There's actually a few other things
that we can determine from the stress strain
curve of a metal that are quite interesting.
And I'd like to just spend a moment looking at that.
So again, we've got the stress here,
on the vertical axis, strain-- typical metal.
We have a curve that looks something like this.
It comes up looking linear elastic, plastic, ultimate
tensile strength, and then fracture.
And so we've determined all of the strengths.
And we have the yield strength, the ultimate tensile strength,
the fracture strength.
But what other properties can we determine?
Well first of all, one of them that you may have heard about
is the ductility.
So ductility has a usage in common language.
You might say, well, what's ductility all about.
If something's very ductile, you might say, well,
it describes how much you can stretch something.
But of course, we know that that's not
really accurate enough.
Stretch, is that referring to elastic or plastic?
So we've got to be better than that.
And in fact, I'll tell you ductility is
a measure of a plastic strain.
So we know clearly it's a strain quantity,
and it refers to plastic deformation, only
plastic strain to fracture.
Now we've got something we can work with-- plastic strain
to fracture.
So let's see well, this is the point of fracture.
That's fracture.
So that fracture, if we unload we'd
have a value here for total strain.
Let me write that in there for you, total strain.
If we unloaded-- if we took the total strain there
at fracture, just a moment before it fractured,
that would be our total strain.
But what if we unloaded it?
Somehow you knew just infinitesimally before it
was going to fracture-- well, we know that the Young's modulus
is structure independent.
So it won't change.
So we would have unloaded that same modulus.
Which means we come back down here to a value on the strain
axis, a finite value corresponding to zero stress.
It's unloaded, there's no stress on it,
but there's still some persistent strain.
That strain has to be plastic.
That's a plastic strain, which means that this strain in here
is elastic.
That's elastic, and that makes sense because what is that?
That's the strain underneath this linear unloading portion.
And the unloading portion, if it's linear,
is governed by Hooke's law.
And we know that's elastic because Hooke's law refers
to elastic behaviour.
So if we unload down and we get plastic strain,
that plastic strain has got to be the ductility.
So ductility, you unload at fracture.
And the remaining strain is the ductility.
Another interesting property that we can determine from this
stress strain behaviour for a metal,
for other material classes as well, is called the toughness.
And the toughness is sometimes not such an intuitive quantity.
You can understand strength, it's force over area.
You got a sense for that.
It's pressure, if you will.
Even modulus you can kind of get a bit of an intuitive
sense for it because it's how hard
is it to bend something elastically.
It's a little harder, but the toughness is-- toughness,
I'll tell you what the tough is.
Toughness is the energy-- it's an energy term.
And it's energy absorbed to fracture.
What we can do actually is integrate.
And that is take the area under the curve.
So if we take the area under this curve here,
it would be this area here, all this area here under the curve
is actually the toughness.
And how do we know that?
We could look at it dimensionally.
If we're taking a product of stress and strain
and look at the dimensions, stress has units of Pascals.
And what's a Pascal?
A Pascal is a Newton per square metre.
Well, I can go on living my life multiplying
whatever I want by 1, and just multiplied this screen by 1.
You didn't even notice.
So here we go, were I multiply Newtons per square metre by 1,
metre over metre, and I end up with a familiar term
in the numerator-- Newton metre.
And of course in the denominator I've got volume units.
But what's the Newton metre?
A Newton metre is nothing more than joule.
So we've now got joules per volume
as units when we integrate under this.
And that's great because we want an energy unit.
So if we integrate under the entire curve up to fracture,
it tells us how much energy went into fracturing that.
And that includes elastic and plastic deformation.
The final thing that we can obtain from the stress
curve, which is another energy unit
and it's quite useful-- it's stored energy unit this time.
We've got stress and strain.
We've got our linear elastic region, plastic deformation,
and fracture-- is the resilience.
And so the resilience is a measure
of the stored elastic strain energy at the yield strength.
So again, we know if it's going to be an energy term, energy
per volume for a given volume of material,
we're going to have to integrate under the curve.
And where are we going to do it from?
Well, we'll go to the yield strength.
And we go down from there.
And if we unloaded at the yield strength--
I'm going to be a little careful about something-- if we
unloaded at the yield strength, you'd
find that you have a little sliver
of permanent or plastic strain.
Maybe it's close to the 0.2% offset strain.
You'd probably have some plastic strain actually accumulated
when we had yield.
For practical purposes, we say it's elastic before yielding
and it's plastic after.
You might have a sliver.
So we're not going to include that if we're
going to be strict with our definition here.
And so that area there is the resilience.
And that area actually is just an area of a triangle.
And we know that the area of a triangle
is 1/2 base times height, which in our case
is 1/2 of-- well, what's the base?
The base is the elastic strain.
And that's good because we're after the stored elastic strain
energy.
So we've got strain elastic.
And what's the height?
Well, the height is the yield strength.
But we can, again, do better than this.
Because if it's elastic, it's the area under this--
or it's the strain underneath this linear unloading portion.
And the linear unloading portion,
we have a mathematical equation for.
We have stress equals E times strain.
It's a straight line.
So that means that the strain is going
to be equal to sigma over E. And we fire that in here,
and we find that the resilience is--
I should erase that-- the resilience is
going to be 1/2 of sigma, and this is sigma yield.
That's what we're using here.
So that's sigma yield over E times sigma yield.
So at the end of the day, the resilience, which we often
use this-- I'll introduced this symbol here.
The full name for this is modulus of resilience.
And modulus is just a fancy word for a special number.
So our special number here is the modulus of resilience.
And we use the uppercase letter U.
Is 1/2 sigma yield squared upon E--
and that's an interest little equation.
It tells you the stored strain energy for a material.
So if you're going to make a material for a spring,
you'd look for something with a high modulus of resilience.
And again, the units here, the dimensions
here of modulus of resilience are going
to be joules per cubic metre.