The damage characteristics of a ribbed cylinder in the torpedo compartment shell is explored. An arbitrary Lagrange–Euler method is used to establish the fluid–structure interaction model for analyzing the ribbed cylinder’s response under near-field underwater explosion while in motion. The influence of detonation direction and standoff distance on the dynamic response of the moving ribbed cylinder is considered. The investigation reveals that the cylinder’s motion causes an uneven distribution of bubble load and secondary load, stemming from cavitation zone collapse, on the shell. This imbalance leads to a notable deflection difference between the shell’s front and rear sections, with maximum deformation concentration at the rear. In addition, in comparison to the lateral condition, static state analysis shows reduced average deflection and increased maximum deflection when the explosion point is above or below the shell, while in the sailing state, both average and maximum deflections increase. Notably, when the charge radius is between 6 and 15 times, the average damage rate in the sailing state consistently remains lower than that in the stationary state, while the maximum damage rate is higher at a specific burst distance.

Torpedoes have played a pivotal role in underwater offensive and defensive confrontations since their inception, significantly contributing to the three-dimensional combat model of naval warfare. The advancement of disciplines such as the marine environment, acoustics, and control technology has enabled modern torpedoes to effectively detect decoys and countermeasures, posing a substantial threat to warships.1,2 Hard-kill anti-torpedo weapons have emerged as an effective means to counter these intelligent and self-guided torpedoes. The Anti-Torpedo Torpedo (ATT), a recent development, exhibits automatic searching and precise guidance capabilities, greatly increasing the likelihood of successful damage infliction compared to conventional hard-kill weapons. Consequently, the ATT has become a focal point of research for numerous navies worldwide.3 The advancements in anti-torpedo technology and torpedo survivability have significant implications for naval development. Therefore, a comprehensive investigation into the damage characteristics of the ribbed cylinder in the torpedo compartment shell amidst near-field UNDEX is indispensable for augmenting the damage capabilities of hard-kill anti-torpedo weapons and bolstering torpedo survivability.

UNDEX loads include shock wave load and bubble load.4,5 The publication “Underwater explosions”6 elucidated various phenomena and principles pertaining to UNDEX. Expanding on this, Zamyshlyaev and Yakovlev7 proposed empirical formulas for UNDEX loads, which remain significant references in the field of UNDEX research. To date, researchers have conducted extensive studies on the dynamic response of structures.8–15 Jamali and Jalili16 investigated the damage modes of a water-filled double-layered cylinder structure after impact through experiments and numerical simulations. Mao et al.17 explored the damage mechanisms of a reinforced cylinder under near-field UNDEX and summarized the influence of shell thickness on structural damage. Praba and Ramajeyathilagam18 employed a combination of numerical simulations and small-volume underwater explosion tests to analyze the anti-explosion mechanism of pressurized cylindrical structures. Regarding the study of interaction effects between UNDEX and dynamic structures, Li19 utilized an auxiliary function method to simulate the interaction between bubbles and rigid bodies, examining how the rigid motion of a cylinder affects the shape of the bubble.

Currently, scholars conduct in-depth research on the response characteristics of cylindrical structures under underwater explosion loads.20–22 However, most related studies are based on the assumption that the movable target is in a stationary state,23 which does not align with engineering reality. For highly maneuverable underwater targets such as torpedoes, studying their dynamic responses solely based on static response results cannot fully capture the actual situations, therefore requiring investigations into the response characteristics of targets under motion. Based on these analyses, this study establishes a fluid–structure interaction model for the ribbed cylinder in motion subjected to near-field UNDEX, investigating the damage characteristics of the cylinder shell under such explosive loads. The results serve as a reference for optimizing the detonation of hard-kill anti-torpedo weapons and designing torpedo survivability.

The structural response to near-field UNDEX constitutes a complex nonlinear dynamic process involving significant deformation and strong nonlinearity.24,25 The ALE method is one of the most established methods for solving this class of dynamics problems. This paper utilizes the ALE method to establish a fluid–structure interaction model for a ribbed cylinder experiencing near-field UNDEX under a sailing condition.

The ALE method combines the characteristics of Euler and Lagrange approaches, offering significant advantages in dealing with transient fluid-structure interaction and large deformation issues arising from near-field UNDEX. In this method, the boundary is treated using a Lagrange approach, while the interior is handled using an Euler approach, minimizing mesh distortion within the material-free boundary and allowing for the tracking of free surfaces. After each time step deformation, the Euler elements are regularized.26,27

The governing equations of the ALE method are shown below:28 

Mass conservation equation:
ρt=ρvxiwiρxi.
(1)
Momentum equation:
vt=(σij,j+ρbi)ρwivixj.
(2)
Energy equation:
Et=(σijvi,j+ρbivj)ρwiExj.
(3)
Here, ρ is the density, E is the specific internal energy, bi is the unit mass tension, and σij is the stress tensor.

Upon detonation in a water environment, the explosive generates shock waves that propagate through the medium, subsequently exerting an impact on structures. Moving torpedoes primarily experience a detrimental effect from shock waves. The peak pressure of the shock wave serves as a crucial parameter for characterizing the magnitude of the load, with its accuracy being a prerequisite for dependable calculation results concerning structural response. To validate the precision of the numerical method employed in determining the shock wave load, a free field UNDEX model is utilized. A 1 kg spherical TNT charge serves as the explosive source for comparing the computed shock wave peak pressure results with empirical and experimental findings.29 

The empirical equation for shock wave load is shown below:6 
pm=44.1×106W3R1.5,6RR012,52.4×106W3R1.13,12<RR0240,
(4)
where pm is the peak pressure, W is the weight of TNT, R is the standoff distance, and R0 is the charge radius.

Figure 1 presents a comparison between the calculated peak pressure and values derived from the empirical formula and experiment. The observed temporal and spatial distribution of impact pressure aligns closely with the trend predicted by the empirical formulas. The maximum relative error between the numerical and empirical/experimental values is 5.91%, with a minimum of 2.32%. These findings affirm the high accuracy of the numerical method employed for UNDEX load calculation, validating its suitability for near-field UNDEX fluid–structure interaction model computations.

FIG. 1.

Pressure validation.

FIG. 1.

Pressure validation.

Close modal

In order to validate the effectiveness of the numerical method for computing the structural response, the calculated results obtained from the one-dimensional model were remapped onto the flow field where the structure is situated and compared with the experimental data presented in Ref. 30. The experiments were carried out in a cylindrical explosion pool of a diameter of 5 m and a water depth of 4 m. The explosive charge used in the near-field UNDEX damage experiments was a spherical TNT with a mass of 0.1 kg. The target structure was a ribbed cylinder with an outer diameter of 100 mm, a length of 185 mm, and a wall thickness of 2.8 mm. A comparison of the experimental results with the numerical simulation results is shown in Fig. 2. The deformations at the center of the blast face, as determined by both the numerical simulation and experiment, were 2.87 and 3.26 mm, respectively, resulting in a relative error of 11.96%. The damage modes observed in the shell were similar. As the simulation did not account for the secondary pressure wave effect, the computed deformation of the shell was slightly smaller than the experimental result. This disparity is reasonable and affirms the reliability of the numerical method in accurately calculating the structural response.

FIG. 2.

Comparison of the experimental result30 with numerical simulation.

FIG. 2.

Comparison of the experimental result30 with numerical simulation.

Close modal

1. Finite element model

The ribbed cylinder model comprises a cylinder shell, ribs, a nose cone, and a bottom cover. The cylinder has an outer diameter of 533 mm, with a thin shell segment length of 1100 mm and a wall thickness of 6 mm. Inside the cylinder, four ribs are evenly distributed, each with a width of 10 mm and a height of 15 mm. There is a rigid connection between the ribs and the cylinder, utilizing ALE elements, totaling 768 elements. The nose cone and bottom cover serve to enclose the air within the cylinder, being rigidly connected to both ends of the cylinder using Lagrange elements, with a total of 2448 elements.

The flow field encompasses both the external water region surrounding the cylinder and the air domain, utilizing Euler elements for modeling. The flow field has dimensions of 3500 × 3000 × 3000 mm3. The basin’s hydro static pressure is equal to 20 m water depth. The flow field is bounded by non-reflecting boundaries. The gravity is 9.81 N/kg. Figure 3 shows the numerical simulation model.

FIG. 3.

Numerical simulation model.

FIG. 3.

Numerical simulation model.

Close modal

The size of the flow field grid influences both the magnitude of the shock wave overpressure and its attenuation rate. To determine an appropriate grid size for the flow field, grid convergence verification is conducted by dividing the flow field grid into sizes of 20, 25, 30, and 40 mm. Figure 4(a) illustrates the temporal pressure evolution and peak pressure at a specific location within the flow field for these grid sizes. Remarkably, when the flow field grid size is 25 mm, the peak pressure exhibits a deviation of 1.33% from that of the 20 mm grid, while the pressure attenuation trend remains similar. Considering computational efficiency, a flow field grid size of 25 mm is selected. Within the ALE method, the grid size of the structure determines the deformation process and outcomes of the elements. The shell structure grid is partitioned into grid sizes of 60 × 120, 80 × 140, 100 × 160, and 120 × 180 (axial grid number × circumferential grid number) for convergence assessment. Figure 4(b) shows the displacement magnitudes and stability duration at the center of the blast face for these grid sizes. Notably, dividing the grid into 100 × 160 yields relatively stable displacement and stability duration. Consequently, it is established that a shell grid has 100 × 160, totaling 16 000 shell elements.

FIG. 4.

Verification of grid convergence.

FIG. 4.

Verification of grid convergence.

Close modal

2. Equation of state and constitutive model

In the numerical calculation process, the polynomial EOS is used to describe the water medium. When the water is under pressure (μ > 0), the EOS is
p=A1μ+A2μ2+A3μ3+(B0+B1μ)ρ0e.
(5)
When water is subjected to tension or cavitation (μ < 0), the EOS is
p=T1μ+T2μ2+B0ρ0e,
(6)
μ=ρρ01,
(7)
e=(ρgH+patm)ρB0,
(8)
where A1, A2, A3, B0, B1, T1, and T2 are constants; ρ is the density of water; ρ0 is the initial density of water; H is the depth of water; patm is the atmospheric pressure; and e is the specific internal energy. The parameters of the water medium are shown in Table I.
TABLE I.

Parameters of water, TNT, and AL7075.

Parameters of waterParameters of TNTParameters of AL7075
ParameterValueParameterValueParameterValue
A1/GPa 2.20 A/GPa 373.77 A/MPa 495 
A2/GPa 9.54 B/GPa 3.75 B/MPa 303.6 
A3/GPa 1.46 R1 4.15 C 0.0097 
B0 0.28 R1 0.9 n 0.39 
B1 0.28 ω 0.35 m 0.77 
T1/GPa 2.20 E/GJ/m3 7.0 Tmelt/K 635 
T2/GPa ρ/g/cm3 1.63 Troom/K 294 
ρ0/g/cm3 1.0 pcJ/GPa 21.0 ρ/g/cm3 2.804 
e/J/kg 933.3 DcJ/km/s 6.93 K/GPa 69.9 
⋯ ⋯ ⋯ ⋯ G/GPa 26.7 
Parameters of waterParameters of TNTParameters of AL7075
ParameterValueParameterValueParameterValue
A1/GPa 2.20 A/GPa 373.77 A/MPa 495 
A2/GPa 9.54 B/GPa 3.75 B/MPa 303.6 
A3/GPa 1.46 R1 4.15 C 0.0097 
B0 0.28 R1 0.9 n 0.39 
B1 0.28 ω 0.35 m 0.77 
T1/GPa 2.20 E/GJ/m3 7.0 Tmelt/K 635 
T2/GPa ρ/g/cm3 1.63 Troom/K 294 
ρ0/g/cm3 1.0 pcJ/GPa 21.0 ρ/g/cm3 2.804 
e/J/kg 933.3 DcJ/km/s 6.93 K/GPa 69.9 
⋯ ⋯ ⋯ ⋯ G/GPa 26.7 
The ideal gas EOS is used to describe the air medium (Table II),
p=(γ1)ρe.
(9)
TABLE II.

Average and maximum deflection under cases 3, 6, and 7.

Case 3Case 6Case 7
Static state Average deflection 183.39 177.57 178.13 
(mm)    
Maximum deflection 219.78 224.38 228.11 
(mm)    
 Average deflection 162.83 164.18 167.18 
Navigational (mm)    
state Maximum deflection 234.71 235.33 241.07 
 (mm)    
Case 3Case 6Case 7
Static state Average deflection 183.39 177.57 178.13 
(mm)    
Maximum deflection 219.78 224.38 228.11 
(mm)    
 Average deflection 162.83 164.18 167.18 
Navigational (mm)    
state Maximum deflection 234.71 235.33 241.07 
 (mm)    

The ideal gas density is represented by ρ = 1.225 × 10−3 g/cm3, the adiabatic index is γ = 1.4, and the specific internal energy is e = 2.068 × 105 kJ/kg.

The JWL EOS is used to describe the pressure of the detonation products:
p=A1ωR1VeR1V+B1ωR2VeR2V+ωEV,
(10)
where E is the specific internal of TNT; V is the relative specific volume of the detonation products; and A, B, R1, R2, and ω are constants. When the explosive expands to a specific volume, its behavior can be accurately described using the ideal gas EOS.
AL7075 is selected as the cylinder and rib material. Its dynamic behavior is described using the linear EOS and the Johnson–Cook model. The linear EOS can be represented as
p=K1u,
(11)
where K is the bulk modulus and u is the compression ratio.
The equation of the Johnson–Cook model can be represented as
σ=(A+BεPn)1+Clnε̇ε0̇(1THm),
(12)
TH=(TTroom)(TmeltTroom),
(13)
where A is the static yield limit, B and n are strain hardening parameters, C is a strain rate-related parameter, ε̇ is the effective plastic strain, ε̇0 is the quasi-static strain rate, Tmelt is the melting temperature, Troom is the room temperature, and m is the temperature softening parameter. The parameters of AL707530 are listed in Table I.

The nose cone and bottom cover material is iron, which is characterized by the linear EOS and the elastic model. The material exhibits a bulk modulus of 15.9 GPa, a shear modulus of 81.8 GPa, and a density of 7.83 g/cm3. These two parts serve as rigid bodies. The specific material parameters will not be listed.

3. Case setting

Torpedoes worldwide typically achieve speeds exceeding 40 knots. Therefore, the numerical simulation is set to an average speed of 25 m/s (∼48.59 knots) for the ribbed cylinder, with the velocity directed along the positive Z-axis. A 5 kg spherical TNT charge is selected as the explosive source. The simulation considers five different standoff distances and three orientations. The dimensionless parameter Le represents the standoff distance, the ratio of the distance from the detonation point to the cylinder surface, and the charge radius (R0). The values of Le are 6, 8, 10, 12, and 15. The positive Y-axis points upward, the positive Z-axis represents the forward direction, and the X-axis indicates the lateral direction. To simulate the presence of ballast and suspend the structure in the flow field, displacement constraints are applied to the nose cone and bottom cover in the X and Y directions. Figure 5 illustrates the case indication.

FIG. 5.

Case indication.

The shock wave load has a significant impact on the damage experienced by a torpedo shell during UNDEX. This study focuses on the dynamic response of the cylindrical shell during the impact phase and the initial bubble expansion phase, disregarding the secondary pressure wave effect. The deformation–time curves and final deformation at different gauge points are presented in Fig. 6. Figure 7 depicts the distribution of Mises stress at classic time within the shell under case 3. Gauge points (hereinafter referred to as P) are evenly distributed along the central line of the blast face, with a 110 mm distance interval between adjacent gauge points. It should be noted that in this paper, the moment when the shock wave is about to reach the shell surface is taken as the starting moment, and the cylinder starts to move from this moment.

FIG. 6.

Deflection on the blast face under case 3.

FIG. 6.

Deflection on the blast face under case 3.

Close modal
FIG. 7.

Mises stress distribution at the classical moments under case 3.

FIG. 7.

Mises stress distribution at the classical moments under case 3.

Close modal

In the static state, the stress distribution on the shell is symmetrical from both ends, while in the navigational state, the stress on the head of the shell is smaller than that on the tail. As shown in Fig. 8, the movement of the cylinder alters the pressure gradient in the flow field, resulting in a localized high-pressure region near the head of the shell. According to fluid flow principles, the fluid tends to converge toward the tail after passing through the midsection of an object, impacting the motion of bubbles. In the static state, the bubble flows around the external surface of the shell toward the back due to the influence of buoyancy and inertia. In the navigational state, the bubble tends to move toward the tail, causing elevated stress at the rear section.

FIG. 8.

Shell response histories under case 3.

FIG. 8.

Shell response histories under case 3.

Close modal

The shell undergoes deformation when subjected to the shock wave, with the deformation speed reaching its maximum level. It can be seen from Fig. 9 that the shell’s air-filled interior releases a significant portion of the energy as a shock wave into the flow field, forming a reflected wave that propagates toward the detonation point. Upon reaching the bubble’s surface, the reflected wave partially transforms into a rarefaction wave and reflects into the flow field. The convergence of these two reflected waves induces a rapid reduction in local pressure, below the cavitation threshold. The collapse of cavitation area results in a secondary load. The secondary load phenomenon predominantly occurs at different regions under two states. In the static state, the secondary loading positions are symmetrically distributed in the front and rear sections of the cylinder. However, the secondary load is greater in the front section of the cylinder under sailing conditions. The sudden change in the local response speed of the shell caused by secondary loading can be seen in Fig. 8.

FIG. 9.

Flow field pressure at 0–0.5 ms.

FIG. 9.

Flow field pressure at 0–0.5 ms.

Close modal

The cylinder’s movement changes the position of the secondary load on the shell over time, causing pronounced response differences between the head and tail regions. Under static conditions, the deformation deflection, velocity, and acceleration peaks occur at the center of the shell. The deformation deflection distribution on the head and tail surfaces of the shell exhibits symmetry. Nevertheless, in the navigational state, the maximum deformation position of the shell is posterior compared to the static state. Furthermore, the deformation in the rear section of the shell surpasses that in the front section significantly.

The damage to ships caused by UNDEX primarily occurs below the hull, while torpedoes face threats from underwater weapons in all directions. The influence of detonation point orientation on the damage characteristics of the cylindrical shell is investigated by combining the computational results obtained from numerical simulations under cases 3, 6, and 7. Figures 10 and 11 illustrate the deformation deflection and plastic strain under cases 3, 6, and 7. Figure 12 displays the histories of shell deformation. It is observed that during the impact phase, the shell’s response processes are similar under different orientation conditions. The deformation velocity and acceleration peaks at the blast face are almost the same, showing that the bubble load is the main factor causing shell damage variations under the same standoff distance but different orientation conditions.

FIG. 10.

Deflection under cases 3, 6, and 7.

FIG. 10.

Deflection under cases 3, 6, and 7.

Close modal
FIG. 11.

Equivalent plastic strain under cases 3, 6, and 7.

FIG. 11.

Equivalent plastic strain under cases 3, 6, and 7.

Close modal
FIG. 12.

Shell response histories under cases 3, 6, and 7.

FIG. 12.

Shell response histories under cases 3, 6, and 7.

Close modal

In the static state, the deformation takes the form of a cone-shaped depression in cases 6 and 7, while in case 3, an arc-shaped depression is observed. When the bubble is positioned directly above or below the cylinder, the uplift of the bubble does not alter its loading position on the cylinder. The combined impact of the delayed flow load and impact load exacerbates the damage to the shell. Particularly, when the cylinder is positioned directly below the bubble, the fluctuating pressure of the bubble during its upward movement shifts toward the shell, resulting in more severe damage from the delayed flow load.

The average deflection and maximum deflection of the shell under cases 3, 6, and 7 are shown in Table II. In the static state, the average deflection for cases 6 and 7 is less than that for case 3, while the maximum deflection increases. In the sailing state, both the average and maximum deflections for cases 6 and 7 surpass those for case 3.

Compared to the static state, the motion of the cylinder leads to a 11.21%, 7.23%, and 6.15% reduction in the average deflection on the blast face but causes a 6.79%, 4.88%, and 5.68% increase in the maximum deflection in cases 3, 6, and 7, respectively. When the explosion point is on the side of the cylinder, the movement of the cylinder significantly influences the response of the shell. The detonation point’s lateral placement leads to a change in the delayed flow load’s position on the shell, caused by the upward movement of the bubble, restricting its effective interaction with the impact load. The contributions of the shock wave load and the delayed flow load to shell damage vary with the standoff distance. Further analysis is necessary to comprehend the damage characteristics of the shell in the lateral orientation at different standoff distances.

According to Ref. 31, the shock wave pressure attenuates exponentially in both time and space. Hence, this study investigates the influence of standoff distance on the damage characteristics of the ribbed cylinder during the navigational state.

Figure 13 presents the deformation deflections at each gauge point for cases 1–5, while Fig. 14 illustrates the distribution of equivalent plastic strain on the shell. The deformation mode of the shell transitions from spherical depression to arc deformation with increasing blast distance. The varying damage modes observed in the shell under different standoff distances represent the distinct influences of the shock wave load and bubble load. The high peak pressure and short duration of the shock wave result in significant local damage to the shell, whereas the relatively lower intensity and longer duration of the bubble load contribute to the overall deformation. Cases 1 and 2 exhibit similar damage modes, with plastic deformation primarily occurring in the center area, resulting in notable localized depressions. The deformation mode of the shell remains consistent under cases 3–5, characterized by arc deformation. In the sailing state, the position of the recessed area of the shell is further back than in the static state.

FIG. 13.

Deflection under cases 1 to 5.

FIG. 13.

Deflection under cases 1 to 5.

Close modal
FIG. 14.

Equivalent plastic strain under cases 1 to 5.

FIG. 14.

Equivalent plastic strain under cases 1 to 5.

Close modal

At close bursting distances, the shell exhibits more pronounced deformation in the longitudinal direction when in the sailing state. In the stationary state, the longitudinal deformation of the shell is symmetrically distributed. As inferred from the above-mentioned analysis, the collapse of the cavitation zone is the primary cause of differences in the shock wave phase of the shell response results. Closer blast distances enhance the coupling effect between the shock wave and the shell, leading to the generation of a large cavitation area near the shell surface. The secondary loading from the cavitation zone collapse induces a sudden change in the shell’s velocity potential, resulting in a larger burst velocity potential at the surface, as illustrated Fig. 15. The shell’s response speed under this secondary loading decreases exponentially with increasing explosion distance. Nevertheless, at the initial expansion stage of the bubble, the movement of the cylinder results in the delayed flow load exerting more influence on the second half of the cylinder. The prolonged impact of the bubble load eventually leads to greater deformation in the second half of the shell than in the front half, as illustrated in Fig. 16.

FIG. 15.

Histories of shell response velocity at P8.

FIG. 15.

Histories of shell response velocity at P8.

Close modal
FIG. 16.

Histories of shell response at P1 and P8 under case 1.

FIG. 16.

Histories of shell response at P1 and P8 under case 1.

Close modal
The average and maximum deflections of the cylindrical shell in cases 1 to 5 are shown in Table III. Analysis of the deformation deflection changes at various gauge points on the shell in cases 1 to 5 reveals characteristic damage modes of the ribbed cylinder under the navigational state. The damage rate is defined as the ratio of deformation deflection to the outer diameter of the cylinder. The preliminary estimation of damage modes within the given range of the involved fitting curves is demonstrated in Fig. 17. The fitted curves are expressed as
Dsa=1.4958eLe10.65620.2453,
(14)
Dsm=2.6652eLe6.01260.0811,
(15)
Dna=0.4173e(Le5.7453)226.8744+0.09636,
(16)
Dnm=0.7767e(Le4.9361)228.9302+0.1171,
(17)
where Dsa and Dsm are the shell average damage rate and maximum damage rate in the static state and Dna and Dnm are the shell average damage rate and maximum damage rate in the navigational state, respectively.
TABLE III.

Average and maximum deflection under cases 1–5.

Case 1Case 2Case 3Case 4Case 5
Static state Average deflection (mm) 323.12 245.31 183.39 125.37 65.13 
Maximum deflection (mm) 479.23 337.08 219.78 152.64 73.87 
Navigational state Average deflection (mm) 272.97 236.57 162.83 104.91 60.01 
Maximum deflection (mm) 460.75 360.84 234.71 134.71 75.43 
Case 1Case 2Case 3Case 4Case 5
Static state Average deflection (mm) 323.12 245.31 183.39 125.37 65.13 
Maximum deflection (mm) 479.23 337.08 219.78 152.64 73.87 
Navigational state Average deflection (mm) 272.97 236.57 162.83 104.91 60.01 
Maximum deflection (mm) 460.75 360.84 234.71 134.71 75.43 
FIG. 17.

Fitted curves of the shell damage rate.

FIG. 17.

Fitted curves of the shell damage rate.

Close modal

Analysis of the fitting curve reveals that, within a specified range, both the average and maximum damage rates of the shell in the two states roughly follow an exponential function distribution, although with distinct trends. The average damage rate in the navigation state consistently remains lower than that in the static state. However, the maximum damage rate surpasses it at a specific explosion distance. In addition, under navigation conditions, the shell damage rate exhibits a slower decrease with increasing blast distance. For the ring-ribbed cylindrical structure considered in this study, characterized by a constant structural size, speed, and charge amount, the damage rate decays more slowly at distances 6 < Le < 10 under the navigation condition. At the structural scale and sailing speed given in this paper, there exists a bursting distance in the range 6 < Le < 10, at which the explosive load of the current charge can be efficiently utilized. Importantly, it is hard to meet the conditions required for the bubble collapse jet to effectively damage a high-speed moving torpedo in practical engineering. Therefore, the predictive model only considers the effects of shock waves and delayed flow load during the initial expansion phase of the bubble.

The dynamic response of the ribbed cylinder subjected to near-field UNDEX during navigation is analyzed. This study seeks to enhance the comprehension of the moving torpedo compartment shell damage under near-field UNDEX. The research reveals the following:

  1. The cylinder’s state affects its damage pattern. Compared to the static state, the cylinder’s movement induces secondary loading from cavitation area collapse and an asymmetric bubble load distribution. It leads to a faster response in the front during the shock wave stage and increased deformation in the rear during the bubble load stage.

  2. The detonation orientation and standoff distance notably affect the shell response, with the deformation decreasing as the standoff distance increases. Compared to the lateral condition, explosions above or below the shell lead to increased maximum deformation and average deflection during navigation. If the explosion point is below the cylinder, stronger coupling of the bubble load and shock wave load causes greater shell deformation.

  3. In the standoff distance range of 6 < Le < 15, the average and maximum shell damage in both states show varying trends with increasing blast distance. The average damage rate is lower under navigation conditions, while the maximum damage rate is higher at a specific explosion distance.

This work was supported by the National Natural Science Foundation of China (Grant Nos. 52179086 and 52269022) and the Central Government Guides’ Local Science and Technology Development Funding Project (Grant No. 23ZYQA0320).

The authors have no conflicts to disclose.

Wei Han: Conceptualization (equal); Resources (equal); Supervision (equal). Yifan Dong: Date curation (equal); Methodology (equal); Software (equal); Writing - original draft (equal). Rennian Li: Supervision (equal); Validation (equal); Writing - review & editing (equal). Yukun Zhang: Methodology (supporting); Writing - original draft (supporting). Lu Bai: Date curation (supporting); Software (supporting).

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

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