A model for shock waves in solids and evidence for a thermal catastrophe

Shock-wave experiments and computer simulations yield a linear relation between shock velocity and particle velocity. We show that a Mie-Grüneisen equation of state combined with the Hugoniot relations matches the computer results and the experiments and leads to an explanation of the generality of the linear velocity dependence. At high shock velocities the onset of a thermal catastrophe, which results in a maximum shock compression, is predicted from our model.     

200 GPa压力范围内铝和铜的等熵压缩线计算

 将冲击Hugoniot线作为Grüneisen物态方程的参考线,以冲击的初始状态为参考状态,推导得到线性和二次曲线表示的冲击绝热线所对应的等熵压缩线方程,计算了200 GPa压力范围内铝和铜两种材料的等熵压缩线,并且计算了以Hugoniot关系为基础的Appy经验物态方程导出的等熵压缩线。计算结果表明,以Appy经验物态方程导出的等熵压缩线与以线性冲击绝热线导出的等熵压缩线接近,在200 GPa压力范围内两者相差不到1.5%。将计算得到的铝的等熵压缩线与美国Sandia实验室ICE实验Z864数据进行了比较,由线性Hugoniot得到的等熵压缩线与实验数据相差不到1%,由Appy经验物态方程得到的等熵线与实验数据几乎重合,说明在200 GPa压力范围内,以Appy物态方程和以线性Hugoniot为参考来计算的等熵压缩线有较高的精度。     

Relation between the Grüneisen ratio and the pressure dependence of Poisson's ratio for metals

The Grüneisen ratio of crystalline solids is shown to be dependent on a parameter n whose values are characteristic of each solid, and can be determined by two independent ways: from experimental shock data and from the pressure derivative of Poisson's ratio. The determinations are made for several metals, using data on the pressure derivatives of polycrystalline elastic moduli or of the second order elastic constants measured on single crystals, and giving the pressure derivatives of Poisson's ratio by means of the Voigt-Reuss-Hill averaging procedure. The values of the parameter n deduced from shock data are found to be in good agreement with those deduced from the pressure derivatives of Poisson's ratio. Positive and negative values of parameter n correspond respectively to increasing and decreasing Poisson's ratio with increasing pressure. Discussion of the results is made using the linear and the quadratic relationships between shock velocity and particle velocity. It is shown that shock wave data cannot yield directly an accurate estimation of the derivative of the initial slope of the Hugoniot.     

New constraints on gruneisen parameters applications to equations of state for pressure markers

Abstract

“Regular” linear relations between shock and particle velocities, u_s = u_s0, + c · u_p, imply a divergence of the Hugoniot pressure p_H at a critical compression V _c/V _o = (c-1)/c, where the thermal pressure Δp _th = (γ/V)· ΔU _th = p _H-p _T becomes very large with respect to the cold (isothermal) pressure p _T at this compression. The Hugoniot relation for the total internal energy U _H results in this case in the relation γ_c = 2(c- 1) for the Grüneisen parameter in this region. Experimental data for the zero pressure value Γ_o and its derivative Γ_o together with theoretical data for the limiting value γ_∞ = 2/3, and the additional expenmental value γ_c are then used to determine the thermal (phonon) pressure with the common Mie-Grüneisen assumption.     

Thermodynamic response of Mie-Grüneisen materials at high pressures

Expressions are derived to give entropy and temperature in a material which obeys the Mie-Grüneisen equation, based on the assumption that both the Hugoniot pressure and the Grüneisen parameter may be expanded as infinite power series in strain. The resulting equations are used to obtain specific expressions for isothermal compression and for isentropic release from a Hugoniot state. A method is suggested for determining the Grüneisen parameter (as a function of strain) from isothermal compression data. Certain limitations involving possible phase transitions are pointed out.     

沿等压路径求解疏松材料Hugoniot关系的微分方程组及其求解

 根据固体材料的三项式物态方程和Grüneisen物态方程,导出了沿等压路径求解疏松材料冲击温度和压缩体积随初始密度变化的微分方程组。从体积的微分方程出发,在假定Wu-Jing参量为常数的前提下,导出了冲击压缩体积和体积-焓物态方程的Wu-Jing表达式。采用数值差分方法求解微分方程组,计算了疏松铜的冲击压缩特性,并与文献中部分实验数据进行了比较,特别强调了热电子对冲击压缩体积、冲击温度和Wu-Jing参数的贡献。还讨论了Grüneisen物态方程与Wu-Jing物态方程的内在联系及后者的适用范围。     

用冲击压缩数据计算物态方程的一种方法——Grüneisen系数最优化

 本文给出了一种用冲击压缩数据计算Grüneisen物态方程参数的方法。它使Grüneisen系数沿实验冲击绝热线的计算值与晶格振动理论的计算值达到最优拟合，给出了23种金属的冷能，冷压公式中的ρ_0K、Q和q以及Grüneisen系数的结果。用这些参数算得的冲击绝热线与实验冲击绝热线之间的平均相对偏差在1%以内。     

40GPa压力范围内铜和铝的准等熵压缩线

 基于自行研制的磁驱动准等熵压缩加载实验装置CQ-1.5,利用全光纤位移干涉仪（Doppler Pins System, DPS）、激光速度干涉计（Velocity Interferometer System for Any Reflectors, VISAR）两种测试手段,以及反积分数据处理方法,实验测量了40 GPa压力范围内T1铜、LY12硬铝和L1纯铝3种材料的准等熵压缩线,将实验准等熵压缩线与基于Grüneisen状态方程的理论等熵压缩线和冲击Hugoniot线进行了比较。结果表明,在该压力范围内,实验准等熵压缩线与理论等熵压缩线相一致,两者偏差小于3%;实验准等熵压缩线靠近冲击Hugoniot线,位于其下方,与国外文献发表的结果相同,进一步表明,实验测量结果正确可靠。     

Hugoniot线上的一个经验常数——来自多孔材料Hugoniot数据的对比研究

 对不同初始密度样品冲击压缩Hugoniot数据的系统分析显示，对每种固体物质而言存在一个表征材料特性的沿其Hugoniot线不随冲击压力而变的经验常数β。β的数值随材料而异。对金属元素而言，该经验特性常数β与初始密实密度ρ₀之间遵循同一个幂指数关系。利用该常数β，可很方便地对不同初始密度的Hugoniot数据进行换算，并可以很方便地获得Grüneisen参数。还对β值为常数的适用范围作了讨论。     

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-Grüneisen 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.     

高岭石的高温高压相图及其地学意义

 用阻抗匹配法和PZT压电探针技术,在100 GPa的冲击压力范围内测量了初始密度分别为1.375 g/cm³和2.001 g/cm³两种孔隙度叙永石样品的Hugoniot状态方程。根据其p_H-ρ_H线所给出的高温高压相变点,用Grüneisen状态方程计算其相变点压力所对应的温度,并结合常压下受热相变的温度值,建立了“高岭石/Al₂O₃+SiO₂+H₂O”的温度-压力相平衡图。通过该相图与线性地热线的交点推断：高岭石至少可在上地幔50 km深处作为一种含水（OH^-）矿物而稳定存在;或在俯冲板块中至少于133 km深处作为一种含水（OH^-）泥质沉积物的过渡相而存在。     

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 U_s ^A and particle velocity u_p was adequate in the through-thickness orientation, damage softening process produces discontinuities both in value and slope in the U_s ^A-u_p 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 U_s ^A-u_p over the range of measurements corresponding to non-linear anisotropic elastic wave shows a value of c₀ ^A (the intercept of the U_s ^A-u_p 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.     

Grüneisen parameter for HCP iron at extreme compression

The Thomas–Fermi approximation gives the Grüneisen parameter γ=γ_∞=1/2 for all materials at extreme compression (P→∞ or V→0). After re-analyzing the existing experimental data of volume dependence of Grüneisen parameter γ of hexagonal close-packed (HCP) iron, we find that γ=1/2+a(V/V ₀)^1/3+b(V/V ₀) ⁿ , where a, b and n are constants. Based on this new form of γ, the second Grüneisen parameter q, the Debye temperature θ_D and the shear sound velocity v _s of HCP iron are discussed in the present work. It is found that the zero pressure second Grüneisen parameter q ₀=0.654, which is consistent with the previously determined value of HCP iron for Earth's core physics from Dubrovinsky et al. The calculations for the Debye temperature and the shear sound velocity are also found to be in good agreement with the experimental data.     

Three points concerning extreme compressions in shocked deuterium

Abstract

Those models, that show agreement with the Hugoniot data of Da Silva et al., over a large pressure range, are fit to the reflected wave experiment of Holmes et al. In these works it is assumed that the reflected wave is a single stable shock, in contradiction to the data. We change the picture to allow for multiwave-reflected waves and show that such is completely consistent with all data. The reflected wave is more accurately approximated by an adiabat than a Hugoniot and the fitted models should be adjusted to account for this difference. This is our first point. We then review from previous work the relevant physics that determines the maximum compression along a Hugoniot. This is our second point. Finally, with this understanding we are able to determine from experimental data on other molecular systems, such as nitrogen and benzene, whether the large compressions seen in the laser Hugoniot data for deuterium are reasonable. They are not.     

C30混凝土冲击绝热关系和Grüneisen型状态方程的实验研究

采用一级气炮加载技术和锰铜压力计测试技术,对含初始空隙的C30混凝土在一维应变条件下的冲击特性进行了实验测量和分析.基于锰铜压力计测量的压力波形,确定了C30混凝土材料的冲击绝热关系,即冲击波速度D与波后粒子速度u之间满足线性关系.再从C30混凝土的冲击绝热数据出发,获得了计及初始空隙度₀影响的多项式形式Grüneisen型状态方程中的各项系数.实测压力波形还显示：不同位置处的压力波形在迅速上升至峰值后均随时间逐渐衰减,而冲击波峰值又随传播 关键词： Grüneisen型状态方程 冲击绝热关系 混凝土     

The equation of state of solids

An equation of state for solids, in reduced variables, is obtained within the context of a system-independent formulation of the thermodynamics of high pressures. This formulation is valid for materials obeying a linear relationship between shock and particle velocities. An adequate set of scaling factors for pressure, compression, specific energy, and temperature, is first introduced. A modified Mie-Grüneisen equation, as well as many other thermodynamic relationships and coefficients, are then expressed in terms of reduced variables. Explicit expressions for the temperature along the Hugoniot, and for the equation of state, are obtained. It is also shown that when given in their reduced form, each of the two thermodynamic coefficients appearing in the equation of state can be considered as having the same constant value for many different materials. The possibility and convenience of using a “standard material” is discussed. Numerical results obtained using this reduced variables formalism are in good agreement with those computed or measured, by different authors, for various materials over a wide range of pressures. This is a good indication of the “universality” of the reduced equations obtained, and of the usefulness of the formalism.     

冲击加载下Zr51Ti5Ni10Cu25Al9金属玻璃的塑性行为

进行了10—27 GPa应力范围内Zr₅₁Ti₅Ni₁₀Cu₂₅Al₉金属玻璃的平面冲击实验以研究其高压-高应变率加载下的塑性行为.由样品自由面粒子速度剖面的分析获得了冲击加载过程的轴向应力,并通过轴向应力与静水压线的比较获得剪应力.实验结果表明,尽管存在明显的松弛效应,但Zr基金属玻璃的Hugoniot弹性极限随着冲击应力的增加而增加.然而,塑性波阵面上的剪应力则显示先硬化而后软化现象,而且软化的幅度随冲击应力的增加而增加.冲击加载下Zr基金属玻璃的上述剪应力变化特征与分子动力学模拟结果比较一致,但与压剪实验结果和一维应力冲击实验结果明显不同.     

Equation of state of fluid helium at high temperatures and densities

Hugoniot curves and shock temperatures of gas helium with initial temperature 293 K and three initial pressures 0.6, 1.2, and 5.0 MPa were measured up to 15000 K using a two-stage light-gas gun and transient radiation pyrometer. It was found that the calculated Hugoniot EOS of gas helium at the same initial pressure using Saha equation with Debye-Hückel correction was in good agreement with the experimental data. The curve of the calculated shock wave velocity with the particle velocity of gas helium which is shocked from the initial pressure 5 MPa and temperature 293 K, i.e., theD ≈u relation,D=C ₀+λu (u<10 km/s, λ=1.32) in a low pressure region, is approximately parallel with the fittedD ≈u (λ=1.36) of liquid helium from the experimental data of Nellis et al. Our calculations show that the Hugoniot parameter λ is independent of the initial density p{in0}. TheD≈u curves of gas helium will transfer to another one and approach a limiting value of compression when their temperature elevates to about 18000 K and the ionization degree of the shocked gas helium reaches 10⁻³.     

Equation of state of fluid helium at high temperatures and densities

Helium, hydrogen, and their isotopes are the simplest monoatomic and diatomic molecules. It is relatively easy to describe their properties using the basic principles of quantum mechanics. In condensed matter physics, hydrogen and helium serve as the models for the condensed matter properties at extreme conditions so that both experi- mental and theoretical physicists pay much attention to the study of their properties[1], especially the insulator-metal transition of hydrogen[2]. The aim to st…     

Lattice-dynamical calculations of the thermodynamic properties of CsBr and CsI

Results of a quasi-harmonic calculation of the normal mode frequencies, mode-Grüneisen parameters, the coefficient of thermal expansion α, the heat capacity C_p, and the macroscopic Grüneisen function γ(itT) for CsBr and CsI are presented. Very good agreement with the experimental results for α and C_p is obtained. A self-consistent procedure is used to parametrize the models so that some often neglected vibrational and configuration-dependent effects can be accounted for. Predictions are made for the values that the potential energy derivatives and the elastic constants would have in the absence of vibrational motion.