 finished Monday, a day when homework's not due. We finished with our now. We have our first look at the material response to these loans, which for us means, I guess, two possibilities. And we haven't yet distinguished how we have some suspicion that one of these possibilities may happen and the other might not. The first thing we're worried about is the material undergoing some kind of deformation. It bends permanently or not, it stretches, it compresses permanently or not, but there's some deformation that the material itself might undergo that could be, in its own right, catastrophic. In other words, the machine that this is a part of might not work as supposed to. It might not work at all, even though it's not a complete failure of the part in terms of actual breakage, which is the other material response that we're worried about. This deformation may or may not be elastic. In other words, once the load is removed, does the piece undeform back to its original shape? It could be that the deformation is no problem, can be easily designed into the piece, can be handled, that's certainly the case with bridges. They're designed to undergo thermal expansion as the temperatures change and once the temperature change is removed, in other words, it returns to the installation temperature, if you will. It just returned back to the original shape and the bridge worked at all times in between. Sometimes deformation, though, just simple deformation, even if it's elastic, could lead to failure not of the part, but of the machine of which the part is a part. We then added to this deal that we were looking at, our new measure of those material responses and we looked at the strain and we had two parts to that, one being the normal strain and I defined that in what way? Remember the letter used? That was an axial, generally an axial response of the piece, some tensile or compressive load would actually change the shape and defined as and del being, the actual change in length from the original length and it's the original length by which we divide. So even though the length is changing as it goes through this deformation, you don't use the new length to divide by, you use just the original length and we're going to see how that adds kind of an odd effect to some of the test data we have coming up, we'll probably get through today. How about the shear strain? Remember what Greek letter symbol we used for that? You got to get used to these. These are pretty standard for all the books I've looked through and any other thing I've seen, these are common among them all, so it's an industry standard and that's defined as not quite, remember? It's the geometric departure of a 90 degree angle to some strain angle and we'll look at another example of that but it could be as simple as actually inscribing a 90 degree angle on a piece, often it's done as either a rectangular or a square and then actually measuring how that angle changes. So we'll do a couple of sample problems to get used to those. They're very geometric problems so be careful when you draw them, there's sometimes answers that look like they're possible in a drawing that are not really the case. Sometimes you'll see 45 degree angles where there aren't really any. Partly depending upon your skill and partly depending upon the fact that some of these strains are very, very small. We're not talking about great large changes as we undergo this material response. So I believe we had a problem on the board for, that we just finished with Friday but we hadn't actually gotten to. All I've done has gotten down to dimensions. So let me re-do those. We have some thin member under some kind of load, 12 kilo Newton's and original dimensions, original length of 500 millimeters and original diameter 16 millimeters and under this load the piece deforms, gets quite a bit longer and since the density is the same it also gets quite a bit narrower. So as you, well you've seen rubber do that. As you pull a rubber band not only does it get longer but it tends to get a bit thinner as well. So greatly exaggerated of course. Let's say it elongates by about 300 micro meters and we haven't talked about what this material is yet. We haven't talked about how different materials respond to those things. We'll start touching on that I believe in the last little bit today and let's say there's a diameter decrease 2.4 micro meters. So the question is what are the strains that this material is experiencing? Which kind of strains am I talking about? Normal strains or shear strains or both? For what's given here we have two normal strains it's undergoing. It's undergoing a length increase and a diameter decrease. So we have two strains in those directions maybe we'll call that epsilon and the x direction of x is the axial direction which will commonly be the case for us in these classes. As often as not we'll take those coordinate directions x, y, and z. But there's also a normal strain going on in the radial direction. There are shear strains certainly however we don't have any original geometry to compare this to. All we're talking about is changes in linear dimensions here. So we just have these two shear strains to compute. Normal. Sorry yeah normal strains. Good catch. I believe I left you with that on Monday and we have a chance to do it. No you're doing homework for me. It's not due to Monday anyway. Now it's due Friday. You've got yourself days. So we need to look at the deformation in the x direction divided by the original length in that direction and the deformation in the radial or dimensional dimension divided by that original diameter. So do that real quick. I want to make sure nothing else that you can work with the appropriate units. So let's have that answer in microbes or micro-ras if you wish. Micro-radius. It's mostly a matter. Certainly I trust you can do simple division but sometimes these units themselves we've got a lot of very very big numbers and we've got a lot of very very small numbers and they're actually going to come into a bit of a conflict for us by the end of the day today just to make things even harder for us. So do a simple division. Make sure the units come out right and let's see what you get. Good warm up. You have to work your powers of ten. One of the days that you weren't here I'm pretty sure there's about eight different ways to show the units on these numbers because they are in a pure sense unitless but we tend to have a very small number over a very large number or a much larger number and so there's lots of different ways to look at these units and you're going to have to be careful of them. So who's got what we need? Joey, are they there yet? Almost. The actual division of the numbers themselves is pretty easy. We've got 300 divided by 500 which is integers of 6. Is it 600 micros or some other? 6,000. We've got 600 here. Split the difference. Unfortunately physics is neither a democracy nor a...this isn't economics. Or you can do it any way you want with the numbers just to make the point. Based on facts. Based on facts for the economics department. Oh well. I haven't been kind to the math department all the years. Is this right? 600 micros? Try it if you think so. Joe? Bill? Micro rads. 600 micros, micro rads if you wish. Whichever. A micro just means it's 600 times ten to the minus six meters per meter or millimeters per millimeter or whatever the units are. What about this one then? What'd you say? Did you squeeze the negative in there at the last minute? Negative because we have a decrease in the size. And that's reflected in the strain as well. So fairly simple calculations when you have fairly simple geometries. Some of the problems are just a little bit more complex, but they're mostly more complex in the geometry than in the physics itself. So you just have to be careful. Take your time. Make a simple drawing and help you think through it before you actually start doing the work. Generally tend to make a few less mistakes, fewer mistakes. Alright. That's all I had for that one. Simple as that for a search. But we're not happy. We want a harder problem. Alright. So let's look at another one. Alright. In some way we start with a square. Whether that's an actual inscription on a piece, just a simple etching on the side of a piece, or whether it's actually a piece itself and we're looking at it as removed from the rest of the gizmo in which it works. But it undergoes some kind of loading such that it stretches, so it strains in something like this. Oh, the original is 10 by 10 inches. Strains to something like this. Of course, not to scale. So due to whatever loading it looks like quite possibly either an axial loading in the y direction, a tensile loading or a compressive loading in the x direction. We're not talking about just what the load is. It's just the response of the material to it. We'll look at what loads do and then figure out what this is going on as we go through the rest of this term. So I have a change in dimension there. It's symmetric on the bottom as well. There's a change in dimension here. Again, obviously not to scale subject to my ability to sketch .3 inches in there. That's pretty rude when we're taping isn't it? Let me make sure it's not my car. Is this the red one out in front, Linda? No, she's not yours either. And it could be any of yours because I'm only looking at the faculty parking lot. By my count, there's at least five different strains that we can calculate on this piece. I think we can find a normal strain in the x direction in the y direction and a strain along one edge of that original square because that length changes as well. I also see that there's no reason we couldn't come up with a shear strain at this corner. I'll call it q and that corner I'll call it p. So let's find those five strains. It depends on what's this a part of, what your directional concerns are with either the greater pieces a whole. Steel is an isotropic material for the most part. It acts the same in any direction. Whereas something like carbon fiber or wood is an anisotropic solid in that its material properties are very, very different depending on which direction we're talking about. You might need these numbers and the diagonal numbers as well. David, what did you say? With the wood, certainly the grain with carbon fiber, obviously the fibers which way the mats are laid down. You can either decrease strains in certain directions or allow them to increase more depending on what your need with the part is. Joe? Is that ten times ten inches in the x direction? That's the original square. Ten by ten. Original square is ten by ten. So again, mostly just a geometry problem. What I'd like is let's look at these numbers this time in percentages. I'll let you choose. It doesn't matter. We're going to have the same digits for the more part. So as much as anything you need to see how those dimensions change, or you can do all the way across the piece and have the two changes on the other side. Ken at the end of class. Oh, I have. While you guys are working, I'll go get one more. If you're getting numbers, check with each other doing the same problem. No reason we shouldn't agree. But then give you time to check your numbers. What do we have? Got some agreement back there. Phil, who'd you talk to? Did you get all five? I got the first two. I'll take notes. Is that working for you? Yeah. Getting used to it? Yeah. You've got to give it a try. Yeah, straight line. Does it make your drawings? Well, your drawings are pretty good anyway. But if you draw a straight line, does it then replace it with a real straight line? No. There's some drawing programs, sketching programs. They're working on a lot of engineering classes that will do that for you. Basic classes. Would you be nice? I'm just doing my best to get you guys to work on paper. See you all fashionably. All right. Most of these, certainly these first 30 over here, the normal stripes are pretty straightforward. What's the length of one edge? I mean, sorry, not one edge, but one part in the X direction. Whether you do it this original length and that change or all the way across original length and both changes, you're still going to get the same proportions. Travis, which way did you do it? The whole thing. So the distance all the way across is, you know, you don't say things that way in engineering. Give me a number and I can register in my brain. So if you go all the way across in the X direction, then the change in the X direction is both of those on either side. So it would be 0.6. If you only did half way and used one of the changes, you're going to get the same proportion. And that came out to be, yeah, not because it was 0.6. And so you've got in whatever form you wish to present it, 0.42 or 2.4. 4.24. And you said what for the dimensions? Inches, yep, inch per inch or nothing. Or you can also, of course, multiply by 100 and make it a percentage, whichever one works. In the Y direction, you get the same thing, only it's an increase in length, a slightly different length. So it's going to be a slightly different strain. Phil, what'd you do with this one? Just use 0.2 inches. So you went just one-half the way in the last. And that would make for about 0.7 inches. And this strain came out to be, what way did you finally leave it? I did 0.808. 0.28. And left it as is. Either inches per inch is meter per meter. Radiance, if you wish, I guess. Or a percentage, 2.8%. And how about the strain along one edge in that direction? The original length in one edge, of course, is given as the original length of the square that we started with. Since this is rather big 10 inches by 10 inches, this is kind of a maybe that might be done on the side of a pressure vessel or something. Then what was the change? The change in the edge direction, you'd have to find that length there. And then subtract 10 inches. What'd you get, Phil? You got it? 0.071 inches. Okay, something like that, I have a slightly different, maybe some round off. Is that an increase? Yeah. Okay. That's about what I got, too. Ballpark, somewhere in there. A much, much smaller strain. Oh, 0.071. Any question on those normal strains? Anybody get something radically different? Could be a little bit of round off? Because we don't want to experience that edge. The original edge was this, 10 inch, and then now it's that. A decrease? Yeah. What I wrote down there, but that was probably four years ago. It's easy to figure out. We've got a triangle now that this was 0.007. It's now 0.3, which is 0.677. And this side was 0.007. It's not 0.2 longer. Simple that edge is 0.27. So what's that give for the hypotenuse? I'm going to have it written down. Now that's a decrease. It was a decrease. So we do need a minus in here. And in there, I noticed. What's this hypotenuse about to be? A decrease. Yeah, clearly a decrease. So I want to take it again next year and we'll make that a mistake. What about the shear strains? The shear strain is the departure of a 90 degree angle, which we have at both places P and Q. So all we need after that is the angle that they make now at those corners. That's the theta prime. And then the difference in those two is the, difference in those two is the, then the shear strain. Purely a geometric strain. David? I'm thinking gamma Q might be theta P and, you know, a P might be theta Q and it would be properties of a rhombus angles. I'm not sure. Gamma Q might be theta P? Yeah. So what's your idea? I can't. I don't think so. Okay. No, I don't think so. There is a relation between these two, but that's not the one I saw. I don't think it's true. Okay. I didn't give it much thought. So we can even use this drawing. Then the angles here are half of the theta angles that we need, this one being Q and this one being P. That's a very exaggerated triangle. Yeah, 86. Yeah. This one's, this one's I think twice 47. This one's twice 43. So the strain at Q would you, how'd you put it then? Negative because it's decreased. Or the angle itself increased, but the strain is a decrease. Sorry. Oh, what? 698, something in that ballpark? That's what I have. I have ratings. I think that's what I have done. Travis, what'd you get? Negative 4.8 degrees. Degrees? Yeah. Oh, okay. Not typically left as degrees, but 4.18? 4.8. 4.8. Should we be in ratings? Yeah. It's more common in ratings. Is that the 0.069, something in there? Does something like that something in there? Is there a 7? That calculator is just not insane. I'm just going to break the three-way tie we have. I got 4.1 and 4.8. Okay, I'm taking that 4.8 from Travis. You got 4.1. Yeah. 4.8. 4.8. Oh, thank you Travis. You guys don't need to plant some mistakes. I'll make plenty on my own, thank you. So that becomes in ratings, yeah, 0.07. 1, 2. Something like that. I had 0.0698 and it's negative like that. What about the other one? The other side? D. Be positive. The angle is decreasing. We're looking at the change from those angles rather than the angle itself. That's going to be positive. This is going to be positive. Wouldn't it be the same number? Yeah. Why is that? 360 degrees. The sum of the angles in a polygon is 360. So whatever decrease you get there, you get the same increase here. So then that changes to 0. All right, any questions on that before we clean up? Let's see, take a quick break if you want while I reset my taper. I should actually do this twice a class, not once. But it's better with some titles. Other than that, it's great. You can't see and can't hear. So it's kind of like class in real. I'm training you for going to RPI where all the professors have very thick foreign accents. You can't understand them anyway. And you're going to sit way at the back of a 400 seat auditorium. It's the same way on RIT. My first calculus teacher in RIT was from Chile. It was easy to understand. You can get used to it. And the third and fourth year classes are rarely in real large settings. The first year calculus and physics classes usually are very large classes. How do they do that? Would you like me to stop? Yeah, you might as well. To the shear strain, imagine a triangular bracket of some kind such that it looks something like this where it's 900 millimeters on that side, 300 on that side. These two angles, 30 and 60. I'll leave it as an exercise for the reader then to figure out what the other angle is. And there's a load of some kind or other. It doesn't matter. Remember in the strain calculation, it's merely the geometric response of the material to a load. The load itself does not come into these yet. We will be attaching those things all together in a little bit here. And it undergoes a strain such that, of course, exaggerated, such that point at the bottom there, point P if you wish, drops a millimeter. So find the strain at P. Find the shear strain at P or this bottom corner. And again, quite a bit of exaggeration in the scale and a purely geometric problem. It's not necessarily true that these sides come in if it's bonded to that upper support. So that the deformation would look something like that. So that's the one millimeter. So I need some teaching aid. Well, this is why professors go to PowerPoint so that they never have to suffer these imperfections and the drawings at the board. Everything's already prepared. But it's rather difficult for you to take notes when they're flying up on PowerPoints. All right, so figure out the strain again, what you need to do. Find out what the current angle is and subtract that from 90. It might be a little bit easier if you figure out these two angles and then just add them together. And don't forget to get the signs correct. Oh, trigonometry problem. Yep. Yeah, it's permanently bonded to the top to a rigid support, meaning there's no change in dimension of this upper piece. It's still 300, which may be prettier though. I forget what the screen there is. I could do the same kind of thing. It's pretty difficult to use it. Sometimes you just use jock. When I take paper notes, I know what kind of paper. Again, probably it's easy to figure out these two final angles. Add them together. That's the final, final angle. What's that? Oh, micro-ings. Don't forget the micro is 10 to the minus 6. So that should be six moves of your decimal place. Joe. Okay, getting it? Second, we'll put the stress, the strain, the load, and the material properties all together. We'll actually be at the start of chapter three. What else is getting there? Do you have something on it? A little bit different than Travis's, but maybe it's round off. Don't round off too much as you're going along. Otherwise, the whole thing disappears. Kind of carry some numbers forward for a ways and then round off a little bit at the end. Okay, it's generally three or four on one more problems. The final angle theta prime should get something a little less than 90. Got something? Bill? I got 89.904. Yeah, exactly what I got. 89.904 degrees. Of course, in radians, you can't subtract that from pi over 2. I hope you realize. And something then like 001796. Am I right, Bill? That's in radians or in radians for places. And we get something like 1796 for radians. Bill, about like that? Okay, Joey. But it's a geometry problem. So maybe a little more time, maybe a little bit bigger drawn. What you can do is find the length of this. I guess it'd be the height of the triangle. And then you can use that height with the other two sides to figure out the angles are and then add them together. That's one way. I don't find them to really know another way to do it. And that, yeah. Questions before we move on? This is actually what we need for chapter two. So we're going to start making into chapter three a little bit as we start actually bringing the material properties into account here because you can imagine different materials. Strain, stress, and deform very, very differently than other materials. Now we can finally bring all of this in together. So any questions about that before we go off? Okay. The most common way all of this is put together is with what's simply called a tensile test. And of course it's reverse partner, the compression test. Usually what's done, though there are variations of course, a test piece of the material supplied usually by the manufacturer is made such that there's two big fat ends where it's threaded and then fastened into the machine, the tensile test machine. And they just simply thread it right into the, I believe they call the mandibles at either end. This test length here generally is about two inches. So they might actually put little marks on the piece itself and then use those as an indicator of how much the piece is deforming in that axial direction. So this is typically about two inches with a round cross section on the piece and then the mandibles move apart, putting the piece under load and the deformation is monitored often. Sometimes with the machine itself they can just tell how much farther apart the jaws are getting. However, that's not very useful because that's over this original length, not over that original length, so they sometimes, not sometimes, they usually will put on an extensiometer of some kind that will actually record the elongation of the test section there in the middle. Most of these test pieces being of very standard sizes and then the machine knows L0, which is usually the two inches there, can then compare it now to the deformed length and from the difference between those two get del and then you have the normal strain. So the machine can record what is being done. I need to actually adjust this piece. The stirred test piece is easily a part of the calculation. So then the output from this test will look something like this. On the y-axis is the load, either in terms of P itself but more likely in terms of P divided by the original area which we know as load is typically normalized over the original area. We know that as, well, this is the cross-sectional area, this area of the piece there. But the ratio of the load to that normal area is the normal stress. So on the y-axis we have the normal stress and then on the x-axis we have the elongation, the difference between the extensiometer measurement of length divided by the original length which we know as, of course, then the normal strain. Then as the piece undergoes this test, those two are plotted against each other for a particular material. And it's really characteristic of the material itself. The only variation from one test to another of the same pieces is just minor differences in the manufacturing of the material itself. All right, all set. All set to turn the machine on. And if you go to Angel, I've got some videos of these very tests going on there. They're not the most spectacular tests to watch other than occasionally the compression tests are pretty exciting because when the piece finally comes apart in compression, it sometimes does it in a Hollywood manner, explodes all over the place, pieces flying, and then you can hear all the test guys in the background whooping and hollering like it was a Mythbusters episode. The test goes then something like this. Starts at very low, low, just start turning on the machine, let it start pulling apart the pieces. So P is very small, A0 is very small, and the piece is not deforming very much. These are at the early loads. What's interesting to us is the fact that this early part is very linear. Most of the parts are like this little, mo-type of structural solids we have. They're very nervous. Put down a tennis bat for low curtain steel. Just so you get an idea of what some of these numbers are. Very common structural steel. So the test goes very linearly, the load increases, and the response of the material, the deformation, is proportional to that load for a good section. If for some reason in here the coffee breaker, the union says they must take a break, they turn off the machine, let the load go back to zero, the material will go right back down that line to its original length. No deformation when there's no load. It retraces this part of the curve almost perfectly. A lot of trust in that happening is a big part of structural design. As the loads change, it's very, very predictable what the response of the material is to those loads. So it's very important. This region is a very, very important region because it is so predictable. So operators get back from their coffee break, turn the machine back on, go right back up this very same curve, and let it keep going a little bit farther this time. Until some point when the linearity of it starts to, it departs from a linear response, it starts to turn over a little bit. In this case, as the load increases only a little bit more, we're starting to get more of an elongation of the solid itself. So we're starting to increase our distance down the y-axis, or x-axis. Put a couple of numbers on these for the material solid. This strain down here is sunk to the load there. Here take a little up 40. The extreme difference in the scale of this is a small number. This is a very large number in comparison. Here, in the linear response, we can make the scales anything we want, but it starts to get in trouble as we go through this. There's two points of interest that happen very quickly in here. The first is this last little bit of linearity. It's called the proportional limit. Soon after that, though, the material starts to get longer, starts to deform much more than it had been before, even with no increase in load. The load can be left the same, but the load is starting with itself. In fact, this region then, and it may actually dip down a little bit as the material really starts to stretch. This piece, this point somewhere in there, and there's different ways to pick that point, but it's a fairly standard point there. It's called the yield stress, because it's here that we're leaving the elastic region. Remember, this was all an elastic response in that the load was relieved, the material went right back down that line. If the load is relieved now, the material tends to go down parallel to that line more or less, but left with a permanent strain in it, even though the load has now been relieved. This first region here is the elastic region. The second region here is the yield zone. The end of that, and this is where the curve then starts to turn up again, as the load increases, now the deformation of the material isn't as increasing quite as fast. This region ends at about .02. The material, as the load increases, tends to go up something like that, then starts to flatten out yet again for low-carbon steel, or materials like low-carbon steel will have very much the same characteristic curve, just the numbers will be different. As part of the designer, the different materials are chosen because we need different responses in here in this elastic region. This is where almost all of the engineering design is, because it's predictable, it's elastic, it's a long way from any kind of catastrophic failure, either by deformation, or by the piece breaking. All the pieces doing so far is just getting longer and longer. This region in here is called the strain-hardening region. It's not going to be of great interest to us, because almost all of our concern is going to be down here in the elastic region. But to material scientists, this can be of a lot more concern. So that's the strain-hardening region. Then after this, and if you've looked at the video you've seen this, the material has not had an appreciable change in the cross-sectional area. So all of this region is pretty much as divided by the cross-sectional area in calculating the stress. But then from here on, the material starts to go from this nice, generally cylindrical shape that it has in here. It starts to do what is called necking, where at a region somewhere in the middle, it actually starts to decrease substantially in its cross-sectional area as the load increases. And you can see that if you look at the videos carefully enough. I think one of the videos, they even stop it right there and say, you can see that the material is necking now, and they even put an exclamation mark after it, because it's a very exciting time. But there's an interesting thing that happens with the graph itself. Since the area is decreasing, if that's not taken into account in the calculation, then the curve tends to go something like that. That is mostly an indicator that the area is decreasing in the real solid, but the calculation is with the original cross-sectional area. So this is still quite big. The material is, it appears as if the load is actually going down. The stress is decreasing until we get to a point where the piece ruptures. There's actually material failure, the piece cracks across there, and the test is over, because now the piece is completely destroyed. This is sometimes called the breaking stress. That comes at a value of about 0.25. Oh, this was about 0.2 back here. This region is called the necking region where that pinching of the area is occurring. If, however, the curve, if the curve had been calculated with this new cross-sectional area, the actual cross-sectional area, the curve would have gone more like this and then ruptured at the same place. So that's using a actual, this is with a original, which is no longer a true cross-sectional area of the piece because it's changing cross-sectional area. This extensiometer, if they've left that on here until it ruptured, that would have destroyed. So this is typically taken off when they notice that the linear region has stopped and they're now into the greater material responses. The piece is no longer on there measuring this difference in length. So things do get more approximate from the norm. It's also no typical then, once this piece is broken, they're put back together on the lab bench and the actual distance between these two original dots is measured to check this strain at rupture, the normal strain at rupture. So that's the typical curve, specifically with numbers to low-carbon steel, but it's the general curve where they're called ductile materials. Ductile materials are those that have this kind of response at generally room temperatures, and that's pretty much where you want to design for things is at room temperatures. Okay, there's an introduction to the stress-strain diagram. We'll talk more about it on Monday then.