 In this video, I want to talk about the maximum sheer stress theory, which is the first of the failure theories that I want to talk about for this class. So maximum sheer stress theory is kind of a basic theory intended or targeted at ductile materials, so specifically for ductile materials. And the basic idea is that, you know, it uses this assumption that ductile materials will fail if they exceed the maximum sheer stress. So a pretty simple idea. One of the ways that we visualize this theory is we can do so graphically. So if I go ahead and draw an axis in here, and I'm going to use sigma 1, sigma 2 axes, where sigma 1 and sigma 2 are the max and min normal stresses that we would see on a part. So we plot on these axes the yield stress of our material, sigma y, sigma y here, in the negative direction on both, and so forth. So we have kind of this square grid of sigma y values. Now on this plot, then we can chart an area that looks like this. We draw lines connecting these, and then we actually go diagonally across here, and the resulting shape is this polygon. It's actually called or sometimes referred to as Tresca's polygon. Tresca was like a French engineer that came up with this theory. And the basic principle is that if my sigma 1, sigma 2 are inside the shaded region, then I don't expect failure. Oops, I just deleted the wrong lines there. I don't expect failure. If I go outside of this polygon, then I do expect failure. And I can represent this mathematically by saying that the absolute value of sigma 1 must be less than sigma y, and the absolute value of sigma 2 must be less than sigma y. And that's if sigma 1 and sigma 2, the min and max, or max and min in that order, have the same sign. So if they're in quadrants 1 or 3 of my chart here, I need to see that they have those two values under sigma y. So again, this is if sigma 1 and sigma 2 have same sign. If they don't have the same sign, then what I'm looking at is that the difference between sigma 1 and sigma 2, the absolute value of that, must be less than sigma y. And that's what's defining these two lines in quadrant 2 and quadrant 4, the kind of connecting lines between there. And it defines the limit if they have opposite signs. And so that's really it. That's the straightforward criteria here for whether or not we predict failure for ductile materials under the max shear stress theory. And there's not much more to it. We can make this simple comparison between sigma 1 and sigma 2, and the difference between those and sigma y plotted on this chart if you really want to. And if it's inside the polygon, it's safe. If it's outside the polygon, you're predicting failure. And that's it.