http://forum.physorg.com/index.php?showtop...ndpost&p=226737

This is a follow-up to the post linked above, where I looked into the equation of propagation of the collapse front during the crush-down phase, as derived in Bazant and Verdure.

I now look into the derivation of the equation for the crush-up phase. This phase begins when the compacted mass of Zone-B created during the crush-down phase reaches the ground (= the bottom of the "bathtub").

Whereas in the crush-down phase the z(t) coordinate designated the distance of the collapse front from the initial position of the roofline; in the crush-up phase, the y(t) coordinate designates the distance of the collapse front from the current moving roof-line. The collapse front is now on top of Zone B. The lowest Part A no longer exists at the start of crush-up.

We thus have that -- in a manner that is similar to the crush-down case -- dy(t)/dt does not represent the speed of the "upper block" (now corresponding the Part C alone) but rather represents the "shortening speed" of this block. After some slice of infinitesimal thickness dy of Part C is crushed to thickness lambda*dy. The height of the rubble pile is increased by lambda*dy, and it follows that the roofline has descended (1 - lambda)*dy. The net speed of the upper block is thus (1 - lambda)*dy/dt. Likewise, its acceleration is (1 - lambda)(d/dt)(dy/dt).

We now use m(y) to represent the (implicitly) time dependent mass of the upper block. The momentum of the upper-block is thus

p = m(y)*(1 - lambda)(dy/dt)

The downward acceleration of the upper block is opposed by upward inertia force

dp/dt = FC_i = -(d/dt){m(y)[1 - lambda]dy/dt}

Calculating the time derivative we get:

[eq. 1] dp/dt = -m(y)(1 – lambda)(d/dt)(dy/dt) – (dm/dt) )(1 - lambda)(dy/dt)

B&V calculate the rate of variation of the momentum that results from mass reducing its velocity to 0 as it hits the top of the rubble pile (and is thus transferred from Part C to Zone B). This is

[eq. 2] dp/dt = (\mu(y)(dy/dt)(1 – lambda)(dy/dt)

where \mu(y) is the linear mass density. (“\mu” is the Greek letter mu.)

Notice that \mu(y)(dy/dt) just is dm/dt. It follows that this expression [eq. 2] just is the second term of the right hand side of [eq. 1].

So, while the first term in [eq. 1] represents the variation in momentum that is due to Part C accelerating, the second term represents the variation in momentum that is due to mass being lost to the rubble pile (Zone B). The inertial reaction from the deceleration of mass that is represented by this term really ought not to figure into the equation of dynamic equilibrium since this mass is being decelerated by the normal force from the surface of the compacted rubble pile and certainly not as a result of Zone C somehow pulling on it. This is a mistake B&V do. Yet, they also claim that the inertial force from this second term acts upward as a reaction, from Zone B onto Part C. They label this FB_i and they make this force figure as well into the equation of dynamic equilibrium. This is a second mistake that exactly cancels the first one! As a result they get the correct equation for crush-up (eq. 17 in the paper.)

In my next post on this subject, I will look at the derivation of the crush-up and the crush-down equations through consideration of potential and kinetic energies that appears in the “Alternative Formulations, Extensions, Ramifications” section of the B&V paper. Notice that there is a typo in eq. 27. There figures one “(z)” to many in both the second and third terms of the LHS of this equation. There is also a significant error in the crush down derivation even though, at the end, they get the correct equation of motion. I’ll offer a free beer to whoever spots the error before I post it here. (In the meantime, I’ll try to work out an analogy with the variable mass "rocket equation".)