Derivation of equations of state for ideal crystals of simple structure

We consider an approach to the derivation of thermodynamic equations of state by averaging the dynamic equations of particles of the crystal lattice. Microscopic analogs of macroscopic variables such as pressure, volume, and thermal energy are introduced. An analysis of the introduced variables together with the equations of motion permits obtaining the equation of state. Earlier, this approach was used to obtain the equation of state in the Mie-Grüneisen form for a one-dimensional lattice. The aim of this paper is to develop and generalize this approach to the three-dimensional case. As a result, we obtain the dependence of the Grüneisen function on the volume, which is compared with the computations performed according to well-known models with experimental data taken into account. It is proved that the Grüneisen coefficient substantially depends on the form of the strain state. Moreover, we refine the equation of state; namely, we show that the Grüneisen coefficient depends on the thermal energy, but this dependence in the three-dimensional case is much weaker than in the one-dimensional case. A refined equation of state containing a nonlinear dependence on the thermal energy is obtained     

Constitutive behaviour of anisotropic materials under shock loading

Thermodynamically and mathematically consistent constitutive equations suitable for shock wave propagation in an anisotropic material are presented in this paper. Two fundamental tensors α_ij and β_ij which represent anisotropic material properties are defined and can be considered as generalisations of the Kronecker delta symbol, which plays the main role in the theory of isotropic materials. Using two fundamental tensors α_ij and β_ij, the concept of total generalised “pressure” and pressure corresponding to the thermodynamic (equation of state) response are redefined. The equation of state represents mathematical and physical generalisation of the classical Mie–Grüneisen equation of state for isotropic material and reduces to the Mie–Grüneisen equation of state in the limit of isotropy. Based on the generalised decomposition of the stress tensor, the modified equation of state for anisotropic materials, and the modified Hill criteria, combined with the associated flow rule, a system of constitutive equations suitable for shock wave propagation is formulated. The behaviour of aluminium alloy 7010-T6 under shock loading conditions is considered. A comparison of numerical simulations with existing experimental data shows good agreement of the general pulse shape, Hugoniot Elastic Limits (HELs), and Hugoniot stress levels, and suggests that the constitutive equations are performing satisfactorily. The results are presented and discussed, and future studies are outlined.     

An equation of state for dense nitrogen

An empirical equation of state for nitrogen at high pressure and density is considered. It is shown that for nitrogen at densities greater than 0.6 g/cm³, by using available data [1–3] on static compression of gaseous nitrogen and shock compression of liquid nitrogen, it is possible to construct a Mie-Grüneisen type equation of state which gives a pressuredensity relationship close to experiment along the shock adiabat of liquid nitrogen and agrees with the calculations of other authors for temperature values beyond the shock-wave front [2–3]. Heat capacity, entropy, and Grüneisen coefficient values beyond the shock-wave front in liquid nitrogen are calculated.     

弹体侵彻干砂的数值模型

基于砂粒的不可压缩性假设,利用球形空腔动态收缩模型和广义Mises强度准则推导了干砂的孔隙压密演化方程;根据Hugoniot冲击突跃条件和Grüneisen系数,推导了干砂考虑孔隙演化影响的状态方程;根据关联流动法则,得到了大变形时砂的弹塑性应力应变关系;基于动力有限元计算平台,采用上述模型分析了弹体高速侵彻干砂的作用过程。结果表明,该模型能够表征高速侵彻时砂的孔隙演化对应力应变状态的反向影响,能够较准确地反映高速侵彻作用下干砂的动力响应过程。     

Experimental-theoretical study of the penetration of rigid projectiles and identification of soil properties

The parameters of the Grigoryan soil model are determined using an experimental-computational method previously proposed and the results of reversed experiments on penetration of projectiles with flat and hemispherical heads at impact velocities of 50–450 m/sec in sandy soil. It is shown that the quasistationary dependences of the resistance force on impact velocity obtained in the reversed experiment can be used to solve problems of deep penetration of projectile in soil with an error not exceeding the measurement error.     

The shock-wave structure in a two-velocity mixture of compressible media with different pressures

The problem of the shock-wave structure in a mixture of two compressible media with different velocities and pressures of components is considered. The problem is reduced to solving a boundary-value problem for two ordinary differential equations that describe the velocity relaxation and pressure equalization of the components. Using methods of the qualitative theory of dynamic systems on a plane, the existence and uniqueness of four types of waves are shown: (a) fully dispersed waves; (b) frozen-dispersed waves; (c) dispersed-frozen waves; (d) frozen waves of two-front configuration. A chart of solutions of the corresponding flow types is constructed in the plane of the following parameters: the initial velocity of the mixture and the initial volume concentration of one of the components. The numerical calculations conducted illustrate the obtained analytical structures of the shock wave. It is shown that the results obtained using the suggested mathematical model are in agreement with experimental data on the dependence of the velocity of the dispersed shock wave on the equilibrium pressure behind the shock-wave front for a mixture of silica sand and water. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 2, pp. 10–19, March–April, 1998.     

Detonation wave initiated by explosive condensation of supersaturated carbon vapor

An experimental study of the influence of condensation of supersaturated carbon vapor formed behind reflected shock waves on the process of propagation of a shock wave and formation of a detonation wave of condensation is carried out. Highly supersaturated carbon vapor was formed from thermal decay of unstable carbon suboxide C₃O₂ → C + 2CO behind a shock wave in mixtures containing 10–30% C₃O₂ in Ar. This reaction was followed by fast growth of condensed carbon particles, accompanied by heat release. Experiments have shown a considerable temperature and pressure increase in the narrow zone behind the wave front, resulting in shock wave amplification and transition to a detonation-like regime. An analysis of the kinetics and heat release in the given conditions and calculations based upon one-dimensional detonation theory have shown that in a mixture of 10% C₃O₂ + Ar, insufficient heat release resulted in a regime of “overdriven detonation”. In a mixture of 20% C₃O₂ + Ar a very good coincidence of measured values of pressure and wave velocity with calculated Chapman–Jouguet parameters is observed. In a 30% C₃O₂ + Ar mixture, an excess heat release caused a slow down of the effective condensation rate and a regime of “underdriven detonation” is observed.     

Re-initiation phenomenon of gaseous detonation induced by shock reflection

The two-dimensional, time-dependent, reactive Navier–Stokes equations including the effects of viscosity, thermal conduction and molecular diffusion were solved to reveal the wave evolution and chemical dynamics involved in the re-initiation process. The computation was performed for hydrogen–oxygen–argon mixtures at the low initial pressure (8.00 kPa), using detailed chemical reaction model. The results show that, the decoupled leading shock reflects on the right wall of the vertical branch. High temperature and pressure behind the reflected shock induce the generation of hot spots and local explosion. Therefore, the re-initiation of gaseous detonation occurs. In the re-initiation area, there exist very high OH concentration and no H ₂ concentration. However, in front of reflected shock, there exist relatively high H ₂ concentration and no OH radicals. Additionally, the shock–flame interaction induces RM instability. This results in the fast mixing between hot reacted gas mixture and the relatively cold unreacted gas mixture and accelerates the chemical reactions. However, the shock–flame interaction contributes much less to the re-initiation, in contrast with shock reflection. The transition of leading shock from regular reflection to Mach reflection happens during the re-initiation. The computed evolution of wave structures involved in the re-initiation is qualitatively agreeable with that from the experimental schlieren images.      

Soil Behavior at the Interface with a Rigid Projectile During Penetration

The motion and state of soil at the interface with a penetrating rigid projectile is studied by numerical solution of the problem of a cylindrical projectile which expands and at the same time moves translationally along its axis in soil. The soil behavior is described using the model of a compressible elastoplastic medium with transition to a plastic state depending on the pressure in it. It is shown that a thin layer of soil at the interface with the projectile nose should be set in motion and move together with the projectile without sliding. An analysis is performed of the validity of using the dry friction law to determine the shear stresses on the projectile surface during penetration. The heat release in the soil layer at the interface due to internal friction and its possible effect on the penetration are estimated. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 6, pp. 116–127, November–December, 2005.     

On shock waves in a special class of thermoelastic solids

This paper describes a thermoelastic model for shock waves in uniaxial strain based on a subclass of the so-called materials of Mie–Grüneisen type. We compare the Hugoniot curve with the isotherms and isentropes for this model, and we construct the shock-wave solution to a simple impact problem.     

钙质砂介质中爆炸波传播规律的试验研究

开展了一系列钙质砂和石英砂的地面爆炸试验，主要对比分析了两种砂土介质中爆炸波的传播规律，包括峰值压力、弹塑性波速及升压时间、爆坑尺寸等。试验结果表明，爆炸波在钙质砂中的传播与在石英砂中存在较大差异：地面爆炸作用下钙质砂爆坑较石英砂爆坑的直径和深度更小，且成坑形状为两阶同心圆；钙质砂中弹性波速为236～300 m/s，石英砂中弹性波速为218～337 m/s，弹性波速和塑性波速均随炸药质量增加而增大；爆炸波在钙质砂中的升压时间随比例距离的增加而增加，而在石英砂中升压时间随比例距离变化不明显，且较钙质砂中升压更迅速；在地面爆炸作用下，低含水率钙质砂的衰减系数为2.86，石英砂为2.79。     

Analogy between soap film and gas dynamics. II. Experiments on one-dimensional motion of shock waves in soap films

This paper presents an experimental investigation of one-dimensional moving shock waves in vertical soap films. The shock waves were generated by bursting the films with a perforating spark. Images of propagating shock waves and small disturbances were recorded using a fast line scan CCD camera. An aureole and a shock wave preceding the rim of the expanding hole were clearly observed. These images are similar to the x-t diagrams in gas dynamics and give the velocities of shock and sound waves. The moving shock waves cause jumps in thickness. The variations of the induced Mach number, M₂ and the ratio of film thickness across the shock wave, δ ₂/δ ₁, are plotted versus the shock Mach number, M _s. Both results suggest that soap films are analogous to compressible gases with a specific heat ratio of γ≅1.0. Published online: 15 October 2002     

Aluminum Particles–air Detonation at Elevated Pressures

The effect of initial pressure on aluminum particles–air detonation was experimentally investigated in a 13 m long, 80 mm diameter tube for 100 nm and 2 μm spherical particles. While the 100 nm Al–air detonation propagates at 1 atm initial pressure in the tube, transition to the 2 μm aluminum–air detonation occurs only when the initial pressure is increased to 2.5 atm. The detonation wave manifests itself in a spinning wave structure. An increase in initial pressure increases the detonation sensitivity and reduces the detonation transition distance. Global analysis suggests that the tube diameter for single-head spinning detonation or characteristic detonation cell size would be proportional to (d ₀: aluminum particle size, p ₀: initial pressure). Its application to the experimental data results in m ~ O(1) and n ~ O(1) for 1 to 2 μm aluminum–air detonation, thus indicating a strong dependence on initial pressure and gas-phase kinetics for the aluminum reaction mechanism in detonation. Hence, combustion models based on the fuel droplet diffusion theory may not be adequate in describing micrometric aluminum–air detonation initiation, transition and propagation. For 2 μm aluminum–air mixtures at 2 atm initial pressure and below, experiments show a transition to a “dust quasi-detonation” that propagates quasi-steadily with a shock velocity deficit nearly 40% with respect to the theoretical C–J detonation value. The dust quasi- detonation wave can propagate in a tube with a diameter less than 0.4–0.5 times the diameter required for a spinning detonation wave.     

RETRACTED ARTICLE: Kuhn–Grün analysis of polarized Rayleigh and Raman scattering experiments to deduce segmental orientation in polymeric systems

The intensity of Rayleigh and Raman scattered light from molecular structural units is proportional to the quadratic polarizability tensor and the derived polarizability tensor, respectively. The orientation of the polymer skeletal backbone is directly related to the orientation of the scattering structural units comprising it. The mathematical structure of the quadratic scattering tensors for a single Kuhn bond are deduced in terms of the unit vector along a Kuhn bond from symmetry considerations alone (Boehler 1987). Subsequent application of the Kuhn–Grün conditional probability analysis (Kuhn and Grün, Kolloid Z 101:248–271, 1942), which uses a freely jointed chain model, yields a general expression for the quadratic Raman and polarizability tensors for a single chain segment with five independent terms. Each term is multiplied by a spectroscopic parameter that is a complex function of the intrinsic spectroscopic tensors and the orientation distribution of monomers within an elementary Kuhn bond. A small stretch analysis of the Kuhn–Grün representation of the quadratic polarizability reveals that independent fourth moments of the segmental orientation distribution function can only be determined experimentally when the deformation or stretch of the flexible polymer is large and finite, thus severely restricting a primary advantage of the Raman and Rayleigh scattering methods. A general segmental additivity theorem is rigorously proven which demonstrates that polarized scattering experiments physically reflect the average orientation and stretch of flexible polymer skeletal backbone segments, or sub-segments, independent of chain architecture or molecular weight. Constitutive equations are fundamentally constructed to determine Kuhn bond orientation and are intrinsically related to the Kuhn–Grün analysis. The decoupling approximation, which is always invoked in Doi–Edwards type models of entangled polymeric liquids, is examined in light of the Kuhn–Grün analysis.     

Experimental study of planar shock wave interactions with dense packed sand wall

A dense packed sand wall is impacted by a planar shock wave in a horizontal shock tube to study the shock-sand wall interaction. The incident shock Mach number ranges from 2.18 to 2.38. A novel device for actively rupturing diaphragm is designed for the driver section of the shock tube. An apparatus for loading particles is machined by the electrical discharge cutting technique to create a dense packed particle wall. High-speed schlieren imaging system and synchronized pressure measurement system are used together to capture the wave structures and particle cloud velocity. The dynamic evolution model from dense packed particles to dense gas–solid cloud at the initial driving stage is established. The blockage and permeation effects of the sand wall work together and influence each other. The high pressure gas behind the incident shock wave blocked by the sand wall pushes the upstream front of the wall forward like a piston. Meanwhile, the high speed gas permeating through the sand wall drags the sands of the most downstream layer forward. The incident shock strength, initial sand wall thickness and particle diameter are varied respectively to investigate the shock attenuation and the wall acceleration. Increasing the sands diameter or mixing in small diameter sands can significantly attenuate the incident shock. The smaller particles or the particles in thinner wall can be dispersed into a larger range in the process of transform from dense packed particles to dense gas–solid cloud. Moreover, the stronger incident shock can disperse the particles into a larger region.     

Shock wave standoff distance for a sphere slightly above Mach one

This paper describes a numerical simulation of bow shock formation ahead of a sphere at steady supersonic flow in the Mach number range of 1.025–1.20. Turbulent viscous flow results are presented using the Spalart–Allmaras turbulence model. The purpose of this study is to determine the shock standoff distance for a spherical projectile at slightly supersonic free flight speeds. Results are compared to experimental data, including double exposure holographic interferograms obtained from a 40 mm polycarbonate sphere launched by a light gas gun. The shock standoff distance was determined from the interferograms. The present numerical simulations were found to agree with previously published data, and reached down to M = 1.025—a range where almost no previously published data exists. The computed flow structure and shock wave locations agree well with recently obtained free-flight interferograms.     

Simulation of converging cylindrical GPa-range shock waves generated by wire array underwater electrical explosions

Results of one-dimensional numerical simulations of the parameters of the converging strong shock wave generated by electrical underwater explosions of a cylindrical wire array with different array radii and different deposited energies are presented. It was shown that for each wire array radius there exists an optimal duration of the energy deposition into the exploding array, which allows one to maximize the shock wave pressure and temperature in the vicinity of the implosion axis. The simulation results agree well with the 130-GPa pressure in the vicinity of the implosion axis that was recently obtained, which strongly indicates the azimuthal symmetry of the converging shock wave at these extreme conditions. Also, simulations showed that using a pulsed power generator with a stored energy of ~200 kJ, the pressure and temperature at the shock wave front reaches ~220 GPa and 1.7 eV at 0.1 mm from the axis of implosion in the case of a 2.5 mm radius wire array explosion. It was found that, in spite of the complicated equation of state of water, the maximum pressure at the shock wave front at radius r can be estimated as P ≈ (P*(r*/r) ^α , where P* is the known value of pressure at the shock wave front at radius r* ≥ r and α is a parameter that equals 0.62±0.02. A rough estimate of the implosion parameters of the hydrogen target after the interaction with the converging strong shock wave is presented as well.     

Experimental solid mechanics in the nineteenth century

Fashion, fortune, dogma, and genius have each played a role in the 19th century history of experimental solid mechanics. The dominant themes are traced and critically evaluated, from the epoch-making experiments of Alphonse Duleau and Pierre Dupin in 1811 that for 50 years stimulated successive studies in various aspects of solid mechanics including giving impetus to the linear theory of elasticity in the 1820's, to the ingenious and definitive experiments of Eduard Grüneisen in 1906, who adapted Michelson-Morley interferometry to solid mechanics, reaching strains as small as 10⁻⁸. The 19th century was rich in experimentists and elegant experiments that in every decade contributed to a fascinating panorama of discovery in the laboratory. Paper was presented at the 1989 SEM Conference on Experimental Mechanics held in Cambridge, MA on May 28–June 1.     

Stress analysis of concrete and reinforced-concrete slab structures under a high-velocity impact

A mathematical model is developed, which describes the behavior of reinforced concrete under high-velocity impact and explosion conditions within the framework of mechanics of continuous media. The problem of a model projectile penetrating into a layered target consisting of two concrete slabs separated by a sand layer and blasting of an explosive charge encased in the embedded projectile is solved in the three-dimensional formulation by the finite-element method. The effect of reinforcement on penetration and failure of reinforced-concrete slabs is studied by means of mathematical simulations. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 165–173, May–June, 2005.     

Experiments on the penetration of thin long-rod projectiles into thick long-cylindrical borosilicate targets under pressure-free polycarbonate,aluminum and steel confinements

It is well established that confinement pressure inhibits comminution and fragment-flow during projectile penetration of ceramics. Here, a high-pressure gas gun is used to investigate the role of confinement wave impedance on the failure kinetics of ceramics during penetration. Tool-steel rods of fixed lengths and L/D ratios of 12, 16 and 24 impact and penetrate unconfined borosilicate cylinders and those under pressure-free polycarbonate, aluminum and steel confinements. The cylinders are all of the same size with projectile–target diameter ratios lying between 12 and 24, and projectile–target length ratio equal to 8. A stress wave controlling confinement is introduced to approximate an elastic waveguide set-up. Penetration depths into the comminuted borosilicate and the corresponding fragment jet diameters are measured between 168 and 1038 m/s impact velocities with high-speed photography and a witness plate. Expectedly, target resistive pressure increases with confinement impedance but decreases with projectile diameter. However, cylinders confined by steel are less resistive to penetration than those confined by aluminum. This anomalous behavior suggests that comminution increases with dynamic compression and it may be related to densification and the failure wave which occur in silica glasses above certain critical pressures. On this basis, comminution threshold conditions are determined and found to depend strongly on the propagation of stress waves across the target–confinement interface. These results are useful for material selection of impact/penetration-resistant structures with ceramic cores.