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STUDENT: Let's talk
about the Mohr's Circle,

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an important tool
in materials science

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and mechanical engineering.

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What is Mohr's Circle,
and what does it do?

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Assume you have a
rod, and you want

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to break it with your hands.

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What can you do about it?

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You can pull it,
compress it, or twist it.

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Each of these will
induce a stress

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in the rod in a different way.

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And the Mohr's Circle
helps you analyze this.

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Mohr's Circle is a
graphical tool, a visual way

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to see how to go from one
state of stress to another.

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Usually you are given
a state of stress

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that you calculated from
using stress equations.

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Then you want to find
a state of stress

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at a new angle of
orientation, or principal

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stresses at that orientation,
or the maximum in-plane shear

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stresses.

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And Mohr's Circle
allows you to draw

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this representative
element to visualize

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the calculations for you.

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Now, let's derive
the Mohr's Circle.

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So now we take a look
at the piece of material

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that we cut from the rod
that we're trying to break.

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And originally we had
this coordinate system.

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And we have some stresses,
where the signals

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represent the normal
stress and the tiles

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represent the shear stress.

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So we draw this
representative pair

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of stresses for convenience.

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And then we want to
analyze the stresses

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of this cross-section area.

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So what we do is, we need to
rotate the coordinate system

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so that x-axis would line up
with the unit normal vector

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to this cross-sectional area.

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And then the y-axis would be
rotated to this direction,

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and would be lined up with the
n prime unit vector shown here.

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So this is done by rotating this
coordinate system by an angle

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theta counterclockwise.

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Then we can obtain the unit
normal vector in matrix form.

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And that is just cosine
theta and sine theta.

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And similarly, the n prime
vector, that's just negative

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sine theta and cosine theta.

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So we can now get
the stress tensor

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by taking the dot
product of the stresses

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in the x-direction with the
unit normal vector, and the dot

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project of stresses in the
y-direction with the unit

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normal vector.

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So now we can write
out the normal stress

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in the new coordinate
system by just

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taking the dot product
of the stress tensor

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with the unit normal vector.

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And similarly, we can
find the shear stress

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in the new coordinate
system by taking

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the dot product of the stress
tensor with the n prime.

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And we can do these
complicated calculations

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on Mathematica, which I
will show you right now.

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So I've already typed in these
vectors into Mathematica.

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And n represents the unit normal
vector, and np is just n prime,

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and t is the stress tensor.

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So we want to find
the normal stress

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in the new coordinate
system, sigma prime.

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And that is just
dotting t with n.

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And we get this
simplified version.

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That.

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So that's our sigma prime.

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Then we want to
find our tau prime.

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And that is just
t dotted with np.

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Again, we want to find
the simplified version.

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So that's that.

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And don't worry about
this thetas and 2 thetas.

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We'll use trig identities
to change them.

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So here's our trig identities.

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And we just plug them back in.

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And we obtain these two
equations for our new stresses.

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Then we move all
the terms containing

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2 theta to the right
side of the equations,

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and all the rest of the terms to
the left side of the equations.

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And we square both
sides of the equations

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to get equation 1 and
equation 2, as shown here.

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We add equation 1 and equation
2, and some terms cancel.

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Finally, we get this really
clean equation shown here.

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Now look closely
to this equation.

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What form of equation
does it look like?

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You might have
guessed that it looks

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like the equation of a circle,
and you're absolutely right.

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This is exactly how
Mohr's Circle is derived.

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We can compare the equation
with the equation of a circle,

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and we see that Mohr's Circle
is centered at the coordinates

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sigma x plus sigma y over 2, 0.

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And sigma x plus sigma y over
2 is the average normal stress

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in the old coordinate system.

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We can also obtain the
radius of Mohr's Circle.

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Now let me draw this
Mohr's Circle for you.

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So we go back to our
old coordinate system

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with some calculated stresses.

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And then we need to construct
this new coordinate system

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to draw this Mohr's Circle on.

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I take the positive
x-direction as the plus sigma

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and the negative y-direction as
plus tau, just by convention.

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Then, in order to
draw the circle,

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we take a pair of normal
stress and shear stress

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and we plot their values
as the coordinates

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in our coordinate system.

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So we find a dot,
and it's represented

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by sigma x and tau xy.

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Then, by our previous
equations that we derived,

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we can find the center
of the circle, which is

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located at sigma average and 0.

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And sigma average is just
sigma x plus sigma y over 2.

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Then we find its values, and
we plot that on our sigma axes.

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That's just our
center of the circle.

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Then we connect these
two dots that we drew,

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and that's just the
radius of the circle.

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Now we can draw this circle,
since we know its center

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and a dot on its circumference.

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We can now confirm the equation
for the radius of the circle

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by doing some geometry here.

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So, this length is
just sigma average.

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And then we have another
length that's just sigma x.

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And the difference
between these lengths

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is just sigma x
minus sigma average.

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Then this line right
here is simply tau xy.

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And by looking at
this right triangle

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and using Pythagoras'
Theorem, we

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can obtain the radius of the
circle, which is written here

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under the square root.

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Actually, it confirms
our original equation

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for obtaining the radius
r of Mohr's Circle.

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So Mohr's Circle is very useful
for visualizing the stresses

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on the material.

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So we can, just by
doing simple geometry,

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find the values for
its principal stresses.

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So in green, you can
see on the circle,

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there is the maximum
normal stress

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and minimum normal stress.

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And now, in red, I'm
labeling the minimum

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and maximum shear stresses.