Dynamic loads on human and animal surrogates at different test locations in compressed-gas-driven shock tubes

Abstract

Dynamic loads on specimens in live-fire conditions as well as at different locations within and outside compressed-gas-driven shock tubes are determined by both static and total blast overpressure–time pressure pulses. The biomechanical loading on the specimen is determined by surface pressures that combine the effects of static, dynamic, and reflected pressures and specimen geometry. Surface pressure is both space and time dependent; it varies as a function of size, shape, and external contour of the specimens. In this work, we used two sets of specimens: (1) anthropometric dummy head and (2) a surrogate rodent headform instrumented with pressure sensors and subjected them to blast waves in the interior and at the exit of the shock tube. We demonstrate in this work that while inside the shock tube the biomechanical loading as determined by various pressure measures closely aligns with live-fire data and shock wave theory, significant deviations are found when tests are performed outside.

Appendix

1.1 Calculation of static, dynamic, stagnation, and reflection pressures during blast

Pressures are measured both in live-fire and shock tubes using pressure gauges with time resolution sufficient to capture rise times of the order of a few microseconds, and hence the sensors and the data acquisition system should be rated with 1 MHz or higher sampling frequency. If blast overpressure $p_0^{\mathrm{gauge}}$ is measured as side-on pressure (either in the field or shock tube walls), then they are typically expressed in terms of either kPa or psi. This $p_0^{\mathrm{gauge}}$ should be converted to bars using $p_0 =p_0^{\mathrm{gauge}}/101.32$ for kPa or $p_0 =p_0^{\mathrm{gauge}}/14.7$ for psi; thus, $p_0$ is in bars. In the literature, the suffix x typically indicates ambient un-shocked conditions, while the y denotes values within the shocked gas. In this Appendix, we are mainly concerned with air blast with ambient condition denoted by $T_x =15\,^{\circ }\hbox {C}=288\,\hbox {K}$; $\hbox {pressure}=1.01325$ bars; $\hbox {density}=1.225\,{\hbox {kg}}/{\hbox {m}^{3}}$ and at sea level.

Some of the basic quantities can be found using the following equations.

$$\begin{aligned} \frac{p_y }{p_x }= & {} \frac{p_0 +1.0132}{1.0132} \end{aligned}$$

(2)

$$\begin{aligned} M= & {} \sqrt{\frac{1}{7}\left( 1+6\frac{p_y }{p_x }\right) } \end{aligned}$$

(3)

$$\begin{aligned} \frac{T_y }{T_x }= & {} \frac{\left( 5+M^{2}\right) \left( 7M^{2}-1\right) }{36M^{2}}. \end{aligned}$$

(4)

Then the acoustic velocity in the shocked medium can be computed. Recall that velocity depends on the atomic vibration which depends on the temperature. If $a_x =340\,{\hbox {m}}/{\hbox {s}}$, then $a_y$ the acoustic velocity in the shock can be computed using

$$\begin{aligned} \frac{a_y }{a_x }=\sqrt{\frac{T_y }{T_x }}. \end{aligned}$$

(5)

The particle velocity in the shock tube which is the main cause of dynamic pressure can also be calculated. The particle velocity $u_\mathrm{p}$ just behind the shock front can be calculated using

$$\begin{aligned} u_\mathrm{p} =\frac{5\left( M^{2}-1\right) }{6M}a_x. \end{aligned}$$

(6)

Stagnation pressure $p_{\mathrm{stag}}$ is the absolute pressure, when a moving flow stream with particle velocity is brought to rest isentropically. It is given by

$$\begin{aligned} p_{\mathrm{stag}} =\left[ {1+\frac{\left( {\frac{u_\mathrm{p} }{a_x }} \right) ^{2}}{5\left( \frac{T_y }{T_x }\right) }} \right] \left( {\frac{p_y }{p_x }} \right) . \end{aligned}$$

(7)

For practical purposes $p_{\mathrm{stag}}$ can be assumed to be equivalent to $p_{\mathrm{tot}}$ the total pressure measured in gauge units using pencil probe. Equation (8) expresses $p_{\mathrm{stag}}$ in absolute units. In order to convert to gauge units for comparison purposes, we need to use $p_{\mathrm{stag}}^{\mathrm{gauge}} =p_{\mathrm{stag}} -1.0132$. Thus, we can compute dynamic pressure as $p_{\mathrm{dyn}} =p_{\mathrm{tot}} -p_{\mathrm{o}} \approx p_{\mathrm{stag}} -p_{\mathrm{o}}$. While $p_{\mathrm{tot}}$ can be measured using pencil probe, this is not the pressure one will measure on a flat object normal to the shock flow. When the fast-moving particles are suddenly brought to rest instantaneously, it initiates a reflected shock that travels against the oncoming incident shock. The pressure acting on the target is given by the reflected pressure and not total pressure. This reflected pressure $p_{\mathrm{ref}} =p_\mathrm{r}$ can be computed using

$$\begin{aligned} p_\mathrm{r} =\left[ {\frac{\left( {4M^{2}-1} \right) \left( {7M^{2}-1} \right) }{3\left( {M^{2}+5} \right) }} \right] p_x. \end{aligned}$$

(8)

Reflection ratio

$$\begin{aligned} \varLambda =\frac{p_\mathrm{r} -p_x }{p_y -p_x }=\frac{\hbox {Reflection overpressure}}{\hbox {Blast overpressure}}=\frac{8M^{2}+4}{M^{2}+5}.\nonumber \\ \end{aligned}$$

(9)