The purpose is to investigate analytical models of accretion, particularly inelastic, which may be the same thing as dynamic plastic deformation in the grand scheme. One is slanted from the physics perspective, the other from engineering mechanics. Ultimately, while the latter description is correct from a continuum material standpoint, the former seems to embrace the notion of free body participation which would exist internally in some fashion in any real collapse.

The models under discussion are 1D unless otherwise indicated. The development here will focus primarily on an analytic treatment in the continuous domain, but the relation to discrete models and their inner workings will be discussed, too. There is much crossover and tie-in, and it's natural to expect one to validate the others here and there. Finally, this can also be useful in working towards models of granular collapse, which is the specific focus of yet another thread.

What follows is the original post in the toy simulacra thread. It's very basic and really gives no indication of the depth I now wish to explore, time permitting.

I **** this out this evening...

Uniform density ρ given total height H and mass M:

ρ = M/H

The change of mass accumulated with respect to time:

dm/dt = ρdy/dt = ρv

Conservation of momentum in incremental inelastic collision:

mv = (m + Δm)(v + Δv)

Multiplying out quantities in parentheses:

mv = mv + mΔv + vΔm + ΔmΔv

Eliminating original momentum from both sides:

0 = mΔv + vΔm + ΔmΔv

Dropping second order quantity:

mΔv = -vΔm

Dividing both sides by Δt, the increment of time:

mΔv/Δt = -vΔm/Δt

Allow increment to go to infinitesimal:

mdv/dt = -vdm/dt

Substitute expressions for instantaneous acceleration and mass accumulation rate:

ma = -ρv

^{2}

Thus the resistive force due to inelastic accretion is:

F

_{r}= -ρv

^{2}

where the negative sign indicates the force opposes the direction of motion.

If the moving portion of the mass is subject to gravitational force, then the total force on the moving portion at a given time is

F = mg - ρv

^{2}

where m, ρ, and v are all functions of time.

Acceleration is therefore:

a = g - (ρ/m)v

^{2}

Since the instantaneous upper block mass m is:

m = ∫ρdy evaluated on [0,y] = ρy

the above relation a = g - (ρ/m)v

^{2}becomes:

a = g - v

^{2}/y

This is Cherepanov's equation of motion. Wow, that was easy! Now, how does one get asymptotic approach of acceleration to g/3 from this? Dr. G seemed to know about five years ago over on physorg, didn't say how. Must check Cherapanov's paper. Or brush up on my math.