## 2 Analysis of Underwater Shock Loading and Bubble Pulse  
The most important element of any submersible body is the pressure hull. It contributes about one-fourth to one-half of the total underwater vehicle weight. Figure 1 presents various pressure hull configurations used in submersible bodies [26; 27; 28]. Exposing the hull structure to shock wave and bubble pulsation upon UNDEX events leads to great damage to the hull structure [2]. Figure 2 presents the different configurations that occur during UNDEX events [29; 30]. The loading mechanisms resulting from UNDEX include incident wave, free surface reflection, shockwave, bottom reflection wave, gas bubble oscillation, bubble-pulse loading and bulk cavitation [31]. The compressive load on the structure of the hull is increased due the reflections from the bottom of the ocean which propagates from the shock wave, while the reflection of the shock wave from the ocean's free surface causes a reduction in the pressure [32; 33]. Equation (1) presents the evaluation technique of pressure time history (\(P_{in}(t)\)) for the pressure profile, as follows [2; 34; 35; 36; 37; 38]:  
\[P_{in}(t)=P_{max}e^{-(\frac{t-t_{1}}{0})}\ \ \ \ \ (\text{Mpa})?\ \ \ \ t\geq t_{1} \tag{1}\]  
where \((t)\) denotes the time elapsed after detonation of charge (ms), (\(P_{max}\)) denotes the peak pressure (Mpa), (\(t_{1}\)) denotes the arrival time of shock wave to the target after the detonation of the charge (ms), and (\(\theta\)) denotes the time decay constant that describes the exponential decay (ms). The peak pressure and decay constant depends on the size of the explosion and the stand-off distance from the charge at which the pressure is measured. The peak pressure (\(P_{max}\)), decay constant of the wave (\(\theta\)), impulse (\(I\)), bubble oscillation period (\(T\)), the maximum radius of the first bubble of explosive gas (\(R_{max}\)), and the energy flux density/energy per unit volume (\(E\)) can be expressed as per Equations (2)-(7) [39; 40; 41]. \[P_{max}=K_{1}\left(\frac{W^{\frac{1}{3}}}{R}\right)^{A_{1}}\ \ \ \ \ (\text{Mpa}) \tag{2}\]  
\[\theta=K_{2}W^{1/3}\left(\frac{W^{\frac{1}{3}}}{R}\right)^{A_{2}}\ \ \ \ (\text{ms}) \tag{3}\]  
\[I=K_{3}W^{1/3}\left(\frac{W^{\frac{1}{3}}}{R}\right)^{A_{3}}\ \ \ \ (\text{Mpa-sec}) \tag{4}\]\[T=K_{5}\frac{W^{\frac{1}{3}}}{\left(D+9.8\right)^{5/6}}\quad\left(\text{sec}\right) \tag{5}\]  
\[R_{max}=K_{6}\left(\frac{W}{\left(D+9.8\right)}\right)^{1/3}\quad\left(\text{m}\right) \tag{6}\]  
\[E=K_{4}W^{\frac{1}{3}}\left(\frac{W^{\frac{1}{3}}}{\left(R\right)}\right)^{A_{4 }}\quad\left(\text{m}\cdot\text{kPa}\right) \tag{7}\]  
where \(\left(K_{1},K_{2},K_{3},K_{4},K_{5},K_{6}\right)\) and \(\left(A_{1},A_{2},A_{3},A_{4}\right)\) are constants that depend on the explosive charge type with values as in [39; 42]. These input constants are illustrated in Figure 2. Phenomena of the underwater explosions (UNDEX): shock wave, high pressure and bubble motion. Figure 1: Various wall structures utilized for pressure hulls. (**a**) Reproduced with permission from [26], Copyright Elsevier, 2011. (**b**,**d**–**f**) reproduced with permission from [27], Copyright Elsevier, 2003. (**c**) reproduced with permission from [28], Copyright Elsevier, 2016. (\(W\)) denotes the mass of the explosive charge in (Kg), (\(D\)) denotes the charge depth in \(m\), and (\(R\)) denotes the distance between the explosive charge and target in (m). In UNDEX events, there are two types of subsequent cavitations that occur: the bulk and the local cavitation. The latter is caused by the reflection of the shock wave at the free surface and must be taken into consideration during the analysis of the surface ship. When the shock wave impinges upon the structure, then the total pressure \(P_{tot}(t)\) at the fluid structure interface can be expressed according to Equation (8) [43]:  
\[P_{tot}(t)=P_{in}(t)+P_{c}(t)+P_{st} \tag{8}\]  
where \(P_{in}(t)\) denotes the incident shock wave, \(P_{c}(t)\) denotes the scattered pressure, and \(P_{st}\) denotes the hydrostatic pressure.