## 2 Spalling Tests
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Diameter [mm]** & \(\mathbf{G_{20}}\) & \(\mathbf{G_{60}}\) & \(\mathbf{G_{40}}\) & **Length [mm]** & **Density [kg/m\({}^{3}\)]** \\ \hline \#1 & 44.51 & 19.93 & 60 & 40.16 & 140 & 2382 \\ \hline \#2 & 44.46 & 20.05 & 60.05 & 39.91 & 140 & 2338 \\ \hline \#3 & 44.47 & 20.04 & 60.1 & 40.05 & 140 & 2384 \\ \hline \#4 & 45.46 & 20 & 59.96 & 40.07 & 140 & 2363 \\ \hline \multicolumn{5}{l}{Average} & & & & 2367 \\ \hline \end{tabular}
\end{table} 
Table 1: Initial dimensions and calculated density of the samples tested in spalling. Figure 2: Spalling sample (**a**) prior to testing and (**b**) after testing. The dynamic tensile strength is evaluated by considering the spall strength provided by [PERSON]'s formula (Equation (1)) [17] and the pullback velocity (\(\Delta V_{pb}\)). The pullback velocity is the difference between the maximum particle velocity on the free end and the particle velocity at the first rebound. However, this quantity is used under the assumption that the constitutive law of the material is governed by linear elasticity before reaching the ultimate tensile strength. \[\sigma_{spall}=\frac{1}{2}\rho C_{0}\Delta V_{pb} \tag{1}\]  
\[C_{0}=\sqrt{\frac{E_{dyn}}{\rho}} \tag{2}\]  
Generally, the wave velocity \(C_{0}\) (Equation (2)) is computed from the difference between the time corresponding to the maximum value of the signal given at the strain gauge \(G_{20}\) and the time corresponding to the maximum value of velocity measured on the free end. The values of \(C_{0}\) listed in Table 2 appear smaller than the expected values deduced from Equation (2) when considering the mean density of the Lingulid sandstone samples (2380 Kg/m\({}^{3}\)) and the mean compression Young's modulus (32.06 GPa, Table 2 [1]) deduced from quasi-static uniaxial compression tests (Table 2 [1]). Finally, given that structural defects may have influenced the wave propagation in spalling tests, the wave speed (\(C_{0}=3670\) m/s) provided by the mean value of Young's modulus and density of uniaxial compression tests was considered in the data processing of the spalling tests (Equation (2)). The experimental data of the spalling tests are shown in Table 2 and depicted in Figure 4. The results pertinent to both strain gauges and the virtual gauges from the full field strain data, at the corresponding positions, are presented. A change of the compressive wave is clearly observed by comparing the compressive part of the signal at gauges \(G_{20}\), \(G_{40}\), and \(G_{60}\) (Figure 4). In addition, there is a certain discrepancy between the strain gauge and virtual gauge measurements due to the presence of a non-uniform failure through the sample cross-section (see failure pattern on Figure 2b). Furthermore, the nominal stress calculated from the rear face velocity as \(1/2\rho C_{0}V\) was compared to the nominal stress obtained from the gauge \(G_{20}\) determined from the product \(Ee\) (Figure 5) considering the density (\(\rho=2380\) Kg/m\({}^{3}\)), 1D wave speed (\(C_{0}=3670\) m/s), and [PERSON]'s modulus (\(E=32.06\) GPa) provided in [1]. It is observed that the rising part of the curve calculated from the rear face velocity is much stiffer than the nominal stress obtained from the gauge. This behaviour can be explained by the closure of pre-existing defects (micro-cracks or structural cracks) during wave propagation [10; 18] which can also be observed as heterogeneous deformation in the compressive part of the loading by analysing the space-time field plot of the section average axial strain in Figure 6. These zones could indicate the locations of pre-existing cracks (structural defects). From the measured pullback velocity (\(\Delta V_{pb}=1.65\) m/s), the dynamic tensile strength is deduced (\(\sigma_{dyn}=7.2\) MPa) to be compared to the quasi-static tensile strength of the tested sandstone (\(\sigma_{static}=2.29\) MPa).