When looking to design armor - how can I determine whether a given ballistic object would completely stop, deflect into another direction, or penetrate another object given complete physical information on the two objects and nature of the collision, including material makeup, shape, or any other variables?
Even if we can't know exactly - due to various limitations - what equations or process would we use to go about making our best guess?
While your question only requires fairly basic kinematics (physics), there are quite a few effects to consider. I can't cover them all, but I'll try to hit the important ones. This will be a longer answer!
Here's what I'm taking as fact before I get into the meat of my answer:
I believe these assumptions are reasonable based on the wording in your question, and the (physics) & (hard-science) tags, but let me know if otherwise, as that could change everything.
So, what matters?
(high school physics)
Before even considering materials, it's important to look at mass and velocity, as they make an enormous difference, no matter what materials you use. For my favorite discussion of the topic, see the very first What If? : Relativistic Baseball.
First, let's look at the kinetic energy, $E_k$, of any projectile with a given mass (m) and velocity (v):
$$E_k = \frac{1}{2} mv^2$$
Informally speaking, the more kinetic energy the projectile has, the harder it can hit. I'll get to that suspicious-looking "can" in a second, but first, a quick observation:
Velocity matters more than mass. Since the velocity (v) term is squared, doubling the velocity will do more than doubling the mass. However, practical considerations mean ballistics designers strike a balance between the two.
Now, again, before considering materials, here are some properties of the armor (or, more generally, of the nature of the collision) that will affect the damage done:
More information: Elastic collision (Wikipedia)
For maximum damage, all of your projectile's energy needs to be deposited into the target. In other words, the projectile needs to be completely stopped by the target. If that isn't the case, a smaller fraction of the kinetic energy will "sink in" to the target, and the projectile will be less effective.
Often deflection is accomplished with geometry: give your armor lots of sharp angles and round corners, and projectiles will tend to take glancing blows, and retain more of their velocity (and hence impart much less of their energy into the target.) A useful analogy here is billiard balls. Hit a ball thinly, the cue ball retains a high velocity, but your object ball doesn't go very far. Hit it dead on, the cue ball stops, and the object ball gets all of that energy (well, almost—nothing's 100% efficient).
In concrete terms, successful deflection increases the projectile's final velocity. That is, after it bounces off (or travels straight through) your target, it retains more speed. First, a numerical example:
Let's say it retains half (0.5) of its velocity and all of its mass (does not fragment). Then we can do some simple rearranging to see how much of that energy stays with your target:
$$\% E_{target} = 1 - \frac{\frac{1}{2}m(0.5 \times v)^2}{\frac{1}{2}mv^2} = 75\%$$
You can generalize or play around with that by changing the 0.5 to some other percentage between 0.0 (worst case for the target—all energy transferred,) and 1.0 (no energy transferred—a miss). Since most of those terms cancel out, this can be written more simply:
$$\% E_{target} = 1 - x^2$$
Where $x$ is the fraction of the original velocity, as a number between 0 and 1.
Ballistics vests used by law enforcement are a good example of this. They have an armor plate that, when struck, is designed to distribute the energy from the projectile over a large (chest-sized) area. Without the vest, the bullet's energy would be focused on a $1cm^2$ area, and hence easily penetrate the body.
It's hopefully obvious that applying the same force over a smaller area will be more likely to result in a penetrating injury, thanks to higher pressure. If you need convincing, try to stab a balloon with a 1 kg baseball bat. Then try again with a 1 gram needle.
Vests and similar armor actually operate on multiple principles, but it's not that complicated. Vests:
Wearers of vests still experience significant pain and (usually) minor injury, but they're a great example of how you can use physics to completely change the nature of a ballistic impact.
As above, armor can attempt to decelerate the projectile by providing some kind of padding, or, like a car, the armor can deform ("crumple") under stress. What it's doing is converting the energy of that projectile into a small amount of heat, which reduces the more harmful penetrating or concussive effects that would injure or damage your target.
In classical physics, this is simply deceleration. If your crumple zone can slow the projectile down, you can plug a smaller $v$ into the earlier $E_k = \frac{1}2 mv^2$ formula. And, as you recall, a smaller $v$ makes a relatively big difference.
To have your projectiles fly straight and true, the shape will be important, and with many projectiles, imparting spin will keep them much more stable during flight. How your projectiles behave in flight, how much they're slowed down by air drag, and how fast they can travel, are all governed by something known as external ballistics. Since I believe you are more concerned with armor design, and this answer is rather long already, I won't go into detail on that, but let me know (or ask another question) if you need to know more about projectile design; it's something I've studied.
I've left this for the end of my answer because, as you'll hopefully agree by now, there's a lot you can do with pure physics that doesn't depend on materials.
But materials do matter, of course! A full review of all known armor-grade materials known to man would of course be beyond the scope, here, but I can give you some general guidelines, especially now that I've gone over some of the essential physics:
Generally speaking, you can have a hard substance that will resist penetration but be more brittle, or a soft substance that isn't as strong, but is able to bend and deform so it doesn't shatter and fail completely. In conventional armor (or blade) design, you'll tend to see a mixture of different types of steel (different carbon content/heat treating process).
If you know what kinds of materials your enemy's ammunition is made of, you can choose armor that is harder than the ammo, to cause the ammo to deform or fragment. (Again, harder ammo (e.g., high carbon steel) will be more likely to fragment, softer ammo (e.g., lead) will more likely deform.) If you can cause the ammo to fragment, the pieces will fly off in different directions, which will reduce the energy your armor needs to absorb. (Again, think of this as a higher final velocity, and/or a lower mass (m), as less of the projectile ends up going forward into the target.)
Thanks for sticking with me through a very long answer-slash-physics review. If you value brevity, I apologize. In this case, I felt the longer answer was the right approach, since you asked for equations, and how all of the factors influence the projectiles and armor performance. Hopefully I've done my job well enough for you to come away with a decent grip on the subject. Do feel free to ask questions in the comments.
I hope to flesh this answer out better later but I only have a few minutes to start on this right now.
As a general rule of them the quality of materials like metals for use as armor can be approximated by the metal's melting point.
Superalloys > Titanium > Aluminum > Lead
Note there are many exceptions to this rule of thumb since some metals with high melting points make lousy armor due to brittleness (e.g. Beryllium) or other factors.
Modern armor is more of a "system" rather than a single material. It was known since before WWII that the best armors had different qualities through the thickness of the material (e.g. high hardness at the front, high ductility at the back).
During WWII armorers worked on heat treatments that gave the often single sheet of steel these qualities. In that regard, the armor on US warships was significantly better per unit thickness than that of the Japanese. Some felt that the US's Iowa class' 12" main armor belt may have provided superior protection to that of the Japanese's Yamato class' 16" main armor belt. No naval engagement got to test that theory.
Even during WWII considerable thought went into defensive systems to harden ships to weapons (e.g. torpedoes). These involved many layers of protection, each providing unique benefits. For instance:
Each side of the ship was protected below the waterline by two tanks mounted outside the belt armor, and separated by a bulkhead. These tanks were initially planned to be empty, but in practice were filled with water or fuel oil. The armored belt tapered to a thickness of 4 inches (100 mm) below the waterline. Behind the armored belt there was a void, and then another bulkhead. The outer hull was intended to detonate a torpedo, with the outer two compartments absorbing the shock and with any splinters or debris being stopped by the armored belt and the empty compartment behind it.
Modern ship defensive systems and Cobham Armor continue using this approach, layering materials each of which provided a specific "strength" to the overall defensive system design.
Although the construction details of the Chobham Common armour remain a secret, it has been described as being composed of ceramic tiles encased within a metal matrix and bonded to a backing plate and several elastic layers. Due to the extreme hardness of the ceramics used, they offer superior resistance against shaped charges such as high explosive anti-tank (HEAT) rounds and they shatter kinetic energy penetrators.
Proposed composite armor
When defending against hyper-velocity rounds the energy involved is usually sufficient to completely vaporize both projectile and armor. This robs the weapon of penetration ability but also robs the defender of many mechanisms of defending themselves.
The best armoring system currently devised for these types of weapons is called a Whipple Shield.
Whipple shields consist of a relatively thin outer bumper placed a certain distance off the wall of the spacecraft. This improves the shielding to mass ratio, critical for spaceflight components, but also increases the thickness of the spacecraft walls, which is not ideal for fitting spacecraft into launch vehicle fairings. The advantage of a bumper placed at a standoff over a single thick shield is that the bumper wall can shock the incoming particle and cause it to disintegrate. This spreads out the impulse particle over a larger area of the inner wall of the spacecraft.
This concept HAS been applied for use in the context of weapons of war.
In many areas, a "first order analysis" can give you a very good understanding of the overall problem. However, in matters of armor and defensive systems, this is not really true. There's much to consider and even the equations involved don't give the researchers and engineers enough information to design these defensive systems. Most of what we now consider "the best solution" has been found by trial and error rather than theoretical work.
