Propagation of impact-induced shock waves in porous sandstone using mesoscale modeling


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Generation and propagation of shock waves by meteorite impact is significantly affected by material properties such as porosity, water content, and strength. The objective of this work was to quantify processes related to the shock-induced compaction of pore space by numerical modeling, and compare the results with data obtained in the framework of the Multidisciplinary Experimental and Modeling Impact Research Network (MEMIN) impact experiments. We use mesoscale models resolving the collapse of individual pores to validate macroscopic (homogenized) approaches describing the bulk behavior of porous and water-saturated materials in large-scale models of crater formation, and to quantify localized shock amplification as a result of pore space crushing. We carried out a suite of numerical models of planar shock wave propagation through a well-defined area (the “sample”) of porous and/or water-saturated material. The porous sample is either represented by a homogeneous unit where porosity is treated as a state variable (macroscale model) and water content by an equation of state for mixed material (ANEOS) or by a defined number of individually resolved pores (mesoscale model). We varied porosity and water content and measured thermodynamic parameters such as shock wave velocity and particle velocity on meso- and macroscales in separate simulations. The mesoscale models provide additional data on the heterogeneous distribution of peak shock pressures as a consequence of the complex superposition of reflecting rarefaction waves and shock waves originating from the crushing of pores. We quantify the bulk effect of porosity, the reduction in shock pressure, in terms of Hugoniot data as a function of porosity, water content, and strength of a quartzite matrix. We find a good agreement between meso-, macroscale models and Hugoniot data from shock experiments. We also propose a combination of a porosity compaction model (ε–α model) that was previously only used for porous materials and the ANEOS for water-saturated quartzite (all pore space is filled with water) to describe the behavior of partially water-saturated material during shock compression. Localized amplification of shock pressures results from pore collapse and can reach as much as four times the average shock pressure in the porous sample. This may explain the often observed localized high shock pressure phases next to more or less unshocked grains in impactites and meteorites.


Porosity and water content are typical properties for rocks of the upper crust of Earth such as sandstone. Regolith breccias on the Moon are characterized by a significant amount of porosity, and the presence of subsurface water on Mars is widely accepted. Bulk density of a number of asteroids is <1.3 g cm−3, which corresponds to a very high amount of empty pore space and porosities of up to 75% (Britt et al. 2002), and comets are known to have very low densities (e.g., Richardson et al. 2007). These are only a few examples of the importance of porosity and water on planetary bodies and other cosmic objects. Impact cratering plays an important role in all of these bodies, and it can be assumed that in particular the formation of small to midsize craters is affected by the presence of porous and water-saturated material. In the case of water-saturated material, it is important to distinguish between fully saturated materials, where all pore space is filled with water, and partially water-saturated materials, where 50% of the pore space is filled with water.

Hypervelocity impact crater formation is characterized by the generation of shock waves. It is well known that porosity affects shock wave propagation, attenuation, and shock heating. The crushing of pore space is an effective mechanism to absorb shock waves (Zel’dovich and Raizer 1967, chapter 11), and the additional plastic work involved in the compaction of pore space causes higher shock temperatures in porous material than in competent materials at the same shock pressures. This may cause an increase in shock-induced melting in porous materials; however, on the other hand, the shock wave attenuates faster in porous material, and low impedance pore contents, i.e., air and/or water, result in lower shock wave velocities and pressures in porous materials so that a smaller volume of material experiences sufficiently high shock pressures for melting. Both processes are competing factors and quantifying their net effect on the production of impact melt can be determined only by numerical modeling (Wünnemann et al. 2008).

Another consequence of lower shock pressures and faster attenuation of the shock waves in porous material is a decrease in crater efficiency (Wünnemann et al. 2006, 2011) and a decrease in ejection velocities (Housen and Holsapple 2003). The effect of porosity and water content on crater formation has been addressed in studies of terrestrial and extraterrestrial craters, in impact experiments, and numerical modeling (e.g., see Kieffer et al. 1976; Love et al. 1993; O’Keefe et al. 2001; Britt et al. 2002; Holsapple et al. 2002; Goldin et al. 2006).

Apparently, the presence of pore space and water affects impact processes on different scales. The overall bulk behavior of porous material can be observed on the scale of natural craters in terms of crater size and the generated melt volume. By means of microscopic observations of shock-induced modifications such as planar deformation features (PDF), high-pressure mineral phases, and melt in rock samples that have undergone shock compression, the amplitude of the shock load, and thus the decay with distance from the point of impact, can be estimated (Stöffler and Langenhorst 1994; Langenhorst and Deutsch 2012). However, initially porous material often shows a somewhat ambiguous picture. Localized high shock pressure phases occur next to more or less unshocked grains in impactites and meteorites (Kieffer et al. 1976; Grieve et al. 1996). Studies on shock metamorphism in porous Coconino sandstone at the Meteor crater (Kieffer 1971) revealed two distinct phenomena that occur during shock-induced pore collapse: Depending on the initial shock pressure, the closure mechanism of a pore can either be described as “shrinking” or “jetting.” In both cases, an amplification of the shock pressure occurs, while “jetting” causes stronger localized pressure amplifications than “shrinking.”Kieffer (1971) described the process of “jetting” as extrusion of material. The open pore space collapses and the material surrounding the pore is then injected into pore space. Although Kieffer (1971) described the process more phenomenologically, a quantitative description of the process is still lacking. Numerical modeling of shock propagation in heterogeneous porous material has been carried out on meso- and macroscales. On the mesoscale, the heterogeneous structure including open or water-filled pores is resolved explicitly (Crawford et al. 2003; Ivanov 2005; Riedel et al. 2008; Borg and Chhabildas 2011). On the macroscale, processes are studied that are affected by the presence of pore space but that occur on a scale several orders of magnitude larger than the actual size of an individual pore, such as the formation of impact craters. In the latter case, porosity is usually treated as a state variable and the change in porosity due to shock compression is taken into account by a so-called compaction model such as the P–α model (Kerley 1992; Carroll and Holt 1972) and the ε–α model (Wünnemann et al. 2006).

Laboratory impact experiments using a sandstone, Seeberger Sandstein, as a target (Poelchau et al. 2013), carried out in the framework of the Multidisciplinary Experimental and Modeling Impact Research Network (MEMIN), provide new data on the meso- and macroscales to further our understanding of the thermodynamic and mechanical response of heterogeneous, porous, dry, or (partially) water-saturated material to shock loading (Kenkmann et al. 2011; Schäfer et al. 2006). Detailed observations on pore space collapse as a function of crater depth (Buhl et al. 2013), the ejecta dynamics (Sommer et al. 2013), the effect on crater morphometry and morphology (Dufresne et al. 2013), and shock-recovery experiments with samples of Seeberger sandstone (Kowitz et al. 2013) are reported in this issue.

The reproduction of any of these observations by numerical modeling implies appropriate material models to describe the thermodynamic behavior of porous material for different degrees of water saturation. Previous approaches used meso-scale models (Crawford et al. 2003; Ivanov 2005; Riedel et al. 2008; Borg and Chhabildas 2011) to develop appropriate material models that describe the bulk behavior of dry porous material and material mixtures in the case of fully water-saturated rocks (Pierazzo et al. 2005). The goal of this work was to bring meso- and macroscale observations into accordance and to quantify processes on different scales. On the macroscale, this includes the reduction in shock pressure and increase in temperature, whereas mesoscale modeling aims at the quantification of localized amplification of peak shock pressures in the vicinity of single pores. Mesoscale modeling will also be used to test a new macroscale model describing the bulk behavior of partially water-saturated material. The goal was to find a universal description of material mixtures of silicates and water that also contain open pore space.

Finally, we will compare modeling results with literature Hugoniot data of sandstones with similar properties as the Seeberger sandstone used in the framework of the MEMIN project. We also include new Hugoniot data for the Seeberger sandstone that were obtained by flyer plate experiments. In the first section, we provide information on the shock physics code iSALE, the setup and range of the numerical experiments (parameter studies), and the material models used. In the next section, we present the results of shock amplification due to pore space collapse considering the crushing of a single pore and a set of pores followed by the investigations of bulk effects of pore space collapse on shock wave propagation. In this section, detailed resolution and dimensionality tests are presented first. Additionally, the section comprises investigations of shock wave propagation through porous material and water-saturated material as well as the effect of strength in porous material. Finally, a comparison of the meso- and macroscale models is carried out. In the last section, we discuss implications of our results.


Numerical Hydrocode

We used the shock physics code iSALE 2-D/3-D (Wünnemann et al. [2006] and references therein; Elbeshausen et al. 2009) for both meso- and macroscale studies. iSALE is based on the original SALE (Simplified Arbitrary Lagrangian Eulerian) code by Amsden et al. (1980). To simulate hypervelocity impact processes in solid materials, SALE was modified to include an elasto-plastic constitutive model and fragmentation model (Collins et al. 2004), various equations of state (EoS), and multiple material handling (e.g., Ivanov et al. 1997; Elbeshausen and Wünnemann 2011). The code includes a porosity compaction model, the so-called ε–α model (Wünnemann et al. 2006; Collins et al. 2011), that enables modeling of shock wave propagation in porous materials. Basically, the code consists of three components: the numerical solver of the equations describing the motion of matter (that are based on the conservation equations of mass, momentum, and energy), the equation of state dealing with the thermodynamic behavior of matter (see the next section), and the constitutive model describing the mechanical response of rocks to elasto-plastic deformation. The equations of motion for a continuous medium can be either Lagrangian or Eulerian. iSALE includes both a Eulerian and Lagrangian numerical solver. In the Eulerian description, the cells of the computational domain are fixed in space and material is advected through the numerical grid. In Lagrangian models, material initially located in a computational cell is fixed and the transport of material is calculated by the movement and deformation of the whole grid in space. The latter numerical approach often faces the problem of extreme deformation of cells, for instance, if in a mesoscale model an open pore is completely closed as a result of shock compression. This usually causes numerical problems as grid resolution (given by the number of cells per reference area) varies significantly and becomes infinitesimally small where empty pore space is erased. To fix the problem, numerical cells have to be eroded if deformation exceeds a certain threshold or some sort of re-gridding is required (e.g., Anderson 1987). In Eulerian models, the grid resolution is constant in space and the closure of a pore is naturally described by the flow of matter through the computational mesh. All models shown in this paper were carried out in the Eulerian mode of iSALE.

Although iSALE contains sophisticated constitutive models describing the mechanical response of a material to large stresses, we employed a simple von Mises yield criterion to account for plastic material failure. The von Mises model defines a constant stress where plastic yielding occurs and matter experiences permanent deformation. We neglected any dependency of yield strength on the deformation history (damage), pressure, temperature (see e.g., Collins et al. 2004), and strain rate. This simplification makes our models less realistic, but enables us to relate macroscopic and mesoscopic strength parameters in a simple way to ensure the consistency of models on different scales. This study aimed at developing a methodological approach that would be applied in follow-up studies for more complex material behavior.

Equation of State for Porous Materials

The equation of state (EoS) describes the thermodynamic behavior of the material and is therefore key for modeling shock wave propagation. In the form used in iSALE, it relates the state parameters internal energy and density with pressure and temperature. Prior to the actual model run, iSALE generates tables of the state parameters by utilizing the Analytic EoS (ANEOS) (Thompson and Lauson 1972). During the simulation, the pressure and temperature for a given density and internal energy are looked up in the table; intermediate states are interpolated among the closest neighbors in the discrete state table. We used the modified version of ANEOS for quartzite (Melosh 2007) taking molecular clusters in the vapor phase into account to calculate the thermodynamic state of the solid component and incorporating the phase transition from quartz to stishovite; however, due to the low-to-moderate shock pressures considered in this study, we did not expect any differences in comparison with the original ANEOS version. Usually, quartz undergoes various phase changes which, however, are too complex to be all considered in the EOS of the numerical code (Melosh 2007). The solid component of the material under consideration is quartzite in all models. In a geological context, quartzite rarely contains any porosity. To avoid confusion, we approximated the material behavior of any quartz-rich, porous water-saturated or partial water-saturated rock, such as the Seeberger sandstone, by the thermodynamic properties of quartzite and water (described by ANEOS) plus an additional procedure to account for the presence of pore space. In the following, we also use the term “dry porosity” if pore space is empty; if water is present, we indicate this by the term “wet porosity.”

To calculate the thermodynamic state of a porous material, three different cases have to be distinguished:

  • 1 In the case of porous material (pores are empty), the thermodynamic state of matter is significantly affected by the crushing of pores. The presence of porosity causes changes in density due to the closure of pore space that has to be taken into account by so-called compaction models (Hermann 1969; Carroll and Holt 1972; Wünnemann et al. 2006).The iSALE code combines the ε–α porosity compaction model (Wünnemann et al. 2006) and the ANEOS to determine the thermodynamic state in porous material. The ε–α model describes the crushing of pore space as a function of compressive volumetric strain εV (which is defined negative in our model). In the compaction function α = f(εV) the distension α is defined by α = 1/(1 − φ), with φ being the porosity. If compression increases (volumetric strain decreases), distension decreases until all pore space is crushed out and the material is fully compacted (α = 1). The compaction function is defined by an elastic-plastic transition strain ɛe to separate the elastic regime, where the decrease in pore space is not permanent (εV > εe), and a compaction regime where changes in porosity remain in the material (εV > εe). In the compaction regime, pore space is crushed out approximately according to an exponential law where the rate of compaction is controlled by the exponent κ. Note, κ and εe are material parameters that need to be determined in compaction experiments. The parameter values for the porosity model that have been considered for the macroscale simulations of a dry porous material are listed in Table 1. More details are given in Wünnemann et al. (2006).
  • 2 In case of water-saturated material (pores are filled with water), we used an approach proposed by Pierazzo et al. (2005). If all pore space is filled with water (100% saturation), the material can be treated as a two-phase material mixture consisting of the matrix (quartzite) and water. The thermodynamic state of such a material mixture can be calculated by assuming that both phases have to be in a thermodynamic equilibrium (same temperature and pressure). With this boundary condition, a new table for a given porous material with a defined water content can be generated by combining the ANEOS for each phase (water and quartzite). The procedure is described in Pierazzo et al. (2005). Note that the water content in the material mixture is fixed and any change in the distribution by flowing or steaming of water through the material is not taken into account. As a consequence, we assumed that the presence of water prevents complete closure of pores, although the used ANEOS for water-quartzite mixtures allows for compression of the water phase in equilibrium with the quartzite matrix.
  • 3 In the case of a partially water-saturated material, with some pores completely filled with water, and others completely dry, we combined the porosity compaction model and the tabulated ANEOS for mixed material according to the relative proportion of dry and wet pore space.
Table 1. Parameters used in the ε–α porosity model.
α 1.25, 1.43, 1.54, 1.67, 2.0
φin %20, 30, 35, 40, 50
εe −1.0 × 10−5, −7.5 × 10−2, −3.0 × 10−2
κ 0.98, 1.0

In summary, homogenized or macroscopic models used for this study are based on the ANEOS for quartzite and water and the ε–α porosity compaction model. This enables the modeling of dry, partially water-saturated, and fully water-saturated porous materials under shock loading on a macroscale. The described procedure is applicable to any other porous material and is not limited to quartzite as matrix and water-filled or empty pores; however, the large contrast in terms of compressibility between the two different phases, such as water and quartzite, may lead to disequilibrium states that cannot be treated by the procedure for mixed materials as described above.

Model Setup

We chose a model setup very similar to typical laboratory shock wave-recovery experiments (Langenhorst and Hornemann 2005) to carry out meso- and macroscopic numerical simulations of shock wave propagation in porous wet and dry sandstone. The principal model setup is shown in Fig. 1. We generated a planar shock wave by impacting a so-called “flyer plate” on a “buffer plate” at velocities ranging from 500 to 4000 m s−1 corresponding to initial shock pressures generated at the interface between the flyer and the buffer plate of 2.6–28 GPa. The flyer plate is resolved in vertical direction by 600 cells and the buffer plate by 100 cells. The radius of the cylindrical setup is resolved by 1400 and 1700 cells, respectively. The resolution of the “sample” varies according to the number of pores (see the Results: Shock Amplification due to Pore Space Collapse section). A well-defined shock plateau propagates through the buffer plate into the sample. The impulse length of the shock wave is given by the thickness of the flyer plate. In all simulations, it is longer than the extent of the sample; we did not model the unloading.

Figure 1.

 a) Illustration of the mesoscale model setup including the flyer (or impact) plate, the buffer plate, and the sample with resolved pores (the shown number of pores is only representative for the actual number of pores that varies in different simulations). The shown pores can be empty or filled with water. The matrix consists of quartzite. b) Plane view of the 2-D cylindrically symmetric computational grid inside the sample with resolved pores represented by rings with rectangular cross-section. In macroscale models, the number of pores is infinite and porosity is considered by the ε–α porosity compaction model.

The impacting flyer plate and the buffer plate as well as the solid matrix material consisted of quartzite in all models in this study.

The models can be looked at as “numerical experiments” where the “sample” either represents a mesoscopically resolved sample containing a single or several pores embedded in a quartzite matrix or a macroscopic sandstone sample where the number of pores is infinite and porosity is described by the state variable distension α. While in the former case pores are resolved directly by the model, the bulk effect of porosity on the thermodynamic state is considered in the latter case by combining the ANEOS for mixed material and the ε–α porosity compaction model as described in the previous section. In the mesoscale models, the resolved pores can be either empty or filled with water and the size, geometry, and distribution can be varied according to the chosen porosity; however, for this study, we used only pores with a quadratic cross section. Schade and Wünnemann (2007) studied the effect of geometry and found that pore geometry has an effect on peak pressure distribution and localized pressure increase due to pore collapse. The highest pressures have been observed for a cubic geometry in contrast to a rhombic (lowest pressure increase) or cubic-rhombic geometry. However, the effect of pore geometry has been neglected in the present study of pore collapse. We presume that the effect of pore geometry on shock wave propagation is negligible if the number of pores located close to one another is high. The variety of possible geometries is infinite and a systematic analysis is beyond the scope of this study.

An overview of all mesoscale models that have been carried out is given in Table 2. We first looked at the collapse of a single pore under different pressure conditions. Then, we systematically increased the number of pores to investigate the propagation of a shock wave through a heterogeneous target. The most desirable setup for this study would have been a random distribution of pores in space over a 3-D sample very much like how pore space distribution looks in a thin section of sandstone. Although such a study is in principle feasible, the requirements on computer power are high and the computation time would be very long. Therefore, we used a simplified approach where pores are represented by rings with a rectangular cross-section on a 2-D cylindrically symmetric grid (Fig. 1). The pores are regularly distributed in the computational grid (“checkered pattern”). We always ensured a symmetric arrangement of pores with respect to the symmetry axis. In the case of a single pore, the pore is located on the symmetry axis. To test whether this simplified setup is sufficient to study the mesoscopic effects of porosity on shock propagation, we compared the 2-D simulations with uniform pore distribution with 2-D and 3-D simulations where pores are represented by cubes that were randomly distributed in the sample (see Table 1). The distribution of pores does not seem to influence the modeling results significantly and the 3-D simulations agree well with the obtained 2-D modeling results (see resolution test in the Results: Shock Amplification Due to Pore Space Collapse section).

Table 2. Overview of numerical mesoscale simulations (including resolution tests).
Flyer/buffe plate material:Quartzite       
Matrix material:Quartzite       
Cross profile of pore geometry:Squares       
No. of poresPore fillingDimensionPressures in buffer plate (GPa)aFlyer plate velocity (km s−1)bCPL (cells per pore length)Number of cells in computational domainPorosityDistribution
  1. aThe pressure in the buffer plate is the mean pressure at the interface between flyer and upper buffer plate.

  2. bEach pressure value represents one experiment with a given flyer plate velocity.

1Empty2-D6, 14, 22, 281,2,3,41201000 × 1400
  2-D6, 14, 22, 281,2,3,4601000 × 1400
  2-D6, 14, 221,2,3301000 × 1400
  2-D6, 14, 221,2,3101000 × 1400
1Water2-D14, 282,41201000 × 1400
3/6/8Empty 61601000 × 1400
   142601000 × 1400
   223601000 × 1400
   284601000 × 1400
12Empty 61601500 × 1400
   142601500 × 1400
∼500Empty2-D2.6, 6, 14, 22, 280.5,1,2,3,48325 × 17000Uniform
  2-D2.6, 6, 14, 22, 280.5,1,2,3,48325 × 170020Uniform
  2-D2.6, 6, 14, 22, 280.5,1,2,3,48325 × 170030Uniform
  2-D2.6, 6, 14, 22, 280.5,1,2,3,48325 × 170035Uniform
  2-D2.6, 6, 14, 22, 280.5,1,2,3,48325 × 170040Uniform
  2-D2.6, 6, 14, 22, 280.5,1,2,3,48325 × 170050Uniform
  2-D6, 14, 221,2,38325 × 170020Random
  2-D6, 14, 221,2,34162 × 85020Uniform
  2-D6, 14, 221,2,34162 × 85020Uniform
  2-D6, 14, 221,2,3281 × 42520Uniform
  3-D6, 141,28168 × 765 × 16220Uniform
  3-D1428168 × 765 × 16220Random
∼500Water 100%2-D618325 × 170020Uniform
  2-D1428325 × 170020Uniform
  2-D2238325 × 170020Uniform
∼500Water 50%2-D618325 × 170020Uniform
  2-D1428325 × 170020Uniform
  2-D2238325 × 170020Uniform

Besides the geometry and dimensionality of the setup, resolution is key to generate quantitatively meaningful results. Although resolution should be as high as possible, we also have to consider computation time and hardware resources available for this study. We measured resolution in terms of the number of cells per pore length (CPL).

We carried out resolution tests yielding an acceptable resolution of 8 CPL for 2-D and 3-D (see resolution test in the Results: Bulk Effects of Pore Space Collapse on Shock Wave Propagation section), if a large number of pores had to be resolved. For single pore models, where peak shock distribution in the vicinity of the pore was studied in much detail, a resolution of 60 CPL was required (see resolution test in the Results: Shock Amplification due to Pore Space Collapse section).

In summary, we carried out systematic numerical experiments varying the number of pores (1–12), the bulk porosity (20–50%), and the initial shock pressure (2.6–28 GPa). We also studied the effect of water saturation and the yield strength of the matrix (quartzite) ranging between 0 and 1 GPa. Finally, we compared the results of the mesoscale numerical experiments with macroscopic models where the sample contained an infinite number of pores with the same bulk porosity (20–50%).

Processing of Model Results

We recorded several thermodynamic parameters as density, pressure, peak pressures, shock, and particle velocities in space, and time during shock wave propagation through the sample.

To study the localized mesoscopic effects of pore collapse in the vicinity of a single pore or an array of pores, we used Lagrangian tracer particles to record peak shock pressures. Tracers are massless and are initially placed in the center of each computational cell. They may be considered as representative for the volume of material initially located in the same cell. Tracers move along with material through the grid and record the thermodynamic path during the passage of the shock wave. We determined the highest pressures each tracer had experienced and obtained the peak pressure distribution. The final pressure distribution is a result of shock loading and superposition of reflecting shock fronts originating from the collapse of pores. Finally, we plotted isobars of peak shock pressures enclosing all tracers that experienced a certain pressure level and calculated the volume by means of the number of tracers enclosed and their representative volume.

By doing so, we need to consider that a small number of tracers experience very high pressures, sometimes up to 100 times the initial shock pressure, which is most likely a numerical artifact. Such high pressures are certainly not representative to define the range of shock pressure amplification due to pore collapse. Therefore, we define some critical volume that we consider to be significant to estimate the range of pressure increase due to the closure of pores. We define the critical volume as 20% of the initial pore volume. In other words, we consider only peak pressures that have been experienced by a volume that is at least 20% of the initial pore volume. This is a bit of an arbitrary definition; however, we noticed that only an insignificantly small volume (or a small number of tracers) undergoes higher pressures and we consider this material fraction as negligible. To work out the thermodynamic bulk behavior of the porous sample in mesoscopic models, we determined the particle velocity up and the velocity of the shock front Us. A series of numerical experiments for different flyer plate velocities, and thus different initial shock pressures, enable us to plot Hugoniot curves in the Us–up space. Hugoniot curves can be compared for different porosities and water contents. In the case of the mesoscale models, the particle velocities vary across the shock front significantly due to the heterogeneities (pores) causing localized pressure amplifications and reflections at pore boundaries. To address this fact, we averaged the shock wave parameters over a row of computational cells (radial direction) at a certain distance the shock front has traveled through the sample (sample depth). Vertical profiles of particle velocity (along columns of cells) at different points in time are used to determine the shock wave velocity Us in the models. Due to the heterogeneities (pores), the shock front was somewhat uneven and we determined the shock wave velocity Us in several parallel profiles and calculated mean values.

We used particle velocity variations to estimate the increase and decrease in shock pressure as a result of pore space crushing or we averaged out the variations to determine the bulk behavior of the heterogeneous material.

By means of the Rankine-Hugoniot equations (e.g., Zel’dovich and Raizer 1967, chapter 11), particle and shock wave velocity, up and Us, can be used to calculate density ρs, specific volume Vs, and pressure P during shock compression in the sample. The initial bulk density for the porous “dry” and “wet” material varied from 1955 to 2297 kg m−3 according to the chosen porosity and degree of water saturation. The density of the matrix material is 2650 kg m−3 given by the ANEOS for quartzite.

Results: Shock Amplification due to Pore Space Collapse

Resolution Test—Pore Collapse

To study the complex interaction of the shock wave with empty pores on a mesoscale, sufficient resolution (sufficiently large computational domain or number of cells covering the area under consideration) is required. Therefore, we first carried out a suite of simulations varying the resolution. We determined the volume of material in the sample which had experienced a maximum peak pressure that was four times the initial shock pressure considering different resolutions. Resolution is measured in terms of the number of cells per pore length (CPL). We used 10, 30, 60, and 120 CPL and varied the initial shock pressure between 6, 14, and 22 GPa. Figure 2 shows how the volume of material that has experienced a pressure four times the initial pressure changes with increasing resolution from 10 to 120 CPL. Independent of the shock wave pressure, all models approach approximately the same volume at 60 CPL, which is considered as the “true” volume. Further increasing the resolution (120 CPL) does not show any significant difference in volume. Therefore, we concluded that 60 CPL poses a reasonable compromise to keep the error in our simulations as small as possible and the computation time reasonable. Similar results regarding the required resolution in iSALE to determine shock volumes have been obtained by Wünnemann et al. (2008).

Figure 2.

 Volume of material in the sample that experiences a maximum peak shock pressure four times the initial shock pressure of 6, 14, and 22 GPa versus resolution in CPL (cells per pore lengths). At a resolution of 60 CPL, the volume is not dependent on resolution anymore.

Collapse of Single Pores

First, we studied the collapse of an isolated pore. The pore has a quadratic cross-section and is located on the symmetry axis of the 2-D cylindrically symmetric grid. The geometry corresponds to a cylinder. At the point in time when the shock front first hits the boundary of the pore, it is turned into a rarefaction wave traveling in the reverse direction unloading the material from shock pressure (Fig. 3a). The volume of the pore decreases gradually as material at the boundary to the pore is set into motion after shock release (Figs. 3b and 3c). When the pore is completely closed, a secondary shock wave is generated that propagates approximately spherically outward from the original center of the pore (Fig. 3d). Note that the whole sample undergoes a relative motion downward in the direction of the shock front. The secondary shock wave superimposes with the release wave and the initial shock wave causing pressure amplifications in the material that was originally surrounding the pore.

Figure 3.

 Snapshot series of the collapse of a single pore (white square) due to shock compression with an initial pressure of 6 GPa. The series shows the different states of pore collapse and the evolving pressure with time. Zero pressure represents unshocked material.

Figure 4a illustrates the maximum pressure distribution relative to the initial shock pressure after the collapse of a single pore for an initial pressure of 6, 14, and 22 GPa. Maximum shock pressures were recorded in the proximity of the pore to quantify the maximum amplification that could be expected due to pore collapse. As a maximum, the pressure increased up to four times of the initial pressure. The highest pressures could be observed in the zone where the pore was initially located. Note that the material, and thus tracers, experienced a relative motion downward as can be seen by the location of the initial pore in Fig. 4a (dashed square). The absolute pressure amplification slightly depends on the amplitude of the initial shock wave: For a shock pressure of 6 GPa (flyer plate impact velocity vi = 1000 m s−1), the maximum observed pressure reached 17.5 GPa (approximately three times the initial pressure); for 14 GPa (flyer plate impact velocity vi = 2000 m s−1), the shock pressure was locally amplified to 61 GPa (approximately 4.4 times the initial pressure), and for 22 GPa (flyer plate impact velocity vi = 3000 m s−1), the maximum pressure went to 65 GPa (approximately three times the initial pressure) always considering that 20% of the initial pore volume experienced the determined pressure. It is possible that smaller fractions of material experienced even higher shock pressures, but according to the given resolution limit (too small number of tracers), the volume was probably not significant to be considered in our analysis.

Figure 4.

 Peak pressure distribution for (a) a single pore, (b) 3 pores, and (c) 12 pores; each pore is resolved by 60 CPL with an initial pressure of 6, 14, and 22 GPa from left to right. The contour lines show pressure distinctions of 1.5 to 4 times the initial pressure. The light gray dashed lines indicate the position of the symmetry axis. The dashed squares indicate the original positions of the pore(s) in the unshocked material. Each frame represents the final stage (after complete closure of the pore) that we define to be reached when the shock wave has propagated through the entire sample. The peak pressures are the final peak pressures the material has experienced.

Other pore-collapse mechanisms despite the here observed “shrinking” mechanism may influence the pressure amplification. Kieffer et al. (1976) distinguished two different mechanisms resulting in the closure of an open pore. “Shrinking” is considered the gradual closure of the pore from all sides and occurs at low-to-moderate shock pressures. If the shock pressure exceeds a certain threshold, Kieffer et al. (1976) described that material is “jetting” into the pore space from the point where the shock front first interacts with the pore boundary that may be considered as a “free surface.” Kieffer’s description of pore closure is based on observations at shocked Coconino sandstone. Schade and Wünnemann (2007) reproduced the process in mesoscale modeling of pore crushing. In all models of the present study, pores were crushed by the “shrinking” mechanism. Apparently, the maximum shock pressure of 22 GPa in the present study was not sufficient to induce the “jetting.” However, the formation of jets entering the pore is very sensitive to the pore geometry and how the shock front is aligned to the sides of the pore. Thus, the lack of jetting at 22 GPa in our models may have been rather caused by the simplified geometry and planar alignment of the shock front to the upper side of the pore than by too low shock pressure.

Similar numerical experiments with water-filled pores have been carried out. As we do not include any opening of cracks in our model, where the pressurized water can escape, the pore cannot be closed and, therefore, we observe only a very minor shock pressure amplification in the vicinity of the pore (at 22 GPa initial shock pressure, 24 GPa were observed as a maximum).

Collapse of a Set of Pores

An isolated pore may occur in a generally dense material. In porous material, such as sandstone, pores are located relatively close to one another, separated only by individual grains, and shock waves originating from pore closure may interact with one another. Therefore, we set up an array of pores separated by one pore length from one another and analyzed the shock wave propagation through the sample. The same 20% volume criterion as described above was applied to determine significant peak shock pressure amplifications. For a sample with three pores each resolved by 60 cells, the maximum pressure ranges from about 10 to 23 GPa for corresponding initial shock pressure amplitudes of 6–22 GPa, respectively. By increasing the number of pores to 12, the observed maximum peak pressures changed only insignificantly (9, 17, 23 for initial shock pressure amplitudes of 6, 14, 22 GPa). Considering the 20% volume criterion, on average, the amplification of shock pressure is less than two times the initial pressure due to interaction of shock and release waves originating from adjacent pores. Figures 4b and 4c, however, depict the distribution of peak shock pressures in the proximity of 3 and 12 pores neglecting that only a certain volume of initial pore space is considered. The amplification varies similar to the single pore case between 1.5 and 4 times of the initial pressure.

For the array of 12 pores, the highest pressures are located in the first row of pores, which explains why the same maximum peak pressures occur in the 3 and 12 pore case. Those maximum pressures occur when the first row of pores is closed. The pressure amplification decreases with the collapse of additional pore space due to energy consumption. Reflections and interferences of the shock and release wave from neighboring pores reduce the pressure amplification. By considering 20% of the initial pore volume that has experienced a certain pressure, we observed only a maximum amplification of less than two times the initial shock pressure. We considered a threshold of 20% of the initial pore volume experiencing a certain pressure to be more representative for shock amplification in porous material.

Results: Bulk Effects of Pore Space Collapse on Shock Wave Propagation

Resolution Test—Shock Propagation

To study the bulk effects of pore space crushing on the propagation of shock waves by mesoscale modeling the number of pores needs to be increased significantly; however, the number of pores resolved in mesoscale modeling can only be representative for very small sample sizes (i.e., few mm). In crater formation models in sandstone, the number of pores is infinite and other methods to describe the bulk behavior of porous material have to be applied as will be discussed further below. To resolve an as large as possible number of pores, the resolution of each pore has to be reduced significantly in comparison with the investigations above on pore space collapse.

We assume that localized pressure amplification due to single pore collapse where a high resolution was necessary does not contribute to the bulk behavior of shock wave propagation. Nevertheless, we have to ensure that resolution is sufficient for the study on shock propagation in this section. We carried out resolution tests for shock wave experiments as described above where the sample was perforated with about 500 pores that were resolved by 2, 4, and 8 CPL. Vertical pressure profiles where lateral variations are averaged out (see processing of model results in the Methods section) are shown in Fig. 5a for all three different resolutions. In all three cases, the pressure oscillates due to the pores in the sample; however, maximum and minimum pressure fluctuations vary depending on the CPL value. The average level of the shock plateau (mean value of the oscillations) was approximately constant for 4 and 8 CPL (3.6 GPa), but significantly shifted to lower pressures for 2 CPL (3.0 GPa) (Fig. 5b). Apparently, a resolution of 2 CPL is insufficient for the given study. For all models (2-D and 3-D) in the study on shock propagation, we used 8 CPL.

Figure 5.

 Resolution tests with a grid containing a large number of pores, each pore is resolved by 2, 4, and 8 CPL representing comparable pressure profiles through the sample (a) where zero depth represents the interface of the flyer and the buffer plate and (b) the corresponding averaged level of shock plateau pressure in dependency on CPL. In Fig. 5b, the three different symbols (resolutions) are all plotted for the same time corresponding to a point in time when the shock wave has traveled through most of the sample material.

Dimensionality and Distribution of Pores

A vast majority of models in this study were carried out on a 2-D cylindrically symmetric grid. Pores have a quadratic cross-section and are arranged in a checkerboard pattern (see Fig. 1). This simplified setup certainly does not reflect a typical natural distribution of pores in porous material such as sandstone. We have already stated that pore geometry does not affect the bulk behavior. We now demonstrate that distribution and dimensionality (3-D cubes instead of 2-D rings) of pores play only a minor role in the quantification of shock propagation in heterogeneous material. Regarding the dimension of the model, we compared pressure profiles of 3-D simulations and 2-D simulations, both resolving pores by 8 CPL (Fig. 6). The 3-D simulations were conducted on a smaller grid to save computation time. So the number of cells of the sample was in total 50,400 in 3-D and 195,000 in 2-D. Additionally, we compared the effect of a random distribution of pores in 3-D with the regular checkerboard pore distribution in 2-D. The size of pores was not varied. The bulk porosity of the sample was kept constant in all three cases shown in Fig. 6. Although the observed oscillations in the vertical pressure profiles differ among the three cases, the mean shock plateau is approximately the same.

Figure 6.

 Comparison of a 2-D simulation (regular distribution (a)) with a 3-D simulation (regular (b) and random distribution (c)). Pores are resolved by 8 CPL. The contour plots (left) show snapshots of the pressure distribution in the sample after 2.4, 6.4, and 8.8 ms. The vertical profiles (right) depict the corresponding shock pressures, which are averaged over a row of computational cells at a certain distance the shock front has traveled through the sample at t = 6.4 ms (for all profiles). The profiles include the mean pressure value in the sample indicated by the dashed lines, which is about the same for all three cases.

In Fig. 7, we compare the pressure profiles for regular and random pore distribution in 2-D for three different initial shock pressures. For 6 GPa, the averaged shock pressure profiles agree well. With increasing initial pressure, the models show slight differences in the average pressure level of the shock plateau, indicating that with an increase in shock pressure, the pore distribution has an increasing effect on pressure distribution and the overall shock propagation in our mesoscale models. We also have to keep in mind that the model with randomly distributed pores represents only one possible assembly of pores in a sample. Other distributions of pores would cause small differences in the pressure profile. Nevertheless, the influence of the distribution of pores is small and we assume that the uniform checkerboard distribution on a 2-D cylindrically symmetric grid, where pores are represented by rings with a quadratic cross-section, is a sufficient approximation for this study where the implementation of a large number of simulations is crucial.

Figure 7.

 Comparison of pressure profiles for a random and regular distribution of pores in 2-D. The initial pressures are 6, 14, and 22 GPa from left to right. The profiles represent snapshots of shock wave propagation at the same point in time after impact of the flyer plate on the buffer plate. Zero depth represents the interface of the flyer and the buffer plate. The thickness of the flyer plate is identical in all cases. The dashed lines represent the random distribution, the solid lines the regular distribution.

Shock Propagation in Dry Porous Materials

If a shock wave propagates from solid material without pores (buffer plate) into porous material (sample), the shock pressure amplitude decreases at the interface due to the impedance contrast as a consequence of the reduced bulk density in the sample. The average shock amplitude is expected to decrease for the same reason, although we observe localized shock pressure amplification as a result of pore crushing. The crushing of pore space consumes shock wave energy; rarefaction waves originate from the pore interface (the pressure in an empty pore is zero) and interact with the primary shock wave. Overall, these processes lead to a decrease in the shock pressure amplitude in the sample.

We quantified this process by modeling shock propagation at different amplitudes through a sample containing approximately 500 pores. The bulk porosity was varied between 0% and 50% by changing the size and number of pores. For a sample of 20% porosity and a shock pressure amplitude in the buffer plate of 6, 14, and 22 GPa, the averaged pressures decreased to 3.4, 9.5, and 17 GPa. An example of a pressure profile for an initial shock pressure of 6 GPa is shown in Fig. 8. The oscillations in the graph of the porous sample are caused by (1) local pressure amplifications and (2) rarefaction waves from the pore interface. The locally observed shock amplifications are smaller compared with an isolated pore or an array of 3 or 12 pores as stated above due to interferences of reflections from adjacent pores and lower resolution. The local pressure amplifications reach values of over 20 GPa; however, this is in particular the case for the “first” pores in the grid that are crushed. The local pressure amplification decreases with the propagation of the shock wave.

Figure 8.

 Comparison of pressure profiles through a nonporous (left) sample and porous (right) sample. Both profiles represent snapshots of shock wave propagation at the same point in time after impact of the flyer plate on the buffer plate. In the porous case (right), shock pressure is significantly decreased in the sample and the shock front propagates slower than in the nonporous case. In the porous case (right), the dashed line represents the mesoscale model; the solid line, the macroscopic model. Zero depth represents the interface of the flyer and the buffer plate.

Different bulk porosities significantly affect shock wave propagation through the sample. The higher the porosity, the faster is the reduction in the shock wave amplitude in the sample. The presence of more pore space requires additional plastic work to crush out open pore space. The shock pressures and shock wave velocities are smaller for higher porosities. Figure 9 illustrates the effect of porosity on shock wave propagation in Us–up space. Each symbol represents the result of a single mesoscale model for a given porosity and initial shock pressure. With an increase in porosity, the shock wave velocities decrease with respect to particle velocities. The discontinuity in the Hugoniot curve in the Us–up diagram represents a solid-state phase transition of quartz. To summarize, the propagation of a shock wave through a sample with a large number of resolved pores results in an overall decrease in shock pressure. In all experiments, the pore space is completely crushed out.

Figure 9.

 Hugoniot curves in the Us–up space for a nonporous quartzite and quartzite with different porosities. The points indicate data derived from mesoscale modeling. The lines indicate the Hugoniot data obtained by the macroscale model. The line for the nonporous material also includes a phase transition of quartz in the quartzite to the high-pressure phase stishovite. All data were obtained for a yield strength of Y = 0.

Effect of Strength on Shock Propagation in Porous Material

For the results in the previous sections, we assumed that the strength of the matrix surrounding the pores is small in comparison with shock pressure and therefore negligible. However, to generate Hugoniot data for lower shock pressures, the resistance of the material against plastic deformation becomes important for the crushing of pores and, thus, for the propagation of shock waves. We carried out a suite of numerical experiments where we assumed a yield strength Y for the matrix of 1 and 0.5 GPa, respectively. The strength of the matrix can be considered as crushing strength of the pores. If strength is zero (Y = 0), the material behaves similar to a fluid allowing pores to close at small amplitudes of a compression wave. For > 0, crushing occurs if compression exceeds the yield strength of the matrix, or more precisely if J2 > Y2, where J2 is the second invariant of the deviatoric stress tensor.

The Us–up plot in Fig. 10 (dashed lines, symbols) shows that the occurrence of an elastic regime depends on the yield strength of the matrix. The onset of crushing pore space and plastic deformation is shifted toward higher particle velocities (higher shock pressures) with increasing strength of the surrounding matrix. The compaction of pores occurs very abruptly once the shock pressure is in excess of the yield strength. There is hardly any transitional regime recognizable where pores are closed only partially. This is due to a very short rise time of the shock front. In the mesoscale models with Y of 1 GPa, plastic deformation and crushing of pore space occur at a particle velocity up of 300 m s−1 corresponding to a pressure P of 3 GPa, and for Y of 0.5 GPa, crushing of pores already starts at up of 80 m s−1 corresponding to a pressure P of 0.8 GPa. The hydrodynamic material shows no resistance to pore crushing; plastic deformation starts immediately. In the plastic and shock wave regime, the curves for different strengths lie very close to one another confirming that strength is almost negligible for high shock pressures. Regarding the previously described pressure amplifications, yield strength has, therefore, no significant effect on the maximum pressure distribution.

Figure 10.

 Hugoniot curves in the Us–up space for different yield strengths Y. The points indicate data derived from mesoscale modeling [EXP]. The lines indicate the Hugoniot data provided by the macroscale model [HUG] where values for εe = −7.5 × 10−2 corresponding to Y = 1 GPa and εe = −3.0 × 10−2 corresponding to Y = 0.5 GPa have been used (see the Comparison of Meso- and Macroscopic Models section).

Shock Propagation in Water-Saturated Material

The presence of water hampers the crushing of pores. Our mesoscale models of the propagation of a shock wave traveling through a sample, where the individually resolved pores are partially or completely filled with water, show that shock waves travel faster and with higher shock amplitudes through water-saturated material compared with a material with “empty” pore space (Fig. 11).

Figure 11.

 Snapshots of pressure profiles at the same point in time after impact of the flyer plate on the buffer plate through a) a nonporous, b) 100% water-saturated (all pore space is filled with water), c) 50% water-saturated (50% of pore space are filled with water, 50% are empty), and d) porous sample (all pore space is empty). The porosity in the sample is 25%. The initial shock pressure is 6 GPa. The dashed lines indicate data derived from mesoscale modeling; the gray solid line represents data from the macroscopic model. Zero depth represents the interface of the flyer and the buffer plate.

Figure 11 depicts a comparison of pressure profiles for the same instance in time for all three cases (100% water saturation, 50% water saturation, 0% water saturation) as well as for the nonporous case. The initial shock pressure at the flyer–buffer plate interface was 6 GPa. In the 100% and 50% water-saturated case, the oscillations are not as prominent as in the dry case. The observed localized pressure amplification is smaller for the water-saturated sample as already stated above. Hugoniot data for the different cases of water-saturation are plotted in the Us–up space in Fig. 12 (symbols represent the results from mesoscale modeling). The particle and shock wave velocities decrease by adding empty pore space. The fastest attenuation of the shock wave is observed in the case of a dry porous material. In all three porous cases shown in Fig. 11, the porosity is 20%. In the case of 100% water saturation (all pores are filled with water), the volume of pore space changes only little and the absorbed energy due to plastic work carried out to close pores is much smaller compared with a dry target (all pores are empty). For an initial pressure of 6 GPa, about 80% and for 22 GPa, only 45% of the initial volume of water remains in the sample. A decrease in water volume as response to shock loading is due to the compressibility of water. Thus, there is no mass-loss of water. An increase in initial pressure results in a larger reduction in pore space leading to an increase in pressure in water-filled pores.

Figure 12.

 Hugoniot curves in the Us–up space for nonporous quartzite and quartzite of different degrees of water saturation. The symbols represent the data obtained with the mesoscale model; the lines indicate Hugoniot data of the macroscopic model. The changes in slopes in the Us–up diagram are associated with a solid-state phase transition of quartz, where the high-pressure phase represents stishovite.

The partially water-saturated case (50% of the pores are filled with water and the other 50% are empty) represents a transition between the completely dry and wet cases. For an initial shock pressure of 6, 14, and 22 GPa, the pressure amplitude decreases to 4.8, 11.8, and 22 GPa in the fully water-saturated material. For shock pressure >22 GPa, the reduction in pressures is not observable anymore which might be caused by the fact that a significant amount of energy is converted into thermal energy.

Comparison of Meso- and Macroscopic Models

We compare the results from mesoscale modeling for different porosities, water saturation, and strength of the matrix with macroscopic (or homogenized) material models describing the bulk behavior of porous and mixed (water + quartzite) material that have been proposed before (Pierazzo et al. 2005; Wünnemann et al. 2006).

Figure 11 illustrates that both model approaches provide similar results for shock wave pressure and propagation velocity through the porous sample by averaging the pressures in the mesoscale simulation. Generally, we find a very good agreement in terms of Hugoniot data in Us–up space between the macroscopically determined data and data obtained from the mesoscale models as shown in Figs. 9, 10, and 12. A similar good match was achieved when comparing other thermodynamic state parameters such as pressure, density, and internal energy.

In the case of 100% water-saturated pores, the models confirm that ANEOS for a quartzite-water mixture assuming thermodynamic equilibrium between the water and the quartzite phase is a reasonably good approximation to describe material behavior. This implies that the crushing of pore space is insignificant, although the water content changes due to the compressibility. However, we do not allow the water to flow or steam into the matrix material.

To compare meso- and macroscale models for low shock pressures where material strength plays an important role, the crushing strength Y of the matrix in the mesoscale models has to be related to the elastic threshold strain εe (material parameter for ε–α model). Crushing first occurs if the stress exceeds the yield strength Y of the matrix. According to the constitutive model as described in Methods section, this is given if the square root of the second invariant of the deviatoric stress tensor is equal to or larger than Y (J2 > Y2). For simplicity, we made use of a von Mises yield criterion (Y is constant and does not depend on deformation history (damage), pressure, or temperature to derive a relationship between the macroscopic and mesoscopic parameters. We are aware of the fact that the von Mises criterion is not typical for rock materials. However, a constant yield strength allows for direct analytic comparison between the elastic threshold parameter in the macroscale model (ε–α-model) and the yield strength of the matrix in the mesoscale model. A more complex material model requires at least three or more parameters that are difficult to compare with the single parameter εe, in the ε–α model. The next step would be to use a more realistic material model to analyze the relationship between εe and other macroscopic strength parameters such as the coefficient of friction and cohesion in intact and damaged state (see Collins et al. 2004). This would require additional extensive parameter studies which are beyond the scope of this paper.

According to Hook’s law, a constant strength of 1 GPa yields an elastic threshold strain of 0.071 using an elastic modulus (Young’s modulus) of 14 GPa that was determined by laboratory experiments for Seeberger sandstone as used in the MEMIN cratering experiments (Kenkmann et al. 2011; Moser et al. 2013). At the onset of plastic yielding (crushing of pores), the strain is given by

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where εe can be considered as the elastic threshold strain, Y is the yield strength of the matrix and corresponds to the crushing strength, and E is Young’s modulus. From a series of numerical experiments fitting the macroscopic and mesoscopic modeling results, we obtained an elastic strain threshold of −0.075 assuming a constant yield strength of 1 GPa. We consider the deviation between the empirically determined value of 0.075 and the analytically derived 0.071 as small. Similar good results were obtained for a smaller yield strength of 0.5 GPa where we obtained an elastic strain threshold of –0.030, which only slightly deviates from the empirically determined value of 0.036.

The results suggest that the volumetric crushing strain εe as an important input parameter in the ε–α compaction model for porous material can be derived from the yield strength of the grains supporting the porosity. However, the yield strength of individual grains does not correspond to the strength of a larger unit of the same material. Yield strength is a scale-dependent parameter. On the scale of the formation of craters in the framework of the MEMIN experiments (Kenkmann et al. 2011; Schäfer et al. 2006), typically tens of centimeters, the resistance of the material against plastic deformation is usually much softer (smaller yield strength of about 50 MPa) than the stated crushing strength. A meaningful relationship for strength on different scales is still lacking.


The MEMIN project focuses on impact cratering in porous, dry, partially or completely water-saturated sandstone. The Seeberger sandstone used in the impact cratering experiments serves as an analog material representative for crater formation, for instance, in sedimentary targets on Earth. The heterogeneous character on the scale of pores and grains is in particular a challenge for numerical modeling of crater formation and shock wave propagation in sedimentary material. Detailed investigation of porous materials with some water content and their response to shock wave loading are lacking so far.

Models of different scales (meso- and macroscales) are complementary. The macroscale models enable the simulation of the entire cratering processes, whereas the mesoscale models enable a detailed analysis of localized processes during shock compression. We find a good agreement between macroscopic and mesoscopic data. The comparison between meso- and macroscopic models demonstrates that for a sufficiently long duration of the shock plateau (covering an area of several pores), the effect of pore size, distribution, and geometry on shock propagation is negligible. For low shock pressures, the yield strengths of the matrix in the mesoscale models can be related to the volumetric threshold strain at the onset of pore crushing in the ε–α macroscale porosity compaction model. We find good agreement for the elastic and plastic regime for the mesoscopic and macroscopic approach (Fig. 10). However, in the mesoscale model, the transition from the elastic to crushing regime is very abrupt, while a much more gradual transition can be observed in the macroscopic model. Any “smoothing” of the transition in the mesoscale models by using a more sophisticated strength model to describe material failure of the matrix was not successful. Once the stress amplitude of the compression wave exceeds the yield strength, pores are crushed out instantaneously, which presumably is caused by a very short rise time of the shock front.

Complementary to our numerical approaches to determine Hugoniot data for sandstone, we compared our modeled Hugoniot data results with experimental data determined from laboratory shock experiments with Seeberger sandstone carried out in the framework of this study and Hugoniot data from the literature for Coconino sandstone (compiled in Ahrens and Gregson 1964; Shipman et al. 1971; Stöffler 1982). Despite the slight variations in porosity and composition, we find a good agreement as shown by the Hugoniot curve in the pressure-specific volume space in Fig. 13. The good correlation among experimental, meso-, and macroscale model data justifies the applicability of our mesoscale approach to test the macroscopic models describing dry and wet porous material and, in particular, partially water-saturated porous sandstone where the ε–α model has been combined with ANEOS for a water-quartzite mixtures.

Figure 13.

 Comparison of numerical modeling results of quartzite material and laboratory shock experiment data showing Hugoniot curves in the P–V space. The numerical modeling results are shown with lines; the shock experimental data are indicated by symbols.

In addition to the Hugoniot data, the MEMIN cratering experiments provide important standards the models can be tested against, and also serve for quantitative explanations for some unexpected observations. The quantification of shock amplification due to pore space collapse using mesoscale modeling is in good agreement with observations in shock experiments in dry sandstone at low shock pressures (5.0–12.5 GPa). The experiments were carried out to identify and calibrate shock features in weakly shocked Seeberger sandstone (Kowitz et al. 2013). Despite low shock pressures (10 GPa), diaplectic glass was observed that forms in single crystal quartz at about 35 GPa (Stöffler and Langenhorst 1994). The mesoscale models show that an amplification by a factor of 3–4 can occur in the vicinity of a pore, which is in excellent agreement with the observation of shock features representative for shock conditions 3.5 times the initial shock pressure. The amplification of shock pressure due to pore collapse as quantified in the mesoscale models also helps to understand the generation of small vapor and melt phases during the very first contact between the projectile and the target and provide important insights on the observation of planar deformation features in the ejecta of the MEMIN experiments (Ebert et al. 2013). For the determination of peak pressure amplification using mesoscale models, we assumed that at least 20% of the initial pore volume has experienced the amplified pressure conditions. The limitation of 20% is chosen to satisfy the required resolution to obtain reliable results. However, very small volumes that undergo even higher pressures are conceivable, but cannot be resolved in our models.

To model the crushing behavior of a given material accurately, direct measurements of either crushing curves (εe in macroscale ε–α model) or the strength of the matrix (yield strength Y in mesoscale models) are required. Detailed observations of pore space compaction underneath the MEMIN craters (Buhl et al. 2013) enable an indirect measurement of the crushing behavior. By fitting the numerical models to porosity curves as a function of depth from the MEMIN experiments, crushing parameters for the modeled porous material can be obtained. However, the numerical simulations consider shock compression only. The release from shock pressure has not been considered in the macro- and mesoscale models in the present work. The change in porosity with depth underneath the crater in the MEMIN experiments shows that tensile bulking causes an increase in porosity at the near subsurface of the crater. To account for the gain in porosity as a result of shock unloading, further work is required. In fact, shear and tensile stresses may yield an increase in porosity. First preliminary results are shown in Collins et al. (2011) and Güldemeister et al. (2013).

Our work also showed that a combination of the ε–α model and an ANEOS for mixtures is applicable and enables the simulation of partially water-saturated materials. The mesoscale modeling approach is a direct way to describe shock wave propagation in porous sandstone with different degrees of water saturation (0, 50, and 100% water filling). The results are thermodynamically consistent with the macroscopic models using the ε–α model (0% water filling), the ANEOS for material mixtures (100% water filling), and a combination of both (e.g., 50% water filling). We always assume that the water content remains constant in the computational cell. This does not account for the subsequent release in material mixtures and effect of flowing or steaming of water through the matrix material. In the framework of the MEMIN experiments, it was not possible to confirm whether this assumption holds true. In fact, the pore water disappeared in the direct proximity of the crater. Apparently, water can escape in case of very high shock pressures. It is unclear whether the process is dominated by thermal expansion of water vapor in the pore or by tensile fracturing and opening of pores. Further improvements for a more accurate treatment of material mixtures at high shock pressures and subsequent release are required.


To simulate laboratory cratering experiments in the framework of the MEMIN project, a detailed description of the thermodynamic behavior of porous sandstone with different water contents was required. We conducted a series of numerical simulations on the meso- and macroscales to study the effect of porosity and water-filling on shock wave propagation. The mesoscale models provide details on single pore collapse and the bulk response of a set of pores to shock propagation. The results from mesoscale models with previously proposed macroscopic models for the crushing of dry porosity (ε–α crushing model) and water-saturated porosity (ANEOS for material mixtures) are in good agreement. Thus, the propagation of shock waves through a porous material sample that is represented by a homogeneous unit where porosity is a state variable agrees very well with the one through a sample which explicitly resolves a finite number of pores. While the former approach allows for direct measurement of Hugoniot data (particle velocity, shock wave velocity), in the latter case, thermodynamic parameters vary due to the heterogeneous distribution of pores and Hugoniot data have to be determined by averaging over a representative area. The models show that the crushing of pore space is an effective mechanism for absorbing shock wave energy leading to a faster decay of the shock wave. In contrast, the closure of pores causes localized amplification of shock pressure in the vicinity of a single pore. The effect decreases slightly if several pores are located close to one another and reflections of shock and rarefaction waves originating from the pores superimpose. Furthermore, shock waves travel faster through water-saturated material than through porous material, and water-saturated pores are only slightly compacted compared with empty pores that are completely closed. Macroscopic models and laboratory experiment data showed a very good agreement in terms of Hugoniot data supporting applicability and accuracy of our modeling approach. Mesoscale models of shock wave propagation in partially water-saturated material are also consistent with a novel approach of combining the ε–α crushing model for dry porosity and ANEOS for water-quartzite mixtures. Therefore, we conclude that this is a valid approach to describe the thermodynamic behavior of partially water-saturated material during shock wave compression.

Acknowledgments— This research is part of the MEMIN program supported by the German Science Foundation DFG (Research Unit FOR-887; WU 355/6-1). We are grateful to N. Artemieva for providing ANEOS data for water–quartzite mixtures and for her help with the ANEOS model. We thank J. Melosh and an anonymous reviewer for their constructive reviews. We also thank A. Deutsch for helpful comments on the manuscript.

Editorial Handling— Dr. Alexander Deutsch