 All real materials deform when subjected to some form of loading. This deformation might be quite large, invisible, or very small and unnoticeable to the naked eye, but it is still there. Take, for instance, daily part of our lives, the bicycle. If you're a big guy like me and you sit down on a bicycle, you very easily notice that the tires deform under your weight. What you may not notice though is that the entire bicycle deforms. The frame, the seat post, even the handlebars will deform slightly under your weight. If we simulate loading of a person on a bicycle in finite element analysis, we can magnify this deformation as shown here to visualize it a little bit better. Later on in the course, we will examine how this deformation can be important with the distribution of load within a structure. But for the time being, we're going to simply focus on deformation itself. Let's illustrate deformation using this slinky. As you can see, we can deform the slinky in many ways using our hands. However, we need to apply a force to the slinky in order to cause this deformation. This illustrates the first important concept that deformation is a consequence of a force. First, let's examine deformation due to a normal force. The slinky has an initial length L0. If we elongate it with a normal force, we can stretch it to length L1. We can see by the spacing of the coils in the slinky that the deformation is relatively uniform along its length. However, for more complicated loading scenarios, it is possible to locally change the intensity of the deformation while maintaining the overall deformation. We call the intensity of this deformation strain. Looking at the undeformed and deformed states of our slinky, we can define several things. First, the total deformation is simply the final deformation in the deformed state L1 minus the initial undeformed length L0. The average intensity of the deformation, or average strain, is given by delta L over the initial length L0. We denote this strain by the Greek letter ε, but it is only an average in this instance. The normal strain is the limit of the average strain as L0 approaches 0. This is the intensity of the deformation at a given point. We can generalize these definitions to an arbitrary deformable solid as shown here in the undeformed state. We can define two points A and B within this material that have a distance delta S between them along the direction N. Now, if we deform this body, points A and B become A' and B' with a new distance delta S'. The normal strain in the N direction thus becomes delta S' minus delta S over delta S, or the change in length over initial length, as point B approaches point A along the direction N. The units for strain are millimeters per millimeter. Now, most people would say that's dimensionless, but we typically write those units to remind us that it is a strain, that it is related to the intensity of a deformation. It is also often expressed as a percentage of strain or micro strain, which is 10 to the minus 6 strain. We will now examine deformation due to a shear force using our slinky. When we apply a shear force, we can see that the angle of the slinky changes. We get a change in shape, and we can apply a much more complicated shear force distribution and result in an even more complicated change of shape. If we look at our slinky in its underformed state, we can say that it has a width W and height H. If we apply a shear force, we see that the slinky deforms. If we overlay the initial shape, what we can see is that when it's deformed by a shear force, the width W and height H do not change. We do not get any elongation in those two directions. Instead, what we get is a change in angle or a change in shape of the slinky. We denote this angle by the Greek letter gamma, and it is precisely the quantity shear strain. We can generalize our definition for shear strain by considering the arbitrary deformable solid pictured here in the underformed state. If we take two orthogonal line segments, AC and AB, that have an initial angle of pi over 2 radians or 90 degrees between them, and we consider that direction of line segment AB is in the N direction, and the direction of line segment AC is in the T direction, and then we subject this to loading, which causes shear. What we see is the angle between line segment AC and AB will change. In this instance, the angle became smaller and became the angle theta prime. Now we can then define our shear strain in the NT coordinate frame as pi over 2 radians minus theta prime in the limit as point B approaches point A in the direction of N, and as point C approaches point A in the direction of T. This will give us the shear strain in the NT plane. We need to discuss one more thing with respect to strain and that sign convention. Now we have seen that deformation requires a force, and if we look on an intensity level, that means that a strain requires a stress. There's a relationship between these two quantities. Therefore, in applying a sign convention to strain, it is easy to adopt the same sign convention as for stress. So a positive stress results in a positive strain. For normal stresses, this means that elongation will be positive and contraction will be negative. For shear strain, it's a little bit more complicated to visualize exactly which direction is which. But if you look at a coordinate system in the deformation, just think about what shear stress would cause that deformation and whether that shear stress is positive or negative. And that will help you determine your sign of your shear strain.