DP_S2012_02

You ready? Okay. So, I think I did for you the other day, um, the fact that you can pull on a material that has no defects in it with a fair amount of force. But if you put a notch in something, particularly a brittle material, you can pull on it — it just takes ounces to break the piece of paper. Now, paper turns out to be a brittle material. If you put it back together, the pieces match. There's no deformation to speak of.

Whereas other materials like rubber — I'll pass this around if you want — if I take a piece of rubber and I cut a notch in it and I pull on it, that crack tip is blunted. Okay? So you can pass this around if you want. Um, if I take a piece of plastic, in this case this is polyethylene, and the polyethylene — unfortunately, it's a little too thick for me to rip, but if I pull on the polyethylene, you can actually see — I can see right up close, and you can pull on it — that it rounds and it starts to deform. And you see that the material starts to turn white. I mean, it's sort of translucent white, but it gets even more white. And that's because you're starting to deform the material, introduce little microscopic defects which scatter the light, and it goes from cloudy white to a more complete white.

So Dr. Belmar will be talking about fracture and fatigue and things like that. And there is a big difference between brittle materials and ductile materials. And I'm going to talk a little bit about that. But first, let me give a brief introduction. Actually, it's going to take most of it today to talk about the many different deformation processes.

But I'll pass that around. But the first thing I put up here is just the contents of your textbook for this part — Hosford and Caddell. And if you — I think this is Hosford and Caddell. Pretty sure it is. No, no, this is — I'm sorry, that's Simone Belmar's. If I put up Hosford and Caddell — their first chapter is stress and strain, and you've got this book so you can go and read that stuff. Dr. Belmar gave you a synopsis. He talked about plasticity. That's chapter two. He then went into some strain hardening. So in one lecture he gave you the first three chapters of Hosford and Caddell.

This is a senior-graduate level course, and we're not going to go through the things that you might have had as a sophomore if you took mechanics or mechanical behavior or mechanics of materials. If you don't understand it, you've got the book, you can go and read it. I will review a few things, but I'm not going to review everything because we're not going to go back to the very beginning. By the same token, this is your area — mechanics of materials — and if you want to learn more about that stuff, there are other books out there like Engineering Plasticity by Johnson and Mellor, um, which is — well, it's one of the ones I used to use when I was teaching this in more depth. And he takes the first 120 pages of this book and gives you a lot of detail on the stress tensor and Mohr's circle and things like that.

I'm going to talk about Mohr's circle a little bit today, but we're not going to dwell on it, because I want to get to things that are descriptive. And in fact, I already told you that I don't like a lot of the things that we do in education. Typically graduate education in my experience is: some professor thinks, "I'm going to teach you my subject at the level that I know it." And that's not what most students need. If you're going to be that professor's doctoral student, then yeah, you need to learn all those things. But most students need something a little more descriptive and not in quite as much excruciating depth. So I'm not looking to give you depth. I'm looking to give you principles that you can understand and try to solve problems on your own.

So the first thing I did — I handed out this thing, metal forming's book or Taylan Altan's book, and the only reason I handed it out is he lists 48 different metal forming operations. And it starts on page nine and goes to page 35, listing two processes a page, 48 different processes. And this is one approach. I mean, if you took a course on manufacturing, they teach you about machining and casting and they try to go through all the different variations of everything. Well, I don't think that's very useful, in part because you could spend your whole time trying to understand these things. But if you want to know what hobbing is or coining, here's your little definition with little pictures to show you. He thought he came up with every metal forming process he could think of, and maybe some beyond that, to try to explain what's going on.

In fact, we're going to see in a little bit that there's a simpler way to look at it, and I'm going to show you one graph that covers everything in deformation processing — all 48 processes in one graph. But first, how many people have had a mechanical behavior class? Okay. How many people have not? So that means — okay. Well, you know, you're going to have to try to pick up some of this from me.

But there's something called a critical resolved shear stress. If I pull on a bar of material — and I just happen to have a tensile bar of copper — this tensile bar of copper came from this type of wire which has got sort of a weird cross-section. I'll pass it around. This is trolley wire. If you take the Amtrak Acela train or any other Amtrak train from here to New Haven, Connecticut, the wire that runs AC above the train — and they have a little graphite pickup that picks up the electrical energy. So you got a, you know, it's an electric train, and it's got to have electricity. It doesn't generate its own electricity because you have to have big heavy motors. And anyway, essentially you have a sliding contact, and if you're talking Acela at 125 or 150 miles an hour, you got sliding contact along this wire. And it's got these little indents so they can grab it like that about every 50 feet they got these things up there.

So there was a big problem or big question about the quality of the wire, and we had to make tensile bars. And in fact — here are some — well, actually let me put the cross-section of the wire up there. Here's a little piece of the wire. I won't blow it up too much because I'll pass it around. But it's basically a round wire with two little notches in it just so you can grab it. It's called trolley wire. It has to have good electrical conductivity. And here are some tensile bars, one untorn, one torn. This is — well actually, it's not the highest purity copper. It's got a little silver in it. It's only got about two — a tenth of a percent silver, so don't steal it on me. It's not that valuable. But it's relatively pure and it has very good ductility.

So if I want to compare it, it doesn't fracture in a brittle manner. It fractures in a ductile manner with a lot of stretch. And in fact, I can simulate that. A lot of times in deformation processing, we use modeling clay — cheap material — and I can do a tensile test right before your very eyes, and I'll get a fracture that looks a lot like that piece of copper. Okay. Now there are some differences, and we'll go through some of those differences of how modeling clay is different from a crystalline material like copper. But in fact, at a certain level in deformation processing, you can simulate any material, not just metals. It's not just metals processing, it's deformation processing of usually ductile materials.

You can process by deformation some brittle materials, but under very special conditions. Anybody know what those conditions are? Those conditions are when you apply a lot of hydrostatic stress. You can suppress fracture if you have a lot of hydrostatic stress. And I'll show you some things on that later.

But in any case, nothing deforms under pure tension or pure compression. It only deforms on the planes of the material in which you have a shear stress. Things deform — I mean, even though I pulled this thing in tension, it actually ended up fracturing on planes that were at 45 degrees, because there's a shear stress. And shear stress is something that will change the shape of something, or it's the easiest direction to change the shape of something. A square goes into a parallelogram when you apply shear. Okay, so far as that goes.

So if we look — and you look in some books on crystal plasticity, they will talk about a critical resolved shear stress, because certain planes and certain crystals will deform more readily than others. And so if you pull on something with some force, it doesn't just pull all apart on one flat plane unless it's a brittle material. It will actually slide apart on a plane that's oblique. And the maximum shear stress, if the material is not crystalline but is a glassy material or something like plasticine here, will be at 45 degrees.

And so if you pull on this with some force per unit area, you find at 45 degrees this area is the square root of 2 times the area up there. It's 1.44 at 45 degrees, because it's the sine of 45 degrees — or it's the area divided by the sine of 45 degrees, which is √2 over 2, .77. But the force that it's slipping in — if it's slipping in the most favorable direction — will be a force that, if you look at this, it'll be √2 over 2 times this force. And so it turns out the shear stress on this plane is half of what we measure as the yield stress of the material. So if I do a simple tensile bar and pull on this thing and measure a force per unit area to break it, just like we did with that piece of copper, in fact the shear stress is half of that value. It's actually failing in shear on some plane, ideally at 45 degrees. The shear stress will be half of the tensile stress.

Okay, so the tensile stress is one of those normal stresses. If I write down the stress tensor that Dr. Belmar did the other day, I've got sigma 1, sigma 2, sigma 3. And these are the three principal stresses. The shear stresses are sigma — you can call this 11, 22, 33 — sigma 12, sigma 13, sigma 23. And these three right here are the same as the three down here. But these principal stresses are not the stresses that cause the material to deform. It's these stresses that cause the material to deform [distort]. And Simone, we call those the deviatoric stresses. Is that right?

Belmar: Yeah. I mean, these shear stresses — I mean, it's everything but the pressure, right? So the —

Okay, you're right. Okay, that's right. There's the invariant. Okay, the I — right. So, uh, the deviatoric is actually the whole stress tensor modified by the hydrostatic. Anyway, so you need to understand one principle: things deform in shear. They don't deform in tension or compression. But there will always be some angle at which tension or compression will give you a shear.

Okay. The next thing is Mohr's circle. Mohr's circle for stress was put forward in 1882. It's still around. They teach it in Unified over in Aero and Astro. I don't know that they teach it in — well, most mechanics courses will talk about it. We will talk about it later. But basically you take these three principal stresses of sigma 11, sigma 22, sigma 33, and you plot them on the horizontal axis. So here's sigma 1, sigma 2, and sigma 3. You plot them on the horizontal axis, and then you find that there's a construction that says that the difference between sigma 1 and sigma 3, which will be the center of the big circle, will give you at any angle — remember this critical resolved shear stress is at some angle that it's actually shearing — you can calculate the angle. The maximum shear stress would be at 45, and this would be 90 degrees.

And you end up — just when you draw the circle around this, the principal stresses, you will get the shear stresses. So the vertical axis becomes the shear stresses. So the shear stress is half — which is the radius is half the diameter of the circle — and they've got one between these two principal stresses. You have three more circles. For any two combinations of these three, you can draw Mohr's circle. Now later on we may use Mohr's circle diagrams to try to explain some things. So I'm just telling you that if you haven't ever heard of Mohr's circles, you can go look in the first three chapters of Hosford and Caddell, and he will discuss them in a little more detail.

Dr. Belmar talked about elasticity. Now in elasticity — Poisson's ratio — and Poisson's ratio is basically: if I take a material and I pull on it, a green material, simple cube, and I pull on it in the red direction, it will shrink in the other two directions. So if I got a tensile specimen here, I have a piece of polyethylene — if I pull on it and it starts to stretch like that copper stretches, it will shrink in the other two directions. And you can see the copper shrunk, it necked down. Okay. But even if I'm pulling on it elastically, there is a microscopic strain where it shrinks in the other two directions. And that's called Poisson's ratio. And Poisson's ratio for a metal, a solid metal, is something on the order of 0.3. It might be 0.25, it might be 0.35. You can go look up in handbooks what the Poisson's ratio is for a given metal.

Okay. Inelastic behavior, the volume change if I pull on it is finite. Now I can't tell you exactly what it is — there are formulas. It's actually epsilon, the strain 1 plus strain 2 plus strain 3. But because Poisson's ratio is not a half, there is a finite volume change in plasticity — which I'm going to be dealing with for the rest of the term. [Tom misspoke: he means elasticity has the finite volume change; plasticity has zero — he corrects this below.]

We don't usually worry about the elastic behavior. We're worried about deforming the thing. And the elastic behavior typically lasts out to about two-tenths of a percent strain. Well, two-tenths of a percent strain is not very much. We're interested in 10, 20, 30, 40 percent strain, maybe even more. So who cares about the elastic component, which is only two-tenths of a percent? We can usually ignore the elastic component in deformation processing. You can't when you get to fracture and fatigue and things like that, because you're at lower strains, so far as that goes.

So it turns out that — r, r, r, r — that ratio for plasticity is a half, because there is no volume change. Okay. The material, if it's a solid material, will not change its volume. Now if it's not a solid material, then that rule is no longer true. So here's a piece of foamed aluminum, and it's clearly — if I deformed this piece of foamed aluminum, it's going to have different Poisson ratios. It's not going to have a zero volume change if I deform it, cuz I'm going to break up the pores.

Anybody ever made pizza dough from scratch? You let it — yep, you let it rise. It gets full of pores, and then when you punch it down before you roll it out, it loses volume. Okay, you're plastically deforming the pizza dough, but because it's got pores in it and you punch out the pores, volume change is not zero. Hopefully, you get it nice and flat and then it rises again, and then it does have pores again, and you're not eating a piece of matzah. Okay.

Elasticity — where's the volume coming from? It's coming from the fact that you're stretching the bonds. Remember, Dr. Belmar talked about elasticity — you're stretching the bonds. Whereas in plasticity, you're rearranging the atoms. There's not a difference in length of the bonds. But when you have the elastic energy in there — if I plastically deform this, I can bend it, but there's no stored energy. There's not a lot of stored energy in this. Theoretically, there's zero stored energy. If I stretch on it, I'm pulling on it, there's stored energy — it will spring back like a rubber band. Okay? So the volume change is the ability — it's storing the energy to spring back, because I've stretched the atomic bonds.

In plasticity, I've rearranged the atoms and I'm not storing energy in that. Now, it turns out there are residual stresses and other things, so it's not quite zero stored energy, and we'll see that as we go on. Good question. Other questions as I'm going through this? So for plasticity, remember, Poisson's ratio is a half, volume change is zero. That's an important principle in this whole thing.

Now, anybody know who Percy Bridgman was? Everybody heard of P.W. Bridgman? Well, he — this is Percy. Okay, there he is. He won the Nobel Prize. He was a professor at a school just north of us here — little school called Harvard. And let's see, he was born 1882 in Cambridge, Massachusetts. He died in 1961 in Randolph, New Hampshire. He committed suicide, by the way. His field was physics and he won the Nobel Prize.

Okay. He won the Nobel Prize because he developed ways — anybody ever heard of a tetrahedral anvil press? Oh, okay. You ever heard of people trying to make artificial diamond? Okay. They put it in a tetrahedral anvil press, and that's what Percy invented back in the beginning of the 19th — 20th century. Basically, you know what a tetrahedron is. And the tetrahedron has got four equilateral triangles arranged with the vertices in an arrangement, uh, like a saddle type of arrangement. Okay, the vertices of each of the four triangles is like that. Okay, like a universal joint in a car. Anyway, if you squeeze from those four directions, you're squeezing all at a point.

And Professor Bridgman took diamond dyes [dies] and he would put different materials in there, and he would do studies at extremely high pressure. And by extremely high pressure we mean gigapascals — thousands of atmospheres, and eventually — I don't know if he ever got to millions of atmospheres of pressure — but he was studying the properties of materials at very high pressure. Because he developed this tetrahedral anvil press, a whole new area of science was opened up. He won the Nobel Prize.

A brittle material — is rock a brittle material? Most rocks are brittle materials as we know them. But you get down to 10,000 feet below the earth, rock is no longer a brittle material. It flows like plasticine. Okay? And they dig mines down to 10,000 feet to look for diamonds. And they can't prop the mine up. Over the next few weeks, that hole they drilled to look for the mines — and the people will be down there — is slowly creeping. The brittle rock is under such hydrostatic pressure — sigma 1 plus sigma 2 plus sigma 3 is so large — that the rock will not fracture in a brittle manner. It will actually plastically flow, and over two or three weeks that little hole they drilled will essentially flatten itself out to nothing. It will heal itself without fracturing.

Okay? So you want to go study the fracture of rocks for earthquakes and stuff — it's a lot more complex than the simple fracture we have of rocks on Earth on the surface, cuz they're brittle and we can set brittle behavior. But the brittleness of rocks is a function of the depth in the earth. The more hydrostatic stress pressure we have, the less the rock deforms — I mean, the more the rock deforms. So that was Percy.

Um, we have this — coming out in the 111 direction for all you material scientists — from axes of sigma 1, sigma 2, and sigma 3, this three-dimensional thing, we have this cylinder, a yield locus. And Simone talked about the maximum shear stress condition. He talked about the von Mises condition. One of them has got sharp corners. Well, for biaxial loading, nature doesn't really like sharp corners for deformation. So the von Mises criterion is probably a better criterion. It's not always correct in crystalline materials. For a BCC metal or an FCC metal in the one — on the 110 plane in the 011 direction — the Tresca condition is correct, and I'll show you that later in the course.

But in fact, what Percy did is he went further out this thing to higher and higher hydrostatic pressure. Actually he went down in compression, okay, rather than tension, and he could suppress fracture. And he could get those things necking down like that piece of copper that I passed around. That thing's necked down to a reduction in area of like 80%. Anybody have an idea of what a typical piece of steel might have as a reduction in area? 50 or 60% before it snaps. Okay. So I can pull on my plasticine, and that's got — wow, that's got about a 90% reduction in area. Okay, that little neck is down to 90%.

Well, Bridgman could get steel to act like plasticine if he put enough hydrostatic stress on it. Well, we're going to be in deformation processing. We're going to be dealing with stresses that will not get to 10,000 atmospheres like Bridgman did, but we are going to get to stresses that are significant enough that we can suppress fracture. And a little bit later I'll give you a story on that.

But Backofen — other people did this before Backofen, but Backofen's book really hammered something called deformation zone geometry. It's chapter 7. It shows up as figure 9-9 in Hosford's book — so it's a lot later. But figure 7.1 of Backofen, which I'm going to show you in a second — this is 7.2, I think. Anyway — oh, it's 5.1. This is 5.1.

Backofen, he defines a dimensionless number. Now if you're — anybody here a chemical engineer, mechanical engineer by undergraduate? So you learned something about dimensionless numbers, the Peclet number and the Reynolds number and fluid flow and stuff. Well, the deformation processing guys have at least one dimensionless number. It's called the deformation zone geometry, delta. And delta is merely the height of what you're deforming — the white thing between the two platens here — divided by the width of what you're squeezing on it with.

So this is about a one-to-one. Delta is — the height is equal to the width. That's delta equals 1. That's sort of a compression test, if you want to think of it that way. This could be wire drawing or strip drawing. That's what those little — there's a delta sub s for a strip drawing. If you're drawing a strip of something long and thin like a piece of paper that has a lot of depth into the board. Or if it's a wire drawing, you're drawing something axisymmetric, it's got a circular cross-section. And you can just do simple geometry. The height — the average height — is approximately equal to the length. And as he's drawn this, the actual height over here is h1, and here it's h0. The mean height is something in between, and you can calculate what delta is as a function of the reduction ratio of h1 over h0.

Okay. So you can come up with different delta values. If you do something like thinning of the wall of a tube, it's one of the most important deformation processes that we use. Anybody know what we use it for? Ever seen a beer can? Okay. They make a beer can by what they call ironing the wall. They have the term ironing because it's just like taking an iron, or rolling out pizza dough from one direction with a rolling pin, and you just push everything until it's thinner. Okay. And so here's rolling out a pizza dough, or it's ironing of a beer can or a Coke can.

Anybody have any idea of what fraction of all the aluminum in the world goes into making Coke cans and beer cans? Pardon? 70%? Forty percent of all aluminum in the world goes to making containers — which is beer cans and Coke cans. And you better believe it's one of the highest tech products to be able to make a Coke can or a beer can. Okay. The tooling — to make that, because they're popping them out about six a second, okay, as I remember. I mean, it's really fast, and you got to wear earplugs when you walk through the plant. But they're just stamping these things out and making and popping out cans. And they have to do that. They have to have several machines.

I remember going through — my first big consulting job in 1976 or '77 was a failure of a beer keg in the second largest brewery in the United States. And so I got to tour this brewery in New York State. It's closed now, but anyway, at the time it was the second largest. The largest is in fact Coors in Golden, Colorado — I think it still is. And they make like 10 or 12 million barrels of beer a year. This one was only making about 9 and a half million barrels. But in fact, their bottling — they had about 15 or 16 lines that were capable of bottling about one case a second. Okay, so that's a lot. I mean, so you're talking — they weren't all running at the same time, but that plant was probably doing about 10 cases a second of bottles or cans. Now, bottles took a little longer than cans, cuz you had a smaller opening to go through, the glass, than you do through the — before you put the top on the can.

And anyway, there's interesting technology in some of those things. So in — this is sheet metal rolling, and there's a delta value for that. Well, there's lots of different delta values. But now let me show you the one plot, the most important plot in this — in my part of the course. Figure 7.1 of Backofen is the yield stress over here — the pressure that you're deforming this thing with, which is the compressive yield strength in compression, okay — divided by two times the shear strength. So this is the uniaxial compressive strength P divided by two times the yield strength.

Well, at a value of one, you're just talking a simple compression test. That's how you measure the compression strength of the material. P is equal to 2Y — Mohr's circle type of thing. Somewhere in between — if I'm doing an indentation, if I've got a piece of material and I want to deform it with a smaller tool than the height of the deformation zone — let's say I want to pierce this piece of plasticine. I can do this. See, that's why I use plasticine — I can do this. I can do deformation in front of you. I don't have to heat it up. Anyway, I got enough strength to do it. That started out with a deformation zone geometry of maybe four or five. So it's my finger pushing into some thick piece of material and deforming it.

If I do that, it turns out I'm going to have some stress that's not quite double the compressive yield strength of the material. Why? Well, think about it. If I do this, in order to do this, I'm not just compressing the material underneath the tip of my finger. I'm pushing everything around it to the side. I'm squeezing it out. I'm essentially making a donut out of something that was small. Okay? And I'm making it larger in diameter as it spreads out. I have to deform more material.

In order to do that, I have non-homogeneous deformation, and I have redundant shear. Some material that starts shearing this direction later in that process has to turn and go up. I mean, the stuff at the very bottom is going down, and then it has to go to the side, and then it has to go up as I'm doing this. And as it does that, it's extruding around my fingertip. And as it does that, I have what we call redundant deformation. And that redundant deformation causes me to shear in one direction, shear another direction, and then shear back. And to get the same initial shape change, I had to shear it three times to get one shape change, because the material had to flow through a complex path to get from shape A to shape B. If I took a little volume element, it had to shear in several different directions to get there.

And it turns out in the worst case of just doing a hardness test — this would be a hardness indenter on a semi-infinite material — it turns out the theoretical value for no work hardening turns out to be 1 plus π/2, which is 2.57, which for all intents and purposes in deformation processing is three. Okay, 1 plus π/2 is three, right? Because π is four approximately, right? Anyway, with no work hardening, it is 2.57.

With work hardening, does anybody know what the rule is for a hardness test in terms of the pressure of the hardness test compared to the uniaxial compressive strength of the material, whether it's steel or aluminum or anything else? There's actually a correlation. The hardness test — everybody knows what a hardness test is. You're just squeezing, making a little indent in the material, and seeing how much it deforms at a given pressure. It turns out a hardness test — the rule of thumb is it takes three times the stress. I know you know the answer. Okay. But anyway, the rule of thumb for a hardness test is the hardness indentation strength is three times the compressive uniaxial strength of the material.

So if I just have a simple little delta equals one cylinder of material — so there's my simple little cylinder, same diameter as it is high — and I do a compressive test, that's going to be P over 2 tau-i is equal to one. But if I took this and I just took a little thing like this and did a hardness test right in the middle, that takes three times the pressure — oops, sorry — takes three times the pressure to make that hole as compared to deforming the whole thing. Why? Because to make this hole, it's constrained by all the material around it. Whereas if I deform the whole thing, it can bulge out, right?

So the deformation zone geometry is the most important thing. If I go back to the last 35 years since I took and taught this course a couple of times out of Backofen's book, this diagram is the one that comes back in my mind most often — a factor of 20 times more often than any other diagram in the book. Okay. Backofen doesn't show it till chapter 7, because he has to go through stress and strain and all this other stuff, which — I'm assuming you guys, if you didn't take it as an undergraduate, or if you're a senior and you're still an undergraduate and you haven't taken it already — you can read the first three chapters of Hosford and Caddell to learn about stress and strain and work hardening and plasticity. Or you could pay attention — you could have paid attention last Friday when Dr. Belmar was kind of giving you an overview. Okay.

But we're not going to go through that. I want to talk about these types of differences. This is a uniaxial compression test at delta equals one. Something like extrusion or forging — in fact, Backofen's got seven processes across the top here. And that's one of the things I wanted to point out. Altan has 48 different processes. Backofen got sheet and strip rolling here, wire and rod drawing, extrusion, breakdown rolling, heavy forging, piercing, and hardness testing. He's only got seven across there. So he's kind of consolidated the 48 into seven.

Coining — let's take a — I don't have any change with me, I don't think. Anyway, I never carry change. But if I had a nickel or a dime or a penny — anybody know how those are made? How's a penny made? They roll it into sheet, which is a delta down here, one or less than one. Okay. And then they stamp out — someone said stamp, right? They stamp out a blank, a circular blank. And then they squeeze that blank, and they call it coining. Well, why would they call it coining? Okay? Cuz the Romans did this, and the Greeks did this. They would hammer out a little disc for a coin and then they would take a tool and they would indent it. Okay? So they would deform — roll a sheet and stamp out circles, or they just deform something like that, and then they would squeeze. Well, the king's seal could be squeezed into it like that.

Is coining a high-delta or a low-delta operation? What's the deformation geometry? In fact, it's sort of like squeezing a single hole into the big thick thing that I was talking about. Coining is just a little bit of deformation across the surface. So coining is a high-delta operation. Okay? It's over here. It's not over here. You might be working on something like that, but you're really working just microscopically on the surface. You're just taking something that's flat, and you're making Lincoln's head on a penny. And so it's a high-delta operation. Lot of stress on the tools when you're coining the surface, or when the king uses this ring to make the seal in the — except he says do it in molten wax. That's pretty easy, right? It's even softer than plasticine.

But anyway, so that's deformation zone geometry. Everything deforms by shear. The next picture in Backofen is actually an example of a delta equals six operation. Here we have a little indenter on the surface, and this comes from a paper from 1932 that's got a width of 5 mm and a height of 30 mm. Okay, 3 cm tall. So yeah.

Student: I had a question about the last figure you were showing.

Yep.

Student: The difference of that — e —

Two Y — that's Greek tau, two Y. Tau Y is the shear stress, the critical resolved shear stress that I talked about before.

Student: Um —

P is the indentation pressure, and tau — so if I look at this — okay, for a critical resolved shear stress: instead of a force in tension, I'm doing a compression test. P is a pressure to compress a cylinder of material. And the shear stress, remember — nothing deforms unless it's in shear, unless it undergoes shear. The shear stress is on some plane, optimally in a non-crystalline material at 45 degrees. In a crystalline material it may not be at 45 degrees, but the maximum shear stress would be at 45 degrees. And that is called tau Y — that's the shear strength of the material. Okay. But the compressive strength is twice that, and here's the formula. Okay, did that answer your question?

So what Backofen had was 2Y — the uniaxial compressive strength divided by 2Y, that dimensionless number is one. That's the yield strength of the material — the pressure to start to deform the material. And this graph is for all non-work-hardening material. Okay, it's also frictionless. We'll introduce friction later. But okay, any question? So stop me — questions like that, if it's not clear. Okay.

Um, now, so to prove this, it turns out someone took a 5 mm wide die, 30 mm — so 1.2 inch thick plate — or that right? No, about 1 inch thick plate. 3 cm — 1.2 — about 1.2 inch thick plate. And they deformed it a little bit. It says mild steel, but back in 1932 mild steel had a lot of nitrogen in it, and there's an etching technique for a high-nitrogen steel that will show you what material has actually started to deform. Okay. And so here's with a little bit of indentation, a little bit more indentation, and a lot more indentation. And you can see — just like Backofen kind of drew on this — you can see the shear lines. The material is actually shearing on those lines.

And back here it shears along straight planes at 45 degrees. Here the material actually has to deform, it has to change its direction. It's curved. The lines are curved. And I will tell you that when you have no curvature of your shear lines, P over 2Y will always be one. The π/2 is actually for the angle of shear. If it sheared 90 degrees — uh, if it sheared 180 degrees, you'll have π over two as the increased stress.

Okay. Something earlier — Backofen went through slip-line flow field, which is a technique before digital computers back in the '30s for people to be able to calculate some of these stresses. We may use it, but I'm not going to teach you how to do it, because today you would do a finite element calculation on a computer. But you can go and look at certain types of deformation.

Okay. Um, here I've got a cylinder that's being deformed between two platens. So this might be forging or breakdown of a great big billet in a steel mill. And you have a problem here in that it may be compressive right here, but because of the curvature of these flow lines, I end up with tension right there in the center. And the material is weak in the center, in a casting — like a great big forging blank, that you cast — the great big piece of steel. It's got a weakness from the solidification geometry right in the center. And if you're not careful, if you don't use the right contact area of your dies, you can create a defect, a crack right in the middle. Okay?

And I've seen them. Okay? We know how to prevent them. In 1985, I saw one. It caused a — in one of the other lectures on my welding course, I describe it. Actually, I may even describe it — I think I describe it on the lectures you'll see this last summer. It's about a tail shaft of a liquid natural gas cargo tanker that stopped dead full of liquid natural gas in the Philippine Sea, because the tail shaft failed in fatigue, cuz it had a defect right in the center, okay, of the casting. It wasn't a casting — it was — started out as a casting, and they got forged into the shaft, but they had a defect right in the center. Now, it's a little more complex than that, but nonetheless, this is the type of thing that can happen.

And sometimes — this just showing different forging directions. If you actually forge from three directions, you actually can help suppress some of that. Okay, so far as that goes. Or if you forge on an anvil die, which is what he's got here with three directions, you can suppress the tendency. So just forging from two directions is not as good as forging from three directions. Okay, in terms of suppressing the internal defects, depending on the delta operation.

Okay. So one principle of deformation processing is not to think like Taylan Altan at Ohio State, where there's 48 different processes and you have to learn each one. You just have to learn what's your dimensionless parameter delta for your forming operation. And once you know that, you're going to understand whether it's going to be a high-force operation, whether friction is going to be important. I haven't taught you about friction yet, but I'll tell you: delta is equal to one or less, friction controls the whole process. Delta is equal to 10, like a hardness test, friction doesn't make a difference.

Anyone's ever done a hardness test in the lab? Did you have to put lubricant on it, or clean the surface to make sure there was no lubricant? The reason is, you can prove that friction has no effect on a hardness test with a delta of 10. Friction has no effect in deforming things at high delta. But at low delta, friction controls everything. And so in between — extrusion and stuff — friction is sort of in between. It can be important, but how important depends on the delta.

Okay. We've only got a couple of minutes, so I'll just finish up here with this graph, which basically talks about lots of different materials processing — not all deformation processing, but we have here — we have cast permanent mold casting, die casting, powder metallurgy. We have investment casting. So we got casting processes up here. We got cold extrusion and rolling here. Cold drawing, blanking, which is stamping out little shapes out of — drilling and punching, finish milling, polishing, lapping and honing. So down here we have the deformation processes. Up here we have the casting processes. And this is a measure of what tolerance can you get in your part — how precisely can I make this — versus the surface finish of the part.

And on Wednesday — well, let me explain while people are here. So I got an email this weekend. I may have to go out of town, and they said, "Can you be in Philadelphia on Friday?" I said, "Yeah, Jeremy can show you a video on Friday. Tomorrow, Simone will be lecturing in the recitation, room 26." And I was going to — if I had to be out of town on Friday, you just watched the next video from last summer's lectures on casting. But then I got an email this morning and said, "What about Wednesday?" — or he said, "What other days are you available?" The only other day I'm available is Wednesday. So it's likely that I will miss either Wednesday or Friday, in which case you'll watch a video. Simone — Dr. Belmar will do Tuesday and Thursday. I will do either Monday and Wednesday — I did Monday today — I'll do Monday and Wednesday, or Monday and Friday. That's the story of my life. Okay? If I knew where I was going to be next week, it would be something different in my life.

And we will talk about some of these other things, so far as that goes. But this gives you some idea of tolerance. I will pass around next time — this is a surface roughness gauge — and you can see the different types of surface roughnesses that you can get, and what types of tolerances. If you look at this, this is a thousandth of an inch. So one human hair is right about here, surface — type of surface roughness. If you want to get down to something that's a mirror finish, you got to polish it, lap it, or hone it. And there are very critical components that you cannot make by deformation processing that have tolerances down 1/10,000 of an inch. But you have to polish them. You can't deform them to that shape.

But if you want to get something in the human hair surface roughness, or surface tolerance on dimension, I can roll something to 1 inch plus or minus one human hair. Okay? And we do that in the steel mills and the aluminum mills every day. Okay? Those are the types of tolerances we can hold by deformation processing. In casting, you can't come — you're off by a factor of five or 10. Anybody know why? It's called volume change on solidification. Okay. If the thing has a 5% volume change on solidification, guess what? You're going to have a hard time holding more than 5% of that dimension or surface roughness.

If you're rolling something, and you have something that's really rigid, you can get down — and you can roll a piece of steel or a piece of aluminum to within 1/1000 of an inch. Now, that's if it's 1/4 inch to 1 inch. You get to 2 or 3 inches thick or 4 inches thick, and you might have bigger problems. And maybe I'll tell you something about that the next time, of a student I had work at the world's largest rolling mill — Alcoa in Davenport, Iowa — where they make the aluminum plates for all the wings of all the aircraft. And they were having like a 5 or 10% variation in thickness, and they were scrapping plates right and left because of this variation in thickness. And Julie, the student who was doing an LFM thesis at the time, had to go out and figure out why. And she did. She figured out why. And I'll tell you that Alcoa never did anything with her thesis, even though it had a one-day payback. And that's just the way big companies work. Okay? They assume that people at a university are a bunch of academics that never came up with any practical idea in their life, and so they discount whatever we do. Okay, but that's another story. Okay.