Shock profile

Fig. 12. (a) Development of the physically unreasonable overbanging concentration profile and the corresponding shock profile for adsorption with a favorable isotherm and (b) development of the dispersive (proportionate pattern) concentration profile for adsorption with an unfavorable isotherm (or for... 

Within the elastic regime, the conservation relations for shock profiles can be directly applied to the loading pulse, and for most solids, positive curvature to the stress volume will lead to the increase in shock speed required to propagate a shock. The resulting stress-volume relations determined for elastic solids can be used to determine higher-order elastic constants. The division between the elastic and elastic-plastic regimes is ideally marked by the Hugoniot elastic limit of the solid. 

Pressure and Ground Shock Profile Measurements , Final Rept, Proj No LN-304, Denver Res Inst, Menlo Park, Ca (1969) 75) D.L. 

The data presented above shows that both the two- and three-dimensional simulations yield similar results in terms of shock profiles and in... 

S. Couturier, T. deResseguier, M. Hallouin, J.P. Romain, and F. Bauer, Shock profile induced by short laser pulses. J. Appl. Phys. 79(12), pp. 9338-9342 (1996)... 

Numerical tanh/shock profiles correspond to case B where the level in the energy production dissipation balance in Equation 4.3 is predominant compared with the viscous-free BKdV terms. 

Figure 5 Shock profile at 8 ps after arrival at the free surface for several incident 110 fs pulse length...

A similar analysis can be performed for the shocks in the model Navier-Stokes equation l(. It is noticeable that the value of the parameter B remains the same, but the Navlei> -Stokes profile coincides with the Boltzmann shock profile for weak shocks only. 

FIGURE 83 (a) Development of physically unrealistic overhanging concentration profile and corresponding shock profile for a favorable isotherm fi< l.O). (6) Development of dispersive (proportionate-pattern) concentration profile for desorption of a uniformly saturated bed when the isotherm is favorable for adsorption, unfavorable for desorption < 1.0 > 1.0) (after... 

For the reasons given above, a number of authors [23-28] have applied MD to shock wave studies in an effort to obtain details of the various shock compression processes that are not easily available from the conventional continuum method. We have carried out calculations of the shock compression of one-, two- and three-dimensional systems in both solid and liquid phases [29,30], using essentially the same model as in Fig. 1. Here I shall first summarize the general features of the shock profile from our studies, then I shall discuss one representative case, with special reference to the thermal relaxation problem, as an illustration of some of the general results. 

Thermal relaxation occurs behind the shock front to restore thermal equilibrium. In a condensed system, this involves the reestablishment of the equilibrium distribution of both the frequencies and the velocities of the particles, i.e., their potential and kinetic energy distributions. In the simplest case without structural relaxation and the accompanying stress relaxation, we expect this process to occur at the appropriate second sound velocity. In the fully relaxed region the kinetic and potential energy distributions are in equilibrium. In the relaxing region these distributions are not steady. The shock profile as a whole is therefore unsteady. 

The foregoing results showing an unsteady shock wave profile apply to the thermal equilibration in the shock profile in condensed systems. If structural relaxation also occurs behind the shock front (as in cases A, B, C in Fig. 5, and in Ref. [33]) and if the relaxation process is slow compared with the propagation of the shock wave, then we would expect to observe additional energy exchange processes accompanied by further thermal relaxation in the shock profile. Even in a dense Lennard-Jones liquid, we found that the liquid structure did not relax quickly to a hydrostatic state under shock compression, and we observed thermal relaxation similar to that displayed in Fig. 6, but with a smaller overshoot in the kinetic energy after the shock front [30]. 

The effect of an unsteady shock profile may be small or large depending on the information being sought. For example, the unsteady profile does not have much effect on the nearly linear relationship between and U. On the other hand, for given and (the quantities actually measured), the derived pressure at a given density may be in considerable error. 

Finally, as we have noted, the non-uniform distribution of the difference in the stress components in Fig.11c indicated that structural relaxation was occurring locally in the shock profile. From a time sequence of these profiles, it was possible to determine that the relaxation process was slow compared with the motion of the shock front. At higher compression, structural relaxation occurred with greater difficulty and therefore more slowly, because of the greater barriers to atomic motion with increased compression. Thus we observed that the dislocations... 

The displaced BKW isentrope that will describe the observed plate dent, aquarium water shock profiles, explosive interface, and the detonation velocity of X0233 exhibits a weak detonation behavior. 

To investigate the effect of the interaction of a matrix of holes with a multiple shock profile, a matrix of 10% air holes located on a hexagonal close-packed lattice in TATB was modeled. The spherical air holes had a diameter of 0.004 cm. The initial configuration is shown in Figure 3.39. The three-dimensional computational grid contained 16 by 22 by 36 cells each 0.001 cm on a side. The time step was 0.0002 /rsec. Figure 3.40 shows the density and mass fraction cross sections for a 40 kbar shock wave followed after 0.045 /rsec by a 290 kbar shock wave interacting with a matrix of 10% air holes of 0.004 cm diameter in TATB. 

The elastic-plastic treatment in 2DE was compared to one-dimensional, Lagrangian SIN calculations of a 0.3175 cm thick Aluminum plate initially traveling 0.0412 cm/nsec driving a 32.5 kbar shock into 1.27 cm of Aluminum. The yield was 2.5 kbar, the shear modulus was 0.25, and FLAP (see Appendix A) was 0.05 for both calculations. The shock profiles at 1 and 2 nsec are shown in Figure C.l using the same number of cells (100) in the SIN and 2DE calculations. The Eulerian calculations are more smeared, but the magnitude of the elastic component of the rarefaction is reproduced. The elastic-plastic treatment in the Eulerian code gives realistic, if more smeared, results. 

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