Hugoniot curves

Thermodynamic Effects of Shock Compression and the Hugoniot Curve... 

The Rankine-Hugoniot curve is sometimes referred to as the shock adiabat (especially in the Soviet literature). This terminology reflects the fact that the shock process is so fast that there is insufficient time for heat... 

By plotting Hugoniot curves in the pressure-particle velocity plane (P-u diagrams), a number of interactions between surfaces, shocks, and rarefactions were solved graphically. Also, the equation for entropy on the Hugoniot was expanded in terms of specific volume to show that the Hugoniot and isentrope for a material is the same in the limit of small strains. Finally, the Riemann function was derived and used to define the Riemann Invarient. 

Constitutive relation An equation that relates the initial state to the final state of a material undergoing shock compression. This equation is a property of the material and distinguishes one material from another. In general it can be rate-dependent. It is combined with the jump conditions to yield the Hugoniot curve which is also material-dependent. The equation of state of a material is a constitutive equation for which the initial and final states are in thermodynamic equilibrium, and there are no rate-dependent variables. 

Hugoniot curve A curve representing all possible final states that can be attained by a single shock wave passing into a given initial state. It may be expressed in terms of any two of the five variables shock velocity, particle velocity, density (or specific volume), normal stress (or pressure), and specific internal energy. This curve it not the loading path in thermodynamic space. 

Rayleigh line A chord that connects the initial state of a material on its Hugoniot curve to the final state on the curve. Most frequently drawn in the P-F plane. 

Figure 4.2. Pressure-volume compression curves. For isentrope and isotherm, the thermodynamic path coincides with the locus of states, whereas for shock, the thermodynamic path is a straight line to point Pj, V, on the Hugoniot curve, which is the locus of shock states.

Hugoniot curves, such as those depicted in Figs. 4.3 and 4.4, may be transformed from the pressure-volume plane to the pressure-particle velocity plane (Fig. 4.5) using... 

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

The major difficulty in applying this hydrodynamic theory of detonation to practical cases lies in the calculation of E2, the specific internal energy of the explosion products immediately behind the detonation front, without which the Rankine-Hugoniot curve cannot be drawn. The calculations require a knowledge of the equation of state of the detonation products and also a full knowledge of the chemical equilibria involved, both at very high temperatures and pressures. The first equation of state used was the Abel equation... 

Extension of the hydrodynamic theory to explain the variation of detonation velocity with cartridge diameter takes place in two stages. First, the structure of the reaction zone is studied to allow for the fact that the chemical reaction takes place in a finite time secondly, the effect of lateral losses on these reactions is studied. A simplified case neglecting the effects of heat conduction or diffusion and of viscosity is shown in Fig. 2.5. The Rankine-Hugoniot curves for the unreacted explosive and for the detonation products are shown, together with the Raleigh line. In the reaction zone the explosive is suddenly compressed from its initial state at... 

The Hugoniot curve shows that in the deflagration region the pressure change is very small. Indeed, approaches seeking the unique deflagration velocity assume the pressure to be constant and eliminate the momentum equation. 

From Eq. (5.13), it can be considered that the pressure differential generated is proportional to the heat release q. If there is no heat release (q = 0), Px = I 2 and the Hugoniot curve would pass through the initial point A. As implied before, the shock Hugoniot curve must pass through A. For different values of q, one obtains a whole family of Hugoniot curves. 

Thus far in the development, the deflagration, and detonation branches of the Hugoniot curve have been characterized and region V has been eliminated. There are some specific characteristics of the tangency point J that were initially postulated by Chapman [7] in 1889. Chapman established that the slope of the adiabat is exactly the slope through J, that is,... 

The proof of Eq. (5.15) is a very interesting one and is verified in the following development. From thermodynamics one can write for every point along the Hugoniot curve... 

It follows from Eq. (5.17) that along the Hugoniot curve,... 

The subscript H is used to emphasize that derivatives are along the Hugoniot curve. Now, somewhere along the Hugoniot curve, the adiabatic curve passing through the same point has the same slope as the Hugoniot curve. There, ds2 must be zero and Eq. (5.18) becomes... 

But notice that the right-hand side of Eq. (5.19) is the value of the tangent that also goes through point A therefore, the tangency point along the Hugoniot curve is J. Since the order of differentiation on the left-hand side of Eq. (5.19) can be reversed, it is obvious that Eq. (5.15) has been developed. 

With the condition that u2 = c2 at J, it is possible to characterize the different branches of the Hugoniot curve in the following manner ... 

Now [d2P2ld( lp2)2 > 0 everywhere, since the Hugoniot curve has its concavity directed toward the positive ordinates (see formal proof later). 

When q = 0, the Hugoniot curve represents an adiabatic shock. Point 1 (Pi, p ) is then on the curve and Y and J are 1. Then [(l/pt) - (Ilp2) = 0, and the classical result of the shock theory is found that is, the shock Hugoniot curve osculates the adiabat at the point representing the conditions before the shock. 

Along the detonation branch of the Hugoniot curve, the variation of the entropy is as given in Fig. 5.4. For the adiabatic shock, the entropy variation is as shown in Fig. 5.5. 

From this equation for the pressure, it is obvious that the Hugoniot curve is a hyperbola. Its asymptotes are the lines... 

The element may enter the wave in the state corresponding to the initial point and move directly to the C-J point. However, this path demands that this reaction occur everywhere along the path. Since there is little compression along this path, there cannot be sufficient temperature to initiate any reaction. Thus, there is no energy release to sustain the wave. If on another path a jump is made to the upper point (1 ), the pressure and temperature conditions for initiation of reaction are met. In proceeding from 1 to 1, the pressure does not follow the points along the shock Hugoniot curve. 

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