Shock velocity

The importance of the distinction between shock velocity and particle velocity cannot be overemphasized. The particle velocity refers to the velocity... 

This statement refers to a frame of reference in which the initial state of the material is stationary. It is not true for all frames of reference, e.g., one which moves with the shock velocity. 

This simple example Illustrates the important kinematic properties of shock waves, particularly the concepts of particle velocity and shock velocity. The particle velocity is the average velocity acquired by the beads. In this example, it is the piston velocity, v. The shock velocity is the velocity at which the disturbance travels down the string of beads. In general, at time n//2v, the disturbance has propagated to the nth bead. The distance the disturbance has traveled is therefore n d -b /), and the shock velocity is... 

It is instructive to collect the important relations here for comparison to the jump conditions derived in Section 2.4. When the bead parameters are replaced with the properties of particle and shock velocities, force and internal energy, the relations can be written as... 

On the other hand, the thickness of the slab in the Lagrangian system is the constant Ah = Axq, so for a Lagrangian shock velocity of C, the transit time is... 

The relative shock velocity t/ = (7 — Uj is the Eulerian shock velocity often used because it is a material property and is independent of the motion of the... 

The term shock velocity usually refers to this Eulerian shock velocity relative to the particle velocity of the unshocked material unless otherwise stated, and is usually written as (/ (without the prime). 

Hugoniot curve A curve representing all possible final states that can be attained by a single shock wave passing into a given initial state. It may be expressed in terms of any two of the five variables shock velocity, particle velocity, density (or specific volume), normal stress (or pressure), and specific internal energy. This curve it not the loading path in thermodynamic space. 

Shock velocity The velocity of the shock wave as it passes through the material. In the limit of an infinitesimally small shock wave it is equal to the bulk sound speed of the material. 

The shock pressures attainable with direct explosive contact depend on the shock impedance (shock velocity times material density) of the specimen material, and on the explosive energy of the contacting explosive. High-energy explosives placed directly on high-shock impedance materials can produce shock pressures of several tens of GPa. 

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. 

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... 

Shock-compression science originated during and after World War II when experimental facilities for creating planar shock waves were developed, along with prompt instrumentation techniques enabling shock velocity and particle velocity measurements to be made. The main thrust of shock-compression science is to understand the physics and to measure the material properties which govern the outcome of shock-compression events. Experiments involving planar shock waves are the most useful in shock-compression science. 

The diagnostics applied to shock experiments can be characterized as either prompt or delayed. Prompt instrumentation measures shock velocity, particle velocity, stress history, or temperature during the initial few shock transits of the specimen, and leads to the basic equation of state information on the specimen material. Delayed instrumentation includes optical photography and flash X-rays of shock-compression events, as well as post-mortem examinations of shock-produced craters and soft-recovered debris material. 

The propagation of a shock wave from a detonating explosive or the shock wave induced upon impact of a flyer plate accelerated, via explosives or with a gun, result in nearly steady waves in materials. For steady waves a shock velocity U with respect to the laboratory frame can be defined. Conservation of mass, momentum, and energy across a shock front can then be expressed as... 

Figure 4.3. Shock velocity versus particle velocity for several standard materials.

Typical U,-Up data for a wide range of materials are given in Fig. 4.3 and Table 4.1. Here Cq is the shock velocity at infinitesimally small particle velocity, or the ambient pressure bulk sound velocity which is given by... 

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). 

Figure 4.10. Type of Hugoniot necessary to produee a two-wave shoek strueture and resulting wave profile. This type of Hugoniot will in general give a loeus as shown, with a flat region of eonstant shock velocity. Point 2 will not be observed with techniques that measure only the first arrival of the shock wave. (After McQueen et al. (1970).)...

Figure 4.12. Shock velocity versus particle velocity for fused quartz. Three regimes are indicated low pressure, fused quartz regime, the mixed phase regime, and the high-pressure phase, stishovite regime.

Calculate the final shock state pressure and density from the measured shock velocity of 5.77 km/s in a sample of glass (initial density 2.204 g/cm ) which is mounted onto a driver plate of pure Cu. The Cu driver plate is impacted at 4.5 km/s by a Ta flyer plate. Use the impedance match methods. 

RuolT, A.L. (1967), Linear Shock-Velocity-Particle-Velocity Relationship, J. Appl. Phys. 38, 4976-4980. 

If we accept the assumption that the elastic wave can be treated to good aproximation as a mathematical discontinuity, then the stress decay at the elastic wave front is given by (A. 15) and (A. 16) in terms of the material-dependent and amplitude-dependent wave speeds c, (the isentropic longitudinal elastic sound speed), U (the finite-amplitude elastic shock velocity), and Cfi [(A.9)]. In general, all three wave velocities are different. We know, for example, that... 

Hugoniot data have been fitted by the equation = Cq + su + qu, where Uj is the shock velocity and the associated particle velocity. Griineisen parameters have been obtained from best estimates of zero pressure thermodynamic parameters, which are sometimes of dubious value. The pressures and velocities describing the valid range of the fits do not necessarily indicate the onset or completion of a transition. 

The dotted segments represent the region of two-wave structure for those materials exhibiting transitions the lines have been drawn on the basis of the shock velocity of the first wave. The dashed curves represent reflected shocks and rarefaction release loci from the 2024 A1 Hugoniot at the pressures listed. The three heavy curves are the Hugoniots of 2024 Al, Cu, and U-3 wt.% Mo alloy which were determined independently. These were used as standards to determine the Hugoniots of the other materials. 

The line connecting the initial state to the shocked state is termed the Rayleigh line characterized in shock velocity as [/ = Io[P — Pq/Vq — F]. Equations (2.1) represent propagation into undisturbed matter, but can be... 

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