Shock front

The relationship between the two conditions is estabUshed by conservation of energy and by conservation of momentum across the shock front. 

Further reductions in reservoir pressure move the shock front downstream until it reaches the outlet of the no22le E. If the reservoir pressure is reduced further, the shock front is displaced to the end of the tube, and is replaced by an obflque shock, F, no pressure change, G, or an expansion fan, H, at the tube exit. Flow is now thermodynamically reversible all the way to the tube exit and is supersonic in the tube. In practice, frictional losses limit the length of the tube in which supersonic flow can be obtained to no more than 100 pipe diameters. 

Thus, the effluent concentration becomes zero at Ti Tp = 1/R. The position of the leading edge (a shock front ) is determined from Eq. (16-132) ... 

Another way of representing shock-wave profiles is in the form of F-t histories of the pressure or another variable at a series of points along its direction of propagation, as in Fig. 2.9. In the above example, the leading part of the shock front arrives first, effectively increasing the pressure instantaneously. The rarefaction arrives later and decreases the pressure over a time... 

One drawback of systems shown in Fig. 3.1 is that the initial peak shock pressure on the specimen is limited and not well sustained. It immediately starts to decrease, which leads to an attenuation of the shock front as it propagates through the specimen. Attenuation of the shock is detrimental to the accuracy of the resulting experimental data. Also, the late-time release of pressure at the specimen surface is nonplanar, which rules out obtaining accurate information on the specimen s unloading behavior. 

In principle, there is no upper bound in measurements of particle velocity (or stress) using laser velocity interferometry. In practice, very high-pressure shock fronts can cause copious jetting of microparticles from the free surface (Asay et al., 1976), obscuring the surface from the laser beam. To alleviate this, optically transparent materials can be bonded to the specimen, and particle velocity measurements are then made at the specimen/window interface. This has the added advantage of simulating in situ particle velocity... 

Gupta, S.C. and Y.M. Gupta (1984), Response of Ytterbium Foils Oriented Parallel and Perpendicular to the Shock Front, in Shock Waves in Condensed Matter— 1983 (edited by J.R. Asay, R.A. Graham, and G.K. Straub) Elsevier Science, New York, pp. 237-238. 

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... 

In the simplest case when a single shock state is achieved via a shock front, the Rankine-Hugoniot equations involve six variables U, u, p, Pi, i — Eq, and Pi) thus, measuring three, usually U, m, and p, determines the shock state, pi, , - A- The key assumption underpinning the... 

We assume that in (4.38) and (4.39), all velocities are measured with respect to the same coordinate system (at rest in the laboratory) and the particle velocity is normal to the shock front. When a plane shock wave propagates from one material into another the pressure (stress) and particle velocity across the interface are continuous. Therefore, the pressure-particle velocity plane representation proves a convenient framework from which to describe the plane Impact of a gun- or explosive-accelerated flyer plate with a sample target. Also of importance (and discussed below) is the interaction of plane shock waves with a free surface or higher- or lower-impedance media. 

The condition that gives rise to multiple shock fronts (i.e., allows a shock wave to bifurcate as indicated in Fig. 4.10(b)) will occur when the second wave propagation velocity (with respect to the laboratory) is given by (4.39). How-... 

In the case of most nonporous minerals at sufficiently low-shock stresses, two shock fronts form. The first wave is the elastic shock, a finite-amplitude essentially elastic wave as indicated in Fig. 4.11. The amplitude of this shock is often called the Hugoniot elastic limit Phel- This would correspond to state 1 of Fig. 4.10(a). The Hugoniot elastic limit is defined as the maximum stress sustainable by a solid in one-dimensional shock compression without irreversible deformation taking place at the shock front. The particle velocity associated with a Hugoniot elastic limit shock is often measured by observing the free-surface velocity profile as, for example, in Fig. 4.16. In the case of a polycrystalline and/or isotropic material at shock stresses at or below HEL> the lateral compressive stress in a plane perpendicular to the shock front... 

Another important method of determining the Gruneisen ratio in the shock state is the measurement of sound speed behind the shock front. The techniques employing optical analyzers (McQueen et al., 1982) piezoresistive (Chap-... 

McQueen et al. (1982) demonstrated that by placing a series of high-impedance transparent fluids (called optical analyzers) over the sample at a series of thicknesses less than d in the target that the overtaking rarefaction (sound) velocity can be accurately obtained. Arrival of rarefaction waves rapidly reduce the shock pressure. These wave arrivals could be very readily detected by the change in light radiance caused by the onset of a decrease in shock amplitude when the rarefaction wave caught up to the shock front. The... 

In this case the shear stress (in the plastic deformation region) depends on how much plastic strain y that has accumulated and the current rate of deformation y. For example, in shock compression the shear stress t behind the shock front (where y is nominally zero) is a function of y only, as given by the implicit relationship... 

When the elastic shock-front speed U departs significantly from longitudinal elastic sound speed c, immediately behind the elastic shock front, the decaying elastic wave amplitude is governed by (Appendix)... 

The shock-induced micromechanical response of <100>-loaded single crystal copper is investigated [18] for values of (WohL) from 0 to 10. The latter value results in W 10 Wg at y = 0.01. No distinction is made between total and mobile dislocation densities. These calculations show that rapid dislocation multiplication behind the elastic shock front results in a decrease in longitudinal stress, which is communicated to the shock front by nonlinear elastic effects [pc,/po > V, (7.20)]. While this is an important result, later recovery experiments by Vorthman and Duvall [19] show that shock compression does not result in a significant increase in residual dislocation density in LiF. Hence, the micromechanical interpretation of precursor decay provided by Herrmann et al. [18] remains unresolved with existing recovery experiments. 

Asay et al. [24] investigate further the effects of nonlinear elasticity on micromechanical interpretation of decaying elastic shock fronts. Values of (Til, I, r/Cj, and which represent the highest Mg" " impurity concentration are shown in Table 7.1 for D = 0.1 GPa. 

Figure 7.4. Dislocation density at the shock front as a function of shear stress on primary slip planes.

Dick et al. [29] present additional data on the <100) shock compression of LiF which further establishes a threshold shear stress of between 0.24 GPa and 0.30 GPa for nucleation of dislocations in the shock front. 

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