameter are 10 mm and 480 mm, respectively. The gap between the granite block and the steel sleeve was filled with C40 concrete, and the sample was cured for more than 28 days to strengthen the constraint on the granite. The bottom of the target was sealed with a 10 mm-thick steel plate.

During the experiment, the projectile was accelerated by the two-stage light gas gun and then separated from the sabot. The launching velocity was measured by using a laser beam interruption system with an error of less than 1% and the velocity ranges from 1100 m/s to 1800 m/s. The projectile penetrated the target sample at normal incidence, and the impact point was located at the center of the rectangular granite surface. After the tests, the macroscopic impacting damage, ballistics, and penetration depths were first visually observed and then measured through splitting the target, and the projectile mass after penetration was also weighed. Part of the experimental data (velocity range of 1196–1600 m/s) was listed in

Table 2. Testing results of penetration.

Test No.	Lunch Velocity (m/s)	Penetration Depth (mm)	Residual Mass of Projectile (g)
1	1196	118.80	31.64
2	1426	146.02	31.41
3	1430	155.79	31.31
4	1600	163.89	30.83

.

2.1.2. Explosion Test

Numerical calibration of the model in the explosion is based upon the laboratory-scale blasting experiments conducted by Banadaki [39,40]. In his experiments, a cylindrical sample (144 mm diameter and 150 mm height) of Barre granite (the Barre Granite Quarry District, Washington County, VT, USA) with a central borehole of 6.45 mm diameter drilled along the sample length was used as shown in Figure 2. In each sample, a copper tube with a wall thickness of 0.6 mm was tightly installed in the drilled borehole, and a detonating cord (DYNO cord) with a penerythritol tetranitrate (PETN) explosive core load of 3 g/m was placed at the center of the copper tube in a straight way. In such a way, the adopted explosive strength ensured that the fracture networks could only be generated within the samples rather than fragmenting them. For the detonating cord, the inner PETN strand was covered by a polyethylene sheath. The diameter of the PETN strand and the outer diameter (OD) of the DYNO cord were 1.65 and 4.5 mm, respectively. Air was taken as a coupling medium, filling the gap between the outer surface of the detonating cord and the inner wall of the copper tube. The combination of detonating cord, air, copper tube, and borehole diameter (equal to the OD of the copper tube) for each sample prepared for the experiments is shown in Figure 2. After running the tests, the blasted samples were sliced to investigate the developed fracture networks. As is shown in Figure 3, 4 mm thick sample slices were cut at 23, 73 and 123 mm from bottom surface of the exploded samples by using a diamond saw to be used for impregnating, photographing, and mapping the crack pattern induced by shock/stress waves, and three rock slices were named Top, Mid and Bot, respectively. More detailed information on the experiment process can be found in [34,35].

2.2. Numerical Models

2.2.1. Penetration Models

According to previous penetration tests and considering the symmetry of the projectile and granite target, 1/4 models were established by using 3D solid elements as shown in Figure 4. In the experiments, the granite rock sample was regarded as a semi-infinite target because its diameter was approximately 30 times the diameter of the projectile sample. Therefore, the constraint of a concrete and steel sleeve around the granite target could be replaced by a non-reflective boundary for simplicity.

The calculations are performed using LS-DYNA R11, and the unit of system is mm-ms-GPa. There are several constitutive models widely used for simulating the impact-induced dynamic response of the projectile and rock target. In the present study, the material model of the projectile was MAT_PLASTIC_ KINEMATIC, which is a bilinear model to describe the isotropic and kinematic hardening plasticity of the material, considering the option of including strain rate effects as shown in Figure 5a. And strain rate is accounted for using the Cowper and Symonds model, which scales the yield stress with the factor f:

$$f=1+{({\displaystyle \frac{\dot{\epsilon }}{C}})}^{{\displaystyle \frac{1}{P}}},$$

(1)

where $\dot{\epsilon }$ is the strain rate, C and P are the strain rate parameters. This material model can be used to simulate the metal behaviors, and the detailed parameters are listed in

Table 3. Projectile material parameters.

Parameter	Value	Parameter	Value
R0 (kg/m3)	7850	E (GPa)	211
PR	0.3	SIGY (GPa)	0.83
ETAN (GPa)	6.1	BETA	1
C	0.219	P	3.3
FS	1.5	VP	0

, where R0, E, PR, SIGY, ETAN and BETA are mass density, Young’s modulus, Poisson’s ratio, yield stress, tangent modulus, and hardening parameter, respectively; FS is effective plastic strain for eroding element; VP is formula for rate effects.

The material model of granite target was MAT_RHT, which was first proposed by Riedel, Hiermaier, and Thoma from the Ernst–Mach Institute (EMI) in Germany in 1998 [41]. After more than 20 years of development, it has been widely chosen to simulate the rock dynamic behavior under penetrating and blasting impact. In this constitutive model, the equations of elastic limit surface, failure surface, and residual surface were all considered to study the change law of initial yield strength, failure strength, and residual strength of rock under dynamic loads. As shown in Figure 5b, three stages of the rock stress–strain curve, which were the elastic stage, the linear strengthening stage, and the damage softening stage, can also be described accurately in the RHT model. Based on the basic physical and mechanical parameters of Wulian granite in Table 1, the initial RHT material parameters of granite rock are determined through the methods of theoretical derivation and citing the literature. These initial parameters were modified by comparing the numerical results with the experimental results, and then the appropriate parameters were obtained, as presented in

Table 4. RHT material parameters for rock.

Parameter	Value	Parameter	Value
Mass density (kg/m3)	2670.0	Tensile strain rate dependence exponent	0.012
Elastic shear modulus (GPa)	21.0	Pressure influence on plastic flow in tension	0.001
Eroding plastic strain	2.0	Compressive yield surface parameter	0.40
Parameter for polynomial EOS B0	1.68	Tensile yield surface parameter	0.70
Parameter for polynomial EOS B1	1.68	Shear modulus reduction factor	0.48
Parameter for polynomial EOS T1 (GPa)	47.1	Damage parameter D1	0.042
Failure surface parameter A	1.60	Damage parameter D2	1.0
Failure surface parameter N	0.56	Minimum damaged residual strain	0.012
Compressive strength (MPa)	150.0	Residual surface parameter AF	1.60
Relative shear strength	0.38	Residual surface parameter NF	0.60
Relative tensile strength	0.10	Gruneisen gamma	0.0
Lode angle dependence factor Q0	0.64	Hugoniot polynomial coefficient A1 (GPa)	47.10
Lode angle dependence factor B	0.05	Hugoniot polynomial coefficient A2 (GPa)	79.13
Parameter for polynomial EOS T2 (GPa)	0.0	Hugoniot polynomial coefficient A3 (GPa)	48.36
Reference compressive strain rate	3.0 × 10−5	Crush pressure (MPa)	50.0
Reference tensile strain rate	3.0 × 10−6	Compaction pressure (GPa)	6.0
Break compressive strain rate	3.0 × 1025	Porosity exponent	4.0
Break tensile strain rate	3.0 × 1025	Initial porosity	1.01
Compressive strain rate dependence exponent	0.0085

.

Considering the effect of mesh size on numerical results, the ratio of projectile radius to target mesh size was defined as a dimensionless scale parameter λ, and mesh size convergence tests were carried out with six kind of element sizes in

Table 5. Mesh size convergence tests.

No.	Mesh Size	Penetration Depth (mm)	No.	Mesh Size	Penetration Depth (mm)
1	1:1	50.10	4	1:4	115.04
2	1:2	94.33	5	1:5	117.29
3	1:3	100.79	6	1:6	117.30

, which were λ = 1, 2, 3, 4, 5, and 6, respectively. Figure 6 shows the penetrating depth–time curves corresponding to different proportional parameters λ when the impact velocity of the projectile was 1196 m/s. Obviously, with the decrease of target mesh size, the penetration depth tends to converge gradually. When λ = 4, 5, and 6, numerical calculation results of penetration depths were converging gradually, which were consistent with the experimental results with small fluctuation and met the accuracy requirements. Therefore, to reduce resource occupancy and improve computing efficiency, the dimensionless parameter λ = 4, that is, the target mesh size is 1.35 mm, is applied to the subsequent numerical simulations.

2.2.2. Explosion Models

Based on Banadaki’s blasting tests, a cylindrical sample including Barre granite, polyethylene, explosive, and air was modelled with Hypermesh 14.0 software as shown in Figure 7. The MAT_RHT material model and the same material parameters in Table 4 are also chosen to simulate the Barre granite behaviors, because its basic physical and mechanical parameters, such as density, Poisson’s ratio, elasticity modulus, and so on, are highly consistent with those of Wulian granite in Table 1. The material model of PETN explosive is Mat_High_Explosive_Burn, and EOS_JWL (Jones–Wilkens–Lee equation of state) was selected to describe the external expansion process of detonation products, where the explosion pressure is defined as the following formula [42]:

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

(2)

where P is hydrostatic pressure; V is the specific volume; E is specific internal energy; A, B, R₁, R₂, ω are material constants. The terms A and B are the pressure coefficients, R₁ and R₂ are the principal and secondary eigenvalues, respectively. ω is the fractional part of the normal Tait equation adiabatic exponent. As a conventional explosive, the following values of PETN material parameters were chosen: A = 5.86 × 10² GPa, B = 21.6 GPa, R₁ = 5.81, R₂ = 1.77, ω = 0.282, E₀ = 7.38 GPa. The detonation velocity and the density of the explosive are 6690 m/s and 1320 kg/m³, respectively.

As for air and polyethylene, the MAT_NULL (MAT_9) material model from LS-DYNA R11 library is used to describe their mechanical properties, and the relationship between pressure, density, and internal energy is defined in the following form:

P = C₀ + C₁μ₁ + C₂μ₂ + C₃μ₃ + (C₄ + C₅ + C₆μ₆),

(3)

where P is the pressure, E₀ is the internal energy, μ defines the compression of air by μ = (ρ/ρ₀) − 1 with ρ and ρ₀ being the current and initial density of air, respectively. Since polyethylene and air are only regarded as a coupling medium with negligible strength during blasting process, the same material parameters are set in both polyethylene and air, and the specific parameters are as follows: C₀ = C₁ = C₂ = C₃ = C₆ = 0, C₄ = C₅ = γ − 1 with γ being the ratio of specific heats, γ = 1.4, ρ₀ = 1.29 kg/m³ and E₀ = 0.25 × 10⁶ J/m³.

As for copper, the Mat_Johnson_Cook, and Eos_Gruneisen (Mie–Gruneisen equation of state) are chosen to simulate its dynamic response, where the flow stress and pressure for compressed material are expressed as:

$${\sigma }_y=\left(A+B{\overline{\epsilon }}^{p^n}\right)\left(1+C\ln {\dot{\epsilon }}^{\ast }\right)\left(1-T^{\ast m}\right),$$

(4)

$$P={\displaystyle \frac{{\rho }_0{C}^2\mu \left[1+\left(1-\frac{{\gamma }_0}{2}\right)\mu -\frac{\alpha }{2}{\mu }^2\right]}{{\left[1-\left(S_1-1\right)\mu -S_2\frac{{\mu }^2}{\mu +1}-S_3\frac{{\mu }^3}{{(\mu +1)}^2}\right]}^2}}+({\gamma }_0+\alpha \mu )E,$$

(5)

where A, B, C, n, and m are material constants; ${\overline{\epsilon }}^p$, ${\dot{\epsilon }}^{\ast }$, and T* are effective plastic strain, normalized effective strain rate, and homologous temperature; C is the intercept of v_s − v_p (cubic shock velocity–particle velocity) curve; S₁, S₂, and S₃ are the unitless coefficients of the slope of the v_s − v_p curve; γ₀ is the unitless Gruneisen gamma; a is the unitless, first order volume correction to γ₀; μ defines the compression of copper.

2.3. Numerical Calibration

2.3.1. Calibration of Penetration Models

Three penetration tests, as, respectively, No. 1, No. 2, and No. 4 in Table 3, were selected for verification of penetration models with the mesh scale parameter λ = 4, and the final penetration depth–time curves obtained by numerical simulations are shown in Figure 8. When projectile velocities are 1196 m/s, 1426 m/s, and 1630 m/s, the corresponding penetration depths are 115.04 mm, 144.53 mm, and 168.96 mm, respectively, which were close to the experimental data. However, with the increase of penetrating velocity, the gap of penetration depth between numerical calculations and laboratory tests is gradually getting wider. Probably the main reason for this phenomenon is that projectile mass loss increases continuously as the impacting speeds increase in these experiments, while in the simulations, macro-wear of projectiles is not large relatively.

Figure 9 shows a comparison of experimental and simulated target damage with the projectile impacting velocity of 1426 m/s. And in the simulations, the target damage is illustrated by using a cloud map, where 0 means the target has no damage and 1 means the target is destroyed completely. From the comparison between experimental and numerical results, it can be found that the penetration depth and target damage obtained by numerical calculations are in good agreement with the experimental results. Therefore, the computational models selected in this study are suitable for the related research about projectile penetrating into a granite target at high velocity, especially in predicting penetration depth and target damage.

2.3.2. Calibration of Explosion Models

Figure 10 shows the crack pattern of granite sample blasting in experiments and simulations, and the explosion cracks can be roughly divided into three regions, named central crushing zone, tensile fracture zone, and annular crack zone, respectively. When the blast-induced detonation waves in the test sample propagated through the coupling medium and borehole wall, the crushed zone was instantly generated by compression and shear in the vicinity of the borehole. As the detonation waves continue to transmit outward, the detonation waves gradually attenuate to stress waves, and when the stress is lower than the dynamic compressive strength of the granite sample, compression–shear failure stops. Then the radial cracks started to propagate and form the tensile fracture zone, because the tensile stress component exceeded the dynamic tensile strength of the rock. When the stress waves arrive at the free surface, the compressive stress waves reflect and change into tensile stress waves. Circumferential spalling cracks, therefore, appear in the annular zone near the free boundary due to the tensile stress waves in the radial direction. According to the comparison of crack evolution in test and simulation, the crack patterns in the three zones are basically the same in terms of the scales and numbers of main cracks. As the location height of the sample slices reduced, the number of micro-cracks increased gradually due to the constant reflection and superposition of stress waves at the bottom of the granite samples. Based on the comparative analysis of experimental and simulated results, the numerical models established in this study can well predict the propagation of stress waves and simulate the damage evolution process of granite rock under blasting loads.

3. Numerical Investigation

A typical underground granite chamber is employed as a numerical application to illustrate the damage evolution of structures under the combination of penetration and explosion. The stability of the chamber is quantitatively analyzed by the PPV damage criterion, and the specific dates are shown in

Table 6. Critical vibrating velocity of wall rock.

Rock Type	Compressive Strength (MPa)	No Damage (m/s)	Slight Damage (m/s)	Intermediate Damage (m/s)	Serious Damage (m/s)
Hard rock	110–180	0.31	0.62	0.96	1.78
	180–200	0.36	0.72	1.11	2.09

[43]. The threshold damage of the chamber wall that corresponds to the incipient damage level occurs if PPV exceeds 0.96 m/s. As shown in Figure 11a, the target model is composed of homogeneous granite with a length and width of 30 m. At a depth of 20 m underground, there is a chamber with a size of 10 m × 10 m × 5 m, surrounded by a 5 m reinforced rock wall. Three target points were set on the chamber roof, respectively in the center and two corners of the roof, to record the PPV values of granite rock during the penetration and explosion. And in Figure 11b, the simplified model of the GBU-28 bunker buster carrying BLU-113 warhead has an ogive nose of Caliber–Radius–Head (CRH) 3.0, density 7830 kg/m³, mass 2130 kg, length 3.2 m, diameter 0.37 m [44]. The EPW carrying PETN explosive with 300 kg TNT equivalent attacked vertically the granite chamber at a velocity of 1196 m/s.

The 1/4 models are established to improve the calculation efficiency in the numerical calculations, based on the axisymmetric property of the EPW and granite target. Besides, the non-reflective boundary is set around the target model to simulate the semi-infinite target. The Lagrange algorithm is used to simulate the vertical penetration process of the ordnance penetrator impacting into the granite target at a velocity of 1196 m/s. The material models of the projectile and granite rock are MAT_PLASTIC_KINEMATIC and MAT_RHT of the LS-DYNA R11 database, respectively. When the penetration process is completed, the EPW model is deleted, and the PETN explosive model is set up, where the initiation point is set at the bottom of the model. Then the mechanical information of the granite element is inherited from the previous penetration process by using the restart method, and the Arbitrary Lagrangian–Eulerian (ALE) algorithm is chosen to simulate the subsequent blasting process. In this way, the damaging effect of the granite chamber under the combined action of penetration and explosion can be studied.

3.1. Analysis of Penetration Process

Figure 12 shows the process of the hypervelocity penetration into the underground granite chamber. At 2 ms, the warhead begins to make contact with the rock surface, and at 14 ms, the missile speed gradually drops to 0 m/s, the rock damage no longer changes significantly, and the penetration depth is approximately 6.67 m. Under the high-speed impact, the rock mass is in a strongly compressed state, and the radial and tangential stress of the rock around the projectile body easily exceeds the ultimate strength of the rock. The surrounding rock is in a state of strong compression, which leads to the generation of the local damage area, and the rock damage range gradually expands with the penetration of the projectile, and the final damage diameter is about 1.94 m. When the projectile penetrates the rock mass, the granite begins to crack due to a strong squeezing effect, and a rock fracture nucleus is formed below the projectile. As the penetration continues, the volume of the rock fracture nucleus is dilated by the accumulation of energy in the fracture nucleus, and new cracks are constantly generated around the fracture nucleus. When the main cracks reach the ground, the accumulation of energy in the fracture nucleus is released, and the broken rocks are squeezed out, forming the taper crater with a diameter of 5.62 m and a depth of 0.92 m on the rock mass surface. There are radial tensile cracks around the crater, and the longest crack length is about 7.57 m. Figure 13 shows the velocity–time curves at the three points of the cavern roof, in which the peak particle velocity at the center of the roof is the largest, up to 0.145 m/s.

The numerical results show that, when the impact velocity is 1196 m/s, the missile can cause local damage to the granite rock mass. The penetration depth is 6.67 m, and the diameter of the rock mass damage area is 1.94 m. But the chamber, located 20 m underground, is almost unaffected, indicating that the granite can be used as a good protective engineering material.

3.2. Combined Effect of Penetration and Explosion

Following the penetration, the projectile model is deleted, and the PETN explosive and air models, which were Mat_High_Explosive_Burn and Mat_Null of the LS-DYNA R11 database, respectively, were built at the location of the projectile. The numerical calculation time of the explosion is 6 ms, and the fluid–solid coupling method was adopted for the explosion simulation.

When the explosive is detonated, the surrounding rock mass is crushed under the actions of blast waves, forming a crushing damage zone. During the shock waves through the crushed zone, most of the explosive energy is used for crushing and compression of the surrounding rock. Therefore, the energy acting on the unit area of the rock mass is reduced, and the shock waves attenuate into compression stress waves. Under the action of stress waves, compressive strain is generated in the radial direction and tensile strain is generated in the circumferential direction. As the stress waves propagate to the free surface of the rock mass, tensile waves are formed by the reflection of compression waves. Due to the high compressive strength and low tensile strength of granite, radial tensile cracks, leading to tensile failure of the rock mass, will be formed around the crushing area and near the free surface when the tensile stress on the wavefront exceeds the peak tensile strength of granite rock.

The complete process of the explosion is shown in Figure 14. The local damage area quickly expands because of the impact of the explosion shock wave and detonation products on the damaged wall rock directly. And the detonation products overflow along the penetration trajectory owing to high pressure, resulting in the damaged area expanding along the channel. In the end, the diameter of the damaged area was approximately 6.04 m, which was larger than the damage caused by the previous penetration. Because the tensile stress component of the blast wave is larger than the dynamic tensile strength of granite rock, radial tensile cracks are formed around the damage zone. At the same time, the explosion of EPW led to the further expansion and fragmentation of the crater near the surface, or even collapse, and the longest radial crack will further expand to 10.18 m.

As shown in Figure 15, the peak particle velocities at the three observation points were 0.85 m/s, 0.51 m/s, and 0.39 m/s, respectively, indicating that there was damage in the center of the chamber roof and no obvious damage around the chamber. From the final failure pattern of the rock mass, it can be concluded that the granite overlay with a thickness of 20 m is effective in protecting shelters from the attack of small and medium high-speed earth penetrator weapons.

In the interest of investigating the effects of the initial penetration damage on the failure mechanism of the rock subjected to internal explosion, the same charge is assumed to be buried in the same position in the rock mass. And then, the numerical simulation of the internal explosion process is carried out without inheriting the previous stress and strain of penetration.

As shown in Figure 16, when the explosive explodes without initial penetration, the strong shock wave and detonation products directly act on the intact rock mass, and the rock medium is destroyed, forming a damage zone with a diameter of about 8 m. As the stress waves propagate to the free surface of the rock mass, tensile waves are formed by the reflection of compression waves. When the tensile stress on the wavefront exceeds the peak tensile strength of granite rock, tensile fractures are caused in the near-surface rocks, forming radial cracks on the surface, with the longest crack measuring approximately 9.21 m. Compared with the previous combined damage, the damage area is larger and mainly concentrated around the blast hole during the process of independent explosion. The initial penetration damage has a significant influence on the damage processes of the rock mass subjected to internal blast loading. Moreover, Figure 17 shows the comparison of the PPV values of the three observation points in the two simulations, where the three PPV values in the internal explosion without the penetration, which are 1.48 m/s, 0.93 m/s, and 0.71 m/s, are relatively higher. The reason for this phenomenon is that the penetration trajectory formed during the penetration process, so that part of the detonation products and energy can escape along this channel under high pressure. However, during the internal explosion without the initial penetration damage, there is no path for the explosive products to overflow. Therefore, when predicting the damage effects of the EPW on underground structures, it is necessary to consider the effect of detonation products overflow and investigate the combination of penetration and explosion.

3.3. Effect of Overlay Thickness

For the purpose of developing the relationship between the stability of underground chambers and the thickness of granite overlays, five underground chamber models are established with the overlay thickness of 12, 15, 18, 20 and 23 m, respectively, under the specific condition that the missile strike speed is 1196 m/s, and the TNT equivalent of the ordnance penetrator is 300 kg. According to the aforementioned thought, numerical simulations of penetration and explosion are carried out using LS-DYNA R11. And the vertical particle vibration velocities at the centre of the chamber’s roof are plotted in Figure 18, where it can be observed that the damage to the chambers becomes more serious with the decrease of the granite overlays. This is because as the decrease of the thickness of the protective layer decreases, the blasting vibration attenuation in the rock mass gradually weakens, leading to the destruction of the underground chamber. In blasting engineering, the attenuation law of blasting vibration is mainly represented by the relationship among particle vibration velocity, charge, and standoff distance [45]. And this attenuation law can be quantitatively described by Sadov’s Vibration Formula:

$$PPV=K(Q^{{\displaystyle \frac{1}{3}}}/R{)}^{\alpha }=K{\rho }^{\alpha }$$

(6)

where PPV is the peak particle velocity, Q is the explosive charge, R is the standoff distance, K and α are, respectively, the coefficient associated with geology and topography, and the attenuation index, ρ is the proportion of explosive charge.

Based on Sadov’s Vibration Formula and the simulated data in

Table 7. The numerical results at different overlay thickness.

No.	H (Overlays Thickness) (m)	PPV (m/s)	Q (kg)	R (m)
1	12	3.52	300	7
2	15	1.26	300	10
3	18	1.07	300	13
4	20	0.85	300	15
5	23	0.74	300	18

, the relationship between the stability of underground caverns and the overlay thickness is quantitatively analysed. As shown in Figure 19, the logarithm of PPV and ρ is taken to fit the simulated data linearly on the basis of the least squares method, and then the fitting formula is verified with the data of No. 4 in Table 7. Thus, the fitting formula of blasting vibration attenuation is calculated under numerical experimental conditions:

lnPPV = 1.5875lnρ + 1.147; R₂ = 0.9054; K = 3.149; α = 1.5875

(7)

$$PPV=3.149({\displaystyle \frac{\sqrt[3]{300}}{H-5}}{)}^{1.5875}=1.67{(\mathrm{H}-5)}^{-1.5875}$$

(8)

According to Formula (8), when PPV is 0.96 m/s, the critical standoff distance is approximately 14 m, and the granite overlay’s thickness is 19 m. As demonstrated in

Table 8. Critical protection thickness corresponding to different explosive charges.

No.	Explosive Charge (kg)	Critical Protection Thickness (m)	PPV (m/s)	Penetration Depth (m)
1	300	19	0.96	6.67
2	600	23	0.96	6.67
3	900	25.5	0.96	6.67
4	1200	27.6	0.96	6.67
5	1400	28.8	0.96	6.67

, when the bunker buster strike speed is 1196 m/s, the critical protective layer thickness corresponding to varying explosive charges can be rapidly calculated. Similarly, adjusting strike velocities requires only computation of penetration depth and parameter substitution in Formula (8) to determine the relationships between the explosive charge and the critical protection thickness. In this way, numerical simulations and PPV damage criteria can be used to quickly predict the critical protective layer thickness of underground structures under the attacks of different earth-penetrating weapons.

4. Discussion

This study quantifies the stability response of underground granite chambers under the combined effect of penetration and explosion. Our numerical results confirm that the initial penetration damage leads to significant attenuation of the blast stress waves in the surrounding rock mass. This attenuation mechanism primarily stems from two factors: the formation of a penetration channel that allows venting of partial detonation products and energy along the trajectory, and the damaged rock surrounding the penetration path acting as a buffer zone that accelerates the blasting stress waves. While this phenomenon aligns with observations in concrete and UHPC targets [1,32,33,34], our research specifically addresses the knowledge gap regarding rock media.

One may note that the previous numerical analysis [35] is mainly focused on the fracture and failure in targets, while propagation of stress waves caused by an explosion in the rock, which is critical for the design of underground protective structures, is ignored. Building upon these findings, we developed an innovative methodology employing the PPV criterion to predict the damage and calculate critical burial depths for underground fortifications against EPW attacks. For instance, given a missile strike velocity of 1196 m/s with 300 kg TNT equivalent, our model calculates a 19 m critical protection thickness, providing actionable engineering parameters for protective design.

Several limitations require acknowledgment. The RHT model, though effective for concrete dynamics, may inaccurately capture the tensile strength and fracturing behavior of rock mass due to fundamental material differences. Furthermore, the establishment and validation of numerical models relied exclusively on laboratory-scale experiments, and the models are continuous and homogeneous, while in field applications, rock masses are anisotropic and heterogeneous geological bodies, with various weak planes, the presence of groundwater, and layered rock structures [46]. These scale effects and complex geological heterogeneities highlight the need for more sophisticated constitutive relationships to accurately describe the EPW damage effects on the underground structures.

Future research will prioritize three directions:

(1)

Developing improved constitutive relationships and equations of state for rock media through uniaxial, triaxial, and impact dynamics experiments coupled with numerical simulations, enabling more accurate prediction of dynamic mechanical responses under the combined penetration–explosion effects;

(2)

Collaborating with military enterprises to acquire unclassified field test data, or utilizing existing reports and literature about the destructive effects of EPWs, for rigorous model validation;

(3)

Conducting numerical simulations incorporating complex geotechnical conditions, like discontinuities, groundwater, and layered structures, in real underground environments to enhance practical applicability.

5. Conclusions

The methods of fluid–solid coupling and restarting in LS-DYNA R11 are used to investigate the damage evolution and dynamic response of an underground granite chamber under the combined action of the penetration and explosion. The validity of the penetration and blasting model is calibrated against penetration tests and blasting experiments in the previous study. Based on PPV damage criteria, the dynamic response and damage of the chamber under the combined action of penetration and explosion are discussed. By comparing the simulation of internal explosion, the influence of the initial damage of penetration on the damage process of the underground chamber subjected to the action of the explosion load is analyzed. Moreover, the relationship curves among peak particle velocity, velocity of the projectile, and burial depth of the chamber are also investigated. The following results are obtained:

(1)

The methods of fluid–solid coupling and restarting can effectively predict the dynamic response and damage evolution of the underground granite chamber under the combined action of penetration and explosion.

(2)

The hypervelocity impact of the earth-penetrating weapons only causes local damage to the granite rock mass, and more serious damage is caused by the subsequent explosion. When the missile strike speed is 1196 m/s and the TNT equivalent is 300 kg, it is enough for the granite overlays with a critical thickness of 19 m to protect the underground chamber from the attack of the EPW. Therefore, it is feasible to construct military fortifications within natural granite formations to shield underground structures from the EPWs’ strike.

(3)

The initial penetration damage subjected to the projectile has a significant influence on the damage of the subsequent explosion. During the explosion, part of the detonation products and energy may escape along the penetration trajectory with the blast loading. Meanwhile, the damaged rock mass surrounding the penetration path can be regarded as a buffer zone, where the attenuation of blast stress waves is generated rapidly. In contrast, if the charge is directly buried in the rock mass, there will be no path for the explosive products to overflow. More blast energy is accumulated to break the rock mass, and the underground chamber is further damaged. Therefore, it is necessary to consider the effects of detonation products overflow and study the combined action of penetration and explosion in analyzing the damage of the EPW on underground structures.

(4)

Building upon numerical simulations and PPV damage criteria, a calculation method is proposed to determine the critical burial depth of underground fortifications, which can prevent destruction from the EPW attacks. This method provides certain reference significance for evaluating the damage effects of underground buildings under the combined action of penetration and explosion, as well as for the designs of anti-penetration and anti-explosion in underground structures.