Shock-Change Equation

The shock-change equation is the relationship between derivatives of quantities in terms of x and t (or X and t) and derivatives of variables following the shock front, which moves with speed U into undisturbed material at rest. The planar shock front is assumed to be propagating in the x (Eulerian spatial coordinate) or X (Lagrangian spatial coordinate) direction, p dx = dX. 

In Eulerian coordinates x and t, the mass and momentum conservation laws and material constitutive equation are given by (u = = particle velocity,, = longitudinal stress, and p = material density) 

If we define D, to be the time-derivative operator following the shock path dxjdt = U t), then 

The jump conditions across a discontinuity moving into undisturbed material at rest are 

We define two quantities specifying the elastic shock response (the subscript H refers to conditions along the elastic Hugoniot curve) 

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Micromechanical Considerations in Shock Compression of Solids Table A.l. Shock-change equations (exact) A = (pcJpoUf. 

A summary of the shock-change equations for D,a is presented in Tables A.l and A.2 with the exact relationships given in Table A.l and the c, = approximation in Table A.2. 

For the simulations performed using the method described in this chapter, the rate at which the pressure at point B (denote this pressure pj) decreases can be determined using the so-called shock change equation. [15, 16] For purposes here, we assume the internal energy can now be expressed as e = ei p[x,i),v(x,t),X x,t) ) where A is a generalized reaction parameter... 

This approximate form of the shock change equation enables the estimation of pressure decay of the first wave using information that can be obtained directly from the simulations. The approximation of the p-v space path by more than one Rayleigh line in the case of volume decreasing reactions is justified when the Rayleigh lines do not change appreciably during the simulation, i.e. 

The Rayleigh line validity condition Eq. (27) can be shown to be valid for long wave propagation times. By considering a reaction rate of the form A = a(pj-p ), the shock change equation, Eq. (24) gives. 

In this chapter we define what is meant by a shock-wave equation of state, and how it is related to other types of equations of state. We also discuss the properties of shock-compressed matter on a microscopic scale, as well as discuss how shock-wave properties are measured. Shock data for standard materials are presented. The effects of phase changes are discussed, the measurements of shock temperatures, and sound velocities of shock materials are also described. We also describe the application of shock-compression data for porous media. 

Estimates of the Copper residual state can be made by assuming the material is at the same state as if it had achieved its final velocity by a single shock and then had rarefied to one atmosphere. The residual temperature of the Copper initially shocked to 830 kbar and then rarefied to a free-surface velocity of 0.3 cm/fisec is 768 K the residual density is 8.688 g/cc, comparable to the initial density of 8.903. Calculations were performed with Copper initial conditions of 8.903 g/cc and 300 K and of 8.688 g/cc and 768 K. Since the changed equation of state results in only slightly changed explosive shock pressure, the calculated results were insensitive to the Copper jet initial conditions. 

Grady and Asay [49] estimate the actual local heating that may occur in shocked 6061-T6 Al. In the work of Hayes and Grady [50], slip planes are assumed to be separated by the characteristic distance d. Plastic deformation in the shock front is assumed to dissipate heat (per unit area) at a constant rate S.QdJt, where AQ is the dissipative component of internal energy change and is the shock risetime. The local slip-band temperature behind the shock front, 7), is obtained as a solution to the heat conduction equation with y as the thermal diffusivity... 

It has been seen in deriving equations 4.33 to 4.38 that for a small disturbance the velocity of propagation of the pressure wave is equal to the velocity of sound. If the changes are much larger and the process is not isentropic, the wave developed is known as a shock wave, and the velocity may be much greater than the velocity of sound. Material and momentum balances must be maintained and the appropriate equation of state for the fluid must be followed. Furthermore, any change which takes place must be associated with an increase, never a decrease, in entropy. For an ideal gas in a uniform pipe under adiabatic conditions a material balance gives ... 

S2 — S is positive when Ma 1 > 1. Thus a normal shock wave can occur only when the flow is supersonic. From equation 4.96, if Mai > 1, then Ma2 < 1, and therefore the flow necessarily changes from supersonic to subsonic. If Mai = 1> Mai — 1 also, from equation 4.96, and no change therefore takes place. It should be noted that there is no change in the energy of the fluid as it passes through a shock wave, though the entropy increases and therefore the change is irreversible. 

Properties of the gas, such as the velocity, pressure, density and temperature, change by large amounts across the narrow shock wave. Although mass, energy and momentum are conserved across a shock wave, entropy is not. Entropy is created by a shock from supersonic to subsonic flow. The above analysis, comprising equations 6.95 to 6.100 and 6.101 to... 

This relationship, of course, will hold for a shock wave when q is set equal to zero. The Hugoniot equation is also written in terms of the enthalpy and internal energy changes. The expression with internal energies is particularly useful in the actual solution for the detonation velocity tq. If a total enthalpy (sensible plus chemical) in unit mass terms is defined such that... 

A shock wave is characterized by the entropy change across it. Using the equation of state for a perfect gas shown in Eq. (1.5), the entropy change is represented by... 

The change of state across the shock front is given by the adiabatic "Rankine-Hugoniot (R-H) Equation ... 

When you balance an equation, you change only the coefficients. Changing subscripts alters the chemical compounds themselves, and you can t do that. If your pencil were equipped with an electrical shocking device, that device would activate the moment you attempted to change a subscript while balancing an equation. 

In a shock wave the compression is tied to a change in entropy, the only source of which are the dissipative forces—viscosity and heat conductivity. In the calculation we obtain a negligible front depth and compression time in the shock wave. We emphasize that this is a result of the calculation, not an assumption necessary to write the conservation equations. 

If in the relevant state the material has the same chemical composition as in the initial state so that the functions I and I0 coincide (a shock wave without a change in the chemical composition), then the curve satisfying equation... 

The beginning of reaction in a detonation wave is related to compression and heating of the gas by a shock wave (the jump from A to C, Fig. 1 or 5). Let us consider the conditions of occurrence of the chemical reaction, accompanied by a change in the state which more or less closely follows the equation of the Todes line. 

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