Shock wave propagation properties in dry and wet sand

Let us start with dry sand and see how the velocity and pressure waves travel through the first 50 mm of attenuation material. The velocity and pressure profiles are depicted at different points in time in Figure 5.1, showing how the wave evolves as it propagates through the damping material.

Figure 5.1 Shock wave propagation in dry sand

We see that the maximum velocity first increases a little bit, but then starts decreasing. In the beginning, the dry sand obtains a velocity of more than 100 m/s. The pressure increases slightly first, but then also falls off. Note that the position of the gas/dry sand interface, initially at x = 0 mm, moves slightly since the sand acquires such a high velocity.

So, how does the wet sand velocity and pressure profiles compare? This is shown in Figure 5.2.

Figure 5.2 Shock wave propagation in wet sand

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It is clear that there is a huge difference in the behaviour of the shock waves in dry and wet sand.

In wet sand, both the pressure and velocity wave have an almost constant amplitude and the wet sand acquires a much lower velocity than the dry sand, so the gas/sand interface does not move much.

If the 50 mm of attenuation material is followed by a steel bar (as in our example in Chapter 4), the properties of the shock wave at the steel bar interface will depend strongly on what kind of material it has passed through. It is not immediately obvious whether the dry or wet sand will transfer the highest stress pulse to the steel bar. However, by going back to Figure 4.12 we see that the low pressure dry sand impacting at high velocity gives a slightly higher stress amplitude in the bar than the high pressure wet sand impacting at low velocity.

But, what if the damping material thickness had been 500 mm? How do the waves continue to propagate in both materials? Let us examine that as well. The results for dry sand are shown in Figure 5.3 and in Figure 5.4 for wet sand.

Figure 5.3 Shock wave propagation in dry sand (500 mm thickness)

Figure 5.4 Shock wave propagation in dry sand (500 mm thickness)

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We see the same tendency. For dry sand, the velocity and pressure decreases, whereas for wet sand the velocity and pressure are roughly constant as the wave propagates. We also notice that for 500 mm damping material, the velocity of the dry sand has dropped to roughly the same as the wet sand wave. However, the pressure in the dry sand is much smaller.

What happens now to the shock wave if the 500 mm attenuation material is followed by a steel bar? Again, this is not obvious, but from Figure 4.12 we see that now the dry sand gives a much lower amplitude in the steel bar than the wet sand.

The propagation properties of the shock waves in dry and wet sand are summed up in Figure 5.5, which shows the velocity and pressure amplitude as a function of how far the waves have travelled.

Figure 5.5 Shock wave properties in dry and wet sand.

So, a picture of what is going on is starting to emerge. A porous material is quite easily

compacted and when impacted by a shock wave it therefore acquires a high velocity (even more so if the initial density is low). However, as the shock wave travels through the porous medium, both the pressure and velocity amplitudes fall quite quickly. This is very different from the situation in a non-porous medium, where the wave will travel at roughly constant amplitude.

(This is of course standard wave propagation theory).

If the damping thickness is small, a porous material acquires a much higher velocity than a non-porous material because the velocity does not “have time to fall”. This answers the earlier question of why we get higher stress for thin layers. If the damping material obtains such a high velocity that the subsequent impact generates a higher stress than the blast wave would have done without the damping material, then we have this situation. (It is not obvious how to calculate analytically the condition for this to happen). However, for thick layers, both velocity and pressure have fallen off considerably before the wave has propagated through the porous damping material, leading to low amplitude stress waves in the steel bar compared with a non-porous material. Pressure amplitude in damping material

Dry sand Wet sand

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One question that remains is why the porous material behaves so differently from the non-porous materials. Why does both the pressure amplitude and velocity amplitude fall as the wave

propagates through the material? This is an important question and we will therefore look into it in some detail.

The behaviour of a shock wave in a porous material is best understood by looking at a simpler wave than what is generated by an explosive. To illustrate things we will therefore examine the propagation of a square pulse through our dry sand. This can easily be set up in a 1D-situation using a velocity boundary condition in AUTODYN. The behaviour of such a wave is shown in Figure 5.6, where the pressure and velocity profiles are shown at different points in time, illustrating how they propagate through the porous dry sand.

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Figure 5.6 Square shock wave propagating through dry sand.

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We see that both the pressure and velocity have a square shape initially. Now, look at the

situation a little later. The important thing is that after the loading has stopped, an unloading wave propagates through the material, reducing the pressure quite significantly and also the velocity slightly. Further, the propagation velocity of this unloading wave is higher than the velocity of the shock front. Thus, the unloading wave will eventually catch up with the shock front. The illustrations show this happening as well as what happens next. The unloading wave is reflected as a loading wave, going back through the material increasing the pressure and further reducing the velocity. The whole process then repeats once this wave reaches the back of the rear end of the material and reflects as an unloading wave (last two plots in Figure 5.6).

The end result, after several reflections, is a wave of long duration and relatively low velocity and pressure. As we saw in Figure 5.5, this is typical for a shock wave in a porous material. It all comes down to the difference in loading and unloading properties for porous materials. As mentioned in Chapter 4.2.2, when a porous material unloads it follows a linear path in a pressure-density diagram instead of going back to the initial state. Since the wave propagation velocity depends on the slope of the loading/unloading in such a diagram, it follows that unloading waves will travel much faster than loading waves in a porous material. This is the big difference with a non-porous material, where loading and unloading have the same slope and loading and

unloading waves therefore travel at the same velocity.

But, what happens if the loading has such a long duration that the wave has already propagated through the whole porous material before unloading begins? An alternative way of saying this is that the damping thickness is too small compared with duration of the loading. This is exactly the situation we have already encountered in Chapter 4, where the whole material moves at a very high velocity and we receive shock enhancement instead of damping.