Wave profile

Figure 1.1 shows a typical stress-volume relation for a solid which remains in a single structural phase, along with a depiction of idealized wave profiles for the solid loaded with different peak pressures. The first-order picture is one in which the characteristic response of solids depends qualitatively upon the material properties relative to the level of loading. Inertial properties determine the sample response unlike static high pressure, the experimenter does not have independent control of stresses within the sample. 

Table 1.1. Solid-state phenomena causing detail in wave profiles.

Figure 2.8. An x-t diagram of a piston interacting with a compressible fluid. At the origin, the piston begins moving at constant velocity, generating a shock wave. At tj, the piston stops abruptly, generating rarefaction fan. Snapshots of wave profiles at times t2 and 3 are shown.

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... 

Figure 3.14. Manganin stress-wave profiles for Arkansas novaculite at 25 GPa (Grady et al., 1974).

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).)...

The structure/property relationships in materials subjected to shock-wave deformation is physically very difficult to conduct and complex to interpret due to the dynamic nature of the shock process and the very short time of the test. Due to these imposed constraints, most real-time shock-process measurements are limited to studying the interactions of the transmitted waves arrival at the free surface. To augment these in situ wave-profile measurements, shock-recovery techniques were developed in the late 1950s to assess experimentally the residual effects of shock-wave compression on materials. The object of soft-recovery experiments is to examine the terminal structure/property relationships of a material that has been subjected to a known uniaxial shock history, then returned to an ambient pressure... 

Recent experiments by Gray et al. [47] have probed the contribution of the Bauschinger effect on real-time unloading wave profiles and postshock... 

Figure 6.17. VISAR wave profiles of copper and silicon bronze at 10 GPa exhibiting differing unloading wave shapes supporting a Bauschinger effect contribution to unloading.

For a given amplitude of the quasi-elastic release wave, the more the release wave approaches the ideal elastic-plastic response the greater the strength at pressure of the material. The lack of an ideally elastic-plastic release wave in copper appears to suggest a limited reversal component, however, this is much less than in the silicon bronze. Collectively, the differences in wave profiles between these two materials are consistent with a micro-structurally controlled Bauschinger component as supported by the shock-recovery results. Further study is required to quantify these findings and... 

G.T. Gray III and C.E. Morris, Influence of Peak Pressure on the Substructure Evolution and Shock Wave Profiles of Ti-6A1-4V, in Sixth World Conference on Titanium (edited by P. Lacombe, R. Tricot, and G. Beranger), Les Editions de Physique, France, 1989, 269 pp. 

Much of what we currently understand about the micromechanics of shock-induced plastic flow comes from macroscale measurement of wave profiles (sometimes) combined with pre- and post-shock microscopic investigation. This combination obviously results in nonuniqueness of interpretation. By this we mean that more than one micromechanical model can be consistent with all observations. In spite of these shortcomings, wave profile measurements can tell us much about the underlying micromechanics, and we describe here the relationship between the mesoscale and macroscale. 

When (7.10)-(7.12) are combined with the expressions for mass and momentum conservation, we are then able to compare assumptions regarding and v with macroscale observations such as wave profiles, for example. The conservation laws are (in Lagrangian form Pq dX = p dx )... 

A typical shock-compression wave-profile measurement consists of particle velocity as a function of time at some material point within or on the surface of the sample. These measurements are commonly made by means of laser interferometry as discussed in Chapter 3 of this book. A typical wave profile as a function of position in the sample is shown in Fig. 7.2. Each portion of the wave profile contains information about the microstructure in the form of the product of and v. The decaying elastic wave has been an important source of indirect information on micromechanics of shock-induced plastic deformation. Taylor [9] used measurements of the decaying elastic precursor to determine parameters for polycrystalline Armco iron. He showed that the rate of decay of the elastic precursor in Fig. 7.2 is given by (Appendix)... 

J.N. Johnson and L.M. Barker, Dislocation Dynamics and Steady Plastic Wave Profiles in 6061-T6 Aluminum, J. Appl. Phys. 40, 4321-4334 (1969). 

A.S. Appleton and J.S. Waddington, The Importance of Shock-Wave Profile in Explosive Loading Experiments, Acta Metall. 12, 956-957 (1964). 

Figure 8.7. Propagation of wave profile in an elastic-plastic material from the spall plane to the monitoring interface. The wave front propagates at a plastic wave speed whereas the wave release propagates at an elastic wave speed and complicates the analysis of the material spall strength.

Graham, R.A. and Asay, J.R., Measurement of Wave Profiles in Shock-Loaded Solids, High Temp-High Press. 10, 355-390 (1978). 

The unloading wave itself provides a direct measure of the strength at pressure from the shape of the release wave. Such a measurement requires time-resolved detection of the wave profile, which has not been the usual practice for most strong shock experiments. 

Conservation relations are used to derive mechanical stress-volume states from observed wave profiles. Once these states have been characterized through experiment or theory they may, in turn, predict wave profiles for the material in question. For the case of a well-defined shock front propagating at constant speed L/ to a constant pressure P and particle velocity level u, into a medium at rest at atmospheric pressure, with initial density, p, the conservation of momentum, mass, and energy leads to the following relations ... 

Wave profiles in the elastic-plastic region are often idealized as two distinct shock fronts separated by a region of constant elastic strain. Such an idealized behavior is seldom, if ever, observed. Near the leading elastic wave, relaxations are typical and the profile in front of the inelastic wave typically shows significant changes in stress with time. 

It should be observed that, in the most general case, interpretation of the mechanical responses requires time-resolved wave-profile measurements. As shown in Eqs. (2.2) and (2.3), direct evaluation of the response requires quantitative description of derivatives of kinetic and kinematic variables. 

Fig. 2.11. Strength behavior of solids at pressure can be probed with reshock or release measurements. The resulting wave profiles of such measurements on a 6061-T6 aluminum alloy with VISAR instrumentation are shown. Strength behavior indicated on many solids reveals behavior not accurately described by simple materials models (after Lipkin and Asay [77L02]).

The effect of such a transformation on a pressure-volume relation and on wave profiles is shown in Fig. 2.12. Above the transformation, its characteristics dominate the wave profile. At sufficiently high pressure, the peak pressure wave will move at higher speeds and a strong shock regime can be encountered. 

When the pressures to induce shock-induced transformations are compared to those of static high pressure, the values are sufficiently close to indicate that they are the same events. In spite of this first-order agreement, differences between the values observed between static and shock compression are usually significant and reveal effects controlled by the physical and chemical nature of the imposed deformation. Improved time resolution of wave profile measurements has not led to more accurate shock values rather. 

Fig. 2.12. If solids undergo a shock-induced polymorphic transformation, the volume change at the transformation causes significant changes in the wave profile produced by shock loading. In the figure, is the applied pressure, Pj is the pressure of the phase transition, and HEL is the Hugoniot elastic limit.

Perhaps the most visible technical problems studied and the most data available on shock-compressed solids are focused on the loading portion of wave profiles. Often, the portion of the wave profile corresponding to the release of pressure to atmospheric, but elevated temperature, values is the more descriptive of solids in the high pressure state. 

GPa, is particularly interesting because of the anomalous slope of the compressiblity to 3 GPa. The wave profile with loading and release wave in Fig. 2.17 shows the anomalous loading and the shock on release from the high stress state. 

Given the various release-wave behaviors summarized above, it is clear that release waves may often dominate wave profiles, and failure to consider their influences can lead to incorrect interpretation of observed materials responses, especially those in which samples are preserved for post-shock... 

---