Introduction

Calcareous sand is widely distributed in subtropical and tropical continents and coastal cities, such as the South China Sea, the Australian Great Barrier Reef and other coral islands1. Calcareous sand is formed by long-term geological weathering of organic matter and coral debris in the ocean, and its content is more than 90% calcium carbonate2. Due to its special formation mechanism, it has the characteristics of irregular shape, large porosity and easy crushing, and has significantly different physical and mechanical properties from quartz sand3,4. Calcareous sand is widely used as backfill material in human living areas on islands5. The explosion of underground pipelines and the use of controlled explosion to reinforce foundations inevitably involve explosions in sand foundations. Therefore, it is of great significance to carry out explosion research in calcareous sand foundations.

The explosion mechanism of the sand is relatively complex, as it involves the interaction of material nonlinearity, geometric nonlinearity and multiphase media6. The explosion crater is a direct response to the explosion loading in the sand7. The energy generated by the explosion and the pressure of the sand above the explosive play a decisive role in the type and size of the explosion crater8.

Methods for the mechanism of explosion crater can be classified into three main categories: Firstly, theoretical analysis enables the study of explosions in sand9. Based on the theoretical analysis of the similarity principle, some calculation parameters can be reasonably assumed and simplified to derive the dimensionless relationship between the diameter of the explosion crater, the energy of the explosive and the burial depth10. Therefore, the theoretical analysis method has the advantages of simple calculation form and broad application range11,12,13.

Secondly, field test remains an effective method for researching the explosions in sand14. Researchers from the University of Argentina and Hohai University conducted field tests in clay and Yangtze River sand respectively, and found that the scaled burial depth of explosive plays a decisive role in the shape and size of the explosion crater15,16. When the explosive is buried shallowly, the energy generated by the explosion is mainly due to the splashing of the sand above the explosive17. As the burial depth of the explosive increases, the suppression effect of the sand above the explosive on the energy generated by the explosion gradually increases under the action of gravity, thus changing the type of explosion crater. And the empirical prediction formula for explosion crater was derived based on field test results.

Thirdly, numerical simulation method can better to explain the explosion problems in sand with the rapid development of computer software and hardware technology18. The Arbitrary Lagrangian-Eulerian (ALE) method uses the characteristics of the Lagrangian method to deal with structural boundary motion, absorbs the advantages of the Euler method for meshing, and can handle the large deformation problem of explosions in sand quickly19. However, ALE method has the disadvantage of non-smooth boundaries, and the element incompatibility leads to non-physical penetration of Euler matter.

The SPH method can effectively improve the calculation accuracy, but the speed of calculation is very slow. SPH-FEM is a method that couples smooth particle hydrodynamic and finite element method20. The SPH particles were placed in area of large deformation and finite elements were set in areas of small deformation. Which can not only accurately represent the disaster mechanism but also enhances computational efficiency. As a result, the SPH-FEM method gained widespread application in geotechnical engineering21.

The SPH-FEM method is used to simulated the mobility of soil particles during slope instability22,23 and migration of soil particles during soil densification under strong compaction24. At the same time, the SPH-FEM method can also effectively simulate the local high stress area formed by the impact pile in the soil and analyze the yield and stress redistribution phenomenon of the soil around the pile25. Besides, the SPH-FEM method can simulate the deformation the underground structure under dynamic loads, and track the location information and motion of soils at any moment26,27,28.

However, existing research on explosions in sand primarily focuses on rock, clay, and quartz sand, with relatively little research conducted on calcareous sand foundations29,30,31. With the rise in production activities involving explosions in calcareous sand foundations, and given that its physical and mechanical properties differ significantly from quartz sand, there is an urgent need for research in this area. The issue of explosions in sand can be better understood through a combination of field tests and numerical analysis.

Considering the ability of SPH-FEM method to effectively handle large deformations and effectively describe the motion trajectory of sand particles under explosion loads. In this study, we investigated the type and size of craters in calcareous sand foundations through field tests and SPH-FEM method. Furthermore, based on the results from both field tests and numerical analysis, an empirical prediction formula for craters in calcareous sand foundations was derived, providing a theoretical reference for explosion-resistant design and foundation reinforcement.

Method

Theory

The scaled burial depth

High-energy chemical explosives generate high-temperature, high-pressure gases, which exert a strong radial compression on the surrounding sand medium, forming a compaction zone32. The TNT equivalent and the burial depth of the explosive have a significant impact on the type and size of the explosion crater33. To assess the destructive effects of different types of explosive sources under the same conditions, the concept of scaled burial depth is typically introduced to evaluate the explosion damage effects across various explosions under consistent conditions34, as shown in Eq. (1):

$$z=\frac{d}{{{{({W_{TNT}})}^{1/3}}}}$$
(1)

where, z is the scaled burial depth, d is the burial depth, WTNT is the explosive’s TNT equivalent.

The buried depth of the explosive determines the type and size of the explosion crater35. When the buried depth of the explosive is small, the detonation gas generated after the explosion quickly squeezes the sand around the explosive, and the explosion compression effect and peeling effect causes the sand around the explosive to gain an upward initial velocity and cause a throwing phenomenon, forming an ejecta-type crater, as shown in Fig. 136. As the burial depth of the explosive increases, the inhibitory effect of the sand covering the explosive on the throwing of the explosion gradually increases. The detonation gas only causes the local sand around the explosive to form a small cavity, and the cavity collapses under the weight of the upper sand, eventually forming a collapsed-crater, as shown in Fig. 237. As the burial depth of explosives continues to increase, the energy generated by the explosion can only provide vibrations to the foundation, there is no obvious change in the surface, and a camouflet is formed38.

Fig. 1
Fig. 1
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The ejecta-type crater.

Fig. 2
Fig. 2
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The collapse-type crater.

Aspect ratio

On the basis of the scaling law, any linear dimension L of the explosion crater can be indicated as a constant multiplied by the equivalent TNT of the explosive Wα, where α is a coefficient dependent on whether neglected the gravity39. Baker et al. presented a dimensional study with six parameters to model the crater formation phenomenon in underground explosion10. Obtained the aspect ratio (rc) functional as shown in Eq. (2), after many analyzes and observations.

$${r_c}=\frac{R}{d}=f\left[ {\frac{{{W^{\frac{7}{{24}}}}}}{{{\sigma ^{\frac{1}{6}}}{K^{\frac{1}{8}}}d}}} \right]$$
(2)

where, W is equivalent TNT of the explosive; d is the depth of the explosive; R is radius of the crater; \(\sigma\) is dimensions of stress, which was related to the sand strength; K is dimensions of a force divided by a cubic length.

As shown in Eq. (2), the specific weight \(\rho g\) is measured for K, \(\rho {c^2}\) is measured for \(\sigma\), where c is the explosion velocity in the sand. If this empirical formula for different density of sand is plotted in a \({r_c}\) versus \(\frac{{{W^{\frac{7}{{24}}}}}}{{{\rho ^{\frac{7}{{24}}}}{c^{\frac{1}{3}}}{\rho ^{\frac{1}{8}}}d}}\) graph, it can be clearly described that there is a small change in the results.

Field test

Calcareous sand

The sand used in the field test is calcareous sand sourced from the South China Sea, formed through long-term geological and weathering processes of coral in the ocean40. The physical picture and electron microscope scanning picture of the calcareous sand are shown in Fig. 3. As shown in Fig. 3, the calcareous sand particles possess unique physical properties, such as an irregular shape and high porosity. The grain size distribution curve of the calcareous sand is shown in Fig. 4. The mechanical parameters of calcareous sand, obtained according to the Chinese Standard for Geotechnical Testing Method, are presented in Table 1.

Fig. 3
Fig. 3
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Calcareous sand.

Fig. 4
Fig. 4
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Grain size distribution curve.

Table 1 The parameters of the calcareous sand.
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Design of field test

Large TNT equivalent explosions in sand release enormous amounts of energy, requiring higher-standard testing equipment and stringent safety measures, which inevitably increase both the cost and duration of testing41. Small TNT equivalent explosions in sand are often more recommended for testing42. In the field test, the explosive used was factory-customized black powder with a TNT equivalent of 5 g, as shown in Fig. 5. In the field tests, the explosive was buried at the test design depth and connected to the fuse, as shown in Fig. 6. To increase the probability of success in field tests and prevent water in the foundation from wetting the fuse, a 4 mm diameter plastic pipe is wrapped around the fuse.

Fig. 5
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The explosive.

Fig. 6
Fig. 6
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Experimental diagram.

By changing the burial depth of explosives, different scaled burial depth conditions were set. The burial depth range for the placement of the explosives is set at 0.03 ~ 0.37 m respectively, as shown in Table 2. Considering the discreteness in field tests, three sets of parallel tests were set up under the same burial depth conditions.

Table 2 The design condition of the field test.
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Numerical simulation

SPH-FEM coupling algorithm

Smooth particle hydrodynamics is a meshless Lagrangian computational method43. It’s adaptability can better handle large deformation problems and can handle material interfaces smoothly and continuously44. The SPH method discretizes the problem domain through a series of particles, each of which can carry field variables such as density, velocity, stress and displacement45. The SPH method uses an integral expression to approximate a certain field function, and discretizes it into a series summation problem, as shown in Eq. (3). Equation (3) is a general SPH approximate equation, and it’s schematic is shown in Fig. 7. As shown in Fig. 7, h is the support domain radius of the kernel function. This equation can be used to estimate any field variable in continuous space.

$$f({r_i})=\int\limits_{\Omega } {f(r)} {W_{ij}}dr \approx \sum\limits_{j} {{V_j}{f_j}} {W_{ij}}$$
(3)

where, \({W_{ij}}\) is the kernel function, a dimensionless function; \(\Omega\) is the support domain of the kernel function; i is the particle studied; j is the particle near i; r is the distance between particles; \({V_b}\) is the volume of particle j.

The full SPH particles model will greatly increase the calculation time, and may even cause problems such as limited boundary problems46. Therefore, during the modeling process, SPH particles are usually established for the explosive and surrounding sand, and finite elements are used at locations farther away from the explosive. The keyword *CONGTROL_TIED_NODE_TO_SURFACE_OFFSET was used to constrain the mutually fixed coupling between SPH particles and the finite elements, as shown in Fig. 8. Set the SPH particles as the slave surface, set the finite element as the master surface, and constrain the slave nodes to the master surface to realize the transfer of particle speed, stress, mass and other information to the finite elements.

Fig. 7
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The schematic of kernel function.

Fig. 8
Fig. 8
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The SPH-FEM coupling algorithm.

The numerical model

To accurately analyze the characteristics of explosion craters, a 3D finite element model was established based on a 1:1 similarity ratio. The overall size of the numerical model is 0.9 m*0.9 m*0.4 m, among which the range of SPH part is 0.6 m*0.6 m*0.3 m. The dimension of the 1/4 of the numerical model is shown in Fig. 9. The size of element plays a decisive role in calculation accuracy and resource consumption. The trade-off between accuracy and calculation resources is one of the important issues to be solved in numerical analysis. After many trial calculations, the size of element in the model was set to 1 cm. The numerical model consists of 223,240 finite elements and 123,068 SPH particles, with 47 SPH particles representing the explosive. By changing the position of the explosive SPH particles, the different burial depth of the explosive in the numerical model is controlled. To simulate the field test as realistically as possible, the top surface of the numerical model is set as a free surface, with normal constraints applied to the sides of the model, and full constraints applied to the bottom.

Fig. 9
Fig. 9
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3D SPH-FEM model of a quarter.

Calcareous sand model

The key to numerical simulation of explosion problems in sand lies in the reasonable selection of sand models and parameters47. By evaluating the constitutive models suitable for describing sand in the LS-DYNA software, the *MAT_005_ SOIL_AND_FOAM model, one of the earliest constitutive models that supports the SPH algorithm was selected48. This model is simple, with easily obtainable material parameters, and is widely used in addressing explosion problems in calcareous sand foundations.

The loading and unloading stress-strain curves of the model are shown in Fig. 10. Ten sets of data are used to perform multi-segment linear approximation of the compression equation. The equation of state parameters recommended by Wu et al.49 and which was suitable for calcareous sand in calculation, as shown in Table 3. The density of calcareous sand is 1370 kg/m3, the shear modulus is 106.3 MPa, the bulk modulus is 587.3 MPa, the shear yield coefficient on the shear plane a0 = 73.85 kPa2, a1 = 13.19 kPa, and a2 = 0.3.

Fig. 10
Fig. 10
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Schematic of pressure versus volume response for sand.

Table 3 Parameters of state equation of calcareous sand.
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The explosive model

The explosive is simulated using the high-energy explosive model and JWL equation in LS-DYNA software37. The JWL equation of the explosive can accurately describe the pressure, volume, and energy characteristics of the explosion during the detonation process, as shown in Eq. (4):

$${P_e}=A(1 - \frac{\omega }{{{R_1}{V_r}}}){e^{ - {R_1}{V_r}}}+B(1 - \frac{\omega }{{{R_2}{V_r}}}){e^{ - {R_2}{V_r}}}+\frac{{\omega {E_0}}}{{{V_r}}}$$
(4)

where A, B, \({R_1}\), \({R_2}\) and \(\omega\) are the explosive parameter; \({P_e}\) is the pressure generated by the explosion; \({V_r}\) is the relative specific volume; \({E_0}\) is the specific internal energy of the explosion products.

The parameters of the explosives used in the paper are shown in Table 450.

Table 4 Material parameters of explosive model.
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Results

Field test results

The formation process of the crater

In the field test, a high-speed camera with a recording frame rate of 60 FPS was used to capture the crater formation process. Taking the scaled burial depth of 0.76 m/kg1/3 as an example, the formation process of crater was shown in Fig. 11. The crater was formed within 300 ms after detonation. The upper calcareous sand on the explosive overcomes the gravity and the bonding forces between sand particles under the explosion loading.

As shown in Fig. 11, within the initial 51 ms, the impact of the explosion induces the calcareous sand above the explosive to move upward, forming a bulge on the surface. Between 51 ~ 136 ms, under the force of the explosion gas, an ejection and splashing phenomenon occurs, forming a pear-like shape. From 136 to 282 ms, the calcareous sand is thrown outward by the explosion impact, and an ejecta-type crater forms under the influence of gravity.

Fig. 11
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The formation process of the crater (Condition BE2).

Effect of burial depth on the crater size

When the scaled burial depth of the explosive is small, the explosion mainly causes the calcareous sand particles above the explosive to gain an upward initial velocity and form an ejecta-type crater. When measuring the depth of the ejecta-type crater, it’s necessary to remove the sand particles that have fallen back into the top of the crater, as shown in Fig. 12. When measuring the diameter of the crater, test once every 60°, and use the average of the three measurements as the final measured diameter of the crater, as shown in Fig. 12. The diameter and depth of crater with different scaled burial depths are as shown in Table 4.

Fig. 12
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Diameter of crater (Condition BE4).

It can be seen from Table 5 that when the scaled burial depth is less than 0.53 m/kg1/3, as the burial depth of the explosive increases, the depth and the diameter of the ejecta-type crater gradually increases. When the scaled burial depth is 0.53 m/kg1/3, the diameter of the crater reaches the maximum value, with the maximum diameter of the crater being approximately 0.437 m. When the scaled burial depth is greater than 0.53 m/kg1/3, the diameter of the crater gradually decreases as the scaled burial depth of the explosive increases. At this point, the gravity of the upper sand particles and the friction between particles increase the inhibitory effect on the explosion energy, causing the diameter of the crater to gradually decrease. This result is similar to the quartz sand studied by Jia et al.16, but there are large differences due to the different parameter of the sand.

Table 5 Results of crater size under different scaled burial depth.
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The influence of burial depth on the type of crater

When the scaled depth of burial is less than 1.11 m/kg1/3, the energy generated by the explosion dissipates more directly through the free face above the sand, causing the calcareous sand above the explosive to acquire an initial upward velocity and ejection, forming an ejecta-type crater as shown in Fig. 13a. When the scaled burial depth is within 1.11 to 2.05 m/kg1/3, the calcareous sand above the explosive exerts a significant restraining effect on ejection, and the energy generated by the explosion is mainly used for rapidly compressing the surrounding sand. A blast cavity is formed inside the sand, which collapses under the gravity, leading to the observation of clear irregular cracks on the ground surface, resulting in a collapse-type crater as shown in Fig. 13b. When the scaled burial depth exceeds 2.05 m/kg1/3, the gravity of the sand above the explosive predominates. The explosive energy mainly acts on the vibration of the sand foundation, with the energy dissipated through the free face almost negligible. There are almost no visible cracks on the surface at this time, indicating a camouflet is formed in the calcareous sand foundation.

Fig. 13
Fig. 13
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The type of the crater. (a) Ejecta-type crater (Condition BE2). (b) Collapse-type crater (Condition BE9).

Numerical simulation results

Verification of the model

Figure 14 shows the formation process of an ejecta-type crater when the burial depth of explosive is 0.29 m/kg1/3. The formation process of an ejecta-type crater lasts about 300 milliseconds. After the detonation of the explosive, the sand above the explosive is violently thrown due to the expansion of the explosion gases, while the sand at the bottom and around the explosive undergoes substantial plastic deformation due to the explosion shock compression. This ultimately results in the expansion of the crater in both depth and lateral directions. By setting sand SPH particles around the explosive, normal calculations can be ensured even under large explosion deformations, the movement of the sand near the explosive can be intuitively described, and a smooth and clear boundary of the ejecta-type crater can be formed. The movement of sand particles observed in Fig. 14 is basically consistent with that captured by high-speed camera in the field test.

Fig. 14
Fig. 14
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The formation process of the ejecta-type crater.

When the scaled buried depth of explosive is 0.29 m/kg1/3, the diameter and depth of the ejecta-type crater is as shown in Fig. 15. The diameter and depth of the crater measured were 0.36 m and 0.13 m respectively. The data obtained from the tests are basically consistent with those obtained from field test. It can be considered that the SPH-FEM coupling algorithm can better describe the ejecta-type crater in calcareous sand foundation.

Fig. 15
Fig. 15
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Diameter and depth of crater.

Analysis of crater flow field

When the scaled burial depth of the explosive is 0.41 m/kg1/3, the velocity and displacement vector diagram of the SPH sand particles around the explosive at t = 50 ms is shown in Fig. 16. It can be seen from Fig. 16 that after the explosive is detonated, part of the sand around the explosive is radially thrown into the air under the action of the explosion, while the sand on the sides and bottom of the explosive is compressed and compacted approximately along the radial-direction of the sphere at a relatively small speed.

Fig. 16
Fig. 16
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Vector field of the velocity and displacement. (a) Vector field of velocity. (b) Vector field of displacement

Analysis of different equivalents

Field tests are often constrained by factors such as technical challenges, cost, risk, and limited repeatability. Proven numerical simulations can effectively complement field tests. To investigate the impact of different TNT equivalents on crater size, explosives with TNT equivalents of 5 g, 10 g, and 20 g were used, and the burial depths were set according to Table 2. The diameter of the crater under different TNT equivalents is shown in Fig. 17. It can be seen from Fig. 17 that when the TNT equivalent of the explosives is the same, the diameter of the crater first increases and then decreases as the burial depth of the explosive increases. At the same burial depth, the diameter of the crater gradually increases with the increase of the TNT equivalent of the explosive.

Fig. 17
Fig. 17
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Diameter of the crater under different TNT equivalents.

As the TNT equivalent and burial depth of the explosive change, the type and size of the crater will show significant variations. Based on the results of field tests and numerical simulations, a comprehensive analysis of the crater diameters at different scaled burial depths has been conducted. The field test and numerical analysis results of calcareous sand as shown in Fig. 18. An empirical prediction formula for the crater diameter as a function of the TNT equivalence and the burial depth of the explosives has been proposed, as shown in Eq. (4):

$$I{\text{n}}(D/d)=0.731+0.623In(1/{\text{z}} - 0.767)$$
(4)

where, z is the scaled burial depth of the explosive, D is the diameter of the crater, and d is the burial depth of the explosive.

Fig. 18
Fig. 18
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Relationship between diameter of crater and scaled burial depth.

Discussion

In this study, the types and sizes of craters in calcareous sand foundations were examined using field tests and the SPH-FEM coupling algorithm. The results obtained from the SPH-FEM coupling algorithm were largely consistent with the field test outcomes, demonstrating that the SPH-FEM coupling algorithm can effectively avoid non-convergence issues due to element distortion when addressing large deformation problems caused by explosions in sand.

Based on the results of field tests and numerical simulations, it is evident that the scaled burial depth of explosives plays a decisive role in determining the type and size of the explosion crater. There are three types of explosion craters with different scaled burial depths, when the scaled burial depth is less than 1.11 m/kg1/3, the crater is ejecta-type; when the scaled burial depth is in the range of 1.11 ~ 2.05 m/kg1/3, the crater is collapse-type crater; when the scaled burial depth is greater than 2.05 m/kg1/3, a camouflet explosion phenomenon occurred.

In the ejecta-type explosion crater, the size of the explosion crater first increases and then decreases as the burial depth of the explosive increases. When the scaled burial depth is 0.53 m/kg1/3, the diameter of the explosion crater reaches the maximum. The main reason is that when the scaled burial depth is less than 0.53 m/kg1/3, the energy generated by the explosion is more dissipated through the ground surface. As the scaled burial depth increases, the energy generated by the explosion acts more in sand. When the scaled burial depth is greater than 0.53 m/kg1/3, the inhibitory effect of the sand above the explosive on the explosion gradually increases, and the size of the explosion crater gradually decreases, and even changes the type of explosion crater from the ejecta-type to the collapse-type.

The results of calcareous sand and other’s results as shown in Fig. 19. The explosion crater in calcareous sand is similar to results observed in other’s researches10,13,19,33. However, the size of the explosion crater in the calcareous sand foundation is quite different. The size of the crater in the calcareous sand foundation is larger than 40% than other’s results while the scaled of burial depth is same. The primary reason is that calcareous sand particles have more surface pores and a lower specific gravity compared to quartz sand.

Fig. 19
Fig. 19
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Relationship between diameter of crater and scaled burial depth compared other’s results.

This study mainly studies the influence of scaled burial depth on the type and size of crater, based on field tests or numerical simulations. However, in fact, when the calcareous sand particles are subjected to high-frequency loading during the explosion, inevitable particle fragmentation will occur. The fragmentation of calcareous sand particles is mainly concentrated in a small number of particles near the explosive, which can be ignored when studying the type and size of explosion craters. This study focus on the type and size of crater in calcareous sand foundations through field tests and SPH-FEM coupling algorithm still has important reference value.

Conclusions

In this paper, we studied the crater’s type and size of calcareous sand foundation, and found that the SPH-FEM algorithm can better reflect the formation process of explosion craters and the movement trajectories of particles. We obtained the critical scaled burial depth for different type of crater, and derived an empirical prediction formula for calcareous sand. The results provide a theoretical basis for anti-explosion design in calcareous sand foundations.