Linear Hugoniot Research Articles

Investigation of dynamic impact responses of layered polymer-graphene nanocomposite films using coarse-grained molecular dynamics simulations

Polymer nanocomposite films have recently shown superior energy dissipation capability by using the micro-projectile impact testing method. However, how stress waves interact with nanointerfaces and the underlying deformation mechanisms have remained largely elusive. This paper investigates the detailed stress wave propagation process and dynamic failure mechanisms of layered poly(methyl methacrylate) (PMMA)-graphene nanocomposite films during piston impact through coarse-grained molecular dynamics simulations. The spatiotemporal contours of stress and local density clearly demonstrate shock front, reflected wave, and release wave. By plotting shock front velocity (Us) against piston velocity (Up), we find that the linear Hugoniot Us−Up relationship generally observed for bulk polymer systems also applies to the layered nanocomposite system. When the piston reaches a critical velocity, PMMA crazing can emerge at the location where the major reflected wave and release wave meet. We show that the activation of PMMA crazing significantly enhances the energy dissipation ratio of the nanocomposite films, defined as the ratio between the dissipated energy through irreversible deformation and the input energy. The ratio maximizes at the critical Up when the PMMA crazing emerges and then decreases as Up further increases. We also find that a critical PMMA-graphene interfacial strength is required to activate PMMA crazing instead of interfacial separation. Additionally, layer thickness affects the amounts of input energy and dissipated energy for nanocomposite films under impact. This study provides important insights into the detailed dynamic deformation mechanisms and how nanointerfaces/nanostructures affect the energy dissipation capability of layered nanocomposite films.

Damage analysis of a CFRP cross-ply laminate subjected to abrasive water jet cutting

The abrasive water jet (AWJ) cutting technique is one of the promising techniques used in machining of composite materials due to lack of thermal damage, lower tool wear and higher productivity. In this study, the AWJ cutting of a cross-ply CFRP laminate was investigated experimentally and numerically. The purpose is to understand the underlying physics of the AWJ cutting of a composite plate, which can be later used to control the process as well as optimizing its parameters. In the performed tests of the plate with a stacking sequence of [0°/90°/0°/90°] and having a total thickness value of 0.84 mm, where the pressure of particles and impact angle were varied, different failure characteristics such as fiber pull-out, fiber breakage, fiber debonding, matrix cracking and delamination were noticed. A three-dimensional FE model of the process was developed using ABAQUS finite element software to understand the underlying physics. In the model, the pure water jet particles following the linear Hugoniot form of the Mie-Grüneisen equation of state and highly rigid abrasive particles were modelled using smooth particle hydrodynamics. While the three dimensional Hashin damage model was used to simulate the intra ply damage, cohesive zone elements were used to predict the delamination. The damage characteristics in the composite plate was investigated for different process parameters. When the speed of the AWJ particles increased from 300 m/s to 600 m/s, the amount of delamination decreased from 6.44% to 5.69% at the top interface with no more delamination observed at the middle and bottom interfaces. The delamination performance of 0°/90°/0°/90° orientation was found to be better than those of 0°/90°/90°/0° and 0°/0°/90°/90° orientations. The impact angle of the particles affected the material removal rate in the process significantly. The failure behaviour of the laminate subjected to AWJ and pure water jet cuttings (PWJ) were also compared.

РЕШЕНИЕ ЗАДАЧИ О РАСШИРЕНИИ СФЕРИЧЕСКОЙ ПОЛОСТИ В СРЕДЕ С УСЛОВИЕМ ПЛАСТИЧНОСТИ МОРА - КУЛОНА - ТРЕСКА

An analytical solution of the one-dimensional problem of a spherical cavity expanding at a constant velocity from a point in a space occupied by a plastic medium has been obtained. Impact compressibility of the medium is described using linear Hugoniot's adiabat. Plastic deformation obeys the Mohr - Coulomb yield criterion with constraints on the value of maximum tangential stresses according to Tresca's criterion. In the assumption of rigid-plastic deformation (the elastic precursor being neglected), incompressibility behind the shockwave front and the equality of the propagation velocities of the fronts of the plastic wave and the plane shockwave defined by linear Hugoniot's adiabat, a boundary-value problem for a system of two first-order ordinary differential equations for the dimensionless velocity and stress depending on the self-similar variable is formulated. A closed-form solution of this problem has been obtained in the form of a stationary running wave - a plastic shockwave propagating in an unperturbed half-space. This solution is a generalization of the earlier obtained analytical solution for a medium with the Mohr - Coulomb plasticity condition. The effect of constraining the limiting value of maximal tangential stresses on the distribution of dimensionless stresses behind the shockwave front has been examined. Formulas for determining the range of cavity expansion velocities, within which a simple solution for a medium with Tresca's plasticity condition is applicable, have been derived. The obtained solution can be used for evaluating resistance to high-velocity penetration of rigid strikers into low-strength soil media.

Hugoniot equation of state of rock materials under shock compression.

Two sets of shock compression tests (i.e. conventional and reverse impact) were conducted to determine the shock response of two rock materials using a plate impact facility. Embedded manganin stress gauges were used for the measurements of longitudinal stress and shock velocity. Photon Doppler velocimetry was used to capture the free surface velocity of the target. Experimental data were obtained on a fine-grained marble and a coarse-grained gabbro over a shock pressure range of approximately 1.5-12 GPa. Gabbro exhibited a linear Hugoniot equation of state (EOS) in the pressure-particle velocity (P-up) plane, while for marble a nonlinear response was observed. The EOS relations between shock velocity (US) and particle velocity (up) are linearly fitted as US = 2.62 + 3.319up and US = 5.4 85 + 1.038up for marble and gabbro, respectively.This article is part of the themed issue 'Experimental testing and modelling of brittle materials at high strain rates'.

A complete equation of state for polyethylene based on Helmholtz free energy

Polyethylene (PE) is an important kind of plastic, which plays a significant role as the shell material of the fuel capsule, light weight structural element subjected to intense mechanical impact and explosion load. And it is well accepted that semi-empirical three-term equation of state (EOS) is one of the most widely used EOSs in practical work. Therefore, studies of semi-empirical three-term EOS of PE are significant for accurately predicting and analyzing the physical processes and experimental results under high pressure compression. A semi-empirical three-term complete EOS of PE based on the model of Helmholtz free energy is established in this work. According to the EOS model, the Helmholtz free energy is composed of cold energy, thermal contribution of atoms and thermal excitation of electrons. The cold energy is calculated by using the Mie potential. The optical frequency branch of atomic vibration and the thermal contribution of electrons are neglected in the calculation at temperatures below 104 K. The parameters of Helmholtz free energy are calculated by using the shock Hugoniot data and thermal parameters at ambient state. And then, the application pressure range and reliability of the semi-empirical three-term EOS of PE are evaluated. Shock Hugoniot, shock wave temperature and Grneisen coefficient of PE are deduced from the EOS. The results show that shock Hugoniot and shock wave temperature are consistent well with the experimental data and the first-principle calculation in a pressure range of 150 GPa. Because the specific volume of PE does not change obviously in the melting and chain dissociation process, the assumption of linear Hugoniot relation of PE is valid for calculating the cold energy parameters. The calculation results deviate from the experimental results at about 150 GPa while the compression lasts up to the chemical bond dissociation pressure of PE. In addition, the value of buck modulus and its derivative with respect to pressure at zero pressure and temperature depend strongly on Hugoniot parameters. Therefore, the parameter of Helmholtz free energy in this work is only valid for compression. In conclusion, the Helmholtz free energy model and parameters can well reproduce the experimental data and reasonably describe the thermodynamic state of PE at its dissociation pressure. Moreover, it should be pointed out that a more refined model of phase transition and thermal contribution of atoms and electrons should be considered when extrapolated to higher pressure.

Determination of the constitutive relation and critical condition for the shock compression of cellular solids

This study aims at understanding the constitutive relation and critical condition for the shock compression of cellular solids. A 2D virtual foam is constructed from the cross-section of a closed-cell aluminium foam imaged by micro X-ray computed tomography, which enables the realistic consideration of meso-scale structural effect in numerical modelling. Quasi-static and shock compressions of the 2D foam are simulated. A series of Hugoniot relations between shock speed (and other mechanical quantities) and impact speed are determined from the finite element (FE) simulations. It is found that the shock speed increases approximately linearly with impact speed, similar to that observed for condensed solids, but the related material constants for cellular solids have different physical implications, whereas the shock strain, stress and energy increase with impact speed nonlinearly, due to shock-enhanced cell compaction and cell-wall plastic deformation. Based on conservation laws in continuum mechanics, other Hugoniot relations are derived from the basic linear one, which agree well with those obtained from the FE simulations. It is thus demonstrated that the unique linear Hugoniot relation can be used to characterise the shock constitutive behaviour which is distinct from the quasi-static one. Furthermore, a new analytical method based on the linear Hugoniot relation is proposed to estimate the critical impact speed for shock initiation, which has reasonable agreement with the present FE simulations and previous experimental and numerical results, and outperforms the existing methods.

An overview on the effect of manufacturing on the shock response of polymers

Scatter and non-linearity of the Hugoniot in the Us-up plane has been seen in a number of polymers including poly(methyl methacrylate) (PMMA), the polymer considered here. In this study the plate impact technique has been used to investigate the shock response of PMMA between particle velocities of 0.13 and 0.77 mm μs−1. From this data no scatter was seen between our data and the experimental data of Barker and Hollenbach, and Carter and Marsh. Also a linear Hugoniot in the Us-up plane was found, with the equation Us = 2.99 + 0.92up. The non-linearity observed by Barker and Hollenbach was not present in this data, probably due to the non-linearity occur at particle velocities of below 0.13 mm μs-1, within their experimental data. Gruneisen gamma has also been briefly considered using a shock reverberation experiment but more work is needed before a value can be ascertained.

Direct shock compression experiments on premolten forsterite and progress toward a consistent high‐pressure equation of state for CaO‐MgO‐Al2O3‐SiO2‐FeO liquids

AbstractWe performed shock compression experiments on preheated forsterite liquid (Mg2SiO4) at an initial temperature of 2273 K and have revised the equation of state (EOS) that was previously determined by shock melting of initially solid Mg2SiO4 (300 K). The linear Hugoniot, US = 2.674 ± 0.188 + 1.64 ± 0.06 up km/s, constrains the bulk sound speed within a temperature and composition space as yet unexplored by 1 bar ultrasonic experiments. We have also revised the EOS for enstatite liquid (MgSiO3) to exclude experiments that may have been only partially melted upon shock compression and also the EOS for anorthite (CaAl2SiO6) liquid, which now excludes potentially unrelaxed experiments at low pressure. The revised fits and the previously determined EOS of fayalite and diopside (CaMg2SiO6) were used to produce isentropes in the multicomponent CaO‐MgO‐Al2O3‐SiO2‐FeO system at elevated temperatures and pressures. Our results are similar to those previously presented for peridotite and simplified “chondrite” liquids such that regardless of where crystallization first occurs, the liquidus solid sinks upon formation. This process is not conducive to the formation of a basal magma ocean. We also examined the chemical and physical plausibility of the partial melt hypothesis to explain the occurrence and characteristics of ultra‐low velocity zones (ULVZ). We determined that the ambient mantle cannot produce an equilibrium partial melt and residue that is sufficiently dense to be an ultra‐low velocity zone mush. The partial melt would need to be segregated from its equilibrium residue and combined with a denser solid component to achieve a sufficiently large aggregate density.

The high strain-rate behaviour of selected tissue analogues

The high strain-rate response of four readily available tissue simulants has been investigated via plate-impact experiments. Comparison of the shock response of gelatin, ballistic soap (both sub-dermal tissue simulants), lard (adipose layers) and Sylgard® (a potential brain simulant) allowed interrogation of the applicability of such monolithic tissue surrogates in the ballistic regime. The gelatin and lard exhibited classic linear Hugoniot equations-of-state in the US–uP plane; while for the ballistic soap and Sylgard® a polymer-like non-linear response was observed. In the P/σX–v/v0 plane there was evidence of separation of the simulant materials into distinct groups, suggesting that a single tissue simulant is inadequate to ensure a high-fidelity description of the high strain-rate response of complex mammalian tissue. Gelatin appeared to behave broadly hydrodynamically, while soap, lard and Sylgard® were observed to strengthen in a material-dependent manner under specific loading conditions at elevated shock loading pressures/stresses. This strengthening behaviour was tentatively attributed to a further polymeric-like response in the form of a re-arrangement of the molecular chains under loading (a steric effect). In addition, investigation of lateral stress data from the literature showed evidence of operation of a material-independent strengthening mechanism when these materials were stressed above 2.5–3.0GPa, tentatively linked to the generically polymeric-like underlying microstructure of the simulants under consideration.

Measurement of the Mie-Grüneisen equation of state for polyimide

An equation of state (EOS) is a thermodynamic equation describing the state of matter under a given set of physical conditions, and the Mie-Grüneisen EOS is a widely used EOS for solid materials. Polyimide is a typical complex high-molecular polymer of imide monomers which is widely used in the manufacture of parts for aerospace technology and for fabricating the inertial fusion energy shells (IFE). When a spacecraft collides with a meteoroids (or orbital debris) or the IFE shells are radiated by high power laser, the polyimide used in the equipment is in high pressure state. In order to study the safety of the spacecraft and IFE shells, the EOS of polyimide should be determined. In our research, the three key parameters of the Mie-Grüneisen EOS based on the shock adiabat for polyimide have been determined with two-stage light-gas gun experiments, i.e. the bulk speed of sound of polyimide is determined as 2.62±0.22 km/s, the linear Hugoniot slope coefficient is determined as 1.25±0.063, and the Grüneisen parameter at initial state is determined as 1.53±0.135. The Mie-Grüneisen EOS for polyimide is finally obtained, and the parameter values are proved reliable via the comparison of Grüneisen parameter value calculated from two different theoretical models using the experimental data.

Effect of Facesheet Thickness on Dynamic Response of Composite Sandwich Plates to Underwater Impulsive Loading

The response of sandwich structures to underwater blast loading is analyzed. The analysis focuses on the effect of varying structural attributes on energy dissipation and deformation. The structures analyzed are planar sandwich plates with polymer foam cores and fiber-reinforced polymer composite facesheets. The thickness of the facesheets is varied under the conditions of constant material properties and core dimensions. The fully three-dimensional finite-element simulations carried out account for underwater blast loading through the use of the Mie-Gruneisen equation-of-state of a linear Hugoniot form and a modified Drucker-Prager core crushing model. The impulse imparted to the panels is varied from 4 to 42 kPa·s. The results show that there exists an optimal thickness of the facesheets which maximizes energy absorption in the core and minimizes the overall deflection of the structure.

On the dynamic behavior of three readily available soft tissue simulants

Plate-impact experiments have been employed to investigate the dynamic response of three readily available tissue simulants for ballistic purposes: gelatin, ballistic soap (both subdermal tissue simulants), and lard (adipose layers). All three materials exhibited linear Hugoniot equations-of-state in the US-uP plane. While gelatin behaved hydrodynamically under shock, soap and lard appeared to strengthen under increased loading. Interestingly, the simulants under test appeared to strengthen in a material-independent manner on shock arrival (tentatively attributed to a rearrangement of the amorphous molecular chains under loading). However, material-specific behavior was apparent behind the shock. This behavior appeared to correlate with microstructural complexity, suggesting a steric hindrance effect.

Shock compression of liquid silicates to 125 GPa: The anorthite‐diopside join

We determined the equation of state (EOS) of three silicate liquid compositions by shock compression of preheated samples up to 127 GPa. Diopside (Di; Ca2Mg2SiO6) at 1773 K, anorthite (An; CaAl2Si2O8) at 1923 K and the eutectic composition Di64An36 at 1673 K were previously studied by shock compression to 38 GPa. The new data extend the EOS of each composition nearly to the Earth's core‐mantle boundary. The previously reported anomaly at 25 GPa for Di64An36 eutectic was not reproduced; rather all data for this composition fit within error a straight line Hugoniot in particle velocity vs. shock velocity. Di also displays a linear Hugoniot consistent with ultrasonic data and a third‐order finite strain EOS. The full anorthite data set is complex; we examine a model with a transition between two structural states and a fourth‐order finite strain model excluding two points that may not display relaxed behavior. We also report an experiment on room‐temperature solid Di64An36 aggregate that clearly demonstrates increase upon compression of the Grüneisen parameter of this liquid, much as experiment and theory have shown for forsterite and enstatite liquids. We construct isentropes and isotherms from our Hugoniots using Mie‐Grüneisen thermal pressure and evaluate the model of ideal mixing of volumes. Volume may mix almost linearly at high temperature, but deviates strongly when calculated along an isotherm; it remains difficult to reach a firm conclusion. We compare the densities of liquids to lower mantle solids. Our results suggest that basaltic liquids rich in CaO and Al2O3 are notably denser than liquids in the MgO‐SiO2 binary and, subject to uncertainties in the behavior of FeO and in corrections for thermal pressure, such liquids may be the most likely candidates for achieving negative buoyancy in the lowermost mantle.

Atomistic simulations of shock waves in cubic silicon carbide

Molecular dynamics simulations were performed to investigate the mechanical properties of shock waves in cubic silicon carbide (3C–SiC, β–SiC). Shock wave was produced and distinctly demonstrated when a moving sample material impacted to the static sample material on one side. At rather small impact velocity, compressive shock wave traveled with a velocity equal to the longitudinal sound speed. A linear Hugoniot relationship for normalized particle velocity was verified from our atomistic simulation results, which is comparable with the recent shock experiments on SiC powders. Part of cubic lattice atoms transforms into amorphous state when the impact velocity exceeds the critical value of 4.91 km/s. The size of the amorphous region is well proportional to the particle velocity.

Shock compression behaviour of titanium diboride

Titanium diboride (TiB 2) has excellent mechanical properties, and is expected to be used as crashworthy material. In this study, the Hugoniot-measurements were performed on the TiB 2 polycrystal by means of the inclined-mirror method combined with the powder gun and two-stage gas gun to study the elastoplastic transition and to search phase transition. The Hugoniot-elastic limit (HEL) stress was measured to be approximately 12 GPa. The shock velocity ( U S) vs. particle velocity ( U P) relation of the final phase was given by U S = 8.79 + 0.78 U P km / s in the plastic region of 0.6–3.0 km/s in particle velocity. The shock velocities were much large compared with the bulk sound velocity, and those in 0.24–0.6 km/s were further larger than the linear Hugoniot relation. This result showed that this material preserved the large shear strength also in the plastic region. No phase transition was observed up to 120 GPa.

Dynamic behavior of selected ceramic powders

An experimental setup has been developed on the continuous recording of the stress profiles in ceramic powders subject to shock loading with manganin gauges. A series of plate impact experiments on highly porous ceramic powders such as Al 2O 3, SiC and B 4C were conducted at the laboratory's single stage powder gun facility. The relationship between shock wave velocity and particle velocity was measured to obtain the Hugoniot data. Plate impact onto powder sample experiments were conducted at loading stresses ranging from 1.6 to 4.2 GPa. The experimental results show that the shock wave speeds in various ceramic powders vary between 1 and 2 km/s. Linear Hugoniot relations between shock velocity ( D) and particle velocity ( u) are observed. The loading stress–specific volume form of Hugoniot relations ( P– V) was constructed using the data from quasistatic compression test results, Hopkinson bar dynamic compression test results and powder gun plate impact experiment results. The P–V diagram shows that the crush strength of ceramic powders is comparable to the loading stress level. The B 4C and SiC powders with bigger particle size more easily reach the solid state Hugoniot than the powders with smaller particle size at the same loading condition. In the case of Al 2O 3, the material shows less sensitivity to particle size difference at the same level of loading rate as compared to B 4C and SiC.

Shock-data constraints on interatomic potentials for condensed matter

It is customary to represent the shock Hugoniot as a linear relation between shock velocity and the particle velocity behind the shock. It has been recently shown that the coefficients appearing in the general expansion of shock velocity as a power series in particle velocity can be calculated, to arbitrary order, from the derivatives of the shock pressure with respect to volume, evaluated at the initial state. The first two coefficients in the expansion have a simple relation to the interatomic potential describing the material. Most such potentials contain just two parameters (roughly, a “strength” parameter and a “shape” parameter), which can be conveniently extracted from the shock parameters. We present results for the values of the potential parameters, for several commonly-used forms of potential, determined by least-squares fitting of the shock data to nonlinear Hugoniots. The results obtained sometimes differ from those obtained on the assumption of a linear Hugoniot.

An analysis of clustering and ordering in multicomponent solid solutions —II. Fluctuations and kinetics

It is customary to represent the shock Hugoniot as a linear relation between shock velocity and the particle velocity behind the shock. It has been recently shown that the coefficients appearing in the general expansion of shock velocity as a power series in particle velocity can be calculated, to arbitrary order, from the derivatives of the shock pressure with respect to volume, evaluated at the initial state. The first two coefficients in the expansion have a simple relation to the interatomic potential describing the material. Most such potentials contain just two parameters (roughly, a “strength” parameter and a “shape” parameter), which can be conveniently extracted from the shock parameters. We present results for the values of the potential parameters, for several commonly-used forms of potential, determined by least-squares fitting of the shock data to nonlinear Hugoniots. The results obtained sometimes differ from those obtained on the assumption of a linear Hugoniot.