## 4 Numerical Analysis of EOI Test and Spalling Test
Several observations can be made:  
* At T = 40 \(\upmu\)s, the amount of damage is strongly reduced in the second part of the target. However, due to impedance breakage, a few short cracks develop in the first part of the target close to the midplane, following a longitudinal (axial) orientation (perpendicularly to the splitting plane). * At T = 48 \(\upmu\)s, whereas in the area far from the structural defect the amount of damage tends to what was observed without structural defect (Figure 13a,b) the area close to the structural defect remains likely undamaged. This demonstrates that structural defects might prevent or at least reduce the growth of damage in their vicinity. Figure 13: Field of damage within the target (in red: \(D_{1}\) = 1, in blue: \(D_{1}\) = 0). (**a**,**c**) T = 40 \(\upmu\)s, (**b**,**d**) T = 48 \(\upmu\)s. (**a**,**b**) Case without structural defect. (**c**,**d**) Case with a structural defect located halfway between the front edge and the rear edge. Finally, it can be concluded that the influence of structural defects cannot be ignored. This concerns the identification strategy of model parameters (Weibull parameters) as well as their explicit representation in the numerical modelling. Finally, a series of numerical simulations of the spalling test was conducted with the Abaqus-explicit Finite-Element code to evaluate the possible effect of one or two structural defects on wave propagation and on the resulting velocity profile measured on the sample rear free surface. To do so, a cylindrical sample, 140 mm in height, 46 mm in diameter, was meshed with \"C3D8R\" eight-node solid brick elements with reduced integration and a mesh size of about 2 mm. The compression pulse depicted in Figure 5 (G20 strain gauge) was used as a pressure pulse applied to the sample surface (left side in Figure 14a). The KST-DFH model was used as the constitutive law for the Lingulid sandstone. In the first numerical simulation, no structural defect was considered, whereas a single structural defect and two structural defects were considered, respectively, in the second and third numerical simulations. The structural defect that corresponds to a gap of 100 \(\upmu\)m is located at 60 mm from the rear face in the second calculation. In the third calculation, two gaps of 50 \(\upmu\)m each, located at 30 and 60 mm from the rear face, are considered (Figure 14a). The numerical results in terms of the velocity profile averaged on the rear surface of the sample are reported in Figure 14b. In the first numerical simulation, a smooth rising front and descending front up to the rebound with a similar shaped profile to the applied pressure pulse are noted. On the contrary, the two calculations performed with one or two structural defects show a sharp rising edge resulting from a shock generated when the two or three blocks come into contact. In addition, a smaller pull-back velocity (difference between maximum velocity and velocity at rebound) is observed in the third calculation (\(\Delta V_{ph}\) = 1.85 m/s) compared to the second calculation (\(\Delta V_{ph}\) = 2.86 m/s). It results from the entrapped wave in the second block, located in between the two structural defects. Finally,  
Figure 14: Numerical simulations of spalling tests with the KST-DFH model. (**a**) Considered position of structural defects in the second and third numerical simulations. (**b**) Particle velocity averaged on the rear face without, with one, or two structural defects. the third calculation gives a result closer to the experimental value (\(\Delta V_{pb}=1.65\) m/s, test n\({}^{\circ}\)2). In conclusion, both calculations with a structural defect enable us to explain that, in spalling experiments, the rising part of the pulse calculated from the rear face velocity was much stiffer than the nominal stress obtained from the G20 strain gauge.