[MUSIC] Hi, and welcome back. In today's module, we're going to go over the Distortion Energy Theory, which is part of unit two, static failure. The learning outcome for today's module is to understand the basic principles behind the Maximum Distortion Energy Theory. Which is also commonly referred to as von Mises Theory. And a couple of assumptions that you have to make in order to utilize the Distortion Energy Theory. Or in order for this theory to correctly model the behavior, and predict behavior of a component. Is the component needs to be relatively homogeneous. It needs to be isotropic and in the linear elastic region on the stress strain curve. The component has to be made out of a ductile material and behaving in a ductile manner. It needs to be in static loading. And generally, your tensile strength at yield needs to be roughly equivalent to your compressive strength at yield, for the material. So, those are the assumptions that you must have, in order to apply the von Mises, or Distortion Energy Theory. So the Distortion Energy Theory has its basis in strain energy. And strain energy is essentially the energy stored by a system undergoing deformation. Or the energy in an elastic component as it gets loaded and deformed. So as you're deforming a component you're putting this external work on the component and it gets converted into this internal strain energy. So what happens when we distort or put loads on an object is it causes the material or the component to change in both shape and in volume. And both of these cause strain energy to be in the material. So there's a hydrostatic component to the deformation and the strain energy, and that's just the change in straight volume. So change, and in a way in size. And there's also a distortion component to this deformation and this energy. And that you can think of as an angular distortion, so a change in the angles or the shape of a material. And so scientists looked, and engineers looked at this, a lot of testing of materials. And what they determined is that materials in hydrostatic loading, where your principal stresses are all equal. Or in other words, you can think of it as your sigma x equals your sigma y equals your sigma z, and there's no shear stress. They can withstand really high loads, so above the yield strength. They can withstand extremely high loads in this hydrostatic loading. What they determined is, materials have this limited capacity for distortion. And that's really the basis of this theory. So if we look at distortion versus hydrostatic, and how this all breaks down. If we have an element in a triaxial stress state. So it's having stress in the x direction, in the y direction, in the z direction, possibly in some of the shear directions as well. That element's going to have a hydrostatic component to it, where the stresses are all equivalent in all directions. And it's also going to have a distortional component. The hydrostatic component is going to, in this case, squish the element into a smaller box. The box will have the same angles but it'll just be a smaller box. Where the distortion component is going to cause this angular distortion to occur. And the distortional component is really more linked to what is resulting in failure or yield of the material. So both of those components have corresponding strain energies. So there's a strain energy for the hydrostatic component and a strain energy for the distortion component. And what the failure criteria for the Distortion Energy Theory, or the von Mises theory. Is that when your component's strain energy per unit volume is greater than or equal to the distortion energy per unit volume, at yield of a tensile test specimen that's the same material as your component, you're going to get yield. So essentially if we take a component, let's take some sort of component made out of aluminum and we load it in all different directions. When the distortion energy in that aluminum, per unit volume, is greater than if I took a bar of aluminum, loaded it in tension, pulled that bar apart, and see when it yields. So when the distortion energy in the component due to loading is greater than the distortion energy per unit volume at yield of a tensile test specimen, that's when we get yield. That's when we get failure, so it's kind of an energy comparison. So the question is, how do you actually compare energies, Right? And the answer is, there's an equation of course, this is engineering, right? So we have the equation which is your sigma effective, or your effective stress, is equal to. And it includes all the different stresses that could possibly be applied to your component. So you have stress in the x direction, y direction, z direction. And shear stresses in the xy, yz, and zx planes. And this equation combines all of those stresses into an effective stress. And you can take that effective stress, sigma prime, and you can compare it to the yield strength. And when your sigma prime is greater than or equal to your yield strength, that's when your component is going to yield. That's what von Mises theory says. You can also, so they've simplified the equation down for principal stresses, remember we talked about principal stresses a bit. And you can also use this bottom equation for principal stresses. And then your factor of safety is n, your factor of safety is equal to your yield strength divided by your effective stress. And when this is less than 1 you're going to get yield on your component. So that is the von Mises equation and the Von Mises theory. So before next time, what I'm going to have you do is work through this example on your own. So here we have this perfect aluminum cube. And note that I've given you stresses, not loads here, to simplify, this is a very simple example. We see that the aluminum cube has a yield strength in tension that's equal to the compression of 75 MPa. And it has a strain at failure of 0.05, so it's ductile, it's behaving in a ductile manner. It has stress in the x direction of 20 MPa, in the y direction of -40 MPa, in the z direction of 10 MPa, and in the shear yx plane of 10 MPa. And so what they're asking is for the effective stress on the cube and the factor of safety. So, go ahead and attempt to work through that equation on your own, or that problem on your own. And then we'll work through it in the next module. I'll see you next time. [MUSIC]