# Scaled Experiment of the Detonation Control System for the High-Speed Penetration on Concrete
## 3 Parameter Analysis of the Deceleration  
### The Deceleration Model of the Deep Penetration  
The spherical cavity expansion model proposed by [PERSON] et al. has been verified by a lot of experiments, and it has been found to be suitable as an analysis model for concrete penetration [14; 15; 16]. When the projectile impacts a semi-infinite target, the following model derivation is based on the incompressibility of the material and the invariability of the projectile. The penetration process of projectile perpendicular to target can be divided into two different stages: the crater and steady penetration. In this section, the crater model and the steady penetration model, respectively, are described for high-speed deep penetration, based on the theory of spherical cavity expansion. For the deep penetration, the crater stage is very short for the whole penetration process, and the impact on the penetration deceleration is very small. The relationships between the deceleration, velocity, penetration depth and time are  
\[a=\sqrt{\lambda_{1}}v_{s}\sin\sqrt{\lambda_{1}}t,t\leq t^{\prime} \tag{1}\]  
\[v=-v_{s}\cos\sqrt{\lambda_{1}}t,t\leq t^{\prime} \tag{2}\]  
\[z=-\frac{v_{s}}{\sqrt{\lambda_{1}}}\sin\sqrt{\lambda_{1}}t,t\leq t^{\prime} \tag{3}\]  
where \(a\) and \(v\) are the deceleration and the velocity of the detonation control system during the crater stage, \(z\) is the penetration depth, \(v_{s}\) is the initial impact velocity, \(t^{\prime}\) is the time at the end of the crater stage, and \(\lambda_{1}\) is a constant related to the initial impact velocity and the projectile diameter. Figure 1: Deceleration-time history of the detonation control system for the high-speed deep penetration. The steady penetration is the main stage of the deep penetration. The working environment of the detonation control system at this stage is extremely bad, and the deceleration signal is extremely complex [3,17]. When a rigid projectile with an arbitrary nose impacts a semi-infinite target, the motion equation can be written as  
\[M\frac{\mathrm{d}v}{\mathrm{d}t}=\frac{\pi d^{2}}{4}\Big{(}A\tau N_{1}+B\rho N_ {2}v^{2}\Big{)} \tag{4}\]  
where \(M\) is the projectile mass, \(d\) is the projectile diameter, \(N_{1}\) and \(N_{2}\) are two dimensionless parameters relating to the nose shape and friction, \(\rho\) is the initial density of the target material, \(\tau\) is the ultimate stress of the target material, and A, B are dimensionless material constants. Derived from Equation (4), the relation between deceleration a, time t and velocity v can be obtained  
\[a=\frac{\pi d^{2}}{4M}\Big{(}A\tau N_{1}+B\rho N_{2}v^{2}\Big{)} \tag{5}\]  
\[t=\sqrt{\frac{16M^{2}}{\pi^{2}d^{4}A\tau N_{1}B\rho N_{2}}}\arctan\left(\sqrt {\frac{B\rho N_{2}}{A\tau N_{1}}}v\right)+c_{1} \tag{6}\]  
In the steady penetration stage, the initial condition (\(t=0\), \(v=v_{1}\)) can be used to get the constant \(c_{1}\)  
\[c_{1}=-\sqrt{\frac{16M^{2}}{\pi^{2}d^{4}A\tau N_{1}B\rho N_{2}}}arc\tan\left( \sqrt{\frac{B\rho N_{2}}{A\tau N_{1}}}v_{1}\right) \tag{7}\]  
where \(v_{1}\) is the projectile velocity at the beginning of the steady penetration stage. It can be directly seen in Equations (5) and (6) that the parameters that affect the deceleration and time history are as follows: the projectile mass and diameter, the nose shape of the projectile, the target density and strength. To describe the basic characteristics of the nose shape, a dimensionless quantity \(\varphi\) (the caliber-radius-head) is usually introduced. The expression is  
\[\varphi=\frac{s}{d} \tag{8}\]  
where \(s\) has different meanings due to different shapes of noses. The value range of \(\varphi\) is \(\varphi\in\left[\frac{1}{2},+\infty\right)\). With the increase in the \(\varphi\), the nose shape changes from the blunt nose to the hemi-spherical nose and then to the ogive nose, and the most common one is the ogive nose.