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Stress-Strain Diagram · Acedra
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Stress-Strain Diagram
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0:00
[AUDIO LOGO]
0:06
Hey gang. Welcome back. We're on the next video for solids. We're talking about mechanical properties of materials. This is really a material science kind of lecture. And what you have to think about is this is talking about all the electrons and the molecules that are making up steel and aluminum, how they're behaving when loads get applied to them. So we're talking about the material properties and how they affect the mechanical properties
0:34
or abilities of a steel or whatever you're building with. So I've listed over here all of the equations-- you've seen that-- that we've talked about so far in this series, all of these things we've done. And that's going to lead us to a new thing today. And that is called-- this is called the stress strain diagram, the stress strain diagram. It's how stress is related to strain. And what is strain? Strain is deformation. What is stress? The intensity of the force.
1:16
So as the intensity of that force increases, so does the deformation increase, right? So we have two things here. On this axis, we have strain, circle E, right? And over here, we have stress. Now, that's normal stress, right, the forces that make things get longer or shorter.
1:35
And if we were to plot how the deformation acts as the stress goes up, this is kind of what that graph would look like for a ductile material.
1:52
That would be something like mild steel, 836 steel, something like that. A brittler material will not look like this. So cast iron does not look like this. Aluminum does not look like this. But most of your ductile materials, which means it's easy to form it or bend it, right? If it's something that bends easily and doesn't break, that's a ductile material. Now, this is actually probably a brittle material because if I bend this too much, it'll just pew. It'll snap in half, right?
2:23
So brittle materials bend until they can't bend anymore, and they snap. Ductile materials, they bend, and they get permanently deformed. And they just don't go back to where they were. This thing won't bend. This little ruler doodad won't bend and go back to where it was. It'll bend until it snaps. This is a ductile material. So for ductile materials, they behave like this. Now, where in the world does this graph come from? And why are there two graphs? OK, I'm going to tell you.
2:53
Number one, we have one for engineering stress and one for true stress. Now, engineering stress, where does that come from? What we do is we go in a lab, and we get a sample. This is called a dog bone sample. You can see why. This is for a ductile material test. And what you do, it's a tensile test. What you do is you clamp on the top. You clamp on the bottom. And you start pulling on this, right? You put it in tension. So why is it little in the middle?
3:20
Because I don't want it to fail where the clamps are grabbed onto the material. I want it to fail in the middle where I can observe it. And so they put it in tension, and they start pulling on it. And what happens to it? Well, number one, it starts to elongate. That's deformation, isn't it? Number two, as I pull on this, what's happening to the middle of it? Ooh, what's happening? It's getting smaller and smaller. It's called-- that's called necking. And it goes until it can't take it anymore.
3:48
And then, that material, uh. You have to pull it. This is silly putty. It snaps, right? And it breaks. And that's called the fracture point. So where this material fractured, I've got a failure there. So how does that look on our graph? Now, what we do as engineers is we pull on that, and then we observe what's going on in the material. We observe the deformation, right? We know how far it is between the top jaw and the bottom jaw. So we can measure that deformation.
4:17
We can also observe the amount of force that it's taking to stretch that material. And then, we can observe the point at which it breaks. The thing that's very hard to observe, that's very hard to observe, is as I stretch on that, the middle is actually changing in area. Poisson's ratio says it's changing this way and that way, right? So the area is changing, and steel does not work like silly putty, right? It hardly looks like it's deflecting at all. And then, bam, it just breaks, right?
4:48
So it's hard to go in there and measure that area. So what we do as engineers is we assume that that cross-section never changes, although we know that it really does. So as the cross-section changes, you get this true stress curve here. The material starts necking, gets smaller and smaller. And then it breaks. But-- because this curve really makes no sense. You're telling me that there's a point where I can actually lower the stress and the thing continues to deform? No, not really.
5:22
But if you hold the cross-sectional area constant all the time, which is really not what happens, but if you do, then you get this curve here. That's why that curve looks like that. And we call that engineering stress because, again, this whole curve is determined or drawn empirically. In other words, we observe the experiment, and we take notes about the experiment most of the time in a computer. It's recording all that information. But one thing, again, that's really hard to change
5:56
is that change in cross-sectional area, very hard to change. So we hold it constant, and we call this the engineering stress curve. So that's what the difference is. So most of the time in the book, you see this curve. You see engineering stress. And you think, well, that doesn't make any sense that it does that. Well, it really-- that's because it doesn't really do that. Way to go, it doesn't make sense. That would be true stress. So let's talk about this curve here. All right.
6:23
So we're going to start on this part of the curve over here, and we'll just work our way across talking about the different features of this curve. This is something that you really need to understand. It's very important because, as engineers, we need to know where on this curve our designs need to live, where-- if we're going to design a beam to hold up some kind of load, where does on our material properties here, where do we need to live? So what's going on?
6:50
As I start adding stress, then I get a strain, more stress, more strain, more stress, more strain, it's occurring very linearly. This is called the elastic region.
7:11
Now, when you think about steel, you don't really think about elastic. When you think about elastic, you think hey my underwear man it's got, like, I can stretch it, and it goes right back, right? That's elastic. Or a rubber, right, very elastic material. But when you're talking about a steel beam, you don't think it's very elastic. But if this was made out of steel here, if I pulled on that, if you have one of those steel rulers-- you know what I'm talking about, right-- You can deform it.
7:35
And then, when you let go of it, what happens? It goes right back, doesn't it? So it does have some elastic properties. And the elastic properties are that springback. I bend it. It deforms. It goes right back. And it's back where it started from. And it's not deformed at all, right? It's perfectly flat steel. So if that's happening, then you are in this elastic region of the material. For instance, any of your-- if you're going down the highway,
8:03
you're driving down the highway, and you pass an 18-wheeler with one of those aluminum flatbed trailers on it, and you look at it and it looks like that, right? It's bowed upwards. And you're like, that looks bent. It is. And it's bent on purpose so that when it gets a load on it, what does it do? [GROWLS] It goes down flat. It's perfectly flat with a load. You don't want it like an old-- you don't want it like this with a load on it going down the road. You want it flat.
8:30
So when they take the load off of it, guess what happens? That trailer goes right back to bowed up again, doesn't it? You've seen that before. So the elastic region of a material is where we want to live as engineers. I can't think of anything. Maybe you can write me a letter. You can put it in the comments of a thing that we design that we intend for it to be permanently deformed.
8:56
Everybody had one of these when they were a kid, right? You had your favorite slinky, right? And then, your little sister played with it, and she pulled it too hard. And your slinky goes like that now, right? It never went back to-- it never went back to straight up, right? Why is that? Because it got stretched and stretched and then stretched a little bit more. And it got stretched beyond its what we call yield point. And yield is here. This is sigma yield.
9:30
That's where the material, the stress in the material, when it gets so much that when I let it go, it doesn't go back where it started, it's permanently deformed. That's a bad thing. You don't want to be increasing the stress in your slinky until you get a permanent deformation. So the shelves at the hardware store, the pallet rack, they put a bunch of weight on that. They can deflect about 3/4 of an inch. And then, when they take the load off of them, they should go back.
10:00
If you put enough weight on there and the shelf gets permanently kinked, it's permanently bent, that's not good. That's what we as engineers would call a failure. So failure doesn't mean it has to tear in half. Failure means that we've got permanent deformation in a place where we didn't want permanent deformation. So we have to be sure that we design where that doesn't happen. Now, that happens here in this region. This is this, and if I drop a line down here,
10:28
this is what we call the elastic region.
10:37
Now, for this guy here, we've got a new property of materials. This is called the modulus of elasticity. And this is going to be a new equation that we're going to use over here. And it's given by the capital E, the modulus of elasticity. And it just tells you about the springiness of a material. How likely is a material to bend and spring back? How elastic is it? And the modulus of elasticity is given by the slope of that line. It's very important. It's the slope of that line.
11:09
And what is slope? Well, it's rise over run. So it's just this, sigma over epsilon. So that's very important. Now, you're going to see this a couple of different ways. You'll see it as the modulus of elasticity.
11:30
And this is a value that is observed by these charts by testing a bunch of materials. And those values, the modulus of elasticity for different materials, is listed in the material table in the back of our book. So we can look that up. If I want to know what the modulus of elasticity is for mild steel, 836 steel, I can look that up, right? You'll also see this called Young's Modulus, another guy that kind of derived this material property. And another one is Hooke's Law.
12:10
And this one is written a little bit different way. You just rearrange the equation. But it's the same law. So you might see all three of those things. They all mean the same thing. Modulus of elasticity is rise over run, sigma over epsilon, stress over strain. Now, if I do stress over strain, stress might be in megapascals. Strain is in inches per inch. It's really a unitless thing, isn't it? So E is going to be in megapascals, in gigapascals, in PSI or KSI depending on which system you're in.
12:44
So this is a point. And then here, down here is strain, yield strain So it's like all the material can take. If I let it go at that point, whoop, it falls back along that curve back to zero. If I go beyond that, if I go over here in this area here, this, this area here, is called yielding.
13:15
That means that it's permanently deforming. It's deforming. So that's kind of the failure area. You don't want to be in there. Up here at the top is called ultimate strength.
13:33
Ultimate strength is there on the top part. If I drop a line underneath there, you've got two areas here. I've got one here, and I've got another one over here.
13:50
So what are these areas called? This one right here is called strain hardening.
14:05
Strain hardening, what is strain hardening? I think the easiest way to think about this is on a molecular level, think about the little crystalline structure of steel. So let's say that this is like the crystalline lattice of steel, what it looks like, right? And the little molecules, they're are all in harmony, and they're like, I'm in the perfect place where I like to live. I'm in the-- all my little molecular forces are even on all sides. I'm all good, right? Now, it's cold, right?
14:35
I'm not hot. If I'm hot, if I'm red hot on steel, I can form it. But once it cools, all those molecules are locked in place. They're like, OK, I'm happy where I am, but I'm locked here, right? Now, what happens is I start to stretch that material in a cold environment when it's not heated up, right? Well, if it goes past the permanent deformation-- as I start to stretch it, that little crystal lattice now might look something like this. So where I used to be in perfect like this,
15:12
now I'm like, uh, right? I'm all bent over there, right? And what happens is those little molecules start sliding past each other. But they're-- When you're talking about permanent deformation you're getting in yielding, and you're getting into strain hardening, then it stays there. It doesn't go back. And now, everybody's out of place. And they're like, I want to go back to where I live. But I'm not right where I am, the little molecules in there.
15:37
And so you get something like cold-rolled steel, which means that it's been deformed while it was cold. Now, cold-rolled steel is a lot harder. If you try to drill a hole in it, it's a lot harder than mild steel or hot-rolled steel because hot-rolled steel is over here. Cold-rolled steel has been strain hardened. So all of those-- Strain means deformation, right? So all of those little molecular crystalline patterns are all out of whack there. And the material actually gets harder.
16:08
The hardness changes. And that's where you get something like cold-rolled steel. Now, it has its advantages in other places. It's more brittle, right? But it's harder. So if you needed a steel that was harder, cold-rolled works good for that. And then, this last part over here is, this region is called necking. And we saw that with my little silly putty example, right? So in this region over here, the cross-sectional area starts to change, right?
16:36
And it starts getting smaller and smaller and smaller until what happens? Bam fracture.
16:45
And that's the end of our graph there. So that's all of the parts of the stress strain diagram. Now, this is important that you remember what all these parts are. But for engineers, we really want to live over here in this elastic region. You don't want to live over here because that's bad news. We've got some kind of permanent deformation. And generally, we would call that a failure. Now, what happens? What happens if I increase the stress and I get to this point right there?
17:18
And then I say, OK, I'll let it go. Well, number one, we know that it's permanently deformed, right? Can we calculate how much it's permanently deformed? The answer is, yes, we can because even-- even when I bend a beam too far and it's permanently deformed, when I let go of it, it's going to bounce back a little bit. We call that springback, right? Or if I have my slinky, and I bend it a little bit too much and I permanently deform it, it still springs back but not to where it started from.
17:48
We call that springback. And that springback is going to follow a particular path. And that path looks like this.
18:00
Guess what? That springback line, this, and this original, the slope of our modulus of elasticity are parallel. But guess what? Here's where it is now. It is no longer returning to zero. It has now some permanent deformation to it. Now, we can look at this because this is given in inches per inch. And if we know the original length of the specimen, we can calculate what that deformation needs to be. So anywhere on this graph, I can let go of it, but it's not going to return to zero.
18:32
Now, if I let go of it there, guess what? Droop, right back down that line back to zero So as long as I'm in the elastic region, I'm going to return back along that same line to zero. But if I'm over here, return back, permanent deformation, permanent deformation. All right, that's called elastic recovery. And we'll and we'll do another video, and I'll show you how to calculate some things off of this curve. Now, there's two more things that I think you ought to know about.
19:03
And those things go like this. Let's talk about this area here. The area under the curve is like the work the material can do or the energy the material can absorb. This right here is called the-- this is the area, that area. And of course, that's a triangle, isn't it? So that's 1/2 the base, which is sigma-- I mean-- epsilon y strain yield times the height, which is sigma yield, equals an area. And I'll just say the modulus of resilience. This is called the modulus of resilience.
19:54
And that's like the energy thing, a thing will bounce back. Well, when would I need to know that as an engineer? Who cares about the area under that curve? Well, you might. Let's say that you're working for an automotive company, and you're trying to design the perfect bumper that when it takes an impact, how much energy can it absorb and then still spring back to where it was where there's no damage on it? Well, I might need to know about the modulus of resilience of that material.
20:23
And the second thing that you need to remember is this. The area under the entire curve, the whole thing, the whole shooting match, that would be this area here. How could you calculate that? I don't even know. Well, yeah, it's an area of a triangle, and then there's an area of a rectangle and then maybe a portion of a circle up there, right? I could kind of-- I could estimate it. And you'll have to do this in the book by just counting. They'll put a grid under here for you.
21:01
And you'll physically just have to go in and estimate how many squares there are in that grid to estimate what that area is. And you can do that. But that entire area under the curve, the whole thing, is known as the modulus of toughness, the modulus of toughness. So how much energy, total energy, can a material take before it fails? Why would I need to know that? Well, what if you were doing something for law enforcement, and you were maybe designing bulletproof vests for the military, right?
21:43
I would be really interested in a material that could absorb all of that energy without failing, without fracture because if it fails, then it's got a hole in it. And probably, I do, too. And that's ungood, right? So maybe for something like that, I'm interested in-- I don't care that it deforms. I just don't want it to fail. Well, then, I need to know about the modulus of toughness of that material. What can it take? How much energy can it take before it fails?
22:10
That's the area under that whole entire curve. And we'll do some examples and utilize these things. So this is something you really need to know about, the stress strain diagram for ductile materials. You're going to see this. And I'll buy you a Coke if this isn't on a test at some point during the semester. You're going to see this. So you need to know this. So I hope this explanation helps you. And I'll see you on the next video.
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