## 4 Numerical Analysis of EOI Test and Spalling Test
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Properties** & **Target** & **Projectile** & **Confinement** \\ \hline \(E\) (GPa) & 32.06 & 70 & 210 \\ \hline \(v\) & 0.2 & 0.3 & 0.3 \\ \hline \(\rho\) (kg/m\({}^{3}\)) & 2380 & 2700 & 7800 \\ \hline Yield strength (MPa) & - & 450 & 1200 \\ \hline DFH model parameters & & & \\ \hline Weibull parameters [1] & & & \\ \hline \(m\) & 2.77 & & \\ \hline \(\sigma_{w}\) (MPa) & 2.01 & & \\ \hline \(Z_{\mathit{off}}\) (mm\({}^{3}\)) & 3377 & & \\ \hline Obscuration volume parameter [20] & & & \\ \hline \(S\) & 3.74 & & \\ \hline \(k\) & 0.38 & & \\ \hline KST model parameters & & & \\ \hline Parameters for hydrostatic behaviour [1] & & & \\ \hline \(K\) (GPa) & 21.6 & & \\ \hline \(\epsilon_{v}\)(i) & 0; \(-5\times 10^{-4}\); \(-2.5\times 10^{-3}\) & \\ \hline \(p\)(i) & 0; 10.8; 25 & & \\ \hline Parameters for deviatoric behaviour [1] & & & \\ \hline \(a_{0}\) (MPa\({}^{2}\)) & \(4\times 10^{14}\) & & \\ \hline \(a_{1}\) (MPa) & \(2\times 10^{8}\) & & \\ \hline \(a_{2}\) & 3.2 & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Material parameters used in the numerical simulation. These parameters and other parameters of the DFH model are summarized in Table 4. Figure 13a,b show the damage variable \(D_{1}\) associated with the first principle stress in the case with no structural defect. A large network of cracks oriented mainly radially in a cylindrical frame, centred on the impact point, is observed mainly in the second mid-part of the target. These numerical predictions appear to strongly differ from observations from the ultra-high-speed camera and from the post-mortem observations made in the sacrophagus configuration (Figures 9 and 10). Indeed, whereas large pre-existing defects favour the opening of cracks in mode I and the sliding of cracks in mode II no evidence of a radial cracking network can be found. In order to explain the gap between experimental and numerical results the following explanations can be suggested:  
* Assume that the structural defects alone contribute to the failures during the 3-point-bending tests. In such a situation, the Weibull parameters related to strength distribution of intrinsic defects (non-structural defects) could be strongly underestimated. This would result (in the numerical simulations) in an overestimation of damage related to these defects (lack of identification). * Structural defects could have played a role of a fuse, preventing the triggering and propagation of cracks from intrinsic defects (obscuration mechanism)
* Structural defects could have prevented the compression wave from propagating through the target (impedance failure mechanism)
* Radial crack openings are too small to be seen  
Figure 11: Field of hydrostatic pressure in [MPa] (cut view). (**a**) T = 4 us and (**b**) T = 8 us. Figure 12: Field of first principle stress in [MPa] (**a**) T = 20 μs and (**b**) T = 40 μs. A combination of several of these assumptions is possible, even if the first three assumptions seem to be the most likely to occur. A final numerical simulation including a single structural defect was developed in an attempt to demonstrate the potential influence of a structural defect (Figure 13c,d). To do so, the mesh of the target was split into two parts, both of which were modelled with the KST-DFH model while considering the parameters used in the previous numerical simulation. The splitting plane corresponds to a gap of 0.1 mm located halfway between the front edge and the rear edge (Figure 13c). This value was defined assuming that the curve foot in uniaxial compression tests (with an initial strain ranging from 7 \(\times\) 10\({}^{-4}\) to 1.4 \(\times\) 10\({}^{-3}\)) would correspond to one or two horizontal structural defects with an opening of 0.1 mm (the sample length being 140 mm). The field of damage variable \(D_{1}\) associated with the first principle stress is illustrated in Figure 13c (T = 40 \(\upmu\)s) and Figure 13d (T = 48 \(\upmu\)s).