## Mechanical Hysteresis?

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Anthyo
Posts: 16
Joined: Tue Oct 26, 2010 2:53 am UTC

### Mechanical Hysteresis?

I'm having some trouble understanding how it is that mechanical hysteresis works, more specifically the ever popular rubber band experiment. For those of you unfamiliar with the phenomenon, it works like this:

A rubber band is hung from some surface and has its "base" length measured. It then has weights attached to it in regular increments. The band is allowed to settle after each weight is attached and its new length is measured. After reaching a max weight, the weights are removed in the same increments, and the lengths again measured. It can be observed that the rubber band is extended further at each weight when removing weights than at the same weight when the weights are being attached. Here's a link that summarizes the process:

The weight, of course, stops moving because the force exerted up is equal to the downward force of gravity. At the molecular level the attraction between the molecules of the rubber band keeps the rubber band from being ripped apart while struggling to return to its unstressed state.

I am willing to accept that Hooke's Law does not work linearly, but it seems to me that each length must have a specific tension/spring force. Shouldn't an extension of three inches exert the same force in all cases (assuming the same materials, sufficiently similar conditions, etc)? Shouldn't an extension of a certain length consistently balance a certain weight? I am told that it has to do with energy being lost in the form of heat, but I don't understand how friction allows a certain extension to exert more or less force.
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jmorgan3
Posts: 710
Joined: Sat Jan 26, 2008 12:22 am UTC

### Re: Mechanical Hysteresis?

It's not really due to friction, it's due to the material itself changing. You ask "Shouldn't an extension of three inches exert the same force in all cases (assuming the same materials, sufficiently similar conditions, etc)?" The answer is yes, all else being equal, a certain extension should cause a constant force. However, in the case of plastic deformation (basically just another word for mechanical hysteresis) all else is not equal. The material itself changes.

You can imagine the microscopic structure of the rubber band as a network of tangled strands (polymers) with strong bonds between the atoms in individual strands and weak bonds between the strands themselves. When you pull lightly on the rubber band, you just kind of straighten out the network. When you pull hard enough to cause plastic deformation, though, you actually undo some of the weak bonds between the strands, causing them to slip against each other. When you let go, the strands form new weak bonds in their new locations. As a result, at a microscopic scale, the rubber band has changed. That is why the elongation-load relationship is different.

In metals, the same general thing (plasticity) happens, but the process is a little different. Pulling (or pushing) on the material a little past its elastic limit causes defects in the material to move around until two defects that "cancel each other out" meet. In steel, the small amount of carbon molecules present in the metal "cancel out" gaps in the iron's crystal structure. Pulling on the metal will also create more defects. At a certain point (the material's yield strength), the weakening from the extra defects "outweighs" the strengthening from the canceled defects.

Disclaimer: I am not a materials scientist, but I did take a class a couple years ago. A couple of the above facts may be wrong, but the overall picture should be accurate.
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Hobbes_
Posts: 29
Joined: Wed Dec 08, 2010 8:22 pm UTC

### Re: Mechanical Hysteresis?

I'm really just familiar with steel, so I can't help with how rubber specifically responds to this, but the general principles should be the same. JMorgan's description was absoltuley right: it's a material change. Most introductory classes treat materials as being totally linear, and that works as long as the stresses on them stay in the "linear range", but when the stress on the material gets too big the material responds differently. In the linear range, the bonds of the material strech to accomadate the change in stress and deformation. When it leaves the linear range and goes into the palstic range, molecules begin to slip by each other: permanently altering the material (this is a simplification of course, but accurate enough).

Here's a nice little diagram of how steel reacts to stress (source: University of Maine, it was the first image that turned up in my Google search):
Spoiler:

On the y-axis you have stress: force per unit area in the material. On the x-axis is strain: the amount a material elongates as a result of the application of force. It's a unit less axis because it's measured in proportion to it's length, so if a 100 foot bar elongates one foot then it's strain is 1ft/100ft, or a strain of 0.01 ft/ft. Conceptually you can think of it as force vs. displacement the same way a graph of a spring might look. This kind of graph is the lifeblood of several different fields, I think I had to look at it for about 4% of my waking college life.

At first you see the steel responding linearly: that almost vertical bit on the far left side. This is, surprise, the linear range. If you remove the force the material returns to the same length. There's several more points of interest that come later if you're designing in steel, but basically once the response curve is no longer linear, you've entered the plastic region. Here, any additional changes you make are permanent. If you unload the material it will unload down the same line as the linear portion, only shifted over to where you unloaded it. In other words, if you displace it past it's linear upper limit (known as the yield point) by some amount 'x' and then remove the load, the end result will be that it is now longer by 'x' amount from when you started. This is the material change: bonds have been broken rather than just stretched in the material and thus removing the load can not return your object to its original length.

Hysteresis itself is basically a material "remembering" what has been done to it. I imagine it shows up in all sorts of fields, but I'm most familiar with it in context of earthquake engineering. Making sure a structure can absorb enough energy to withstand the earthquake knowing that it will have to enter the plastic response realm. Here's a good graph showing the impact of hysteresis:

Spoiler:

Each time the structure is pushed in one direction it's weakened, and so another push causes it to deform even more, despite the fact the push is of the same magnitude. This is equivalent to bending a paperclip back and forth: you'll notice it gets weaker and eventually breaks.

You can break or weaken something without applying stresses beyond the linear range (less than the yield stress), that's the "fatigue" process, but it's a very different mechanism than plastic deformation which is what the OP was referring to.