# Analysis of Strength Effects on the Dynamic Response of a Shaped-Charge Under Lateral Disturbances
## 3 Dynamic Impact Process of the Jet Under the Influence of Target Strength
### Establishment of Prediction Model  
Dimensional analysis is an effective tool through which to solve contemporary engineering problems, and it has a wide range of applications. Usually, although the same physical quantity can be expressed in different units, its dimension is fixed. Physical quantities have dimensional properties, so it is necessary to ensure that the dimensions on both sides of the equation are consistent when using mathematical formulas to describe the potential laws governing the corresponding problems. This principle is called the principle of dimensional consistency. Based on the principle of dimensional consistency, the relationship between physical quantities in the research problem can be subjected to dimensional analysis. [PERSON] et al. [28] conducted research on the dynamic impact behavior response of the jet by using dimensional analysis but did not focus on considering the influence of the target plate strength on the dynamic penetration depth of the jet. In their approach to analyzing the dynamic impact behavior response problem of shaped charges, especially when considering the simultaneous influence of the lateral disturbance velocity and the target plate strength on the dynamic penetration depth of the jet, the physical quantities influencing the results were examined using dimensional analysis. For the analysis of the conditions examined in this paper, the charge composition remains constant under different working conditions. Therefore, factors such as charge density, charge detonation velocity and charge energy density can be ignored. Due to the complexity of the dynamic impact process of the jet, the density and strength of the jet and the target material play important roles and cannot be ignored. Therefore, the liner height \(l\), caliber \(\phi\), liner thickness \(\xi\), cone angle \(\theta\), charge length \(L\), standoff height \(h\), jet density \(\rho_{\mathrm{j}}\), target plate density \(\rho_{\mathrm{t}}\), target plate elastic modulus \(E_{\mathrm{ty}}\), liner yield strength \(Y_{\mathrm{p}}\), target plate yield strength \(Y_{\mathrm{t}}\), target plate hardening modulus \(E\)th, target plate ultimate strength \(Y_{\mathrm{ts}}\), target plate failure strain \(\varepsilon_{\mathrm{fs}}\), and target lateral velocity \(v_{\mathrm{t}}\) were selected for analysis. \(\mu_{\mathrm{p}}\) and \(\mu_{\mathrm{t}}\) represent the Poisson's ratio of the liner material and the Poisson's ratio of the target material, respectively. This study focused only on metallic materials and does not cover non-metallic materials. As this paper focused on the influence of relative velocity \(v_{\mathrm{t}}\) on the jet impact depth, the target materials used were the same, so the elastic modulus, density and Poisson's ratio of the different targets are very similar and the influence of \(\rho_{\mathrm{t}}\), \(E_{\mathrm{ty}}\) and \(\mu_{\mathrm{t}}\) can be ignored. For the same material liner, the material Poisson's ratio \(\mu_{\mathrm{p}}\) is consistent. Therefore, after simplification, the dynamic penetration depth of shaped-charge jet \(P_{\mathrm{j,dyn}}\) was obtained as follows:  
\[P_{\mathrm{j,dyn}}=f(l,\phi,\xi,L,\rho_{\mathrm{j}},h,Y_{\mathrm{p}},Y_{ \mathrm{t}},E_{\mathrm{th}},Y_{\mathrm{ts}},\theta,\epsilon_{\mathrm{fs}},v_{ \mathrm{t}}) \tag{2}\]  
The jet yield strength \(Y_{\mathrm{p}}\), the standoff \(h\) and the jet density \(\rho_{\mathrm{j}}\) were selected as the reference physical quantities. The determinant was not zero, and the dimensions of the three reference quantities were independent, indicating that any one of the reference quantities could not be represented by the other two reference quantities, so the three reference quantities could be used as a set of basic quantities. This means that the remaining physical quantities could be obtained by linear transformation of the reference physical quantities. In the Plastic-Kinematic model, the Cowper-Symonds model has good effectiveness.