 Hello everyone. In this video I'm going to be talking about another failure theory and that is the maximum distortion energy theory. The max distortion energy theory is another theory for ductile materials and it's basically using this distortion energy principle which is if we looked at how much energy was being absorbed by the molecules within the material and their capacity for absorbing that material or that energy we could predict when they would fail. And obviously we're not going to go into all the details of what constitutes failure on that molecular level. But the basic principle if we are to look at this theory is that again if I draw out an axis system which is a sigma 1, sigma 2 axes, sigma max, sigma min and I mark on it where the yield stress would be. And if you recall when we talked about the max shear stress theory we basically said there's this polygon inscribed within these points and it looks something like that we call the Tresca's polygon. And the max distortion energy theory is very similar but it's slightly more precise which is to say the shape no there's probably no way I draw this accurately. The shape might look something like that where it's actually an ellipse. And being that it's an ellipse we can define it pretty easily mathematically and you can kind of see it touches all those same points where the yield stress is crossing the axes but it's just giving a little bit more definition to that shape. So to evaluate this we calculate what's called the equivalent stress and we're going to call that sigma sub e and this equivalent stress then is equal to sigma 1 squared plus sigma 2 squared minus sigma 1, sigma 2 all under a square root. And if you looked at this equation you'd say hey basically that's the equation of an ellipse because it's defining this elliptical shape. And this equivalent stress then is what we compare against our yielding criteria to determine whether or not it's safe. So before I talk about that in a tiny bit more depth I want to mention that we can also write this equation in terms of sigma x and sigma y you know if we if we're just looking at those stresses as they apply are applied to a stress element and we can substitute those in based on the the math that you know we've previously looked at when it comes to like more circle and stuff and the equation ends up looking like this sigma x squared plus sigma y squared minus sigma x sigma y plus 3 tau x y squared all under a square root and that ends up being the same thing. And then the criteria is simply that if this equivalent stress exceeds our yield stress then we predict that the part will yield. So same general principle to the max shear stress theory in that as long as we're inside the ellipse everything's good if we plot our sigma 1 sigma 2 points outside the ellipse then we're predicting failure. The max shear stress theory is slightly more conservative than the max distortion energy theory you can see it's it's smaller the inside area is smaller than the area of the ellipse and that means that it's going to predict failure more frequently than the max distortion energy theory. So the max distortion energy theory is more accurate than the max shear stress theory but again max shear stress theory is more conservative meaning it'll predict failure more often max distortion energy theory more accurate. And that's the basic criteria that we would use then for for ductile materials. Thanks.