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# Mindlin-Herrmann Rod Theory

## Mindlin-Herrmann Rod Theory

(OP)

I don’t know if this is the right forum for this……but we’ve discussed elastic wave propagation theory here in the past and it wasn’t a issue. Lately I’ve been reading this text on wave propagation that discussed the Mindlin-Herrmann Rod Theory. This theory predicts transverse motions from longitudinal waves (due to Poisson’s ratio). But reading this theory……it appears that the wave is dilatational, i.e. there is no net motion laterally.

For those of you familiar with this theory…..is that accurate or am I misunderstanding something?

### RE: Mindlin-Herrmann Rod Theory

Never heard this - I spent time doing research on waves so here is a small summary of my conclusions from this research :).

The full set of modes that can propagate in a finite structure are given by the superposition of the fundamental longitudinal and transverse waves (in infinite domain).

With finite boundaries we get (due to reflections) Lamb Waves - this can be for plates and then also for cylinders. In plates these are the Symmetric (S modes - S0 is a longitudinal type of mode) and Anti symmetric (A modes - A0 is a bending mode), similar theory and names are given for waves in cylindrical structure. I would suggest books on wave motion by Graaf and Achnebach if you are interested.

The motion you mention is a longitudinal mode or a quasi-long. mode (e.g., S0) which will as you say due to Poisson's effect have a contract/expansion of the cross section, but the wave is a longitudinal type of wave (S0 type, thus the main in plane displacements are in the same direction as the wave propagation along the rod), and not a bending wave where the main disp. is transverse to the wave propagation direction.

This is a video if this:

(OP)
Thanks Erik.

### RE: Mindlin-Herrmann Rod Theory

Glad to help - good to be able to share this and perhaps help a bit otherwise it sits there for nothing taking up space (have not used this for 10 over years) :).

### RE: Mindlin-Herrmann Rod Theory

(OP)
One thing that occurred to me today: since the strains developed under compression and tension (in steel for example) are typically not identical (and since this lateral displacement under these longitudinal waves is proportional to longitudinal strains)......I would think a net effect would develop along the length of a rod that would give a net lateral displacement.

Food for thought.

### RE: Mindlin-Herrmann Rod Theory

(OP)
Ok, I think I've figured this out: There is no net lateral motion via this theory. Looking at one of the texts you mention ('Wave Motion in Elastic Solids', by; Graff, p.511), it appears it is a pure dilatational wave.

Ergo, lateral vibration should be expected only when the equations of motion take into account initial imperfections, other (antisymmetric) wave types (including those from boundary reflections), etc.

In other words what you said was quite right Erik. Thanks again.

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