Command manual

This command is used to define material-specific properties of a spall cone generated by secondary fragments. The cone apex is located at $\mathbf{x}_c$ where the threat exits the back face of the armour. The cone axis $\mathbf{n}_c$ is aligned with trajectory of the threat.

A vital volume that lies within the spall cone and has a clear line of sight to its apex may be damaged by fragment impacts. The damage probability $p_D$ is calculated from the contributions of its exposed external element faces.

Consider an exposed face $i$ with centroid $\mathbf{x}_i$, area $A_i$, and outward unit normal $\mathbf{n}_i$. Its individual probability of full damage is

$\displaystyle{ p_i = \mathrm{min} \left[1, s_f \cdot i_f \cdot \frac{A_i \cdot \mathrm{max} (0, \mathbf{n}_i \cdot \hat{\mathbf{x}}_{ic})}{\pi R_i^2} \right] }$

where $s_f$ is a fragment sensitivity parameter defined in VAA_VITAL_VOLUME and

$\displaystyle{\hat{\mathbf{x}}_{ic} = \frac{\mathbf{x}_c - \mathbf{x}_i}{\vert \mathbf{x}_c - \mathbf{x}_i \vert}}$

$\displaystyle{z_i = (\mathbf{x}_i - \mathbf{x}_c) \cdot \mathbf{n}_c}$

$\displaystyle{R_i = z_i \mathrm{tan}\left( \frac{\theta}{2} \right)}$

$z_i$ is the axial distance from the cone apex to the face centroid and $\pi R_i^2$ is the cone cross-sectional area at that position. The formulation applies to faces located inside the cone, with $0 \lt z_i \leq h$.

$p_i$ is proportional to the fraction of the cone cross section covered by the projected face area. A coarse surface mesh may reduce the accuracy of this approximation.

Assuming that the contributions from $N$ exposed faces are statistically independent, the damage probability of the vital volume is:

$\displaystyle{ p_D = 1 - \prod_{i=1}^N (1 - p_i) }$

Spall cone defining region with secondary fragments

Secondary fragment example

Simple example with two vital volumes, where one is sensitive to secondary fragments and one is not.