Shock-Wave Equations of State

In this chapter we define what is meant by a shock-wave equation of state, and how it is related to other types of equations of state. We also discuss the properties of shock-compressed matter on a microscopic scale, as well as discuss how shock-wave properties are measured. Shock data for standard materials are presented. The effects of phase changes are discussed, the measurements of shock temperatures, and sound velocities of shock materials are also described. We also describe the application of shock-compression data for porous media. 

Equation (4.8) is often called the shock-wave equation of state since it defines a curve in the pressure-volume plane (e.g.. Fig. 4.4). 

In order to relate the parameters of (4.5), the shock-wave equation of state, to the isentropie and isothermal finite strain equations of state (discussed in Section 4.3), it is useful to expand the shock velocity normalized by Cq into a series expansion (e.g., Ruoff, 1967 Jeanloz and Grover, 1988 Jeanloz, 1989). 

Ruoff (1967) first showed how the coefficients of the shock-wave equation of state are related to the zero pressure isentropic bulk modulus, and its first and second pressure derivatives, K q and Kq, via... 

The study of shock-wave equations of state of porous materials provides a means to expand knowledge of the equation of state of condensed materials to higher temperatures at a given volume than can be achieved along the principal Hugoniot. Materials may be prepared in porous form via pressing... 

Jeanloz, R. (1989), Shock Wave Equation of State and Finite Strain Theory, J. Geophys. Res 94, 5873-5886. 

Jerry Wackerle, W. L. Seitz, and John C. Jamieson, Shock Wave Equation of State for High-Density Oxygen , Symposium on the Behavior of Dense Media Under High Dynamic Pressures (1967). 

The jump conditions must be satisfied by a steady compression wave, but cannot be used by themselves to predict the behavior of a specific material under shock loading. For that, another equation is needed to independently relate pressure (more generally, the normal stress) to the density (or strain). This equation is a property of the material itself, and every material has its own unique description. When the material behind the shock wave is a uniform, equilibrium state, the equation that is used is the material s thermodynamic equation of state. A more general expression, which can include time-dependent and nonequilibrium behavior, is called the constitutive equation. 

For simplicity, we have shown an expansion wave in which the pressure is linearly decreasing with time. This, in general, is not the case. The release behavior depends on the equation of state of the material, and its structure can be quite complicated. There are even conditions under which a rarefaction shock can form (see Problems, Section 2.20 Barker and Hollenbach, 1970). In practice, there are many circumstances where the expansion wave does not propagate far enough to fan out significantly, and can be drawn as a single line in the x t diagram. 

Prompt instrumentation is usually intended to measure quantities while uniaxial strain conditions still prevail, i.e., before the arrival of any lateral edge effects. The quantities of interest are nearly always the shock velocity or stress wave velocity, the material (particle) velocity behind the shock or throughout the wave, and the pressure behind the shock or throughout the wave. Knowledge of any two of these quantities allows one to calculate the pressure-volume-energy path followed by the specimen material during the experimental event, i.e., it provides basic information about the material s equation of state (EOS). Time-resolved temperature measurements can further define the equation-of-state characteristics. 

Equation-of-state measurements add to the scientific database, and contribute toward an understanding of the dynamic phenomena which control the outcome of shock events. Computer calculations simulating shock events are extremely important because many events of interest cannot be subjected to test in the laboratory. Computer solutions are based largely on equation-of-state models obtained from shock-wave experiments which can be done in the laboratory. Thus, one of the main practical purposes of prompt instrumentation is to provide experimental information for the construction of accurate equation-of-state models for computer calculations. 

These are some of the oldest, yet still the most useful gauges in shock-wave research. They contribute mainly to shock-velocity measurements. In some cases, these gauges alone can provide accurate Hugoniot equation-of-state... 

McQueen, R.G., S.P. Marsh, J.W. Taylor, J.N. Fritz, and W.J. Carter (1970), The Equation of State of Solids from Shock Wave Studies, in High Velocity Impact Phenomena (edited by R. Kinslow), Academic Press, New York, pp. 293-299. 

Understanding such interaction is important both in predicting the amplitudes of shock waves transmitted across interfaces (in the case where the equations of state of all materials are known), and in determining release isentropes or reflected Hugoniots (when measurement of the equation of state is needed). Consider first a shock wave in material A being transmitted to a... 

Ahrens, T.J., and O Keefe, J.D. (1977), Equation of State and Impact-Induced Shock-Wave Attenuation on the Moon, in Impact and Explosion Cratering (edited by Roddy D.J. et al.), Pergamon Press, New York, pp. 639-656. 

Jeanloz, R., and Grover, R. (1988), Birch-Murnaghan and Us-Up Equations of State, in Proceedings of the American Physical Society Topical Conference on Shock Waves in Condensed Matter, Monterey, CA, 1987 (edited by Schmidt S.C. and N.C. Holmes), Plenum, New York, pp. 69-72. 

Walsh, J.M., and Christian, R.H. (1955), Equation of State of Metals from Shock Wave Measurements, Phys. Rev. 97, 1544-1556. 

Duvall, G.E., Shock Waves and Equations of State, in Dynamic Response of Materials to Intense Impulsive Loading (edited by Chou, P.C. and Hopkins, A.K.), US Air Force Materials Laboratory, Wright-Patterson AFB, 1973), pp. 89-121. 

In extreme cases, very high pressure waves are encountered in which the time to achieve peak pressure may be less than one nanosecond. Study of solids under the influence of these high pressure shock waves can be the source of information on high pressure equations of states of solids within the framework of specific assumptions, and of mechanical, physical, and chemical properties under unusually high pressure. 

The pressure is to be identified as the component of stress in the direction of wave propagation if the stress tensor is anisotropic (nonhydrostatic). Through application of Eqs. (2.1) for various experiments, high pressure stress-volume states are directly determined, and, with assumptions on thermal properties and temperature, equations of state can be determined from data analysis. As shown in Fig. 2.3, determination of individual stress-volume states for shock-compressed solids results in a set of single end state points characterized by a line connecting the shock state to the unshocked state. Thus, the observed stress-volume points, the Hugoniot, determined do not represent a stress-volume path for a continuous loading. 

Shock-compressed solids and shock-compression processes have been described in this book from a perspective of solid state physics and solid state chemistry. This viewpoint has been developed independently from the traditional emphasis on mechanical deformation as determined from measurements of shock and particle velocities, or from time-resolved wave profiles. The physical and chemical studies show that the mechanical descriptions provide an overly restrictive basis for identifying and quantifying shock processes in solids. These equations of state or strength investigations are certainly necessary to the description of shock-compressed matter, and are of great value, but they are not sufficient to develop a fundamental understanding of the processes. 

It has been seen in deriving equations 4.33 to 4.38 that for a small disturbance the velocity of propagation of the pressure wave is equal to the velocity of sound. If the changes are much larger and the process is not isentropic, the wave developed is known as a shock wave, and the velocity may be much greater than the velocity of sound. Material and momentum balances must be maintained and the appropriate equation of state for the fluid must be followed. Furthermore, any change which takes place must be associated with an increase, never a decrease, in entropy. For an ideal gas in a uniform pipe under adiabatic conditions a material balance gives ... 

A shock wave is characterized by the entropy change across it. Using the equation of state for a perfect gas shown in Eq. (1.5), the entropy change is represented by... 

For a transmitted shock wave advancing into any gas at an initial pressure pe of 1 atm, the RH (Rankine-Hugoniot) equation defines a functional relationship between pressure p and particle velocity w behind the wave S3, involving initial pressure, initial specific volume v, and equations of state of the target medium. Similarly, the conditions behind the reflected wave S2 and close to the product-target interface are expressible by means either of the shock wave equations or the Rie-mann adiabatic wave equations in terms of any one such variable and the conditions... 

The various shock-producing systems were calibrated by using free-surface velocity measurements of specimen plates and corresponding shock-wave velocities obtd from the known equations of state of the specimen plate materials. Accdg to Footnote 4 on p 1931 of Ref 15a, the free-surface velocity for a plane shock wave is almost twice the particle velocity"... 

The Equation of State of Detonation Products Behind Overdriven Detonation Waves in Composition B (Ref 15, pp 47-52), stated that experiments conducted in England have shown that in many high explosives the shock compressions and adiabatic expansions of... 

Pure shock waves) 4) G.B. Kistiakowsky, p 951 in Kirk Othmer 5 (1950), pp given in the text (Not included in the 2nd edition) 5) Corner, Ballistics (1950), 100-01 (Corner Noble-Abel equations of state) 6) SAC MS, Ballistics (1951), 18 (Covolume and equation of state of propint gases) 7) Taylor(1952), 34 (Boltzmann and Hirschfelder Roseveare equation of state for the expln products) 69-72 (Rankine-Hugoniot equation of state) 87-98 (Abel, Boltzmann and other equations of state applicable to deton of condensed expls yielding only gaseous products) 114 (Equations of state applicable to deton of condensed expls whose products contain a condensed phase)... 

Dewey (Ref 2) described determination of detonation parameters from photographic observations. Cowan Fickett (Ref 4) determined the effects of the various Kis-tiakowsky-Wilson equation of state parameters on the calculated D - pQ curve for 65/35-RDX/TNT expl mixture. Baum, Stanyukovich Shekhter (Ref 5) presented some parameters of shock waves. Stein... 

Refs 1) J.M. Walsh et al, "Shock-Wave Compressions of Twenty-Seven Metals. Equations of State of Metals , PhysRev 108, 196(1957) 2) Cook (1958), p 206... 

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