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Results Project 5

Numerical modeling of impact cratering processes

The objective of the project is divided into two parts:

  • The development of numerical models of impact crater formation.

    Among other aspects, this includes the development of a material model for sandstone (dry and water saturated) based on existing EoS for quartzite and water, and a porosity compaction model. Meso-scale models are used for a detailed analysis of shock-induced closure of pore space and multi-phase material behavior. Results from meso-scale modeling are used as benchmarks to callibrate macro-scale models of the bulk behaviour of sandstone.

  • The application of models to further our understanding of impact crater formation and to validate the models against experiments.

    Here we focus on the effect of target properties such as porosity and water content on crater dimensions and shock wave propagation.


Numerical models are an important tool to support the interpretation of observations gained in other subprojects.

Mesoscale modeling

Sandstone has a polygranular structure, predominantly consisting of quartz grains, wih pores that may be partially or totally filled with water. The influence of the filling is a big challenge to our models. It can be investigated by so-called “meso-scale modeling” that explicitly resolve the internal granular structure of the material. We employ the Eulerian hydrocode (or shock physics code) iSALE and the Smoothed Particle Hydrodynamics (SPH) code SOPHIA.

In a first, simplified approach, we approximated sandstone as a homogeneous quartzite matrix, i.e. a compact conglomerate of quartz grains with regularly distributed and geometrically simplified pores.

In order to characterize the material response of sandstone under shock, simple loading conditions need to be applied. The so-called flyer plate test configuration, we use here, considers a quartzite rod striking a laterally confined sandstone rod to generate a one-dimensional strain wave propagating through the sample.


Simulation setup of flyer plate impact. A compact quartzite rod impacts a sandstone rod surrounded by quartzite layers. The configuration was chosen to ensure a smooth planar wave signal.

As the shock wave passes through, pores are compacted. Depending on the strength of the quartzite matrix and the pore content, they exhibit different crushing behavior. When shear strength of quartzite is neglected or shock pressure significantly exceeds the strength of the matrix, quartzite behaves like a fluid and jets from the upper side into the pore generating another shock pulse at opposite side of the pore.. The presence of water hinders the closure of the pore.


Pore crushing in dry sandstone and water-saturated sandstone, quartzite strength is neglected.

Click here for a video of pore crushing.

Pore space collapse due to shock loading results in an amplification of the shock pressure. The localized amplifications can reach up to 4 times the initial pressure depending on pore space distribution, the size of the pore relative to the length of the shock pulse, pore filling, and surrounding material properties.

Click here for a 3D animation of pore space crushing

The animation shows the propagation of a shock wave through a porous sample in 3D. We see a complete crushing of pore space. Although the overall shock amplitude decreases during the propagation through the porous sample, the shock pressure is ramped up locally as a result of pore space collapse. The localized amplification of shock pressure can be seen as oscillation in the animation.

Macroscale modeling

In macro-scale models sandstone is considered a homogeneous material whose porosity is a macroscopic “state” parameter. The so-called ε-α model enables to describe how porosity evolves under the compressive volumetric strain of the bulk material.

The use of macro-scale models compared against and calibrated with simple numerical meso-scale loading experiments and constraints from laboratory shock experiments, can be extended in a second step to study cratering phenomena. For this purpose, an appropriate strength and damage model to describe the brittle/ductile behaviour of geological rocks has been adjoined to the macro-scopic equation of state model. Additionally, a dilatancy model, quantifying the opening of cracks and flaws as a result of shear and tensile bulking, has been developed and implemented. In the following the effect of porosity and water content as well as the influence of dilatancy is presented. The extent of the damage zone in the numerical models corresponds approximately with the experimental observations.


Effect of different target properties on crater formation and crater damage. Contours are representing damage: blue is undamaged material, red is fully-damaged material. The profiles in a) represent the final crater shape in experiments. The diameter of the final crater in the experiments is enlarged due to spallation which is not considered in the numerical model so far.

As shown on the upper figure that captured the point in time when the transient crater was approximately reached, the extent of the damage zone is larger in non-porous targets than in porous material. The presence of water mainly acts as a mechanism preventing the crushing of pores, and thus the shock wave is less attenuated. Crater diameter increases for 100 % water-saturation in comparison to the dry porous target, and the 50 % water-saturated target. Crater depth in the water-filled target is smaller than in dry targets. The model predictions are confirmed qualitatively by comparison with experiments as far as data were available.

3D impact simulations show that damage at different depth is not fully symmetric to the axis along the vertical impact trajectory. The cross section in plane depicts concentric fractures that are interrupted and deviate from circular shape.


3D simulation of cratering in iSALE: damage and fractures can be observed. 

Click here for a video of damage growth


Mesoscale modeling of strength effects

In order to further improve the description of strength effects and inter-granular cracking in sandstone on the mesoscale, a more realistic 3D granular structure of sandstone has been computationally generated. From a back-scatter electron image analysis of a sandstone slice, a statistical distribution of quartz grain diameters was generated. Subsequently, the so-called “structure generator” places spheres according to the given size distribution in a virtual box and adjust the size to satisfy an arbirtraily defined porosity. In a final step, a mesh is generated using tetrahedrons. All interfaces and pore space between grains are resolved in the mesh.


Sphere placement in box, growing/shrinking phase and meshing. 

From Hopkinson-Bar measurements with sandstone, macroscale failure strength parameters can be derived. They are used as reference values for simulations with the Lagrangian Finite-Element code MESOFEM. This code is used to determine meso-scale strength parameters by comparison with macro-scale observations, and to perform high-resolution meso-scale simulations of sandstone strength. First results of simulations of uniaxial tensile loading indicate that both tensile and shear stresses occur at grain interfaces and lead to inter-granular cracking (represented as blue interfaces in the figure).



Mesoscale sandstone structure under uniaxial tensile loading in its initial state and after crack initiation and propagation. Grain interfaces undergo a combined shear and tensile loading.

Cooperation with other MEMIN projects

The meso-scale models of shock compression and the macro-scale models of crater formation provided valuable information for the interpretation of observations at the experiments. On the other hand, observational data such as damage and compaction of pore space were essential to calibrate the numerical models. The determination of the velocity of the propagating shock wave in project 4  is used to refine the numerical calculations regarding the material parameters and the extent of the damage zone. A comparison of the numerical model and ultrasound tomography measurements shows a good agreement of the size of the zone where material is completely damaged.



Comparison of a numerical model (left) and ultrasound tomography measurements (right). For both figures, blue indicates undamaged material and red indicates zones with totally damaged material. The latter corresponds to lower sound velocities. Both models agree well regarding a zone of totally damaged material (low velocity) near the crater floor and a zone of partly damaged material (middle velocity).


The interpretation of low-shock pressure experiments in project 7 was supported by meso-scale modelling of such experiments. The localized shock amplification of 4 times the initial pressure as a consequence of pore closure predicted by meso-scale modelling may explain the observed high shock pressure modifications such as the formation of diaplectic glass at a nominal shock pressure of only 12.5 GPa. The quantification of pressure amplifications and induced temperatures during and after shock compression by meso-scale modelling can also be used to estimate the thermodynamic conditions at the first contact of the projectile with the porous sandstone target (project 8). The obtained pressures can be directly related to post-shock temperatures. For example, in a compact sandstone (quartzite) a determined pressure during shock of 70 GPa results in post shock temperature increase of ~ 3000K which would facilitate the formation of melt phases. These post shock temperatures can be additionally increased due to plastic work involved in the compaction of pore space.


Pressure distribution during shock compression at the mesoscale at the interface between the projectile and the target.

In the numerical modelling project we also collaborate with other institutions. The development of the material models was supported by Gareth Collins (Imperial College London) and Natalia Artemieva (Planetary Science Inst., Tucson; Inst. for Dynamics of Geospheres, Russian Academy of Science, Moscow).


Nathanael Durr, Nicole Güldemeister

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