I think a good way to appreciate the nature of fracture toughness is to look first at how very brittle materials behave. Because a very interesting phenomenon in ceramics that again, is worth knowing about as kind of a subset of our concept of fracture toughness, and that is ceramics are very weak in tension, but strong in compression. You might think of reinforced concrete as an example of that, in which it's maintained under a steady state of compression for maximum structural integrity. Well, if you look at the stress, strain curves for a common ceramic, like aluminum oxide, used widely in furnace applications and a number of industrial processes, that material, unlike the stress curve for typical metal, does not have a plastic component. It simply has that straight line, elastic deformation component, at the end of which is fracture. So, we term that situation a Brittle Fracture, one in which the material fails abruptly and catastrophically, with no warning of overall structural deformation that would come from plastic deformation. Now in order to understand why ceramic is weak in tension but strong in compression, we have to borrow from the analysis of an aeronautical engineer from the early part of the 20th century. Arnold Griffith, a British engineer in studying the structure of materials for Aerospace applications at that time found that sharp surface cracks in a brittle material lead to stress amplification. And a simple schematic of that shows us that the amount of amplification at the tip of the crack, is enough to lead to the fracture of the material at an overall stress it might think to be quite modest. So, again a structural aluminum oxide when pulled in tension, will break at a fairly low stress level, whereas if you're pressing on it in compression, not pulling against that Griffith crack, not leading to stress amplification, then its ultimate strength would be quite high. In fact, about an order of magnitude higher in compression than in tension. So that concept is one that leads to our analysis of fractured toughness. And one in which we'll see in our video segment, our video lecture, is a very similar analysis that the definition of fractured toughness is very reminiscent of the Griffith analysis from almost 100 years ago. And so let's take a look at that video and see in some detail how that stress amplification can lead us to very careful design considerations and how to avoid certain defects, especially on the surface of a material larger than some critical size. >> So now let's see the pay out for this Fractured-toughness concept. And what's sometimes called a design plot, fundamentally important. This is another one of those slides that I like to call out as we go along as one that's especially important and one that really should be one of the take aways from the course. It's that same thing when we introduced that one slide that took us from the atomic scale up to the macroscopic tensile test scale, and relating atomic bonding to the elastic modulas. That's one of those important images that we like you to take away from the course, one of the highlights. Well here's another highlight. And this deserves some detailed conversation because it really has a tremendous amount of our practical information tucked away that's perhaps not obvious at first glance. So let's emphasize the critical point. Now first of all, as I said, the fracture toughness is a stress parameter, very similar to the stress amplification that we defined in the brittle material case for the Griffith criterion. But here we're talking about this in a more general sense and this applies to a wide range of structure materials as we'll see. So it says K But I'm gonna call K 1c variable. This is the fracture toughness. And we see in the, read about this in detail, in the discussion in the text, is that this is a parameter that represents the degree of stress amplification associated with a crack in the material, specifically a surface crack or an internal crack in the material. We'll generally describe this in terms of those surface cracks, just like we had in the sketch on the previous slide. Just tidy up that. So again this is a t. So K 1c then is a material parameter just like the elastic modulus, just like the tensile strength. It's a term that, is essentially telling us about the susceptibility of a particular material to fail. So again, it's another material parameter, just like the big four. We'll now call these the big five. This is the fifth important parameter that tells us a lot about the potential utility of a particular material in a given structural application. But this is a parameter unlike the previous four that focuses on the propensity for the material to fail with a pre-existing flaw. So the equation that our colleagues in continuing mechanics have given us to make this analysis, is tucked away right here, and it's basically describing the stress at which failure will occur, depending upon the length of that surface flaw. So again, keep in mind we're talking about a surface flaw. And as in the previous sketch, that surface flaw is a microns or millimeters deep. So, that's the size of that flaw. That's the scale of that surface flaw. That obviously appears in the denominator of this expression. We could rearrange this to give us our formal definition of K1c. K1c then is equal to, if you just rearrange terms, sigma times radical pi A. So what it's saying is that the K1c material parameter is simply the value of stress at which catastrophic failure will occur, the material will break in two, given the presence of a particular surface wall.