Shock propagation

These equations are integrated from some initial conditions. For a specifiea value of. s, the value of x and y shows the location where the solution is u. The equation is semilinear if a and h depend just on x and y (and not u), and the equation is linear if a, h, and/all depend on X and y, but not u. Such equations give rise to shock propagation, and conditions have been derived to deduce the presence of shocks. Ref. 245. For further information, see Refs. 79, 159, 192, and 245. 

W. Band and G.E. Duvall, Physical Nature of Shock Propagation, Amer. J. Phys. 29, 780(1961). 

When an isotropic material is subjected to planar shock compression, it experiences a relatively large compressive strain in the direction of the shock propagation, but zero strain in the two lateral directions. Any real planar shock has a limited lateral extent, of course. Nevertheless, the finite lateral dimensions can affect the uniaxial strain nature of a planar shock only after the edge effects have had time to propagate from a lateral boundary to the point in question. Edge effects travel at the speed of sound in the compressed material. Measurements taken before the arrival of edge effects are the same as if the lateral dimensions were infinite, and such early measurements are crucial to shock-compression science. It is the independence of lateral dimensions which so greatly simplifies the translation of planar shock-wave experimental data into fundamental material property information. 

Figure 4.11. Diagrammatic sketches of atomic lattice rearrangements as a result of dynamic compression, which give rise to (a) elastic shock, (b) deformational shock, and (c) shock-induced phase change. In the case of an elastic shock in an isotropic medium, the lateral stress is a factor v/(l — v) less than the stress in the shock propagation direction. Here v is Poisson s ratio. In cases (b) and (c) stresses are assumed equal in all directions if the shock stress amplitude is much greater than the material strength.

Gupta [28] presents results on the effect of crystal orientation on shock propagation in LiF crystals. This work supports earlier studies and shows... 

A one-dimensional mesh through time (temporal mesh) is constructed as the calculation proceeds. The new time step is calculated from the solution at the end of the old time step. The size of the time step is governed by both accuracy and stability. Imprecisely speaking, the time step in an explicit code must be smaller than the minimum time it takes for a disturbance to travel across any element in the calculation by physical processes, such as shock propagation, material motion, or radiation transport [18], [19]. Additional limits based on accuracy may be added. For example, many codes limit the volume change of an element to prevent over-compressions or over-expansions. 

In this chapter the regimes of mechanical response nonlinear elastic compression stress tensors the Hugoniot elastic limit elastic-plastic deformation hydrodynamic flow phase transformation release waves other mechanical aspects of shock propagation first-order and second-order behaviors. 

In a numerical exercise described in section 4.2.2, it was shown that, for a stoichiometric, hydrocarbon-air detonation, the theoretical maximum efficiency of conversion of heat of combustion into blast is equal to approximately 40%. If the blast energy of TNT is equal to the energy brought into the air as blast by a TNT detonation, a TNT equivalency of approximately 40% would be the theoretical upper limit for a gas explosion process under atmospheric conditions. However, the initial stages in the process of shock propagation in the immediate vicinity of... 

Fig. 3.30 is believed to be in the Sedov phase, with the shock propagating at a high velocity (730 km s-1) in the rarefied intercloud ISM ( h 0.9 cm-3) and at a much slower velocity (< 140 km s-1) in nearby dense molecular clouds (nu 150 cm-3) where it gives rise to optical and UV emission. The X-ray continuum is a sum of recombination radiation from various ions plus thermal bremsstrahlung, aka free-free emission, the latter being given for an ion with charge Ze by... 

Fj) G.K. Adams M. Cowperthwaite, Explicit Solutions for Unsteady Shock Propagation in Chemically Reacting Media , Ibid, pp 502-11... 

The commonly accepted model for LVD is the so-called cavitation model. It is believed that precursor shocks, propagating thru the charge container, produce cavitation zones in the liq ahead of the LVD. The vapor bubbles (cavities) then act as sites for ignition and growth of chemical reaction to support the LVD wave (Ref 15)... 

For the accurate determination of detonation pressure (Pq), a technique of impedance mismatch is applied. The explosive is detonated in contact with water (its equation of state is known and is transparent which facilitates record of shock propagation by shadowgraphy technique). Then, after measuring the transmitted shock velocity in water, detonation pressure is calculated by Equation 3.12 ... 

W.W. Hillstrom (Ref 21) attempted to correlate the pyrophoricity of bulk metals with the kinetic energy at impact on a variety of targets, both projectile and target being characterized by Brinnel hardness. Such a study would be of value in the opinion of this author, if the shock propagation properties of the materials at the moment of impact had been used instead. Hillstrom s findings may be instructive (Table 5) ... 

Most of the simulations reviewed were performed in two dimensions, although some simulations of three-dimensional detonations are discussed in Sec. 3.1.2. Equilibrated molecular crystals for AB Model I in both two and three dimensions are shown in Fig. 2. This model has slightly longer bond lengths than either Model II or III, but otherwise the starting crystal structures are similar for all three models. To model an infinite crystal, almost all detonation simulations were carried out with periodic boundary conditions enforced perpendicular to the direction of the shock propagation. 

The sole exceptions axe discussed in Sec. 3.2, where free boundary conditions perpendicular to the direction of shock propagation were assumed so that critical widths in two dimensions could be studied. Spot checks were made to ensure that increasing the size of the repeated simulation cell did not significantly affect the results obtained with periodic boundary conditions. When the model material was taken as infinite in the direction of the... 

Fig. 19. Top Illustration of a nanocrack in an otherwise perfect AB crystal. The two types of atoms are shown as open and solid circles. The piston driven shockfront approaches the crack from the left. Periodic boundary conditions are used perpendicular to the direction of shock propagation. Bottom Front positions versus times for a series of crack widths. Piston velocity is 1.4 km/s in all cases.

Fig. 23. Components of the kinetic temperature. The diamonds and crosses are for the components parallel and perpendicular to the direction of shock propagation, respectively.

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