Properties relevant to impact absorption? Assuming you are creating or evaluating a material used as padding. An approximation of the padding's use would be wrapping a lightweight bat so as to make it as close to harmless on a person as possible, with the minimal layer possible.
The question is, what properties of the padding's material matter in how well it would do, and how so?
 A: This is a really complicated question. There are several things that can cause damage during impact - so when you test padding, you always do so with a specific purpose in mind. Broadly, you can think of the following mechanisms:


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*Acceleration of the target. The heavier the projectile, and the lighter the target, the greater the final velocity of the target. The maximum acceleration is sometimes a source of internal stress, and damage (think concussion in the head). A very compliant pad (thick, "squishy") will create lower acceleration over longer time, and reduce the damage due to acceleration. This is why air bags work.

*Local compressive stress. A sharp point will penetrate the target; sometimes, you want to spread the load. This might require a "pad" that is itself very puncture-resistant, and that has good lateral rigidity. Again - air bags spread the load (you hit a big bag of air instead of the edge of the steering wheel).

*Shear stress. Impact with a large cylindrical object (hitting with the circular end, like a blunt bullet) would produce large shear stresses in the target (between the area under the projectile, and immediately adjacent). Your "pad" would need to be very stiff to spread the load. This is a key feature in bullet proof vests - some of which are made of a ceramic that fractures in a cone to spread the load, then the cone is backed by a thick layer of Kevlar that prevents the edge of the fragment from penetrating the skin). See for example this diagram from US patent 5970853



*Energy absorption. If you have an elastic collision between two objects, the projectile will "bounce back"; from conservation of momentum, this means the target will have the greatest momentum (thus velocity, and acceleration). A perfectly inelastic collision minimizes the velocity after the impact.


There are other potential things that go into this, but typically it means that you want your padding to have good elasticity, good rigidity, and the ability to absorb / dissipate as much of the impact energy as possible (this makes a pad built from springs relatively poor).
This is not a full treatise - but I hope you get the general idea.
