Review of Mechanics of Materials: Strain, Stress, Shear, Thermal Expansion by @mes on 3Speak
In this video, I review the Mechanics of Materials needed to solve Problem 1 of the EM 304 test that Dr. Judy Wood gave her students on February 6, 2003, five days after the Space Shuttle Columbia ...
Tags: #math #science #engineering #materials #physics #nasa #threespeak #scifimultiverse #chessbrothers
Review of Mechanics of Materials for those who have never taken a course in it. And again, it's been over 10 years since I took mine in university. And also just good for overall review and so on, because it's pretty useful to know. So let's first cover some of the mechanics of materials terms first. So strain and thermal expansion. So we are given the thermal strain for the support plates and aluminum tube via their coefficients of thermal expansion. So we're given that the... this thermal expansion coefficient of the plate is equal to 0.00, and this is actually microstrain, so micro, and then this is epsilon symbol, microstrain per degree Fahrenheit. So the plate is rigid, so it's not going to be straining at all. And then the coefficient of aluminum, the coefficient for the aluminum tube is 12.8 microstrain per degree Fahrenheit. strain is the relative deformation compared to a reference position and has SI units or standard base units of meter per meter or M slash M like that. Basically, it means if you have a length like this, that's your reference position, let's say L, yeah, let's just say L and it's equals to one meter. And then let's say you increased as you were, the strain basically, it was expanding by delta L equals to 0.1 meters. So then the strain is going to be, this is going to be like this, and that's going to be, I mean, the strain is going to be just, the units are going to be by themselves, it's going to be just basically delta L over L, which equals to 0.1 meters over one meter. And this is going to be 0.1. Well, you can cancel those are out, this is to be 0.1, or meter per meter. Yeah, because divided by 1 just doesn't change anything. Or basically 10% of it. So basically 10%. of it, yeah, it just shows you the degree that it's expanding or contracting. So alpha, the symbol right here, Greek symbol alpha, is the coefficient of linear thermal expansion. or a tendency of matter to increase in size with increasing temperature. So if you have, here's an example, so figure expansion joint in a road bridge used to avoid damage from thermal expansion. Yeah, because the materials of concrete steel and all that expands with temperature and contracts with colder temperature. So then you're going to need these right here so that it basically in hot weather, these expands and basically it connects in cold weather and so on, make this bit bigger. All right, so the units, this micro, yeah, this. mu and epsilon divided by degrees Fahrenheit equals to microstrain per degree Fahrenheit, which equals to, this is a unit is 10 to the negative 6 strain per degree F. So instead of 0, now we're going, so basically the decimal place you need to go 1, 2, 3, 4, 5, 6. So 5, 0, and then there's 1. So that is microstrain. And that's typically what you're dealing with in these, it's not going to be expending 10 % in or so. Yeah, because that'll be a bridge I will not drive over. Anyways, and this is per degree, yeah, Fahrenheit. So basically, what this means is micro-strain, basically, every degree's Fahrenheit increase, you're going to have a change in the strain by this amount. So, yeah, so 10 to this negative 6 strain corresponds to one unit of length changed per 1,000, yeah, well, I mean 1 million unit of length, aka one part per million. So, for example, aluminum expands at 12.2 millionths of an inch, per inch of length for every 1 degree Fahrenheit temperature increase. Yes, if you increase temperature by 10, this would be increased by 10 fold. So the plates are assumed to not expand at all with temperature, hence you have this alpha of the plate is equal to 0. And I'll put a plate here. Alright here, I just put the plate, a subscript like that, equals here, for the plates, sir. So, and then now let's look at the 2014 aluminum oil, a T6 aluminum oil. So, 2014 aluminum oil or aluminum in USA slash Canada. So this is, and Wiki has this aluminum, aluminum, or aluminium, aluminium, yeah, but there's an extra eye there. This one doesn't have the eye, so that's, I've been used to this one here. So, ore aluminum and USA slash Canada is an aluminum-based alloy, often used in the aerospace industry. The T6 refers to the tempering process used, which is a process of heat treating, yeah, which is a process of heat treating. used to increase the toughness of typically iron -based alloys. So there's tempering, basically, apply heat, and so on. Here's a figure right here. Differentially tempered steel, so you basically heat up at different spots and so on. The various colors produced indicate the temperature of the steel is heated to, so this would be colder, it would be hotter, I assume. Light straw indicates, yeah, this light one right here. indicates 204 degrees Celsius or 399 degrees Fahrenheit. And light blue indicates, oh yeah, this is hotter, hotter blue, this one's lighter, so lighter and hotter, indicates 337 degrees Celsius and 639 degrees Fahrenheit there. And now the T6 temper, temper, so there's a type of, yeah, the preferred type of tempering process, the type of heat treatment. The T6 temper involves solution heating, heat treating, and artificially aging until the metal meets... standard mechanical properties and here's mechanical properties of it you can click over here and view this mechanical properties of this from this website make it from.com shows you the type of here bernel hardness and all these and so on your elastic youngs modulus and other structures above this is from that web page said below but it was a compare to 2014 t6 aluminum to, yeah, two other ones, two other 2 ,000 series on top, all aluminum alloys, and the entire database bottom, and a full bar means this is the highest value in the relevant set. A full, a half, full bar means it's 50% of the highest, and so on, for example, the Bernel hardness and elastic modulus of 2014 T6, aluminum is very high compared with other aluminum alloys. So basically, what this means is this is going to be the 2000 series, the top one is 2000. So these are 2,000, et cetera, type of all alloys. And then this is middle is going to be all aluminum. And this is all, just everything. I think it's other metals and so on. Everything, just for example. And this is all aluminum. The middle one's all aluminum, everything, and so on. And basically, it shows you where it is on it. So in terms of this Bernel hardness, how hard it is, this is going to be on this range. This is the 2014. T6, yeah, so here's the 2014 T6 right here. That's the one we're dealing with, and then basically, Young's modulus is very, very high compared to all the other 2000 series, and again, very high here, but then compared to just all the whole data set, it's not that high, you get other stuff with, they're more stiff and so on. Yeah, like other materials, steel and other alloys, and so on. And it shows you, yeah, there's a whole bunch of other stuff, you can go and look at that. and elongation at break how long it goes yet, so there's a lot of mechanical properties that are catalogued. All right, so now let's take a look at elastic constants. So we are given the elastic constants for the support plates and aluminum tube, and we're given this right here, Young's model list of the plate is equal to infinity, the shear model list equals infinity, and the pazon's ratio, V is equal to zero, basically it's rigid, so I'm moving, and that's why all these infinite stuff here, so it's infinitely rigid. in shear, in compression, and in this Poissons ratio there, we'll get to that. But that's basically transverse expansion or compression. So, and then the, for the aluminum plate, we're given, Young's model list for the aluminum plate is 10.6 MSI, or millions pounds per square inch. And this is, G, is the shear modelist, 3.9 MSI, and then the Poisson's ratio is 0.34. So basically, this means this, if you're pulling something, Yeah, if you're pulling something here, it's going to be decreasing, actually. Yeah, the relative to the expansion, the actual expansion is going to be a compression, yeah, compressing point, yeah, 34% of that value. Yeah, we'll double check when we get into that section. So Young's moddless, or the Young Modlis, E is a measure of tensile, compressive stiffness when a force is applied lengthwise. Young's modeless is given by the equation of stress, force per unit area, divided by the strain. So this is basically, young, yeah, yeah, basically just this young model is equals to the stress, how much force per area, divided by the strain, how much is actually being adjusted, how much is actually being compressed or expended. So stress, the sigma symbol. Yeah, the Greek symbol sigma for actual stress describes the forces present during deformation. So here's in tension, that's what's typically used for, tension and compression, tension and compression like that. Yeah, basically expanding, and yeah, this is compression. So compression, yeah, you can see how it's compressed there. This is expanding like that. And, yeah, notice how when you compress, it gets bigger. So Byzant's ratio indicates how much is this versus this. this one as you pull it, it's getting smaller. So that's very interesting. And then that's compression. There's shear, shear is this way, and then there is bending, you have bending like that, and there's torsion, you turn it around. And then this one, you turn opposite, so you're just twisting it like that. That's mechanical stress, and... Yeah, and going further, just to show you that sigma sign, so a figure here is idolized stress in a straight bar with uniform cross-section. So if you take this bar right here, if you look at inside there, so each one's going to be experiencing the stress as you're pulling it out. And this is basically the stress over the full area. So F divided by A, it's like an average value. It's literally just an average value of that force, but along the entire setup there. There's sigma there between. So basically the entire section of it is under, yeah, under, this is tension. because that's being pulled. All right, going further, and here's another chart right here. And this is a stress versus strain chart. It's very, yeah, it's very common in engineering. So basically, Young's module is slope, yeah, just equals to the slope of this. So this is rise of a run, yeah, equals to that E there. So this equals to E. So you have the stress. As you're applying stress, for example, a tension one, you're applying it, it's going to be straining, and this is the elastic zones. If you let go, it typically can go all the way back there. And if you apply stress until you get to this point right here, after this, it starts just expanding with that same stress without needing to increase. That's the yield strength, so it's basically going to be just expanding without even adding any more stress. And then there's going to be strain hardening where now you need to apply even more stress to go, but it's at a curve right here. And here's the ultimate strength right here, where now it's called necking, and it's going to be expanding at less stress. Yeah, and there's the fracture, and it fails like that. So that's quite a, so I should have this typical chart. Yeah, so this is the slope of leaner part of the stress strain curve for material under tension. And there's the elastic region, and then, because it can typically go back to where it started if you let go of the load. And then there's the, yeah, plastic region. Then here, just change my pen. Yeah, so this one, if you let go, it's not going to go back. If you let go, it should be back from here, but then you're going to have a... permanent deformation that you can't ever get back. All right, there's a plastic region, and there's a fracture. So when the load applied is removed, the component's original dimension strain will be fully restored. Yeah, this one here, should be fully restored on this zone. So figure, a relationship between Yang's modulus and deformation under stretching aluminum. So they're stretching aluminum. That's very fascinating. And now the sheer modulus, or modulus of rigidity, G, or sometimes S, or a mu, is a measure of the elastic shear stiffness of the material and is defined, hence this dot right here. I never knew that's mathematics. This means definition. By definition, these two colon, the colon there, the colon equal sign, as the ratio of sheer stress to sheer strain. So how much, yes, sheer stress you're applying. And that's going to be, instead of like this one, force of area, it's going to be on the side force applying there and across this entire area. So that's a sheer. set up like that and then that's the shear stress f over a and then divided by the strain in shear is how much it separates over here so that's going to be this delta x divided by this the original length right here is going to be l yeah so basically how much it just moves transversely like that divided by its original length and then when you get that you divide these out you're going to get now you're going to go just put them together f over a and then this is going to put this l on the top l over x like that and this is just going to be sigma also actually not sigma it's going to be just a stress actually we already have that there so it's just to be sigma xy l over a delta x but anyways yeah that's just uh like that so f over a anyway so that is what we get All right, yeah, so there's the figure of the sheer strain. And now shear stress, this tau symbol, is just a component of stress co-planner on the same plane with material cross-section. So it's co-planner on the same exact plane. So that's going to be divided by it, and we get the basically the average force across the entire area. So it arises from the sheer force, the component of force of the force vector. Now let's put the force vector parallel to the material cross section. Yeah, and then compare that with normal stress. Normal stress, on the other hand, arises from the force vector component perpendicular to the material cross section on which it acts. And now here is, yeah, just tau is equal to f, a divided by a, so if you have a section like this, this is a, just a, that's a cue boy, that's what they call this, I think. the rectangular cuboid. So as you pull this, as you strain it to the right, you're going to have a separation right here. But then across that, yeah, across this area there, it's going to be the, yeah, tau is going to be acting upon this top of the area. So here's figure side view of a parallel pipe. This parallel pipe basically means this is just a cuboid. Imagine it's like in 3D like that, so it's in 3D like that. Yeah, and then basically this full area where you're... or um but initially it's going to be this area yeah it's going to be the same it should be the same area in idlas format yeah so it's going to be like that you're pulling it uh i'm straining it and that's the side view of a parallel pipe where shearing forces apply to the top of a rectangular cuboid while the bottom is held in place uh this resulting sheer stress a tau deforms the cuboid's rectangular side into a parallelogram Yeah, so it goes aside from a rectangle to parallelogram. The area involved would be the top of the parallel pipe, yeah, so it's just top of this parallel pipe. This is the same as that a cuboid as well. So look like that. That's fascinating stuff. I just erase this. Just use that, think of that, and I have calculated for this as well. We're going for this, so Pazon's ratio, V, or see what the actual term is. And yeah, here, quickly, I just went to my new MESFM slash symbols. So you typed that in. Yeah, it's actually a new. Yeah, new, all right, that's pretty cool. Anyways, play around with this, I made a whole bunch of calculators and so on. So, absolutely amazing setup there. And let's go further. Yeah, so boson's ratio, new is a measure of the pazon effect or poisonophy. Let's see how it's pronounced. Here is a website here, just has pronunciations, and yeah, you best believe I'll be making my own, and yeah, this one here, try this right here. Poisson. Yeah, so that one says Poisson, interesting. This has four or five rating and it's 65 votes. Yeah, so it'll make something like this, but better. And if you like mine, let me know any suggestions and feedback and so on, but I'll be putting up there, probably the URL MES at FM slash pronounce. I'll be, yeah, something good. Anyways, let's go further. So, yeah, that's how you pronounce it. So, yeah, Poisson's ratio, new. or this V symbol, is a measure of the Poisson effect, the deformation, expansion, or contraction of a material in directions perpendicular to the specific direction of loading. So if you're loading in this Y direction, this vertical loading, what's going to happen is it's going to squish this way, but it's going to expand laterally. And if you pull it, it's going to be the reverse. It's going to be expanding, yeah, expanding axially, but then compressing laterally. So that's the boson's ratios. It's basically take a ratio of those. and figure this above one, Poisson's ratio of a material defines the ratio of transverse strain or lateral strain that's in the X direction to the axial strain Y direction. This one here, we had the, this is the lateral is in the delta X, and this one is delta Y is the actual. But for this figure right here, so figure right here, figure one, I'll just erase this figure one. as in the wiki link. So a cube with sides of length L of an isotropic linearly elastic material, basically the same in all directions, and so on, same type. It's subject to tension along the x-axis right here. So if you're applying tension here, so you're pulling it along this axis, this x-1 now. Now it's actually loaded on the x-1, not the y-1, and it's that out of figure. with a peasant, yeah, this one has a poisons ratio of 0.5. The green cube is unstrained, and then the red is expanded in the X direction by Delta L due to tension, and contracted in the Y and Z directions by Delta L, yeah, Delta L prime. So I guess L prime, yeah. So this is the expanding here. This is basically isotropic same material everywhere, and it's expanding. This is the green, this is unexpended. and now it's going to expand delta L in the actual. And now you have the transverse delta L prime like that. Yeah, so the ratio is going to be, well, delta L prime over delta L. That's the, it puts on its ratio. But then this one's be taking this D over D setup. Basically, it's going to be looking at the strain. So there's a formula, mu equals to D. This is infinitely small, basically, back to our derivative setup. And it's used to a negative value, just so that this becomes positive. because these are going to be always opposite of each other. So if the actual one is positive, this one's going to be minus, if this one is minus, this is going to be plus. Yeah, so you're always going to be having a negative value, so you're going to be changing. Because notice here, if this is expanding, these are compressing. Yeah, if this one's expanding, it's expressing, and if you squish it, it's the other way around. So if you squish it, this expands. So if you have a negative strain on one end, you can have a positive strain on the other, hence put a negative sign, so you always get a positive value for the strain number there. So mu equals the D, the strain of the trans direction divided by D, this is the strain or the epsilon of the actual one, and then equals to also minus, this is the strain of the y, or the strain of the z divided by, yeah, strain in the actual direction. These ones are equal in this case. Yeah, so that's the actual, in the Z direction, and in Y direction, divided by the actual one, which is the X direction. All right, so fascinating stuff. And finally, before we get to the actual solution, let's take a look at the MES mechanics and materials calculators, so go to MES .m slash materials, play around…