2.2 Solids
2.2.6 Phase Transitions
2.2.6.1 Background
Phase transitions are a critical feature in any complete description of the material behavior. Shock waves, in particular, provide a valuable tool for accessing the intricacies of high pressure, high temperature phase transitions. Theories on the mechanics, thermodynamics, and kinetics of these transitions have been the focus of a great deal of research [39]. In the simplistic summary presented here, the focus will be on first-order polymorphic or melting transitions. In general, the Hugoniot curve is not as simple as what has been idealized previously, such as that shown in Figure 2.8.
V
P
V
0A
B
P
1P
2 (a)t
P
A
B
P
1P
2 (b)Figure 2.23: Hugoniot curve in which the Hugoniot elastic limit is shown at point A, and a new phase forms at point B.
Instead, a real material will generally exhibit a Hugoniot such as the one represented in Figure 2.23(a), where the material reaches the Hugoniot elastic limit (HEL) at point A and begins to transition to a new phase starting at point B.
Properly measuring this type of Hugoniot, however, is complicated by the stability of the shock wave, where stability is defined by whether or not the shock wave will divide into multiple waves. Combining Eqns. 2.37 and 2.38, it is possible to relate the Rayleigh line, which is the chord con- necting the initial and final states, to the shock speed.
U2 s V2 0 = P +−P− V−−V+, (2.85)
Thus, shock stability can be determined by examining the slopes of the chords connecting the states of interest. Consider the possibility of jumping from an initial state (0), to an intermediate state (1), and finally to an end state (2). Assume the slope of the chords are such that the following inequality is satisfied: P2−P1 V1−V2 < P1−P0 V0−V1 . (2.86)
In this case, the higher pressure shock continually falls further behind the first shock and a two- shock system is stable. On the other hand, if the inequality is reversed, the high pressure shock is the fastest traveling wave in the system and overdrives everything else, such that only a single shock is observed.
Examples of these situations are given in Figure 2.23, where two final shock pressures are achieved. The first, higher pressure state, P1, is achieved through the largest sloping Rayleigh line which directly connects the initial and final states. Thus, the expected wave profile in this situation is a single shock up toP1. In considering a second state, which is associated withP2, the Rayleigh
line connecting the initial and final states now lies below the alternative of using multiple Rayleigh lines. The multiple Rayleigh lines, plotted in Figure 2.23(a), have monotonically decreasing slopes as the pressure is increased. As a result, the multiple shock solution becomes stable, and results in the multi-shock profile shown in Figure 2.23(b). In this case, the transition pressures along the Hugoniot are directly related to each step in the wave profile. Generally, the Hugoniot elastic limit of most metals is small compared to the shock pressure, and is neglected when calculating principal Hugoniot points. This assumption, however, cannot be made for higher pressure transitions. This can make the behavior of the principal Hugoniot between phase transition states of this nature extremely difficult to determine.
2.2.6.2 Effect on Mach Lens Configuration
The types of phase transitions discussed above involve discontinuities in the shock speed of the material. This brings to light the question of how converging shocks, and, specifically, the waves in the Mach lens configuration are affected by a discontinuity of this nature. As discussed in the previous section, the Mach wave produces a continuous regime of shocked states between the interface and normal Mach stem pressures. As such, the configuration provides a sensitivity to phase transitions over a large range of pressures, which is lost with one dimensional plane shock waves. While the resulting reflections appear to be very complex and depend on the nature of the phase transition, numerical simulations suggest a steady state is reached and it may be possible to detect phase transitions that conventional methods currently struggle to measure. As will be shown in Chapter 4, a complicated wave structure can arise in which a phase transition wave precedes the Mach reflection. In this case, proper interpretation of the properties of the reflections can be used to gain insights into the nature of the transition. Further discussion of the exact nature of the reflections and behavior of a material containing a phase transition will be relegated to future chapters where specific examples are given in detail.
Chapter 3
Experimental Method
The experimental techniques used to examine the material behavior in the Mach lens configuration are presented in this chapter. The first section details the loading system used to launch projectiles to velocities on the order of 2 km/s. Two different systems were used, both of which utilize gun powder to provide the necessary force to achieve these high velocities. The systems are deemed the Caltech and Sandia powder guns. The second section provides the details on the diagnostic used to monitor the propagation of the shock waves through the target. The VISAR and ORVIS techniques utilize optical interferometry to provide high resolution velocity information and are the primary methods used to provide quantitative information. The third and final section provides details on the target configuration. The targets are designed to provide the means to study the configuration in both the strong and weak confinement regimes and are optimized to obtain a steady state Mach reflection. The logistics of how the target is used in conjunction with the loading system and the diagnostics are also presented.