# Development of Energy-Based Impact Formula--Part I: Penetration Depth
## 3 Development of New Energy-Based Penetration Formula
### Concept of New Formula
#### 3.1.3 Force on the Nose of a Projectile  
To calculate the elastic deformation of a panel, a projectile, and a strain rate, Forrestal's study for normal stress on the nose of a projectile is referred [19; 24; 25; 36]. The force (\(F_{nose}\)) is represented as shown in Equations (46) to (51). \[F_{nose}=\pi r_{p}^{2}\left(\tau_{0,st}A+N_{p}B\rho_{c}V_{imp}^{2}\right) \tag{46}\]  
\[A=\frac{2}{3}-2\left(\frac{\tau_{m,st}}{\tau_{0,st}}\right)\ln\left[\frac{ \gamma}{1+\tau_{0,st}/2E_{c,st}}\right] \tag{47}\]\[ \begin{array}{l}B=\frac{2}{2(1-\eta)}+\frac{\left[3\tau_{0,st}/E_{c,st}+\eta \tau(1-3\tau_{0,st}/2E_{c,st})^{2}\right]}{\gamma^{2}}\\ -\frac{\gamma}{2(1+\tau_{0,st}/2E_{c,st})^{4}}\bigg{[}1+\frac{3(1+\tau_{0,st}/2E _{c,st})^{3}}{(1-\eta\ast)}\bigg{]}\\ \\ \gamma=\left[\left(1+\frac{\tau_{0,st}}{2E_{c,st}}\right)^{3}-(1-\eta\ast) \right]^{1/3}\\ \\ \eta\ast=1-\rho_{0}/\rho\ast\\ \\ N_{p}=\frac{8 CRH-1}{24 CRH^{2}}\end{array}  \tag{48}\]  
where \(r_{p}\) is the radius of a projectile (mm), \(\tau_{0,st}\) is the static shear stress without uniaxial load (MPa); \(\tau_{m,st}\) is the static shear stress under triaxial loads (MPa); \(N_{p}\) is the nose shape factor of a projectile; \(\rho_{c}\) is the density of concrete (kg/m\({}^{3}\)); \(V_{imp}\) is the impact velocity (m/s); \(E_{c,st}\) is the elastic modulus of concrete under the static loading (MPa); \(\eta\ast\) is the function related to the density of concrete; and CRH is the caliber-radius-head. \(\eta\ast\) can be transferred to another form using the bulk modulus and the relation between the change in volume and other physical quantities. When inducing dynamic loading on an object, the body exhibits high hydrostatic pressures on short timescales [37]. Assuming that equal forces (\(F_{cube}\)) act on all surfaces, the original volume (\(Vol_{ori}\)) can be changed to the deformed volume (\(Vol_{d\!f}\)). The reduced volume (\(\Delta Vol\)) is defined as the original volume minus the deformed volume (Equation (52)). It is related to the bulk modulus (\(K\)), force, area, and original volume, and \(\eta\ast\) is newly derived from Equations (52) to (57). \[\Delta Vol=Vol_{ori}-Vol_{def}=\frac{1}{K}\frac{F_{cube}}{A_{cube}}Vol_{ori} \tag{52}\]  
\[Vol_{def}=Vol_{ori}-\frac{1}{K}\frac{F_{cube}}{A_{cube}}Vol_{ori}=Vol_{ori} \bigg{(}1-\frac{1}{K}\frac{F_{cube}}{A_{cube}}\bigg{)} \tag{53}\]  
\[K=\frac{E_{c,st}}{3(1-2v_{c,st})} \tag{54}\]  
\[Vol_{def}=Vol_{ori}\bigg{(}1-\frac{3(1-2v)}{E_{c,st}}\frac{F_{cube}}{A_{cube }}\bigg{)}=Vol_{ori}\bigg{(}1-\frac{3(1-2v)}{E_{c,st}}\sigma_{c,st}\bigg{)} \tag{55}\]  
\[\eta\ast=1-\rho_{0}/\rho\ast=1-\frac{\text{weight/original volume}}{\text{ weight/deformed volume}}=1-\frac{Vol_{def}}{Vol_{ori}} \tag{56}\]  
\[\eta\ast=1-\frac{Vol_{def}}{Vol_{ori}}=1-\frac{Vol_{ori}\bigg{(}1-\frac{3(1-2v _{c,st})}{E_{c,st}}\sigma\bigg{)}}{Vol_{ori}}=\frac{3(1-2v_{c,st})}{E_{c,st}} \sigma_{c,st} \tag{57}\]  
where \(\Delta Vol\) is the volume obtained by subtracting the changed volume from the original volume (mm\({}^{3}\)); \(Vol_{ori}\) is the volume of the original cube (mm\({}^{3}\)); \(Vol_{def}\) is the changed volume of the cube under triaxial loads (mm\({}^{3}\)); \(K\) is the bulk modulus (MPa); \(F_{cube}\) is the acting force on the surface of the cube (N); \(A_{cube}\) is the area of the one cube's face (mm\({}^{2}\)); \(v_{c,st}\) is [PERSON]'s ratio of the concrete under the static loading; \(\sigma_{c,st}\) is the compressive strength of the concrete under the static loading (MPa); and \(E_{c,st}\) is the elastic modulus of the concrete under the static loading (MPa). Using the newly derived \(\eta\ast\), the expressions of \(\gamma\) and A can be changed to Equations (58) and (59), respectively.