Mechanical properties of steel 7: evolution of plastic strain
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Cooman, Bruno C. de
The seventh in a series of lectures given by Professor Bruno de Cooman of the Graduate Institute of Ferrous Technology, POSTECH, South Korea. This particular lecture develops further the concepts of plasticity with a focus on the formability of steels, particularly those for automotive applications.
 So, well, I left this person finished. So, again, about strategy of answering the quizzes. So, we have a quiz every week, yes? There are about, each time there are 10 questions. That means that over the semester, you will answer, you know, close to 150 questions about, yes? So, what you basically do every time you answer a question correctly, you accumulate points, right? So, that's how you have to look at it, yes? So, you know, if you're absent, you know, you don't accumulate points, right? I don't, I will not take that into account, because I will normalize your grade, yes? But the idea is not that you make one very successful test and then don't ever show up for the quizzes again, right? So, if you're not there to take the quiz, you know, you miss out more than one or two quizzes, you know, will have a serious talk, because you're supposed to be here, okay? Then when you answer, please write your name. You don't have to write very much, right? You have to write your name, yes? And then, 10 letters, either a T or an F. So, you know, there's still people who just, you know, I get someone who writes this, right? And, you know, and if it's not well written, it can be anything, right? So, don't do this to me, okay? So, this is a T and this is an F, right? And if you can't write these two letters, you know, it's going to be an error, okay? It's very simple and I'm not asking too much. And write your name nicely also, you know, so I can, because there are two people who have very similar names and they scribble their names, so I can't see the difference. I think there are two Mr. Shin's, okay? So, please write your name carefully, so I don't give your very high grade to your friend. Okay, good. So, let's collect the papers. Let's just bring them here in front and let's move on with what we were doing. Right, and the best way to do well in this course is just, you know, participate to all the quizzes. And I really stick to what I've said in class. So, if you go through the slides and the materials you had, you know, you should be okay in the course. Whatever level of difficulty we go into during the course. Okay, so we had arrived at the point on Monday, on Tuesday, excuse me, that, you know, there's this whole body of theory of plasticity and, you know, and we're all happy about having a flow rule and everything like this and know about r factors and delta r's and what the impact is on strain increments and things like this, but it also leaves you with a feeling that, you know, what can I do with this in practice? And, well, in practice, when you go into real practice of formability and plasticity, you realize that actually stresses, there's not much discussion about stresses in that case. In practice, very often when there is discussion of formability, it's about strains that the discussion goes. So, because formability is a really nice subject in basically two areas, one of them is press forming, that's really complex, you know, there's a lot of interest in that and then massive forming, which is basically, you know, where you have big blocks of material and, you know, you shape them in a specific form, so you have large deformations, the areas, in these areas people are interested in plasticity and of course there's a lot of work in plasticity which is related to fracture, you know, and what goes on before fracture, certainly when we'll be talking about plastic fracture, you know, this plasticity will pop up again as a subject. But, so we'll focus on sheet forming and in sheet forming, you know, you have these big presses, you know, where you basically form objects like these. Many things around us are made like this by presses and the material you start with is sheet steel, it's just flat sheet and that can be, of course, steel, but, you know, we have also sheet, other metals and alloys are available as sheet to make parts, aluminum alloys, magnesium alloys, titanium alloys, zinc are to name a few. So you can see it here, it's a nice picture, you see these, the sheet material is cut into blanks, so he's got a stack of blanks here, you see, you see they're oval shaped here and so he puts it in the press, you know, and then outcome these parts, yes? And, you know, and then you have to cut the trim the edges, you may have to punch holes in this, you may have to do some surface coating, yes, to get the final part, yes? And so this is a manual press, you know, and, you know, for automotive applications, you have big, huge transfer presses where you have huge presses, one after the other and they each do part of the deformation till you get the final part, hmm? Okay, so, and obviously we don't see this guy here, you know, what he does is he presses two knobs and there it goes, so he's not doing much stress measurements or wondering about von Mises' yield criteria, yeah? So there is a little bit, you know, a big divide between making a real-life parts, yes, like these side panels of the car and then when you're making a material, yes, and trying to evaluate how it will behave in this application, right? And so people will do research, they will do quality control on sheet products to evaluate how this material will work in this situation, right? And obviously, so they'll do some, they'll use special parts, you know, laboratory parts here, this looks very nice, it's a nice part, it's been designed, so it tests the material in all possible difficult forming situation, but it's not used for anything else, you know, it's not a real part. And it's also part that people use, develop, for instance, to see if when they do a finite element simulation, whether their calculations mimic reality, to see how it's basically validating, they call this validating a code, yeah, or basically a program. And so that's one thing, but this is already the domain of research, but in practice, you know, when you want to quickly evaluate the formability of a material, you have simple tests, and we'll talk about three simple tests today, and one of them is the cup drawing test, it looks like a very simple test, and we'll see how this test helps us to define formability in practice, and then we'll also, we'll talk about two other tests, the so-called hold expansion test and the spring back test. And the reason why I want to say this is because when you do a tensile test, I've already told you you can determine the r value, yeah, the r value, and that's, we already know, that's a measure of formability, but that's not the only measure of formability, there are other things people are interested in, and so that's what we'll be talking about, and go a little bit into detail, so you get an idea of the complexity. So this particular, excuse me, cup here that I just showed you on the right, yes, okay, that's made with a sheet forming test machine, yes, and there are many laboratories, people use this machine, this is built by Erickson, but there are other ones, yes, and which are basically, there are two basic parts in this machine, yes, and that is a punch and a die, yes, and what do these parts do? Right, yes, I'll just, well, okay. So look at the cross section on the upper right here, so you have your die, yes, you put on this die a small circular blank, yes, you clamp it between the so-called blank holder, that's the top ring there, yeah, and then you punch, you use a punch to push the sheet into the die, and so if I interrupt the test on its way to making a full cup, you see, you have this hat shaped, yes, and what happens when you push down the punch, the diameter of the blank decreases as you punch down the material, right, and so the material is pulled into this circular cylindrical hole, okay, and so we have three parts to this cup, a flange part, yes, a wall part and a bottom part, yes, and it eats undergo a different, what we call a strained path, yes, okay, so this is the sideways, you see, you have a die, yes, you have a blank holder who keeps this flange flat, yes, and you have a punch, okay, good. So first of all, when you do the simple test, one of the first things you see after you make the cups is that, well, some cups have a nice flat edge, yes, other cups have edges with waves, which may be very pronounced or not very pronounced, usually four, and that tells you already the appearance of this, we call them ears, yes, tells you that this material is anisotropic, yes, it's got r values, the change with direction, okay, we already see things that tell us something about the formability, okay, okay, what happens during the test here, okay, so first of all, the test is, you just don't go about and just, you know, put this thing in and go and make a cup, you have specific test conditions, yes, which you have to determine what are the good test conditions, yes, and these test conditions, once you have determined it, will also tell you something about formability of the material, so let's just look at a little bit more detail in this flow of the material, so say we have interrupted the test, we have this hat shape, yes, what happens in the flange part, you see, the original blank is basically a circular, like a 12 centimeter or whatever, that order of magnitude, 10 centimeter diameter sheet material, yeah, and then as you punch it in, yes, you make, this edge moves towards the center, okay, so the material at the edge here, yes, the radius or the outer radius here is squeezed into this radius, yeah, so if you squeeze material, yeah, into, you know, that's this wide, into something this wide, yes, what has to happen? The material gets thicker, yes, so you get a thickening of the flange part of the blank, so the thickness changes, so you can see it here, what we've done is, we've taken a cup, we cut, just cut it, just cut it, yes, and it's about 1.3 in thickness here, at the bottom, yes, and if we go up, we can see that it, you clearly see that it becomes thicker, yes, and that's because we, in the flange, you squash the material and you make it longer, yes, okay, another thing that's really interesting here is as we cut the wall loose, yes, it curls, yes, what does this tell you? That tells you that there is, that you have very high internal stresses in your materials, in your material, so we won't be talking about this, but it's something I want you to be aware of, because you can see it here on the image, is that any time you deform things, yes, and it looks perfectly flat, you know, perfectly straight, there are very high, there may be very high internal stresses in this material, yes, and they appear when you cut it off, and you cut parts out, and then the cup is no difference, okay? We'll talk this in more detail, so you have a thickening of the flange, the wall here is in what we call plain strain conditions, I really told you that that's a rather scary condition, every lengthening is directly translated in thinning of the wall, and then at the bottom, yes, we have thinning of the bottom part of the flange because we have stretching conditions, so we can already see different stress states in different parts of the cup, right, let's see, so if we look at the stress situation during the cup forming, it's actually rather complex, and it changes, so... and, okay, so these would be the circles of the vomisus circles, right, and you could actually determine what is the stress situation, for instance, at the bottom of the cup, as you do the pressing, yes, so this is shown here, you have here is the change of the stresses in this area, and this is the change of the stresses in the flange, so in the flange, you have different stress conditions, different stress state, than in the flange here, right, and in the bottom, yes, and it also changes, the severity of the stresses change during the test, you can see here that you have plane strain, yes, excuse me, the plane strain refers to this, the wall part, not the flange part, which is in this so-called deep drawing stress state, in deep drawing, we have compression in one direction and pulling in the tension in the other perpendicular direction, and you can see that as you go, the stresses in this part here increase very appreciably during the test, okay, good, so let's go and see, say something about the flange part, yes, the flange part, so we know that this thing, this outer ring, has to go move in here, yes, what happens if, when you do this, if you try to do this with a piece of cloth, for instance, you try to push a piece of cloth into this hole, for instance, you will see, the piece of cloth will start developing wrinkles, yes, the same thing happens here, if I don't apply a blank holder force here, yes, to keep the flange flat, the flange will develop wrinkles, yes, and your cup will not look like this, but will look like this, yes, so these wrinkles, yes, that were under the blank holder were, you know, were basically deep drawn, and you get this obviously really bad cup, so the point I'm trying to make is that the blank holder, the force we use on this blank holder, and the size of the blank relative to the diameter of the punch are important parameters, ok, so we need to look at the ratio of the blank diameter relative to the punch diameter, and this ratio is called the drawing ratio, so that's an important parameter of formability, another important parameter is of course the blank holder force, ok, let's see how we work this out in practice, ok, and you will already see that formability is also visible in this type of test, ok, on this graph, it's schematic, I show on the x-axis the so-called drawing ratio, the drawing ratio is the ratio of the blank diameter on the punch diameter, so when the blank, the ratio is 2, it means that I have a diameter, a blank diameter, that's twice the diameter of the punch, ok, and on the y-axis I have the blank holder force, right, so when it's 0, I have no blank holder force, there's no blank holder force, yes, ok, so now let's imagine I have a not-too-wide blank, and I have no blank holder force, no blank holder force, and I'm punching a cup without blank holder force, the flange will start to form waves, we get wrinkling, yes, as I increase the blank holder force, yes, I can suppress this wrinkling, I can suppress the wrinkling, and I can make a nice cup, yes, ok, but how high can I go, how strong can I press the blank holder, ok, well let's imagine we continue increasing the blank holder force, we'll make it very high, yes, at one time the force on the blank holder will be so high that the material will be unable to flow, right, the material will just be stuck, it'll just be stuck between the die and the blank holder, and the cup will break, it'll just break, yes, so there is a maximum, a maximum, yes, where the, where I get fracture, yes, ok, so, now what happens to this maximum, if I make the blank larger and larger, yeah, say I make a larger blank here, ok, same, same punch, larger blank, right, because I have a larger blank, I will need less force, less blank holder force to make the cup break, yes, so this upper line decreases as I make the blank holder, the blank size larger, in other words, when I increase the drawing ratio, yes, this curve goes down, yeah, and eventually the wrinkling line, that's pretty much horizontal, yes, it moves up slightly, and the fracture line meet, yes, and it means that I can only form a cup within this range, yes, within this range of blank holder force and drawing ratios, yeah, what is an important parameter of formability is the maximum drawing ratio, the maximum drawing ratio is the maximum black diameter, that where I will still be able to make a cup divided by the punch diameter, yes, ok, now say you're, you know, you don't know anything about plasticity, yes, but you've done this test, and you have two materials, and one material gives you this maximum drawing ratio, and another material, another material gives you this, I'm playing this about the same, yeah, ok, which one would you rather use in your press shop? Well, the one that's not going to, you know, that's going to not break over a larger domain of stresses and drawing ratios, yeah, because this is the range of blank holder force, yes, for one of the materials, and this is the range of formability for the other material, so the other material is, will be easier to form than the first material, the one that has the red forming range, yes, so this parameter here, the beta max, is a very important technical evaluation of formability, yes? You can also already see that having, you know, a stronger material, yes, can improve your formability, because it allows you to use larger forces, yes, without risk of fracture, ok? Right, of course, and here I want to come back to the fact that you have to take the idea of force in a broad sense, right, because if you increase formability, as I've told you, it has an impact on the stresses at which the deformation and eventual fracture will occur, so in the bottom you have a biaxial stretching, in the flange you have drawing, because the material is squashed in this direction, and pulled in this direction, so that's a drawing situation, or pure shear if you want, and then in the wall here we have plane strain, yes? And we know we have plane strain, because the material can get longer in this direction, but it cannot change its width, and the reason why it cannot change its width, is because it's constrained by the presence of the punch, the punch surface is there, so it's just basically stuck against, it cannot get smaller, or larger, yes? So it's plane strain, and we know, and this was really important, that you can have texture hardening, and that at higher r values, you can delay the yielding, okay? It makes the material basically stronger, in the important places, you don't want fracture here, you don't want fracture here, and you want the material to be able to flow here, yes? And this is not, the drawing part is not very dangerous, yes? And we'll see in a moment why that is, okay? But before we can do that, we need to talk a little bit about strains, yes? Strains in more detail, okay? So say I have a piece of material, yes? And in this piece of material, I have, I imagine a little sphere of material, yes? And now I deformed this material, and in this particular case, I've, what did I do? I squashed it, I squashed it by half, and so because I squashed it, I have to elongate it, because I have to keep my volume in plastic deformation, yes? And then I moved it to the right, and I turned it around. So moving, translation and rotation doesn't do anything, right? To the deformation, right? So this original little circle will turn into an ellipse, yes? It will always do this. I mean, you can do this at home with like plastic foil, or any time you have it, it has to be small enough, right? Small enough volume. What's small enough? Well, millimeter-sized, yes? Millimeter-cube size, or this circle will turn into an ellipse. And it's really nice, because we automatically know our principal directions, yes? You always know your principal directions, yes? And that makes the analysis of strains easy, yes? In sheet-forming, instead of looking at a little volume of material, we just etch a little circle on the surface. And if we do the deformation, this little circle will turn into an ellipse, yes? And if we deform it, it will turn into an ellipse, which may change its shape, yes, through its... eventually it will end being a final ellipse, yes? And so this allows us to look at this... define the strains in terms of principal strains, yes? We don't have to worry about shear stresses, shear strains, et cetera, because we automatically have a very simple way of analyzing our observations. What do I mean? Well, let's just look at a situation, the situation that happens in the flange of a cup test, or in any flange of a part where you do dip drawing. Your circle, little circle that you put on the surface, changes, for instance, into this ellipse. So it's been squashed in one direction and elongated in another direction. Where does this type of deformation happen? Well, it happens here. If I draw a little circle here, and on my original sample, I have a little circle here, and I track this little circle during deformation, I will find that this original little circle that looked like this is now elongated in this direction and squashed in the other direction, yes? So it's a little circle at the beginning, so the diameter is L0. I do this type of straining, which we call this particular type of straining, it's called drawing, yes? And now this is now an ellipse shape, and I can measure this ellipse. I can measure the length this direction, I can measure the length in that direction, yes? So if I make the natural logarithm of n length in the principal direction, divided by the starting length, for one of the major direction, and then for the other major direction, principal direction, L2 divided by L0, and then take the natural logarithm, and basically have the two principal strains, two principal strains. Okay, and because it's plastic deformation, I automatically have the third principal direction, which is perpendicular to the sheet, yes? And we know, because the volume is constant in plastic deformation, that the strain in the one, the second, and the third principal direction must be zero. So as soon as I measure two of the principal directions, I have the third one, yes? Okay. So these two are in the plane, and this is perpendicular to the plane, in the plane of the sheet. Okay. But of course, when you deform a sheet, the circle can do other things, right? For instance, it can just expand isotropically. If I pull by actually this circle, as I increase the strains, I can have a perfect circle after the deformation, right? So that is stretching. That's called stretching, yes? Or I can have plane strain, where one of the strains doesn't change. One of the strains is zero, and I stretch the material only in one direction, hmm? Yeah. And the nice thing with this approach is that any deformation you can give a sheet, yes, can be represented on this graph, yes? You don't need a third dimension, yes? We'll see how we can put in information about thickness on this graph in a moment, yes? But that's basically it, yeah? And of course, why would we do this? Because if I deform a material, for instance, in this direction, eventually it'll fracture, yes? And I can indicate at what point this fracture happens, I obtain that way, I obtain what is called a forming limit diagram, okay? All right, but just again to simplify things, let's look at our tensile test, because, you know, tensile test is, certainly if you do a tensile test on a flat sheet, yes, a flat material, yeah? What happens in this case? Well, if you had a standard tensile specimen like this for sheet, yeah? And you would put some circles on this, a circle pattern, yes? We know that we have a length strain, yeah? We have a width strain, and we have a thickness strain, okay? And so these circles would turn into ellipses of this nature, yes? Because there is no force that prevents a reduction of the width, yes? And I get ellipses that look like this, yes? So a tensile test on this diagram, in this diagram, would look like this. So you would start having a circle here, and then this circle would become longer and longer, like this, okay? But it would become... so the strain in this direction here is negative, and the strain in this direction is positive, yes? So you would have a strain path along this direction, yes? Eventually, you reach the... a uniform elongation, right? You reach the moment where the sample starts to neck, right? So you can indicate this. At this moment, the sample starts to neck, yes? And at this moment, the sample breaks, okay? And this would be another way of presenting the formability, as it were, for your sample, yes? Okay. All right, so... okay, so I hope that's clear. How... okay, and so, again, what we get is this two-dimensional graph where you put the major strain, so the larger of these two on this axis and the smaller one on this axis, yes? And so the major strain is always larger than the minor strain, yes? That's why it's called the major strain. Both of them are principal strains, yes? And the minor strain in sheet form can be either positive or negative, okay? All right, good. The way you can study the behavior of the material in terms of how far can I deform before it starts to neck and fracture, yes? By using different type of samples, okay? For instance, you remember this bulge sample here, yes? When you put a punch here under the sample, you basically stretch into direction, yeah? So you obviously go... your strain path goes in this direction. Both the major strain and the minor strain are equal, yes, and positive, no? I can change the shape of this sample, yes? For instance, get this sample and get the material to deform in such a way that they get plain strain, yes? And then look when does this break, okay? So now let's add a... the dimension of the thickness strain. So it's very simple, as I said. We know that the sum of the principal strains is zero. That's basically expressing mathematically that the volume doesn't change, yeah? So the thickness strain is minus major and minor strains, yeah? Okay, so that means that at every place here in the graph, I can draw a line of constant thickness strain, okay? Okay? So for instance, this is an interesting line here, okay? And let me... you'll be very clear in a moment why that is an interesting line, yeah? The line here at 45 degree, what's special about this? Along this line, the major strain is equal to the minor strain, yes? Okay, so... excuse me. The major strain is minus because this is negative, right? And this is positive. The major strain is... is equal to minus the minor strain, right? So the thickness strain plus the major strain plus the minor strain is zero, right? If these two are equal in size and different size, it means that thickness strain is zero, right? So... okay? Thickness strain is zero. So this line is the line of no change in thickness. So if you can do a deformation with a strain path along here, yes? Your material will not bend, yes? Right, and so... right, so... Now, you remember what R was, yeah? R is the ratio of the width strain over the thickness strain, right? Okay, so the... the line here, yes? This slope, if you use this equation, you can show that you have this when R is very large. R is infinite. Why is that? Because width strain, thickness strain zero, yeah? That means R is infinite, okay? Infinite R, we don't have that, you know? In the best case, you can have R equals to two for steels, yes? Or slightly larger, yes? So strain paths along this, you know, you'd never find. But that's kind of the limit. And of course, when you do deformations, deep drawing deformation, that's one of the reasons why you like to have R large, yes? It's because it allows you to do the deformation with strains that have a minimum thickness reduction. And of course, thickness reduction is what happens on the way to fracture, right? So you want to avoid thickness reduction. Okay, so if you look at the... use this again, this equation here, for instance, you say, okay... Well, you plot, yeah, you basically plot constant, yeah? So, because... Excuse me, yeah? So if you say the thickness strain is a constant, say we assume the thickness strain is constant, then it also means that the sum of the minor and the major strains is a constant, or the major strain is equal to constant minus the minor strain, yes? And so this basically means these are lines with a negative slope, yeah? So these lines here are lines of thickness strain, of negative thickness strain. Here it's zero, yes? In this case, it's 10% or 0.1 thickness strain, 0.2 thickness strain, 0.3 thickness strain, etc. So when I move from here to here, I get thinning of the material, yes? The material gets thinner, deformations get thinner. In this diagram also, there are areas where you cannot... where there are no solutions, basically, that's this area here, yes? The minor strain can never be larger than the major strain, right? So there's no... this is a wide region of this plot, right? Because the major strain is always larger than the minor strain. So this we don't have. In this area, we can't usually come because of the thickening. Although we have seen that it can... it does occur at the flange region, right? You can have situations where the deformation path is such that you get into this region where the material gets thicker. So that's not impossible. And we'll see some examples in a moment. But this region is really a wide region. Nope, there cannot be data points there, okay? Right, so you have this feeling that on this diagram, I can plot four certain strain paths, yes? I can plot combinations of strains of major and minor strains where the fracture will occur. This is an example here. I have this thing, this sample here, and I push a punch underneath it so the material gets stretched in two directions, yes? In positive amounts. So I have minor strain and major strain are equal, yes, in size. So I make this circle, my circle grids expand, expand, expand, yes? I start to get a crack and then it breaks, right? And I can measure, I can say, okay, well, that moment, yes, I make a point here, yes? And then I try to build this for the rest of the deformation modes, yes? And I can determine a range where the deformations, yes, will be safe, will not lead to necking and fracture. And of course, the larger I can make this range, yes, the better my formability is, yeah? Right, so, okay. So how would you do this? Because it looks like, my God, how can I do different strain paths? It's obvious, what I showed you here, you can make strain paths. Or if I do biaxial, I can make also a very well-defined strain path, yes? Well, there is what's called the Nakazima approach, what you basically do is you basically take samples, yes? You use the same device, the same bulge test, yes? Where the punch is basically a dome, yes? And when you want to do the biaxial test, you have a full dome, yes? If the full square or round blank, yes? If you want to do the other strain paths, you reduce the width of the sample. You can see, it becomes progressively narrower and narrower and narrower. And here, it's very narrow strip, yes? And when you do this, basically go from stretching to deep drawing or uniaxial tension, yes? Just for those who are not familiar with it. In this test, you don't make a cup, yes? The edges of the sample are firmly gripped, yes? You really deform the material, yes? It doesn't flow towards the center, right? It's not like in the cup, when you make cup-forming tests, you really want the material to flow and form the cup, yes? In this case, when you're doing this FLD, you firmly keep the edge from moving going anywhere, OK? All right, so what you do is, on all these samples, you make little circle grids. You put little circle grids and you measure. You basically measure, for instance, OK? This is an example of some measurements. So let's just look at biaxial strain. This would be, for instance, a sample here of a relatively small sample of 30 centimeter by 30 centimeters, yes? A small sheet like this. And I've put a circle grid over this sample and I measure the strain, yes? The strains over this debulch, yes? And this is what I get, yes? This is biaxial situation. So the stresses, the strains, excuse me, in one direction and the perpendicular should be about equal, right? And you can see the squares fall on top of the other symbols, yes? They're the same, yes? The thickness strain is very large and minimum because when you stretch in two directions, everything has to come from thickness, OK? How much is it? Well, you have about 0.4 strain for each one of them, each direction. So volume is constant, so the thickness strain should be the sum of the two, 0.4 plus 0.4 minus, yeah? So about 0.8 and that's what you get, yeah? Plain strain, yes? We're in a situation where we're in this situation here. So we have a positive major strain and a zero minor strain, yes? So positive major strain, positive major strain, negative minor strain. The reason why this change is here is because I'm measuring different positions on the sample, right? OK, and this, excuse me, this point here is where it starts to break or where it starts to fracture, OK? So in this case, my minor strain is zero, yeah? So all the lengthening is translated into thickness strain, thinning, OK? And this is the situation here where I'm at the other end, so in this case, this specimen here, if I measure the strains there, OK? I find a positive major strain, a negative minor strain, and that's where you get the smallest thickness strain. And that's to be expected because we get closer to this line, yeah? Of the zero thickness strain. Now, in this figure here, this is data, this is actually measurements here, and you can see these points here, this point here in particular, right? And this point here, that's where the fracture occurs, yes? These are measurements made very close to the crack, yes? So these are the points that I will put on my diagram, you see? So this is, for instance, for low-carbon steel and for so-called jewel-face steel, data, yes? So in, for instance, this point here corresponds to a measurement for major and minor strain where I will have fracture, OK? And so this allows me, yes, to define a region of deformations and strains where it's safe to deform the material. Up to that amount of deformation, I can deform the material without problems, OK? And so obviously, you want this line to be as high as possible, yeah? As high as possible, OK? And the, so the minimum in this line, yes? And, yes, this is called the FLC0 point here, this point here is very much determined by the strain hardening, yes? The strain hardening, OK? So the larger our strain hardening, our N value, yes, if you want, is the higher our FLD forming limit curve comes, OK? I always get confused with the words. This forming limit diagram, forming limit curve, this curve here is the forming limit curve, yes? And this point is very dependent on the N value. So you want this point to be as high as possible because it gives me a lot of space as it were in strain space where I can deform without having fracture, OK? And then I can reach the end of the lecture time. So we'll meet on Tuesday next week. And in the meanwhile, have a good day. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you. Thank you.
 So, well, I left this person finished.  So, again, about strategy of answering the quizzes.  So, we have a quiz every week, yes?  There are about, each time there are 10 questions.  That means that over the semester, you will answer, you know,  close to 150 questions about, yes?  So, what you basically do every time you answer a question  correctly, you accumulate points, right?  So, that's how you have to look at it, yes?  So, you know, if you're absent, you know,  you don't accumulate points, right?  I don't, I will not take that into account,  because I will normalize your grade, yes?  But the idea is not that you make one very successful test  and then don't ever show up for the quizzes again, right?  So, if you're not there to take the quiz, you know,  you miss out more than one or two quizzes, you know,  will have a serious talk, because you're supposed to be here,  okay?  Then when you answer, please write your name.  You don't have to write very much, right?  You have to write your name, yes?  And then, 10 letters, either a T or an F.  So, you know, there's still people who just, you know,  I get someone who writes this, right?  And, you know, and if it's not well written,  it can be anything, right?  So, don't do this to me, okay?  So, this is a T and this is an F, right?  And if you can't write these two letters, you know,  it's going to be an error, okay?  It's very simple and I'm not asking too much.  And write your name nicely also, you know,  so I can, because there are two people  who have very similar names and they scribble their names,  so I can't see the difference.  I think there are two Mr. Shin's, okay?  So, please write your name carefully,  so I don't give your very high grade to your friend.  Okay, good.  So, let's collect the papers.  Let's just bring them here in front and let's move on  with what we were doing.  Right, and the best way to do well in this course is just,  you know, participate to all the quizzes.  And I really stick to what I've said in class.  So, if you go through the slides and the materials you had,  you know, you should be okay in the course.  Whatever level of difficulty we go into during the course.  Okay, so we had arrived at the point on Monday,  on Tuesday, excuse me, that, you know,  there's this whole body of theory of plasticity and, you know,  and we're all happy about having a flow rule and everything  like this and know about r factors and delta r's  and what the impact is on strain increments and things  like this, but it also leaves you with a feeling that,  you know, what can I do with this in practice?  And, well, in practice, when you go into real practice  of formability and plasticity, you realize  that actually stresses, there's not much discussion  about stresses in that case.  In practice, very often when there is discussion  of formability, it's about strains that the discussion goes.  So, because formability is a really nice subject  in basically two areas, one of them is press forming,  that's really complex, you know, there's a lot of interest  in that and then massive forming, which is basically,  you know, where you have big blocks of material and,  you know, you shape them in a specific form,  so you have large deformations, the areas,  in these areas people are interested in plasticity  and of course there's a lot of work in plasticity  which is related to fracture, you know,  and what goes on before fracture, certainly when we'll be talking  about plastic fracture, you know,  this plasticity will pop up again as a subject.  But, so we'll focus on sheet forming and in sheet forming,  you know, you have these big presses, you know,  where you basically form objects like these.  Many things around us are made like this by presses  and the material you start with is sheet steel,  it's just flat sheet and that can be, of course, steel,  but, you know, we have also sheet, other metals  and alloys are available as sheet to make parts,  aluminum alloys, magnesium alloys, titanium alloys,  zinc are to name a few.  So you can see it here, it's a nice picture,  you see these, the sheet material is cut into blanks,  so he's got a stack of blanks here, you see,  you see they're oval shaped here  and so he puts it in the press, you know,  and then outcome these parts, yes?  And, you know, and then you have to cut the trim the edges,  you may have to punch holes in this,  you may have to do some surface coating, yes,  to get the final part, yes?  And so this is a manual press, you know,  and, you know, for automotive applications,  you have big, huge transfer presses where you have huge presses,  one after the other and they each do part of the deformation  till you get the final part, hmm?  Okay, so, and obviously we don't see this guy here,  you know, what he does is he presses two knobs  and there it goes, so he's not doing much stress measurements  or wondering about von Mises' yield criteria, yeah?  So there is a little bit, you know,  a big divide between making a real-life parts, yes,  like these side panels of the car  and then when you're making a material, yes,  and trying to evaluate how it will behave  in this application, right?  And so people will do research,  they will do quality control on sheet products  to evaluate how this material will work in this situation, right?  And obviously, so they'll do some,  they'll use special parts, you know, laboratory parts here,  this looks very nice, it's a nice part, it's been designed,  so it tests the material in all possible difficult forming situation,  but it's not used for anything else, you know, it's not a real part.  And it's also part that people use, develop, for instance,  to see if when they do a finite element simulation,  whether their calculations mimic reality,  to see how it's basically validating,  they call this validating a code, yeah, or basically a program.  And so that's one thing, but this is already the domain of research,  but in practice, you know, when you want to quickly evaluate  the formability of a material, you have simple tests,  and we'll talk about three simple tests today,  and one of them is the cup drawing test,  it looks like a very simple test,  and we'll see how this test helps us to define formability in practice,  and then we'll also, we'll talk about two other tests,  the so-called hold expansion test and the spring back test.  And the reason why I want to say this is because when you do a tensile test,  I've already told you you can determine the r value, yeah, the r value,  and that's, we already know, that's a measure of formability,  but that's not the only measure of formability,  there are other things people are interested in,  and so that's what we'll be talking about,  and go a little bit into detail,  so you get an idea of the complexity.  So this particular, excuse me, cup here that I just showed you on the right, yes,  okay, that's made with a sheet forming test machine, yes,  and there are many laboratories, people use this machine,  this is built by Erickson, but there are other ones, yes,  and which are basically, there are two basic parts in this machine, yes,  and that is a punch and a die, yes,  and what do these parts do?  Right, yes, I'll just, well, okay.  So look at the cross section on the upper right here,  so you have your die, yes,  you put on this die a small circular blank, yes,  you clamp it between the so-called blank holder,  that's the top ring there, yeah,  and then you punch, you use a punch to push the sheet into the die,  and so if I interrupt the test on its way to making a full cup,  you see, you have this hat shaped, yes,  and what happens when you push down the punch,  the diameter of the blank decreases  as you punch down the material, right,  and so the material is pulled into this circular cylindrical hole,  okay, and so we have three parts to this cup,  a flange part, yes, a wall part and a bottom part, yes,  and it eats undergo a different, what we call a strained path, yes, okay,  so this is the sideways, you see, you have a die, yes,  you have a blank holder who keeps this flange flat, yes,  and you have a punch, okay, good.  So first of all, when you do the simple test,  one of the first things you see after you make the cups is that,  well, some cups have a nice flat edge, yes,  other cups have edges with waves,  which may be very pronounced or not very pronounced, usually four,  and that tells you already the appearance of this,  we call them ears, yes, tells you that this material is anisotropic, yes,  it's got r values, the change with direction, okay,  we already see things that tell us something about the formability, okay, okay,  what happens during the test here, okay,  so first of all, the test is,  you just don't go about and just, you know, put this thing in and go and make a cup,  you have specific test conditions, yes,  which you have to determine what are the good test conditions, yes,  and these test conditions, once you have determined it,  will also tell you something about formability of the material,  so let's just look at a little bit more detail in this flow of the material,  so say we have interrupted the test, we have this hat shape, yes,  what happens in the flange part, you see,  the original blank is basically a circular,  like a 12 centimeter or whatever,  that order of magnitude, 10 centimeter diameter sheet material, yeah,  and then as you punch it in, yes, you make,  this edge moves towards the center, okay,  so the material at the edge here, yes,  the radius or the outer radius here is squeezed into this radius, yeah,  so if you squeeze material, yeah, into, you know, that's this wide,  into something this wide, yes, what has to happen?  The material gets thicker, yes,  so you get a thickening of the flange part of the blank,  so the thickness changes,  so you can see it here, what we've done is,  we've taken a cup, we cut, just cut it, just cut it, yes,  and it's about 1.3 in thickness here, at the bottom, yes,  and if we go up, we can see that it,  you clearly see that it becomes thicker, yes,  and that's because we, in the flange, you squash the material  and you make it longer, yes, okay,  another thing that's really interesting here  is as we cut the wall loose, yes,  it curls, yes, what does this tell you?  That tells you that there is,  that you have very high internal stresses in your materials,  in your material,  so we won't be talking about this,  but it's something I want you to be aware of,  because you can see it here on the image,  is that any time you deform things, yes,  and it looks perfectly flat, you know, perfectly straight,  there are very high, there may be very high  internal stresses in this material, yes,  and they appear when you cut it off,  and you cut parts out, and then the cup is no difference, okay?  We'll talk this in more detail,  so you have a thickening of the flange,  the wall here is in what we call plain strain conditions,  I really told you that that's a rather scary condition,  every lengthening is directly translated  in thinning of the wall,  and then at the bottom, yes,  we have thinning of the bottom part of the flange  because we have stretching conditions,  so we can already see different stress states  in different parts of the cup, right, let's see,  so if we look at the stress situation  during the cup forming,  it's actually rather complex, and it changes,  so...  and, okay, so these would be the circles  of the vomisus circles, right,  and you could actually determine what is the stress situation,  for instance, at the bottom of the cup,  as you do the pressing, yes,  so this is shown here,  you have here is the change of the stresses in this area,  and this is the change of the stresses in the flange,  so in the flange, you have different stress conditions,  different stress state,  than in the flange here, right,  and in the bottom, yes,  and it also changes,  the severity of the stresses change during the test,  you can see here that you have plane strain,  yes, excuse me,  the plane strain refers to this, the wall part,  not the flange part,  which is in this so-called deep drawing stress state,  in deep drawing, we have compression in one direction  and pulling in the tension in the other perpendicular direction,  and you can see that as you go,  the stresses in this part here  increase very appreciably during the test, okay,  good, so let's go and see, say something about the flange part,  yes, the flange part,  so we know that this thing, this outer ring,  has to go move in here, yes,  what happens if, when you do this,  if you try to do this with a piece of cloth,  for instance, you try to push a piece of cloth into this hole,  for instance, you will see,  the piece of cloth will start developing wrinkles, yes,  the same thing happens here,  if I don't apply a blank holder force here, yes,  to keep the flange flat,  the flange will develop wrinkles, yes,  and your cup will not look like this,  but will look like this, yes,  so these wrinkles, yes,  that were under the blank holder were,  you know, were basically deep drawn,  and you get this obviously really bad cup,  so the point I'm trying to make is that the blank holder,  the force we use on this blank holder,  and the size of the blank relative to the diameter of the punch  are important parameters, ok,  so we need to look at the ratio of the blank diameter  relative to the punch diameter,  and this ratio is called the drawing ratio,  so that's an important parameter of formability,  another important parameter is of course the blank holder force,  ok, let's see how we work this out in practice,  ok, and you will already see that formability  is also visible in this type of test, ok,  on this graph, it's schematic,  I show on the x-axis the so-called drawing ratio,  the drawing ratio is the ratio of the blank diameter  on the punch diameter,  so when the blank, the ratio is 2,  it means that I have a diameter, a blank diameter,  that's twice the diameter of the punch, ok,  and on the y-axis I have the blank holder force, right,  so when it's 0, I have no blank holder force,  there's no blank holder force, yes, ok,  so now let's imagine I have a not-too-wide blank,  and I have no blank holder force,  no blank holder force,  and I'm punching a cup without blank holder force,  the flange will start to form waves,  we get wrinkling, yes,  as I increase the blank holder force, yes,  I can suppress this wrinkling,  I can suppress the wrinkling,  and I can make a nice cup, yes, ok,  but how high can I go,  how strong can I press the blank holder, ok,  well let's imagine we continue increasing the blank holder force,  we'll make it very high, yes,  at one time the force on the blank holder will be so high  that the material will be unable to flow, right,  the material will just be stuck,  it'll just be stuck between the die and the blank holder,  and the cup will break, it'll just break, yes,  so there is a maximum, a maximum, yes,  where the, where I get fracture, yes, ok,  so,  now what happens to this maximum,  if I make the blank larger and larger,  yeah, say I make a larger blank here, ok,  same,  same punch, larger blank, right,  because I have a larger blank,  I will need less force, less blank holder force  to make the cup break, yes,  so this upper line decreases  as I make the blank holder, the blank size larger,  in other words, when I increase the drawing ratio, yes,  this curve goes down, yeah,  and eventually the wrinkling line,  that's pretty much horizontal, yes,  it moves up slightly,  and the fracture line meet, yes,  and it means that I can only form a cup within this range,  yes, within this range of blank holder force  and drawing ratios, yeah,  what is an important parameter of formability  is the maximum drawing ratio,  the maximum drawing ratio is the maximum black diameter,  that where I will still be able to make a cup  divided by the punch diameter, yes,  ok, now say you're, you know,  you don't know anything about plasticity, yes,  but you've done this test,  and you have two materials,  and one material gives you this  maximum drawing ratio,  and another material,  another material gives you this,  I'm playing this about the same, yeah,  ok, which one would you rather use in your press shop?  Well, the one that's not going to, you know,  that's going to not break over a larger domain  of stresses and drawing ratios, yeah,  because this is the range of blank holder force,  yes, for one of the materials,  and this is the range of formability for the other material,  so the other material is,  will be easier to form than the first material,  the one that has the red forming range, yes,  so this parameter here, the beta max,  is a very important technical evaluation of formability, yes?  You can also already see that having, you know,  a stronger material, yes,  can improve your formability,  because it allows you to use larger forces,  yes, without risk of fracture, ok?  Right, of course, and here I want to come back to  the fact that you have to take the idea of force  in a broad sense, right,  because if you increase formability,  as I've told you, it has an impact on the stresses  at which the deformation and eventual fracture will occur,  so in the bottom you have a biaxial stretching,  in the flange you have drawing,  because the material is squashed in this direction,  and pulled in this direction,  so that's a drawing situation, or pure shear if you want,  and then in the wall here we have plane strain, yes?  And we know we have plane strain,  because the material can get longer in this direction,  but it cannot change its width,  and the reason why it cannot change its width,  is because it's constrained by the presence of the punch,  the punch surface is there,  so it's just basically stuck against,  it cannot get smaller, or larger, yes?  So it's plane strain,  and we know, and this was really important,  that you can have texture hardening,  and that at higher r values,  you can delay the yielding, okay?  It makes the material basically stronger,  in the important places,  you don't want fracture here,  you don't want fracture here,  and you want the material to be able to flow here, yes?  And this is not, the drawing part is not very dangerous, yes?  And we'll see in a moment why that is, okay?  But before we can do that,  we need to talk a little bit about strains, yes?  Strains in more detail, okay?  So say I have a piece of material, yes?  And in this piece of material,  I have, I imagine a little sphere of material, yes?  And now I deformed this material,  and in this particular case, I've, what did I do?  I squashed it, I squashed it by half,  and so because I squashed it,  I have to elongate it,  because I have to keep my volume in plastic deformation, yes?  And then I moved it to the right,  and I turned it around.  So moving, translation and rotation doesn't do anything, right?  To the deformation, right?  So this original little circle will turn into an ellipse, yes?  It will always do this.  I mean, you can do this at home with like plastic foil,  or any time you have it, it has to be small enough, right?  Small enough volume.  What's small enough?  Well, millimeter-sized, yes?  Millimeter-cube size, or this circle will turn into an ellipse.  And it's really nice,  because we automatically know our principal directions, yes?  You always know your principal directions, yes?  And that makes the analysis of strains easy, yes?  In sheet-forming, instead of looking at a little volume of material,  we just etch a little circle on the surface.  And if we do the deformation,  this little circle will turn into an ellipse, yes?  And if we deform it, it will turn into an ellipse,  which may change its shape, yes, through its...  eventually it will end being a final ellipse, yes?  And so this allows us to look at this...  define the strains in terms of principal strains, yes?  We don't have to worry about shear stresses,  shear strains, et cetera,  because we automatically have a very simple way of analyzing our observations.  What do I mean?  Well, let's just look at a situation,  the situation that happens in the flange of a cup test,  or in any flange of a part where you do dip drawing.  Your circle, little circle that you put on the surface,  changes, for instance, into this ellipse.  So it's been squashed in one direction  and elongated in another direction.  Where does this type of deformation happen?  Well, it happens here.  If I draw a little circle here,  and on my original sample,  I have a little circle here,  and I track this little circle during deformation,  I will find that this original little circle that looked like this  is now elongated in this direction  and squashed in the other direction, yes?  So it's a little circle at the beginning,  so the diameter is L0.  I do this type of straining,  which we call this particular type of straining,  it's called drawing, yes?  And now this is now an ellipse shape,  and I can measure this ellipse.  I can measure the length this direction,  I can measure the length in that direction, yes?  So if I make the natural logarithm of n length  in the principal direction,  divided by the starting length,  for one of the major direction,  and then for the other major direction, principal direction,  L2 divided by L0,  and then take the natural logarithm,  and basically have the two principal strains,  two principal strains.  Okay, and because it's plastic deformation,  I automatically have the third principal direction,  which is perpendicular to the sheet, yes?  And we know, because the volume is constant in plastic deformation,  that the strain in the one, the second,  and the third principal direction must be zero.  So as soon as I measure two of the principal directions,  I have the third one, yes?  Okay.  So these two are in the plane,  and this is perpendicular to the plane,  in the plane of the sheet.  Okay.  But of course, when you deform a sheet,  the circle can do other things, right?  For instance, it can just expand isotropically.  If I pull by actually this circle,  as I increase the strains,  I can have a perfect circle after the deformation, right?  So that is stretching.  That's called stretching, yes?  Or I can have plane strain,  where one of the strains doesn't change.  One of the strains is zero,  and I stretch the material only in one direction, hmm?  Yeah.  And the nice thing with this approach  is that any deformation you can give a sheet, yes,  can be represented on this graph, yes?  You don't need a third dimension, yes?  We'll see how we can put in information about thickness  on this graph in a moment, yes?  But that's basically it, yeah?  And of course, why would we do this?  Because if I deform a material,  for instance, in this direction,  eventually it'll fracture, yes?  And I can indicate at what point this fracture happens,  I obtain that way,  I obtain what is called a forming limit diagram, okay?  All right, but just again to simplify things,  let's look at our tensile test,  because, you know, tensile test is,  certainly if you do a tensile test on a flat sheet, yes,  a flat material, yeah?  What happens in this case?  Well, if you had a standard tensile specimen like this  for sheet, yeah?  And you would put some circles on this, a circle pattern, yes?  We know that we have a length strain, yeah?  We have a width strain, and we have a thickness strain, okay?  And so these circles would turn into ellipses of this nature, yes?  Because there is no force that prevents a reduction of the width, yes?  And I get ellipses that look like this, yes?  So a tensile test on this diagram, in this diagram,  would look like this.  So you would start having a circle here,  and then this circle would become longer and longer, like this, okay?  But it would become... so the strain in this direction here is negative,  and the strain in this direction is positive, yes?  So you would have a strain path along this direction, yes?  Eventually, you reach the... a uniform elongation, right?  You reach the moment where the sample starts to neck, right?  So you can indicate this.  At this moment, the sample starts to neck, yes?  And at this moment, the sample breaks, okay?  And this would be another way of presenting the formability, as it were, for your sample, yes?  Okay.  All right, so... okay, so I hope that's clear.  How... okay, and so, again, what we get is this two-dimensional graph  where you put the major strain, so the larger of these two on this axis  and the smaller one on this axis, yes?  And so the major strain is always larger than the minor strain, yes?  That's why it's called the major strain.  Both of them are principal strains, yes?  And the minor strain in sheet form can be either positive or negative, okay?  All right, good.  The way you can study the behavior of the material  in terms of how far can I deform before it starts to neck and fracture, yes?  By using different type of samples, okay?  For instance, you remember this bulge sample here, yes?  When you put a punch here under the sample,  you basically stretch into direction, yeah?  So you obviously go... your strain path goes in this direction.  Both the major strain and the minor strain are equal, yes, and positive, no?  I can change the shape of this sample, yes?  For instance, get this sample and get the material to deform in such a way  that they get plain strain, yes?  And then look when does this break, okay?  So now let's add a... the dimension of the thickness strain.  So it's very simple, as I said.  We know that the sum of the principal strains is zero.  That's basically expressing mathematically that the volume doesn't change, yeah?  So the thickness strain is minus major and minor strains, yeah?  Okay, so that means that at every place here in the graph,  I can draw a line of constant thickness strain, okay?  Okay?  So for instance, this is an interesting line here, okay?  And let me... you'll be very clear in a moment why that is an interesting line, yeah?  The line here at 45 degree, what's special about this?  Along this line, the major strain is equal to the minor strain, yes?  Okay, so... excuse me.  The major strain is minus because this is negative, right?  And this is positive.  The major strain is... is equal to minus the minor strain, right?  So the thickness strain plus the major strain plus the minor strain is zero, right?  If these two are equal in size and different size, it means that thickness strain is zero, right?  So... okay?  Thickness strain is zero.  So this line is the line of no change in thickness.  So if you can do a deformation with a strain path along here, yes?  Your material will not bend, yes?  Right, and so... right, so...  Now, you remember what R was, yeah?  R is the ratio of the width strain over the thickness strain, right?  Okay, so the... the line here, yes?  This slope, if you use this equation, you can show that you have this when R is very large.  R is infinite.  Why is that?  Because width strain, thickness strain zero, yeah?  That means R is infinite, okay?  Infinite R, we don't have that, you know?  In the best case, you can have R equals to two for steels, yes?  Or slightly larger, yes?  So strain paths along this, you know, you'd never find.  But that's kind of the limit.  And of course, when you do deformations, deep drawing deformation,  that's one of the reasons why you like to have R large, yes?  It's because it allows you to do the deformation with strains that have a minimum thickness reduction.  And of course, thickness reduction is what happens on the way to fracture, right?  So you want to avoid thickness reduction.  Okay, so if you look at the...  use this again, this equation here, for instance, you say, okay...  Well, you plot, yeah, you basically plot constant, yeah?  So, because...  Excuse me, yeah?  So if you say the thickness strain is a constant, say we assume the thickness strain is constant,  then it also means that the sum of the minor and the major strains is a constant,  or the major strain is equal to constant minus the minor strain, yes?  And so this basically means these are lines with a negative slope, yeah?  So these lines here are lines of thickness strain, of negative thickness strain.  Here it's zero, yes? In this case, it's 10% or 0.1 thickness strain, 0.2 thickness strain, 0.3 thickness strain, etc.  So when I move from here to here, I get thinning of the material, yes?  The material gets thinner, deformations get thinner.  In this diagram also, there are areas where you cannot...  where there are no solutions, basically, that's this area here, yes?  The minor strain can never be larger than the major strain, right?  So there's no... this is a wide region of this plot, right?  Because the major strain is always larger than the minor strain.  So this we don't have.  In this area, we can't usually come because of the thickening.  Although we have seen that it can... it does occur at the flange region, right?  You can have situations where the deformation path is such that you get into this region where the material gets thicker.  So that's not impossible.  And we'll see some examples in a moment.  But this region is really a wide region. Nope, there cannot be data points there, okay?  Right, so you have this feeling that on this diagram, I can plot four certain strain paths, yes?  I can plot combinations of strains of major and minor strains where the fracture will occur.  This is an example here.  I have this thing, this sample here, and I push a punch underneath it so the material gets stretched in two directions, yes?  In positive amounts.  So I have minor strain and major strain are equal, yes, in size.  So I make this circle, my circle grids expand, expand, expand, yes?  I start to get a crack and then it breaks, right?  And I can measure, I can say, okay, well, that moment, yes, I make a point here, yes?  And then I try to build this for the rest of the deformation modes, yes?  And I can determine a range where the deformations, yes, will be safe, will not lead to necking and fracture.  And of course, the larger I can make this range, yes, the better my formability is, yeah?  Right, so, okay.  So how would you do this?  Because it looks like, my God, how can I do different strain paths?  It's obvious, what I showed you here, you can make strain paths.  Or if I do biaxial, I can make also a very well-defined strain path, yes?  Well, there is what's called the Nakazima approach, what you basically do is you basically take samples, yes?  You use the same device, the same bulge test, yes?  Where the punch is basically a dome, yes?  And when you want to do the biaxial test, you have a full dome, yes?  If the full square or round blank, yes?  If you want to do the other strain paths, you reduce the width of the sample.  You can see, it becomes progressively narrower and narrower and narrower.  And here, it's very narrow strip, yes?  And when you do this, basically go from stretching to deep drawing or uniaxial tension, yes?  Just for those who are not familiar with it.  In this test, you don't make a cup, yes? The edges of the sample are firmly gripped, yes?  You really deform the material, yes?  It doesn't flow towards the center, right?  It's not like in the cup, when you make cup-forming tests, you really want the material to flow and form the cup, yes?  In this case, when you're doing this FLD, you firmly keep the edge from moving going anywhere, OK?  All right, so what you do is, on all these samples, you make little circle grids.  You put little circle grids and you measure.  You basically measure, for instance, OK?  This is an example of some measurements.  So let's just look at biaxial strain.  This would be, for instance, a sample here of a relatively small sample of 30 centimeter by 30 centimeters, yes?  A small sheet like this.  And I've put a circle grid over this sample and I measure the strain, yes?  The strains over this debulch, yes?  And this is what I get, yes?  This is biaxial situation.  So the stresses, the strains, excuse me, in one direction and the perpendicular should be about equal, right?  And you can see the squares fall on top of the other symbols, yes?  They're the same, yes?  The thickness strain is very large and minimum because when you stretch in two directions, everything has to come from thickness, OK?  How much is it?  Well, you have about 0.4 strain for each one of them, each direction.  So volume is constant, so the thickness strain should be the sum of the two, 0.4 plus 0.4 minus, yeah?  So about 0.8 and that's what you get, yeah?  Plain strain, yes?  We're in a situation where we're in this situation here.  So we have a positive major strain and a zero minor strain, yes?  So positive major strain, positive major strain, negative minor strain.  The reason why this change is here is because I'm measuring different positions on the sample, right?  OK, and this, excuse me, this point here is where it starts to break or where it starts to fracture, OK?  So in this case, my minor strain is zero, yeah?  So all the lengthening is translated into thickness strain, thinning, OK?  And this is the situation here where I'm at the other end, so in this case, this specimen here, if I measure the strains there, OK?  I find a positive major strain, a negative minor strain, and that's where you get the smallest thickness strain.  And that's to be expected because we get closer to this line, yeah?  Of the zero thickness strain.  Now, in this figure here, this is data, this is actually measurements here, and you can see these points here, this point here in particular, right?  And this point here, that's where the fracture occurs, yes?  These are measurements made very close to the crack, yes?  So these are the points that I will put on my diagram, you see?  So this is, for instance, for low-carbon steel and for so-called jewel-face steel, data, yes?  So in, for instance, this point here corresponds to a measurement for major and minor strain where I will have fracture, OK?  And so this allows me, yes, to define a region of deformations and strains where it's safe to deform the material.  Up to that amount of deformation, I can deform the material without problems, OK?  And so obviously, you want this line to be as high as possible, yeah?  As high as possible, OK?  And the, so the minimum in this line, yes?  And, yes, this is called the FLC0 point here, this point here is very much determined by the strain hardening, yes?  The strain hardening, OK?  So the larger our strain hardening, our N value, yes, if you want, is the higher our FLD forming limit curve comes, OK?  I always get confused with the words.  This forming limit diagram, forming limit curve, this curve here is the forming limit curve, yes?  And this point is very dependent on the N value.  So you want this point to be as high as possible because it gives me a lot of space as it were in strain space where I can deform without having fracture, OK?  And then I can reach the end of the lecture time.  So we'll meet on Tuesday next week.  And in the meanwhile, have a good day.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.  Thank you.
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