[MUSIC] Hi, and welcome back. In today's module, we're going to go through the beginning of Static Failure Theories. And the learning objectives for today's module are to understand the definition of static loading, fatigue loading, brittle behavior, and ductile behavior. And then understand the criteria for using specific static failure theories. So there are some assumptions that go into static failure theories. In this course we're going to be assuming, or in order for these theories to work, your material needs to be homogeneous, isotropic, and you need to be in the linear elastic region of the material. Static loading means that the load is not changing over time. So it's a stationary force or a stationary couple or torque. And it's unchanging in direction, in magnitude, and the point of application. So for example, if you have a bridge, the weight of the actual bridge on its support pillars, that's a static load. And when we talk about failure, failure is a rather broad term, and it can refer to a number of different mechanisms that are occurring. So failure could mean permanent distortion or yield of the component, it could mean fracture, it could mean that the component has compromised functionality. Or it could just mean that there's been a reliability downgrade in the component. For most of the failure mechanisms we're going to be looking at, we'll be looking at yield or fracture as failure. So in order to determine which failure theory to use, for static failure theories, it's really critical that you're able to classify the material as a brittle material or a ductile material. Now ductile materials, it's in their stress strain curve, it's very clear where the plastic deformation begins. So, there is this clear yield point on the curve. There's considerable elongation at failure, so your strain at failure is generally considered to be above 0.05 if the material is ductile. And the failure is the yield point, in this case we'll be classifying it as the yield point. In other applications, failure could mean something different. For a brittle material, there's no clear beginning of plastic deformation and there's little elongation at failure. So your strain at failure will be less than 0.05, and typically you don't see a yield, you just see a fracture at failure. We talked a little bit about this, how brittle failure is so dangerous because it occurs without warning. Where ductile materials, you often see that yield first before you get to the fracture. So the first static failure theory that students typically learn is called the Maximum Principal Stress Theory. And this is a theory you should have learned in your mechanics of materials, or deformable bodies class. And essentially what you're learning here is that, at each point in an object, there's a stress and the stress varies depending on the direction. So in the Principal Stress Theory, you calculate the highest principal stress and you compare it with the yield stress. So just a reminder, if you have an element in tension, so our little square here is going to be our stress element. Typically you tend to calculate, or students tend to calculate, stresses along the x and y axes, which have been somewhat arbitrarily set. So there's no guarantee that since stress at each point varies with direction, or stress is directional, that you're calculating the maximum stresses. They might not be lying along the x and y axis. So your stress element might actually look, if we looked in all the directions, it might look something like this. And what principal stresses do, is they go ahead and they figure out which stresses are the largest and at what direction they're acting. And they determine the magnitude and direction of the maximum stress at that point. So this is a great failure theory, it's an excellent one to learn first. There's a couple of problems with it. So principal stresses do not safely predict failure in ductile materials. In some types of loading situations they're very conservative. And then in other types of loading situations, they actually are not conservative at all and you can get failure before it's predicted. So where you tend to see principal stresses used is in the Coulomb Mohr theory, either for brittle materials or for materials that have very different strengths and compressive or tensile strengths. One thing, if you didn't learn principal stresses you can learn them here in Dr. Whiteman's Mechanics of Materials, MOOC, The Fundamentals of Stress & Strain and Axial Loading, and Modules 17 through 26 give a great overview of principal stresses. And you'll need to understand them in order to go into Coulomb-Mohr Theory. So we're going to break down the theories by the types of loading and if the material is ductile or brittle. So right now we're learning failure theories for static materials, where the load does not vary over time. So for static materials that are ductile, there's a couple of different failure theories. There's the Maximum Shear Stress Theory, and the von Mises Theory. And these both hold if your yield strength in tension is equal to or close to your yield strength in compression. Now if your yield strength in tension is vastly different than your yield strength in compression, there's a Ductile Coulomb Mohr Theory that works well for materials like that. For brittle materials, if your yield strength, now typically in brittle materials they're much stronger in compression than they are in tension, so your tension and compressions strengths are almost never the same. So Brittle Coulomb Mohr Theory works well for brittle materials and there is also a Modified Mohr theory. So the conservativeness of these theories varies a bit, so what you'll find in industry is von Mises theory is used the most. It turns to be a little bit less conservative than the Maximum Shear Stress Theory, but it really predicts the yield of components very well. So you'll see von Mises Theory used throughout industry, and that's what we're going to focus on learning today. For brittle materials, Brittle Coulomb Mohr Theory is more conservative, it tends to be a bit more popular. And we're going to focus on that in this class, so we'll also learn Brittle Coulomb Mohr Theory. Once you understand Brittle Coulomb Mohr Theory, it's very easy to understand the Ductile Coulomb Mohr Theory or the Modified Mohr Theory, and you guys can pick up those out of any textbook. So next time what were going to do, is start learning von Mises Theory, which is also called the Distortion Energy Theory. It's a really essential static failure theory, it dominates the industry and I look forward in seeing you next time. [MUSIC]