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The Richtmyer–Meshkov instability (RMI) arises at an impulsively accelerated interface between two materials of different density. Historically, this instability was studied in fluids. Recently, RMI studies have been extended to investigate material properties of solids. Material strength at high strain-rates in solids have been extracted from the amplitude and growth of the RMI spike in an untamped environment, specifically, the metal-vacuum interface. This technique has also been shown to elucidate material properties in a distended tamping media, metal-porous solid interface. Here, a bridge to understanding the nonlinear mechanical behavior of copper into a liquid tamping media is investigated experimentally and computationally. We show the RMI growth rate and resulting profile are dependent on initial shock strength, as well as the nondimensional perturbation, with an initial Atwood number of $$-0.78$$ . Data collected from a tamped liquid environment range in metal breakout pressures up to ten GPa. This information is used to calibrate and validate numeric model parameters. The oscillatory shock front in the liquid tamping media is used to approximate the viscosity from a transient 1-D analytic approximation. The viscosity is found to be in agreement with other experimental work, however is not determined to be the only dissipative force in the experiment. Hydrocode simulations of our experiments show reasonable alignment with current and previously published work.

Dynamic interface instabilities such as Rayleigh–Taylor, Kelvin–Helmholtz, and Richtmyer–Meshkov are important in a number of physical phenomena. Besides meriting study because of their role in natural events and man-made applications, they can also be used to study constitutive properties of materials in extreme conditions. Both RTI and RMI configurations have been used to study the strength of solids at high strain rates, though RMI has largely been limited to zero or ambient pressure. Recently, advances in imaging have allowed tamped RMI experiments to be performed in which the pressure is maintained above ambient. In this study, we examine the tamped RMI for determining material strength. Through simulation, we explore the behavior of the jetting material and examine the sensitivity of jetting to material properties. We identify simple scaling laws that relate the key physical parameters controlling jetting, which are compared to previous results from the literature. We use these scaling law and other considerations to examine issues associated with tamped RMI experiments.

The shock wave perturbation decay experiment is a technique in which the evolution of a perturbation in a shock wave front is monitored as it propagates through a material field. This tool has recently been explored to probe the high-rate shear response of granular materials. This dynamic behavior is complicated due to inter- and intra-granular phenomena involved. Mesoscale modeling can give insight into this complexity by explicitly resolving the interactions and deformation of individual grains. The peridynamic theory, which is a nonlocal continuum theory, provides a suitable framework for modeling dynamic problems involving fracture. Prior research has focused mostly on the continuum, bulk response, neglecting any localized material failure, of granular materials. A systematic investigation of the effects of grain fracture and frictional contact forces between grains on the continuum behavior of granular materials is carried out by peridynamic simulations of a shock wave perturbation decay experiment. A sensitivity assessment of dominant factors indicates that grain fracture, a phenomenon ignored in most computational investigations of granular materials, plays a large role in the bulk dynamic response. Our results show that the wave propagates faster with an increase in the toughness of the material and the inter-particle friction. Also, the shock amplitude is shown to decay faster in tougher materials. It is further confirmed that under strong compression self-contact among fractured grain sub-particles cannot be neglected.

At low pressures, of the same order as the strength of the underlying material, the removal of porosity is the dominant aspect controlling the behavior of shock-loaded porous materials. In this chapter, we discuss experimental approaches used in this regime including sample configuration and characterization, instrumentation, and error analyses for the results. We follow this with discussion of modeling approaches utilized at both the continuum and the mesoscale. Building upon these fundamentals, we consider several phenomena that are specific to the low-pressure dynamic behavior of porous materials. Finally, we discuss some of the outstanding issues in the behavior and treatment of porous materials in this regime.

A correction to this paper has been published: https://doi.org/10.1007/s40870-021-00302-x

The strength of brittle porous media is of concern in numerous applications, for example, earth penetration, crater formation, and blast loading. Thus, it is of importance to possess techniques that allow for constitutive model calibration within the laboratory setting. The goal of the current work is to demonstrate an experimental technique allowing for strength assessment of porous media subjected to shock loading, which can be implemented into pressure-dependent yield surfaces within numerical simulation schemes. As a case study, the deviatoric response of distended α-SiO2 has been captured in a tamped Richtmyer–Meshkov instability (RMI) environment at a pressure regime of 4–10 GPa. Hydrocode simulations were used to interpret RMI experimental data, and a resulting pressure-dependent yield surface akin to the often employed modified Drucker–Prager model was calibrated. Simulations indicate that the resulting jet length generated by the RMI is sensitive to the porous media strength, thereby providing a feasible experimental platform capable of capturing the pressurized granular deviatoric response. Furthermore, in efforts to validate the RMI-calibrated strength model, a set of Mach-lens experiments was performed and simulated with the calibrated pressure-dependent yield surface. Excellent agreement between the resulting Mach-lens length in experiment and simulation provides additional confidence to the RMI yield-surface calibration scheme.

Understanding and quantifying the uncertainties in experimental results are crucial to properly interpreting simulations based on those results. While methods are reasonably well established for estimating those uncertainties in high-pressure shock experiments on homogeneous materials, it is much more difficult to treat relatively low-pressure experiments where shock rise times are significant and material strength is not negligible. Sample heterogeneity further complicates the issue, especially when that heterogeneity is not characterized in each sample. Here, we extend the Monte Carlo impedance matching approach used in high-pressure Z experiments to low-pressure experiments on heterogeneous porous materials. The approach incorporates uncertainties not only in the equation of state of the impedance matching standard but also those associated with its strength. In addition, we also examine approaches for determining material heterogeneity and evaluate its effect on the experimental results.

The pertrubation decay technique has been recently applied to probe granular material dynamic response under high-rate shear deformations. A vast majority of the simulations of this experiment, however, has only considered the bulk response of the granular materials and neglected grain-scale phenomena such as grain-to-grain contact and friction, as well as fracture. Mesoscale modeling can be used to address these shortcomings. We utilize a peridynamic modeling framework to explicitly model each individual granular particle and the contact between them, in addition to considering material failure. Results indicate that these factors play a large role in the event.The pertrubation decay technique has been recently applied to probe granular material dynamic response under high-rate shear deformations. A vast majority of the simulations of this experiment, however, has only considered the bulk response of the granular materials and neglected grain-scale phenomena such as grain-to-grain contact and friction, as well as fracture. Mesoscale modeling can be used to address these shortcomings. We utilize a peridynamic modeling framework to explicitly model each individual granular particle and the contact between them, in addition to considering material failure. Results indicate that these factors play a large role in the event.

Plate-impact techniques are used to investigate the equation of state and constitutive relations of copper at high pressures. The parameters are inferred from the velocity history of the surface of a shocked and then released copper plate in contact with a window. The initial shock interacts with the window interface sending a release wave back into the copper, thus, delaying any subsequent shock or release waves traveling toward the window. A backwards-characteristics code is used to regenerate the in situ particle velocities and correct for the presence of the window material. Given these corrections, the Lagrangian sound speed, pressure, and volume (density) are found, and from the sound speed, the strength parameters of the copper are calculated. This investigation finds a shear modulus of ∼100 GPa and a yield strength of ∼1 GPa for copper, at impact pressures of ∼130 GPa. The inferred shear modulus is in excellent agreement with the work of Hayes et al. The yield estimates are in broad agreement with, but are distinct from, previous theoretical and experimental studies.

Development of models to describe the shock states of distended mixtures is motivated by the need to understand how these materials respond over large compression ranges starting from mechanical crush and ending in extreme thermodynamic states. The engineering and physical science communities have dedicated much effort in understanding and modeling the compressive response of distended materials. Unfortunately, the endeavor to understand becomes more complicated when the material of interest is actually a heterogeneous mixture of individual components rather than a single distended solid. The mixture may inherently have components with widely disparate densities, moduli, and strengths, thus adding to the challenge. A material modeling approach is presented which is comprised of a thermodynamically consistent Hugoniot equa- tion of state (EOS) built into a mixture-modeling framework. This combination enables the user to describe a hetero- geneous combination of material components. The modeling approach can describe the dynamic response of distended mixtures over very large compression ranges.

Understanding the impact, penetration and cavity formation of heterogeneous granular systems is of fundamental importance to a wide variety of research endeavors. In this work a series of experiments were conducted in order to investigate the penetration dynamics of loose dry sand. High-speed photography coupled with a particle image velocimetry (PIV) technique was used to capture both the grain and bulk response of the penetration event, while buried quartz gages simultaneously recorded transmitted stress wave profiles. Depth of penetration was measured via postmortem examination. Experiments were conducted over a velocity range of 30–200 m/s using both cylindrical and spherical projectiles in a unique semi-infinite experimental configuration in order to directly observe a cross-section of the impact and penetration event. The experimental results are compared to simple continuum Eulerian hydrocode simulations and an analytic penetration model. The simulations are not able to resolve both stress and velocity measurements. However, the simulations do reproduce the depth of penetration for a wide variety of penetration experiments.

There have been a variety of numeric and experimental studies investigating the dynamic compaction behavior of heterogeneous materials, including loose dry granular materials. Mesoscale simulations have been used to determine averaged state variables such as particle velocity or stress, where multiple simulations are capable of mapping out a shock Hugoniot. Due to the computational expense of these simulations, most investigators have limited their approach to two-dimensional formulations. In this work we explore the differences between two- and three-dimensional simulations, as well as investigating the effect of stiction and sliding grain-on-grain contact laws on the dynamic compaction of loose dry granular materials. This work presents both averaged quantities as well as distributions of stress, velocity and temperature. The overarching results indicate that, with careful consideration, two- and three-dimensional simulations do result in similar averaged quantities, though differences in their distributions exist. These include differences in the extreme states achieved in the materials.

The evolution of spall for a brittle material is investigated under variance of anisotropy, grain boundary fracture energy, and loading. Because spall occurs in the interior of the specimen, fundamental studies of crack nucleation and growth are needed to better understand surface velocity measurements. Within a cohesive approach to fracture, we illustrate that for anisotropic materials, increases in the fracture energy cause a transition in crack nucleation from triple-points to entire grain boundary facets. Analysis of idealized flaws reveals that while crack initiation and acceleration are strong functions of the fracture energy, flaws soon reach speeds on the order of the Rayleigh wave speed. Finally, simulated surface velocities of spalled configurations are correlated with microstructural evolution. These fundamental studies of nucleation, growth, and spall attempt to link atomic separation to the macroscopic spall strength and provide a computational framework to examine the evolution of spall and the impact on the simulated surface velocity field.

Under shock loading, metals typically increase in strength with shock pressure initially but at higher stresses will eventually soften due to thermal effects. Under isentropic loading, thermal effects are minimized, so strength should rise to much higher levels. To date, though, study of strength under isentropic loading has been minimal. Here, we report new experimental results for magnetic ramp loading and impact by layered impactors in which the strength of 6061-T6 aluminum is measured under quasi-isentropic loading to stresses as high as 55 GPa. Strength is inferred from measured velocity histories using Lagrangian analysis of the loading and unloading responses; strength is related to the difference of these two responses. A simplified method to infer strength directly from a single velocity history is also presented. Measured strengths are consistent with shock loading and instability growth results to about 30 GPa but are somewhat higher than shock data for higher stresses. The current results also agree reasonably well with the Steinberg–Guinan strength model. Significant relaxation is observed as the peak stress is reached due to rate dependence and perhaps other mechanisms; accounting for this rate dependence is necessary for a valid comparison with other results.

Mesoscale calculations have been conducted in order to gain further insight into the dynamic compaction characteristics of granular ceramics. The primary goals of this work are to numerically determine the shock response of granular tungsten carbide and to assess the feasibility of using these results to construct the bulk material Hugoniot. Secondary goals include describing the averaged compaction wave behavior as well as characterizing wave front behavior such as the strain rate versus stress relationship and statistically describing the laterally induced velocity distribution. The mesoscale calculations were able to accurately reproduce the experimentally determined Hugoniot slope but under predicted the zero pressure shock speed by 12%. The averaged compaction wave demonstrated an initial transient stress followed by asymptotic behavior as a function of grain bed distance. The wave front dynamics demonstrate non-Gaussian compaction dynamics in the lateral velocity distribution and a power-law strain rate–stress relationship.

Strength is an important aspect of material behavior for armor performance, planetary science, and accurate property measurement under quasi-isentropic loading and static high pressure. The advent of time-resolved diagnostics allowed the elastic–plastic behavior of solids under shock loading to be detected for the first time. These early experiments revealed a two-wave structure of elastic and plastic shock waves for stresses above the elastic limit. However, the full stress state in the shocked state remained unknown because the conservation equations provide only the longitudinal stress. Since then, a variety of techniques has been used to determine strength in the shocked state. One of the most successful has been using shock/release and shock/reload to find the stress state and strength in the shocked condition. This technique has been applied to a variety of metals and ceramics at stresses of 100 GPa or higher. These experiments have revealed a rich set of behaviors, many of which are still not fully understood. Recently, there has been significant interest in isentropic loading where the material can be significantly cooler and strength even more important. In this paper, we present a current perspective on strength under high pressures, particularly in the dynamic regime, with emphasis on the techniques used to measure strength and their advantages and disadvantages. Results of strength measurements on several materials will be discussed, and interesting aspects of behavior will be highlighted. In addition, shock results will be contrasted with those under isentropic loading. Finally, some new directions in the study of strength will be outlined.

Velocity-time histories of free- or windowed-surfaces have been used to calculate wave speeds and hence deduce the shear moduli for materials at high pressure. This is important to high velocity impact phenomena, e.g. shaped-charge jets, long rod penetrators, and other projectile/armour interactions. Historically the shock overtake method has required several experiments with different depths of material to account for the effect of the surface on the arrival time of the release. A characteristics method, previously used for analysis of isentropic compression experiments, has been modified to account for the effect of the surface interactions, thus only one depth of material is required. This analysis has been applied to symmetric copper impacts performed at Sandia National Laboratory's Star Facility. A shear modulus of 200GPa, at a pressure of ~180GPa, has been estimated. These results are in broad agreement with previous work by Hayes et al.

The dynamic compaction of sand was investigated experimentally and computationally to stresses of 1.8 GPa. Experiments have been performed in the powder's partial compaction regime at impact velocities of approximately 0.25, 0.5, and 0.75 km/s. The experiments utilized multiple velocity interferometry probes on the rear surface of a stepped target for an accurate measurement of shock velocity, and an impedance matching technique was used to deduce the shock Hugoniot state. Wave profiles were further examined for estimates of reshock states. Experimental results were used to fit parameters to the P-Lambda model for porous materials. For simple 1-D simulations, the P-Lambda model seems to capture some of the physics behind the compaction process very well, typically predicting the Hugoniot state to within 3%.

In theory, a shock wave traveling through a material gives rise to a well defined Hugoniot state. However, in practice, the measurement technique used to probe the shocked state imparts on the data a unique set of experimental artifacts which can affect interpretation of this data. Two commonly used methods for acquiring shock wave data, VISAR and inclined‐mirror measurements are examined to determine the effects of the measurement technique on the final Hugoniot determination. Recent plate impact experiments on the ceramic silicon carbide are used to calibrate a one‐dimensional computer model, which is then used to simulate experimental VISAR and inclined mirror data. The results, which highlight potential pitfalls in interpretation of experimental data, will be discussed and solutions to the discrepancies will be proposed. Further, this work is extended to include ceramics that undergo phase transitions.

Summary form only given. A capability to produce quasi-isentropic compression of solids using pulsed magnetic loading on the Sandia Z accelerator has recently been developed and demonstrated. The accelerator is capable of producing >20 MA current pulses with ~300 ns rise-time into a short circuit load, generating intense magnetic fields within the anode cathode (AK) gap. With proper design of the short circuit load and suitable shaping of the current pulse, this technique allows planar, continuous compression of materials several hundred microns in thickness to stresses approaching 350 GPa. Development of a loading capability for direct measurement of material properties along an isentrope has been a long standing goal in the equation of state (EOS) community. Combined with more traditional Hugoniot compression data, the principal isentrope provides a more comprehensive description of the complete EOS. Furthermore, direct measurement of the isentrope, which lies very close to the room temperature isotherm, has the potential to improve the accuracy of pressure standards used to calibrate pressure scales in static high-pressure diamond anvil cell experiments. Isentropic compression experiments (ICE) are also well suited for the study of pressure induced phase transformations in materials. Since the loading path followed in the experiment is the material isentrope, even relatively small volume change transitions, which might be extremely difficult to probe in shock wave experiments, are readily observed. Furthermore, due to the low temperature increase associated with isentropic compression it is possible to study the kinetics of rapid solidification. Examples of such experiments will be discussed. As a final example, the use of ICE techniques to infer material strength at high pressures and densities will be discussed. Comparison of the loading and subsequent unloading wave profiles in a material provides a means to directly infer the strength of the material at pressure. Feasibility experiments performed on aluminum to stresses of ~150 GPa will be presented

To address known shortcomings of classical metal plasticity for describing material behavior under shock loading, a model which incorporates a distribution in the deviatoric stress state is developed. This distribution will translate in stress space under loading, and growth of the distribution can be included in the model as well. This proposed model is capable of duplicating the key features of a set of reshock and release experiments on 6061‐T6 aluminum, many of which are not captured by classical plasticity. The model is relatively simple, is only moderately more computationally intensive, and requires few additional material parameters.

Unloading from a plastically deformed state has long been recognized as a sensitive measure of a material's deviatoric response. In the case of a ramp compression and unload, time resolved particle velocity measurements of a sample/window interface may be used to gain insight into the sample material's strength. Unfortunately, measurements of this type are often highly perturbed by wave interactions associated with impedance mismatches. Additionally, wave attenuation, the finite pressure range over which the material elastically unloads, and rate effects further complicate the analysis. Here, we present a methodology that overcomes these shortcomings to accurately calculate a mean shear stress near peak compression for experiments of this type. A new interpretation of the self-consistent strength analysis is presented and then validated through the analysis of synthetic data sets on tantalum to 250 GPa. The synthetic analyses suggest that the calculated shear stresses are within 3% of the simulated values obtained using both rate-dependent and rate-independent constitutive models. Window effects are addressed by a new technique referred to as the transfer function approach, where numerical simulations are used to define a mapping to transform the experimental measurements to in situ velocities. The transfer function represents a robust methodology to account for complex wave interactions and a dramatic improvement over the incremental impedance matching methods traditionally used. The technique is validated using experiments performed on both lithium fluoride and tantalum ramp compressed to peak stresses of 10 and 15 GPa, respectively. In each case, various windows of different shock impedance are used to ensure consistency within the transfer function analysis. The data are found to be independent of the window used and in good agreement with previous results.

Large amplitude steady waves in materials have been observed to display certain scaling relationships between the strain rate and the stress amplitude. In many homogeneous materials, strain rate scales with stress to the fourth power. However, scaling of strain rate with stress to the first, second, and fourth power has been found for different classes of heterogeneous materials. We examine wave structures for three classes of heterogeneous materials through mesoscale simulations that resolve the scale of heterogeneity explicitly. We utilize these simulations to gain insight into the scaling phenomena observed and to identify the critical non-dimensional parameters for the phenomena. These parameters are then applied to the available experimental data for the three classes. The same set of non-dimensional groups is found to be appropriate for layered and particulate composite materials, while somewhat different groups are found for granular materials. Two different types of simulations lead to different conclusions on the need for the inclusion of strength in the non-dimensionalization for granular materials. The groups formed are found to collapse the experimental data quite well when the strength parameter is not included. Finally, a simple model for granular materials demonstrates that the crucial aspect of their behavior that controls the scaling of waves is the need for mass transfer to close voids in the material.