## 5 Numerical Investigation on DOP
### Modification of Young Equations for DOP Prediction  
Based on an extensive experimental database of projectile impact, Sandia National Laboratories proposed the empirical Young penetration equations to predict the DOP into natural earth materials and concrete. With the expansion of the experimental database, Young equations have been constantly updated, and the latest version for concrete is as follows (in SI units):  
\[DOP=0.00000153\ \ N\,K_{\rm e}K_{\rm h}(t_{c}T_{c})^{-0.06}(11-P)(35/f_{c})^{0.3}(m\ /A)^{0.7}(V_{0}-30.5),\ \ \ \ V_{0}\geq\ 61\ m/s \tag{12}\]  
\[N=0.18(CRH-0.25)^{0.5}+0.56,\ \text{for tangent give nose shapes} \tag{13}\]  
\[K_{\rm e}=(F/\ W_{1})^{0.3} \tag{14}\]  
\[K_{h}=\ 0.46\ (m)^{0.15},\ \text{when}\ m\ <\ 182\ \text{kg};\ \text{else},\ K_{h}=1.0 \tag{15}\]  
where \(N\) is the nose performance coefficient of the projectile, \(K_{e}\) is the correction coefficient for edge effects in concrete targets, and \(K_{h}\) is the correction coefficient for the lightweight projectile. More details about the three coefficients are presented in Reference [32]. \(t_{c}\) is the cure time of concrete, \(T_{c}\) is the thickness of the target, \(P\) is the volumetric percentage of rebars in concrete targets, \(f_{c}\) is the compression strength of concrete, \(m\) is the mass of the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Time** & **Internal Damages of Target U-2** & **The Axial Force of Steel Fibers** \\ \hline \hline \(t=630\ \upmu\)s & \(t=2700\ \upmu\)s & \(t=2700\ \upmu\)s \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Development of internal damages and steel fibers axial force in target U-2. projectile, \(A\) is the cross-sectional area of the projectile, \(V_{0}\) is the striking velocity of the projectile. The Young equations comprehensively consider the parameters that affect the DOP, which makes them applicable in many penetration scenarios. For normal concrete and high-strength concrete with compression strength under 100 MPa, the accuracy of the Young equations is well validated. However, according to the above parametric analysis, the functional relationship between the DOP and the striking velocity of the projectile for UHPC is different from the linear relationship described in the original Young equations. At the same time, the original Young equations do not consider the influence of steel fibers on the penetration depth. Consequently, the original Young equations are not applicable to predicting the DOP into UHPC with ultra-high strength and steel fibers and should be modified. The parametric analysis shows that the functional relationships between the DOP and striking velocity \(V_{0}\), compression strength \(f_{c}\), and volume percentage of steel fibers \(V_{f}\) can be described by an exponential function, power function, and quadratic polynomial, respectively. Hence, the modified Young equation for UHPC without rebar is expressed as:  
\[DOP=\alpha\cdot NK_{c}K_{h}\cdot(t_{c}T_{c})^{-0.06}(f_{c}^{\beta})(k_{0}+k_{1} V_{f}+k_{2}V_{f}^{2})(m/A)^{0.7}(e^{\gamma V_{0}}) \tag{16}\]  
where \(N\), \(K_{c}\), and \(K_{h}\) are the same as the original equations, and \(\alpha\), \(\beta\), \(k_{0}\), \(k_{1}\), \(k_{2}\), and \(\gamma\) are the undetermined coefficients. According to the numerical results of the DOP listed in Table 6 and the equation form expressed as Equation (13), multivariate nonlinear fitting is performed in MATLAB to determine the pending parameters. The values of \(\alpha\), \(\beta\), \(k_{0}\), \(k_{1}\), \(k_{2}\), and \(\gamma\) are determined to be 0.00344, \(-\)0.43, 292.8, \(-\)5048, 77946, and 0.003, respectively.