# Double-Bend and Semi-Spherical Energy Deflectors' Blast Load Efficiency--A Comparative Study
## 3 Numerical Modelling
### Material Parameters  
A simplified Johnson-Cook model was used to simulate the behavior of a deflector subjected to detonation wave loading. The Johnson-Cook constitutive model is the first model that considers strain hardening, strain rate effect, and metal thermal softening [40]. Although developed in 1983, it remains the most commonly used constitutive model in numerical analyses of dynamic processes and phenomena. A flow equation describing stress-strain behavior beyond the elastic limit has the following form:  
\[\sigma_{flow}=\big{(}A+B(\varepsilon^{p})^{n}\big{)}\Big{(}1+Cln\dot{\varepsilon }^{p*}\big{)}\big{(}1-(T*)^{m}\big{)} \tag{1}\]  
where \(\sigma_{flow}\)--equivalent flow stress, \(A\)--yield stress, \(B\)--plastic strain hardening coefficient, \(\varepsilon^{p}\)--effective plastic strain, \(n\)--strain hardening exponent, \(C\)--plastic strain rate coefficient, \(m\)--thermal coefficient, \(\dot{\varepsilon}^{p*}\)--normalized plastic strain rate, \(T*\)--homologous temperature. The homologous temperature is expressed by following equation:  
\[T*=\frac{T-T_{r}}{T_{m}-T_{r}} \tag{2}\]  
where \(T\)--sample temperature, \(T_{r}\)--reference temperature, \(T_{m}\)--melting temperature of the materials sample. Such effects as strain hardening, strain rate effect, and thermal softening are included in the Johnson-Cook equation in the form of parts presented in the brackets and multiplied  
Figure 8: Pendulum with modelled air domain. Figure 7: Numerical model of the ballistic pendulum with mounted energy deflector. 1—Steel ropes; 2—pendulum; 3—counterweight; 4—front plate; 5—spacers; 6—mounting plate; 7—deflector. by itself. The first part--in the first bracket--shows the influence of the strain hardening on the equivalent flow stress in the form presented originally by [PERSON] [41]. The strain rate effect's influence in logarithmic form is presented in the second part of the equation, and the thermal softening is in the last part. The only difference between the simplified and the full Johnson-Cook model is that the simplified model does not consider the influence of thermal effects on the material properties. Thus, such a model can be expressed as:  
\[\sigma_{flow}=\big{[}A+B(\epsilon^{P})^{n}\big{]}\Big{(}1+Cln\epsilon^{p*} \Big{)} \tag{3}\]  
In order to determine the necessary JC material constants, a curve-fitting process was carried out on experimental data [42; 43; 44]. Initially, a yield strength equal to 353.3 MPa was taken as the value of material constant A, identical to that of the experimental study. The variability of parameter B was studied in the interval <0, 1400> with a step of 1, n in the interval <0, 1> with a step of 0.1, and C in the interval <0, 0.1> with a step of 0.0003. For such input parameters, a fit expressed as R\({}^{2}\) of 0.964 was obtained. Due to the fact that this value was relatively low, it was decided to remove the constraint on parameter A. As a result, values of constants deviating from those commonly found in the literature were obtained, but characterized by an R\({}^{2}\) of 0.9924 (Figure 9). The constitutive model values used are presented in Table 2.  
The steel rope's material was described with the use of CABLE DISCRETE BEAM. In this material model, force is generated by rope only during stretching [45]:  
\[F=\max(F_{+}+K\AL,0). \tag{4}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Description** & **Unit** & **Value** \\ \hline \(\rho\) & Density & g/cm\({}^{3}\) & 7.89 \\ \(E\) & [PERSON]’s modulus & MPa & 209,600 \\ \(\upsilon\) & [PERSON]’s coefficient & - & 0.307 \\ \(A\) & Material constant & MPa & 14 \\ \(B\) & Material constant & MPa & 1214 \\ \(n\) & Material constant & - & 0.28 \\ \(C\) & Material constant & - & 0.0936 \\ \(\epsilon_{f}\) & Plastic strain at the failure & - & 0.308 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Johnson–Cook material constants. Figure 9: Results of the Johnson-Cook material curve fitting.