Calculating force absorbed by compression of a material (i.e. rubber)

Question

Say I was trying to design padding to absorb impacts using polymers or similar materials, such as rubber. I have a good understanding of any relevant physics, most likely, but I wouldn't know where to start in calculating the degree of impact that can be absorbed by a material as a function of the dimensions, density, properties, velocity, forces, etc. of or acting on the material. Note the following points:

• The material should not be permanently deformed by the impact more than negligibly.
• The material is expected to be either one chunk or a series of thin layers, whichever is more effective, a total of less than 10cm thick.
• The object being protected will have a smooth but curved surface, to some degree. Any calculus this fact introduces is fine by me.
• Impacts are vertical.

Answer 1

let me assume that there is a force due to impact from a falling object mass(M) from certain height-h

Once it hits the padding, it continues to descend by an amount (u) until all the impact energy is absorbed. If you know the mass of the falling structure, you can calculate the strain required to arrest it. The deformation of the structure, through the distance (h) is determined by the elastic deformation of the structure it falls upon. I presume that, once the elastic limit of the structure is exceeded, the structure will fail, likely through buckling, or certainly through plastic deformation. There certainly can be a large amount of energy absorption through plastic deformation, but the forces we will calculate, up to the proportional limit, this deflection u after the mass contacts the structure it deforms the structure by an amount that depends upon the stiffness of the structure. Treat the structure like a spring of stiffness k. Then we know that the force is related to deflection, u, through

the spring stiffness using: F=K*u If we divide by the cross sectional area, we obtain:

F/A=k/a*u

We obtain a relation between u, the height/thickness of the padding, L, and the strain, ε:

U= εL we have the deflection of the padding, under the influence of a force, F, we can relate the impact energy to the deformations to extract the impact force, F. In order to arrest the falling mass we must absorb all the energy of that mass. The mass falls through a total drop height of h+u, so the energy that must be absorbed is given by:

U=mg(h+u) Replace u by εL

U=mg(h+ εL)

This energy is absorbed thru deformation of the structure, approximately up to elastic limit.

U(elastic)=1/2*σ* ε*A*L

σ = F/A

mg(h+ εL)= 1/2*F/A* ε*A*L

if we simplify by eliminating A, and multiply by 2, and dividing by ε and L, we get

F=2*m*g((d/ εL)+1)

This is the anticipated impact force for strains up to a chosen value. It is reasonable to assume that strains that are very near yield represent the end of recoverable deformation. the notation * indicates multiplication(product).