DP_S2012_07

Aeronautical Engineering, nuclear engineering — those were the first of their type in the country, were at MIT. In any case, Harvard, between 1872 and 1917, tried to purchase MIT three times, because they wanted an engineering school. People were building up engineering schools and whatnot. I handed out the little thing on leadership at MIT where I quote Thomas Edison, where he says MIT was the greatest school in the country and blah, because businesses were learning that engineers were useful to their professions or to their economy and stuff.

So in any case, in 1917 — actually between 1914 and 1917 — MIT and Harvard merged for three years, and students were getting a degree from both schools in engineering. Professors had joint appointments at both departments, both schools. But in 1917 the Supreme Court of Massachusetts knocked apart the whole deal, and MIT was then forced to have to try to pay for these buildings you're sitting in right now. They had started building them — this building was built in 1917 — and they had been building it and they were trying to do the merger and Harvard was going to pay for it, but all of a sudden they weren't going to be able to merge because the Supreme Court of Massachusetts said that private university couldn't buy the land grant college.

And so MIT was bankrupt because they weren't going to get some of the Harvard money, and an anonymous donor bailed them out with $6 million. Anybody know who the anonymous donor was? Building 6 is named after him. It's not Mr. 6, it's Mr. Eastman. George Eastman of Eastman Kodak gave MIT $6 million. Over his life he gave about $30 million to MIT, which would be like three or four hundred million today. He was very generous to MIT. He really liked MIT, just like Thomas Edison. Anybody know what company Thomas Edison started? General Electric, right.

So in any case, in the meantime, Harvard didn't have an engineering school, so they took the land Andrew Carnegie bought them, which was right across the river from down from Harvard Square, and they built their own engineering school. And today you can drive by — it's known as the Harvard Business School, because back then businesses and engineering were sort of synonymous. So the Harvard Business School is technically Harvard's engineering school. But that wasn't why they came in [first] in the country in 1990s [1900s] in engineering.

Okay, questions people have? One question came up, one student emailed me and he wanted to do his project on touchscreen displays. And of course Wednesday's when you're supposed to tell me what your project is about, and you don't have to write up a whole thing, you just scratch out in pencil on that little form I gave you what it is you want to do. I just want you to know that you thought about it. I wrote back, the touchscreen displays — if you talk about the whole display, too complex. It would be a superficial presentation to talk about the lighting and the LEDs and the stuff. If he wanted to talk about the coating on the touchscreen display, that would be fine. What's the coating, anybody know? It's indium tin oxide. And what's unique about indium tin oxide? It's transparent. It's a transparent conductor. It conducts electricity and it's transparent, and usually those two things don't go together.

In fact, it's the only material in the world that I know of that both conducts electricity — electrons — and is transparent. Ordinarily if you have free electrons it will absorb light and therefore it's not transparent. Are there any other devices, or are there any other — it's the only material I know of. But I'm sure people are trying to develop other materials. Why? One's the wear resistance. No, it's not. When you're making touchscreens there might be a dollar's worth of indium on there. Tin's not that expensive, about the price of silver. But the problem is, it doesn't have great wear resistance. And you ever go into an automatic teller and down where they have the button in the same corner, and people hit it with their fingernail, and it wears out? So wear is a problem.

And so for example, if this particular student who will be making a presentation wants to talk about the wear resistance of indium tin oxide, he wants to talk about the properties of indium tin oxide and why it is both an electronic — and an electronic conductor and transparent, if he wants to talk about how you process it, which is probably vapor deposition so far as I know, I'm not sure — any one of those topics or a combination of a couple would be fine for a project. But you can't take something as complex as a display. You couldn't take something as complex as that projector, because what are you going to say in 10 minutes? Well, projector projects light. Okay, well I want something a little more depth than that. So the reason I'm asking you to pick something — I want it picked, I want something very simple, because something very simple, even a paperclip, if you get into it, there's problems of corrosion resistance, making it low cost. Design of a paperclip — how many designs of paperclips are there, right? You got to make it out of steel or plastic or what, and what are the tradeoffs. A very simple product will bring up lots and lots of questions. So I don't want you to come up with some complex product. Everybody understand? Keep it simple. KISS, right. Okay, so that's my thoughts on the projects that you're supposed to talk about.

So today, just to kind of review where we were, we are, we've talked about forming methods. And generally you can think of, if you're going to use a temperature difference — which is what the blacksmith used to do — you have a cold tool and a hot workpiece. And so you can form it with the same material, but the tool is cold and the workpiece is hot, and therefore the workpiece deforms and the tool doesn't. It's not very useful if the tool and the workpiece have the same hardness. If I tried to deform clay with clay, what's going to happen? They're both going to deform and I end up destroying my tool, right. So that's what the blacksmith did.

There's also isothermal forming, which has been used for years, where you have a hard tool and a soft workpiece. Well, if I'm using a drill, I've got a hard tool, the drill, and the workpiece hopefully is not as hard as the drill. You can't drill diamond except with another diamond. You can drill diamond with diamond powder, but it's very slow process. The tool is often heat treated. They learned how to heat treat steel back thousands — several thousand years ago. It really became an art about a thousand years ago. They didn't have a lot of steel — they had some steel, they got it from meteorites and other things. They could make it as wrought iron back a thousand years ago. But they had to forge it, which is basically the — top cold, a cold workpiece in a hot tool — hot workpiece, a cold tool in a hot workpiece — and then they had to heat it and quench it and temper it.

And one of the things, anybody know about the Damascus swords? The Damascus swords, they had nice little patterns on them that they etched on the surface. Anybody know — they felt that the best swords were not just quenched in water, they were quenched in a slave. And not a dead slave but a live slave. And that gave them power against the enemies, to kill one of your slaves to quench your sword. I'm glad I wasn't a slave back then.

Anyway, so you anneal the workpiece, and there may be multiple anneals and multiple dies, and we've talked about different shaping dies and all the steps you have to go through. And you have to anneal in between — not on every step do you have to anneal — but you can only get so much work into something before it becomes too hard and starts approaching the hardness of the tool.

The other thing you can do is to use a die that has high hot strength and use something today that's super plastic, and we're going to talk about that. You can do everything essentially in a single die. Now, super plasticity back — often came up with — uh, he reinvented it, rediscovered it, if you will, back in the early 60s. We've talked about it. I'm not going to talk — we're probably going to be tomorrow or so before we get to it. But let's see, did I — oh, here it is. No wonder I can't find it. This is an example of a part. This was made in 1988, but in some of the super plasticity stuff, people were forming out of titanium super plastic parts by the mid 70s. That's one of the quickest adoptions of technology — less than 15 years — of any technology I can think of. There's only a couple that I can think of. The average time to adopt a new technology in materials processing is about 20 years. To do it in 10 or 12 years is incredible. Just demonstrates the need.

But this basically shows, in the darkest gray here, is the final part. And I don't know specifically, but it's probably some part of aircraft part that has a bunch of ribs and kind of an eggshell or egg crate type of construction. And so here's one of the ribs, stiffening ribs, and this is what they finally machine. The conventional forging would have been something in the white outline here, and the super plastically formed part would be something that would be in the gray part, and the actual machined part. And if you go up here, conventional versus hot die — which means isothermal super plastic forming — the machined part weight is 28 kg. The formed weight for conventional was 154. That gives you a buy-to-fly ratio of about 6:1. The amount of machining was 126 kg. Well if you're paying $300 a kilogram for titanium, you can multiply that out. This is not a cheap part. And if you can save 50 kg by going to isothermal forging at $300 a kilogram for your titanium, plus the machining cost, which probably brings it up to $600 a kilogram, you're saving a lot of money in the overall part. So the net savings of kilograms was 45 kg for that particular part. So there can be some big savings if you go to isothermal forging.

Now, any questions on that? Today I want to start getting in a little more technical. A lot of my stuff has been kind of descriptive. I want to talk about plasticity, and this is actually in the very beginning of Hosford, first chapter. You have a stress tensor — this is generalized stress tensor — and for plasticity, it doesn't matter what your orientation is. σ₁₁, σ₂₂, σ₃₃ are the tensile stresses or compressive stresses in the three directions. If they are actually principal stresses, where the shear stresses go to zero — you can always find an orientation where the shear stresses can go to zero — then we just call it σ₁ rather than σ₁₁. That's the notation in the book, most people use. So principal stresses are σ₁, σ₂, and σ₃. Shear stresses, we have subscripts that are not equal, or not the same — 1 1 or 2 2 — 1 2, 1 3, and 2 3. And these three up here are the same identical as 2 1, 3 1, and 3 2, and you can prove that from symmetry and other things if you go to the whole stress tensor.

But the important thing to remember is, the three tensile stresses or compressive stresses, if they're principal stresses, equal — when summed together — actually I should divide by three there — divided by three is the hydrostatic stress. And this is all in your book. So the hydrostatic stress is one-third of the sum of those. Now the strain tensor is essentially the same thing, except we write it down for strain. But the sum of the strains in plasticity is zero, because nu is equal to a half. And we're going to talk today about this R value, which is just the incremental strain in the two and three direction if I'm pulling in my one direction or compressing in my one direction. This is all in your book.

That leads to biaxial straining, which is not something most of you have studied, but is critical to understanding deformation processing, particularly in sheet metal. Very rarely are we processing something by just pulling on it in one direction. There is one example I can think of. We make millions of — well, millions of miles — millions of feet every year of something that we pull on in one direction. Can anybody think of it? Optical fibers, very good. We take optical fibers, Corning takes optical fibers, they build up this composite layered glass and then they just pull it. They just, without a die, they just pull it into long thin fibers, and they put miles and miles of that all around the country every year. That's because glass is Newtonian flow, which is not all that different than super plasticity. But we'll get to that a little bit later.

But in general, if we're going to form something of more complex shape, it's going to be stressed in two directions if it's sheet metal. So we're interested in biaxial flow, biaxial plasticity. And we have — you've seen this before — we have either the Tresca condition, which is this kind of straight-line yield criteria. And all we're doing is, we're plotting σ₁ versus σ₃, or σ₁ versus σ₂. Usually the horizontal axis is our highest stress. If this is σ₁, σ₃, these would be principal stresses, but they don't have to be — it could be σ₁₁, σ₃₃. In any case, we have the Tresca condition. Anybody remember — I think Dr. Belmar told you — what the criteria for the Tresca condition is? It's called MSC, maximum shear stress condition. So if you're talking about a non-crystalline material, the maximum shear stress would be on the 45° plane, at an angle 45° to your pulling direction, and the yield stress would be half of the pulling stress. That's P over 2 to Y, right, where τ would be the shear stress in the shear direction on the shear plane, for a glass. That's not true of a crystalline material, and we're going to get to that hopefully today.

The ellipse around this is the von Mises criteria. And the von Mises criteria is also known as the strain energy criteria. And the strain energy — maximum strain energy — there's only so much strain energy the material can support. And this is — see, the hexagonal prism is the maximum shear stress criteria. This is the general statement of the von Mises, and everything's a square of the strain — I mean, of the stress. That's a strain energy — strain energy is proportional to σ². And so we have the differences between the three principal stresses happens to be 2σ_y², so far as that goes, and that's the equation for the von Mises ellipse. There's the von Mises ellipse, and using that equation σ₁, σ₂, you can come up with certain little criteria here. You don't have to worry about those things — they're in the book somewhere.

But in fact that is just a two-dimensional plot of our three-dimensional σ₁, σ₃, σ₂. This is a three-dimensional cube here, axes. And we have a cylinder going right straight up the 111 direction out of that cube, right, if you're a Miller indices materials person. And all you're seeing — the ellipse is the cross-section of a cylinder at some angle. And you have the Tresca condition, which actually is a pure hexagon if you think about it in the 111 direction.

So that's just sort of a review of things we've talked about. Now it turns out today we're going to talk about the fact this doesn't have to be a cylinder. It can be an elliptical cylinder. Who says that the strength in the through-thickness direction of a thin sheet is the same as it is in the sheet, pulling in the direction of the sheet? And who says that pulling in the one direction is the same as pulling in the two direction, if one is the rolling direction of a piece of sheet, and two is the transverse direction, and three is the through-thickness direction? Why would anyone think that all three of those have the same strength, particularly when you know that we get this grain structure, this texture, that makes it look like the same type of structure as wood?

Wood is clearly an anisotropic material, right. You can split it with a wedge if you split in the right direction. Anyone ever try to split wood with a wedge in the transverse direction? It's not so easy, right. It'll split along the other direction, right, because of Poisson's ratio, right. So next time you're splitting wood in the wrong direction, you say "forgot Poisson's ratio," right, and everybody will look at you a little funny. But anyway, so it turns out, to understand deformation processing, we have to get into biaxial straining and biaxial stresses so far as that goes. So that's what we're going to talk about today.

Now, one of the things that Hosford got — and this is the old edition of Hosford, but he basically points out, for plasticity — Hooke's law is stress is proportional to strain, and the proportionality is Young's modulus. This is Hooke's law. And ε strain to the 1.0 power is just the strain. So stress is proportional to strain — with a, you know — you learned that as a sophomore in mechanics. But in plasticity, stress has a different constant of proportionality, and it has an exponent that's less than one. And we know that because if you run a tensile test — and this is figure 3-4 in an old edition of Hosford, it's also in Backofen, and comes from a paper in 1969 — where they plotted true stress versus true strain on a log plot. And anyone knows that anything plotted on a log plot is a straight line. Well, almost anything. The straight lines could be different. But in fact for elasticity it's just 45° slope, because Hooke's law — and Hooke goes back to the 1650s — stress is proportional to strain, and so it's a 45° slope even on a log plot. This is elasticity.

But when something goes plastic, even though we know that stress-strain curve in an engineering stress-strain curve looks like that — and here's your — it's not true stress and true strain, but here's your curvature. And it turns out that if you put it on a log plot, becomes a straight line, and it has a slope of n. Now this is 1100 aluminum. 1100 is 99.9% aluminum. And the n value is a quarter, that this person measured. So we have this relationship that we can use, of σ is equal to K ε to the n. And n is going to be one of the important variables when we characterize different materials and how they're going to behave in different types of stress and strain.

And in fact, if I go to typical sheet metal properties — let's see, aluminum alloys, this guy says — and this guy actually is the super plasticity book, but no, I'm sorry, this is Hosford on page 289 of your edition, third edition. So you don't have to copy this down, you just make a note if you want to know — it's on page 289. Aluminum alloys, instead of 0.25, he gives a range. Here's an R value, we're going to define that a little bit more and show you what it means. But R values can go from 0.6 all the way up to 5. And that's the strain in the two direction versus the three direction. That's what R is — it's that ratio. m is in fact — if σ is equal to K ε to the n, σ is also equal to — they usually use S — ε-dot to the m power. And ε-dot is just the strain rate, right. People always use ε-dot — use dot quite often for the derivative with respect to time. So this is just the stress proportional to strain, this is the stress proportional to the strain rate.

A Newtonian material has an m of one. It's not dependent on strain rate. It will just pull — you know — and will not neck down, it'll just pull forever. He doesn't have glass up here. Glass has an m value of like 0.8, and that's why Corning can take these optical fibers and just pull them for miles in tension without a die. Because whatever shape it starts out with, you can just pull it forever and thin it down. You can pull for tens of thousands, if not hundreds of thousands of percent, and you'll end up with something — if it started out uniform cross-section, it'll end up a uniform cross-section. However, the m values for most materials are very low. They're very strong functions of the strain rate. If you can get m up to 0.3, it will start to behave super plastically. So that's the difference. Super plastic materials have large m values like 0.3. Most metals have super plastic — most metals have m values on the order of 0.01 or 0.02 maybe. So most materials neck down. We'll talk about necking once I review it and try to remember how necking gets to be fairly complex, so I got to review it and get to that. But we'll get to it eventually.

By the way, I haven't talked about schedule. I forgot to talk about schedule because I was telling the story of Harvard and MIT and engineering schools. The schedule is, I'll lecture Monday and Wednesday this week, and Dr. Belmar will do Tuesday, Thursday, and Friday. And right now we might even finish a week before spring break, but nonetheless you can come and watch the other videos which you haven't really watched a whole lot of. I think you've only done one as a class, right. Although the others are online, and you may decide that you're going to do those offline rather than making Jeremy come in and show it all to you. But we'll talk about that when we finally finish the live lectures. And then sometime before spring break, after I get your things — your assignment on Wednesday of what it is you want to do a project on — I'm going to try to group things a little bit. You know, if one person is doing touchscreen displays and another person is doing something else on displays, or one person's going to be doing forging of titanium and another one is going to do some titanium sheet metal part, I may try to combine things. So we'll do two a day. You're going to have about 10 to 12 minutes for your presentation, and then we're going to have 10 minutes of discussion, and so we'll be able to get about two in a day. And we'll do six or seven of those. Probably won't start until like mid-April, but hopefully we'll finish by the end of April. So then you'll be done with the course, ooh, and you can do all your exam studying.

Lucky you. You know, I'm so glad I don't have to — when I left Bethlehem Steel, the thing that made me most happy was to no longer have to write a trip report. Whenever I travel for Bethlehem Steel, I'd have to come back and I'd have to write a trip report. Well, I went to Pittsburgh to see Westinghouse and met with you know, blah blah. Anyway, I was so happy when I came back here as an assistant professor, I didn't have to write trip reports. I was also very happy when I graduated from school and no longer had to take exams. Although, anyway, that's another story. There's still some types of exams you have to take.

Now, this is the Tresca criteria. If you just want to think of some little volume element with axes x₁, x₂, x₃, uniaxial tension — the maximum shear stress will be on some planes at 45°. Now, this is assuming it's a non-crystalline material. It's something like clay. The maximum shear plane is going to be — without any crystalline preferred orientations from the crystal structure, it's just going to fail on the 45° plane in a direction that would be in the direction of these lines. And that's going to give you a shear stress on that plane, and things deform in shear. Remember that, shear stress is going to be 1/2 of σ_y. The shear stress will be 1/2 of σ_y, and that's the basis of the construction of Mohr's circles, which I haven't gone through, but that's the Tresca criteria. And we've already talked about that.

Now, well actually I can put this up very briefly. This is out of your book, Hosford, when he's talking about plasticity, and this is just coming out in the 111 direction. There's the circle, there's the Tresca hexagon, the circle becomes an ellipse. That ellipse has interesting properties, and the ellipse is more physical. Nature doesn't like corners. Why would nature want to have sharp corners? If you're pulling in a slightly different orientation, if you're pulling something in directions by biaxial stretching, why would you have corners? But anyway, the ellipse itself has the property that the normal to the ellipse is the strain. So if I take, at this point, I actually am now in what we call a plane strain condition. If I make a vertical tangent here, then obviously the strain — and this strain will be dε₁ — there will be some vector of whatever the strain is in the one direction there. If I did the other direction, there is no strain in the two direction. This particular position where I have a vertical tangency is called the plane strain condition, and it becomes very important in a lot of our processing.

And the reason it's important is, if you go back — and this is out of Backofen, where he actually writes down Hooke's law, not like Hooke did. Hooke said stress is proportional to strain, but the modern version of Hooke's law says the strain in the x direction is one over Young's modulus times the stress in the x direction minus Poisson's ratio times the stresses in the other two directions. The sum of the stresses in the other two directions. So most people don't write this down, but if you're going to study plasticity, you need to start thinking about stresses and strains in multiple directions. Everything you learned in mechanics before this was always sort of uniaxial tension or uniaxial compression. You got to start thinking about biaxial tension and compression.

So Backofen's got this little plot, that for isotropic material we got our von Mises ellipse. We would have plane strain right here, where there is no component in the two direction. Down here, in uniaxial tension, where I'm just pulling a stress in the one direction — so this is just your tensile test — I'll have some strain in the one direction. I'll have some strain in the two direction. If it's plasticity, that is minus dε₁/2, because Poisson's ratio is 1/2. So, and dε total, the total strain, is going to be some vector down here. What is dε₃ for uniaxial tension? If you remember, ε₁ + ε₂ + ε₃ is equal to zero. That's one of my criteria for plasticity — there is no volume change, and Poisson's ratio is a half. And so it turns out that dε₃ is the same as dε₂, and the sum of the three is zero. So if dε₁ is positive ε₁, dε₂ is negative 1/2 of that, and dε₃ is negative — is the same thing as dε₂ — and minus a half plus minus a half plus one is equal to zero. Duh. Okay, did that in my head, didn't even have to do that in my calculator before coming to class.

Now down here, we have a condition where I'm compressing in biaxial compression. I'm compressing in — I'm sorry, this is biaxial tension. I'm sorry, this is balanced biaxial tension. I'm sorry, it's not balanced biaxial — it's tension in the one direction and it's compression in the two direction. σ₁ is minus σ₂, and I get a total strain vector like that. And I get dε₁ is positive, dε₂ is negative of that. What's dε₃? It's zero, right. So this is also a plane strain condition. This is plane strain right here. This is plane strain right here, when you have a 45° slope.

And in fact, what is plane strain? You can produce a plane strain condition by loading a wide sheet in compression — a wide thin sheet in compression. So this is a plane strain compression test. Why is it zero strain in one of the directions? Plane strain just means one of the strain components is zero — either one or two or three has a zero component. If I wrote down my strain tensor, one of these three is zero. That's what plane strain is. That doesn't mean that one of the stresses is zero, because in simple uniaxial tension, this stress is twice these two stresses, which are negative of that. I'm sorry, they're Poisson's ratio times — σ₂₂ is minus nu σ₁₁ and σ₃₃ is minus nu σ₁₁. Neither — none of those three are zero. In fact, the three of them add up to a hydrostatic stress. So when I pull a simple tensile test, I'm actually getting a triaxial component of that, because all three of them have some finite value. But if I'm talking a strain tensor, if any one of these three is zero, then I have a plane strain condition.

And you can sort of see why a plane strain condition is something that I get sort of when I'm doing rolling. Okay, big wide sheet. Why is it zero in this direction? This is compression in this direction and compression in this direction, and no strain in the other direction. Because this material out here is not yielding, if you want to think of it as — after I've gotten a little bit of deformation in the center here, this material is still elastic, it's never deformed. The only thing that's deformed in here is the material in the groove, and it's lost thickness and it's gained width in the longitudinal direction. It has gained no width in the transverse direction. If no width gain in the transverse direction means the strain in the transverse direction is zero, this is plane strain. And this is equivalent — a plane strain compression, which is σ₃ compression and half of nu times σ₃ for σ₂. If I add a σ hydrostatic component equal to σ₃, I get balanced biaxial — uh, tension. Not balanced biaxial, I get a biaxial tension. That's the same as this.

So this is plane strain where I have σ₁ is minus σ₂. This is plane strain where I have σ₁ is equal to minus σ₃/2. So that's an equivalence of two plane strain conditions on that Tresca, and if you go around all the way around, you'll have two other conditions as well. So there's four conditions on the Tresca ellipse that are plane strain processing, and many of those occur in service.

Okay, this is another drawing from Backofen, when he shows the equivalence between through-thickness compression — if I add a hydrostatic component is balanced biaxial tension. Where's balanced biaxial tension on this diagram? It's right up here. Okay. So a through-thickness compression, which would be down here, is the same as that point up there, for an isotropic material. You're looking askance. Let me show you again. A simple compression test, if I add a hydrostatic stress to it, will end up giving me balanced biaxial tension. Balanced biaxial tension on the von Mises ellipse is up here. σ₁ σ₂ are equal — that's right here — and that's going to be the same as a simple compression test. These are not exactly intuitive. You have to play with them a little bit, but they are equivalent. So if you will, this point and this point are equal, and if I went around the whole thing I would have other equal points on the other half of this — just tension versus compression. So, I've said it, you're supposed to believe it. You haven't challenged it. Anyone want to challenge me? Okay.

Now let's talk a little bit about work hardening. And work hardening basically — because I don't have something that comes up and is just perfectly plastic with no work hardening — I actually have some work hardening, which means there's some slope to the stress-strain curve. Basically the Mises ellipse just sort of grows. But who says that the ellipse is going to grow with the same amount of work hardening in the one direction as the two direction? The ellipse doesn't have to be just a larger version of the smaller ellipse. You can have a change in your major and minor axis, if the material is not perfectly isotropic in the way it deforms. And in fact we find that materials, real materials, are not isotropic. They actually can have ellipses that have large major axes versus minor axes, or they can have the opposite. And you can have positive or negative deviations from isotropy.

It turns out that if I want to do sheet metal forming, I like to have long-axis ellipses, things that are good in balanced biaxial tension, that become stronger. Look at this — this material is strengthened by pulling it at the same time in the two direction. This is the yield strength for uniaxial, this is the yield strength for balanced biaxial. It's higher. It's stronger when I pull it in both directions. And if I'm going to do some deep drawing of cups, we're going to see that becomes important. You can get better deep draws if you have an anisotropic stress-strain curve. And for the last 50 years you've been able to buy steel and other metals based on an R value.

So again, here's isotropy, here's your plane strain value. If you have anisotropic where the σ₃ is greater than σ_y · 1/2, you actually are going to have plane strain at not a strain of a half — you know, one of them being a half of the other — and being balanced. Your strain vector is now still going to be perpendicular to that 111. Now what was a cylinder, now it's an ellipse. And still the strain vector still can be proven to be perpendicular to that three-dimensional ellipse coming up in the one direction. But your strains in the different directions are different, and Poisson's ratio still has to average out to a half, but it's not the same in all directions. Poisson's ratio is not 1/2 in all directions. But they all have to work out so there's no volume change.

Okay, so I realize some of this is sort of counterintuitive. I'll give you an example later — maybe not later today — but what gives rise — why can we produce sheet metal that gives me non-isotropic behavior? And the reason is, mostly we're dealing with face-centered cubic or body-centered cubic metals. Aluminum and steel — 98% of all metal made, aluminum, steel, and copper together, those three are 98% of all metals made. And you only get to BCC [HCP] if you start talking about zinc or magnesium or titanium. Anyway, these things are all in the noise. Not all titanium — some titanium is BCC if that's what you're thinking of. But the alpha titanium at low temperature, pure titanium at room temperature, is hexagonal close packed.

Okay, but these materials tend to deform — on the cubic materials, whether FCC or BCC, on the close-packed plane in the close-packed direction. It turns out FCC, which looks like one of these — ta-da — that's face-centered cubic crystal structure. And this is also face-centered cubic, just a little more complex, more atoms. The white lines here are actually the face-centered cube, so you're just looking at a different orientation. But face-centered cubic — so there's the cube and there's the face center. There's the cube and there's the center. What is the close-packed plane, or what's close-packed direction — that's easier. In FCC, where do the atoms touch? Along here, right. In the 110 direction. So FCC has got a 110 direction, and the close-packed plane turns out to be the 111 plane. The 111 plane is the one that — well, it's harder to see, but it's the one that goes — it goes from here, it's down — it contains two close-packed directions, actually three close-packed directions, if you were to draw — that's the 111 plane.

Okay, and it's got three 110 directions. So I got an atom here, an atom here, an atom here, and they're all touching along these three lines. So that's FCC — is a 111 plane in a 110 direction. And body-centered cubic — oh, by the way, this is aluminum, you can tell, the atoms are gray. This is copper, because you can tell the atoms are copper color, and it says that by the people who make these old models. Anyway, body-centered cubic looks like this. What's the close-packed direction? It's 111, where they touch all along the 111 direction. And the close-packed plane turns out to be the 110. This plane is the same as this plane. There's an atom density of one on this plane, which is a quarter atom on each corner — four corners times a quarter is one — did that one in my head too. This plane, the 110 plane, is also one atom per plane, but there's only one atom, but it's completely enclosed by the 110 plane. One atom density per plane. One atom density per plane. That's the highest density plane.

And so it turns out FCC and BCC are just sort of mirror images of each other in terms of what direction they want to deform. The easiest way for me to think about it is, the direction in which the atoms are closest is the strongest direction. And the fibers will align in the strongest direction. The crystals are going to rotate as I deform this polycrystal sample, and I'll get a grain orientation, all other things being equal. And they're not all necessarily always equal. But if I was doing wire drawing, where they are all equal, I will get a texture with the close-packed direction as the axis of my wire. As I deform it, the crystals will rotate. So a wire — an FCC crystal that's drawn with lots of deformation will end up with a 110 texture in the axial direction, and a 111 texture for the BCC. And I'll prove that to you on —