0:04

In this lesson we're going to introduce the concept of a critical resolved

Â shear stress.

Â And a critical resolved shear stress is a particular property of the material

Â that we will describe in this lesson.

Â Let's go back to the figure that appeared earlier In this module,

Â where I'm looking at a cylinder.

Â Uniform cross section and the material is being stressed along the Z-axis.

Â The cylinder has an original cross sectional area A1.

Â Now if I happen to be interested in a particular plane,

Â as the one I have shown on the diagram.

Â What I can do is I can look at the various stresses

Â that are associated with this plane as a result of the applied stress

Â that I have made remotely, in other words along the Z-axis.

Â So, I can determine what the normal to this is.

Â I can determine what the shear in this plane is, and so

Â I have all the stresses then completely established.

Â Now, the other thing that I need to recognize is that

Â the cross-sectional area that's illustrated in this diagram

Â is different then the actual cross-sectional area A1 that I began with.

Â And you would expect that as you change the angle

Â of the plane that you're interested in looking at.

Â You will find that that cross-sectional area is going to change.

Â So, when we look at the cross-sectional area, what we find is that as

Â we rotate through various degrees of phi, in terms of our rotation axis.

Â We are going to get different cross-sectional areas.

Â It's going to become infinite at one value.

Â And ultimately, it's going to become a maximum when we look at

Â the plane when it's perpendicular to the Z-axis.

Â Then, when we look at the shear stress on the plane,

Â we see that the shear stress is going to follow the cosine function.

Â And what we're going to see is that it's going to vary as a function of

Â the angle phi.

Â And we're going to see in particular, that that shear stress is going to be a maximum

Â when the value of the cosine or phi is 45 degrees.

Â 2:21

So here is our function which is going to be

Â shear stress that we're trying to calculate.

Â And when we divide by the original cross-sectional area

Â we can turn this into the stress.

Â And what we see is that the cosine and the sine are both involved in

Â the calculation of what the critical resolved shear stress is.

Â So if we take those two functions and plot them as a function of the angle theta.

Â What we see is, at a maximum value, we have the ET 45 degrees.

Â So we'll have then, if we look in a material that's being deformed in this

Â cylinder, the plane of maximum shear will occur at 45 degrees to the stress axis.

Â Now, this is ultimately important to us when we begin to talk about

Â what happens in a material that has a slip plane.

Â And a particular value for

Â the direction in which deformation is going to occur by dislocation motion.

Â 3:22

So we're going to define something called the critical resolved shear stress.

Â And it's resolved in the slip plane in a single crystal.

Â So, we're going to look at the orientation of that particular slip plane.

Â With respect to the axis on which the stress is being applied, or

Â the force is being applied.

Â 3:42

And what we're going to determine is what is the stress that is necessary,

Â that's the remote stress.

Â To initiate slip in a single crystal, and

Â to determine what that value is for a specific material.

Â 3:59

Then since the threshold to initiate the plastic deformation

Â is going to be a critical value we turn our

Â normal terminology here Is to call this the critical resolved shear stress.

Â So we're looking at the shear stress in the plane that will wind up beginning

Â to initiate slip as a result of having a remote stress signal along the Z-axis.

Â And when we look at the critical resolved shear stress calculations.

Â What we see is tau critical resolved shear stress is going to be equal to sigma.

Â Sigma is along this stress axis and cosine phi and cosine theta

Â 4:54

Now, when we look at sigma, sigma is the magnitude of the applied stress.

Â The angle five represents the angle between the slip plane normal and

Â the direction of the applied force.

Â So consequently, if we have slip on a particular plane, and we'll see for

Â example that in single crystals of face center cubic materials,

Â the slip is going to be on 1-1-1 planes.

Â And we know that the normal to the 1-1-1 plane is the 1-1-1 direction.

Â So as a consequence of that, we know we can determine the angle

Â five by simply taking the dot product of those two vectors.

Â Then we look at theta, and theta then represents the angle between the slip

Â direction and the direction again of the applied stress.

Â 6:04

And we're going to then define the slip plane.

Â And then once we've defined the slip plane, we define the slip plane normal.

Â So if you're given a slip plane, and for example if I were to give you the slip

Â plane 1-1-1 in a face center cubic material.

Â What you would immediately know is the slip plane normal happens to be the 1-1-1.

Â Now the force, we can

Â align that force to a crystallographic axis inside of the crystal.

Â So now we can take the relationship between the direction of the force and

Â the normal.

Â In order for us then to calculate the angle phi between those two.

Â 6:50

When we look at the slip direction, we're now looking at the angle theta, and

Â we're interested in the orientation of the tensile or

Â the force axis, and the slip direction.

Â Which in crystals such as face center cubic materials,

Â it's in the close packed directions, and it will be vectors of the type A0 over 2,

Â onto the 1-1-0 type direction.

Â 7:18

So, we start out with our expression and

Â we take the force along the axis, divided by the original cross-sectional area.

Â And when we do that,

Â what we're going to do is we're going to calculate then an intensive variable.

Â We start out with force which extensive and

Â now we get the intensive variable stress.

Â And it is now the product of cosine theta and cosine phi.

Â And then we know that the relationship for

Â the critical resolved shear stress is just then simply sigma.

Â That's again, the intensive variable and the applied stress on the material.

Â And cosine theta and cosine phi have their definitions based

Â upon the orientation of the slip plane and the orientation of the slip direction.

Â