Frontiers of Biomedical Engineering

BENG 100 - Lecture 18 - Biomechanics and Orthopedics

Chapter 1. Introduction [00:00:00]

Professor Mark Saltzman: So, this week we're going to talk about biomechanics and the reading is Chapter 10. I think on your syllabus it says Chapter 10 and 13 but we're really only going to talk about Chapter 10 during class. Then, our section on Thursday will also involve two aspects of biomechanics, talking about materials properties, which is what I'm going to focus on in the lecture today, and thinking about some elementary gait analysis; gait, G-A-I-T, or how humans walk and run and the mechanics of that particular kind of motion.

This is like many of the other topics that we've covered in the course, a very big subject. Some of you, I have no doubt, are going to go onto study biomechanics in greater detail. I've picked a couple of things that I thought were interesting to sort of teach you sort of the elements of mechanics, how do I use physics to think about biological materials, and how organisms live in a physical world. Today we're going to talk about properties of biological materials, and in particular, two properties elasticity and viscosity. We'll talk about how most real biological materials, your skin, your brain, the gel that makes up the interior portion of your eye have both of those qualities. They're both viscous and elastic, and they exhibit something that we call viscoelasticity.

By the end of the lecture today you should a little bit about what viscoelasticity is and how materials that have that property behave, and most biological materials have that property. Then on Thursday we're going to talk about locomotion: how do animals move, in particular how do humans move. We'll talk about three kinds of motion. The most interesting one may be flying that humans don't do unaided but it's interesting to think about how flight is possible from a mechanical perspective. How do organisms release themselves from the gravitational field and learn how to move in flight? We'll talk about swimming, and we'll talk about running or walking.

First the mechanical properties of biological materials and these are the two questions that I want to focus on during the lecture today, and the two questions that you should be able to answer by the end. What is elasticity? How would I measure elasticity? How would I quantify elasticity? There might be other ways to think about that question, and what is viscoelasticity? How do viscoelastic materials differ from elastic materials? How does that make them different in biological--when they're part of a biological organism?

Chapter 2. An Experiment on Elasticity [00:03:17]

We're going to think about this simple experiment that's shown schematically on the top here in Panel A of this diagram. We assume we have some material, and this could be anything, but let's think about it in the simplest possible way. It's some uniform material, at least initially. Maybe it's steel, maybe its rubber, plastic; maybe it's some biological material like a muscle or some other kind of tissue, skin, an artery. Let's just think about rubber for the beginning, maybe even a rubber band. You hang the rubber band from some solid support that's unmovable like the ceiling here. So, we have some way of affixing this solid material to the ceiling, and we can measure some properties of the material, mechanical properties of the material. For example, it has a length and it has cross-sectional area.

When we do mechanical testing we're trying to understand how that material behaves when it's exposed to forces, when it's exposed to forces it encounters in the physical world. We want to set up an experiment where we can apply a force uniformly and measure the result of that force simply. The experiment I show here is just to hang a weight on that material that's suspended from the ceiling, that rubber band, for example. I hang a weight on it, and so that weight applies some force to the material because gravity is going to pull it down with some force based on its mass. For most materials, certainly for rubber bands, if I put them on the--affix them to the ceiling that way and attach the weight, what would happen? They would lengthen; they'd get longer because they're exposed to that weight.

I've applied a force and I've experienced a deformation. I've applied a force and the material has deformed. I'm going to quantify that, and I can quantify that fairly easily because I chose a simple material that had a constant cross-sectional area from the top to the bottom. I'm going to assume that that cross-sectional area doesn't change when it deforms, but I hang the weight on it and the length increases. The length is now L + ΔL. I can quantify the deformation here by defining a property called strain. The strain is just the change in length divided by the total length. Now, important what length we use here, and we're going to use the initial length as the length. So, the strain is the change in length divided by the length in the unstressed position, in the unstressed orientation, unstressed condition, that's the right word. Does that make sense?

This is a measure of how much it deforms and we're going to define the stress and stress is just force per unit area. You can imagine if I hang the same weight on this material, if it's as material that has a large cross-sectional area, it's very thick, or it's a material that's very thin, then its deformation is going to be different because that force is going to be distributed over either a big area or a small area. So, the stress is just equal to the force per area, and we're going to assume that we arranged it in some way that it's uniform over the area. So the experiment makes sense? Hang it from the ceiling, suspended, one end is held rigid, I put a weight on it, I watch it deform, I measure the deformation that occurs as a result of applying that stress.

Now, to analyze how this material behaves under different stress conditions. So, I might want to know how does the muscle of my thigh behave when it experiences different kinds of deformations. What stress builds up when I stretch it out, for example, because I'm interested in the mechanics of running. What I want to know is, "What is the deformation that occurs in this particular muscle when I've applied different forces, or different stresses to it?" The way that I would measure that is by just applying different forces. Start with a small force, add a larger force, add a larger force, and then measure the deformation at each of those conditions. If I did that I could plot, on this diagram here, stress versus strain. Now, the strain, remember, just ΔL/L, the stress just the force divided by the area. I would get different points on this curve and each one would represent a different experiment where I added a different force and eventually I could draw a line through it.

Now, think back again to the first example I said, this simple example, just a rubber band. If you did this experiment with a rubber band, you hung different forces on it, what would you see? As you put more weight on, the rubber band would stretch more and more, and more so the strain would go up. In fact, up to a certain amount of force, if I measured the strain that results from forces, I would get a straight line. I could plot stress versus strain and I would get a straight line. That--a material that behaves that way is called an elastic material, that's an elastic material. A perfect elastic material, I could apply those forces, it would stretch out, and if I took the forces away it would go right back up again. At the end of the experiment of adding more weight, then taking the weight off, the material would be exactly the same as how it started. That's an elastic material that's perfectly reversible, a perfect elastic material.

Now, I could tell you everything you need to know about the elastic properties of that material by giving you one number, the slope of this line. Because once you knew the slope of this line, you could draw the curve that represents all of its stress strain behavior. Instead of someone else having to do this experiment again, you could just tell them, "Oh for those rubber bands I measured the elasticity and the elasticity or the slope of this line is some number E." This letter "E" is called the elastic modulus or the Young's modulus and it's a characteristic of the material. For this material in this particular geometry, that's the elastic modulus. Someone who knew that would know how to predict how that material behaves, at least under the conditions in which you did the experiment. Does that make sense?

Now, if you've studied physics in the past, I assume most of you have in high school, then you've thought about perfect elastic materials. What did you call a perfect elastic material in physics? Anybody remember studying a perfect elastic material in physics? A spring, that's exactly what a spring is. When you talked about springs, you talked about a spring constant, which was how to relate the force to the elongation. This is exactly the same thing but it's sort of described in a different way because now you're talking about a real material that has cross-sectional area, not an imaginary spring. When you thought about springs and spring constants, you were thinking about this exact same physical process. We're just describing it in a different way now. In fact, by the end of the lecture we'll come back to thinking about springs as a way of idealizing or describing materials that have complex properties.

The spring, the imaginary object that you talked about in physics is just an idealization of real materials and how they really behave when forces are applied to them. If you continue to add force, then at some point the material's not going to behave perfectly elastically anymore. You'll see that because it's not a straight line anymore, it's not linear. It still might be elastic but it's not linearly elastic so you can't describe it by just the slope anymore. At some point, for any material, you could enough weight that it's going to fail. You could define the stress at failure, when the material physically breaks, or the strain at failure when it physically doesn't function as a material anymore; for the rubber band you add so much force that it breaks.

There's some weird behavior that starts to happen when it's in that mode of starting to fail, in that some of the assumptions that we made don't hold any longer. For example, the assumption that the cross-sectional area stays the same. If you stretch out a rubber band, if you stretch it so, so, so much the cross-sectional area is going to start to cave in at one point, not along the whole rubber band but at some point. So, it doesn't even behave in the way that stress continuously goes up with strain anymore, and that's because of the way we've defined it. The simple experiment makes sense.

Let's imagine, then, that we're doing this experiment where we've got the material attached to the ceiling. It's a rubber band, still, and we're adding weight to it. One way to think about that is that we're going the point zero, the totally unstressed material, and we're adding more weight. We're going up towards Point A here, and so we're moving towards Point A by adding more weight to the material and it's stretching out. As we're adding more weight, we're applying more stress; we're experiencing or observing more strain. If you go up to Point A, I can come back down from Point A by taking the weights off and the material will follow exactly the same path back along the same stress strain curve. I could stretch the material, I could return the material to its original state, I could stretch it, I could return it to its original state. That's a reversible elastic material. At the beginning it has one property, you stretch it out, you let it come back and it's the same as it was at the beginning. That's a perfectly elastic material.

If you overstretch a material, that is, you go instead of Point B you add so much weight--instead of Point A you add so much weight that you go up to Point B. When you take the weight off it might now follow the same stress strain behavior coming back down. In this case, it follows this different curve, the dotted line here. Now, what does that mean? What would you really have observed when you did this experiment that ended up being plotted this way? I was adding weight, I was stretching out the material, and I stretched it so much that I actually changed the material. It can't go back to its original shape anymore, but when I take the weight off, it's permanently deformed. When I take the weight off it follows this line back and even at zero stress it still has some deformation, meaning it's longer than it was when I started. You've had this experience with rubber bands or balloons. If you play with it enough, you stretch it, you stretch it, you really stretch it and now it doesn't go back to its original shape anymore, and if you look at the rubber band it physically looks different. There might be a part that's narrower or the texture of it looks different because it's stretched in one direction.

You physically deformed the material beyond its elastic limit, and you've created what's called a plastic deformation. You've changed the material so it's not the same anymore. That's going beyond the elastic limit, which in this case might be A, and you've created not just an elastic deformation but a plastic deformation. That material's now forever changed. Maybe there some ways that you can get it back, but probably not. It's like--does anybody have a Slinky? That's a 60's toy, you didn't play with Slinky's but maybe you had. They used to wire and you could cut yourself on them and now they're plastic but they're these--I need a nod that somebody else besides me knows what a Slinky is. You're supposed to be able to make it walk down the stairs but it doesn't ever really work. If you take a Slinky you can stretch it out and it'll go back, stretch it out and it goes back, but what if you have your brother or your sister grab it and you pull it all the across the room. The material gets physically deformed, it doesn't go back any longer, that's a plastic deformation of a Slinky. Phone cords the same way, so you have a lot of experience with this in your real life. You've used a phone that has a wire on it, right? Coil--

If we're just talking about elastic materials, that means materials that have the property of elasticity and they haven't been deformed beyond their elastic limit, then I can stretch them, they'll come back, stretch them, come back, and I can do that as many times as I want and I'll always get the same material. You could classify materials that are elastic into different properties. Now, we're not just thinking about rubber bands but we're thinking about every material. Most materials have some elasticity and over a certain range of stresses they will deform elastically. Steel deforms elastically, you don't think about it doing that. You've got to apply really high stresses in order for it to deform but it does deform elastically. Wood does under some conditions, bones do; all materials do under at least some range of stress conditions.

Some are going to be what's called more compliant, that is they're--at a given stress, at a given force application they deform more. That's a compliant material, one that's very stretchy or very elastic. Or stiff, one that doesn't deform very much when a stress is applied, that's a stiffer material, a less compliant material. Or brittle, that is deforms even less and will break rather than stretch very much. You could define different classes of materials based on this property of elasticity.

Chapter 3. Viscosity [00:18:21]

That's one part of what I wanted you to understand today, 'What's an elastic material, how would I characterize it, and what does it mean to be an elastic material?' The second thing I want to think about is viscosity. Viscosity is a property that we associate with fluids. We've talked about viscosity, we talked about the viscosity of blood flowing through arteries, for example, and how viscosity contributed to the resistance of a vessel. It's in that relationship between pressure drop and flow. Well, let's think about another experiment where we have a fluid and we want to measure its viscosity. That is, we want to measure what's its resistance to flow of the fluid?

A way to do that experiment, and it turns out that there's a lot more practical ways than this, but here's one way to do the experiment is to take a table like this one, a solid surface that's not going to move and spread the liquid all over the surface. Now, let's forget for a minute about the fact that this table is only so wide and so fluid's going to flow off the edges, and let's assume that I could put a layer of fluid on here and it would stay. Then, I could put another desktop on top of it, so what I've created is a space between two desktops, one of which is fixed to the floor so it won't move and the other which is floating on top of a layer of fluid. Does this make sense?

So, now you could imagine if you could that, if I had a layer of fluid here and another desktop sitting on top, I could apply a force this way parallel to the surface and the thing would move easily. If it was water it would move very easily, if it was alcohol the fluid, it would move very easily. If it was molasses it would move but I'd have to apply more force in order to move the desktop on the surface, but I could do it. What you're really doing in that experiment I just described is the same thing as the experiment we did when we hung the rubber band from the top of the ceiling here, is that I've applied a force and I've measured a deformation. Now, the force and the deformation are different in this case because now I'm applying a force not in tension, before I was talking about only applying forces in tension to create a tension force in the material. Now, I'm applying a force in shear. I'm trying to move one plate relative to another plate and I'm trying to move them across each other so that's a shear force.

The deformation is not a deformation like it increases in length, the deformation is that the wall on top moves and it moves continuously. Imagine if I start to apply--I have this setup here, the desktop is on top of the layer fluid, on top of the desktop below, and I somehow apply a constant force to that top plate, that top desktop, it's going to continue to move as long as I apply a constant force it'll move, it'll move, it'll move, eventually I'd fall off the desk here. What's created is not a strain, what's created is a velocity. The top plate is going to move, and if I apply a constant force, a constant shearing force to the top plate I will measure a velocity. Does that make sense?

What's really happening in the fluid here is that within this fluid, when I apply the force on the top plate, the molecules of fluid that are right next to the top--let me call it a plate now instead of a desktop because that's a shorter word, that molecules of fluid that are right next to this top plate, when I start to move it, they move along with the top plate. Those molecules of fluid experience some friction with the top plate, and when I move the top plate they move along with. The molecules of fluid that are just below that also want to move along with it, because there's some friction between the molecules inside the liquid. There's some friction but it's not perfect friction, it slips. Viscosity is a measure of how slippy the fluid is; different layers of fluid, how easily they slide by one another when they're sheared.

Water, because it has a low viscosity, compared to molasses, those layers of water are going to move easily over one another, whereas in molasses, there's more friction between the layers and so it's harder to move. In either case, what will happen is that when I apply the force, the packets of fluid up near the top are going to move along with the top plate, the ones a little below it are going to move a little slower, the ones below it are going to move a little slower than that, slower than that, slower than that, slower than that until the ones at the very bottom don't move at all because they're experiencing friction from the bottom plate. If I could look microscopically at what happens when I shear this top plate over the bottom plate within the layer of fluid between, I've created a velocity gradient where the fluid is moving very fast near the top plate, slower in the middle, and not at all at the surface of the bottom plate. By doing this I can measure the force that's required in order to establish that velocity gradient.

I'm going to make the same kind of plot here that I made in stress versus strain for an elastic material, but I'm going to plot different things. Instead of stress--tensile stress I'm going to plot shear stress, and that is the amount of force that I've applied on this wall in order to get it to move divided by the cross-sectional area of the wall. It's the amount of force I had to apply in order to set this thing in constant motion divided by its cross-sectional area. What I'm going to plot down here is the rate of deformation. Because there is no limit to the deformation here, as long as I apply the force it's going to continue to slide. I can't measure the strain because it will continuously deform, there's no end to it. What I can measure, instead, is the maximum velocity divided by the thickness, the gap between the two walls, or the height of the fluid. This velocity divided by height is a measure of the rate of deformation. Does that make sense?

Kind of the same thing; I did an experiment set up a different way, applied a force, measured a deformation. In this case, shear stress and rate of deformation. What I'd find here is that over a certain length--over a certain range of forces, I would get a straight line. The slope of that line is this property called the viscosity. For fluids that have a high viscosity like molasses or honey, I have to apply a lot of force in order to achieve a certain velocity. For fluids that are less viscous, like water, I don't need to apply so much force in order to change that same velocity, in order to achieve that same velocity. Does this make sense?

Here's another kind of material, and there are biological materials that behave this way, their mechanics are like this; blood is one of them. Blood is a viscous material. If I put blood between this here and I did this experiment, I would see roughly this behavior. As I increase the force I increase the velocity, and it's linear and I could measure the viscosity. Now, it turns out that blood is a little bit different, in that it turns out that unlike water if I apply even the smallest force the plate's going to move when it's water between here. If it was blood, it turns out that for small forces it doesn't move at all, that I have to apply a certain amount of force in order to get it to move.

Why do you think it would be different for blood than for water? What can you imagine that's different for blood that creates this situation that at very small forces it doesn't move at all, but it has some yield force or yield stress that one has to get over in order to start it in motion. What's different about blood than water? It's thicker than water, you know that from the Bible, isn't that right? Blood is thicker than water, I don't even know. Justin?

Student: Large molecules in red blood cells.

Professor Mark Saltzman: It has large molecules and red blood cells, and has red blood cells at a fairly high volume fraction, about 50% is blood cells. It turns out that to get those blood cells rolling, to get them moving out of each other's way requires a certain amount of force. In fact, when blood is stagnant, even if it's anti-coagulated, the blood cells tend to form structures where they align with one another, and so it's hard to get that moving. For most of its--for most of the shear stresses that are biologically important, blood behaves as a viscous liquid, it's about three times as viscous as water. So that's viscosity, so you know about elasticity and viscosity.

Chapter 4. Deformation and Viscoelasticity [00:28:47]

Let's go back now to the material I talked about at the beginning and let's assume now that it's not a perfect elastic material but it's a material that is different than that. Let's do the experiment in a different way or look at it in a different way, where I'm still hanging the material from the ceiling and I'm still applying a force to it, and I'm asking how it deforms. Now instead of just looking at its total deformation I want to look at how the deformation changes over time. Time is an important variable in most biological processes. You care not only about how much deformation I get when I apply a force, but you care about how quickly I achieve that deformation, and how quickly materials can physically move when forces are applied.

If this is a perfect elastic material like a rubber band or like the springs you learned about in physics, when I apply a force here, it instantaneously changes its length. Deformations are instantaneous; they take no time at all for a perfect elastic material. How much would the deformation occur? Well it would depend on the elasticity, but as soon as I added that force it's going to stretch out--bang--to that length and it's going to stay there. You know real materials don't behave like this but our prefect elastic material does. If the material is viscoelastic, meaning it has properties of both elasticity and viscosity like a muscle, say, like the fluid that's inside your eye. If I apply a force it's going to exhibit two different behaviors. First it's going to elongate instantaneously just like the perfect elastic material would and then it's going to continue to elongate slowly. It's going to continue to elongate. I've shown that here, if I look at the length versus time it's going to instantaneously elongate to some extent and then it's going to slowly keep deforming.

Now, that behavior, where I apply a force and it goes initially out to here looks like elastic material; the behavior where it continually starts--continues to stretch out over time and gets longer, and longer, and longer that looks like a viscous material. That's how a viscous material would behave, I apply a force and it continuously deforms with some rate. This material is behaving like an elastic solid and like a viscous liquid, both. It's what we call a viscoelastic material, and most biological materials have that property. They have some viscosity that is they deform continuously under stress. They have some elasticity, they deform elastically under stress.

Another way to do this experiment would be not to hang a weight and then measure what deformation occurs, but to instead do the experiment as is shown in this diagram here, which is stretch the material out suddenly and then measure the force that's required to hold this apart. For an elastic material, you'd get the same result. If you apply a strain, that is, I just pull this up to some length, it's going to take some force to do that. All the force gets applied instantaneously and then after that I just maintain the force and I hold the material there. For a viscoelastic material it behaves differently, in that its initial response will be like an elastic solid. It will--you stretch it out to its strain and it's going to take the same force to open it up, to stretch it out, to deform it. Then, the materials going to relax and start to flow which would--what you would measure is that the force required to hold it at this deformation starts to drop over time as the material itself rearranges and deforms in order to accommodate that strain that you've put in it. Does that make sense?

Two ways to do the experiment, two different ways of looking at viscoelasticity. One is you apply a force and now you can see continuous deformation, the other is that you apply a deformation and you see a change in force required to hold that deformation over time. It turns out that not all viscoelastic materials exhibit the same behavior but there's a range of viscoelastic properties. Some viscoelastic materials behave more like elastic solids, some behave more like viscous liquids, and some behave somewhere in between. Physicists have figured out ways, models that they can use to explain how viscous materials behave. You could imagine that this would be important in biomedical engineering. For example, if I want to understand how the leg works, then an important part of how the leg works as a machine is understanding the properties of the leg muscles. How they deform under stress and how, when stresses are applied to them, how that changes their shape. That's essential for if a muscle is going to contract and apply a force to your leg, then you need to know how the strain or deformation of that muscle relates to its ability to create a force, to pick up the leg for example, or to move it.

I'd like some way to describe that mathematically. I'd like some way to describe the stress strain behavior of complicated biological materials like muscles. One way to do that is to build models based on elements that I understand. So, I understand the spring. I understand the spring and if it was a perfect--if the muscle was a perfect elastic material I could describe all of its behavior by its spring constant or its elastic modulus. I would know everything about its stress-strain behavior, but muscles don't behave like strings because they have some viscous properties.

What's a simple example of a viscous material? Well, I gave you this example of the plate sliding over the layer of fluid. Well, there's another model that physicist's use to describe viscous materials and maybe you've seen this, it's an element that you don't really encounter in life. You encounter things like springs, but you don't encounter things like dashpots very often. That's what this is called; this is a dashpot. A dashpot is an idealized physical model of a viscous liquid. One way to imagine it is that it's a cylinder, the black cylinder here, inside of a hollow cylinder. So, it's a cylinder inside of a hollow cylinder and there's some friction between the outside wall of the black cylinder and the inside wall of the hollow cylinder. There's some well-behaved, well-known friction between the inside of the hollow cylinder and the outside of the inner cylinder.

Now--then if I apply a force, apply a force down here in that direction to this dashpot what's going to happen? The inner cylinder is going to slide over the outer cylinder, slide through the outer cylinder. For a constant force, since this is a well-behaved--the friction between here is well-known and well-behaved, it's going to just slide with a uniform velocity the inner cylinder over the outer cylinder. Does that make sense? So, this dashpot is an idealized physical model for how a viscous liquid behaves. I apply a force and it continuously deforms. Now, it's better as a thought experiment than a real experiment because you can imagine making things that behave like real springs. Real dashpots would be hard to make. Why? Because eventually no matter how long your outer cylinder was, you'd eventually come out of it. A real dashpot, if it's going to deform for a long time, would have to be infinitely long as well, so it's better as a thought experiment than a real thing to build but you get the thought experiment, right?

I have a spring here, how will it behave when I force on it? It's going to deform instantaneously to a certain length. The length will depend on the force. The dashpot, I apply a force to it, how will it behave? It's going to slowly deform over time, continuously with a constant rate but it will never stop deforming. What would happen if I put a spring and a dashpot together like this? That is, I have a solid wall here, I first attach a spring, and I attach a dashpot to the spring, and now I apply a force. Well, this spring-dashpot combination is going to have elastic properties, it's going to have viscous properties, but it's going to behave in a certain way.

When I apply the force, the force is going to be applied over the whole structure at once. The spring is going to fully experience that force and it's going to stretch out. Then, the dashpot is also going to experience that force and it's slowly going to deform. What I would observe experimentally, if I could build this spring dashpot combination, is I'd see an initial deformation and then continuous deformation after that. This is an example of one kind of viscoelastic behavior. If I took my muscle, and I did measurements on it, and I found experimentally that it behaved like this, then I would say, 'Oh I can describe how the muscle behaves by this simple thought experiment, this simple mathematical construct of a spring and a dashpot connected together.' Does that make sense?

Well that's not the only way that you could create a viscous--that you could create an imaginary viscoelastic material. I could also attach the spring and the dashpot in series, that is, attach both the spring and the dashpot to the wall, have a little strut connecting them, and then apply the force that way. How would this material behave? It would behave differently because when the force is applied the spring, would like to stretch out but it can't. It can't because it's constrained by the dashpot which only deforms slowly. The spring would like to deform, would like to take all of the force and deform, but it can't because the dashpot physically takes time to move. It can only deform at some rate. In the initial period the dashpot is taking all of the force and what you would see initially is slow deformation. Now, while the dashpot would like to keep deforming forever, this thing can't because the spring has a limit to how much it can stretch. When you get up to the elastic limit of the spring, or the elasticity of the spring, the spring is going to be taking all of the force. The dashpot, then, will have no force on it, and it will stop deforming.

A viscoelastic material that behaved like a spring and a dashpot in series would have this behavior. It would not have any initial deformation. It would deform slowly at first and then it would reach some maximum deformation like an elastic solid, but it would be evolving over time. Does this make sense? Now, these are two simple models of materials that have viscous and elastic properties. I could measure real materials and I could see, 'Well, do they act more like this, do they act more like this?' I could use these thought constructs in order to describe those materials. I could learn about how the materials behave under unusual conditions that way, I could start to predict how they behave.

It turns out that most real viscoelastic materials don't obey either this model or this model, but you can build more complex ones. You could imagine having a spring and dashpot on this side and just a dashpot over here, or you could imagine having three elements in a series. You could imagine all sorts of different ways of putting viscous elements and elastic elements together to define an imaginary material. You could then predict how that imaginary material is going to behave and you could see how close it is to real materials.

Chapter 5. Conclusion [00:42:13]

What I want you to remember from this is a couple of different things. One, we talked about elasticity at the beginning, how do purely elastic materials behave, you know that. How do purely viscous materials behave? You understand that. Real materials, real biological materials have--are viscoelastic. They have both elastic and viscous property, but that doesn't mean just one thing. That can mean a whole range of behaviors. It's possible in some cases to define that whole range of behaviors by building simple thought experiments or simple models like these spring-dashpot combinations. In general, the viscoelastic--real viscoelastic materials we see are more complex than any of the--we have more complex behaviors than any of the idealized models we put together, although sometimes they come very close. Questions?

If you looked for the homework assignment over the weekend, and then you look at it today, or yesterday, you will have noticed that it changed. It became simpler. I decided that first I had a homework assignment that had two parts. The first part of it was to write an introductory paragraph for your paper and an outline for the rest and turn it in. Then, there was some problems related to the material for this week from Chapter 10. What I decided to do instead was ask you just to focus on your papers this week, and just turn in that introductory paragraph and the outline. That's all the homework for this week, and then we'll put off the homework assignments on this topic, and what we'll talk about Thursday that won't be due until next Thursday.

I'd like for you to really focus on your research papers this week. We'll come back and we'll do the homework set next week. Then, the week after that there won't be any homework except just to finish your research papers and that will be the week that it's due. I just wanted you to keep that in mind, because I know that timing and balancing all of your responsibilities to different courses is critical at this time of the year. Questions about that or about what I talked about at the beginning: instructions for the paper, peer-reviewing? Good, see you on Thursday.

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