0:04

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

Â slip in hexagonal close pack systems.

Â If you recall,

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

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

Â Remember the way the FCC structure worked was we had

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

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

Â 0:45

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

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

Â that takes you from the A through the B and

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

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

Â 1:23

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

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

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

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

Â the distance of c is two-thirds of that.

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

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

Â And that's going to be equivalent to the packing

Â distances that we see in the HCP structure as well.

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

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

Â So that's the distance in the FCC structure.

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

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

Â 2:39

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

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

Â Now we've talked about this previously and the module

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

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

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

Â meaning that the spheres are perfectly or

Â the structures are perfectly spherical for the atoms.

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

Â When we look at actual materials,

Â materials like cadmium through beryllium as illustrated on this table,

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

Â systems to the ideal hexagonal closed pack structure we see

Â deviation with respect to that c over a ratio.

Â When we look at zinc, for example, and

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

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

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

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

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

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

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

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

Â we are now below what happened in terms of the packing

Â sequence as compared to the ideal case.

Â 4:32

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

Â We're looking at the ABA packing sequence.

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

Â we refer to these planes as the basal planes.

Â Those are the planes in the hexagonal.

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Packing plane.

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

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

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

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

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

Â 5:24

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

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

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

Â on the prism plane.

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

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

Â 6:16

the deformation in HCP.

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

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

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

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

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

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

Â 7:14

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

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

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

Â that close packed basal plane of the HCP structure works.

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

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

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

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

Â We will then be on the plane of maximum share and

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

Â Now if we take data by recognizing and

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

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

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

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

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

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

Â the critical resolve sheer stress would be.

Â And that's the solid line.

Â So the solid line, then,

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

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

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

Â or we measure the stress necessary to determine the deformation

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

Â satisfying the condition of Schmidt's Law.

Â 9:22

And so, here is our diagram again.

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

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

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

Â by recognizing that we can determine at 45 degrees.

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

Â 9:52

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

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

Â normal, and the deformation access, and

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

Â In this lesson we talked a little bit about

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

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

Â namely the basal planes, in the HCP system.

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

Â have in FCC.

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

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

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

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

Â Thank you.

Â