Hugoniot States Research Articles

Shock response of a model structured nanofoam of Cu

Using large-scale molecular dynamics simulations, we investigate shock response of a model Cu nanofoam with cylindrical voids and a high initial porosity (50% theoretical density), including elastic and plastic deformation, Hugoniot states, shock-induced melting, partial or complete void collapse, nanojetting, and hotspot formation. The elastic-plastic and overtaking shocks are observed at different shock strengths. The simulated Hugoniot states can be described with a modified, power-law P−α (pressure–porosity) model, and agree with shock experiments on Cu powders, as well as the compacted Hugoniot predicted with the Grüneisen equation of state. Shock-induced melting shows no clear signs of bulk premelting or superheating. Voids collapse via plastic flow nucleated from voids, and the exact processes are shock strength dependent. With increasing shock strengths, void collapse transits from the “geometrical” mode (collapse of a void is dominated by crystallography and void geometry and can be different from that of one another) to “hydrodynamic” mode (collapse of a void is similar to one another); the collapse may be achieved predominantly by flow along the {111} slip planes, by way of alternating compression and tension zones, by means of transverse flows, via forward and transverse flows, or through forward nanojetting. The internal jetting induces pronounced shock front roughening, leading to internal hotspot formation and sizable high speed jets on atomically flat free surfaces.

Using Time-resolved Wave Profile Measurements to Determine Elusive Phase Transitions in Molybdenum

The purpose of this study was to investigate the dynamic strength of molybdenum (Mo) at high pressures, particularly at the location of a solid-solid phase transition within Mo prior to entering the melt regime. The intent was to build on previous work by conducting both symmetric and non-symmetric impact experiments using a two-stage light gas gun and VISAR diagnostic system to examine molybdenum behavior up to pressures of 305GPa. The approach required compensating for the wave interaction due to the low impedance LiF window, but provided detailed information regarding the release state. The main effort consisted of a series of Mo symmetric impact experiments having a fixed ratio of impactor thickness (1mm) to target thickness (4.25mm) and variation of the impact velocity which resulted in changes in both shock and release wave velocity. Based on the geometry, any structural change resulting in a change in release wave velocity would be noticeable through changes in the amount of time at the Hugoniot state or dwell time. As expected, the shock velocity increased proportionally with increased impact velocity, but a noticeable change in dwell time occurred at approximately 190GPa. Additional analysis of the strength variation of Mo showed an increase in strength from 1.3GPa to 3GPa at approximately 190GPa the location of phase transition, followed by a dramatic drop in strength for stresses above 190GPa. Strength increased again for stresses of 300 and 305GPa. The data acquired in this effort indicate the phase transition occurs at a slightly lower pressure than previously indicated and there appears to be a mixed phase region. This paper describes the strategy, experimental method, and corresponding results which are used to draw conclusions about the dynamic strength of molybdenum at high pressure.

High-fidelity Hugoniot analysis of porous materials

An experimental technique and analysis methodology for obtaining high-fidelity Hugoniot measurements with defined uncertainty bounds on powder compacts using optical velocimetry is presented. Impedance matching is used to calculate the shocked state in the powder from the measured initial compact density, ρ(00), impact velocity, V(Imp), and shock velocity, U(S). Detailed characterization of the powder thicknesses at precise locations results in improvements in characterization of the initial density state and accurate measurements of the powder thickness at locations corresponding to shock velocity measurements. These measurements result in high accuracies in the equilibrium Hugoniot state and reduced uncertainties in the measured and calculated Hugoniot parameters. Assumptions in this analysis include a constant and homogeneous initial porous density, and steady state wave propagation. The approach is applied to a system of CeO(2) powder pressed to 4.0 g/cm(3) (55% theoretical maximum density), and results indicate a complex dynamic response.

Mesoscale Simulation of Mixed Equations of State with Application to Shocked Platinum-doped PMP Foams

Hydrocarbon polymers and foams are utilized in high energy-density physics (HEDP) and inertial confinement fusion (ICF) experiments as tampers, energy conversion and radiation pulse shaping layers in dynamic hohlraum Z-pinches, and ablators in ICF capsule implosions. Shocked foams frequently are found to be mixed with other materials either by intentional doping with high-Z elements or by instabilities and turbulent mixing with surrounding materials. In this paper we present one-dimensional and three-dimensional mesoscale hydrodynamic simulations of high-Z doped poly-(4-methyl-1-pentene) (PMP or TPX) foams in order to examine the validity of various equation of state (EOS) mixing rules available in two state-of-the-art simulation codes. Platinum-doped PMP foam experiments conducted at Sandia's Z facility provide data that can be used to test EOS mixing rules. We apply Sandia's ALEGRA-MHD code and the joint LLNL/SNL KULL HEDP code to model these doped foam experiments and exercise the available EOS mixing methods. One- dimensional simulations homogenize the foam with platinum dopant and show which EOS mixing methods produce results that are consistent with measured Hugoniot states. These simulations produce sharp shock fronts that are well described by traditional Hugoniot relations. Three-dimensional mesoscale simulations explicitly model the foam structure embedded with discrete platinum particles. The heterogeneous structure of the foam results in diffuse shock fronts and an unsteady post-shock state with large fluctuations about an average state. We compare shock propagation through pure foam and Pt-doped foams (50-50 mixture by weight) at equal average initial density, and examine how well the results compare to the experimentally measured Hugoniot states.

Shock Waves Propagation and Phase Transition in Single Crystal Iron under Ramp Compression: Large Scale Parallel NEMD Simulations

The influences of micro-structures of materials on shock wave have long been recognized, but a few are reported. The evolution of micro- structure during shock waves pass through materials is essential to many practical applications, such as the field of blast and ballistic protection. Traditional continuum theory cannot involve any atomic information of materials when shock waves pass through. Non- equilibrium molecular dynamics (NEMD) simulation is thought to be a promising way to bridge microscopic atomic processes with macroscopic continuum behaviors. The α→ɛ transition of iron at about 13GPa has been recognized for more than half a century, but no essential progress have been make until the recent decades. Experiments still have difficulties to reach temporal and spatial scales and NEMD can make an essential complements. Unfortunately, the former atomic interaction potentials of iron are more or less not appropriate to the high pressure area. We have performed large-scale parallel NEMD simulations with recently developed modified analytic embedded-atom-model (MAEAM) potential on the ramp compression of single crystal iron with different orientations at different ramp time. Our results reveal the relations of profile structures and microstructures. And the phase transition mechanisms of iron are examined to be no difference with the case of shock. Different ramp time have no obvious effects on the Hugoniot states a shock could reach after phase transition though it may reach a higher stress state when compared with the situation of shock compression. However, the grains formed after transition incline to contain fewer defects or grow bigger with the increasing ramp time.

Modeling the effect of orientation on the shock response of a damageable composite material

A carbon fiber-epoxy composite (CFEC) shock response in the through thickness orientation and in one of the fiber directions is significantly different. The hydrostatic pressure inside anisotropic materials depends on deviatoric strain components as well as volumetric strain. Non-linear effects, such as shock effects, can be incorporated through the volumetric straining in the material. Thus, a new basis is required to couple the anisotropic material stiffness and strength with anisotropic shock effects, associated energy dependence, and damage softening process. This article presents these constitutive equations for shock wave modeling of a damageable carbon fiber-epoxy composite. Modeling the effect of fiber orientation on the shock response of a CFEC has been performed using a generalized decomposition of the stress tensor [A. A. Lukyanov, Int. J. Plast. 24, 140 (2008)] and Mie-Grüneisen's extrapolation of high-pressure shock Hugoniot states to other thermodynamics states for shocked CFEC materials. The three-wave structure (non-linear anisotropic, fracture, and isotropic elastic waves) that accompanies damage softening process is also proposed in this work for describing CFEC behavior under shock loading which allows to remove any discontinuities observed in the linear case for relation between shock velocities and particle velocities [A. A. Lukyanov, Eur. Phys. J. B 74, 35 (2010)]. Different Hugoniot stress levels are obtained when the material is impacted in different directions; their good agreement with the experiment demonstrates that the anisotropic equation of state, strength, and damage model are adequate for the simulation of shock wave propagation within damageable CFEC material. Remarkably, in the through thickness orientation, the material behaves similar to a simple polymer whereas in the fiber direction, the proposed in this paper model explains an initial ramp, before at sufficiently high stresses, and a much faster rising shock above it. The numerical results for shock wave modeling using proposed constitutive equations are presented, discussed, and future studies are outlined.

One-dimensional plate impact experiments on the cyclotetramethylene tetranitramine (HMX) based explosive EDC32

Eight one-dimensional plate impact experiments have been performed to study both the Shock to Detonation Transition and Hugoniot state in the cyclotetramethylene tetranitramine (HMX) based explosive EDC32. The experiments covered shock pressures ranging from 0.59 to 7.5 GPa with sustained shocks, double shocks, and short pulse shocks. Experiments were instrumented with embedded magnetic particle velocity gauges. Results include; (1) wave profiles of particle velocity vs. time vs. depth in the explosive, (2) time-distance coordinates for onset of detonation vs. initial shock pressure (aka the Pop-plot), (3) a reactants Hugoniot, and (4) measurement of the Hugoniot Elastic Limit of 0.22.GPa.

Plastic behavior of Zr51Ti5Ni10Cu25Al9 metallic glass under planar shock loading

Planar shock compression experiments are performed on a Zr-based bulk metallic glass (BMG), Zr51Ti5Ni10Cu25Al9 at peak shock stresses from 10 GPa to 27 GPa to investigate its plastic behavior under high pressure and high strain-rate. The particle velocity profiles measured at the free surface of the samples are analyzed to estimate longitudinal stresses of the Zr-based BMG in the shock loading process,and then shear stresses are obtained by comparing longitudinal stresses with a hydrostat. Though there is an obvious relaxation effect after elastic front, the Hugoniot elastic limit of the Zr-based BMG is found to increase with shock stress increasing. However, the shear stresses across the plastic shock front display stress hardening above the Hugoniot elastic limit followed by a stress relaxation (softening) to Hugoniot state, and the relaxation level also increases with shock stress increasing. The changes of shear stresses under planar shock compression are consistent with the results from molecular dynamic simulations, but obviously different from the pressure-shear impact experimental results or uniaxial stress impact experimental results.

Effect of fiber orientation on the structure of shock waves in carbon fiber-epoxy composites

Using an analytical relation between the Hugoniot (anisotropic and isotropic) states and other thermodynamic (anisotropic and isotropic) states at high pressures, the effect of fiber orientation on the structure of shock waves in carbon fiber-epoxy composites of various symmetry is investigated. A correct nonlinear model of propagation of shock waves in anisotropic materials is proposed, which employs the conception of total generalized pressure and the pressure corresponding to the thermodynamic response, i.e., to the equation of state. The equation generalizes the nonlinear Hugoniot equation to anisotropic materials and is reduced to the classical variant in the case of isotropy. Invoking the relations of nonlinear anisotropic solids and the generalized decomposition of stress tensor, the double structure of shock waves, consisting of nonlinear anisotropic and isotropic elastic parts, is examined. The numerical calculations of Hugoniot levels of stress agree well with experimental data for a carbon fiber-epoxy composite selected.

Microscopic Calculations of Hugoniot Curves of Neat Triaminotrinitrobenzene (TATB) and of Its Detonation Products

We compute the Hugoniot curves of both neat triaminotrinitrobenzene (TATB) and its detonation products mixture using atomistic simulation tools. To compute the Hugoniot states, we adapted our sampling constraints in average (SCA) method (Maillet et al. Appl. Math. Res. eXpress 2009, 2008, abn004) to Monte Carlo simulations. For neat TATB, we show that the potential proposed by Rai (Rai et al. J. Chem. Phys. 2008, 129, 194510) is not accurate enough to predict the Hugoniot curve and requires some optimization of its parameters. Concerning the detonation products, thermodynamic properties at chemical equilibrium are computed using a specific reaction ensemble Monte Carlo (RxMC) method (Bourasseau et al. Phys. Chem. Chem. Phys. 2011, 13, 7060), taking into account the presence of carbon clusters in the fluid mixture. We show that this explicit description of the solid phase immersed in the fluid phase modifies the chemical equilibrium.

General construction of mean-field potential and its application to the multiphase equations of state of tin

The mean-field potential (MFP) approach is an efficient way to evaluate the free energy contribution of ion motions for both solid and liquid states. In this paper the MFP is generally constructed with a volume-dependent term and a shape function. The former is derived in accordance with quasi-harmonic approximation. The latter is given semi-empirically. Application to multiphase equations of state for β-, γ- and liquid-tin has been examined. The theoretical phase diagram and thermodynamic properties of isotherm, thermal expansion, heat capacity, Hugoniot states as well as phase transitions are all in excellent agreement with experiments.

Window corrections of z-cut quartz at 1550 nm under elastic, uniaxial compression up to 10 GPa

Z-cut α-quartz single crystals were shocked up to 10 GPa by symmetric impact to carefully examine their performance as an optical window in new velocity interferometers operating at 1550 nm, such as displacement interferometer system for any reflector (DISAR) [Appl. Phys. Lett. 89, 111101 (2006)]. In situ particle velocity measurements allowed the shock Hugoniot data and velocity corrections to be obtained accurately by a simple probe. The results give the elastic Hugoniot state equation as Ds = 6.270 + 1.651up, and the dependence of velocity correction (Δus) on the true particle velocity (up) at 1550 nm is well described by exponential relation Δus = 0.0948 up1.1876. By combining the velocity correction and shock velocity measurements, the relation of the refractive index (n) versus relative density (ρ/ρ0) for z-cut quartz is also determined to be n = 1.0890 + 0.4393 ρ/ρ0. The wave profile measurements also indicate that the pressure limit where z-cut quartz’s response is simple is 10.04 GPa, and it can be used as an optical window within this limit in shock experiments.

Modeling Shock Waves in Composite Materials

An accurate extrapolation of high-pressure shock Hugoniot states to other thermodynamics states for shocked Carbon Fiber Composite (CFC) materials is presented. The proposed anisotropic equation of state (EOS) represents mathematical and physical generalization of the Mie-Grüneisen EOS for isotropic materials. A generalized decomposition for separation of material volumetric compression (compressibility effects−equation of state) from deviatoric strain effects is formulated. Using an anisotropic nonlinear continuum framework and generalized decomposition of a stress tensor, the shock waves propagation in CFC materials is examined. A numerical calculation shows that Hugoniot Stress Levels (HELs) agree with the experimental data for selected composite material.

Dynamic response ofCu46Zr54metallic glass to high-strain-rate shock loading: Plasticity, spall, and atomic-level structures

We investigate dynamic response of ${\text{Cu}}_{46}{\text{Zr}}_{54}$ metallic glass under adiabatic planar shock wave loading (one-dimensional strain) with molecular dynamics simulations, including Hugoniot (shock) states, shock-induced plasticity, and spallation. The Hugoniot states are obtained up to 60 GPa along with the von Mises shear flow strengths, and the dynamic spall strengths, at different strain rates and temperatures. The spall strengths likely represent the limiting values achievable in experiments such as laser ablation. For the steady shock states, a clear elastic-plastic transition is identified (e.g., in the shock velocity-particle velocity curve), and the shear strength shows strain softening. However, the elastic-plastic transition across the shock front displays transient stress overshoot (hardening) above the Hugoniot elastic limit followed by a relatively sluggish relaxation to the steady shock state, and the plastic shock front steepens with increasing shock strength. The local von Mises shear strain analysis is used to characterize local deformation, and the Voronoi tessellation analysis, the corresponding local structures at various stages of shock, release, tension and spallation. The plasticity in this glass, manifested as localized shear transformation zones, is of local structure rather than thermal origin, and void nucleation occurs preferentially at the highly shear-deformed regions. The Voronoi and shear strain analyses show that the atoms with different local structures are of different shear resistances that lead to shear localization (e.g., the atoms indexed with $⟨0,0,12,0⟩$ are most shear-resistant, and those with $⟨0,2,8,1⟩$ are highly prone to shear flow). The dynamic changes in local structures are consistent with the observed deformation dynamics.

An equation of state of a carbon-fibre epoxy composite under shock loading

An anisotropic equation of state (EOS) is proposed for the accurate extrapolation of high-pressure shock Hugoniot (anisotropic and isotropic) states to other thermodynamic (anisotropic and isotropic) states for a shocked carbon-fibre epoxy composite (CFC) of any symmetry. The proposed EOS, using a generalised decomposition of a stress tensor [A.A. Lukyanov, Int. J. Plasticity 24, 140 (2008)], represents a mathematical and physical generalisation of the Mie-Grüneisen EOS for isotropic material and reduces to this equation in the limit of isotropy. Although a linear relation between the generalised anisotropic bulk shock velocity Us A and particle velocity up was adequate in the through-thickness orientation, damage softening process produces discontinuities both in value and slope in the Us A-up relation. Therefore, the two-wave structure (non-linear anisotropic and isotropic elastic waves) that accompanies damage softening process was proposed for describing CFC behaviour under shock loading. The linear relationship Us A-up over the range of measurements corresponding to non-linear anisotropic elastic wave shows a value of c0 A (the intercept of the Us A-up curve) that is in the range between first and second generalised anisotropic bulk speed of sound [A.A. Lukyanov, Eur. Phys. J. B 64, 159 (2008)]. An analytical calculation showed that Hugoniot Stress Levels (HSLs) in different directions for a CFC composite subject to the two-wave structure (non-linear anisotropic elastic and isotropic elastic waves) agree with experimental measurements at low and at high shock intensities. The results are presented, discussed and future studies are outlined.

Dynamic behaviour of dry and water-saturated sand under planar shock conditions

Plane shock-wave experiments were performed on dry and partially water-saturated sand, using three water contents, in order to validate predictive models of material behaviour at stress levels between 1 and 10 GPa. Gas and powder guns were used to load the sample under uni-axial strain conditions at low and high stress levels, respectively. Wave motions were detected by piezoelectric pins in the samples and a VISAR (Velocity Interferometer System for Any Reflector) recorded the free-surface velocity on the back target. This study presents both experimental and simulated results. Experimental data are used to determine shock Hugoniot states. Significant differences are observed in the dynamic response of the materials under various water-saturated conditions, and are reproduced with good agreement by numerical simulations using the ARMORS (A Rheological MOdel of Rocks Saturated) model.

Interaction of a thin shock with turbulence. I. Effect on shock structure: Analytic model

A two-dimensional magnetohydrodynamical model describing the interaction of thin shock waves with turbulence is developed by adopting a multiscale perturbation analysis. The interaction is found to be governed by a two-dimensional inviscid Burgers’ equation that includes “perturbation terms.” Initially prescribed perturbation profiles are explored with numerical simulations to show how the shock front is modified by turbulence. Our numerical simulations show that magnetic field perturbations play a very important role in modifying the structure of perpendicular and parallel shocks. While turbulence can balance the nonlinear steepening of a shock wave at some regions, it can also help to create a larger jump in physical quantities such as the magnetic field at other regions. The plasma medium in these regions can therefore experience a higher compression, which will result in a downstream state that differs from the usual Rankine–Hugoniot state.

Shock compression of preheated molybdenum to 300 GPa

Shock compression experiments on samples at elevated initial temperature provide a test of the Mie-Grüneisen method of predicting off-principal Hugoniot states. Pure molybdenum, preheated to 1673 K, was shocked to peak pressures up to 300 GPa, double the compression range previously studied for this material at elevated temperature. The data lie strictly below the cold Hugoniot of Mo in shock velocity vs. particle velocity space, with some downward curvature. Previous approximations and extrapolations from lower-compression data fail to match these results, but the data are well fit by a Mie-Grüneisen correction to the cold Hugoniot. However, the data are insufficiently precise and the phases obtained in the shock experiments too uncertain to discriminate among different functional forms of the density dependence of the Grüneisen parameter.

An equation of state for anisotropic solids under shock loading

An anisotropic equation of state is proposed for accurate extrapolation of high-pressure shock Hugoniot states to other thermodynamics states for shocked single crystals and polycrystalline alloys. The proposed equation of state represents mathematical and physical generalization of the Mie-Gruneisen equation of state for isotropic material and reduces to this equation in the limit of isotropy. Using an anisotropic nonlinear continuum framework and generalized decomposition of a stress tensor [Int. J. Plasticity 24, 140 (2008)], the shock waves propagation along arbitrary directions in anisotropic solids of any symmetry can be examined. The non-associated strength model includes the distortion effect of the yield surface which can be used to describe the anisotropic strength differential effect. A numerical calculation showed that the general pulse shape, Hugoniot Elastic Limits (HELs), and Hugoniot stress levels for aluminum alloy 7010-T6 agree with the experimental data. The results are presented and discussed, and future studies are outlined.

PHASE TRANSITION AND CURRENT PROPERTIES OF PZT 95/5-2NB FERROELECTRIC CERAMIC UNDER DYNAMIC LOADING

The high density ferroelectric ceramic with composition near Pb 0.99( Zr 0.95 Ti 0.05)0.98 Nb 0.02 O 3 ( PZT 95/5-2 Nb ) has been studied under dynamic loading. To characterize its ferroelectric-to-antiferroelectric(FE/AFE) phase transition under shock wave compression, reverse-impact experiments were conducted to determine the Hugoniot states of poled and unpoled PZT 95/5-2 Nb , and the shock-induced depoling currents were examined in an external circuit under short-circuit conditions for the poled PZT 95/5-2 Nb . The Hugoniot data and current results identify that the phase transition of FE/AFE has occured in the stress range of 0.4 GPa~1.8 GPa. In the phase transition stress range, the measured short-circuit currents show the increasing of rise time and the decreasing of final levels with the increase of shock stress.