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This screencast about the elastic behavior of materials.
We have a block of material here with a dimension x0 in the x direction and
y0 in the y direction. And if we apply a stress, what you know to
happen is that this body will get a little longer in the direction we apply
the stress. So in other words, it will have some new dimension,
x, in the x direction, and a smaller dimension in the y direction.
We define the strain in the x direction as just the ratio of the
change in dimension in the x direction divided by the initial dimension.
So this is sort of a fractional change in dimension.
And the same is true in the y direction. So this is just going to be y minus y0 over
y0. Now, you'll notice this one, this strain in
the x direction is going to be greater than 0.
And the strain in the y direction is going to be less than 0.
It's going to be a negative. Because y is actually smaller than y0.
Now, if these strains are small, we can show that there's a relationship
between the stress and the strain. So the stress that I apply in the x direction
is pretty close to being proportional to the strain in the x direction.
And that proportionality constant, E, is just the Young's modulus.
It has units of stress because the strain has no units.
It's a fraction. And the left-hand side is the stress.
So this guy has to have units. The Young's modulus has to have units of stress.
So if we were to plot the stress versus the strains, a stress versus
strain, for any material, you typically get something
that looks like this. As long as the strain is small, you'll get
a line. And the slope of that line is the Young's
modulus. Now, you might think the magnitude of the
Young's modulus would vary with the type of bonds that the solid is made of.
Stronger bonds would make it more difficult to stretch an object.
So the modulus would increase. In other words, for the same stress, I would
get less strain. Let's see how the modulus varies with materials.
So as I mentioned, modulus is in units of gigapascals.
So I'll make a little table here, gigapascals. That's 10 to the ninth pascals.
So the first one I'll list is diamond. Obviously single carbon bonds in the diamond
structure between carbon atoms.
That has a whopping 1,200 gigapascal modulus. Sapphire is another good one, Al2O3, that's
got a Young's modulus of about 400.
Steel, usually nickel-based steel, this is around 200 gigapascals.
How about carbon fiber composite? That's not obviously a homogeneous material.
It's got carbon fiber in an epoxy. Use it for things like your tennis racket
or kayak or the skin of the fuselage on a 787 Dreamliner.
That's got 180 modulus. You can see it competes quite well with steel,
obviously being a fraction of the density.
Density of steel is around 8 grams per cc, whereas carbon fiber composite is
less than 2 grams per cc. Now, let's go to some polymers.
How about high density polyethylene, things milk jugs are made out of?
This has a much lower modulus, around 0.8 gigapascals.
Low density polyethylene, that's like the plastic rings that hold 6 packs of
Coke together, that's down at 0.2 gigapascals. And if you're down here at rubber, that's
0.05 Young's modulus in units of gigapascals.
So you can see materials span a very broad range of Young's modulus, very
broad range of elastic behavior. And we'll see how this is related to the bonding.