To investigate the dynamic response of a hollow cylindrical shell structure subjected to a near-field underwater explosion, underwater explosion experiments were conducted in a 2 × 2 × 2 m water tank, and high-speed cameras were used to record the interactions between the bubbles generated by the underwater explosion and the hollow cylindrical shell. The high-speed photography results showed that the cylindrical shell experienced a minor degree of deformation during the shock-wave stage. However, during the bubble-pulsation stage, the cylindrical shell experienced significant deformation that surpassed the deformation observed during the shock-wave stage. On this basis, combined with the damage results for the cylindrical shell, a numerical model for the hollow cylindrical shell subjected to an underwater explosion was established using LS-DYNA software. The dynamic process and damage mechanism of a hollow cylindrical shell that was subjected to a near-field underwater explosion were revealed by analyzing the pressures and strains of the shell elements, the velocities and displacements of the nodes, and the variations in the energy.

In recent years, the threat of underwater explosions to submarine gas pipelines has gradually increased due to the intensification of regional conflicts and terrorism. In September 2022, the Nord Stream gas pipeline explosion significantly impacted the ecological environment1 and the global energy supply pattern.2,3 Based on their structural characteristics, subsea gas pipelines can be simplified into single-walled cylindrical shell structures. Therefore, the study of the dynamic responses of single-wall cylindrical shell structures under near-field underwater explosions has practical significance for the protection of submarine gas transmission pipelines.

The effects of underwater explosions on structures primarily consist of shock waves and bubble pulsations resulting from these events.4 According to the damage characteristics of structures subjected to underwater explosions and the characteristics of the explosives used during these events, such explosions can be divided into contact (e.g., near-field and near-boundary explosions) and noncontact explosions (e.g., near-field and far-field explosions).5 For far-field underwater explosions, the shock waves produced by underwater explosions often act on structures within the elastic range, and the effects of bubble pulsations on the structures can be ignored.6–8 For medium- and near-field underwater explosions, damage to structures often results from the combined action of the shock waves and bubble pulsations. For near-field underwater explosions, the characteristics of the shock-wave loads and bubble-pulsation loads are different. Specifically, the shock-wave loads produced by underwater explosions have high peak pressures and short action times, while the bubble-pulsation loads have low peak pressures and long action times.9 Because the shock-wave loads and structural interactions occur much earlier than the bubble-pulsation loads, these characteristics can be explored separately.

In recent years, many scholars have carried out experiments and numerical simulations on cylindrical shell structures subjected to shock waves generated by underwater explosions, mainly focusing on the explosion resistance of composite materials,10 reinforced structures,11–14 covered structures,15,16 and internally filled structures.17,18 However, in near-field underwater explosions, the dynamic responses of structures during these explosions are significantly different from those subjected to underwater explosion shock-wave loading. He19 studied the impact of the explosion distance on the dynamic response of a hull structure that was subjected to a near-field underwater explosion. The results showed that when the distance increased or decreased from a threshold value, the level of deformation decreased; thus, a new concept of impact resistance design for offshore structures was proposed. Yan20 carried out experimental and numerical simulation research on the dynamic responses of reinforced concrete piles subjected to underwater explosions. The research results showed that when the bubble-pulsation stage was fully developed, the damage to the structure was generally more significant than that during the shock-wave stage of a near-field underwater explosion. Tong21 studied the interactions between bubbles and a suspension plate during underwater explosions with small equivalent charges at different explosion distances through experiments and numerical simulations; in addition, this scholar studied the influence of the distance between the bubble and plate on the jet morphology and structural load and revealed the physical mechanism that affected the bubble rupture mode. Chen22 studied the transient response of a polymer-coated circular plate to near-field underwater explosions through a series of underwater explosion tests. The results showed that the foam coating could reduce the shock-wave strength and change the bubble shape and direction of the water jet, but it could not provide the same protection during the bubble loading stage due to the need for a relatively large deformation space. Wang23 used the three-dimensional boundary integral method to simulate the physical characteristics of bubbles near a ship plate, and the results showed that the free surface elevation could significantly affect the bubble dynamics and transient response of the ship. However, there is little research on the dynamic responses of hollow cylindrical shells under the combined action of two loads (e.g., bubbles and shock waves).

In this study, damage tests of hollow cylindrical shells under 2 g explosions using 8701 explosives in a square water tank with side lengths of 2 m were conducted. The interactions between the bubbles and cylindrical shells subjected to the underwater explosions were recorded by high-speed photography. LS-DYNA numerical simulation software was used to simulate the test conditions. By analyzing the variations in the pressures and strains of the elements, the velocities and displacements of the nodes, and the energies of the cylindrical shells, the damage processes and mechanisms of the hollow cylindrical shells subjected to near-field underwater explosions were determined, and the damage characteristics of the hollow cylindrical shells under the combined action of the underwater explosion shock-wave loads and bubble-pulsation loads were revealed.

The tests were carried out in an explosion water tank with a length, width, and height of 2 m. The test object was a hollow circular tube with an outer diameter of 140 mm, a length of 500 mm, and a thickness of 0.5 or 1.5 mm. The two ends of the cylindrical shells were sealed. The small, combined charges used in the test were composed of an LG microelectric detonator and 8701 spherical main charge (Fig. 1). The charge of the LG microelectric detonator was only 15 mg; thus, the energy output of the explosive in each direction was considered to be unaffected by the detonator. As shown in Fig. 2, the charge axis was located directly above the center point of the explosive surface of the circular tube, and the support frame was placed at the bottom of the explosion water tank. The center point of the cylindrical shell was aligned with that of the glass window for high-speed photography. Two groups of tests were carried out, and the explosion distances were determined to be different depending on the wall thickness of the cylindrical shell. The stand-off distance r of the cylindrical shell with a wall thickness of 1.5 mm was 10 cm and that of the shell with a wall thickness of 0.5 mm was 25 cm.

FIG. 1.

Small combined charge.

FIG. 1.

Small combined charge.

Close modal
FIG. 2.

Physical diagram of the test assembly.

FIG. 2.

Physical diagram of the test assembly.

Close modal

Figure 3 shows high-speed photographs of pulsations between a cylindrical shell with a wall thickness of h = 0.5 mm and an explosive bubble. Figure 3(a) shows that the explosive charge detonated at t = 0 ms, emitting bright white light and generating a shock wave. The shock wave first reached the cylindrical shell, which experienced minor deformation. During the initial stage of bubble expansion, due to the distance between the bubble and the cylindrical shell, the bubble was unaffected by the shell, and it expanded in a spherical manner. The cylindrical shell underwent plastic deformation, as shown in Fig. 3(b). During the expansion stage, the bubble was affected by the cylindrical shell to a reduced degree and expanded in a spherical manner. At t = 17.4 ms, the bubble reached its peak size. When the internal pressure was lower than the external pressure, the bubble began to contract. Due to the Bjerknes force on the cylindrical surface, the contraction of the lower part of the bubble was suppressed, and the entire bubble formed a spindle shape, as shown in Figs. 3(f)3(h). Notably, the plastic deformation of the cylindrical shell occurred before t = 4 ms until the bubble shrank to its smallest size, and there were no evident signs of enlargement, as shown in Fig. 3(f). At the beginning of the second pulsation, after the first pulsation, the plastic deformation of the cylindrical shell began to intensify, as shown in Fig. 3(j). With the re-expansion of the bubble, the deformation of the cylindrical shell continued to intensify. According to a comparison of the deformation results of the cylindrical shell in Figs. 3(k), 3(l), and 4, the deformation process was complete at t = 36.4 ms. Based on a comparison of Figs. 3(b) and 3(k), the deformation of the cylindrical shell during the initial stage of the second pulsation was much larger than that during the shock-wave stage (Fig. 4).

FIG. 3.

High-speed camera images of the interactions between the cylindrical shell and pulsating bubbles. Times: (a)–(l) 0, 4, 11, 17.4, 25.6, 30.6, 31.6, 32.8, 33.2, 34, 36.4, and 43.5 ms, respectively. (a) t = 0 ms, (b) t = 4 ms, (c) t = 11 ms, (d) t = 17.4 ms, (e) t = 25.6 ms, (f) t = 30.6 ms, (g) t = 31.6 ms, (h) t = 32.8 ms, (i) t = 33.2 ms, (j) t = 34 ms, (k) t = 36.4 ms, (l) t = 43.5 ms.

FIG. 3.

High-speed camera images of the interactions between the cylindrical shell and pulsating bubbles. Times: (a)–(l) 0, 4, 11, 17.4, 25.6, 30.6, 31.6, 32.8, 33.2, 34, 36.4, and 43.5 ms, respectively. (a) t = 0 ms, (b) t = 4 ms, (c) t = 11 ms, (d) t = 17.4 ms, (e) t = 25.6 ms, (f) t = 30.6 ms, (g) t = 31.6 ms, (h) t = 32.8 ms, (i) t = 33.2 ms, (j) t = 34 ms, (k) t = 36.4 ms, (l) t = 43.5 ms.

Close modal
FIG. 4.

Damage characteristics of the cylindrical shell. (a) Top view and (b) side view.

FIG. 4.

Damage characteristics of the cylindrical shell. (a) Top view and (b) side view.

Close modal

To explore the dynamic process and damage mechanism of the hollow cylindrical shell under the combined action of the shock waves and bubble pulsations generated by an underwater explosion, the finite element software LS-DYNA was used to establish a numerical model of the hollow cylindrical shell when subjected to underwater explosions. The pressures and strains of the elements on the cylindrical shell were computed. The variations in the velocities and displacements of the nodes and the energy of the cylindrical shell over time were analyzed.

1. Equation of state for fluid media

With respect to underwater explosions, many scholars24–26 have used the Grüneisen equation of state to describe the mechanical characteristics of water, which has the following forms.

When water is compressed, the following equation is valid:
P = ρ 0 C 2 μ [ 1 + ( 1 ( γ 0 / 2 ) μ ( a / 2 ) μ 2 ) ] 1 ( S 1 1 ) μ S 2 ( μ 2 / μ + 1 ) S 3 ( μ 3 / μ + 1 ) 2 + ( γ 0 + a μ ) E .
When the water had expanded, the following expression is valid:
P = ρ 0 C 2 μ + ( γ 0 + a μ ) E ,
where P is the pressure; ρ0 is the initial density of the material; S1, S2, and S3 are the slope coefficients of the shear compression wave velocity curve; C is the intercept of the shear compression wave velocity curve; E is the initial internal energy of the explosive; γ0 is the Grüneisen constant; and a is the γ0 first-order volume correction.

2. Equation of state for explosives

The standard Jones–Wilkins–Lee (JWL) equation of state was adopted for the 8701 explosives,27 
P = A ( 1 ( ω / R 1 V ) ) e R 1 V + B ( 1 ( ω / R 2 V ) ) e R 2 V + ω E V ,
where P is the pressure; V is the relative volume of the explosive product; E is the initial internal energy of the charge; and A, R1, B, R2, and ω are constants related to the charge properties. The material parameters were as follows: ρ0 = 1.68 g/cm3, E0 = 8.5 × 106 J/kg, C–J pressure = 37 GPa, D = 8800 m/s, A = 852.4 GPa, B = 18.02 GPa, R1 = 4.55, R2 = 1.3, and ω = 38.

3. Equation of state for steel

The steel material used was Q235 steel, which was described by the simplified Johnson–Cook material model,
σ y = ( A + B ε ¯ P n ) ( 1 + C ln ε ˙ ) ,

The simplified Johnson–Cook material model ignores thermal effects, thereby reducing the computational time. Since the cylindrical shell did not break due to the underwater explosion, the impacts of thermal effects on the damage results were ignored.

The numerical simulation of the cylindrical shell subjected to an underwater explosion involved the arbitrary Lagrangian–Eulerian (ALE) method and the Lagrange coupling algorithm, where the detonation products, air, and water were considered to be fluids, and the steel cylindrical shell was considered the solid. The fluids and solid were coupled for the calculations. The g–cm–μs unit system was used for the calculations, and SOLID164 units were used for the grid division. The three fluids used Eulerian grids and shared common nodes during the modeling process. The fixed device, cover plate, and cylindrical shell used Lagrangian meshes. Because the deformations of the fixed support and cover plate were not considered, the material model used was *MAT-RIGID. To reduce the computational complexity, a 1/4-scale model of the actual model was selected and analyzed, and the two symmetry planes were defined as symmetric constraints, while the other boundaries were nonreflective. The model is shown in Fig. 5.

FIG. 5.

Numerical model.

Figure 6 displays the experimental results that were captured by high-speed photography on the left side of the cylindrical tube shell, while the right side shows the numerical results that were obtained from LS-DYNA. According to a comparison of the experimental results at different times with the numerical simulation results, the numerical simulation results were in good agreement with the experimental results, and the numerical model could accurately simulate the interaction process between the cylindrical shell and the pulsating bubbles. Figure 7 shows a comparison of the results for cylindrical shells and numerical simulations, indicating that numerical simulations could provide reliable results at the macroscopic level. Based on the comparison of high-speed photography, cylindrical shell deformation, and numerical simulation results, it was concluded that the numerical model could effectively simulate the processes and results for cylindrical shells subjected to underwater explosions under these conditions.

FIG. 6.

Experimental and numerical simulation results for the pulsating bubble and cylindrical shell interactions. Times: (a)–(h) 0, 3.4, 17.4, 25, 28.8, 32, 34, and 39 ms, respectively. (a) t = 0 ms, (b) t = 3.4 ms, (c) t = 17.4 ms, (d) t = 25 ms, (e) t = 28.8 ms, (f) t = 32 ms, (g) t = 34 ms, (h) t = 39 ms.

FIG. 6.

Experimental and numerical simulation results for the pulsating bubble and cylindrical shell interactions. Times: (a)–(h) 0, 3.4, 17.4, 25, 28.8, 32, 34, and 39 ms, respectively. (a) t = 0 ms, (b) t = 3.4 ms, (c) t = 17.4 ms, (d) t = 25 ms, (e) t = 28.8 ms, (f) t = 32 ms, (g) t = 34 ms, (h) t = 39 ms.

Close modal
FIG. 7.

Comparison of the damage results and numerical simulation results of cylindrical shells.

FIG. 7.

Comparison of the damage results and numerical simulation results of cylindrical shells.

Close modal

As shown in Fig. 8(a), the shock wave reached the shell before 0.2 ms. At this time, the maximum strain near the center of the upper face of the cylindrical shell was 1.283 × 10−2. With increasing time, the strain concentration area decreased, but the maximum value of the strain increased, as shown in Fig. 8(b). Until t = 0.8 ms, the strain concentration area further decreased, and the maximum strain peaked at 2.987 × 10−2 in the shock-wave stage. At this time, the effect of the shock-wave load from the underwater explosion on the hollow cylindrical shell was basically complete. At t = 33.8 ms, the volume of the bubble decreased to its minimum value, as shown in Fig. 8(e). The strain distribution and the maximum strain in the cylindrical shell at this time did not change significantly compared to those at t = 0.8 ms. The expansion and contraction of bubbles did not cause any new plastic deformations of the cylindrical shell from the end of the shock wave to the end of bubble pulsation. However, when the bubble-pulsation level decreased to a minimum value, the strain distribution in the cylindrical shell changed dramatically. The strain distribution area was divided into left and right segments, and the maximum strain increased by 4.987 × 10−2. At this time, the cylindrical shell underwent significant deformation. After the first pulsation, the bubble began to expand again. With the expansion of the bubble, the deformation of the cylindrical shell gradually increased. At t = 38.2 ms [Fig. 8(i)], the deformation of the cylindrical shell peaked, and the maximum strain was 6.24 × 10−2.

FIG. 8.

Strain nephograms for the cylindrical shell. Times: (a)–(i) 0.2, 0.4, 0.6, 0.8, 33.8, 34, 34.4, 35.2, and 38.2 ms, respectively. (a) t = 0.2 ms, (b) t = 0.4 ms, (c) t = 0.6 ms, (d) t = 0.8 ms, (e) t = 33.8 ms, (f) t = 34 ms, (g) t = 34.4 ms, (h) t = 35.2 ms, (i) t = 38.2 ms.

FIG. 8.

Strain nephograms for the cylindrical shell. Times: (a)–(i) 0.2, 0.4, 0.6, 0.8, 33.8, 34, 34.4, 35.2, and 38.2 ms, respectively. (a) t = 0.2 ms, (b) t = 0.4 ms, (c) t = 0.6 ms, (d) t = 0.8 ms, (e) t = 33.8 ms, (f) t = 34 ms, (g) t = 34.4 ms, (h) t = 35.2 ms, (i) t = 38.2 ms.

Close modal

To further analyze the specific process for the hollow cylindrical shell that was subjected to an underwater explosion, as shown in Fig. 9, six elements, namely, E1–E6, located at different positions on the hollow cylindrical shell, and six nodes, namely, n1–n6, corresponding to these six elements were selected. These parameters were used to examine the damage mechanism of the hollow cylindrical shell that was subjected to both an underwater explosion shock wave and bubble pulsations. Figures 10 and 11 show the peaks in the element pressure and velocity curves, respectively, for nodes at different positions. The acting loads of the two peaks were different (the shock-wave load generated by the underwater explosion caused the first peak, while the bubble-pulsation load caused the second peak), and the time difference between the two peaks was significant. Therefore, the deformation process of the cylindrical shell was divided into two stages: shock wave and bubble pulsation.

FIG. 9.

Schematic diagram of the element locations.

FIG. 9.

Schematic diagram of the element locations.

Close modal
FIG. 10.

Pressure time-history curves at different positions in the cylindrical shell. (a) Whole process; (b) shock-wave stage; (c) bubble-pulsation stage.

FIG. 10.

Pressure time-history curves at different positions in the cylindrical shell. (a) Whole process; (b) shock-wave stage; (c) bubble-pulsation stage.

Close modal
FIG. 11.

Velocities of the nodes at different positions in the cylindrical shell. (a) Whole process; (b) shock-wave stage; (c) bubble-pulsation stage.

FIG. 11.

Velocities of the nodes at different positions in the cylindrical shell. (a) Whole process; (b) shock-wave stage; (c) bubble-pulsation stage.

Close modal

As shown in Fig. 10(b), an underwater blast wave was generated after the charge was detonated. The wave first reached the center of the upper face of the cylindrical shell (element E1), and the arrival time was 63.4 μs after initiation. After 20.1 μs, the pressure of element E1 increased to 232.8 MPa, and the arrival time of the incident wave passing through the air medium inside the cylindrical shell was 103.4 μs. The time to pass through the air medium was 40 μs. The average velocity of the shock wave in the air medium was 350 μs, while the peak pressure of element E6 at the center of the lower surface of the cylindrical shell was 116 MPa, which was far less than the peak pressure on the upper surface of the cylindrical shell. The times and peak pressures of the shock wave that reached the elements at different positions were related to the distances from the elements to the charge center. The greater the distance was, the earlier the arrival time of the incident wave, and the greater the peak pressure of the element. Because the differences in the distances between the elements on the central points of the upper and the lower faces of the cylindrical shell were slight, the time differences between the incident wave and each element on the upper face were minimal, with values of 2 μs. The peak pressures of E1 to E5 were between 228 and 232.8 MPa.

As shown in Fig. 11(b), the velocity of node n1 at the center of the upper face at t = 74 μs peaked, and the maximum speed was 45.6 m/s. Among all the selected nodes, the maximum speed of node n1 was the largest, while the maximum speed of node n6 in the center of the lower surface was the smallest at 4.5 m/s. The maximum velocity difference was not significant at the top of the cylindrical shell for any of the selected nodes. The maximum value was 45.6 m/s at node n1, and the minimum value was 30 m/s at node n5. The displacement of each node shown in Fig. 12 corresponds to the results shown in Fig. 11(b). During the shock-wave stage, the maximum displacement was 0.74 cm at node n1, and the minimum displacement was 0.06 cm at node n6.

FIG. 12.

Time-history curves of the node displacements at different positions.

FIG. 12.

Time-history curves of the node displacements at different positions.

Close modal

At the end of the shock-wave process, the selected node velocity and element pressure values gradually stabilized in specific regions. The node velocities were stable at approximately 0, while the pressures of the elements stabilized at different values. This difference arose because the deformation of the cylindrical shell that was caused by the shock wave from the underwater explosion was not completely stable after the shock wave. During this period, the bubble was still expanding and contracting, and the bubble always acted with the cylindrical shell, resulting in the pressures of some elements stabilizing at negative values. However, some elements stabilized at positive values. The pressures of elements E3 and E5 remained at approximately 0 due to their distances from the deformation area. As shown in Fig. 12, the displacement of each node in the region gradually stabilized, and the expansion and contraction of the bubbles did not lead to any effective plastic deformation of the cylindrical shell.

After the stable period, the pressures of the elements and the velocities and displacements of the nodes suddenly changed between the end of the first pulsation and the beginning of the second pulsation. These changes occurred due to the combined effect of the water jet during bubble collapse and the pressure of the bubble pulse generated at the beginning of the second bubble pulsation.28,29 As shown in Fig. 10(c), at t = 33.756 ms, the pressure of element E1 at the center of the upper face changed first, and after 76 μs, the peak pressure reached 378.35 MPa at t = 33.831 ms. This value was 1.625 times greater than that of element E1 in the shock-wave stage. During bubble pulsation, the element with the maximum peak pressure was E3, which reached a value of 437.43 MPa at 33.904 ms, which was 59 MPa higher than the peak pressure of element E1. The reason for this phenomenon could have been that the bubble jet acted on the central area of the cylindrical shell, and the pressure of element E1 first changed. At this time, element E3 was in tension (the pressure was negative) due to the deformation of element E1. Because the center of the top surface moved downward, the position of element E3 was above that of element E1, and the pressure of element E3 increased when the bubble expanded and impacted the cylindrical shell. From the perspective of the pressure distribution, except for element E3, the peak pressures of the other elements were significantly lower than that of element E1. From the perspective of the axial distribution, the greater the distance from element E1 was, the lower the peak pressure of the element. From the velocity distribution, the maximum velocity was 54.04 m/s at node n5, which occurred at t = 34.2699 ms, while the maximum velocity of node n1 was less than the maximum velocities of nodes n5 and n4. The displacement of central node n1 on the underside of the cylindrical shell was significantly greater than that of the other nodes.

The deformation of the cylindrical shell in the bubble-pulsation stage exhibited the following characteristics that differed from the deformation that occurred in the shock-wave stage. The differences between the peak pressures due to the bubble pulsation in each element were significant, and the characteristics of the peak pressures were independent of the distance from the charge center. During the bubble-pulsation stage, the time differences of the sudden changes in the element pressures were significant, and the maximum difference was 53 μs, which was much larger than the maximum difference during the shock-wave stage of 2 μs. In the velocity distribution, the node with the largest peak velocity in the bubble-pulsation phase was n5, and the peak velocity of node n1 was less than the peak velocities of nodes n5 and n3. In terms of the displacement distribution, during the shock-wave stage, the node displacements in both the radial and axial directions decreased as the distances from the center point of the cylindrical shell top surface increased or decreased. However, the maximum displacements of the nodes located on the cylindrical shell top surface were much larger than those in the shock-wave stage.

Under the near-field action of underwater explosions, the energy of the cylindrical shell changes dramatically. Figure 13 shows the curves for the internal energy, kinetic energy, and total energy (total energy = internal energy + kinetic energy) of the cylindrical shell with time. When the cylindrical shell was subjected to the shock wave, the internal energy and kinetic energy suddenly changed at t = 74 μs. The kinetic energy reached the first peak at t = 80 μs, corresponding to the results shown in Fig. 11, with a value of 11.7 J. In contrast, the internal energy peaked simultaneously at 231 J. After the shock wave, the internal and kinetic energies decreased. The internal energy gradually decreased and stabilized at 203 J, while the kinetic energy gradually decayed to 0 J. Between the end of the first bubble pulsation and the beginning of the second bubble pulsation, the kinetic and internal energies had secondary peaks. Then, the kinetic energy decayed to 0 J, while the internal energy decayed and stabilized at 789 J. As shown in Fig. 14, the strain changes for each element indicated that the expansion and contraction of the bubbles did not lead to effective plastic deformation of the elements during this period. The plastic deformation of the cylindrical shell was 29.3% due to the shock-wave load and 70.7% due to the bubble-pulsation load. In terms of energy, the bubble-pulsation load was the main load that damaged and deformed the cylindrical shell.

FIG. 13.

Variations in the energy of the cylindrical shell with time.

FIG. 13.

Variations in the energy of the cylindrical shell with time.

Close modal
FIG. 14.

Effective strains of elements at different positions.

FIG. 14.

Effective strains of elements at different positions.

Close modal

Based on the analyses of the pressures and strains of the selected elements, the velocities and displacements of the selected nodes, and the energy of the cylindrical shell, the dynamic response of the hollow cylindrical shell that was subjected to a near-field underwater explosion under the specified conditions was divided into the following three stages.

In the first stage (shock wave), the cylindrical shell was plastically deformed under the shock-wave load. The peak pressure of each element and the velocity and displacement of each node were closely related to the distance from the center of the explosion source. The element pressures decreased with increasing distance from the explosive. The peak pressure of each element on the explosion face was approximately 230 MPa because the differences in the distances between the elements on the explosion face and the center of the explosion source were small. The velocity and displacement of each node decreased with increasing distance from the center of the explosion source. The peak velocity of each node on the upper face was in the range of 30–50 m/s, and the maximum displacement was in the range of 0.2–0.75 cm.

In the second stage (stability), during this period, the pressure and strain of each element were stable, the velocity of each node was 0, the displacement was constant, the kinetic energy of the cylindrical shell was 0, and the internal energy was stable at 203 J. Moreover, the expansion and contraction of bubbles did not produce an effective plastic deformation of the cylindrical shell.

In the third stage (bubble pulsation), at the end of the first pulsation and the beginning of the second pulsation, the cylindrical shell exhibited significant plastic deformation, and the degree of deformation was much greater than that of the first stage. During this period, the pressure distributions of all elements significantly differed from those in the first stage. The peak pressure difference of each element on the upper face of the cylindrical shell was relatively large, and the time differences at which sudden changes in each element occurred were significantly larger than in the first stage.

The dynamic response of a hollow cylindrical shell subjected to an underwater explosion shock wave and bubble pulsation was studied via model tests and simulations by using LS-DYNA finite element software. Underwater explosion tests were carried out in a 2 × 2 × 2 m explosion water tank. Images of the interaction process between the underwater explosion bubbles and the hollow cylindrical shell were obtained via high-speed photography, and a fluid/structure coupling model was constructed via the ALE method using LS-DYNA software. The objectives of this paper were to explore the interaction between pulsating bubbles and a cylindrical shell and to evaluate the deformation and energy characteristics of the cylindrical shell. The main conclusions were as follows:

  1. The high-speed camera images showed that under the combined effect of an underwater explosion shock wave and bubble pulsation, the cylindrical shell was slightly deformed in the shock-wave stage. In the bubble-pulsation stage, the cylindrical shell began to deform greatly. The degree of deformation was much greater than that in the shock-wave stage.

  2. A finite element model was established using LS-DYNA software. Based on comparisons of high-speed camera images with cylindrical shell deformation results, the differences between the numerical simulation results and the experimental results were determined to be within an acceptable range. This numerical model could effectively simulate the dynamic process of near-field bubble interactions with cylindrical shells and hollow cylindrical shells under the effect of underwater explosions.

  3. The numerical simulation results showed that the bubble-pulsation load was the most important load for the deformation of the cylindrical shell under the near-field effect of an underwater explosion, where the bubble-pulsation load accounted for 70.7%, and the underwater explosion shock-wave load accounted for only 29.3% of the deformation.

This research was supported by the National Natural Science Foundation of China (NNSFC) (Grant Nos. 12072372 and 52108483), to which the authors are most grateful.

The authors have no conflicts to disclose.

Wen-sheng Mao: Writing – original draft (lead); Writing – review & editing (lead). Ming-shou Zhong: Supervision (equal). Xing-bo Xie: Investigation (equal). Hua-yuan Ma: Software (lead). Gui-li Yang: Validation (lead). Lei Fan: Validation (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
M.
Jia
,
F.
Li
,
Y.
Zhang
et al, “
The Nord Stream pipeline gas leaks released approximately 220,000 tonnes of methane into the atmosphere
,”
Environ. Sci. Ecotechnol.
12,
100210
(
2022
).
2.
Kóczy
,
D.
Csercsik
, and
B. R.
Sziklai
, “
Nord Stream 2: A prelude to war
,”
Energy Strategy Rev.
44,
100982
(
2022
).
3.
J. W.
Goodell
,
C.
Gurdgiev
,
A.
Paltrinieri
et al, “
Global energy supply risk: Evidence from the reactions of European natural gas futures to Nord Stream announcements
,”
Energ. Econ.
125,
106838
(
2023
).
4.
R. H.
Cole
and
R.
Weller
, “
Underwater explosions
,”
Phys. Today
1
(
6
),
35
37
(
1948
).
5.
H.
Yue
,
Warship Survivability
(
Science Press
,
Beijing
,
2022
).
6.
Nagesh
and
N. K.
Gupta
, “
Response of thin walled metallic structures to underwater explosion: A review
,”
Int. J. Impact Eng.
156,
156103950
(
2021
).
7.
W.
Wu
,
A.-M.
Zhang
,
Y.
Liu
et al, “
Local discontinuous Galerkin method for far-field underwater explosion shock wave and cavitation
,”
Appl. Ocean Res.
87
,
102
110
(
2019
).
8.
P.
Wang
,
Z.
Zhang
,
Q.
Yan
et al, “
A substructure method for the transient response of vertical cylinders subjected to shock wave of underwater explosion
,”
Ocean Eng.
218,
108128
(
2020
).
9.
W.
Yang
, “
Numerical simulation of damage to ship structures caused by shock waves from contact underwater explosions
,”
J. Nav. Sci. Technol.
44
(
15
),
12
15
(
2022
).
10.
Y.-Z.
Liu
,
S.
Li
,
X.
Lin
et al, “
Dynamic response of composite tube reinforced porous polyurethane structures under underwater blast loading
,”
Int. J. Impact Eng.
173,
104483
(
2023
).
11.
R.
Surya Praba
,
K. J. S.
Ramajeyathilagam
, and
O.
Structures
, “
Microstructural damage and response of stiffened composite submersible pressure hull subjected to underwater explosion
,”
Ships Offsh. Struct.
18
(
8
),
1116
1131
(
2023
).
12.
Y.
Zhen-Xia
and
C.
Yong
, “
Shock mitigation of circular tubular stiffeners subjected to underwater explosion loads
,”
Noise Vib. Control
43
(
1
),
239
(
2023
).
13.
L.
Renrong
,
L.
Chen
,
Z.
Qingming
et al, “
Instability study on deep-underwater-explosion of convex ring-stiffened cone-cylinder combined shell
,”
J. Beijing Univ. Technol., Nat. Ed.
43
(
3
),
240
251
(
2023
).
14.
R. S.
Praba
and
K.
Ramajeyathilagam
, “
Numerical investigations on the large deformation behaviour of ring stiffened cylindrical shell subjected to underwater explosion
,”
Appl. Ocean Res.
101,
102262
(
2020
).
15.
C.
Yin
,
J.
Liu
, and
Z.
Jin
, “
Experimental and numerical study of the near-field underwater explosion of a circular plate coated by rigid polyurethane foam
,”
Ocean Eng.
252,
111248
(
2022
).
16.
C.
Yin
,
Z.
Jin
,
Y.
Chen
et al, “
Effects of sacrificial coatings on stiffened double cylindrical shells subjected to underwater blasts
,”
Int. J. Impact Eng.
136,
103412
(
2020
).
17.
W.-S.
Mao
,
M.-S.
Zhong
,
X.-B.
Xie
et al, “
Research on the dynamic response of pressurized cylindrical shell structures subjected to a near-field underwater explosion
,”
AIP Adv.
13
, 025046 (
2023
).
18.
S.
Iakovlev
, “
Structural analysis of a submerged fluid-filled cylindrical shell subjected to a shock wave
,”
J. Fluids Struct.
90,
450
477
(
2019
).
19.
Z.
He
,
Z.
Chen
,
Y.
Jiang
et al, “
Effects of the standoff distance on hull structure damage subjected to near-field underwater explosion
,”
Mar. Struct.
74,
102839
(
2020
).
20.
Q.
Yan
,
C.
Liu
,
J.
Wu
et al, “
Experimental and numerical investigation of reinforced concrete pile subjected to near-field non-contact underwater explosion
,”
Int. J. Struct. Stab. Dyn.
20
(
06
),
2040003
(
2020
).
21.
S.-Y.
Tong
,
S.-P.
Wang
,
S.
Yan
et al, “
Fluid–structure interactions between a near-field underwater explosion bubble and a suspended plate
,”
AIP Adv.
12
(
9
),
095224
(
2022
).
22.
Y.
Chen
,
F.
Chen
,
Z. P.
Du
et al, “
Protective effect of polymer coating on the circular steel plate response to near-field underwater explosions
,”
Mar. Struct.
40,
247
266
(
2015
).
23.
J.-X.
Wang
,
Z.
Zong
,
K.
Liu
et al, “
Simulations of the dynamics and interaction between a floating structure and a near-field explosion bubble
,”
Appl. Ocean Res.
78,
50
60
(
2018
).
24.
Y.-X.
Peng
,
A.-M.
Zhang
, and
F.-R.
Ming
, “
Numerical simulation of structural damage subjected to the near-field underwater explosion based on SPH and RKPM
,”
Ocean Eng.
222,
108576
(
2021
).
25.
T.-H.
Phan
,
V.-T.
Nguyen
,
T.-N.
Duy
et al, “
Numerical study on simultaneous thermodynamic and hydrodynamic mechanisms of underwater explosion
,”
Int. J. Heat Mass Transf.
178,
121581
(
2021
).
26.
Z.-L.
Tian
,
Y.-L.
Liu
,
A.-M.
Zhang
et al, “
Jet development and impact load of underwater explosion bubble on solid wall
,”
Appl. Ocean Res.
95,
102013
(
2020
).
27.
Corporation L S T.
,
LS-DYNA Keyword User’s Manual
(
Livermore Software Technology Corporation
,
Livermore
,
CA
,
2007
).
28.
A. P.
Mouritz
, “
Advances in understanding the response of fibre-based polymer composites to shock waves and explosive blasts
,”
Compos., Part A
125
,
105502
(
2019
).
29.
P.
Wanchoo
,
H.
Matos
,
C. E.
Rousseau
et al, “
Investigations on air and underwater blast mitigation in polymeric composite structures—A review
,”
Compos. Struct.
263
,
113530
(
2021
).