**Figure 1.**
The collision of exploration under water. (**a**) Exploration vehicle with ship; (**b**) Exploration vehicle under the sea.

**Figure 1.**
The collision of exploration under water. (**a**) Exploration vehicle with ship; (**b**) Exploration vehicle under the sea.

**Figure 2.**
The verification of the model and the result. (**a**) The geometry and finite element model (FEM) of structural shell for analysis; (**b**) The material constitutive model for FEM simulation; (**c**) The plot of reaction load vs. defection (0.3 m).

**Figure 2.**
The verification of the model and the result. (**a**) The geometry and finite element model (FEM) of structural shell for analysis; (**b**) The material constitutive model for FEM simulation; (**c**) The plot of reaction load vs. defection (0.3 m).

**Figure 3.**
The typical plot of reaction force vs. displacement.

**Figure 3.**
The typical plot of reaction force vs. displacement.

**Figure 4.**
The plot of reaction force vs. pressure with fitting function. (**a**) Case 1; (**b**) Case 2; (**c**) Case 3; (**d**) Case 4.

**Figure 4.**
The plot of reaction force vs. pressure with fitting function. (**a**) Case 1; (**b**) Case 2; (**c**) Case 3; (**d**) Case 4.

**Figure 5.**
The geometry and mesh model for the cylinder and spherical models. (**a**) The plan view of the cylinder model; (**b**) The plan view of the spherical model.

**Figure 5.**
The geometry and mesh model for the cylinder and spherical models. (**a**) The plan view of the cylinder model; (**b**) The plan view of the spherical model.

**Figure 6.**
The geometry and mesh model for the cylinder and spherical models. (**a**) The mesh of the cylinder model; (**b**) The mesh of the spherical model.

**Figure 6.**
The geometry and mesh model for the cylinder and spherical models. (**a**) The mesh of the cylinder model; (**b**) The mesh of the spherical model.

**Figure 7.**
The initial state of X-stress (Pa) with R = 2.0 m, t = 0.03 m, q = 20 MPa, v = 4 m/s. (**a**) Cylinder shell model; (**b**) Spherical model.

**Figure 7.**
The initial state of X-stress (Pa) with R = 2.0 m, t = 0.03 m, q = 20 MPa, v = 4 m/s. (**a**) Cylinder shell model; (**b**) Spherical model.

**Figure 8.**
The plot of displacement vs. time with R = 2.0 m, t = 0.03 m, q = 20 MPa, and v = 4 m/s.

**Figure 8.**
The plot of displacement vs. time with R = 2.0 m, t = 0.03 m, q = 20 MPa, and v = 4 m/s.

**Figure 9.**
The plot of contact force vs. time with R = 2.0 m, t = 0.03 m, and q = 20 MPa.

**Figure 9.**
The plot of contact force vs. time with R = 2.0 m, t = 0.03 m, and q = 20 MPa.

**Figure 10.**
The plot of energy vs. time with R = 2.0 m, t = 0.03 m, q = 20 MPa, and v = 4 m/s. (**a**) Kinetic energy; (**b**) Internal energy and total energy.

**Figure 10.**
The plot of energy vs. time with R = 2.0 m, t = 0.03 m, q = 20 MPa, and v = 4 m/s. (**a**) Kinetic energy; (**b**) Internal energy and total energy.

**Figure 11.**
The typical plot of force vs. time.

**Figure 11.**
The typical plot of force vs. time.

**Figure 12.**
Effect of velocity on forces and impulse, with t = 0.04 m and 0.06 m. (**a**) t = 0.04 m; (**b**) t = 0.06 m.

**Figure 12.**
Effect of velocity on forces and impulse, with t = 0.04 m and 0.06 m. (**a**) t = 0.04 m; (**b**) t = 0.06 m.

**Figure 13.**
Collision force vs. time with velocities of 12 m/s, 14 m/s, and 16 m/s, and t = 0.06 m.

**Figure 13.**
Collision force vs. time with velocities of 12 m/s, 14 m/s, and 16 m/s, and t = 0.06 m.

**Figure 14.**
Effect of shell thickness on forces and impulse at impact velocities of 4.0 m/s and 8.0 m/s. (**a**) 4.0 m/s; (**b**) 8.0 m/s.

**Figure 14.**
Effect of shell thickness on forces and impulse at impact velocities of 4.0 m/s and 8.0 m/s. (**a**) 4.0 m/s; (**b**) 8.0 m/s.

**Figure 15.**
Effect of elastic modulus and external hydraulic pressure on forces and impulse. (**a**) Force vs. elastic modulus ratio; (**b**) Force vs. external pressure.

**Figure 15.**
Effect of elastic modulus and external hydraulic pressure on forces and impulse. (**a**) Force vs. elastic modulus ratio; (**b**) Force vs. external pressure.

**Figure 16.**
Effect of structural mass on forces and impulse. (**a**) Thickness 0.06; (**b**) Thickness 0.04.

**Figure 16.**
Effect of structural mass on forces and impulse. (**a**) Thickness 0.06; (**b**) Thickness 0.04.

**Figure 17.**
Collision contact force vs. time with fitting result, R = 2.0 m, t = 0.06 m, q = 15 MPa, and E = 210 GPa.

**Figure 17.**
Collision contact force vs. time with fitting result, R = 2.0 m, t = 0.06 m, q = 15 MPa, and E = 210 GPa.

**Figure 18.**
The initial state of shell in the X-stress (Pa). (**a**) 2.8 m/s; (**b**) 2.7 m/s.

**Figure 18.**
The initial state of shell in the X-stress (Pa). (**a**) 2.8 m/s; (**b**) 2.7 m/s.

**Figure 19.**
The X-stress during contact procedure (Pa). (**a**) 2.8 m/s at t = 0.04 s; (**b**) 2.7 m/s at t = 0.04 s; (**c**) 2.8 m/s at t = 0.05 s; (**d**) 2.7 m/s at t = 0.05 s; (**e**) 2.8 m/s at t = 0.06 s; (**f**) 2.7 m/s at t = 0.06 s; (**g**) 2.8 m/s at t = 0.07 s; (**h**) 2.7 m/s at t = 0.07 s.

**Figure 19.**
The X-stress during contact procedure (Pa). (**a**) 2.8 m/s at t = 0.04 s; (**b**) 2.7 m/s at t = 0.04 s; (**c**) 2.8 m/s at t = 0.05 s; (**d**) 2.7 m/s at t = 0.05 s; (**e**) 2.8 m/s at t = 0.06 s; (**f**) 2.7 m/s at t = 0.06 s; (**g**) 2.8 m/s at t = 0.07 s; (**h**) 2.7 m/s at t = 0.07 s.

**Figure 20.**
The X displacement time history at velocities of 2.8 m/s and 2.7 m/s. (**a**) The case of 2.8 m/s velocity; (**b**) The case of 2.7 m/s velocity.

**Figure 20.**
The X displacement time history at velocities of 2.8 m/s and 2.7 m/s. (**a**) The case of 2.8 m/s velocity; (**b**) The case of 2.7 m/s velocity.

**Figure 21.**
The comparison of prediction and numerical simulation.

**Figure 21.**
The comparison of prediction and numerical simulation.

**Table 1.**
The result of criteria force with R = 5.0 m.

**Table 1.**
The result of criteria force with R = 5.0 m.

Case 1 | ${\mathit{p}}_{\mathit{c}}/{\mathit{q}}_{\mathit{c}}$ | p (MN) | q (MPa) | Case 2 | ${\mathit{p}}_{\mathit{c}}/{\mathit{q}}_{\mathit{c}}$ | p (MN) | q (MPa) |
---|

t = 0.05 m E = 210 GPa ν = 0.3 | ${p}_{c}$ = 11.0 ${q}_{c}$ = 11.5 | 3.5 | 7.8 | t = 0.10 m E = 210 GPa ν = 0.3 | ${p}_{c}$ = 88 ${q}_{c}$ = 48 | 78 | 8.0 |

5.0 | 5.2 | 53 | 14 |

8.9 | 2.0 | 39 | 24 |

12 | 1.6 | 31 | 26 |

15 | 1.35 | 26 | 32 |

21 | 1.05 | 22 | 35 |

26 | 0.72 | 18 | 42 |

**Table 2.**
The result of criteria force with R = 3.0 m.

**Table 2.**
The result of criteria force with R = 3.0 m.

Case 3 | ${\mathit{p}}_{\mathit{c}}/{\mathit{q}}_{\mathit{c}}$ | p (MN) | q (MPa) | Case 4 | ${\mathit{p}}_{\mathit{c}}/{\mathit{q}}_{\mathit{c}}$ | p (MN) | q (MPa) |
---|

t = 0.05 m E = 200 GPa ν = 0.23 | ${p}_{c}$ = 18 ${q}_{c}$ = 32 | 49.2 | 1.6 | t = 0.05 m E = 500 GPa ν = 0.23 | ${p}_{c}$ = 45 ${q}_{c}$ = 80 | 75 | 10 |

31.5 | 3.3 | 49 | 12 |

25.1 | 4.2 | 28 | 23 |

16.2 | 5.3 | 23 | 34 |

12.3 | 7.6 | 20 | 41 |

9.6 | 12 | 13 | 60 |

6.3 | 20 | 9.6 | 71 |

**Table 3.**
Fitting results of Equation (4).

**Table 3.**
Fitting results of Equation (4).

Case | 1 | 2 | 3 | 4 |
---|

$\alpha $ | 0.190 | 0.219 | 0.226 | 0.198 |

$\beta $ | −0.919 | −0.789 | −0.811 | −0.981 |

**Table 4.**
The statistic of the contact forces under different cases (unit: MN, GPa, meters, s).

**Table 4.**
The statistic of the contact forces under different cases (unit: MN, GPa, meters, s).

Case | Vel | $\mathrm{\Delta}\mathit{t}$ | ${\mathit{f}}_{\mathit{p}}$ | ${\mathit{f}}_{\mathit{m}}$ | ${\mathit{k}}_{\mathit{s}}\text{}$ | $\mathrm{Im}\text{}$ | Case | Vel | $\mathrm{\Delta}\mathit{t}$ | ${\mathit{f}}_{\mathit{p}}$ | ${\mathit{f}}_{\mathit{m}}\text{}$ | ${\mathit{k}}_{\mathit{s}}\text{}$ | $\mathrm{Im}\text{}$ |
---|

R = 2.0 E = 210 ν = 0.15 t = 0.04 | 2 | 0.044 | 8.3 | 5.0 | 1.67 | 0.22 | R = 2.0 E = 210 ν = 0.15 t = 0.06 | 6 | 0.030 | 41.0 | 23.5 | 1.74 | 0.71 |

3 | 0.044 | 13.1 | 7.4 | 1.74 | 0.33 | 8 | 0.029 | 55.4 | 33.7 | 1.64 | 0.98 |

4 | 0.042 | 18.3 | 10.4 | 1.75 | 0.44 | 10 | 0.028 | 73.4 | 41.9 | 1.75 | 1.17 |

6 | 0.044 | 24.5 | 14.7 | 1.67 | 0.65 | 14 | 0.027 | 105 | 60.8 | 1.73 | 1.64 |

**8** | **0.052** | **26.1** | **16.5** | **1.59** | **0.86** | **16** | **0.028** | **113** | **69.2** | **1.63** | **1.93** |

**Table 5.**
The result of regression analysis.

**Table 5.**
The result of regression analysis.

Thickness | ${\mathit{k}}_{\mathit{v}}\text{}$ | $\mathit{k}{\mathit{m}}_{\mathit{v}}\text{}$ | ${\mathit{k}}_{\mathit{s}}\text{}$ |
---|

0.06 m | 7.0 | 4.2 | 1.67 |

0.04 m | 4.6 | 2.6 | 1.73 |

**Table 6.**
The statistics of the contact forces in different thickness cases (unit: MN, GPa, m).

**Table 6.**
The statistics of the contact forces in different thickness cases (unit: MN, GPa, m).

Case | t/v | $\mathrm{\Delta}\mathit{t}$ | ${\mathit{f}}_{\mathit{p}}$ | ${\mathit{f}}_{\mathit{m}}$ | $\mathrm{Im}\text{}$ | Case | t/v | $\mathrm{\Delta}\mathit{t}$ | ${\mathit{f}}_{\mathit{p}}$ | ${\mathit{f}}_{\mathit{m}}\text{}$ | $\mathrm{Im}\text{}$ |
---|

R = 2.0 E = 210 ν = 0.15 q = 15 | 0.03/1 | 0.063 | 3.07 | 1.91 | 0.12 | R = 2.0 E = 210 ν = 0.15 q = 15 | 0.03/2 | 0.062 | 5.7 | 3.5 | 0.22 |

0.04/4 | 0.042 | 18.3 | 10.4 | 0.44 | 0.04/4 | 0.042 | 18 | 10 | 0.42 |

0.05/4 | 0.034 | 21.2 | 12.8 | 0.44 | 0.05/8 | 0.035 | 44 | 26 | 0.91 |

0.06/4 | 0.031 | 27.0 | 15.0 | 0.47 | 0.06/8 | 0.029 | 55 | 32 | 0.93 |

**Table 7.**
The regression analysis into the effect of shell thickness.

**Table 7.**
The regression analysis into the effect of shell thickness.

Velocity | ${\mathit{k}}_{\mathit{t}}\text{}$ | ${\mathit{C}}_{\mathit{t}}\text{}$ | $\mathit{K}{\mathit{m}}_{\mathit{t}}\text{}$ | $\mathit{C}{\mathit{m}}_{\mathit{t}}\text{}$ | ${\mathit{k}}_{\mathit{t}}/\mathit{k}{\mathit{m}}_{\mathit{t}}\text{}$ | ${\mathit{C}}_{\mathit{t}}/\mathit{C}{\mathit{m}}_{\mathit{t}}\text{}$ |
---|

4 m/s | 449 | −0.33 | 265 | −0.38 | 1.69 | 1.69 |

8 m/s | 950 | −2.5 | 556 | −1.5 | 1.67 | 1.72 |

**Table 8.**
Results with different elastic ratios and external pressures (GPa, MN, meters).

**Table 8.**
Results with different elastic ratios and external pressures (GPa, MN, meters).

Case | e | $\mathrm{\Delta}\mathit{t}$ | ${\mathit{f}}_{\mathit{p}}$ | ${\mathit{f}}_{\mathit{m}}$ | $\mathrm{Im}\text{}$ | Case | q | $\mathrm{\Delta}\mathit{t}$ | ${\mathit{f}}_{\mathit{p}}$ | ${\mathit{f}}_{\mathit{m}}\text{}$ | $\mathrm{Im}\text{}$ |
---|

R = 2.0 t = 0.04 q = 15 v = 3.0 | 0.7 | 0.055 | 6.8 | 4.1 | 0.23 | R = 2.0 t = 0.06 e = 1.0 v = 3.0 | 1.0 | 0.043 | 8.4 | 5.2 | 0.22 |

1.0 | 0.046 | 8.2 | 5.0 | 0.23 | 1.5 | 0.046 | 8.2 | 5.0 | 0.23 |

1.5 | 0.036 | 10.5 | 6.4 | 0.23 | 2.0 | 0.047 | 8.0 | 4.9 | 0.23 |

2.0 | 0.033 | 12.2 | 7.2 | 0.24 | 2.5 | 0.048 | 7.8 | 4.8 | 0.23 |

**Table 9.**
The regression analysis into the effect of elastic moduli and external pressure.

**Table 9.**
The regression analysis into the effect of elastic moduli and external pressure.

Thickness | ${\mathit{k}}_{\mathit{e}}\text{}$ | ${\mathit{C}}_{\mathit{e}}\text{}$ | $\mathit{K}{\mathit{m}}_{\mathit{e}}\text{}$ | $\mathit{C}{\mathit{m}}_{\mathit{e}}\text{}$ | ${\mathit{k}}_{\mathit{e}}/\mathit{k}{\mathit{m}}_{\mathit{e}}\text{}$ | ${\mathit{C}}_{\mathit{e}}/\mathit{C}{\mathit{m}}_{\mathit{e}}\text{}$ | Remark |
---|

0.04 m | 4.2 | 4.0 | 2.4 | 2.4 | 1.75 | 1.67 | Elastic |

0.06 m | −0.4 | 8.8 | −0.26 | 5.4 | 1.67 | 1.72 | Pressure |

**Table 10.**
The statistic of force in different cases (unit: MN, GPa, meters).

**Table 10.**
The statistic of force in different cases (unit: MN, GPa, meters).

Case | $\mathit{m}$ | $\mathrm{\Delta}\mathit{t}$ | ${\mathit{f}}_{\mathit{p}}$ | ${\mathit{f}}_{\mathit{m}}$ | $\mathrm{Im}\text{}$ | Case | $\mathit{m}\text{}$ | $\mathrm{\Delta}\mathit{t}$ | ${\mathit{f}}_{\mathit{p}}$ | ${\mathit{f}}_{\mathit{m}}\text{}$ | $\mathrm{Im}\text{}$ |
---|

t = 0.04 R = 2.0 E = 210 v = 4.0 | 0.5 | 0.033 | 13.1 | 7.8 | 0.26 | t = 0.06 R = 2.0 E = 210 v = 4.0 | 0.5 | 0.027 | 20.3 | 12.2 | 0.33 |

1.0 | 0.042 | 18.3 | 10.4 | 0.44 | 1.0 | 0.031 | 26.7 | 15.2 | 0.47 |

1.5 | 0.050 | 21.4 | 12.5 | 0.63 | 1.5 | 0.034 | 32.2 | 19.4 | 0.67 |

2.0 | 0.057 | 24.4 | 14.2 | 0.81 | 2.0 | 0.039 | 36.4 | 21.6 | 0.84 |

**Table 11.**
The result of regression analysis on Equation (12).

**Table 11.**
The result of regression analysis on Equation (12).

Thickness | ${\mathit{k}}_{\mathit{M}}\text{}$ | ${\mathit{C}}_{\mathit{M}}\text{}$ | $\mathit{K}{\mathit{m}}_{\mathit{M}}\text{}$ | $\mathit{C}{\mathit{m}}_{\mathit{M}}\text{}$ | ${\mathit{k}}_{\mathit{M}}/\mathit{k}{\mathit{m}}_{\mathit{M}}\text{}$ | ${\mathit{C}}_{\mathit{M}}/\mathit{C}{\mathit{m}}_{\mathit{M}}\text{}$ |
---|

0.04 m | 7.4 | 10 | 4.3 | 5.9 | 1.72 | 1.69 |

0.06 m | 10.8 | 15.5 | 6.5 | 9.0 | 1.67 | 1.72 |

**Table 12.**
The criteria velocity statistic (MPa, MN, meters, m/s).

**Table 12.**
The criteria velocity statistic (MPa, MN, meters, m/s).

Case | q | V_{cr} | Max Force | Case | q | V_{cr} | Max Force | Case | q | V_{cr} | Max force |
---|

t = 0.04 R = 2.0 E = 210 | 10 | 16 | 45/3.0 | t = 0.04 R = 1.5 E = 210 | 30 | 9.3 | 30/1.5 | t = 0.04 R = 2.5 E = 210 | 10 | 7.1 | 23/2.0 |

15 | 9.8 | 26/1.7 | 45 | 6.9 | 20/1.0 | 15 | 4.2 | 16/1.3 |

20 | 6.7 | 20/1.3 | 20 | 15 | 40/2.0 | 20 | 3.3 | 11/1.0 |

25 | 5.1 | 17/1.0 | 25 | 10 | 34/1.7 | 25 | 2.2 | 7.0/0.6 |

**Table 13.**
The ratio of criteria in the random parameters (unit: meters, GPa).

**Table 13.**
The ratio of criteria in the random parameters (unit: meters, GPa).

Case | q^{’} | p^{’}/p^{e} | Ratio | Case | q^{’} | p^{’}/p^{e} | Ratio | Case | q^{’} | p^{’}/p^{e} | Ratio |
---|

t = 0.04 R = 2.0 E = 210 | 0.10 | 3.0/1.40 | 2.12 | t = 0.04 R = 1.5 E = 210 | 0.18 | 1.5/0.73 | 2.12 | t = 0.04 R = 2.5 E = 210 | 0.16 | 1.9/0.84 | 2.26 |

0.15 | 1.7/0.90 | 1.97 | 0.26 | 1.0/0.47 | 2.22 | 0.23 | 1.3/0.55 | 2.36 |

0.20 | 1.3/0.65 | 1.99 | 0.12 | 2.0/1.15 | 1.74 | 0.31 | 0.9/0.37 | 2.43 |

0.25 | 1.0/0.50 | 2.03 | 0.14 | 1.7/0.97 | 2.12 | 0.39 | 0.6/0.27 | 2.22 |