# 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
In this model, the dynamic yield strength \(\sigma_{\mathrm{y}}\); the strain rate \(\dot{\varepsilon}\); the hardening parameter \(\beta\) and the plastic hardening modulus \(E_{\mathrm{p}}\) are determined by the elastic modulus \(E_{\mathrm{ty}}\) and the hardening modulus \(Eth\). Thus, it can be found from Equation (2) that the peak value of dynamic yield strength considering strain rate is related to yield strength, hardening modulus and failure strain. In this paper, the liner height \(l\), caliber \(\phi\), liner thickness \(\xi\), cone angle \(\theta\) and charge length \(L\) were not changed. Therefore, by incorporating the Cowper-Symonds model, after dimensional analysis, the expression of the dimensionless parameter \(P_{\mathrm{j,dyn}}/l\) was obtained as follows:  
\[\frac{P_{\mathrm{j,dyn}}}{l}=f(v_{\mathrm{t}}\sqrt{\frac{\rho_{\mathrm{j}}}{Y_ {\mathrm{p}}}},\frac{\sigma_{\mathrm{y}}}{Y_{\mathrm{p}}},\epsilon_{\mathrm{fs}}) \tag{3}\]  
In order to analyze the influence of each variable on the dynamic impact performance of the jet, the relationship between the dimensionless variable \(P_{j,dyn}/l\) and \(v_{\mathrm{t}}\sqrt{(v_{\mathrm{j}}/Y_{\mathrm{p}})}\), obtained by numerical simulation, is illustrated in Figure 6.  
The findings show that there is a strong exponential relationship between \(P_{j,dyn}/l\) and various factors. Therefore, the variables are coupled in exponential form. Applying the \(\Pi\) theorem, let Equation (4) be transformed as follows:  
\[\frac{P_{\mathrm{j,dyn}}}{l}=\chi\Bigg{(}v_{\mathrm{t}}\sqrt{\frac{\rho_{ \mathrm{j}}}{Y_{\mathrm{p}}}}\Bigg{)}^{a}\left(\frac{\sigma_{\mathrm{y}}}{Y_{ \mathrm{p}}}\right)^{b}\epsilon_{\mathrm{fs}}{}^{c} \tag{4}\]  
where \(a\), \(b\) and \(c\) are undetermined exponential coefficient terms. In order to determine the value of the undetermined coefficients in the above expression, the simulation data results were statistically analyzed. The results of the fitted functional relationship are as follows:  
\[\frac{P_{\mathrm{j,dyn}}}{l}=1.213\Bigg{(}v_{\mathrm{t}}\sqrt{\frac{\rho_{ \mathrm{j}}}{Y_{\mathrm{p}}}}\Bigg{)}  
It can be seen from Figure 7 that the dynamic penetration depth of the jet shows a negative correlation with both the strength of the target plate and the lateral disturbance velocity. The surfaces in the Figure 7 represent the predicted values of the established dimensional analysis model; the scattered points are the original results of the numerical simulations; and the surfaces represent the predicted results of the established dimensional model. The \(R^{2}\) value of the equation is 0.946, and the expression obtained by dimensional analysis is in good agreement with the numerical simulation results, and has high consistency with the numerical simulation results. At the same time, the prediction model has a good characterization of the effects of lateral disturbance velocity and target strength on jet dynamic penetration. Combined with the dimensional analysis model, it can be seen that with the increase of the lateral disturbance velocity, the penetration depth of the jet presents an exponential downward trend, which is highly consistent with the conclusion given in the existing relevant literature [7]; For targets with different intensities, the penetration depth under the same lateral disturbance is inversely proportional to the target intensity. Compared with the lower lateral disturbance velocity, the difference of the impact of different target intensities on the dynamic penetration depth of the jet is significantly higher than that of the higher lateral disturbance velocity, that is, compared with the target intensity, the lateral disturbance velocity has a greater impact on the dynamic penetration depth of the jet.