MT,
And the foolishness continues...
OK, tfk, you caught me in three errors.
1) a typo - W1 should've been W2 in one instance
Wrong.
I told you in my last response that NEITHER "W1 + W1" NOR "W1 + W2" = the total energy dissipated.
So, repeating that you should have written "W1 + W2" isn't very, uh, astute.
Care to try again?
2) I claimed W1 was not present in figure 4C, it is
Yup, it is.
3) W1 + W2 != total dissipated energy
Right, it's not. So it wasn't really a "typo", was it?
For figures 4a and 4b, W1 + W2 + mgh(1-λ) is the total energy dissipated in crushing as the load displacement curve is integrable over the entire drop.
Now you're saying that W1 + W2 + mgh(1-λ) equals the dissipated energy.
Nope. Still wrong.
Rather than go round & round & round, here's the right answer.
Here's a slightly elaborated version of Bazant's curve:
The work done by any force F(z) acting over a distance ∆z is defined as W = ∫F(z) dz, i.e., the area under a F(z) vs z curve.
So the work done exerting a force curve F(z) over a distance (1-λ)h (=z2) is:
Not W1.
Not W2.
Not W1+W2.
Not W1 + W2 + mgh(1-λ)
In the graph above, the work done crushing (whatever) from 0 to z2 using the force curve F(z) is the sum of the area W1 + CA. (Where CA = the common area = the orange area.)
The work done crushing the same (whatever) from 0 to z2 using a constant force Fc is equal to W2 + CA.
Equating the work done (essentially a statement of conservation of energy) by either force curve is equivalent to saying:
W1 + CA = W2 + CA
Which means that W1 = W2.
This is exactly why W1 is set equal to W2. It is a consequence of the physical laws:
1. Conservation of energy
2. Equivalence of work & energy
But you have an error, too, tfk. Very fundamental. You say
- W1 is associated with the portion of the fall when the upper block decelerates.
- W2 is associated with the portion of the collapse when the upper block accelerates.
This is incorrect. Note the block accelerates while still in the W1 region in 4a, and it also accelerates while in W2 region in 4b, therefore the association you claim is false.
The block decelerates whenever F(z)> mg.
It accelerates whenever F(z)<mg.
I asked you which of the values described on Bazant's graphs was incorrectly marked. No answer...?
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In my estimation, you got as much business "correcting" Bazant as you do "correcting" Edward Witten's string theory.
Just my opinion, of course...