## 2 Problem Definition
### Penetration Model  
In UXO cleanup practice at formerly used military sites, it is usual to utilize empirical or semi-empirical physics-based equations, known as phenomenological models, to estimate the final depth of burial (DoB) before the cleaning process [16]. The model utilized in this study for penetration is the well-known Poncelet model [29], which is a phenomenological model based on [PERSON]'s second law of motion [30]. It was chosen for its well-established applicability in modeling high-speed projectile penetration in soils. Unlike purely empirical models that require extensive calibration with experimental data, the Poncelet equation provides a physics-based approach by accounting for resistive forces that vary with velocity and depth. This makes it particularly suitable for probabilistic analyses, where variations in soil properties and impact conditions significantly influence penetration outcomes. Furthermore, its relative simplicity allows for efficient Monte Carlo simulations, enabling a systematic exploration of uncertainty in UXO burial depth predictions without the computational cost of more complex numerical models. [PERSON]'s third law requires that the stress experienced by the target material equals the stress exerted on the projectile. This fundamental requirement leads to the formulation of the [PERSON] (1969) [31] equation (Equation (1)). \[P=\frac{1}{2}\rho_{p}(v-u)^{2}+Y=\frac{1}{2}\rho u^{2}+R \tag{1}\]  
Figure 1: Vertical penetration of 60\({}^{\circ}\) cone projectile in soils. where \(P\) is the pressure at the projectile-target interface, \(v\) is the projectile's impact velocity, \(u\) is the projectile's penetration velocity, \(\rho_{P}\) is the projectile density, \(\rho\) is the target density, \(Y\) is the projectile strength, and \(R\) is the target strength. Equation (1) may result in various scenarios based on the relation between the projectile's strength, \(Y\), and the target's strength, \(R\), as follows:  
* If \(Y<<R\), the projectile collapses on the surface of the target, and no penetration occurs. * If \(Y\cong R\), penetration occurs. The projectile decelerates and fails gradually during penetration. * If \(Y>>R\), penetration occurs. The projectile decelerates but does not deform during penetration. This is the case of rigid UXOs penetrating soils, as considered for this study. For the case of \(Y>>R\), Equation (1) is replaced by the Poncelet (1839) equation [26]. The studied projectile with mass, \(M\), cross-section area, \(A\), and velocity, \(V\), is rigid, so it does not absorb any energy. Consider that in time, \(\Delta t\), the projectile loses velocity, \(\Delta V\). In time, \(\Delta t\), the kinetic energy, \(K.E.\), of the projectile changes by:  
\[\Delta K.E.=\Delta\left(\frac{MV^{2}}{2}\right)=M\;V\Delta V \tag{2}\]  
In that time, material in the path of the projectile must be moved out of the path of the projectile. Using the approximations that the mass flow rate of soil moved in time, \(\Delta t\), is \(\rho AV\) and it acquires velocity, \(V\), as it is pushed away from the projectile's surface, and equating the change in momentum gained by the lateral motion of the soil to that lost by the projectile, we have:  
\[M\Delta V=\rho AV^{2}\Delta t \tag{3}\]  
or  
\[M\frac{\Delta V}{\Delta t}=\rho AV^{2} \tag{4}\]  
where, according to [PERSON]'s second law, \(M\frac{\Delta V}{\Delta t}\) is the force of the projectile. The situation is also analogous to stagnation pressure in a fluid, where it is customary to modify Equation (4) with a drag coefficient to account for geometric simplifications. Hence, the inertial penetration resistance is expressed by Equation (5). \[F=-M\frac{dv}{dt}=C\rho Av^{2} \tag{5}\]  
where \(C\) is the Poncelet drag. The velocity-dependent resistance encountered during penetration is captured by Equation (5). However, in soils there exists an additional frictional resistance to penetration that depends on the mean stress at the penetration front.