## 5 Conclusions--Dynamic Testing and Numerical Modelling  
In the present study two types of dynamic experiments have been developed to improve the understanding of wave propagation and damage modes in the Lingulid sandstone rock when subjected to impact or impulsive loading. The experimental results can be summarized as follows:  
* Spalling tests have been conducted to characterize the dynamic tensile strength of the sandstone rock. However, structural defects are observed to play a major role during compression wave propagation by strongly stiffening the compression wave and reducing the time corresponding to the rising edge. The wave velocity cannot be computed from the difference between the time corresponding to the maximum value of the signal given from the strain gauge near the contact surface and the time corresponding to the maximum value of velocity measured on the free end. In the present work, strain rate was estimated from full-field measurements using DIC-processing applied to pictures from an ultra-high-speed camera. Further, the strength provided by [PERSON]'s formula is observed to be much higher than the static strength provided by bending tests.
* Edge-on-impact tests have been performed in open and sarcophagus configurations. Pre-existing structural defects are again observed to favour the opening of macro-cracks in mode I and the sliding of macro-cracks in mode II leading to energy dissipation by friction. The pre-existing cracks are also restricting the fragmentation of the material originating from less critical defects.  
To complement the experimental results, a numerical investigation was developed to better understand the state of confinement and tensile loading applied to the EOI target when subjected to impact and to the spalled sample. In particular, the numerical investigation aims to evaluate the expected amount of damage (EOI test) and the rear face velocity profile (spalling test) with and without structural defects. The numerical results can be summarized as follows:  
* The numerical simulation of the EOI test involving only the KST model illustrates the large increase of pressure present beneath the confinement system. This justifies the calibration procedure of the KST model based on quasi-oedometric compression experiments (pressure-range up to 360 MPa).
* The numerical simulation of the EOI test also predicts that a large network of radial cracks can be expected in the absence of structural defects.
* The result above is confirmed based on a numerical simulation in which the target of the EOI test is modelled with the coupled KST-DFH plasticity-damage model. Indeed, in the absence of a structural defect, a large amount of damage is predicted, whereas the growth of damage is clearly reduced when a structural defect is implemented in the form of a small gap of 0.1 mm in thickness.
* Finally, numerical simulations of a spalling test performed considering one or two structural defects represented by one or two small gaps of total thickness 0.1 mm allow explaining the sharp rising edge observed on the velocity profile measured on the sample rear face.  
In conclusion, according to the dynamic experimental results and the numerical analysis, pre-existing structural defects play a major role in wave propagation and damage modes. These defects are notorious in the material at issue and will have to be accounted for in any numerical simulation of industrial processes involving Lingulid sandstone (such as percussive drilling), as well as in the identification strategy of constitutive models.  
**Author Contributions:** Methodology, D.S., B.L. and F.S.; Formal analysis, P.F., M.S. and P.-L.L.; Investigation, P.F., M.S., D.S., B.L., F.S., K.W. and P.-L.L.; Writing--original draft, P.F., M.S. and P.-L.L. All authors have read and agreed to the published version of the manuscript.  
**Funding:** The research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.