Theory of shock waves and introduction to gas dynamics/Chapter 22

by Yakov Zeldovich

Chapter 22

Simulation and Similarity of Destructions Caused by Shock Waves

Destruction occurs when the stress in a material reaches limit values. Similarity will therefore be achieved if we use the same material in the model as in the actual case, and, of course, if the model is geometrically similar to the actual object.

By using the same material we will be sure to have a similarity in the propagation of the shock wave, in its transition from one medium to the other, and so on. We have seen that the characteristic pressure amplitude is constant. In similar explosions the pressures are identical at similar points.

The regions in which the stresses caused by the explosion exceed the permissible values and bring about the destruction of the material will also be similar.

Destruction requires that a specific deformation be reached, i.e., that certain particles of the body be shifted with respect to other particles. Inertial forces and elasticity prevent deformation and destruction from occurring instantly. Could it be that the existence of a specific deformation time will lead to a violation of similarity?

But we can easily see that similarity will be maintained. It is precisely the inertia of the substance, which depends on density, and its elasticity that determine the speed of sound in the substance. It can be formally shown by means of analysis that from density and elasticity we can plot deformation time only on the basis of the dimensions of the body, and this will be the time required by the wave to pass through the body. The time will turn out to be proportional to the size of the body. If we change the scale, deformation time changes following the same law as the one governing the shock wave action time, and the relationship between the times will remain constant. This ensures similarity of the phenomena.

Similarity is also applicable to the more complex type of destruction, in which the shock wave momentum is decisive (see Chapter 20) rather than peak pressure.

Let us take an elastic beam, the oscillation period of which exceeds shock wave action time.
By reducing the dimensions of the charge, the beam and the distance between them by a factor of $n$, the oscillation period of the beam will decrease by a factor of $n$, and the frequency will increase by a factor of $n$. This can readily be verified with the aid of elasticity theory for any specific practical method of securing the beam.

The mass has decreased by a factor of $u^{3}$, at a similar point the shock wave momentum per unit of surface has decreased by a factor of $n$ on account of a decrease of the shock wave width and a decrease in shock wave action time at a constant peak pressure, and the surface receiving the pressure has decreased by a factor of $n^{2}$. Thus linear velocity reached by the beam as a result of the effect of pressure momentum will be independent of the size of the beam. The amplitude of the oscillations will be of the order of the product of velocity $\times$ period, i.e., it will be proportional to the size of the beam. Hence we see that the relative deformation and density of elastic energy proportional to the square of initial velocity are identical in the model and in actuality. The result will also be identical, namely, the presence or the absence of destruction. Let us note that similarity will not be violated by friction which depends on velocity and on the load in the case when the load is also assigned in the fundamental shock wave action, since velocity and pressure are the same in similar systems.

A less trivial case is the one frequently encountered in structural mechanics. It is the case in which the stability of the structure and the effort required for its destruction depend on the structure's weight. The simplest instance of this kind is the sandy area without cohesion. Another instance is a stack of bricks, the solidity of which depends on the weight of the bricks and on the friction produced by the pressure of one brick on the other. Khariton emphasizes that such a type of stability very frequently determines the resistance of a structure to destruction. The stack of bricks represents one extreme example in which the weight determines internal cohesion. A solid steel box, which is easier to topple as a whole than to destroy, is another extreme example in which the explosion works against the force of gravity.
Here the impossibility of a strict similarity is obvious. The theory now includes acceleration of gravity $g$ expressed in terms of length/time$^{2}$. Together with the characteristic velocities of the explosion process, e.g., $c_{0}$, the presence of $g$ permits the plotting of the length, e.g., $c_{0}^{2}/g$ and time $c_{0}/g$. The absence of similarity is obvious: if we compare two charges of different size buried in the sand at an appropriate depth, we can see that the pressure of the soil at the level of the charge is proportional to the depth, and to the size of the charge. Likewise, minimum pressure required for the toppling of a wall in the second example is also proportional to the size. However, atmospheric pressure and blast pressure do not depend on the size.

Thus, with a change in size there is also a change in the ratio of soil pressure or the pressure required for the beginning of destruction to blast pressure, and similarity is therefore violated.

An excellent simulation method was proposed by Pokrovskiy [109]. To obtain similarity as we change the scale of the experiment, we must also change the length proportionally. Pokrovskiy obtains this by changing acceleration, and replacing gravity with centrifugal force. The model is exploded on a centrifuge and the dimensions are reduced with respect to nature at the same ratio of centripetal acceleration to acceleration of gravity. We can readily verify that soil pressure at similar depths will be similar.

Pokrovskiy made extensive use of his method for the purpose of modeling large scale explosions for excavation, and also for the purpose of studying the effect of various soils and different positions of the charge on the result of explosions. The linear modeling scale in his experiments reached 29, i.e., all the dimensions of the model were reduced bya factor of 29 as compared with the dimensions of the real object. The weight of the charge, which characterized the cost of the experiment, was reduced by a factor of 25,000.

Zel'dovich and Khariton proposed an approximate method for simulating the work of explosives against the forces of gravity. It is based on the fact that the new criterion on which depends the absence of similarity in the case of a change in scale, differs appreciably from unity. Thus, if we write this criterion as a ratio of characteristic length $c_{0}^{2}/g$ to the size of the charge $R$, then for a charge weighing 1 kg we get $c_{0}^{2}/gR=2\times 10^{5}$. The ratio of static soil pressure to blast pressure yields, at a crater depth of several meters, a quantity of the order of $10^{-4}$-$10^{-5}$. Thus, the criterion in the most varied formulations turns out to be sharply different from unity. This means that we are dealing here with the case in which not all the quantities are of the same order. It is obvious that we find ourselves in the domain of extreme or critical laws, in a domain, that is, in which we may expect self-simulation in the same way as self-simulation arises in hydrodynamics at very high or very low, Reynolds Numbers.

We now have to find the physical nature of this self-simulation.

Let us give a closer look to the toppling of a wall (see Fig. 56). At the beginning of the preceding chapter we brought it up as an instance for a process which lasts considerably longer than the action of the shock wave (in this case the time ratio yields another criterion which sharply differs from unity), i.e., a process in which the decisive role is played by the general wave momentum. We divided the process into two stages: 1) the action of the wave on the object which determines its momentum, and 2) the motion of the object by inertia, which overcomes the force of gravity, and we readily find the conditions for similarity.

In fact, the object's momentum $K$, equal to the force momentum, (for a geometrically similar change of the system, in which the dimensions of the object and the distance between the object and the charge change proportionally to the dimension of charge $R$) is proportional to

$K\sim Fi\sim R^{2}{\frac {p_{0}}{c_{0}}}R\sim {\frac {\rho _{0}}{c_{0}}}R,$

(XXII-1)

where $F$ is the area on which the wave acts, $i$ is the pressure momentum per unit of surface. The momentum of the object sufficient for its toppling will be determined as follows. The object's kinetic energy is equated to the work required for lifting the gravity center of the object to a height proportional to the size of the object,

$E\sim {\frac {K^{2}}{M}}\sim MgR.$

(XXII-2)

Into $K$ of Eq. (XXII-1) we substitute the expression of the object's mass $M$ by the characteristic dimension $R$ and the object's density $\varphi$, and get

$M\sim \rho R^{3};\quad {\frac {p_{0}^{2}}{c_{0}^{2}}}{\frac {(R^{3})^{2}}{\rho R^{3}}}\sim \rho R^{3}gR;\quad {\frac {c_{0}^{2}}{p_{0}^{2}}}\rho ^{2}gR={\text{idem}}.$

(XXII-3)

The sign idem adopted in similarity theory signifies that similarity will take place if the term on the left remains constant. For all explosions in the air under normal conditions $c_{0}=$const, $p_{0}=$const, the criterion is simplified and $\rho ^{2}gR=$idem.

This criterion also includes exact simulation - the change in acceleration $g$ is inversely proportional to the size $R$ (centrifugal simulation). But on the basis of the approximations made earlier we obtained a criterion which also admits another solution: the change in density is inversely proportional to the root of its dimensions. This method was proposed by Khariton and this author [105]. This method allows for a sufficiently wide change in the scale. By substituting a material with density 2 (stone) with a material with density 11 (lead) it becomes possible to reduce $R$ by a factor of 30, which corresponds to the reduction of the charge by a factor of 27,000, i.e., it is possible to simulate the explosion of 1 ton of explosive by the explosion of 40 g of the same substance.

In Khariton's many experiments, the edgewise standing bricks turned out to be convenient indices for the distance at which the momentum of a shock wave drops to a specific value.

It is obvious that centrifugal simulation is necessary in more complex cases in which, along with a rigid structure, the soil also plays a role. The approximate simulation by changing density, as proposed by Khariton and this author, is considerably narrower in scope and the advantage of this method is only the simplicity of experimentation.

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