Discussion Forum

Does anyone have any experience or knowledge of how best model inelastic recoil due to adhesion?

Suppose I model force as a function of position. Then for any closed trajectory the change in potential energy is zero, and the kinetic energy (excluding any external force) will be unchanged. So if I'm modeling a bouncing ball as long as the repulsive or attractive force at impact is a function of position only then neglecting numerical issues or damping or transmission of energy into the floor (acoustic waves) then the ball will return to its original position (I'm assuming the ball has no oscillation modes, as exciting these would act as an additional energy reservoir).

However, if I model the force as a function of position and velocity, then the recoil can be lossy. Typically recoils are modeled in terms of an elasticity: the fraction of kinetic energy retained in the recoil.

Now there is a contact pair model in COMSOL but I've had terrible luck with it and so have switched to defining my own coordinate-based contact force, of the following type:

(x>x0)*((x-x0)/1[nm])*10 [GPa]

===== begin digression =====

This may lead to the simplistic assumption that the penetration will be proportional to force, but this isn't true: it's a volume force and so the volume experiencing the force (x>x0) is an increasing function of the penetration. For example, suppose a protrusion is a parabolic surface: z = –(x² + y²) . Then the cross-sectional area proportional to distance from the tip. This results in an integrated force proportional to the following:

F ∝ ∫ (r - z) z dz (from 0 to r)
= r z² / 2 – z³ / 3 (evaluated r, 0)
= r³ / 6

The elastic energy is then proportional to the integral of this with respect to distance, which yields

E ∝ r⁴/ 24

So the contact force is portion to the cube of the penetration for this sort of tip.

If the trajectile then has a kinetic energy 1/2 Mv², then the maximum penetration rmax is proportional to the square root of the velocity. The total energy is thus:

½ Mv₀² = ½ Mv² + A x⁴/ 24 (where were x us the position relative to contact at x = 0), and v₀ is the velocity at first impact.

v = √ [ v₀² – (A/12 M) x⁴ ]

===== end digression =====

Anyway, I've digressed (thinking out-loud). This is a position-dependent force. I need to introduce an energy loss mechanism which will emulate to some degree an elastic coefficient for recoil.

One simple way to do this would be to have the contact force be a sigmoidal function of velocity.
Sigmoidal functions: en.wikipedia.org/wiki/Sigmoid_function

Then I can define the bulk contact force to be:

(x>x0)*((x-x0)/1[nm])*10 [GPa]*(1-α*S(-v/vref))

where S(t) is a sigmoidal function (0 @ t≪0; 1 @ t ≫0), and vref is small and α is an inelastic coefficient.

Does this seem reasonable? Has anyone attempted this sort of thing or know how it's generally done? Most impact modeling I've seen assumes the impact is nonlocal and instantaneous (macroscopic). This is modeling the impact locally and on a detailed time basis (microscopic modeling).