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In this lesson, we're going to be talking about qualitative discussions of

slip in hexagonal close pack systems.

If you recall,

one of the ways that we can describe packing in the hexagonal close pack is to

compare it to packing that we see using the hard sphere approximation for FCC.

Remember the way the FCC structure worked was we had

ABCABCABC packing, where as in the case of the hexagonal clothes

pack as shown on the right, the stacking sequence is ABAB.

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with respect to the hexagonal close pack structure, that arrow direction,

the magnitude of that is two thirds of the magnitude of the vector

that takes you from the A through the B and

the C back to the A again, which is in the face centered cubic structure.

So what we've done in the past is to relate those two directions,

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Now, what we were able to do then is to relate in the C axis

how long the direction is or repeat direction is in the HCP.

So we start out and recognize that, that arrow that's given the unit of one.

The magnitude of that is a zero under the square root of three, so

the distance of c is two-thirds of that.

Then what we can do is to use the close pack directions

in the playing of the one, one, one in the face center cubic structure.

And that's going to be equivalent to the packing

distances that we see in the HCP structure as well.

And what we're then able to do is to recognize that a in the hexagonal system

is going to be related to A0 under the square root of 2, divided by 2.

So that's the distance in the FCC structure.

So now we have both c and a expressed in terms of

the dimensions of the unit cells with respect to the face center cubic system.

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And when we take a ratio, what we get is the ratio of C

over A is the square root of eight over the square root of three.

Now we've talked about this previously and the module

that had to do with crystallography but I wanted to bring this to your attention.

And remember, what this C over A Is in this particular case,

it happens to be the ideal packing arrangement for the ATP structure,

meaning that the spheres are perfectly or

the structures are perfectly spherical for the atoms.

And as a consequence, we have this ideal c over a ratio.

When we look at actual materials,

materials like cadmium through beryllium as illustrated on this table,

what we see is that when we compare the c/a ration in this particular system.

systems to the ideal hexagonal closed pack structure we see

deviation with respect to that c over a ratio.

When we look at zinc, for example, and

we're going to be looking at zinc a little bit further along in this lesson.

But when we look at zinc, it has a c/a ratio which is greater.

This would suggest that we're stretching a bit in the c direction.

It may also indicate that, rather than having perfect spheres,

we have structures in terms of the atoms as being deviating from that,

maybe an ellipse, so that the deviations lead to this increase in

the c axis between those Densely packed plains in the HCP system.

And then again when we start looking at elements like magnesium through barilium,

we are now below what happened in terms of the packing

sequence as compared to the ideal case.

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So let's take a look at the ideal system with respect to the HCP structure.

We're looking at the ABA packing sequence.

And you can see how those planes are related to one another and

we refer to these planes as the basal planes.

Those are the planes in the hexagonal.

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Packing plane.

And they are the densest packed planes in the Hexagonal System.

And as a consequence, what we would expect to see is,

we would expect to see only slip on those densely packed planes.

And they would be in the directions of the vectors indicated on the diagram.

As into the direction of A1, A2, and A3 which described the HCP system.

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Now, it turns out if we deviate the C over A ratio, we begin to look at

the process on other planes and of on the case that we have up here on the diagram.

What we're actually looking at is the possibility of having slip

on the prism plane.

So, what we would have to do is to look at the atoms and

how they are distributed on this plane and calculate the relative planer density.

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the deformation in HCP.

But I want to point out that when you look at the close pack directions and

the close pack planes, what you see is when you're talking about the number of

potential slip systems, what you have is effectively slip on one plane,

which is that densely packed plane and in the three directions a1, a2 and a3.

So we're limited into the amount of slip that we can have in this system.

What we're going to do is to use this to an advantage to show the possibility

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So when we take a look at deformation in the hexagonal system,

what we can do is we can look at the critical resolve sheer

stress that we have and now what we want to think about is how

that close packed basal plane of the HCP structure works.

One of the things that we know from an earlier lesson is that the plane

of maximum shear that we can have in a cylinder of material Is it 45 degrees?

So what that's going to help us do is if we were to take our single crystal

of zinc and orient it, so that the basal plane was at 45 degrees.

We will then be on the plane of maximum share and

in this particular case, this is where the deformation is going to occur.

Now if we take data by recognizing and

determining what the critical resolve shear stress is at 45 degrees,

what we can then do is to go back and use the equation

that's at the top of the slide that describes Schmidt's Law.

And then what we're able to do, by keeping the angle cosine theta, which is

the slip direction and the deformation direction, keeping that constant.

And then what we can do is to calculate what the variation and

the critical resolve sheer stress would be.

And that's the solid line.

So the solid line, then,

represents the values that we would calculate using Schmidt's Law.

Now if we look at the red dots, the red dots actually represent the real data.

So what this is telling us is that if we measure the critical resolve shear stress,

or we measure the stress necessary to determine the deformation

at different orientations, what we find is that we are in effect

satisfying the condition of Schmidt's Law.

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And so, here is our diagram again.

Here is our slip plane, we're looking at variations with respect to the slip plane.

That is oriented at different orientation with respect to the stress axis and

consequently were able to reproduce the curve on the left hand side

by recognizing that we can determine at 45 degrees.

What the critical resolve sheer stress is by looking at the minimum in this curve

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So that's our angle fined that we'd be interested in, and

it is related to the angular relationship between the slip plane,

normal, and the deformation access, and

theta then represents the angle associated with the Force and the slip direction.

In this lesson we talked a little bit about

the process of deformation in a hexagonal close-packed system.

We recognized that there is one densely packed set of planes,

namely the basal planes, in the HCP system.

And that packing density is going to be similar to the packing density that we

have in FCC.

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law to in fact show that the critical result of sheer stress that we measure

in a material where we can restrict slip like the zinc single crystal.

Then what we're able to do is to make some calculations and

in fact show that the slip process does follow Schmidt's rule.

Thank you.