 Hi everyone. So in this video I want to talk about the Morse theory and the modified Morse theory, which is a failure theory for brittle materials. And it's a little bit more complicated than the failure theories that we've talked about so far, but we'll walk through it and I think it'll be all right. So it's four brittle materials and before I go ahead and draw the Sigma 1, Sigma 2 diagram, I do want to show something else, which is the plot of a Morse circle, which kind of demonstrates what we're talking about here. So I have a Sigma tau axis and basic principle, if we have a material, we might say that it fails out here at Sigma U T and it fails back here in compression at Sigma U C. And that's pretty common that the the ultimate compressive strength for brittle materials would be higher than the tensile strength. And if we draw the Morse circle for these, we'd you know, we'd have something like this and something like this. And then we would say that we're safe as long as our stress falls within this envelope, which encompasses both of those plots. So as long as we're under those ultimate tensile or compressive strengths that we'd be okay and kind of in between there is covered as well. This is assuming that for our brittle material, the only information we have is tensile test machine type data that would give us ultimate tensile strength or ultimate compressive strength. However, if we have available to us the ultimate torsional strength, tau sub U, from a torsion test, then that gives us a little bit more information. And if we did a pure torsion test, then what we'd be finding is the ultimate torsion strength, which is going to fall somewhere in here. And we didn't scribe that circle around the origin like that. And now we have this this third bit of information that we need to take into account. So in this case, we didn't scribe the envelope around to include that circle, because that also would constitute a failure point that we need to watch out for. And I would kind of give this this shape around here. Now obviously, my circle didn't go down far enough down here. But the general principle is we inscribe all three of those those circles in order to be considered safe. Now, if I go ahead and draw this on the axes that we've been looking at, which is a sigma one, sigma two axis, then I can go ahead and mark my ultimate tensile and ultimate compression strengths. And then I'm going to connect the dots more or less between these points. And this basic version here is what we call, in this case, the simplified Morris theory, which is to say it's kind of the basic form that encompasses that that tensile, and that impressive component. Now, if I want to take in the the additional bits of information and kind of really flesh out the theory, or the the full analytical expression, I would actually have curves that look something like this. And I'll see if I can draw these in so kind of these blue dashed lines. And this would be what we call the true Morris theory. However, very few calculations that we might do would actually use that that true version of it's mathematically a little bit more complicated. So instead, what we might do is we can dash in a line this way, and this way. And the slope of this line is sigma one over sigma two is equal to minus one. So it's got a 45 degree angle passing down there. And then we go ahead and find where that line crosses our true Morris theory. And we draw in lines that intersect with those two points. And lines here. And we get this kind of red shape. Well, okay, black here, red in the quadrants two and four and black on this side. And that shape is what we call the modified Morris theory, which would be what would be more commonly used. And now let's talk about how we actually encompass these areas. So if I go ahead and analyze what's going on, I'm going to write these in terms of safety factor, which is basically saying the same thing as what we were doing before. But in quadrant one, it's pretty simple. I'm looking at whether or not my stress exceeds my ultimate tensile. So my safety factor in this case is sigma ut over sigma one. So whether or not sigma one exceeds that ultimate tensile strength determines whether or not I fail. In quadrant three, same general idea, safety factor equals sigma uc over sigma two. So same general principle. If sigma two exceeds that compressive strength, then we would predict failure. In this region in the middle, quadrants two and four, now I'm still talking about simplified Morris here, my safety factor is written one over safety factor is equal to sigma one over sigma ut minus sigma two over sigma uc. And that gives us a measure of what's inside those those two circles. Now if I want to get the area under the red curve on either of these quadrants, which I'm going to go get a different color pen, then my equation ends up looking like sigma ut sigma uc divided by sigma uc sigma one minus sigma ut times sigma one plus sigma two. And so this kind of complicated looking equation is what defines the safety factor if we're in quadrants two or quadrant four under the modified Morris theory. So it's a little bit uglier, a little bit, you know, messier equation, but in general it's all it's doing is trying to mathematically describe the shape that we've drawn using this this 45 degree angle line and where it intersects with the true Morris theory. And it gives us an approximation right if we if we look at our lines we see that you know the black line in quadrant two as an example compared with the blue line is a little bit off whereas the red line which has these two straight segments is much closer to the true Morris theory than than the simplified version. So it gives us something to start with and I'll go ahead and stop there.