Up until this point, we've been looking at deformation that occurs in single crystals and how this location behavior operates with respect to single crystals. What we can do is, before we leave the topic, one of the things that we can do is to see how the relationship between critical resolved shear stress and what we see in polycrystalline materials can be related back to what we see in single crystal materials. For metal system, when we look at a polycrystalline material what we recognize is that we have the potential of several slip systems that are favorably oriented for slip. When I say favorably oriented what I mean is they are close to 45 degrees, which represents the plane of maximum shear. So, if we look at the one slip system that is most favorably oriented, it has the largest resolved shear stress acting on that plane. So, here is our little picture over here and when we are at 45 degrees we are then on a plane of maximum shear. So, what we're going to assume is because we have many grains and they are randomly oriented, we are going to look at the one that is the closest, or appears to be at 45 degrees, with respect to the stress axis. And now we'll go back and we'll take a look at the critical resolved shear stress equation. And in this particular case what we're looking for is the relationship between the maximum critical resolved shear stress and the value of cosine theta and cosine phi when it's at a maximum which of course is at 45 degrees. So, yielding will occur on that most favorably oriented system first. And what that means is, that the stress that's necessary for slip through a current system then is given by the expression that we have on the slide. Where we are dividing the critical result shear stress by that maximum angular relationship that is in the denominator. When we put in that value, we know that theta and phi when they're both equal to 45 degrees, that's going to give us a maximum value. We make that substitution into the expression that we have up there, on the slide and we see that the yield strength of a polycrystalline material is approximately equal to twice the value of the critical resolve shear stress. So consequently, some of the data that's generated from single crystals can be used to evaluate what is occurring with respect to polycrystalline materials. Thank you.