There are
a few other things that we can determine
from the stress-strain curve of metal that
is 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. We have the yield strength,
the ultimate tensile strength, and 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 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
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 a 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 Young’s modulus is structure-independent. So it won’t change. So we would have unloaded
that same modulus. This 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 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 behavior. 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
behavior for a metal, for other material classes as
well, is called 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
is integrate. And that is to 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 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 looking at the dimensions,
stress has units of Pascals. And what’s a Pascal? A Pascal is a Newton
per square meter. Well, I can go on living
my life multiplying whatever I want by 1 and just
multiply this screen by 1. You didn’t even notice. So here we go, where I multiply
Newton per square meter by 1, meter over meter, and I
end up with a familiar term in the numerator– Newton meter. And of course in the denominator
I’ve got volume units. But what’s the Newton meter? A Newton meter is
nothing more than a 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, is
another energy unit and it’s quite useful– it’s
a 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. Thus 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 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 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 the 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 introduce
this symbol here. The full name for this
is the 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 interesting
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 meter.As found on YouTubeExplaindio Agency Edition FREE Training How to Create Explainer Videos & SELL or RENT them! Join this FREE webinar | Work Less & Earn More With Explaindio AGENCY EDITION
So we’ve sketched stress-strain. A curve like this
for typical metal. And we know an equation within
the linear elastic region. That’s before the
proportional limit. And that’s, of course,
Hooke’s Law– sigma equals E times epsilon. But what do we have after
plastic deformation? How can we perform calculations
after plastic deformation? If, for example,
we had something that– a bolt in
the ceiling– oh, that’s a horrendous drawing. Let’s fix that. Let’s fix that
before I lose my job. So here’s– that’s even worse. Oh, my goodness. OK. This is good. No, that’s still awful. But there’s a hook
and, I don’t know, something is hanging from it. I was going to draw
a force so I don’t have to draw something else. There’s a hook hanging
from the ceiling and you apply a
force to it and you want to know,
well, how much does that tie– that’s
what that is– it’s called intention– that
tensile tie– elongate? What’s its distance? What’s its length? And, well, you can only, at this point, do calculations
if the force results in a stress that’s
less than the yield strength. If it’s more than that, while
it’s classically deformed, we couldn’t deal
with it unless we had some kind of
way of describing the shape of the
curve after it leaves the linear elastic region. And it’s nice. We do have an
equation that fits the curve. But it’s going– we can’t
use engineering stress. We have to use what’s
called true stress. So that’s what I’d like to
introduce to you right now. So again, we’re going to take
a look at a generalized sample here. And the idea– the
goal– is of course that we’re going to go to
be able to this– calculate and understand
plastic deformation. So here’s our sample with its
initial cross-sectional area as we’ve discussed before. And we said, well,
when you load it, it gets longer
and gets narrower. And it’s that reduction in
the cross-sectional area that we’re interested in. This area here now, we could
call it an instantaneous area. Whereas this area, the white
one, was the initial area. The initial area was
what we started with. But then while the load
is applied– sorry, let me draw the force in. While the load is applied,
the cross-sectional area has decreased. So the first thing
you could do is we could say, all right, well, that
means that this material itself is experiencing a
force over a smaller area. So we could define
the true stress as the force over that
actual cross-sectional area. This is the true stress. This is the stress that
the material itself is feeling. OK? And that subscript
I am telling us this is the instantaneous
cross-sectional area. Instantaneous. I can’t do more than one thing
at the same time. Instantaneous– I missed a
U– cross-sectional area. Instantaneous cross-sectional area. And I’ll show you
what the plot would look like in just a moment. We could also do the same
thing for the strain, although that’s going to be
just a little bit– require a little bit more thinking. The true strain has to
account for the fact that what we’re doing is we’re
applying a change in length. Right? We’re elongating it
over a certain length. The very first little
bit of elongation is elongation over l0. But then, after
that, the elongation is elongated over the
previous length which was l0 plus that little delta l. And so if you do that
for infinitely small– infinitesimally small
changes in length, the way we would write that
is we’d have to say, the true strain, what we’re
doing is we’re integrating. We’re integrating those
infinitesimally small changes in length– that’s dl– by l
from l0– the initial length– to the instantaneous length. And so if we do that,
you find that you have ln of l instantaneous
minus length 0, which is ln of l instantaneous over l0. So we have another
equation there. I’ll put a box around that. So this is the true strain. True strain. And if we take that true
stress and we plot it against the true strain,
I’ll show you what we get. Let me just
plot stress and strain and I’ll show you what
we’ve already seen. That’s the engineering
stress-strain curve. And then what I’ll do is I’ll
plot for you– after it starts to plastically deform the–
ran out of space there– the true stress– so this one
here– continues to increase. It doesn’t have that
decrease at the UTS. That’s the true stress
true strain curve. And this one is, of
course, engineering. The nice thing
about this plot is once you’ve got true
stress and true strain, we can fit that data
quite nicely for most metals with a simple equation. And that is true stress
is equal to this coefficient times the true strain
raised to the power n. So that’s an equation that fits
that true stress true strain data quite nicely. And what’s useful about
this is these are constants. That’s the constant n– I’ll
define it for you in a moment– and this K is also a constant. Those are material properties. We can look those up in
an engineering handbook. So n is called the–
well, this equation is called the
strain-hardening equation. Strain hardening equation. And strain hardening– hardening
correlates to– hardness correlates to strength. So really this is
the equation that’s telling us that we’re
strengthening the material and we’ve got the
strain hardening exponent and the strain
hardening coefficient K.As found on YouTubeExplaindio Agency Edition FREE Training How to Create Explainer Videos & SELL or RENT them! Join this FREE webinar | Work Less & Earn More With Explaindio AGENCY EDITION
And how do we know that? We could look at
it dimensionally. If we’re taking a product
of stress and strain and looking at the dimensions,
stress has units of Pascals. And what’s a Pascal? A Pascal is a Newton
per square meter. Well, I can go on living
my life multiplying whatever I want by 1 and just
multiply this screen by 1. You didn’t even notice. So here we go, where I multiply
Newton per square meter by 1, meter over meter, and I
end up with a familiar term in the numerator– Newton meter. And of course in the denominator
I’ve got volume units. But what’s the Newton meter? A Newton meter is
nothing more than a 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, is
another energy unit and it’s quite useful– it’s
a 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. Thus 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 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 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 the 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 introduce
this symbol here. The full name for this
is the 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 interesting
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 meter.As found on YouTubeExplaindio Agency Edition FREE Training How to Create Explainer Videos & SELL or RENT them! Join this FREE webinar | Work Less & Earn More With Explaindio AGENCY EDITION
And so if you do that
for infinitely small– infinitesimally small
changes in length, the way we would write that
is we’d have to say, the true strain, what we’re
doing is we’re integrating. We’re integrating those
infinitesimally small changes in length– that’s dl– by l
from l0– the initial length– to the instantaneous length. And so if we do that,
you find that you have ln of l instantaneous
minus length 0, which is ln of l instantaneous over l0. So we have another
equation there. I’ll put a box around that. So this is the true strain. True strain. And if we take that true
stress and we plot it against the true strain,
I’ll show you what we get. Let me just
plot stress and strain and I’ll show you what
we’ve already seen. That’s the engineering
stress-strain curve. And then what I’ll do is I’ll
plot for you– after it starts to plastically deform the–
ran out of space there– the true stress– so this one
here– continues to increase. It doesn’t have that
decrease at the UTS. That’s the true stress
true strain curve. And this one is, of
course, engineering. The nice thing
about this plot is once you’ve got true
stress and true strain, we can fit that data
quite nicely for most metals with a simple equation. And that is true stress
is equal to this coefficient times the true strain
raised to the power n. So that’s an equation that fits
that true stress true strain data quite nicely. And what’s useful about
this is these are constants. That’s the constant n– I’ll
define it for you in a moment– and this K is also a constant. Those are material properties. We can look those up in
an engineering handbook. So n is called the–
well, this equation is called the
strain-hardening equation. Strain hardening equation. And strain hardening– hardening
correlates to– hardness correlates to strength. So really this is
the equation that’s telling us that we’re
strengthening the material and we’ve got the
strain hardening exponent and the strain
hardening coefficient K.