Let's continue on with our discussions of various strengthening mechanisms. And the second one we're going to look at is dislocation strengthening. In the case of dislocation strengthening, we can look at the diagram that we have illustrated in the slide that is on the board. What we see is that when we look at a material like a whisker, that we just described previously in an earlier lesson, where that dislocation, where that structure contains a single dislocation. What we find is that the flow stress that we measure in a material like a whisker is going to be very, very high, extraordinarily high, close to the theoretical strength of the material. Now, as more dislocations are added to the material, what we find is, the strength of the material winds up reducing. And the reason that this is occurring is, that when we have dislocations present, and energetically we're always going to have a certain number of dislocations present. What we find is, if we just have the right number of dislocations, what we find is they will help the slip process. Eventually, as the concentration of dislocation's winds up changing and increasing. What we find is that the material now begins to increase in strengths, so that whole region between the minimum point and as we go to higher dislocation contents, what we're seeing is an increase of the strength of the material. And that increase in strength material is due to the interaction of the dislocations with one another. So the more dislocations we have in the system, then what we find is the higher the strength of the material, because that dislocation, dislocation interaction winds up reducing the mobility of the individual dislocations and hence, the deformation process. It requires more force in order to have the dislocations move through this high density of dislocation arrangements. We can examine the behavior that we've been describing with respect to increasing the dislocation content by examining what happens during a stress strain curve where we test the material. Remember that when we look at the result of the stress strain curve, we have a region where we first begin to load the material. And in the region, we go up to position A, that's at the red circle, and what happens at this particular point is we deviate beyond A over to B. We deviate from the elastic portion and now we're into the plastic regime, and what you can see is as you go from A to B, there is a corresponding increase in the stress of the material. And now if we would unload it and take it back to our original position, of course what we would find is, we would see that the material has undergone some permanent plastic damage, and as a result of unloading from B down to C, what we will wind up with is a recoverable elastic deformation. Now if we were to reload the specimen from C back up to B. Effectively, rather than talking about the strength being at position A, it's now at position B. Position B is now our new yield strength of the material. If we were to continue on beyond B up to larger and larger amounts of deformation, what we are in effect doing is increasing the amount of deformation we put in. And that deformation is manifested in an increase in the dislocation density. So that dislocation density now is causing a reduction in the mobility of individual dislocations. And hence, when we reduce the mobility of dislocations, we wind up decreasing or increasing the strength of the material. So one of the mechanisms by which we can strengthen a material is to go through a deformation process like a rolling process. So for example when we look at material that undergoes rolling, what can be done is you can go through a series of steps during rolling, increasing the dislocation content and thereby increasing the strength of that sheet that is, that has undergone the rolling process. Now there's only a limited amount of increase in strength that you can achieve by doing this because at some point and time, the more deformation you put in you'll ultimately wind up reaching the point where the material can no longer withstand all the deformation and it fails. But this is a mechanism by which we can in fact increase the strength of a alloy. Now, the way we consider this is there's a relationship that says that the flow stress of the material is going to be related to some value which is inherent of the particular structure that we're working with, and will add the contribution to that parameter, tau zero. And that is going to be related. The additional strength is going to be related to the dislocation content. So what I have here is sum constant k which is characteristic of the particular material. On to the square root of rho, which represents the density or the dislocation density. So as the dislocation density goes up, then what we find is an increase in the strength of the material. And people have done experimental observations looking at the electron through the electron microscope. And you can see increases in the dislocation density that correspond to this increase in strength. And here are some examples that show you these large arrays of dislocation networks that are associated with the increase in deformation that goes into the material. In addition to showing those networks of dislocations, I've also indicated the presence of dislocations that happened to be around particles or second phase. We'll see more of this coming up. But all around those particles, we have a number of dislocations that are interacting with those particles. Over here, we see another clumping or distribution of those particles of the second phase. So, when we begin to look at the microscope images, we see that they're really quite complex when we're dealing with large dislocation contents. But, certainly what we're seeing is, with this increase in dislocation density, we're seeing this corresponding increase in strength so this now represents the third mechanism by which we can improve the strength of an alloy. Consequently, as a result of adding these dislocations during the deformation process we have in fact shown the second mechanism by which we can strengthen material, and that is by increasing the dislocation density through the deformation process. Thank you.