## Abstract

Explosion containment vessels (ECVs) are important equipment used for underwater explosion experiment research. In this paper, a method is proposed to improve the blast resistance of the ECV by placing it in the water medium. The shock response caused by the underwater explosion in a spherical ECV is studied. The single-layer thin-walled spherical vessel shell submerged in an infinite domain water medium is deduced and simplified as a single-degree-of-freedom elastic vibration system with viscous damping. The underwater explosion shock wave is simplified to an exponential shock loading, and the analytical solutions of the radial shock vibration of the spherical shell are obtained using Duhamel's integrals. Compared with the numerical simulation results, the accuracy of the theoretical model is verified. The study results show that the water medium radiates out vibration energy and plays an important role in eliminating vibration through damping. It is found that the vibration damping ratio can be controlled by adjusting the vessel radius and thickness, so that the vibration of the shell can be controlled within three periods and the impact fatigue can be reduced. In addition, the radiation damping of the water medium greatly reduces the maximum radial displacement of the spherical shell, which significantly improves the blast resistance of the spherical shell.

## 1 Introduction

Since World War II, underwater explosions have been widely used in national defense and industry fields, such as the design of underwater weapons, the protection of ships and submarines, underwater rock blasting excavation, and underwater explosion processing [1–3]. To effectively protect the structure and rationally destroy the target with underwater explosion energy, it is necessary to deeply study the formation and propagation of shock waves in water, the pulsation of bubbles, and the interaction mechanism between shock waves and structures. There is no doubt that experimental research is an essential approach to studying underwater explosions. However, field experiments such as lakes and oceans are expensive, long-period, poorly repeatable, and have strict requirements for experiment sites and supporting equipment.

Due to economy, safety, and operability, explosion containment vessels (ECVs) are an attractive alternative for underwater explosion experimental research [4,5]. ECV is a special kind of closed protective container, which has been widely used in the fields of national defense, industry, public safety, and scientific research because it can effectively limit the shock wave, fragmentation, and toxic gases generated by the internal explosion, ensuring the safety of personnel, equipment, and the surrounding environment [6]. Due to the high shock pressure and high-power output produced by the explosion, the ECV is in an ultimate shock state with a high strain rate. And the stress wave repeatedly propagates and superimposes inside the container, resulting in the phenomenon of strain growth, which makes the explosion container face the risk of impact damage. In addition, for reusable explosive containers, the risk of container fatigue damage increases due to the superposition of multiple vibration shocks. To improve the load-bearing capacity of explosive containers, many methods have been proposed, such as increasing the wall thickness of the container, placing blast-resistant materials inside the container, covering explosives with water, designing double and discrete multilayered structures, winding the container with fibers and steel ribbons, and burying the container in the ground, concrete, and rock formations [7–15].

In the underwater explosion experiment, the ECV is filled with a water medium, in which hydraulic pressure is loaded to simulate the water depth environment [5]. Due to the high pressure of the shock wave generated by the underwater explosion and the need to load the hydrostatic pressure, the experimental charge is generally small for the large underwater explosion container. For example, the largest ECV in China is 7 m in diameter, and the maximum test charge is only 1 kg TNT under the condition of simulating a water depth of 600 m [16]. Moreover, the shell is designed to be thicker to resist the impact of large amounts of explosives. However, thicker steel plates are difficult to process and weld, and their impact resistance and fatigue resistance will decrease. The problem of bearing strength can be solved by adopting a double-layer and multilayer shell structure, wrapping the container with fibers and steel strips, and burying the container in the concrete or surrounding rock. But these methods create challenges for shell maintenance and inspection, increasing the risk of using ECVs.

Therefore, this paper proposes a new method to improve the blast resistance of the ECV used for underwater explosion experiment research by placing it in water. The feasibility of the proposed method is analyzed through theoretical calculations and numerical simulations, and the effect of the damping ratio on the dynamic response of water-protected ECVs is studied, which can provide a reference for the design and protection of ECVs in the future.

## 2 Theoretical Analysis

### 2.1 Motion Equation of Spherical Shell.

A common spherical ECV can be simplified into a thin shell as shown in Fig. 1, with an inner radius of $R$ and a wall thickness of $\delta $. The inside of the container is filled with water medium, of which the spherical charge explodes at the center, and the outside is infinite water. To simplify the analysis, the following assumptions are made: (1) The radial stress of the spherical shell is negligible compared with the stress in other directions. (2) Stress is evenly distributed along the wall thickness. (3) There is no shear force on the cross section.

*P*(

*t*). As the spherical shell vibrates submerged in water, the surrounding fluid will be driven to move together, which is equivalent to adding a part of the mass to the original mass of the structure, making the structure's inertia increase. This part of the mass is the additional mass. In certain scenarios, the additional mass will be of the same order of magnitude as the mass of the structure itself and will have a significant influence on the natural frequency of the structure. The inertia force

*f*

_{i}per unit area of the spherical shell can be expressed as

where $\rho $ is the density of shell material; $ma$ is the additional mass per unit area of the shell; and $ur$ is the radial displacement.

*f*

_{d}is caused by the movement of the shell in the water medium. When the shell vibrates in water, due to its small range of motion, it can be simply considered that the breathing motion of the shell only causes simple sound radiation in the medium. The dynamic force exerted by the radiated sound field on the surface of the shell is

*F*=

*pA*

_{a}(here,

*p*=

*ρ*

_{0}

*C*

_{0}d

*u*/d

_{r}*t*denotes the sound pressure; and $Aa=4\pi Ra2$ denotes the surface area of the additional mass layer). The dynamic resistance per unit area of the shell can be expressed as

*f*

_{d}=

*F*/

*A*=

*F*/4

*πR*

^{2}. Therefore, the resistance on the surface of the shell is

where $\rho 0,\u2009C0$ are the density and the speed of sound of the water medium, respectively, $\rho 0C0$ is the sound impedance; and $Ra$ is the radius when the additional mass layer is considered as a pressure transfer layer.

where *σ _{θ}* is the stress in the

*θ*direction, and

*σ*is the stress in the

_{φ}*φ*direction.

where $E$ is Young's modulus, and $\nu $ is Poisson's ratio.

where $C$ is the longitudinal sound velocity of the spherical shell material, satisfying *C*^{2} = *E*/*ρ*.

which is a differential equation of motion with damping controlled by radial displacement.

*ω*of the breathing mode is therefore

### 2.2 Dynamic Response to Blast Shock Wave

#### 2.2.1 Pressure Load on the Inner Wall.

in which $Pm1$ is the peak shock wave pressure, MPa; $t$ is the time, ms; $\tau $ is the time constant, ms; $W$ is the TNT equivalent, kg; and $R$ is the distance between the detonation point and the observation point, m.

*v*(

*t*), when the incident pressure of the shock wave in water satisfies $0\u2264Pi(t)\u2264120\u2009MPa$, the reflected pressure acting on the obstacle can be expressed as

*P*

_{m}is the peak reflected pressure, which satisfies

#### 2.2.2 Dynamic Response Solution of Spherical Shells.

The response of a single-degree-of-freedom system under arbitrary excitation can be solved by Duhamel's integral method, the basic idea of which is to consider the excitation as a series of successive micro-impulses. First, the response of the system to each micro-impulse is calculated, and then the response of the system to any excitation can be obtained according to the principle of linear superposition [20]. Assuming that the dynamic displacement and velocity of the shell are 0 at the initial moment of $t=0$, the exponential excitation load $P(t)$ in Eq. (18) is substituted into the Duhamel's integral equation and simplified, and the solutions of Eqs. (11) and (12) can be obtained as shown in Eqs. (20) and (23).

## 3 Numerical Simulation

### 3.1 Finite Element Models.

As shown in Table 1, there are six spherical shells of different dimensions. To verify the damping effect of water outside the ECV, simulations are carried out for the free spherical shell and the spherical shells in water, respectively. To reduce computational scale and time, ls-dyna [21] is used to establish 1/8 three-dimensional numerical models as shown in Fig. 2, taking advantage of the symmetry of the structure and the loads. The model of the free ECV includes only steel vessel elements, and the ECV in water includes both steel vessel elements and water medium elements. The numerical simulation models are all constructed using eight-node hexahedron elements. Lagrangian method is adopted for steel vessel element and multimaterial arbitrary Lagrangian–Eulerian method is adopted for water medium element. To balance the calculation accuracy and calculation time, the vessel element sizes of spherical shells 1–6 are taken as 10 mm, 10 mm, 9 mm, 7 mm, 10 mm, and 9 mm, respectively. The thickness of the water domain is twice that of the shell, and the element size is the same as that of the spherical shell. The fluid–structure interaction algorithm based on penalty function coupling is used between the vessel and the water medium.

Shell no. | Inner radius R (m) | Thickness δ (mm) | Radius-thickness ratio R/δ |
---|---|---|---|

1 | 2.0 | 50 | 40 |

2 | 1.5 | 50 | 30 |

3 | 1.5 | 45 | 33 |

4 | 1.5 | 35 | 43 |

5 | 1.0 | 50 | 20 |

6 | 1.0 | 45 | 22 |

Shell no. | Inner radius R (m) | Thickness δ (mm) | Radius-thickness ratio R/δ |
---|---|---|---|

1 | 2.0 | 50 | 40 |

2 | 1.5 | 50 | 30 |

3 | 1.5 | 45 | 33 |

4 | 1.5 | 35 | 43 |

5 | 1.0 | 50 | 20 |

6 | 1.0 | 45 | 22 |

Symmetric boundary conditions are applied to the symmetric faces of the model. To simulate infinite water in a limited space, the outer surface of the water is set as the transmission boundary to prevent the reflection of artificial stress waves at boundaries. In this paper, various conditions are given with a maximum TNT equivalent of 0.5 kg and a minimum of 0.2 kg. The reflected pressure peaks and time constants are calculated according to Sec. 2.2.1, as shown in Table 2. The pressure time curve is defined and applied to the inner wall of the shell. It is worth noting that the six conditions in Table 2 and the six spherical shells in Table 1 correspond to each other.

Condition no. | TNT equivalent W (kg) | Reflected pressure peak P_{m} (MPa) | Time constant τ (ms) |
---|---|---|---|

1 | 0.5 | 36.81 | 0.1056 |

2 | 0.4 | 46.84 | 0.0928 |

3 | 0.3 | 42.03 | 0.0864 |

4 | 0.2 | 36.08 | 0.0781 |

5 | 0.2 | 57.05 | 0.0704 |

6 | 0.2 | 57.05 | 0.0704 |

Condition no. | TNT equivalent W (kg) | Reflected pressure peak P_{m} (MPa) | Time constant τ (ms) |
---|---|---|---|

1 | 0.5 | 36.81 | 0.1056 |

2 | 0.4 | 46.84 | 0.0928 |

3 | 0.3 | 42.03 | 0.0864 |

4 | 0.2 | 36.08 | 0.0781 |

5 | 0.2 | 57.05 | 0.0704 |

6 | 0.2 | 57.05 | 0.0704 |

### 3.2 Material Properties.

^{3}. The state change of water is described by the well-known Mie–Grüneisen equation of state, and as given by

where *ρ*_{0} is the initial density of the material; *C* is the intercept of the *v _{s}* (shock velocity)–

*v*(particle velocity) curve;

_{p}*S*

_{1},

*S*

_{2}, and

*S*

_{3}are the dimensionless coefficients of the slope of the

*v*–

_{s}*v*curve;

_{p}*γ*

_{0}is the dimensionless Gruneisen gamma;

*a*is the first-order volume correction to

*γ*

_{0};

*E*

_{0}is the initial specific internal energy; and $\mu =\rho /\rho 0\u22121$. The parameter values are shown in Table 3.

C (m/s) | S_{1} | S_{2} | S_{3} | γ_{0} | a | E_{0} (GPa) |
---|---|---|---|---|---|---|

1480 | 2.56 | −1.986 | 0.2268 | 0.5 | 0 | 0 |

C (m/s) | S_{1} | S_{2} | S_{3} | γ_{0} | a | E_{0} (GPa) |
---|---|---|---|---|---|---|

1480 | 2.56 | −1.986 | 0.2268 | 0.5 | 0 | 0 |

where $\sigma 0$ is the initial yield stress; $\epsilon \u02d9$ is the strain rate; $Ep$ is the plastic hardening modulus, determined by the elastic modulus *E* and the tangential modulus *E*_{t}; $\epsilon effp$ is the effective plastic strain; $\beta $ is the hardening constant; and $C$, $p$ are the strain rate constants. The material parameters used are summarized in Table 4.

## 4 Results and Discussion

### 4.1 Comparison Between Theoretical and Simulation Results.

Figure 3 presents the radial displacement time curves obtained by theoretical analysis and finite element method (FEM) for the free spherical shell and the spherical shell submerged in infinite water, respectively. From Fig. 3, we can draw the following conclusions: (1) The radial displacement of the spherical shell reaches the maximum value in the first motion period. Subsequently, the free spherical shell vibrates with equal amplitude, while the spherical shell submerged in infinite water vibrates with decaying amplitude. (2) The radial displacement of the spherical shell submerged in infinite water is smaller than that of the free spherical shell.

Table 5 indicates that for free spherical shells, the errors of the maximum radial displacements between theoretical and numerical simulation results are within 3%, and the errors of the periods in the free vibration phase between theoretical and numerical simulation results are also within 3%. Small deviations between theory and simulation may be due to approximate assumptions of numerical simulation.

Maximum radial displacement | Period | |||||
---|---|---|---|---|---|---|

Shell no. | Analysis (mm) | FEM (mm) | Error (%) | Analysis (ms) | FEM (ms) | Error (%) |

1 | 2.091 | 2.081 | −0.48 | 1.451 | 1.476 | 1.70 |

2 | 1.709 | 1.698 | −0.64 | 1.088 | 1.107 | 1.71 |

3 | 1.606 | 1.586 | −1.25 | 1.088 | 1.101 | 1.16 |

4 | 1.629 | 1.613 | −0.98 | 1.088 | 1.107 | 1.68 |

5 | 1.026 | 1.013 | −1.27 | 0.726 | 0.742 | 2.22 |

6 | 1.140 | 1.127 | −1.14 | 0.726 | 0.742 | 2.22 |

Maximum radial displacement | Period | |||||
---|---|---|---|---|---|---|

Shell no. | Analysis (mm) | FEM (mm) | Error (%) | Analysis (ms) | FEM (ms) | Error (%) |

1 | 2.091 | 2.081 | −0.48 | 1.451 | 1.476 | 1.70 |

2 | 1.709 | 1.698 | −0.64 | 1.088 | 1.107 | 1.71 |

3 | 1.606 | 1.586 | −1.25 | 1.088 | 1.101 | 1.16 |

4 | 1.629 | 1.613 | −0.98 | 1.088 | 1.107 | 1.68 |

5 | 1.026 | 1.013 | −1.27 | 0.726 | 0.742 | 2.22 |

6 | 1.140 | 1.127 | −1.14 | 0.726 | 0.742 | 2.22 |

Table 6 shows that for spherical shells immersed in infinite water, the errors of the maximum radial displacements between theoretical and numerical simulation results are less than 7%, and the errors of the periods in the free vibration phase between theoretical and numerical simulation results are less than 10%. The deviations between the theoretical and numerical simulation results are within acceptable limits, indicating that the theoretical model is reasonable. The reasons for the deviations may be: (1) The calculation according to the theoretical model does not consider the effect of the additional mass. (2) When the spherical shell contracts inward, the theoretical results overestimate the ability of water to attenuate shell vibrations because the water cannot withstand the tension.

Maximum radial displacement | Period | |||||
---|---|---|---|---|---|---|

Shell no. | Analysis (mm) | FEM (mm) | Error (%) | Analysis (ms) | FEM (ms) | Error (%) |

1 | 1.203 | 1.121 | −6.79 | 1.612 | 1.697 | 5.27 |

2 | 1.102 | 1.065 | −3.36 | 1.152 | 1.228 | 6.60 |

3 | 0.996 | 0.954 | −4.24 | 1.168 | 1.246 | 6.66 |

4 | 0.908 | 0.880 | −3.02 | 1.231 | 1.248 | 1.41 |

5 | 0.753 | 0.719 | −4.44 | 0.744 | 0.812 | 9.21 |

6 | 0.812 | 0.801 | −1.33 | 0.748 | 0.804 | 7.52 |

Maximum radial displacement | Period | |||||
---|---|---|---|---|---|---|

Shell no. | Analysis (mm) | FEM (mm) | Error (%) | Analysis (ms) | FEM (ms) | Error (%) |

1 | 1.203 | 1.121 | −6.79 | 1.612 | 1.697 | 5.27 |

2 | 1.102 | 1.065 | −3.36 | 1.152 | 1.228 | 6.60 |

3 | 0.996 | 0.954 | −4.24 | 1.168 | 1.246 | 6.66 |

4 | 0.908 | 0.880 | −3.02 | 1.231 | 1.248 | 1.41 |

5 | 0.753 | 0.719 | −4.44 | 0.744 | 0.812 | 9.21 |

6 | 0.812 | 0.801 | −1.33 | 0.748 | 0.804 | 7.52 |

The duration of the underwater explosion shock wave is short, which is far less than the vibration period of the spherical shell, as evidenced by the fact that the period in Table 5 is more than ten times the time constant in Table 2. Therefore, the spherical shell vibrates under the pressure of the underwater shock wave in the early stage of the first period, and the shell vibrates freely when the pressure load decays to 0, as shown in Fig. 3. According to Eqs. (20) and (23), the vibration frequency of the free spherical shell is *ω*, while the vibration frequency of the spherical shell in water is *ω*_{d}. The vibration period of a spherical shell immersed in water is slightly longer than that of a free spherical shell.

where $ur\u2212max$ and $ur\u2212max\u2212water$ represent the maximum radial displacement without water and with water protection, respectively.

The reduction factors for spherical shells 1–6 calculated by the theoretical solution and the FEM solution are shown in Fig. 4. The maximum and minimum values of the reduction factors for the theoretical analysis are 44% and 29%, respectively, while those for the finite element are 46% and 29%, respectively. The reduction factors obtained from the FEM are higher than those obtained from the theoretical analysis, which is mainly because the additional mass of water is not considered in the theoretical calculation.

To sum up, the spherical shell will vibrate under the action of the internal explosion shock wave, and the external water medium exerts a damping force on the vibrating shell. The vibration energy in the vessel is propagated to the water in the form of acoustic energy to realize the energy dissipation, thus improving the blast resistance of the vessel.

### 4.2 Effect of Damping Ratio.

Equation (9) shows that the damping ratio mainly depends on four factors: the material of the container, the ratio of diameter to thickness, the density of the medium, and the velocity of wave propagation in the medium. When other conditions are constant, the damping ratio is directly proportional to the diameter-thickness ratio, as shown in Fig. 5. The damping ratio increases with the radius-thickness ratio. The damping ratios calculated from spherical shells 1 to 6 are 0.22, 0.24, 0.33, 0.36, 0.44, and 0.47 in descending order.

As shown in Fig. 6, the peak value of dimensionless radial displacement decreases and the vibration period increases as the damping ratio increases. In the case of $\zeta \u22640.22$, spherical shells in water will not stop vibrating until four periods, while in the case of $\zeta \u22650.24$, the vibration amplitudes of the dimensionless radial displacement will gradually decrease to 0 within 2–3 periods.

Strain growth refers to that the maximum value of the vessel response that does not occur during the first vibration period, but in the later stage. Since Buzukov [23] first observed the strain growth phenomenon experimentally in 1976, many scholars have studied the mechanism of strain growth. Currently, the explanations fall into three main categories: resonance due to explosive loading, linear superposition of vibrational modes, and nonlinear coupling of vibrational modes [24]. In experiments, strain growth generally occurs after three motion periods [6,25]. In the design of the water-protected explosion vessel, the density and wave velocity of the medium are known, and after the vessel material is determined, the damping ratio can be adjusted by adjusting the ratio of vessel radius and thickness so that $\zeta \u22650.24$ which can control the shell vibration within three motion periods. In this case, the strain growth phenomenon will be eliminated. At the same time, when a single explosion occurs, the water-protected ECV will decay below the fatigue limit after several vibration periods. The fatigue problem caused by a single explosion can be ignored, and only the fatigue caused by multiple explosions needs to be considered. In this way, the service life of the water-protected ECV is at least ten to dozens of times longer than that of the free ECV.

## 5 Conclusions

The spherical explosion containment vessel submerged in infinite water is simplified as a damped single-degree-of-freedom vibration system. The underwater explosion shock wave is simplified as an exponential decay load. The radial displacement response solution of the vessel shell is obtained by Duhamel integral.

The maximum radial displacement error between the theoretical analysis and numerical simulation for the water-protected spherical shell is within 7%, and the period error in the free vibration stage is not more than 10%, which verifies the accuracy of the theoretical model.

The maximum radial displacement of the spherical shell in water is at least 29% smaller than that of the free spherical shell. In the free vibration stage, the free spherical shell vibrates with equal amplitude, while the spherical shell in water with gradually decaying amplitude. The vibration period of the former is slightly larger than the latter.

The radius-thickness ratio of the spherical shell can be adjusted to control the damping ratio. When the damping ratio exceeds 0.24, the vibration will stop within three motion periods, which is essential for eliminating strain growth and improving impact fatigue life.

## Funding Data

National Natural Science Foundation of China (Grant Nos. 12072067 and 12172084; Funder ID: 10.13039/501100001809).

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.