## 4 Numerical Results and Discussion  
Finite element simulations of the high-strain-rate compression tests were performed by LS-DYNA 971. As mentioned before, the length of the steel striker bar was 0.5 m, while the length of the aluminum  
Figure 31: The quasi-static and high-strain-rate test results of \(8\times 8\times 8\) specimens.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Strain Rate** & **Modulus** & **Fracture Stress** & **Fracture Strain** \\ \(\mathbf{s^{-1}}\) & **MPa** & **MPa** & \\ \hline \(4.180\times 10^{-3}\pm 3.030\times 10^{-7}\) & 985.73404 \(\pm\) 3.5528 & 13.7369 \(\pm\) 2.8497 & 2.167 \(\times\) 10\({}^{-2}\)\(\pm\) 2.740 \(\times\) 10\({}^{-3}\) \\ \(1813.5829\pm 48.4181\) & 8553.5254 \(\pm\) 85.4232 & 35.0658 \(\pm\) 1.9018 & 6.520 \(\times\) 10\({}^{-3}\)\(\pm\) 7.650 \(\times\) 10\({}^{-4}\) \\ \(2146.7858\pm 12.5833\) & 8391.0523 \(\pm\) 398.3287 & 37.0466 \(\pm\) 1.5355 & 7.360 \(\times\) 10\({}^{-3}\)\(\pm\) 6.435 \(\times\) 10\({}^{-4}\) \\ \(2309.9629\pm 30.2695\) & 8351.9092 \(\pm\) 417.6605 & 38.0406 \(\pm\) 1.7206 & 8.820 \(\times\) 10\({}^{-3}\)\(\pm\) 3.495 \(\times\) 10\({}^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 5: The fracture stress, strain, and modulus in the in-plane direction under increasing strain rates.  
Figure 32: The fracture stress, strain, and modulus of the quasi-static and high-strain-rate tests of \(8\times 8\) specimens: (**a**) fracture stress and strain; (**b**) modulus.  
striker bar was 0.4 m. The numbers of elements used in these striker bars were 60,000 and 48,000, respectively. The length of both the input and the output bars was 1.5 m. The number of elements used in these bars was 180,000. Their diameters were all 14.5 mm. The number of elements on the cross-section was 300 for all these bars. The *MAT_ELASTIC material model was used for all the bars. The values of density, Young modulus, and Poisson's ratio for the steel and aluminum bars were \(7.85\times 10^{3}\) kg/m\({}^{3}\), 206 GPa, 0.27 and \(2.8\times 10^{3}\) kg/m\({}^{3}\), 72 GPa, 0.32, respectively. The eroding surface-to-surface contact was defined between the specimen and input/output bar. The eroding single-surface contact was defined between the composite sub-laminates [28; 52]. The \"soft\" contact option type 2 was utilized for the eroding contact [52]. The automatic surface-to-surface contact was defined between the striker bar and input bar. An initial velocity was defined for the striker bar according to the impact velocity measured in the experiments under a certain impact pressure [17].  
The cohesive element was applied to simulate the cohesive failure using the *MAT_COHESIVE_MIXED_MODE material model [24; 26; 52]. The traction-separation relationship is shown in Figure 33. EN and ET are the initial slopes of the stress-displacement relationship before separation in the normal and tangential directions, with units of stress/length. T and S are the peak stress in the normal and tangential directions. \(\Delta\)N and \(\Delta\)S are the total displacement at complete separation in the normal and tangential directions. In addition, GIC and GIC are the energy release rates for the normal and tangential directions, with units of stress - length. The value of EN/ET was chosen to be sufficiently large, so as to essentially prevent separation before the unloading [24].