Understanding Uniaxial Monotonic Plasticity

Alright, so in this lecture we're going to talk a little bit about uniaxial monotonic plasticity. And just as kind of an introduction, we can say that plasticity is a branch of mechanics that deals with the inelastic behavior of materials and structures. There's lots of different sub-branches, so things can be rate-dependent, some can have some time-dependent recovery, and different things like that. The focus that I'm going to take in this particular class is metal plasticity, which is... In some instances can be regarded as rate dependent, but we're going to initially look at rate independent loading. And we're not going to worry about time dependent recovery, although we might have situations that we're going to examine where we have material properties that may change a little bit over time. So we may have some work hardening or some work softening going on and some cyclic loading. But for now, we're going to keep things simple. And so when I talk about plasticity, it's probably easiest if we first kind of mention what. elasticity is just to kind of give a counter to that. And so one way to define elastic material behavior is that when you load a body and unload it, its deformation is recovered. So if we were to take a bar of steel or whatever and load on it, we know that we're going to stretch it out. I'm going to exaggerate this a little bit. And when we let go of that load, if it goes back to its original dimensions, then we would say that we have had elastic deformation. All of that deformation has been recovered. Now there's an alternative definition to elastic behavior that we can examine as well, and that is that we have a complete energy recovery through a loading and unloading cycle. It's basically just as I said, but thinking of it in terms of energy allows us to open up this concept to some different applications. So if we have a stress-strain curve, and we have elastic material behavior and it can be even nonlinear elastic that's okay it gets a little bit curvy right here where at this point if we release this and we get back here trying to show this following the same path to zero stress and strain we do not have any area contained inside of that curve that I just drew We'll eventually learn more about this. That's a hysteresis loop. If it is of a width where we do have some permanent deformation. But if we don't have any deformation, we have complete elastic recovery, then we can think of that as elastic material behavior. Now, I will say this. Elastic material behavior is an assumption that we make, and it's very good in a lot of instances, particularly for a lot of metals. However, we do know that... I can't guarantee, no one can guarantee, that every single atom in that piece of metal goes back to its original position. So there may be some development of slip systems and so forth. In fact, that's the basis for metal fatigue when we're below the yield stress of the metal. There are permanent changes that occur in the material. So elasticity, in a lot of instances, is an assumption that we're going to make to make our math and our analysis a little bit easier. But it is just that, an assumption. And just like any assumption in engineering, we have to be cognizant of when those assumptions break down. All right, so now we can talk about a plastic material. And in this case, we do have permanent deformation that is experienced by the material when it's loaded. And it's typically... observed when we go through our unloading cycle. So again, if we start with our bar of a certain initial length, we put enough stress and enough force on it, we stretch it out, that's our change in length at maximum load. If we release the load, We end up with a final length that is greater than the original length. And we have some permanent change in length. So again, we typically can't observe that while we're loading it. While we're loading it, it's very hard to distinguish a nonlinear elastic material from an elastic plastic material. But when we unload it, we can see that we do. have some permanent deformation. And by a plastic material I'm not talking about a polymer that's made out of you know polymer chains and so forth. A lot of times people use the word plastic to mean things that are polymers but in this case the word plastic in theory of plasticity means permanent deformation. So these definitions can be regarded as very good at the macroscopic length scale. And as we mentioned, at small length scales, there may be some movement of atoms or planes of atoms that can be observed even for macroscopically elastic loading. So let's take a little bit of a look at stress strain curves. And to understand stress strain curves, we need to know where the data comes from. So we start with a specimen. I have it drawn there with intention. And we can either do a tension test and what's called load control or displacement control so P is the load and Delta is the displacement so either one we can do if we want but the idea is typically this is done on a either constant load or a constant displacement rate and with time this just increases and eventually this thing will break now we do get some different looking stress-strain curves depending on if we are in Load control or displacement control, and we'll talk about that as we make our sketch. But in many common ways to do a tension test, we use displacement control, especially if we're using electromechanical testing machines. So with a servo hydraulic testing machine we can test in load control or displacement control very easily. It's a computer controlled testing machine that will make sure that you achieve a certain level of load or a certain level of displacement when you want it to. Electromechanical testing machines are older, an older style, but they're very useful and often used to construct stress strain curves. Now the difference then is if we were to look at our load versus displacement. for which we would construct our stress train curve. If we are in displacement control, we're controlling the separation of the grips along this horizontal axis. So it may increase, and it may just keep increasing, and the load is allowed to decrease during our tension test or displacement control. If we had our load control instead of our displacement control, we would be controlling along this vertical axis. And once we were at our peak load, then our specimen would fail. We would not be able to apply any higher load than the maximum load that the specimen can take. So we would not get this downward behavior of the stress-strain curve or the load-displacement curve. There are There are safety reasons why you'd probably want to do a displacement control test, but there's other reasons why you might want to do a load control test. You just have to be a little bit careful with it. So with a displacement control test, is what we're going to talk about now, we have our specimen, and before we test our specimen, we measure our original length and our original diameter of our specimen in the gauge section. We would take and we would hook up an extensometer. Which is a device that measures strain. And that extensometer has some initial gauge length. Now depending on what unit system you're in, it might be a 1 inch gauge length. And these blades that are attached to the specimen separate as the material is pulled. There are wires. it go off to the testing machine and the load is recorded the displacement is controlled as is recorded in the extensometer readings recorded along with time what I'm going to do is I'm going to supply you with a data set from a tension test where we have load crosshead displacement extensometer displacement in time all in the same file and I'm going to have you construct the stress strain curve Alright, so we have this load displacement data, and we're going to look at it. If we want to construct a stress-strain curve, we know that, you know, if you're interested in this class, you probably already know there's different ways to define stress and strain. We can use engineering stress and true stress, engineering strain and true strain, and there are actually a lot of different measures for stress and strain, some of which we'll talk about in a little bit more detail later. But to make an engineering stress-strain curve... What we're going to do is we'll take our, define our stress. I'll put a little subscript E and G to emphasize that's engineering stress. We'll take that stress. That will be our force that's applied to the particular instance divided by our original cross-sectional area. Of course, if it's a round bar, the A naught is pi over 4 times the original diameter squared. And our engineering strain is going to be defined as our extensometer displacement divided by our original extensometer gauge length. which amounts to the change in length divided by the initial length for that extensometer gauge section. Now when I draw these sketches, you know, like in this example here, this is load versus displacement. I have a large displacement with a little bit of load. Really, if you were to look at a piece of metal, this would be very nearly parallel to this axis. It has a little bit of a slope, but compared to the rest of the curve, the rest of the curve kind of really takes off. So these are just kind of pictures or pictograms. When we get to look at our actual data, then we'll have a better idea. of what that behavior looks like. So for all the data up until fracture I can compute engineering stress versus engineering strain curve. And so the shape is really the same. We're just kind of scaling a little bit. We're taking out the effect, the size, the specimen. And, you know, again, if you've had classes in stress analysis and stress-strain curves, you talked about that before, basically what you're doing is you're taking the effect, that size, out of the... specimen to get the material behavior. So a large specimen or a small specimen should have pretty nearly the same stress-strain curve. You'd use different size specimens depending on what testing machine capability that you have. Now there are some limits. If you get to a microstructural detail, you have a specimen so small that you get into microstructural length scales, well then you have to think about what effect that may have on your stress-strain curve. Or if you have things like casting porosities or things like that, macroscopic features, you may want to be careful about the size of your specimen in that case as well. But in the ideal case, the engineering stress-strain curve effectively takes the size of the specimen out of consideration. We're only looking at the material response now. Well, the stress-strain curve has some typical features to it. This point right here, where it failed, where that X is, we call that the fracture stress. That would be the engineering fracture stress. That would be our final load. divided by our original area. The highest point on the stress-strain curve, that's going to be kind of important for us. Even in this class, that's known as the ultimate stress. And the point of deviation from linearity, maybe it's around this point right here, it also has a specific name, and I'll write these down in just a moment. Let's call it the proportional limit stress, sigma sub PL. It's the highest point on the stress-strain curve where we no longer have a straight-line relationship between stress and strain. And that's typical of metals. Other materials might be a little bit different. Polymers may have more of a curvy or nonlinear initial part to their stress-strain curve. But most metals have a very straight-line relationship to their stress-strain curve. And so those are the definitions there over on the right for those different terms on the curve. Now, different materials have different stress strain curves. They may have like a low-carbon steel stress strain curve, has an upper and lower yield point. So these are some common features, but you may see some unusual things. And some materials have what's called serrated flow, and so things aren't very smooth. But again, this is just kind of a pictogram to kind of describe kind of a generic behavior of a material. Now another place that we want to investigate a little bit is right near the proportional limit for most materials is where we're going to define as our yield stress. Now our yield stress will probably be around in this area somewhere. Our yield stress is going to be defined as the point at which we begin to have permanent deformation. Now that's a very important location in plasticity because we use that as our demarcation between Elastic behavior, whether it's linear or nonlinear, and plastic behavior or permanent deformation. Well, this is kind of interesting. The way that we've defined our permanent deformation and our plastic material behavior already is by loading a specimen and unloading it and see if we have any change in dimension. This is not very practical to do in a tension test. In a tension test, we keep loading it until it breaks. So we need some way to estimate where we might have the beginning of permanent deformation. And even if we could load it and unload it, the point at which we start to have permanent deformation really depends on the sensitivity of our measuring equipment anyway. So we're going to use an approach called the 0.2% offset yield stress method to give us an estimate of where our yield stress might be for this kind of material. So let me... make a little bit of room and we're going to talk about the 0.2% offset yield stress. Before we do that, let's just go ahead and make a note. Maybe at this red spot right here, we get the sigma yield. That would be our yield stress. And we can find that as the point at which we begin to have permanent deformation. Like I said, some materials are very distinct where you start to have permanent deformation, particularly for low-carbon steels that have not been tested before in a monotonic tension test. You have an upper and lower yield point, but in some materials, titanium, aluminum, copper, things like that, we need to come up with an estimate for this yield stress point. So let's talk a little bit about this 0.2% offset method to estimate that yield stress. So again, what I'm going to do now is I'm going to zoom in on the initial portion of the stress-strain curve. On our engineering stress-strain curve, I'm trying to draw a straight line here. A lot of metals will have an initial straight line slope. We know this slope in the region where we're below the proportional limit stress. as the modulus of elasticity. Rise over the run of the stress-strain curve in the 0.2% offset method what that refers to the 0.2% refers to the strain now strain is Delta L over L naught. It's kind of like a percentage calculation except without the times 100 Well, you can actually represent strain as a percent. We were to convert this into inches per inch or millimeters per millimeter then this value would be 0.002 as a unitless quantity. And the idea with the 0.2% offset method, let me change colors, is that we go on our strain axis and we find the location of 0.002, and we draw a line that's parallel to the initial slope of the stress-strain curve. I'll do my best to try to get that slope the same. And I'm continuing and I'm dotting it up here. And the point of intersection of that newly dotted line with our stress-strain curve is known as our 0.2% offset yield stress. So it's pretty easy to show on a picture. It's sometimes a little more difficult to describe in words. So let me take a moment here, and I will write down some notes about how to find that 0.2% offset yield stress. All right, now if you're student in my class, I'm going to ask you to do a homework assignment where you examine some 6061 T6 aluminum data and plot the engineering stress-strain curve and some other things. But I want to talk about that data just a little bit first so we have a good understanding of what that data is all about. So this was taken... with an electromechanical testing machine. And I've got some other video on YouTube here that shows a tension test of a couple different materials, so it's very similar to that. I wouldn't say it's the same material. I don't remember if it is or not. But it's definitely not the same specimen, but it may be the same material. But on a text file, and I'll make this available to everyone on my personal website. Here's a readme file that describes the data just a little bit. Let's see if I can find that here. Okay, so it was a 1-inch gauge length for the extensometer. It had a 0.25-inch diameter. We're using English Standard Units. And also the strains need to be adjusted using the determined modulus of elasticity. And so what I mean by that, so when we put a... specimen in a testing machine. The specimen looks something like this. It has a large grip section, kind of is turned down in the middle and flares back out through some radius of curvature. I'm sorry that's not a very good looking specimen, but that's the way it is. It's gripped over on these surfaces. And when you use what's called electromechanical testing machines with mechanical wedge grips. The grips are in here, these are the jaws, and they are got an angle to them. And they sit inside a grip housing that has a matching angle, something like that. As these grips are tightened, as the specimen is pulled, these wedges and tension tighten up in this matching slope right here. And this is the same thing down on the bottom. But in order for you to start the test, in order to initially get it started, you have to apply a little bit of preload. So there may be 100 to 200 pounds preload, depending on... the how much force it takes to bite into that specimen now usually that is done before you apply the extensometer so if you think about it you're starting here at a stress level that's your initial force but you haven't measured any strain yet you usually zero your extensometer after everything is all set up and then you do your test now that preload is real causes real stress and it will affect your stress-strain curve. And really, the starting point for the strain is not at zero, but it is whatever strain corresponds to that preload. And so I want my students in this class to account for the preload strain. Well, how do you get that? Well, if you know the modulus of elasticity of the material, you can extend the slope downward and calculate this offset. And so you should apply... That offset to all of your data in your stress-strain curve to make an adjusted stress-strain curve. I do have an entire other video on this. It goes through in great detail. how to use Excel to calculate this and the equation of the line that does this and everything so I'll refer you to that I won't spend a lot of time on that but I but I do want everyone to adjust that strain because it's not an insignificant amount of strain when you're talking about a quarter inch diameter specimen and a couple hundred pounds the stress can easily get to be a pretty high number so now in this data file see if I can find that and open that up for you here's a a 36 deal here 60 61 we're going to look at that in this data file we have extension time load and strain in the extension is the extensometer reading both of these should be the same value I believe as we go through here Time is time and that happens to be in minutes in this data file. Here's the load. So actually we had 274 pounds of preload, units of pound. And then here's the strain again. So this is unadjusted strain is what this is. Now another point that I want to make about this data is There's a certain point where I've gotten to the ultimate stress, and this is starting to come down, where my extensometer only extends so much. And then I have to take it off. I really don't want to have my specimen break while an extensometer is on there. When I take the extensometer off, what happens is your strain values will look like they do something weird like this. This data right here for the strain channel after the extensometer removal isn't good. It's good for the load, but it's not good for the strain. Okay, so, you know, I'd like for you to go and edit your data to get rid of that extensometer removal point and anything after that on the strain side. Now, if you do need to find your final fracture stress, you can look at the final load before it breaks, and that's still a good value. Now, lastly, the other point I want to make about our stress-strain curve, then, is that the highest point on a stress-strain curve, we know that is the ultimate stress. There's a certain physical phenomenon that occurs around this ultimate stress and it's called necking of the specimen. N-E-C-K-I-N-G. And basically what happens is before necking what we have is our specimen starts out at a certain length and it increases in length uniformly. That means every point along this has the same strain. Let's make a note, uniform axial strain until necking, which is also known as the instability point. After necking, we have very localized deformation. You start to see kind of a little dent in our specimen that grows. And, you know, really we can't use epsilon is equal to delta L over L anymore because that strain isn't constant over our gauge length. We can still compute the number, but it may not make a lot of physical sense once we have that non-uniform deformation. In fact, I have a picture that I want to show you of a specimen. And so here's the specimen grips. I have here, you see the indentation where it was biting in, and here's the gauge section, here's my radius curvature, and you can kind of see it in this picture where I stopped this test at necking, I didn't break this specimen. Let's see if I have another picture where I've zoomed in on it. There we go. You see the surface is kind of modeled. It's beginning the failure process. In fact, we no longer have a uniform strain distribution along the axis of the bar. We no longer have a uniform stress distribution either. Because of this indentation, we start to get... at a more multi-axial stress state. It's very much like a notch in a bar. You can do different types of corrections. There's like a Bridgman correction and different things that you might do. purposes, once we get to this point of instability, then we don't have that uniform stress or strain distribution anymore. So that's going to be an important point for us, and it's going to be particularly important when we go to convert our engineering stress and strain to a true stress-strain curve. So for the homework assignment, let me see if I can find I want you to get this data, construct an engineering stress strain curve, find the modulus of elasticity account for this offset. We should find a 0.2% offset yield stress, construct a true stress strain curve for the material, and then plot the true stress strain curve and the engineering stress strain curve on the same set of axes. Now we need to talk about the true stress and true strain now at this point. Let me make a little bit of room and I'll pull up a new page. Okay, so here's our definition of our engineering stress and strain. Force divided by the original cross-sectional area and the change in length divided by the initial length. Both the force and the change in length are instantaneous quantities as we do our test measurements. But let's talk about definitions of true stress and strain. We know that when we apply tension to a material, its cross-section dimensions don't stay constant. So, here's the original diameter and the original length of this bar. When I pull on this, and I'll exaggerate this a lot, it gets a little bit skinnier, and it gets a little bit longer. So the final length is greater than the original length. The final diameter is going to be less than the original diameter. So to define our true stress, and maybe let's use a different color to emphasize that. I may not use this black color all the time. For now, our true stress can be defined as our force divided by our instantaneous cross-sectional area, a sub i. And we can define our true strain as an integral from our original gauge length to our final gauge length of dl. over L, where L is the current gauge. So here I've used the lowercase L-naught, but that's the same as my capital L-naught in this image. Now when we evaluate this integral, what we end up with is a natural log, and we have a natural log of L over L-naught. So L is the instantaneous length. L0 is the same as what I had called capital L0 before. That's the original length. And this AI I'm calling the instantaneous area. We can also call that A. so we can convert our engineering stress and strain to true stress and strain and let me go ahead and Let's get you another picture. It will be very similar to the one I've already done. Again, to exaggerate what's going on here, here's our specimen. It has some sort of initial area and some initial length. And it changes. It gets a little bit skinnier and a little bit longer. Again, very exaggerated. It has some instantaneous area and some instantaneous length. And we can define this difference in length as delta L. so that the final length is equal to the initial length plus the change in length. Well, if we use this then, and we go back to our definition of our logarithmic strain, I'm just going to call this epsilon now. A logarithmic strain is our current length, which is our initial plus our change in length. divided by our initial length. Take the logarithm of that quantity. We have 1 plus the change in length divided by the initial length. That's our engineering strain. So the true strain is related to our engineering strain by the change in length. this method. One plus the engineering strain but with that logarithm there in front of it. Now if the engineering strain is really tiny then they're going to be about the same quantities. This equation for true strain and the conversion from true strain from engineering strain is only valid up until the point of instability or necking. And the reason for that is because of that discussion we had about uniform and non-uniform strain distributions. Let's turn our attention now to stress. We're going to make an assumption here, and we'll talk about the validity of this assumption in a moment. We're going to make an assumption that for our analysis of true stress as related to engineering stress, that the material maintains a constant volume through its deformation process. So to compute the volume, of course, we'd have A0 times L0. And over here, we would have A times L. Well, if we are in elastic loading, we usually have very, very tiny strains. The dimension of the cross-section instantaneously and the dimension of the original cross-section dimension aren't too far different from one another. And so the true stress and true strains are very close to one another. In the elastic region, the assumption of constant volume is not a very good one. We do have some non-conserving volume in the elastic region of materials due to Poisson's contraction. This is known as the dilatational strain. However, in fully developed plastic flow, conservation of volume is pretty good. So, let's just make a note of that. So it's not so good in the elastic range, but quite frankly it doesn't matter in that part of the stress-strain curve. But it is pretty good when we have a lot of plasticity. So when the plastic strains, much greater than the elastic strains. All right, so if we can use this idea that volume is conserved then, then we can come up with a ratio. Let's see. We have A-naught, L-naught is equal to A-L. I can rewrite this as L over L-naught. is equal to a naught over a. I'm just doing some cross-multiplying. And with that, since the strain is the logarithm of L over L naught, it's also equal then to the logarithm of a naught over a. Now since the engineering stress is equal to the instantaneous force over the original cross-sectional area. We can say that the force, then, is equal to the engineering stress times the original cross-sectional area. Our true stress, I'll just call that sigma, our true stress is equal to the instantaneous force over the instantaneous area. And if we look at this, since epsilon is equal to ln of a0 over a, we can also write this then, we know that we can write this then as the ln of 1 plus the engineering strain. So if we take the logarithm and get rid of that on both sides, the argument of the logarithm has to be equal. So a-naught over a should be equal to one plus the engineering strain. And so let's do this now. Our force then is equal to the engineering. stress times A naught over A, but we just showed that A naught over A is equal to 1 plus the engineering strain. So the true stress in the material, I'll do this in another color, the true stress in the material is equal to the engineering stress times one plus the engineering strain at that particular load level. And just to have it written down here, the true strain is equal to the logarithm of one plus the engineering strain. And both of these are only valid until necking, or until we reach our ultimate stress of our metal. Alright, so what I want to do next, and I'll do this in a separate video, is I want to go through the data. I know this is a homework set, but my students should have finished this by now. I want to go through the data and work on my solution to come up with my engineering stress-strain curve and then my true stress-strain curve for this particular material.

The focus that I'm going to take in this particular class is metal plasticity, which is... In some instances can be regarded as rate dependent, but we're going to initially look at rate independent loading. And we're not going to worry about time dependent recovery, although we might have situations that we're going to examine where we have material properties that may change a little bit over time. So we may have some work hardening or some work softening going on and some cyclic loading.

But for now, we're going to keep things simple. And so when I talk about plasticity, it's probably easiest if we first kind of mention what. elasticity is just to kind of give a counter to that. And so one way to define elastic material behavior is that when you load a body and unload it, its deformation is recovered.

So if we were to take a bar of steel or whatever and load on it, we know that we're going to stretch it out. I'm going to exaggerate this a little bit. And when we let go of that load, if it goes back to its original dimensions, then we would say that we have had elastic deformation. All of that deformation has been recovered.

Now there's an alternative definition to elastic behavior that we can examine as well, and that is that we have a complete energy recovery through a loading and unloading cycle. It's basically just as I said, but thinking of it in terms of energy allows us to open up this concept to some different applications. So if we have a stress-strain curve, and we have elastic material behavior and it can be even nonlinear elastic that's okay it gets a little bit curvy right here where at this point if we release this and we get back here trying to show this following the same path to zero stress and strain we do not have any area contained inside of that curve that I just drew We'll eventually learn more about this. That's a hysteresis loop.

If it is of a width where we do have some permanent deformation. But if we don't have any deformation, we have complete elastic recovery, then we can think of that as elastic material behavior. Now, I will say this.

Elastic material behavior is an assumption that we make, and it's very good in a lot of instances, particularly for a lot of metals. However, we do know that... I can't guarantee, no one can guarantee, that every single atom in that piece of metal goes back to its original position. So there may be some development of slip systems and so forth.

In fact, that's the basis for metal fatigue when we're below the yield stress of the metal. There are permanent changes that occur in the material. So elasticity, in a lot of instances, is an assumption that we're going to make to make our math and our analysis a little bit easier. But it is just that, an assumption.

And just like any assumption in engineering, we have to be cognizant of when those assumptions break down. All right, so now we can talk about a plastic material. And in this case, we do have permanent deformation that is experienced by the material when it's loaded. And it's typically...

observed when we go through our unloading cycle. So again, if we start with our bar of a certain initial length, we put enough stress and enough force on it, we stretch it out, that's our change in length at maximum load. If we release the load, We end up with a final length that is greater than the original length. And we have some permanent change in length. So again, we typically can't observe that while we're loading it.

While we're loading it, it's very hard to distinguish a nonlinear elastic material from an elastic plastic material. But when we unload it, we can see that we do. have some permanent deformation.

And by a plastic material I'm not talking about a polymer that's made out of you know polymer chains and so forth. A lot of times people use the word plastic to mean things that are polymers but in this case the word plastic in theory of plasticity means permanent deformation. So these definitions can be regarded as very good at the macroscopic length scale.

And as we mentioned, at small length scales, there may be some movement of atoms or planes of atoms that can be observed even for macroscopically elastic loading. So let's take a little bit of a look at stress strain curves. And to understand stress strain curves, we need to know where the data comes from.

So we start with a specimen. I have it drawn there with intention. And we can either do a tension test and what's called load control or displacement control so P is the load and Delta is the displacement so either one we can do if we want but the idea is typically this is done on a either constant load or a constant displacement rate and with time this just increases and eventually this thing will break now we do get some different looking stress-strain curves depending on if we are in Load control or displacement control, and we'll talk about that as we make our sketch.

But in many common ways to do a tension test, we use displacement control, especially if we're using electromechanical testing machines. So with a servo hydraulic testing machine we can test in load control or displacement control very easily. It's a computer controlled testing machine that will make sure that you achieve a certain level of load or a certain level of displacement when you want it to.

Electromechanical testing machines are older, an older style, but they're very useful and often used to construct stress strain curves. Now the difference then is if we were to look at our load versus displacement. for which we would construct our stress train curve. If we are in displacement control, we're controlling the separation of the grips along this horizontal axis.

So it may increase, and it may just keep increasing, and the load is allowed to decrease during our tension test or displacement control. If we had our load control instead of our displacement control, we would be controlling along this vertical axis. And once we were at our peak load, then our specimen would fail.

We would not be able to apply any higher load than the maximum load that the specimen can take. So we would not get this downward behavior of the stress-strain curve or the load-displacement curve. There are There are safety reasons why you'd probably want to do a displacement control test, but there's other reasons why you might want to do a load control test. You just have to be a little bit careful with it.

So with a displacement control test, is what we're going to talk about now, we have our specimen, and before we test our specimen, we measure our original length and our original diameter of our specimen in the gauge section. We would take and we would hook up an extensometer. Which is a device that measures strain. And that extensometer has some initial gauge length. Now depending on what unit system you're in, it might be a 1 inch gauge length.

And these blades that are attached to the specimen separate as the material is pulled. There are wires. it go off to the testing machine and the load is recorded the displacement is controlled as is recorded in the extensometer readings recorded along with time what I'm going to do is I'm going to supply you with a data set from a tension test where we have load crosshead displacement extensometer displacement in time all in the same file and I'm going to have you construct the stress strain curve Alright, so we have this load displacement data, and we're going to look at it. If we want to construct a stress-strain curve, we know that, you know, if you're interested in this class, you probably already know there's different ways to define stress and strain. We can use engineering stress and true stress, engineering strain and true strain, and there are actually a lot of different measures for stress and strain, some of which we'll talk about in a little bit more detail later.

But to make an engineering stress-strain curve... What we're going to do is we'll take our, define our stress. I'll put a little subscript E and G to emphasize that's engineering stress. We'll take that stress. That will be our force that's applied to the particular instance divided by our original cross-sectional area.

Of course, if it's a round bar, the A naught is pi over 4 times the original diameter squared. And our engineering strain is going to be defined as our extensometer displacement divided by our original extensometer gauge length. which amounts to the change in length divided by the initial length for that extensometer gauge section. Now when I draw these sketches, you know, like in this example here, this is load versus displacement.

I have a large displacement with a little bit of load. Really, if you were to look at a piece of metal, this would be very nearly parallel to this axis. It has a little bit of a slope, but compared to the rest of the curve, the rest of the curve kind of really takes off.

So these are just kind of pictures or pictograms. When we get to look at our actual data, then we'll have a better idea. of what that behavior looks like. So for all the data up until fracture I can compute engineering stress versus engineering strain curve.

And so the shape is really the same. We're just kind of scaling a little bit. We're taking out the effect, the size, the specimen. And, you know, again, if you've had classes in stress analysis and stress-strain curves, you talked about that before, basically what you're doing is you're taking the effect, that size, out of the... specimen to get the material behavior.

So a large specimen or a small specimen should have pretty nearly the same stress-strain curve. You'd use different size specimens depending on what testing machine capability that you have. Now there are some limits.

If you get to a microstructural detail, you have a specimen so small that you get into microstructural length scales, well then you have to think about what effect that may have on your stress-strain curve. Or if you have things like casting porosities or things like that, macroscopic features, you may want to be careful about the size of your specimen in that case as well. But in the ideal case, the engineering stress-strain curve effectively takes the size of the specimen out of consideration.

We're only looking at the material response now. Well, the stress-strain curve has some typical features to it. This point right here, where it failed, where that X is, we call that the fracture stress.

That would be the engineering fracture stress. That would be our final load. divided by our original area. The highest point on the stress-strain curve, that's going to be kind of important for us.

Even in this class, that's known as the ultimate stress. And the point of deviation from linearity, maybe it's around this point right here, it also has a specific name, and I'll write these down in just a moment. Let's call it the proportional limit stress, sigma sub PL.

It's the highest point on the stress-strain curve where we no longer have a straight-line relationship between stress and strain. And that's typical of metals. Other materials might be a little bit different.

Polymers may have more of a curvy or nonlinear initial part to their stress-strain curve. But most metals have a very straight-line relationship to their stress-strain curve. And so those are the definitions there over on the right for those different terms on the curve. Now, different materials have different stress strain curves. They may have like a low-carbon steel stress strain curve, has an upper and lower yield point.

So these are some common features, but you may see some unusual things. And some materials have what's called serrated flow, and so things aren't very smooth. But again, this is just kind of a pictogram to kind of describe kind of a generic behavior of a material.

Now another place that we want to investigate a little bit is right near the proportional limit for most materials is where we're going to define as our yield stress. Now our yield stress will probably be around in this area somewhere. Our yield stress is going to be defined as the point at which we begin to have permanent deformation.

Now that's a very important location in plasticity because we use that as our demarcation between Elastic behavior, whether it's linear or nonlinear, and plastic behavior or permanent deformation. Well, this is kind of interesting. The way that we've defined our permanent deformation and our plastic material behavior already is by loading a specimen and unloading it and see if we have any change in dimension. This is not very practical to do in a tension test.

In a tension test, we keep loading it until it breaks. So we need some way to estimate where we might have the beginning of permanent deformation. And even if we could load it and unload it, the point at which we start to have permanent deformation really depends on the sensitivity of our measuring equipment anyway.

So we're going to use an approach called the 0.2% offset yield stress method to give us an estimate of where our yield stress might be for this kind of material. So let me... make a little bit of room and we're going to talk about the 0.2% offset yield stress.

Before we do that, let's just go ahead and make a note. Maybe at this red spot right here, we get the sigma yield. That would be our yield stress.

And we can find that as the point at which we begin to have permanent deformation. Like I said, some materials are very distinct where you start to have permanent deformation, particularly for low-carbon steels that have not been tested before in a monotonic tension test. You have an upper and lower yield point, but in some materials, titanium, aluminum, copper, things like that, we need to come up with an estimate for this yield stress point. So let's talk a little bit about this 0.2% offset method to estimate that yield stress.

So again, what I'm going to do now is I'm going to zoom in on the initial portion of the stress-strain curve. On our engineering stress-strain curve, I'm trying to draw a straight line here. A lot of metals will have an initial straight line slope. We know this slope in the region where we're below the proportional limit stress. as the modulus of elasticity.

Rise over the run of the stress-strain curve in the 0.2% offset method what that refers to the 0.2% refers to the strain now strain is Delta L over L naught. It's kind of like a percentage calculation except without the times 100 Well, you can actually represent strain as a percent. We were to convert this into inches per inch or millimeters per millimeter then this value would be 0.002 as a unitless quantity.

And the idea with the 0.2% offset method, let me change colors, is that we go on our strain axis and we find the location of 0.002, and we draw a line that's parallel to the initial slope of the stress-strain curve. I'll do my best to try to get that slope the same. And I'm continuing and I'm dotting it up here. And the point of intersection of that newly dotted line with our stress-strain curve is known as our 0.2% offset yield stress.

So it's pretty easy to show on a picture. It's sometimes a little more difficult to describe in words. So let me take a moment here, and I will write down some notes about how to find that 0.2% offset yield stress.

All right, now if you're student in my class, I'm going to ask you to do a homework assignment where you examine some 6061 T6 aluminum data and plot the engineering stress-strain curve and some other things. But I want to talk about that data just a little bit first so we have a good understanding of what that data is all about. So this was taken...

with an electromechanical testing machine. And I've got some other video on YouTube here that shows a tension test of a couple different materials, so it's very similar to that. I wouldn't say it's the same material.

I don't remember if it is or not. But it's definitely not the same specimen, but it may be the same material. But on a text file, and I'll make this available to everyone on my personal website.

Here's a readme file that describes the data just a little bit. Let's see if I can find that here. Okay, so it was a 1-inch gauge length for the extensometer. It had a 0.25-inch diameter.

We're using English Standard Units. And also the strains need to be adjusted using the determined modulus of elasticity. And so what I mean by that, so when we put a... specimen in a testing machine.

The specimen looks something like this. It has a large grip section, kind of is turned down in the middle and flares back out through some radius of curvature. I'm sorry that's not a very good looking specimen, but that's the way it is.

It's gripped over on these surfaces. And when you use what's called electromechanical testing machines with mechanical wedge grips. The grips are in here, these are the jaws, and they are got an angle to them. And they sit inside a grip housing that has a matching angle, something like that. As these grips are tightened, as the specimen is pulled, these wedges and tension tighten up in this matching slope right here.

And this is the same thing down on the bottom. But in order for you to start the test, in order to initially get it started, you have to apply a little bit of preload. So there may be 100 to 200 pounds preload, depending on...

the how much force it takes to bite into that specimen now usually that is done before you apply the extensometer so if you think about it you're starting here at a stress level that's your initial force but you haven't measured any strain yet you usually zero your extensometer after everything is all set up and then you do your test now that preload is real causes real stress and it will affect your stress-strain curve. And really, the starting point for the strain is not at zero, but it is whatever strain corresponds to that preload. And so I want my students in this class to account for the preload strain. Well, how do you get that?

Well, if you know the modulus of elasticity of the material, you can extend the slope downward and calculate this offset. And so you should apply... That offset to all of your data in your stress-strain curve to make an adjusted stress-strain curve.

I do have an entire other video on this. It goes through in great detail. how to use Excel to calculate this and the equation of the line that does this and everything so I'll refer you to that I won't spend a lot of time on that but I but I do want everyone to adjust that strain because it's not an insignificant amount of strain when you're talking about a quarter inch diameter specimen and a couple hundred pounds the stress can easily get to be a pretty high number so now in this data file see if I can find that and open that up for you here's a a 36 deal here 60 61 we're going to look at that in this data file we have extension time load and strain in the extension is the extensometer reading both of these should be the same value I believe as we go through here Time is time and that happens to be in minutes in this data file.

Here's the load. So actually we had 274 pounds of preload, units of pound. And then here's the strain again. So this is unadjusted strain is what this is.

Now another point that I want to make about this data is There's a certain point where I've gotten to the ultimate stress, and this is starting to come down, where my extensometer only extends so much. And then I have to take it off. I really don't want to have my specimen break while an extensometer is on there. When I take the extensometer off, what happens is your strain values will look like they do something weird like this.

This data right here for the strain channel after the extensometer removal isn't good. It's good for the load, but it's not good for the strain. Okay, so, you know, I'd like for you to go and edit your data to get rid of that extensometer removal point and anything after that on the strain side.

Now, if you do need to find your final fracture stress, you can look at the final load before it breaks, and that's still a good value. Now, lastly, the other point I want to make about our stress-strain curve, then, is that the highest point on a stress-strain curve, we know that is the ultimate stress. There's a certain physical phenomenon that occurs around this ultimate stress and it's called necking of the specimen.

N-E-C-K-I-N-G. And basically what happens is before necking what we have is our specimen starts out at a certain length and it increases in length uniformly. That means every point along this has the same strain. Let's make a note, uniform axial strain until necking, which is also known as the instability point.

After necking, we have very localized deformation. You start to see kind of a little dent in our specimen that grows. And, you know, really we can't use epsilon is equal to delta L over L anymore because that strain isn't constant over our gauge length.

We can still compute the number, but it may not make a lot of physical sense once we have that non-uniform deformation. In fact, I have a picture that I want to show you of a specimen. And so here's the specimen grips.

I have here, you see the indentation where it was biting in, and here's the gauge section, here's my radius curvature, and you can kind of see it in this picture where I stopped this test at necking, I didn't break this specimen. Let's see if I have another picture where I've zoomed in on it. There we go.

You see the surface is kind of modeled. It's beginning the failure process. In fact, we no longer have a uniform strain distribution along the axis of the bar.

We no longer have a uniform stress distribution either. Because of this indentation, we start to get... at a more multi-axial stress state.

It's very much like a notch in a bar. You can do different types of corrections. There's like a Bridgman correction and different things that you might do. purposes, once we get to this point of instability, then we don't have that uniform stress or strain distribution anymore.

So that's going to be an important point for us, and it's going to be particularly important when we go to convert our engineering stress and strain to a true stress-strain curve. So for the homework assignment, let me see if I can find I want you to get this data, construct an engineering stress strain curve, find the modulus of elasticity account for this offset. We should find a 0.2% offset yield stress, construct a true stress strain curve for the material, and then plot the true stress strain curve and the engineering stress strain curve on the same set of axes. Now we need to talk about the true stress and true strain now at this point. Let me make a little bit of room and I'll pull up a new page.

Okay, so here's our definition of our engineering stress and strain. Force divided by the original cross-sectional area and the change in length divided by the initial length. Both the force and the change in length are instantaneous quantities as we do our test measurements. But let's talk about definitions of true stress and strain.

We know that when we apply tension to a material, its cross-section dimensions don't stay constant. So, here's the original diameter and the original length of this bar. When I pull on this, and I'll exaggerate this a lot, it gets a little bit skinnier, and it gets a little bit longer. So the final length is greater than the original length.

The final diameter is going to be less than the original diameter. So to define our true stress, and maybe let's use a different color to emphasize that. I may not use this black color all the time. For now, our true stress can be defined as our force divided by our instantaneous cross-sectional area, a sub i. And we can define our true strain as an integral from our original gauge length to our final gauge length of dl.

over L, where L is the current gauge. So here I've used the lowercase L-naught, but that's the same as my capital L-naught in this image. Now when we evaluate this integral, what we end up with is a natural log, and we have a natural log of L over L-naught. So L is the instantaneous length.

L0 is the same as what I had called capital L0 before. That's the original length. And this AI I'm calling the instantaneous area. We can also call that A. so we can convert our engineering stress and strain to true stress and strain and let me go ahead and Let's get you another picture.

It will be very similar to the one I've already done. Again, to exaggerate what's going on here, here's our specimen. It has some sort of initial area and some initial length.

And it changes. It gets a little bit skinnier and a little bit longer. Again, very exaggerated. It has some instantaneous area and some instantaneous length. And we can define this difference in length as delta L.

so that the final length is equal to the initial length plus the change in length. Well, if we use this then, and we go back to our definition of our logarithmic strain, I'm just going to call this epsilon now. A logarithmic strain is our current length, which is our initial plus our change in length.

divided by our initial length. Take the logarithm of that quantity. We have 1 plus the change in length divided by the initial length.

That's our engineering strain. So the true strain is related to our engineering strain by the change in length. this method.

One plus the engineering strain but with that logarithm there in front of it. Now if the engineering strain is really tiny then they're going to be about the same quantities. This equation for true strain and the conversion from true strain from engineering strain is only valid up until the point of instability or necking. And the reason for that is because of that discussion we had about uniform and non-uniform strain distributions. Let's turn our attention now to stress.

We're going to make an assumption here, and we'll talk about the validity of this assumption in a moment. We're going to make an assumption that for our analysis of true stress as related to engineering stress, that the material maintains a constant volume through its deformation process. So to compute the volume, of course, we'd have A0 times L0. And over here, we would have A times L.

Well, if we are in elastic loading, we usually have very, very tiny strains. The dimension of the cross-section instantaneously and the dimension of the original cross-section dimension aren't too far different from one another. And so the true stress and true strains are very close to one another. In the elastic region, the assumption of constant volume is not a very good one.

We do have some non-conserving volume in the elastic region of materials due to Poisson's contraction. This is known as the dilatational strain. However, in fully developed plastic flow, conservation of volume is pretty good.

So, let's just make a note of that. So it's not so good in the elastic range, but quite frankly it doesn't matter in that part of the stress-strain curve. But it is pretty good when we have a lot of plasticity. So when the plastic strains, much greater than the elastic strains.

All right, so if we can use this idea that volume is conserved then, then we can come up with a ratio. Let's see. We have A-naught, L-naught is equal to A-L.

I can rewrite this as L over L-naught. is equal to a naught over a. I'm just doing some cross-multiplying.

And with that, since the strain is the logarithm of L over L naught, it's also equal then to the logarithm of a naught over a. Now since the engineering stress is equal to the instantaneous force over the original cross-sectional area. We can say that the force, then, is equal to the engineering stress times the original cross-sectional area. Our true stress, I'll just call that sigma, our true stress is equal to the instantaneous force over the instantaneous area.

And if we look at this, since epsilon is equal to ln of a0 over a, we can also write this then, we know that we can write this then as the ln of 1 plus the engineering strain. So if we take the logarithm and get rid of that on both sides, the argument of the logarithm has to be equal. So a-naught over a should be equal to one plus the engineering strain.

And so let's do this now. Our force then is equal to the engineering. stress times A naught over A, but we just showed that A naught over A is equal to 1 plus the engineering strain. So the true stress in the material, I'll do this in another color, the true stress in the material is equal to the engineering stress times one plus the engineering strain at that particular load level. And just to have it written down here, the true strain is equal to the logarithm of one plus the engineering strain.

And both of these are only valid until necking, or until we reach our ultimate stress of our metal. Alright, so what I want to do next, and I'll do this in a separate video, is I want to go through the data. I know this is a homework set, but my students should have finished this by now. I want to go through the data and work on my solution to come up with my engineering stress-strain curve and then my true stress-strain curve for this particular material.

Transcript for:Understanding Uniaxial Monotonic Plasticity

Transcript for:
Understanding Uniaxial Monotonic Plasticity