In this lesson, we're actually going to calculate some information about the critical resolved shear stress or given a critical resolves shear stress, we're going to calculate something about the magnitude of the stress necessary to initiate slip. So, once again, we're going to continue working with the FCC crystals. And what we're looking at then is a problem where we have a metal single crystal, it has a particular orientation. The crystal is oriented with the axis 101 of the crystal parallel to the axis in which the material is being deformed. The other thing that we have from the problem is that slip is occurring on the 111 plane. And we also are given the fact that slip direction is a0 over 2 in the 1 bar 1 0 direction. So what we want to do now is we want to calculate w hat the amount of stress we need to apply remotely is to initiate slip if we know that the critical resolved shear stress for this particular FCC material is 0.34 megapascals. So, we have the slip plane, we have the slip direction, we have the orientation of the crystal with respect to the stress axis, we're given the critical resolve, shear stress. Now we want to know how much stress is going to be required to initiate slip in this material. So here's our box that has the slip plane, the slip direction, and of course we see that the direction and the plane are coincident. So the slip direction lies on that plane. And then what I've done is to describe the vectors and the normal to the plane. So first of all, the stress axis is along the 101 direction, and that's indicated in the figure to the left. Now when we look at the Slip plane normal because the slip plane was 111. The Slip plane normal is also 111. So now we have that. And then the next is the Burgers vector. The Burgers vector lies on the Slip plane and it is a0 over 2 onto the vector 1, 1 bar 0. So now we begin to have all the information necessary to make our calculations. We know that this is the equation that allows us to relate this orientation to the critical resolve shear stress. And then all we have to do is to rearrange the equation, because now what we're looking at is how much stress do I need to apply along the 101 direction in order to begin to see deformation occurring in this particular single crystal. So now what I've done is, I have reoriented it so that my cube lies inside of my cylinder. And this cylinder now is what I am deforming. So, my crystal is now in the shape of a cylinder, but you can see I have the embedded crystallography. Inside of the cylinder I have the tensile axis, I have the crystal, I have the fact that I have oriented the various directions and now I'm ready to make my calculations. So, here we begin and now what we're going to do is to calculate that stress. Now the first think we want to do is to calculate the angle of cosine theta. And the way we do that is we simply take the orientation of the Burgers vector along with the orientation of the stress axis. And as a result, in order to make that calculation, we're looking at the dot product of those two vectors along with the magnitude of one vector and the magnitude of the other vector. And when you do that you come out with a value for cosine theta is one half. Now what we're going to do is to look at the angle fey and the angle fey then represents the angle between the Slip plane normal and the structure axis which is the 101 direction. So once again we take a.b. And as a result of that we get now another number. And then we look at the value of the magnitude of the 111, and the magnitude of the 101. And what we will wind up with is value of 2 over the square root of 3, or the square root of 2 over the square root of 3, for the value of the cosine phi. We put those together and then ultimately what we wind up doing is calculating the stress. And that stress, sigma, is the amount of stress I need to apply remotely to my crystal in order to get plastic deformation. So these are the kinds of calculations that we will do and they all follow the same procedure. We need to know what the Slip plain is, the Slip plain normal. We need to know what the Slip directions are. And once know all of those, we can calculate those interaxial angles based upon what is the direction of the applied stress. Thank you.