Tag: résilience

 There are a few other things that we can determine from the stress-strain curve of metal that is quite interesting. And I’d like to just spend a moment looking at that. So again, we’ve got the stress here, on the vertical axis, strain– typical metal. We have a curve that looks something like this. It comes up looking linear elastic, plastic, ultimate tensile strength, and then fracture. And so we’ve determined all of the strengths. We have the yield strength, the ultimate tensile strength, and the fracture strength. But what other properties can we determine? Well, first of all, one of them that you may have heard about is the ductility. So ductility has a usage in common language. You might say, well, what’s ductility all about? If something’s very ductile, you might say, well, it describes how much you can stretch something. But of course, we know that that’s not accurate enough. Stretch, is that referring to elastic or plastic? So we’ve got to be better than that. And in fact, I’ll tell you ductility is a measure of a plastic strain. So we know it’s a strain quantity, and it refers to plastic deformation, only plastic strain to fracture. Now we’ve got something we can work with– plastic strain to fracture. So let’s see well, this is the point of fracture. That’s a fracture. So that fracture, if we unload we’d have a value here for total strain. Let me write that in there for you, total strain. If we unloaded– if we took the total strain there at fracture, just a moment before it fractured, that would be our total strain. But what if we unloaded it? Somehow you knew just infinitesimally before it was going to fracture– well, we know that Young’s modulus is structure-independent. So it won’t change. So we would have unloaded that same modulus. This means we come back down here to a value on the strain axis, a finite value corresponding to zero stress. It’s unloaded, there’s no stress on it, but there’s still some persistent strain. That strain has to be plastic. That’s a plastic strain, which means that this strain here is elastic. That’s elastic, and that makes sense because what is that? That’s the strain underneath this linear unloading portion. And the unloading portion, if it’s linear, is governed by Hooke’s law. And we know that’s elastic because Hooke’s law refers to elastic behavior. So if we unload down and we get plastic strain, that plastic strain has got to be the ductility. So ductility, you unload at fracture. And the remaining strain is the ductility. Another interesting property that we can determine from this stress-strain behavior for a metal, for other material classes as well, is called toughness. And the toughness is sometimes not such an intuitive quantity. You can understand strength, it’s force over area. You got a sense for that. It’s pressure, if you will. Even modulus you can kind of get a bit of an intuitive sense for it because it’s how hard is it to bend something elastically. It’s a little harder, but the toughness is– toughness, I’ll tell you what the tough is. Toughness is the energy– it’s an energy term. And it’s energy absorbed to fracture. What we can do is integrate. And that is to take the area under the curve. So if we take the area under this curve here, it would be this area here, all this area here under the curve is the toughness. And how do we know that? We could look at it dimensionally. If we’re taking a product of stress and strain and looking at the dimensions, stress has units of Pascals. And what’s a Pascal? A Pascal is a Newton per square meter. Well, I can go on living my life multiplying whatever I want by 1 and just multiply this screen by 1. You didn’t even notice. So here we go, where I multiply Newton per square meter by 1, meter over meter, and I end up with a familiar term in the numerator– Newton meter. And of course in the denominator I’ve got volume units. But what’s the Newton meter? A Newton meter is nothing more than a joule. So we’ve now got joules per volume as units when we integrate under this. And that’s great because we want an energy unit. So if we integrate under the entire curve up to fracture, it tells us how much energy went into fracturing that. And that includes elastic and plastic deformation. The final thing that we can obtain from the stress curve, is another energy unit and it’s quite useful– it’s a stored energy unit this time. We’ve got stress and strain. We’ve got our linear elastic region, plastic deformation, and fracture– is the resilience. Thus resilience is a measure of the stored elastic strain energy at the yield strength. So again, we know if it’s going to be an energy term, energy per volume for a given volume of material, we’re going to have to integrate under the curve. And where are we going to do it from? Well, we’ll go to the yield strength. And we go down from there. And if we unloaded at the yield strength– I’m going to be a little careful about something– if we unloaded at the yield strength, you’d find that you have a little sliver of permanent or plastic strain. Maybe it’s close to the 0.2% offset strain. You’d probably have some plastic strain accumulated when we had yield. For practical purposes, we say it’s elastic before yielding and it’s plastic after. You might have a sliver. So we’re not going to include that if we’re going to be strict with our definition here. And so that area there is the resilience. And that area is just an area of a triangle. And we know that the area of a triangle is 1/2 base times height, which in our case is 1/2 of– well, what’s the base? The base is the elastic strain. And that’s good because we’re after the stored elastic strain energy. So we’ve got strain elastic. And what’s the height? Well, the height is the yield strength. But we can, again, do better than this. Because if it’s elastic, it’s the area under this– or it’s the strain underneath this linear unloading portion. And the linear unloading portion, we have a mathematical equation for. We have stress equals E times strain. It’s a straight line. So that means that the strain is going to be equal to sigma over E. And we fire that in here, and we find that the resilience is– I should erase that– the resilience is going to be 1/2 of sigma, and this is the sigma yield. That’s what we’re using here. So that’s sigma yield over E times sigma yield. So at the end of the day, the resilience, which we often use this– I’ll introduce this symbol here. The full name for this is the modulus of resilience. And modulus is just a fancy word for a special number. So our special number here is the modulus of resilience. And we use the uppercase letter U. Is 1/2 sigma yield squared upon E– and that’s an interesting little equation. It tells you the stored strain energy for a material. So if you’re going to make a material for a spring, you’d look for something with a high modulus of resilience. And again, the units here, the dimensions here of modulus of resilience are going to be joules per cubic meter.As found on YouTubeExplaindio Agency Edition FREE Training How to Create Explainer Videos & SELL or RENT them! Join this FREE webinar | Work Less & Earn More With Explaindio AGENCY EDITION