In this lesson we're going to introduce the concept of a critical resolved shear stress. And a critical resolved shear stress is a particular property of the material that we will describe in this lesson. Let's go back to the figure that appeared earlier In this module, where I'm looking at a cylinder. Uniform cross section and the material is being stressed along the Z-axis. The cylinder has an original cross sectional area A1. Now if I happen to be interested in a particular plane, as the one I have shown on the diagram. What I can do is I can look at the various stresses that are associated with this plane as a result of the applied stress that I have made remotely, in other words along the Z-axis. So, I can determine what the normal to this is. I can determine what the shear in this plane is, and so I have all the stresses then completely established. Now, the other thing that I need to recognize is that the cross-sectional area that's illustrated in this diagram is different then the actual cross-sectional area A1 that I began with. And you would expect that as you change the angle of the plane that you're interested in looking at. You will find that that cross-sectional area is going to change. So, when we look at the cross-sectional area, what we find is that as we rotate through various degrees of phi, in terms of our rotation axis. We are going to get different cross-sectional areas. It's going to become infinite at one value. And ultimately, it's going to become a maximum when we look at the plane when it's perpendicular to the Z-axis. Then, when we look at the shear stress on the plane, we see that the shear stress is going to follow the cosine function. And what we're going to see is that it's going to vary as a function of the angle phi. And we're going to see in particular, that that shear stress is going to be a maximum when the value of the cosine or phi is 45 degrees. So here is our function which is going to be shear stress that we're trying to calculate. And when we divide by the original cross-sectional area we can turn this into the stress. And what we see is that the cosine and the sine are both involved in the calculation of what the critical resolved shear stress is. So if we take those two functions and plot them as a function of the angle theta. What we see is, at a maximum value, we have the ET 45 degrees. So we'll have then, if we look in a material that's being deformed in this cylinder, the plane of maximum shear will occur at 45 degrees to the stress axis. Now, this is ultimately important to us when we begin to talk about what happens in a material that has a slip plane. And a particular value for the direction in which deformation is going to occur by dislocation motion. So we're going to define something called the critical resolved shear stress. And it's resolved in the slip plane in a single crystal. So, we're going to look at the orientation of that particular slip plane. With respect to the axis on which the stress is being applied, or the force is being applied. And what we're going to determine is what is the stress that is necessary, that's the remote stress. To initiate slip in a single crystal, and to determine what that value is for a specific material. Then since the threshold to initiate the plastic deformation is going to be a critical value we turn our normal terminology here Is to call this the critical resolved shear stress. So we're looking at the shear stress in the plane that will wind up beginning to initiate slip as a result of having a remote stress signal along the Z-axis. And when we look at the critical resolved shear stress calculations. What we see is tau critical resolved shear stress is going to be equal to sigma. Sigma is along this stress axis and cosine phi and cosine theta are going to have some meanings that we are going to describe in just a moment. Now this relationship is referred to as Schmidt's law. Now, when we look at sigma, sigma is the magnitude of the applied stress. The angle five represents the angle between the slip plane normal and the direction of the applied force. So consequently, if we have slip on a particular plane, and we'll see for example that in single crystals of face center cubic materials, the slip is going to be on 1-1-1 planes. And we know that the normal to the 1-1-1 plane is the 1-1-1 direction. So as a consequence of that, we know we can determine the angle five by simply taking the dot product of those two vectors. Then we look at theta, and theta then represents the angle between the slip direction and the direction again of the applied stress. So here is our picture and now what will do is will take a look at the fact that were describing now a single crystal. And we have a force which ultimately we convert to a tensile stress by simply dividing through by the cross-sectional area A. And we're going to then define the slip plane. And then once we've defined the slip plane, we define the slip plane normal. So if you're given a slip plane, and for example if I were to give you the slip plane 1-1-1 in a face center cubic material. What you would immediately know is the slip plane normal happens to be the 1-1-1. Now the force, we can align that force to a crystallographic axis inside of the crystal. So now we can take the relationship between the direction of the force and the normal. In order for us then to calculate the angle phi between those two. Now we can look at the slip direction. When we look at the slip direction, we're now looking at the angle theta, and we're interested in the orientation of the tensile or the force axis, and the slip direction. Which in crystals such as face center cubic materials, it's in the close packed directions, and it will be vectors of the type A0 over 2, onto the 1-1-0 type direction. So, we start out with our expression and we take the force along the axis, divided by the original cross-sectional area. And when we do that, what we're going to do is we're going to calculate then an intensive variable. We start out with force which extensive and now we get the intensive variable stress. And it is now the product of cosine theta and cosine phi. And then we know that the relationship for the critical resolved shear stress is just then simply sigma. That's again, the intensive variable and the applied stress on the material. And cosine theta and cosine phi have their definitions based upon the orientation of the slip plane and the orientation of the slip direction. So, what we've then been able to do is to define something called the critical resolved shear stress. And it is a particular property of the microstructure, and of the material itself. And this ultimately will be demonstrated in a later lesson. Thank you.