Estimated Thickness of Armour Required for Immunity against AP shell with X-mm of pen.

[Peasant]

When analyzing data from historical ballistic testing of armor, it can be very useful to be able to covert the results between US Army ballistic limit (safety limit) and the US Navy limit (or equivalent of other nations) (limit velocity for perforation of the armour).

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After thinking about it for a while I've come to a realization that the amount of kinetic energy an AP shell expends to move between just touching the rear of the armour plate to perforating it completely, does not change with the overall total thickness of the plate (at least for 0° impact).

When a rigid projectile is penetrating through the armour the friction between them creates a thin coat (1/100th of mm) of molten steel that acts as a lubricant, reducing the friction at it's outer surface to a negligeble amount. Most penetration models disregard the contribution of friction to the overall resistance of target, and so shall we.

Here is a schematic to better illustrate this idea:

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Mathematically speaking:

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The most important factors that determine the value of C constant are the penetrator nose sharpness and plate hardness and ductility, but for most WW2 artillery AP shells with ogival noses of around 1.4 CHR striking MQ RHA it's value can be taken as a constant equal to about 250 m/s.

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For very sharp projectiles, like the steel cores of small arms AP bullets the C is higher, about 325 m/s, while on the other hand, for very blunt projectiles, like soviet blunt nosed shells or uncapped AP attacking face hardened armor, the value of C is somewhere around 150 - 200.

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This is a pretty simple equation, but it's not immediately useful when trying to answer a question of a more immediate usefulness: how much armour would be safe from a shell that can pen X mm of armour?

I've taken a crack at this problem and after some hard work, I have found a solution.

Let's start from the DeMarre's equation:

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First, we'll introduce some simplifications into this formula:
We'll name the (T/D) variable as just T (but remember that from now the thickness is
measured relative to diameter of the shell and it's units are "calibers" not "decimeters".

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then we set the angle to 0° in order to remove the cosine term from the equation
and name exponent of the variable T, "n".

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and so we get to this form of this equation that connects a specific thickness of armor and it's respective limit velocity:

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so, the Navy limit of a plate of thickness T1 will be:

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now we can use the relationship we developed earlier between the Navy and Army limits:

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we substitute the previous equation into the one before and get:

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this equation gives us the US Army limit for the thickness T1, which is still not what we wanted.
Now, I was stumped at this point, not knowing how to proceed, but eventually I've realized that what I needed is to formulate the question in mathematical terms: "What is the thickness T2 that has it's Army BL equal to the Navy BL of T1"?

From here on, it's easy:

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So, there we have it. The value of T2 is the minimum thickness (expressed in calibers) that would be required to be nominally safe from a shell with mass m(kg) and diameter D(dm) striking the armour at 0°.

Here is the output of this formula:

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You can download the EXCEL spreadsheet yourself, if you want to play with the values:

DeMarre Safety Thickness Calculation.xlsx

Edited September 17 by Peasant

[Peasant]

Questions some people might have:

Q: "Can this relationship be used in reverse, to go from the immunity limit to perforation limit?" 

A: "Yes, you can." 

 

Q: "I don't understand any of the math." 

A: "Don't worry, you don't have to. Just use the values from the table whenever you need." 

 

Q: "How did you come up with this?" 

A: "There is no single source for this. I've observed this relationship to hold when analyzing data from many different penetration testing data from different time and countries."

 

Q: "Can you give us an example of some testing data where this rule holds?"

A: "Sure, here is a comfy little table of ballistic limits for .50cal AP bullet from this document here."

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To be updated with more questions and answers...

Edited September 17 by Peasant