[SOUND] Hi, this is Rick Neu. In this lesson, we're going to focus on the instrumented indentation test methodology. The learning outcome is to explain the principle, methods and analysis of instrumented indentation tests. So let's look at a conventional hardness test versus an instrumented indentation test. What is so different about instrumented tests versus what you might think of as just a hardness test that is well-known? All right, in instrumented indentation tests, the key difference is we measure many more things. We measure the complete load, displacement and use all that information. In addition, we'll see that we actually look at the compliance of the interface instantaneously as we load and unload, and we'll see how that's done in a later slide here. In a conventional hardness test, and I'm going to refer to something like a Rockwell hardness test, a very standard hardness test. In that sort of test, what you do is you apply a load, a minimum load, that's where I put a little dot, let's say you apply that load. Then you apply a heavy load and that's where this dot is, where the maximum is and then you unload it back to that minimum load, and then you look at the displacement that occurred between the start and end at the minimum load. And that displacement correlates to the hardness. And all the other data is thrown away, you don't use it. So, in other words conventional hardness tests ignore lots of useful material information that could tell us something about the stress strain response in material. And also most conventional hardness tests are designed to go well beyond yield. So, they're really a measure of the flow strength, something closer to the tensile strength of the material. In fact, usually you correlate hardness to tensile strength. Now one other thing that's important to consider is the shape of the tip of your indenter. And so there's two types. You could have a sharp indenter or a not so sharp indenter and the most logical one would be a spherical indenter. And when you indent with a sharp indenter, almost as soon as you touch your material, you start to plastically deform what you're touching. That's assuming you're indenter's much harder than what you're indenting. And so I'd noted is this orange region, plastic deformation. And then as you indent further, that plastic field grows. Well if it's a spherical indenter, you actually get some elastic deformation that occurs first. And in fact where yielding occurs first, where you meet the von Mises or Tresca yield criterion is some point below the surface along the center line. And then as you deform more, then the plastic deformation region grows. And so when you do a hardness test, you indent it enough that you reach that third stage shown here. But the thing about using a spherical indenter is it can tell you something about the elastic properties too, because we know the relationships between the geometries of a sphere in contact with another flat body. And that can tell us something about, it can allow us to get the whole stress-strain curve, and we'll see how that's done. So when you do a test like this, you measure the load and displacement of the indenter. And you typically will get a curve that might look like this. You start at 1, at the point 2, when you're just getting yielding in that spherical indenter, you're still not that far, it's still a rather low load. And then it becomes non-linear and goes up to 3. By the time you're at 3, you have a fairly large area of plastic deformation and then you unload. And the slope of that unload, as we'll see, can be related to the effective modulus of the indenter and that substrate material. And it turns out we can take that information, and create what's known as an indentation stress-strain curve. So it's not exactly the same as an uniaxial stress-strain curve but it has the same elements, and as we'll see, can be correlated to a uniaxial stress-strain curve. So we have to call it something different, either effective stress, effective strain, or perhaps better just to say indentation stress, indentation strain, and we'll use those terms later on. By using a spherical indenter we can capture the elastic plastic transition during the initial loading so we can obtain that elastic response. With a sharp indenter, you really can't obtain that elastic response because it plastically deforms from almost the instant you touch your sample you're probing. And so we normalize the load and displacement into some effective stress and strain, and we'll look at the mechanics that allows us to do that. And spherical tips are the logical choice here. So we're going to focus on spherical tips here. Now one other thing I said, with instrumented indentation testing, we also measure the stiffness continuously as we load up. And this is done by super imposing a sinusoidal waveform on the loading part of your load displacement curve. And the slope of that tells us something about the stiffness of the contact and so we have a continuous stiffness measurement by using this sort of methodology. This can also be done manually by periodically unloading during a loading of a curve, if your equipment doesn't have this capability, and get the same sort of data. Now this gets to the meat of things, in the indentation stress-strain analysis using spherical indenters. So everything's based on Hertz theory. Hertz theory describes the elastic, frictionless contact between two isotropic materials with quadratic surfaces. Basically a special case is a sphere on a flat contact. Now when you deform the bodies, what happens is you deform the body you're indenting. And the maximum deformation is shown as the solid circular curve on this plot. And then you remove the load and it springs back and that's the elastic spring back. And the dashed curve shows the position after you've remove the load, so the two bodies are just in contact right along one point at that point, and that distance is this h sub e. So it's essentially the displacement between the two bodies. And we'll see that come up in the equations. And so first of all, there's two things you need to know. The geometries of the bodies, so the geometries of the bodies are related to the effective radii. And the thing is, as you deform it, you change the radius of the sample you're deforming too, so you have to take that into account. And so that can be taken into account in your two radii. The other thing we have to take into account is the effective modulus. The modulus, because it's a multi-axial state of stress, depends on both Poisson's ratio and Young's modulus of both the indenter and the material being indented. Presumably we know the properties of the indenter, it's usually tungsten, carbide or diamond in most cases. And we make some assumptions about the Poisson's ratio. And if you do that, you can get the modulus. Now, the key relations to get that indention stress-strain curve. From Hertz theory, there's two key relationships, the size of the contact. So the size of the contacts a here is related to the load that's applied, that's p, this effective radii of the two bodies and the effective modulus by this equation. The other thing you get from Hertz theory is that displacement is related to the size of the contact squared divided by the effective radii. And this just comes from the mechanics of the analysis of theory of elasticity. Now if you take that first equation of Hertz and rearrange it, you can put it in the form as shown here. And this form we're putting it into a stress-strain like relationship, looks kind of like Hooke's law, stress equals Young's modulus times strain. Now we're going to do one other thing, we're going to take that displacement relation and replace the a over R effective with h, the displacement, over a, and there's a good reason for doing that. If you look at the schematic on the left side, you see that the displacement is basically the h, and the reference length, the gauge length Is essentially 3 pi a over 4. And that's shown in that hash region. So it's basically a measure of strain. So it's a clever way of measuring strain, a change in length over some gauge length. And so this allows us to come up with some effective indentation strain measure. And then indentation stress is just load over the projected area, standard way to measure stress. And then they're related by effective modulus, so then we come up with our indentation stress-strain relation. And what we do is compute this stress, this strain and plot it for every indent and we can get an effective stress-strain. Now there's one other thing that we have to consider is how do you determine the size of the contact? What's obviously related to the profiles of the bodies. So you can get that from Hertz theory too. Actually just taking those equations I had in the previous slide and rearranging them again and then taking the derivative of it where load and displacement are changing and you get this equation at the bottom that says the size of the contact is related to 1 over 2 times the effective radii, times the slope, or stiffness of the contact. Now, in a traditional indentation test, you might measure that stiffness just upon unloading after you reach the peak load. But if you're using continuous stiffness measurement, you actually know that stiffness at all points during the loading. So here's just an example of an analysis and data you might generate from doing this. So the initial loading segment where it's elastic is actually just, it's just the beginning of that load displacement curve. But since we know it follows Hertz theory, we can do a regression analysis knowing what functional form that should follow and use that to establish the elastic response. Once you've done that, then you can compute the indentation stress and strain once you go into the plastic regime and plot on the bottom right which is our indentation stress-strain curve. And see, from this curve we can get a modulus value. We can get the yield strength when it becomes non-linear. And you can even get some elements of the hardening behavior. So quite a bit of information from just a simple indentation test. So in this lesson we've explained the principles, methods, and analysis for instrumented indentation testing. So what's next? We're going to look at measurements using high-throughput indentation methods. Thank you. [SOUND]
