A fully dynamic analysis will utilize calculus to make dynamical evolution mathematically precise. If you study, e.g., the Ari-Gur/Singer paper, the question of subtracting a net plastic energy before or after a net elastic energy doesn't arise. Not in a simple way such as being discussed, anyway.
From their experimental results, seen in Fig. 2a,
http://metamars.i8.com/ , at the particular spot in the rod that measurements were made, we can see that an elastic deformation and plastic deformation only overlap for about 1.5 msec. The elastic pulse preceeds the plastic pulse by about .5 msec. So, at least for the Ari-Gur/Singer scenario, it looks like you can say that, in some sense, you can subtract a
portion of elastic energy before plastic energy.
I took a look at the Ari-Gur paper to try and quantitatively determine energy dissipation due to elastic vs. non-elastic dissipation.
The following is inexact - I am eyeballing a graphs to derive figures from it. I hope somebody else will follow up by studying the Ari-Gur paper, then giving either one of the authors a call, or else somebody else who actually works in the field of impact studies. I make assumptions that I'm not really sure about, so take with a grain of salt. I would appreciate it if other people studied this paper....
p. 622 of their paper (which I have uploaded) gives 2 compressive strain vs. time plots (side-by-side). The peak strain for the first plot is about 1000 mu, and the peak strain for the second plot is about 1500 mu. I'm not sure which test sample this is for. A pair of graphs on the bottom of p. 633 show experimental and theoretical graphs for one of the steel samples. Oddly enough, the theoretical graph seem to match the experimental graphs on p. 622 better (ignoring the inversion about the time axis.)
In any event, I referred to the theoretical graph, since I am eye-balling things, and this is the easiest to eyeball.
From the graph, the main elastic pulse is (very) roughly equal in area to a box 1000 mu high by .35 msec wide. Since the speed of sound in steel is 5,100 m/s, that means that the initial "pulse" is felt over a length of about 1.5 m. Since all the steel specimens were under 1 meter (max 380 mm), I take this to mean that it's a good approximation to consider the strain as being representative of the specimen as a whole, and furthermore that the rod can transfer elastic energy (to be calculated below)
through the rod in an amount equal to 1.5 m/ 230 mm = 6.5 times the elastic strain energy corresponding to a 230 mm rod, compressed 1000 microns,
statically. (Actually, this can't be correct, as I will show.)
Please note that I had previously thought that the pulses were smaller than the sample. Thus, I thought the pulses were mostly exiting out the bottom of the impacted rod, never to return.
Now, that seems to be definitely wrong! The fact that nobody ever questioned me on this point just shows, I think, why we need to talk to experts in the field. I don't know that anybody is looking at the relevant studies, besides myself, and I am clearly not a domain expert.
Anyway, rolling right along....
p. 623 has dimension and measurements of the various steel specimens being impacted. Using a middling one, Y5, I have a length of 230 mm, width 19.05 mm, and thickness 1.6 mm.
From Hooke's Law, using a constant of 10^9 N/m^2, I find that the elastic static strain energy is
(19.05 x 10^-3 m) * ( 1.6 x 10^-3 m ) (10^9 N/m^2) * (1,000 / 1,000,000) * (2.3 x 10^-1 m)
= 7.01 Newton
However, the Kinetic Energy is just 1/2 m v^2, where the striking mass has mass 180 grams and velocity 10.05 m/s
so KE = .5 * .18 kg * 100 m^2/s^2
= 9 Newton
Earlier, we saw that the "pulse" would have exceeded the length of the struck object. If we interpret this as meaning that 6.5 x the static strain energy is passing through this object, we end up with an energy sink greater than the energy source! Obviously, a contradiction.
That being the case, it seems unintuitive that the strain energy would "sit around" in the struck rod until after (9 Newton - 7.01 Newton) got dissipated in plastic strain, and
then leave through the bottom of the rod. The plastic strain begins after the elastic loading begins, and it's oscillations "of consequence" continue after the main elastic pulse has terminated.
If the main elastic pulse had lasted about as long as the plastic strain oscillations, I would think it likely that all of the 'extra' (i.e., exceeding the static case) elastic energy goes into plastic strain. However, this isn't the case.
All of which means,
I'm really not sure what becomes of the elastic energy.
As we are discussing energy dissipation in a collapse, and we want to try and get a handle as to how much elastic strain energy can pass through the bottom of the building via the columns, I would hope that I'm not the only one interested in quantitative experiments which can shed light on this.
N.B. : Ari-Gur never responded to my email inviting him to participate.
