RD-V: 0540 Free Air Burst

Since the experimental data (UFC charts) use the unit system {cm, g, µs}, this study is conducted using the same unit system.

The free air burst corresponds to a blast wave model characterized by spherical wave propagation, without ground reflection (as shown in the animation below).

Here the scaled distance $Z$ (cm/g^1/3):

The scaled distance suggests that two charges with the same geometry, explosive composition, and ambient conditions will generate self-similar blast waves if their scaled distances match.

This allows the pressure to be computed from experimental data for a given charge ( $W$ ) and a given location ( $R$ ).

The equation used to compute the pressure in /LOAD/PBLAST is the modified Friedlander equation:

The profile of the modified Friedlander equation at a given location is:

In Radioss, the negative phase is ignored and $P^{-}$ is replaced by a $P_{min}$ value (defaults to 10^-20). In this case, and until the end of this study, the pressure is described only by:

• $P_{max}$ which replaces $P^{+}$ .
• $d{t}_0$ which replaces $t^{+}$ .
• $P_{min}$ which replaces $P^{-}$ .
• $t^{-}$ and $I^{-}$ vanish.
• $t_a$ is still the same

These two quantities result in the following formula for the pressure applied to the surface:

$$P_{blast}\left(t\right)=\left\{\begin{array}{c}{\cos }^2\left(\theta \right).{\text{P}}_{\text{r}}\left(\text{t}\right)+\left(1+\text{cos}\left(\theta \right)-2.{\cos }^2\left(\theta \right)\right).{\text{P}}_{\text{so}}\left(t\right)\\ P_{so}\left(t\right)\end{array}\right.\begin{array}{c}if\cos \left(\theta \right)>0\\ if\cos \left(\theta \right)\le 0\end{array}$$

Where, $\theta$ is the angle between the surface normal and the direction of the detonation point.

This corresponds to the measurement data obtained with the measuring instruments (UFC charts).

Previously, $I^{+}$ was introduced as the integral of the pressure with respect to time (positive or negative phase). The same process is applied to ${\text{P}}_{\text{r}}$ and $P_{so}$ , with the impulses denoted by ${\text{I}}_{\text{r}}$ and $I_{so}$ . In the charts, ${\text{i}}_{\text{r}}$ and $i_{so}$ are introduced as the impulse divided by $W^{1/3}$ ; the same applies to $d{t}_0$ and $t_a$ .

Model Description

The boundary conditions are represented below:

The two models are tested with 2D shell elements using the /PROP/TYPE1 (SHELL) property and the /MAT/LAW1 (ELASTIC) material, with a rigid body attached. The elements are placed at distance of $R$ from the detonation point ( $W$ is imposed).

The first goal is to compare ${\text{P}}_{\text{r}}$ , $P_{so}$ , ${\text{I}}_{\text{r}}$ , $I_{so}$ , and $d{t}_0$ and compare UFC charts and Radioss outputs. To achieve this, $W$ is imposed $\left(W={500.10}^{3}g\right)$ and only $R$ is varied. The second goal is to evaluate formula Equation 3 for every 10 degrees from 0 degrees to 180 degrees for different $Z$ values.

Points Considered

UFC Quantities Evaluation

For each point in the table, $P_{so}$ , ${\text{P}}_{\text{r}}$ , $I_{so}$ , ${\text{I}}_{\text{r}}$ and $d{t}_0$ are measured from the pressure profile on the 2D shell elements.

The maximum error obtained is 3.82% on ${\text{I}}_{\text{r}}$ for $Z$ = 0.5 and the average error across all data is 0.06%.

These errors are due to interpolation between charts data points.

Formula Evaluation

For point B, D and H, the formula is evaluated every 10 degrees from 0 degree to 180 degrees:

$$P_{blast}\left(t\right)=\left\{\begin{array}{c}{\cos }^2\left(\theta \right).{\text{P}}_{\text{r}}\left(\text{t}\right)+\left(1+\text{cos}\left(\theta \right)-2.{\cos }^2\left(\theta \right)\right).{\text{P}}_{\text{so}}\left(t\right)\\ P_{so}\left(t\right)\end{array}\right.\begin{array}{c}if\cos \left(\theta \right)>0\\ if\cos \left(\theta \right)\le 0\end{array}$$

The average error obtained is 0.04% with a maximum error of 0.10%.

Again, these errors are due to interpolation between charts data points.

I_z Option Evaluation

To evaluate I_z, a displacement is imposed from $Z = 1 cm/g^{\left(1/3\right)}$ (point B) to $Z = 2 cm/g^{\left(1/3\right)}$ (point V) and the detonation is launched after the displacement.

For I_z = 1, since $R$ is constant and $Z$ is evaluated at time = 0, the pressure ( ${\text{P}}_{\text{r}}$ ) obtained should correspond to $Z = 1$ , even after the displacement.