To compare the damage zones of spherical and cylindrical charges in rock and soil, a quasistatic spherical model was established to predict the characteristic dimension of the cavity. The results indicated that the damage zones of cylindrical charges were larger than those of spherical charges. Furthermore, the cavity development of two charges with different shapes was obtained by numerical simulation, and a comparison of the prediction results between the quasistatic model and numerical simulation was made. The comparison showed that the model could predict the damage zones exactly and faster than numerical simulation. Ultimately, the influence of explosions and soil media was discussed by the quasistatic model. It was observed that larger damage zones were generated by smaller values of the product of pressure and exponential expansion. However, the influence of soil media was complex, and larger damage zones were usually generated by the harder soil media.

The problem of the damage zones caused by an explosion in the rock and soil has been widely investigated. The size of the damage zones affects the seismic wave field generated by the explosion; thus, the damage zones with different sizes are encountered in different engineering fields. In practical engineering, spherical charge and cylindrical charge are two commonly used charge forms; therefore, it is necessary to conduct a comparative study on the damage zones to provide a basis for the choice between these two charge forms.

The research methods for the damage zones caused by these two kinds of charge mainly include the analytical method, numerical analysis, and explosive test method in soil. Henrych, respectively, proposed the cavity analytical expressions for the rock and soil under quasiempirical, quasistatic, and dynamic conditions [

Based on the quasistatic theory, this study established a prediction model for the damage zones of the spherical drug package, which was compared with the column-symmetric model established by Drukovanyi et al. [

The process of explosives from blasting to finally forming elastic waves in the distance is accompanied by a series of chemical and physical changes. The energy keeps attenuating during the transmission process due to various dissipation mechanisms, showing the evolution from the powerful blast waves to elastic waves finally. This process is composed of four stages, which are hydrodynamic stage, geomaterial crushing stage, dynamic expansion stage, and elastic wave propagation stage. Meanwhile, the medium would have irreversible deformation under the strong explosion effects during the attenuation process, so that the geomaterial near the explosive source shows certain fluid properties under the impacts of huge energy. At the moment of explosion, a wave front could be pushed out from inside the explosion cave, which is in a shape the same as the explosive. With the development of the blast wave, its peak stress attenuates rapidly during the outward propagation process until below the ultimate failure strength of the geomaterial. At this moment, the geomaterial turns from the fluid stress state to the elastic-plastic stress state. And the attenuation of the stress wave continues until its peak value falls below a certain value, and the geomaterial transforms from the plastic state to the elastic state. In this case, the explosive source forms the explosion cavity area, plastic area, and an elastic area in sequence when it blasts in geomaterial along the energy transformation direction (Figure

Distribution of different damage zones.

Since the explosion wave generated by the spherical charge explosion is a spherical wave and the cylindrical charge explosion produces a cylindrical explosion wave, therefore, under the same charge quantity, the two charge forms will produce different damage zones (Figure

Explosive wave characteristics produced by two charge forms.

To predict the size of damage zones of different charge structures, a quasistatic model for damage zones prediction was established. In the model, there are several assumptions as follows: (1) The rock and soil medium are incompressible. (2) The change in the density of rock and soil caused by the explosion is ignored. (3) The formation of the explosive cavity and damage zones is instantaneous. (4) The detonation of explosives is instantaneous. As shown in Figure _{m} is the cavity radius, _{0} is the radius of the fracture zone. The whole region can be divided into three parts:

Elastic zone:

Cracked zone:

Crushed zone (cavity):

Based on the above assumptions, the shear stress is zero and the annular stress is equal under the condition of spherical cavity expansion in the elastic medium; thus, the equilibrium equation can be expressed as follows:

Therefore, in the elastic region, the displacement

According to the above assumptions, the equilibrium equation under the condition of cylindrical charge can be expressed as follows:

The displacement

The annular stress is zero in the radial fracture zone; thus, the equilibrium equation is presented as follows:

At the boundary of region 3,

Then, the boundary at region 2 is expressed as follows:

From the stress-strain relationship of

Similarly, the simplified equilibrium equation of cylindrical charge is expressed as follows:

At the boundary of region 3,

From the stress-strain relationship of

In the crushed zone, the rock and soil are in close contact with the explosive, and the instantaneous pressure generated by the explosion is far greater than the compressive strength of rock and soil. Therefore, in this region, significant plastic deformation, fracture, and other phenomena will appear on the medium. At present, the prediction of this area is very different, and it is generally believed that the radius of this area is about 3 to 5 times the charge radius. To describe the destruction of rock and soil in the crushed zone, a granular incompressible medium model with cohesive force is used. The simplified Mohr–Coulomb failure criterion is as follows:

By substituting equation (

The pressure on the wall of the chamber is expressed as follows:

After applying the incompressible conditions and calculating the boundary

Because

When

Then,

Thus, the detonation pressure

According to the adiabatic law, the pressure relationship on the blasting chamber is

We can obtain the size of the maximum crushed zone

The range of radial fracture zone

Introduce the linear radial strain and hoop strain according to the spherical symmetry and linear theory to simplify the motion equation as follows:

The solution of this equation can describe the forced vibration of the particle under viscous damping, and its general form is

By substituting equation (

The pressure on the wall of the chamber is expressed as follows:

By applying the incompressible conditions, we have

Now, let us consider:

Therefore, the detonation pressure

According to the adiabatic law, the pressure relationship on the blasting chamber is

We can obtain the size of the maximum crushed zone

The range of radial fracture zone

According to the calculation model of damage zones under two charge forms obtained in this study, TNT explosion in ordinary sandy clay under two charge forms was calculated, among which the characteristic parameters of TNT and siltstone are shown in Tables

Characteristic parameters of TNT.

Classification of explosive | Explosive velocity ( | Density ( | Initial pressure (GPa) | Expansion index |
---|---|---|---|---|

TNT | 6900 | 1650 | 9.82 | 3.15 |

Characteristic parameters of sandy clay.

Categories of medium | ||||||
---|---|---|---|---|---|---|

Sandy clay | 11.6 | 2 | 0.16 | 0.2 | 50 | 1600 |

Contrast of damage zones between spherical and cylindrical charges.

Type of charge | Radius of charge (m) | Radius of cavity (m) | Crushing zone (m) | Fracture zone (m) |
---|---|---|---|---|

Cylindrical | 0.0580 | 0. 155 | 0.361 | 2.093 |

Spherical | 0.0527 | 0.141 | 0.295 | 1.713 |

It can be seen from the calculated results that when the charge amount was the same, the size of the destructed zone produced by the two charge structures in the radial direction was different, and the radius of the destructed zone produced by the spherical charge was smaller than that of the cylindrical charge. This was because of the obvious directivity of the cylindrical charge when it exploded. Hence, the strong pressure was generated in the radial direction, and thus, the damage zones generated in the radial direction were greater than those caused by the spherical charge.

To further compare the size of the damage zones caused by the two charge forms, the software AUTODYN was employed to simulate the explosion process of spherical and cylindrical charges in the solid medium. The initial dosage of the two charge forms was 1 kg. The linear material model and von Mises intensity model were selected for the geotechnical medium. The model parameters were obtained through literature. The JWL equation of state that exists in the AUTODYN software was utilized to describe the explosives. The comparison between the calculated results and the predicted results of the quasistatic model is shown in Figure

Scope of damage zones of cylindrical and spherical charges.

The results of the radius of damage zones obtained by the quasistatic analytical method were similar to the numerical simulation method (Figure

Stress distribution of different charge structures in the medium.

Figure

In engineering practice, the damage zones with different sizes are encountered in different engineering fields; thus, it is necessary to master the control method of the damage zones. According to the analysis of the analytical model, the scope of the damage zones was primarily affected by the properties of explosives and rock and soil. The detonation pressure and expansion index of explosives certainly affected the size of the damage zones. Table

Characteristic parameters of different explosives.

Classification of explosive | Explosive velocity ( | Density ( | Pressure (GPa) | Expansion index |
---|---|---|---|---|

RDX | 8300 | 1700 | 14.64 | 3.4 |

TNT | 6900 | 1650 | 9.82 | 3.15 |

TL | 4500 | 1500 | 3.80 | 2.6 |

BP | 3100 | 1500 | 1.81 | 1.8 |

Considering sandy clay as the rock and soil medium, the damage zones formed by four different explosives are shown in Table

Calculation results of different explosives.

Categories of explosives | Cylindrical charge | Spherical charge | ||
---|---|---|---|---|

RDX | 6.13 | 35.57 | 5.53 | 23.32 |

TNT | 6.24 | 36.17 | 5.60 | 23.50 |

TL | 6.45 | 37.43 | 5.75 | 23.85 |

BP | 8.40 | 48.73 | 7.32 | 27.63 |

The initial pressure of the explosion increased with the detonation speed, but the calculation results from Table

Change curve of the crushed zone under different

Change curve of the crushed zone under different

Figures

The change of the properties of rock and soil would also affect the size of the damage zones. Table

Characteristic parameters of different rock and soil types.

Categories of medium | ^{2}) | ^{2}) | ^{2}) | ^{3}) | ||
---|---|---|---|---|---|---|

Siltstone | 2000 | 143 | 1.13 | 0.267 | 800 | 2600 |

Diabase | 3090 | 560 | 5.76 | 0.67 | 1000 | 2800 |

Concrete | 950 | 160 | 1.49 | 0.44 | 395 | 2070 |

Limestone | 620 | 63 | 2.10 | 0.42 | 420 | 1900 |

Scope of damage zones of two charge forms under different medium conditions.

Categories of medium | Cylindrical charge | Spherical charge | ||
---|---|---|---|---|

Siltstone | 5.22 | 72.9 | 2.51 | 9.37 |

Diabase | 9.15 | 50.56 | 3.02 | 6.99 |

Concrete | 10.36 | 59.7 | 3.57 | 8.7 |

Limestone | 14.0 | 137.6 | 4.50 | 14.12 |

The calculation results in Table

An explosive cavity test was carried out to test the accuracy of the quasistatic model in predicting the size of the explosion cavity. The experimental results are compared with the quasistatic results. 1 kg TNT and 2 kg TNT were exploded in silty clay; the explosion cavity of the experiment is shown in Figure

Explosive cavity in the soil.

Result of the explosion cavity test.

Soil type | Explosive quality (Kg) | Explosive radius (cm) | Radius of explosive cavity (cm) | Radius of quasistatic method | |
---|---|---|---|---|---|

Vertical | Horizontal | ||||

Silt clay | 1 | 5.3 | 40.8 | 38.5 | 36.43 |

Silt clay | 1 | 5.3 | 41.5 | 39.5 | 36.43 |

Silt clay | 2 | 6.7 | 49.5 | 50.5 | 42.04 |

Silt clay | 2 | 6.7 | 48.5 | 48.0 | 42.04 |

The calculation results in Table

The quasistatic analytical method was used to predict the damage zones of spherical and cylindrical charges. The results showed that the radial damage zones of the spherical charge were smaller than those of the cylindrical charge under the same conditions. By comparing with the results of numerical simulation, it was proved that the analytical model can quickly and easily solve the damage zones caused by different drug packages.

The analytical model was utilized to calculate the stress distribution of the two charges in the medium. Because the damage zones of the spherical charge were small, the stress peak in the medium was larger than that of the cylindrical charge.

This model was employed to analyze the impact of explosives and rock and soil on the damage zones. We found that the smaller the expansion index of explosives, the greater the initial explosion pressure, and the larger the damage zones. Therefore, the influence of

In different engineering fields, the size requirements of the damage zones are also different. For example, in the field of seismic exploration, it is necessary to control the scope of the plastic damage zones to obtain stronger seismic waves, while in the mine blasting, it is necessary to strengthen the rock-breaking ability of the charge package and to reduce the blasting seismic effect. According to their different requirements, the spherical charge is more suitable for seismic exploration and the cylindrical charge is more suitable for mine blasting.

The data are available and included within the article; the data supporting the conclusions of this study are also available from the corresponding author.

The authors declare no conflicts of interest.

The authors thank the State Key Laboratory of Explosion Science and Technology in Beijing Institute of Technology. This work was supported by the National Natural Science Foundation of China (no. 51678050) and the National Key R&D Program of China (no. 2017YFC0804702).