RD-V: 0550 Surface Burst

Evaluation of the /LOAD/PBLAST option for surface burst blast waves. This analysis focuses on evaluating the /LOAD/PBLAST keyword by comparing Radioss results with surface burst experimental data (Exp_data = 2).

Figure 1. UFC quantities evaluation model schematic for `Ishape` =1

Figure 2. UFC quantities evaluation model schematic for `Ishape` =2

Figure 3. UFC quantities evaluation model schematic for `Ishape` =3

The analysis is carried out using 4 different FE Radioss models:

Two 2D elements /SHELL with /PROP/TYPE1 (SHELL), with a Rigid Body attached, and correctly positioned for a given scaled distance, for Ishape =1.
Three 2D elements /SHELL with /PROP/TYPE1 (SHELL), with a Rigid Body attached, and correctly positioned for a given scaled distance, for Ishape =2.
One 2D element /SHELL with /PROP/TYPE1 (SHELL), with a Rigid Body attached, and correctly positioned for a given scaled distance, for Ishape =3.
Nineteen 2D elements /SHELL with /PROP/TYPE1 (SHELL), with a Rigid Body attached to all of them, correctly positioned at the same scaled distance, each rotated successively by an angle $θ$ of 10 degrees.

Options and Keywords Used

Material: /MAT/LAW1 (ELAST)
Property: /PROP/TYPE1 (SHELL)
Element: /SHELL and /RBODY
Boundary conditions: /BCS and /LOAD/PBLAST
Sets: /SET/GENERAL and /SURF/SEG

Input Files

Before you begin, copy the file(s) used in this problem to your working directory.

RD-V-0550_Surface_Burst.zip

Surface Burst

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

The surface burst corresponds to a blast wave model characterized by different types of wave propagation, taking into account ground reflection (as shown in the animations below):

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

Z = \frac{R}{W^{\frac{1}{3}}}

Where,

$R$: Distance from the center of the detonation charge to the target (cm)
$W$: TNT-equivalent mass (g)

The scaled distance suggests that two charges with the same geometry, explosive composition, and ambient conditions will generate self-similar blast waves when 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:

P (t) = P^{+} . e^{- b . \frac{t - t_{a}}{t^{+}}} (1 - \frac{t - t_{a}}{t^{+}})

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

Where,

$I^{+}$: The impulses (integral of pressure with respect to time on the interval of time $t^{+}$ )
$t_{a}$: The propagation time of the blast wave from the charge to the target

In Radioss, the negative phase is ignored and

P^{-}

is replaced by a

P_{m i n}

value (defaults to 10^-20). In this case and until the end of this study, the pressure is only described by:

$P_{m a x}$ which replaces $P^{+}$ .
$d t_{0}$ which replaces $t^{+}$ .
$P_{m i n}$ which replaces $P^{-}$ .
$t^{-}$ and $I^{-}$ vanish.
$t_{a}$ is still the same

Figure 10. Blast wave pressure profile implemented in Radioss

The blast pressure computed in /LOAD/PBLAST is fitted using experimental data (UFC charts) which are categorized in two quantities:

The incident wave referred to as the “side on” wave (abbreviation: so), $P_{s o}$ , where the captor/surface normal is perpendicular to the direction of the blast wave.
The reflected wave $P_{r}$ , where the captor/surface normal points towards the blast wave.

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

P_{b l a s t} (t) = \{\begin{matrix} \cos^{2} (θ) . P_{r} (t) + (1 + cos (θ) - 2. \cos^{2} (θ)) . P_{so} (t) \\ P_{s o} (t) \end{matrix} \begin{matrix} i f \cos (θ) > 0 \\ i f \cos (θ) \leq 0 \end{matrix}

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

From Equation 3:

If $θ = 0$ , $P_{r}$ is obtained.
If $θ \geq \frac{π}{2}$ or 90 degrees, $P_{s o}$ is obtained.

This corresponds to the measurement data obtained with the captors (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

P_{r}

and

P_{s o}

, with the impulses denoted by

I_{r}

and

I_{s o}

. In the charts,

i_{r}

and

i_{s o}

are introduced as the impulse divided by

W^{1 / 3}

; the same applies to

d t_{0}

and

t_{a}

Figure 12. Surface burst quantities graph

Model Description

The boundary conditions are represented below:

Figure 13. All boundary conditions applied to the shells for the first three models

Figure 14. All boundary conditions applied to the fourth model

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).

Shell and material properties:

Properties: Value
Density: 7.8 g/cm³
Young's modulus: 2.1 Mbar
Poisson ratio: 0.3
Thickness: 1 cm

The first goal is to compare $P_{r}$ , $P_{s o}$ , $I_{r}$ , $I_{s o}$ , and $d t_{0}$ from the Radioss outputs and the UFC charts, for each value of Ishape. To achieve this, $W$ is imposed $(W = {500.10}^{3} g)$ 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.

Results

Points Considered

The UFC quantities are evaluated for every point listed below. The formula (Equation 3) is evaluated for Points B, D and H and the I_z option is evaluated for Points B and H:

Table 1. List of Points Considered
Point	$R$ (cm)	$W$ (g)	$Z$ (g/cm^1/3)
A	142.866095	500000	1.8
B	190.488126	500000	2.4
C	253.984168	500000	3.2
D	380.976252	500000	4.8
E	555.590368	500000	7
F	793.700526	500000	10
G	1269.92084	500000	16
H	1984.25131	500000	25
I	2936.69195	500000	37
J	3968.50263	500000	50
K	5952.75394	500000	75
L	7937.00526	500000	100
M	9921.25657	500000	125
N	12699.2084	500000	160
O	15874.0105	500000	200
P	20636.2137	500000	260
Q	25398.4168	500000	320
R	31430.5408	500000	396

Ishape = 1

UFC Quantities Evaluation

For each point in the table,

P_{s o}

P_{r}

I_{s o}

I_{r}

and

d t_{0}

are measured from the pressure profile on the 2D shell elements located on the ground and above the charge (Figure 1):

Figure 19. $d t_{0} / W^{\frac{1}{3}}$ evaluation

The maximum error obtained is 0.45% and the average error across all data is 0.06%.

These errors are due to interpolation between charts data points and the discretization of the time pressure response.

Formula Evaluation

For point A, F and L, the formula is evaluated every 10 degrees from 0 degree to 180 degrees:

P_{b l a s t} (t) = \{\begin{matrix} \cos^{2} (θ) . P_{r} (t) + (1 + cos (θ) - 2. \cos^{2} (θ)) . P_{so} (t) \\ P_{s o} (t) \end{matrix} \begin{matrix} i f \cos (θ) > 0 \\ i f \cos (θ) \leq 0 \end{matrix}

Figure 20. Formula evaluation for point A

Figure 21. Formula evaluation for point F

Figure 22. Formula evaluation for point L

The average error obtained is 0.08% with a maximum error of 0.26%.

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

`I`_z Option Evaluation

The I_z option is an option for the evaluation of

Z

If I_z = 1, $R$ is constant and computed during Starter at time = 0 and; therefore, $Z$ is not updated each cycle.
If I_z = 2, $R$ is not constant ( $R (t)$ ) and; thus $Z$ is updated for each cycle during Engine computation only before the wave reaches the surface. After, no update is done.

To evaluate I_z, a displacement is imposed from $Z = 3.2 c m / g^{(1 / 3)}$ (point C) to $Z = 4.8 c m / g^{(1 / 3)}$ (point D) 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 ( $P_{r}$ ) obtained should correspond to $Z = 3.2$ , even after the displacement.

For I_z = 2, $R$ is updated as $Z$ at each cycle; therefore, the pressure ( $P_{r}$ ) obtained should correspond to $Z = 4.8$ after the displacement.

Table 2. `I`_z Evaluation for Point B
`I`_z =	$P_{r}$ expected (Mbar)	$P_{r}$ Radioss (Mbar)	Error (%)
1	8.70e-4	8.71e-4	0.01%
2	4.27e-4	4.26e-4	0.14%