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Elastic Modulus

Teacher/Instructor Jonathan Osbourne
Jonathan Osbourne

PhD., University of Maryland
Published author

Jonathan is a published author and recently completed a book on physics and applied mathematics.

Elastic modulus is a quantitative measure of how much something wants to return to its original shape and size. Generally it can be thought of as stress over strain. We calculate the elastic modulus by using the formula applied pressure / fractional change in size. Young's modulus is an elastic rod stretched one dimensionally, only expanding in length, which behaves like a spring and can be calculated using Hooke's Law.

Okay so today we're going to talk about elastic modulus, the elastic modulus is a property of solids that tell us how much the solid wants to come back to its original shaped and size if we try to deform it. So elastic modulus is always associated with some sort of restorative force which acts to return that solid to its original shape and size. Now the way that we're going to define this, is we're going to say e is equal to stress divided by strain. So it's the amount of stress that we put on the material, which is associated with a force that we're putting on the material. So we're going to write that as a applied pressure divided by the amount of strain that the material shows as a result of that stress that we put it under.

The amount of strain we're going to call the fractional change in size. Alright so let's look at these 2 things. Pressure is equal to applied force divided by area as always, so if I'm going to apply a force I've to apply it over a certain area of that solid. I don't apply it in a pin point. I kind of have to apply it over a big area, so the numerator of this fraction is the force divided by the area. Now what's the unit of that, well we're doing Physics so it's got to be SI, so force is measured in Newtons, area is measured in square meters. So the unit of pressure is Newtons per square meter, which we'll also call Pascal's okay. Alright what about the denominator? Well the denominator is the fractional change in size, so this is defined as how much did the size change divided by how big is it. Now what's the unit of this? Well a change in size divided it's size it looks like kind of the same thing. So this guy is going to be dimensions, so that means that the unit of my elastic modulus will be the same as the unit of pressure Pascal's.

Alright let's look at a specific example of an elastic modulus. There's actually several of them but a lot of times when people refer to an elastic modulus and they don't say anything else, they really mean Young's modulus. Young's modulus is tensile modulus; it has to do with what happens when we stretch a rod of material out okay. So it's like a one dimensional stretching. So here I've got rod of material it's got cross sectional area a it's got length l and I'm going to put a force on it and I'm going to pull it in this way and make it longer by the amount delta l. The way that I'm going to do that is by applying a force f, okay so that means my elastic modulus must be the applied pressure, force over area divided by the fractional change in size delta l over l. So this is my elastic modulus and I can actually look up in a table, what's the elastic modulus of copper? Or what's the elastic modulus of tungsten? Or what's the elastic modulus of steel?

Now what we'll usually use this for, is to solve for the force I need to give me a certain change in length of a rod of steel or iron or whatever. So we'll solve this for f, so we'll have f equals the a is going to go over next to the e and then the delta l over l will come up. So it'll be ea delta l over l just like that, now this is kind of interesting because we've seen it before. Let's see if maybe you can remember, we've got a force that we're applying that's proportional to because look at all this business here, that's just a constant. Remember I looked up that number that number I measured, that's a cross sectional area and this is just how long did it start off being right? So we've got force equal to a number times how much we changed the length. But this is Hooke's law, so that means that this solid actually behaves like a spring as long as we're only making a small change in it's length it will come back just like a spring.

Now of course if I try to pull it too hard I'm going to break the thing. Alright and that's what called the elastic limit, so as long as we're only pulling at a small amount within the elastic limit, then Young's modulus is going to tell us that this rod of material will behave like a spring. Alright so that's elastic modulus.