How to calculate elastic deformation due to kinetic energy (impulse)?

Question

I am a Mechanical Engineering student and I am a little confused on how to find deformation of a material given an amount of energy.

I have a mass falling from a certain height to punch an indent into a piece of metal. I tried to use Young's Modulus in the following way to see if I could fudge it from fundamentals, but I got an answer I am assuming is way off:

mgh = W = F*d

E = stress/strain

strain = F/A

stress = l/l_0

=>

"The work is applied over the thickness of the material? Seems like a bad assumption to make. Oh well."

E = ((W/l)/A)/(l/l_0)

solve l

Please let me know what direction I should go in to understanding this interaction better.

Answer 1

When punching an indent into a piece of metal, the residual stress that is concomitant with the indentation implies yielding and so you have a non-linear interaction.

If you still want to do the linear calculation just to see what it would be, you set the potential energy of the Earth-mass system equal to the energy absorbed by your material during impact:

Note here that we are considering the deflection, x, to be non-negligible in comparison to h, but in reality it almost certainly would be in the small-strain (linear) case. F is the maximum force felt during impact, which happens when deformation is a maximum and the mass isn't moving momentarily.

The equation for the maximum force during impact is derived from the small-strain case where the static deflection quantifies the stiffness of the material:

and

after substituting for F and solving for x, it can be shown that:

All system properties are essentially captured in the static deflection, mg/k. K is the stiffness of the material, which can be thought of as a spring constant. It depends on the shape of the part and for rods loaded axially it is AE/L, where A is the cross sectional area, E is the Young's Modulus, and L is the length of the bar. Estimating K is the hardest part of your problem.

For more information, Ch. 7 of Juvinall's Fundamentals of Machine Component Design is a good starting point.

Answer 2

if we are talking elastic deformation then the problem is easy. you take the kinetic energy of the falling weight at the instant it first touches the object and equate it to the stored strain energy in that object when the falling weight's velocity is zero.

Plastic deformation is a far more complex problem.