 So what is strain? Well, when we're talking about engineering terms and the mechanics of materials, we recognize that materials respond to forces either with changes in motion, that's dynamics, or with deformation. Even though we've talked a lot about static equilibrium and the balance of things, usually there is some small change on a material when you apply forces on both sides of it. Consider, for example, a spring weight system. If I took a spring and hung it from a ceiling and then I decided to attach a weight to that spring, what would happen to the spring? Well, the spring wouldn't just stay there. It would stretch a little bit. It would add something to its length until it reached a point where it was stretched out but was still able to hold on to the weight. The simplest way to describe a spring's response and the response of most springs is a linear response. This means that more weight applied, the longer the spring. Let's consider an equation for that. The length of the spring might be equal to the original length of the spring plus the weight that's added times some constant value. Notice when there's zero weight added, this term would go away and we would get L0, which is the original length of the spring. So this relationship leads to the basic design of a spring scale. Scales that have a little spring inside, that you use and then have gradations on the side that the longer the spring is stretched, the more weight is being held by the spring scale. Let's move this around with a little algebra and generalize. So if we instead solve for the weight, we can recognize that there is a change here. This little change is the difference between the original length of the spring and the final length of the spring when it's weighed down. This term is something called displacement. And again, we can think about it as being the difference between the original length and the final extended length of the spring. I'll use the term delta L, the change in length, to represent that value, delta meaning the change in. Notice we can generalize this equation to something studied in a physics class known as Hooke's Law, which says f equals kx, where f is the force, our weight being a certain type of force, the one pushing down. Our x being this displacement we're talking about being the same as the delta L, the change in the length. And then finally, this k being a constant, this is called the spring stiffness. Notice, because our original value here, c, was a constant, 1 over c is just another constant that we'll go ahead and call k. What would happen to our spring system, thinking about this f equals kx, what would happen to our spring system if we doubled the weight? Well, if we doubled the weight, let's see what would happen. Let's actually apply two springs. Here's one spring. And here's a second spring. If we add, well, let's not double the weight, first of all. We'll go ahead and we'll start by adding twice the weight. And we can think of each weight as extending each spring by the amount x. The first weight extends the first spring by x. The second weight extends the second spring by x. We could think about it as being two times the weight would be equal to k two times the displacement. However, if we instead supported it with two springs that were organized differently and we placed the weights on each of those springs, notice that organization would change things a little bit. Effectively, we would have two times the force that would only move that distance x. But because we have two springs, we can consider it as having twice as much stiffness. This sort of arrangement or organization, the first one would be called arranged in series. And the second one would be called arranged in parallel because the two springs would be parallel. Notice the arrangement sort of leads to different sort of response. One of the things about the arrangement is it makes it a little bit harder to determine what's going on. If I take a longer spring, notice I get the same kind of response. Whereas if I put the spring side by side, I get a different response, which means the response depends not only on perhaps how long the springs were originally because one long spring is essentially the same sort of thing just lined up. It has the same stiffness. Or if I place the spring side by side, it's sort of like the area of the springs comes into the count. So that initial length and the area both change the response. So it's in our interest to perhaps change Hooke's law so it doesn't have such a dependency upon our initial length or this area, the side by side layout of the springs. Another thing to notice is that the initial length of a spring might be hard to determine because the spring may never be completely unloaded. It may always have something connected to it. If it's being pulled or something sitting upon it, it might be hard to know what it was when it's unloaded. You may never be able to unload it. So we want a description that can apply to the spring of any length. So this is where we define the quantity called strain. We represent strain with a Greek letter. The Greek letter is the small letter epsilon. And that's a representative for strain. And we define it as being the change in the length divided by that initial length. This is how we define strain. Notice a change in length is going to have some units of length, for example, centimeters. And the initial length will also have units of length, for example, centimeters. And those units will cancel out. So this ends up being what we call a unitless value. We see unitless value sometimes, particularly when we think about percentages. A percentage is some fraction of some larger thing. And we'll often express it as percentage. But usually that's a unitless value as well. For example, the number of apples that go bad divided by the total number of apples would give you a percentage that was unitless apples divided by apples. Let's look at an example. Say we have a spring where the initial length was 10 centimeters. And when some sort of weight is applied, it stretches to 10.1 centimeters. So in this case, our change in L is 0.1 centimeter. And our strain is equal to the 0.1 centimeter divided by the 10 centimeters. Or 0.01. Notice again that the centimeter units cancel out. The other way we could express this would be 1%, a strain of 1%. Another term for strain is percent elongation, if it's expressed in terms of percent. Our usual convention for strain is the same as the convention for stress. Generally, if we're stretching something, that's considered to be positive strain. And if we're compressing something, that's considered to be negative strain. So let's look at an example. Let's think about a metal rod as a spring. And on that rod, we're going to apply an axial force on both ends. So the system is an equilibrium and a magnitude p. This rod will have a cross-sectional area, represented by a not an initial cross-sectional area, and an initial length, L not. So if we think about Hooke's law, F equals k. And now let's replace some of these pieces. Our force here is the axial force p that we're applying. And our displacement, x, will rewrite as delta L, the chain in the length. I'm going to multiply the left side by the value a naught, and then immediately divide it again by the value a naught. I'm going to do a similar thing on the right side. Notice this doesn't change the value of the equation, because I'm essentially multiplying each side by 1. I recognize that this value here is p over a, which we recognize as being our stress. So I will define our stress here. And I'll recognize that this value here is what we just defined as being our strain. I'll gather my L and my a into one place and represent the delta L over L naught with my strain. Well, our initial length is some value that we're going to consider to be a constant that doesn't change. Our initial area is a value that doesn't change. And our initial stiffness is a value that doesn't change. So this is all, again, one large constant, which we actually have a special letter for. This is the letter E. Let's actually represent it in blue. This is the letter E. And our relationship here says that stress is proportional to strain with a constant value of this letter E. This has a name. This is called the modulus of elasticity, also known as Young's modulus. Elasticity has a special meaning. It means that the material returns to its original shape when force is released. In other words, when the material is unstressed. So this relationship we have here, that the stress is proportional to the strain, only applies for a certain range of stress in the strain. Once we get to the point that the material starts to fail, this is no longer applicable, as we'll see in a later discussion.