Simulation of internal mechanics of materials of high restitution

[xortdsc]

Hello,

just for some basic understanding I wrote a (1D) numerical simulation to investigate the internal mechanics of collision impact in different materials.
For a start I work with a really simple model which consists of:
- basic elements (circles of fixed radius and mass and their current position and velocity)
- links between elements (hookes springs with fixed k and desired distance)
- when 2 elements overlap a dynamic link is added between them for the current timestep to simulate repulsive forces on collision

All computations are to be performed on each element or link individually, so it is like simulating "huge simplified atoms". After all I want to see how macroscopic effects emerge from microscopic ones.

Now the scenario is simple. The starting condition is:
- An "object" on the left consisting of a single element moving with velocity 1 to the right
- An "object" on the right consisting of 2 linked elements (with initial separation of 1) which is stationary
- the masses of all elements are set to 1, so the left object has a total mass of 1 and the right object has a total mass of 2

Left Object
  Element x=-0.1, v=1.0
  Overall x=-0.1, v=1.0
  Energy pot=0.0, kin=0.5, vib=0.0, tot=0.5
Right Object
  Element x=1.0, v=0.0
  Element x=2.0, v=0.0
  Overall x=1.5, v=0.0
  Energy pot=0.0, kin=0.0, vib=0.0, tot=0.0
Total energy=0.5, impulse=1.0

Since there is no internal friction of any sort, energy and impulse (in any case) should be conserved and it is. The simulation goes mostly as expected: the left body moves towards the right one, collides, bounces off and returns with some of its momentum transferred to the right body.
Looking at the numbers I get the following at the end of the simulation (post collision):

Left Object
  Element x=0.01978705, v=-0.13026643
  Overall x=0.01978705, v=-0.13026643
  Energy pot=0.0, kin=0.008484671, vib=0.0, tot=0.008484671
Right Object
  Element x=1.4102714, v=0.56778127
  Element x=2.4689417, v=0.56248504
  Overall x=1.9396065, v=0.56513315
  Energy pot=0.0, kin=0.3193755, vib=0.1721171, tot=0.49149257
Total energy=0.49997723, impulse=0.9999999

(This is just a snapshot as the velocities and relative displacements of the elements of the right object would oscillate, while the overall velocity of both elements is constant)

Now as you can see energy as well as impulse is conserved (disregarding the tiny numerical error). Also some of the energy is transformed into vibrational energy inside the right object (its 2 elements are vibrating relative to each other). This is also the reason, why the final velocities are not the expected -1/3, +2/3 as an impulse-based collision response with CoR=1 would suggest as some kinetic energy is converted into vibrational energy of the right body.
Changing the stiffness of the springs will yield the same results, the only difference is the vibrational frequency, which seems okay.

Now the actual question would be: How do high restitution materials (e.g. a superball with CoR of about 0.95 or so) reflect almost all energy without converting it into vibrational energy ? What would be the property of such material in that simple model ? What would need to change in the simulation to get something close to a fully elastic collision ? Using internal friction does not solve the problem, because as it reduces the internal vibration that energy is lost.

Does anybody have some insight into this ?

 

[xortdsc]

I figured it out. So to answer my own question: High restitution happens if the internal links are very stiff compared to the stiffness of the collision.
Restitution appears to depend solely on the ratio k_internal/k_collision. The remainder of the energy, if restitution is less then one, goes into internal vibrational energy.

Is there an equation which translates these quantities ? So I could compute the ratio of internal to collision stiffness from restitution and vice versa ?