DP_S2012_08

So we've been talking about sheet metal properties, and I'd put this up — this is in Hosford's book, chapter 19, and it is the title of sheet metal properties. He gives you n values, R values, and m values. I've sort of told you n is just the work hardening, the power law hardening, so stress is proportional to strain raised to the n power. n's the exponent. m is the strain rate exponent — that's going to be important when we get to superplasticity. So stress is proportional to strain rate raised to the m power. And R is just the ratio. I don't remember if I gave you the definition of R, but the definition of R is that R is the strain, if I'm pulling — usually my primary pulling direction is the one direction — R is the ratio of the strains in the two and three directions. So you can think of it as kind of Poisson ratio divided by Poisson ratio in two different directions.

And in fact, an average R — if you go buy a piece of sheet metal, and this goes back fifty years now, we've known, and automotive companies will buy from steel companies to a minimum specified R value. We're going to talk about what the values are later, but to do it, they basically measure R in the zero direction, which is the longitudinal rolling direction, at 2 × 45° and 1 × 90°, which is the transverse direction, and divide by four to get an average value of R. So if I have a piece of sheet, they would cut out like a 1 inch by — I think it's about 8 inch — sample from the sheet at 0, 90, and 45°. They would take twice the value of this zero, of this one, times this, and one times that, add them up, divide by four, and that will give you the average R value for the sheet coming out of the rolling mill. When you're going to put that into a stamping press, you're going to make the side fender of a truck or something, and you want to know if you can draw it. We're going to get to drawability, but you have to understand what R is.

They actually had US Steel in the 1960s make a little machine they called modul-R. You basically could take these little 1-inch strips, slip them into this machine, and it would use ultrasonics to measure the speed of sound in that little strip of steel in that direction. So it basically was thumping it ultrasonically. Yeah, very much directions. Yeah, and we're going to talk about that — that's part of today's lecture, I'm giving you sort of an introduction. So anyway, you could buy this little machine from them. I think at that time it cost $40,000. Today it would probably cost $2,000. I don't even know if people — I guess they still make them, I haven't seen one for years. But in any case, you can measure the R value, because it turns out the speed of sound is a function of the crystal orientation. And what we're going to see today is the R value is also a function of crystal orientation. So if you measure the modulus, which is a function of crystal orientation, which you can do with ultrasonics — so modul-R machine was nothing more than something that would measure the modulus, which I guess is where modul-R came from. You measure the speed of sound by ultrasonics, which is proportional to the elastic modulus, in these little strips of steel, and you could calculate your R value, hooray. And you could purchase the steel to a minimum R value because the R value is going to determine how well something forms. We're going to talk about that today.

Now where we left off the last time is I showed you the fact that, for the types of materials we're usually dealing with, which are either body-centered cubic — what's a body-centered cubic metal that people use all the time? Come on, folks. What's my favorite metal? Actually my favorite metal is platinum, but anyway. Steel. It's body-centered cubic, right? Low carbon steel. What's a face-centered cubic metal? Second most commonly used metal is aluminum, and third most commonly used metal is copper. They're face-centered cubic.

So while I'm talking about this, I will pass around this sample, which you can't see very well, but this is a tensile specimen. [Tom passes a sample around.] I was hired — this is probably twenty, twenty-five years ago — by Varian, up here on the North Shore. They make what's called a crossfield amplifier. Carl, you know what a crossfield amplifier is, you're an Air Force guy, you should know. When the planes are flying very low over the ground, they have to have radar so that the pilot doesn't run into a hill. It's a good idea when you're going Mach 1 at 500 feet over the ground — hills get to be a problem. So they actually have radar. And a crossfield amplifier is just a little electron beam that is surrounded by a bunch of copper fingers, and you can put a — the crossfield amplifier — you put a pulse electron beam through there, and the field you put on the little copper fingers that run perpendicular to this electron beam will modulate that beam, the electron beam, and give you all kinds of radar radio waves. So it's basically a transistor for radio waves. You can have an electron beam that gives you your power, and then the signal you want to put in there, which is for phased array radar, will basically amplify that little signal, and the electron beam gives you a big signal.

But on the end of these little copper fingers, when you're pulsing that electron beam, it generates so much heat that it would melt the tips of the copper. So they have to put molybdenum on the end of the tips. Why do you put molybdenum on there? Molybdenum has a thermal conductivity that's sixty percent that of copper, and it doesn't melt until nearly 3000° Centigrade. So they had like a millimeter of molybdenum that would just be sort of a thermal shield for the copper when the electron beam pulses through. They'd been brazing it with a copper gold — a gold copper alloy — it was actually gold nickel. No, it wasn't copper, it was gold nickel alloy. Eighty-two percent gold. And the price of gold had spiked at around $1,000 an ounce in '82, and this was like '85 or '86, and they were still trying to figure out how to save some money on that alloy. And they came up with an alloy — they didn't tell me, they just said, well, we'd like you to come up and talk to us about brazing, how we can get better joints, more reliable joints. Because if you took that gold nickel brazing alloy, where they're brazing copper to molybdenum, and you tried to do a bend test on it, like a 5T bend, before it just snap — brittle fracture. And the reason is molybdenum and nickel form an intermetallic which is brittle. And the braze alloy had eighteen percent nickel in it. Moly and nickel don't like to get together.

They didn't tell me. They put me in a room with a bunch of other engineers — well, I was the only outsider, but there were like ten people in this room, and we had this philosophical discussion of how to bond copper to molybdenum, two dissimilar metals. And I said, you know what, you really need a transient liquid phase diffusion bond. And they said, what's that? So I explained to him what it was. If you take my welding module and diffusion bonding you'll hear about that, and I may tell this part of the story. But in any case, they said, well, we actually have tried a sixty-five percent copper, thirty-five percent gold, three percent nickel alloy — obviously that's 103%, but I can't remember where the 3% was — and we get very good results. In fact we get results where we can bend at 180 degrees on itself and it doesn't break. I said, hmm. And they said, do you think you could analyze this for it? So I brought it back, gave it to one of my graduate students, said, here, go do metallography on this. He comes back and he says, I can't find the braze alloy. It turns out they had stumbled on a transient liquid phase diffusion bond by pure dumb luck, and they had a perfect solution. But the Air Force didn't want to buy it — actually it wasn't the Air Force, I think it was the Navy — didn't want to buy it because they thought they were just trying to reduce the cost. And so I had to write a letter and explain what TLP bonding was and stuff.

But the reason I'm handing this out is, when you braze this to the copper, you had to do it at a fairly high temperature, and therefore you get a lot of grain growth. And when we did the tensile pull, you can see the necking down on this tensile specimen, but you also see a rough surface. Remember I talked to you about orange peel before? That's orange peel. Large grains don't deform uniformly. Same thing is my trolley wire that had the bumps, and large grains in the copper. Anyway, we're going to explain some of that type of stuff in a little more detail as we go through sheet metal forming, and grain size and crystal basis. So anyway, I got a small consulting fee by doing some tests and writing a letter that they could give to the Navy and tell them a scientific reason why sometimes you can save money and get a better product at the same time. Which basically the government didn't believe — that you could both save money and get a better product. They seem to be opposites.

Anyway, so if you take sheet material and you pull it biaxially — here's our yield locus with an anisotropy of one, that's your isotropic von Mises ellipse. With an R value of zero, you end up with a circle. And with an R value of 2, 3, or 5, you can get these long major axis ellipses. There's something called a beta value, we won't go into that, it's just a variation. You can look in Backofen if you want to see what beta is — it's just another variation on things. Well, how do you get an isotropic behavior? Well, if I went back, um — I have to apologize, this morning I got waylaid by a couple of other things. But if I go back to — there actually is a reason sometimes when I show you some things — I showed you this, the stress tensor and strain tensors before. [Tom searches through his papers.] I can't find my stress tensor and strain tensor.

Now I'll just write it down. Remember you've seen this before. If I have ε₁₁, ε₂₂, ε₃₃, ε₁₂, ε₁₃, ε₂₃ — these are my shear strains, these are my tensile or compressive strains, principal strains. There are six — only six — independent of the nine strains in the tensor here. There's three down here, three up here, and three here, nine total, but only six of them are independent. So I have six independent strains to get a random shape change. If I want to do an arbitrary shape change I have to have six independent — but if I'm doing it elastically. If I'm doing it plastically, the sum of these three has to equal zero, because there's no volume change. So to get a crystalline material, or any material, to plastically change shape from one shape to another arbitrary shape, I have to have five slip systems. It just comes out of the math of the tensor. Six of these independent variables plus the one restriction that the volume doesn't change, which says these three have to sum to zero, means I have to have five independent slip systems.

Now in an amorphous material like clay, that's not hard. But in a crystalline material, such as face-centered cubic or body-centered cubic, it turns out you've only got four independent slip systems, for example. And this comes out of Backofen — this is not in Hosford. At Michigan this is too complex. We can teach this at MIT but not, you know — Hosford was an MIT student, he was Backofen's student, and he either didn't like the graduate student who was studying with him at the same time who did this work, which is why he didn't include it in his book, or it's too complex for the average Michigan student. So this is for a cube-on-face texture for the crystal. This is what we call a 101 texture. If you look at the plane stress yield locus in the RT plane for 101 texture — and if you want to look at the slip planes, he's actually doing this for one one one slip planes, in this particular case, which would be face-centered cubic. But it turns out because of the symmetry between directions and planes between FCC and BCC, it works the same way if you do the math.

If you're pulling in this direction, just a uniaxial tension, your slip systems will be these four planes in these four directions. And you have symmetry top and bottom, so it's not really eight slip directions, it's basically — this plane down here is the same as the back plane up there, this plane here is going to be into the board underneath. We don't have to get into all of this, but if you followed that, some of these eight faces are not all independent. There's only four independent planes here, and there are four independent directions. That's four slip systems. That's not five. And without five I cannot get an arbitrary shape change. And so if I have real FCC material or BCC material and the grains are large enough, when I pull on it I'm going to get an orange peel surface, just like that piece I passed around. Right? It's inherent in the crystalline behavior of the material. And it turns out for this texture you just get the Tresca condition, whether you're pulling here, or whether you have a side pull in tension or not. You don't get any effect, you don't introduce anything else.

At this point you actually will have eight slip systems operating. He calls them 2, 5, 8, and 11; –3, –6, –9, and 12. At that one point, balanced biaxial stress, you would actually have eight slip systems and you could get an arbitrary shape change. But anywhere along here you've only got four. Anywhere along here you've only got four. I used to, when I taught this course thirty years ago, I actually had a model with this tetrahedron that I made and soldered up in aluminum, which is another story. And used to try to — I spent two days explaining all this to the students, and they just go to sleep, so I'm not going to explain it to you beyond what I've done. But just believe that Backofen and his students did this right, did this correctly, sixty years ago. And I can verify, because to spend two hours in lecture, I spent about a half a week studying the four or five pages in Backofen to understand the details of this. And I'm sure you can do it too. As one of the students and I discussed after Monday's class, I'm not going through a lot of math with you because I assume you're MIT students, and if you understand the physics you can pick up the math out of the textbook. Most of you don't really care about the math. If I started doing the math with you, A, I would make a mistake — that's one reason, the main reason why I don't do math on the board, because I make mistakes — and two, you'll fall asleep.

Anyway, if I have a cube-on-edge — not cube-on-face — texture, so I've taken my cube and I've just rocked it up on its edge, and I have this texture, it turns out I still have those same planes here, but with this edge texture I don't start yielding until double the stress. So if I had a pure single crystal sheet with this texture, I would get a yield locus that looks like this, and I get this extra strength in this whole triangle up here above the Tresca, which would be cutting across here. All this extra is extra strength, because I don't activate those extra slip systems because of the change in orientation of the crystal. So that's the 110 texture. The 111 texture looks like this — and this is a cube-on-corner — and not only do I get an increase in stress above my uniaxial, I'll get it in both directions. And so I can push this out to 150%. So with 110 [001 (cube-on-face)] I get 100% of my uniaxial yield strength in biaxial. In 111 I get 150%. And if I have 110 [cube-on-edge] I would get 200% of my stress.

So what can we actually get in a real material if I don't have pure texture? I actually have a real material with something less than a perfect texture. [Tom looks for a sheet.] So where did I lose my sheet from Hosford that gave me my n, my R values? Here it is. Okay, here's my sheet from Hosford with values for low carbon steel. I can get from 1.4 to 2.0. Well, 2.0 is my limit for a 110 [110] texture in theory. Interstitial-free steel I actually can get something a little higher, and that actually has to do with work hardening, but let's not get into that right now. Some of these values get influenced by work hardening. What I showed you before was just essentially an elastic yielding without any work hardening. If you get work hardening you can push things up a little bit more. [Pointer battery dies.] Lousy batteries and that stupid pointer.

For high strength low alloy steels it's really around one, which means there's not much anisotropy. You can only get up to 1.2 for extended steels. Copper is actually a little bit weaker, it's a little more like a circle. You don't get good textures. Same thing with brass. Aluminum alloys don't give you what you want. Zinc doesn't give you what you want. But look at alpha titanium. It will give you a texture of five — an R value of four to five. So in fact you can get fantastic sheet drawability in titanium. Alpha titanium, which is hexagonal close packed. And how can you get all the way up to five just because you're hexagonal close packed? I suspect that no one went to the faculty candidate seminar yesterday afternoon at 4:00, right, except me. Well, the faculty candidate was talking about doing these very elaborate experiments. She's a researcher at Lawrence Berkeley Laboratory, University of California, Berkeley, and they have a national center for electron microscopy. So she would make little single crystals, nano crystals, and she had a little tensile machine in the electron microscope, and she could pull the single — this little nano crystal — and actually see the dislocations as they form as the material is deforming. And this is work supposedly funded by General Motors to try to end up with something that has both high strength and high ductility at the same time.

So anyway, you can get fantastic R values. But before I get into — oh, well, how do you do that? You do it by twinning. Now what's twinning in hexagonal close-packed? This is hexagonal close-packed. Anybody know a material that's hexagonal close-packed besides alpha titanium? Alpha titanium is the low-temperature phase of titanium. High temperature phase of titanium is body-centered cubic. So what's — yeah? Graphite. You can tell this is graphite, atoms are black, right. This also is — this could be diamond, where the atoms are also black, but this is face-centered cubic. It's actually not diamond cubic, but nonetheless — where actually is it? This is diamond cubic, isn't it? Or is this face-centered cubic? It's diamond. This actually is diamond. So they're both carbon atoms, they're black. This is diamond, this is hexagonal close-packed graphite.

And the hexagonal crystals can deform on three different planes called the — so here's your sheet, and this would be a pure basal texture. So you have these little hexagons making up the sheet. You can deform on the basal — you can twin on the — twinning is basically a mirror image of the crystal, I'll show you that in a second. You can deform on the basal plane, the prismatic plane, or the pyramidal planes. So you've got a plane here, you've got a plane here, this face over here, this angular face — plane in here — and the basal plane. So you have three different twinning planes.

Now in hexagonal crystals you like to think about what they call the c-over-a ratio. a is the edge face width between atoms on the base plane, c is the distance in the vertical direction going up the hexagon. If the c-over-a ratio is √(8/3) [√3 — actually the ideal hcp c/a is √(8/3) ≈ 1.633], you can prove that this is a perfect — the atoms are spherical atoms. If for some reason the c-over-a ratio is less than that, then the basal planes are scrunched together. If it's greater than that, then they're stretched apart. So if the atom bonding is sort of elliptical, long axis in c direction, you're going to be greater than √3 [the ideal]; if it's short axis in the c direction you're going to be less than √3 [the ideal].

If the crystal twins — and this is the twinning of the crystal — the crystal basically will have a plane. This is twinning on a pyramidal plane, and basically you start out with a hexagon here and then a hexagon here. It's just sort of rotated around, and it's called twinning. It occurs in copper alloys, it occurs — can occur — in steels, not very often, but it occurs. It's the primary deformation mechanism in materials like magnesium or titanium, alpha titanium. And so what does this do? Most of these — and if I had time this morning I would have looked it up — most of the metals, common metals, that are BCC or that are hcp, have a c-over-a ratio less than √3 [the ideal], and that means that they will promote twinning in compression. Okay, I'm sorry — promote twinning, that must be greater than √3. If you compress it, it will want to twin, because it can accommodate the strain by twinning. If you have something c-over-a less than √3, if you pull it, it will accommodate the strain by twinning. So the twinning tendency, if it's not a perfect √3 crystal, will be dependent on the c-over-a ratio. For titanium the c-over-a is greater than √3, and so you will tend to get twinning at lower stresses than you would ordinarily on your von Mises ellipse. And so what happens is, now it's not only anisotropic, it's also dependent — the strength is direction dependent on whether it's compression or tension — because of twinning in these hcp crystals.

So if you actually look at titanium — and this is Dion Lee. Dion Lee was the doctoral student in '65 that worked all this stuff out for Backofen, and when I first graduated he was working for General Electric. He interviewed me here at MIT, and I did not get an interview at General Electric. He thought I was a turkey, I thought he was a turkey, I probably conveyed that to him in the interview. That's this story of my life, I tell people what I think. So anyway, Dion — I think he's retired from General Electric now, but bright guy, pretty much of a jerk. But anyway, just because you're bright doesn't mean you're not a jerk. Even bright people can be jerks, right? So let that be a lesson to you all you bright people out there.

Anyway, so Dion was looking at titanium — alpha titanium sheet — had a uniaxial yield strength of 95 ksi right here. Isotropy is this dashed ellipse here. If he actually did biaxial stretching, which he did in the laboratory, he could get in biaxial stretching strengths up to 145 ksi. So it really makes a difference. This is not a small change. This is real material here. The R value is 2.5, and remember the R value could go up as high as five. So this was not some special sheet that he got — that's for alpha titanium, that's not an exceptional value. It was Ti-4Al with a quarter percent oxygen, which is a lot of oxygen — not a huge amount, but it's a little high.

So why do we care? Well, before I do that I want to show you — I have a pressure vessel down in — was in Mississippi — it was 4 ft in diameter, couple inches thick steel, at the time that it blew up and killed one guy and sandblasted another guy. The door went flying off, the door weighed 4,000 lb, it was found about a half a mile away. This thing was being pressurized at 2200 psi nitrogen. Ordinarily you would do a hydro test according to the ASME code, and this is being tested before it went into service at 1.5 times the operating pressure of 1500 psi — 1480 exactly, but anyway. That's the distribution pressure if you got natural gas coming from Texas and you want to transport it up to New Jersey or wherever, so people up in New Jersey have some gas. You will be — today modern pipelines are operating at 1500 psi pressure. Anybody have an idea what the gas pressure in your house is, if you have gas heat or hot water or something? It's a quarter psi. If I go in the lab and turn on the gas jet and light it — my old high school physics teacher, Captain Hood, he was captain of the Navy, he used to do that for us and get about a two-foot flame shooting out of that gas jet. That's only a quarter psi.

If you went back forty, fifty years ago they were only distributing it at 1,000 psi. And about twenty years ago I worked on the Edison New Jersey pipeline failure. It was a 42-inch diameter pipe. It was transporting gas at 1,000 psi. When they first laid it in the 1960s through Edison New Jersey, it was farmland. By the 1990s, when this thing let go, it was more built up — all of northern New Jersey is just a big suburb of New York, right? And so when it let go there were flames shooting 600 ft in the air from 1,000 psi 42-inch diameter pipe. The radiant heat just melted the roofs of the buildings next door. But that's another story.

So this pipeline, they're testing for 1500 psi — they're testing it with liquid — not liquid nitrogen, well, it came from liquid nitrogen — but testing at 2200 psi just to verify that everything's sound. And it turns out the door lets go. This 2-ton door goes a half a mile away. The wind storm — basically the guy who was standing close to it, he basically got sandblasted by all the — I mean this was a construction site, and so there wasn't a lot of grass around, and he got sandblasted, and he's about to go blind now. Another guy got killed. In any case, so we're testing this stuff and we did tensile tests. [Tom holds up a sample.] I found this on my shelf the other day. I can't even remember all the samples I have sometimes. This is a cup and cone fracture. I don't even remember where it came from — I think it might have been from some testing we did on a gas well in Louisiana, but anyway. That's a cup and cone fracture, which is a typical neck down fracture. This is what a cup-cone fracture looks like. These are actually some of the test results from last fall, when we cut specimens out of this big pressure vessel.

I'm sorry, this is a different pressure vessel — this is another gas distribution line where the turbine that pressurizes the whole system ran in reverse because of a check valve that failed and other stuff. So different problem — I guess I got my problems in Louisiana and Mississippi and Texas mixed up. But anyway, this is just a neck down piece of steel from one of those. This is the ends of the cup-cone fracture. And this is the cup. And you have a 45° shear — this is what we call plane stress. I think Dr. Belmir [Bel'misov?] has talked about plane stress. And he's probably talked about — this region in here is plane strain, and this is the cone. This is the cup. That's nice. How about — this is the cup-cone failure from another specimen. And it started out as a circle, and when it necked down it's no longer a circle, it's an ellipse. There was texture in this piece of steel. How do I know? Because it has an R value, it's not one, right? D over D₃ — you know, ε₃ and ε₂ are different. There's a much bigger ε₃ value than an ε₂ value, or vice versa. And from the shape of that ellipse I could get the R value pulling on that. Now this is probably cut out of — this one is probably one inch thick piece of steel.

Let's see if I got anything else in here I want to show you. Well, I can show you — has Dr. Belmir talked about Charpy tests? Talked about Charpy tests? Well, I'll save it till later. Well, no I won't, I might as well show you. [Tom shows samples.] These are some of these same steels from this particular failure at this other gas compressor station in Texas, and some of them were very brittle. You take a 1 cm square bar of steel that's 10 cm long, and we're looking at the fracture surface. You put a very precise 2 mm deep notch in this 10 mm square of steel, and then you hit it with a hammer, and you measure the energy dissipation in the hammer. It's just a great big swing pendulum, and don't get in the way of the pendulum, it'll kill you if it hits you. Because a good one will have 200 — well, the one we used to have at Bethlehem Steel that I used was 264 foot-pounds of energy. That's a lot more than even a good horse can kick you with.

This is a very tough steel. The other one was a brittle steel — there's no deformation on the fractured surface. We couldn't even break them. Look, the hammer hit back here and just — the steel just — this is some of the toughest steel I've ever seen. This was actually — their machine was almost 300 foot-pounds, and 300 foot-pounds couldn't break the steel. You know, twenty, thirty-five years ago when I worked for the steel company, I was developing some better steels, and I was one of the only guys who would send samples over to the test lab that would stop the machine, stop the hammer. They didn't like that because it cost $500 every time you stop the hammer — you had to recalibrate the machine to make sure you didn't deform the bearings and stuff. They just got to the point where they just tested all my samples and they wouldn't do a recalibration in between. So anyway, they couldn't afford —

Now, other examples of two-dimensional deformation from a three-dimensional object. This is the structure — this comes out of Backofen, and although this might be in Hosford, he may have a picture of this. This is the cross-section of a niobium wire, and I can tell you this is a niobium wire after drawing it to 88.2% — so nearly a 90% reduction in the area. Why does Backofen have this? Well, this is actually Bill Hosford in his thesis, I guess. So Hosford should have a picture of this in his book — if he doesn't, I got it out of Backofen. But this is — niobium is body-centered cubic, and if you draw a wire, it's axisymmetric deformation. You're going from big diameter to small diameter and everything's circular. But it will have a 111 texture. If I think of BCC, it's going to want to align so the close pack direction is in the axis of the wire drawing, and that means the 110 is at a slight angle, 110 plane. And when it deforms, it's going to deform like that tensile specimen that deformed into an ellipse. So individual crystals want to become more plate-like rather than equiaxed, but they can't do that, so they actually form those little plates, but those plates then get bent because they have to be accommodated by all the other grains around them, and you get something — it's called wavy slip. Does he call this wavy slip? Anyway, he doesn't call it wavy slip, but it is wavy slip. And if you go look at a piece of piano wire steel — piano wire, make springs out of it and everything else — you do a cross-section of it, you'll see wavy slip, because to get the high strength of the steel piano wire — lots of reduction to get the high strength, and it's a BCC structure — you'll get the wavy slip. Just another further evidence the microstructure changes.

Now, since yesterday someone asked me — they had some wire that was breaking in a cable, an elevator cable that broke with someone in it. Anybody know what usually happens when an elevator cable breaks with someone in it? All the redundant safety brakes work, and so you drop a couple of feet, and you're shocked, and the fire department has to come and get you out of the elevator. But it doesn't usually go to the bottom. Can you imagine if we had elevators going to the bottom, how many people would take elevators? So about a hundred years ago when Otis started making elevators for the big buildings in New York — and why did we get the big buildings in New York? Because Bethlehem Steel learned how to make I-beams, right? I mean, it all fits together, folks. Anyway, Otis made a lot of money off elevators in those big tall buildings that Bethlehem Steel was building, and they had to develop a number of safety mechanisms. Have any of you ever been in elevator shafts? You have? What, are you an MIT hacker or something?

Oh, you have? Actually, why have you been in an elevator shaft? Most people haven't been in an elevator. Okay, fine. Now some of the elevator shafts you can see — you go to the mall, it's a little two-story elevator and they actually have a piston, so you can see the piston. But the big tall buildings don't have a piston that goes up 100 stories, right? You have to dig a pretty deep hole to do that. So they actually have cables. And anyway, there are all kinds of safeties such that if the thing starts to drop at too high speed, these little brakes swing out and stop you. And that's what — well, actually I've never heard of an elevator falling all the way to the ground. I've had several elevator failures but no one ever got hurt. In fact, you go from a regular elevator, you'll get about 1.2 G's of deceleration when it stops. You can feel — you know, your weight goes up. Don't weigh yourself on the elevator when it stops, weigh yourself when it's starting up. Anyway.

But one of them in a prison — this was an interesting one, happened in a prison in New Jersey, and they were going down the elevator, and actually it wasn't the cable that broke, it was one of the bearings in one of the sheaves, the pulleys, broke. And the brakes operated — the safety brakes operated — and afterwards they simulated the failure, and it was 1.6 G's. Now, ordinarily 1.6 G's shouldn't knock someone to the ground. But the prison guard who was going down the elevator with three great big guys, one of whom used to be a New York Giants lineman — and these were the prisoners — and there was a camera in this prison elevator so that the other guards could be watching to make sure no hanky-panky went on. The guard transporting the prisoners — that he didn't get roughed up by the prisoners in the elevator when the doors closed, right — they had a camera there. Well, when it stopped, all of a sudden the camera went black because one of the prisoners puts his hand over the camera, and another one basically puts his knee in the back of the guard, takes him to the ground. When it's all done, the guard's got a slipped disc. These prisoners didn't really like the guard very much. Of course they said it was because of the 1.6 G's of deceleration. But you're not going to — even if you fall down you don't usually get a slipped disc. This guy had pretty bad back damage. He ended up getting $1.6 million from the court, because we were not allowed to speculate about what the lineman did to him. He was standing behind him, and all of a sudden the television picture went black, and it was black for about 10 or 15 seconds. When it came back on, the lineman and the guard were on the floor of the elevator, the other two guys were standing there, and this guy was injured. So what do you think happened? But we couldn't convince the jury because we couldn't tell the jury that story. That's just the way the law works.

Anyway, so you get wavy slip. And I guess I was going to tell you the story of — so this guy was looking at failures of this elevator cable, and he wanted to know — he had looked at the failures and they were somewhat brittle failures on some of the wires. And one of the things you always suspect when you have brittle failures in steel is hydrogen embrittlement. And he says, but we didn't see intergranular fracture. And I said, of course not, the thing's been deformed so much, and those grains are all wavy-shaped grains. You're not going to see intergranular fracture on a piano wire. So you know, this is only the second time in a month I've had to explain to somebody. I can't tell you about the other one because it had to do with an aircraft, where people were looking and they couldn't find — they thought it was hydrogen embrittlement, but they couldn't find intergranular fracture, which is characteristic of hydrogen embrittlement. It's not characteristic in piano wire because you don't have grains that are well enough shaped to give you intergranular fracture. And this other thing, which is basically a very high strength steel similar to what they use in aircraft landing gear — and I won't — it wasn't an aircraft landing gear, it was an aircraft application, I can't tell you more about it right now — but it probably was hydrogen embrittlement, but you can change the chemistry of the steel and you don't always get intergranular fracture with hydrogen. Ninety-eight percent of the time you do, and so these people are thinking, oh well, I don't see intergranular, it can't be hydrogen. Well, you need to know about the other 2% sometimes.

Okay, so now let's just start talking about sheet metal forming. And this is, how do you draw a cup? I keep using these cups. I won't have to keep bringing them once we get into — now we're getting into sheet metal forming. So I start out with a blank. I start with a sheet and I punch out a blank, and I draw it with a punch and a die. And when it's being drawn, it sucks in the sides, unless I clamp it and hold it tight, and I'll show you something about that later. It sucks in the side and I end up with a cup that looks like this. Now, you can do a thought experiment and just think, I'm going to think about one pie shape out of this thing, and it's going to look like this — it's going to be conical shape but bent and have straight sides along the side. So that's one way to think about drawing of a simple cup.

And now I want to get to — this is not a Backofen, I guess. Should be something like this in Hosford. Here's my anisotropic and my isotropic von Mises yield criteria. In wire drawing — in cup drawing, rather — I'm clearly biaxial loading, and there's two places, right here along the flat wall of the cup, where I have plane strain. And up here in the flange, both of those are plane strain. Down on the wall of the cup, it's basically plane strain condition, where I'm pulling in tension and I have to have a tension along the circumference that is roughly half. If it were isotropic, it would be exactly half. If it's anisotropic and I have an R value greater than one, it's actually going to be a biaxial tension that's greater than half of the downward tension. But in any case, it's biaxial loading with unequal tension. And I just use the construction — remember the strain has to be perpendicular to the tangent to the ellipse — and the place I get plane strain is where I have nothing in the ε direction. The ε direction would be the circumferential direction here — the two direction strain in the two direction would be circumferential. The other place where I have plane strain is where I'm squeezing in one direction and pulling in the other direction. So this is the same drawing tension — this drawing tension here, as I'm drawing the cup, and that same tension is sucking in the flange — and I will get plane strain here, but that's down in this loading condition.

And remember, a couple days ago or whenever, I told you that this is plane strain and this is plane strain, and the two of them are equivalent. They are equivalent except when you're anisotropic. And here the strengthening is minor, here the strengthening is major. But that anisotropy in anything we're dealing with usually is due to the crystalline structure of the material. If I want to think about this a little bit further, here's my little pie-shaped wedge in my disc before I draw it, and it has this shape. And Backofen used to like to — he's the only one I know that does this — he used to say, well, that's sort of like drawing a sheet between some conical dies, and I'll pull something and make straight side. It's like wire drawing, if you will, but starting with a conical-shaped specimen.

And the load — as you do all this, the drawing stress increases and then finally decreases — strain hardening is very important in the beginning, and then the decreasing amount of reduction as you're finally pulling the flange all the way through. You have two typical failure points. One is right up here at the base of the flat wall, so that's the top of the cup up here. And the other is at the bottom of the cup. You don't usually get a fracture in the middle of the wall. You'll get it at the corners, partly because you have stress concentrations at the corners. But when you do this, quite often you end up with a cup that looks a little funny — it has ears on it, and it's called earring. This cup was drawn out of copper, I think. Backofen doesn't usually worry about what the material is. And this has planar isotropy — no preferred crystalline orientation in one direction in the plane of the sheet. This one has a very sharp cube texture, and you can see I have different deformation in the 10 direction than the 110 direction. So I get ears on my cup. And of course that's going to have to be cut off — I have to use more material, and I have scrap if I get earring. So earring is something that people will often try to specify that you don't — you have anisotropy to give you a good R, but you don't want anisotropy that gives you planar anisotropy, which gives you ears, which wastes material.

So here's a failure at the bottom of the cup. Here's wrinkling — you didn't have enough pressure holding these down, and so up here where everything's going into compression, the cup wrinkles on the flange. Here's one where you get earring — this actually comes out of Hosford. This is another one from Hosford, an earring example — these are copper sheets. So earring's a problem.

And so we will continue on. And for a week and a half now I've wanted to tell you the story of the woman in California who was texting and rear-ended another vehicle at 70 mph, and the woman died — the woman in the vehicle that was stationary was stopped when this other woman was texting at 70 mph in her car and she rear-ended the woman in the vehicle that she hit. Died in the fire that occurred. But did I tell — I didn't tell you about this before, but anyway. The judge was not pleased because two months later the woman who had killed the other woman because of she was texting, she got stopped by the state police because she was texting. Two months later after she had killed someone. So he sent her to jail for 5 years. But anyway, I need your proposals sometime today.