Math formulas

[Marek_Tucan]

Where's the mass of the projectile or density or whatever?

[Pyros]

Marek_Tucan said:
Where's the mass of the projectile or density or whatever?

Here...

The Odermatt equation

The Odermatt equation allows the comparison of results from firing tests with differ-ent target inclinations/thicknesses/material properties, penetrator geometries and impact velocities. It also allows the evaluation of influences of penetrator aspect ratio, obliquity and other parameters. Penetrator performance is predictable if penetrator geometrie, density of penetrator and target, ultimate tensile strength of the target, NATO-obliquity and impact velocity are well-known. Reliable extrapolations from a single test result are possible.

In a finite target penetration limit means that the projectile reaches the rear face of the target and spalling opens the penetration channel. The penetrator residue will have a length of approximately one and a half diameter.


Governing parameters

D penetrator diameter [mm]
L overall tungsten penetrator length [mm]
Lw working length of penetrator [mm]
vT impact velocity [m/s]
NATO angle of obliquity [°]
rho_P penetrator density [kg/m3]
rho_T target density [kg/m3]
d target plate thickness [mm]
UTS target material ultimate tensile strength [MPa]
P penetration channel length [mm]


The Odermatt equation is composed of four dimensionless terms with separate representation of the influences of length to diameter ratio A, target oliquity B, density ratio of penetrator to target C as well as material properties and impact velocity D.

Odermatt equation: P/L = A * B * C * D

aspect ratio influence
a good approximation is obtained with the following equation

A = 1+a1*D/Lw*(1-tanh((Lw/D-10)/a2))

with a1 = 3.94 and a2 = 11.2 - valid for Lw/D = 10 up to infinite
Definition of the working length Lw :
The conical tip is replaced by a cylinder of equal mass and diameter D and the remainig length is reduced by 1.5*D.
A good approximation is: Lw = L - 2/3*L_con - 1.5*D

some values of A:

Lw/D A
10 1.394
12 1.270
14 1.185
16 1.126
18 1.085
20 1.057
25 1.020
30 1.007
50 1.000
100 1.000

target obliquity influence
B = (cos °_NATO)^m
the best fit for the exponent is m = -.225

penetrator/target density influence
C = sqrt(rho_P/ rho_T)
this is the well known root density law

Target properties and impact velocity influence
D = exp(-c*UTS/(rho_P*vT^2))

c = 22.1 + 0.01274*UTS - 0.00000947*UTS^2