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Originally published in 1984.
The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.
Product Details
| ISBN-13: | 9780691612171 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 07/14/2014 |
| Series: | Princeton Legacy Library , #870 |
| Pages: | 306 |
| Product dimensions: | 6.10(w) x 9.10(h) x 0.70(d) |
Read an Excerpt
The Mechanical Adaptations of Bones
By JOHN D. CURREY
PRINCETON UNIVERSITY PRESS
Copyright © 1984 Princeton University PressAll rights reserved.
ISBN: 978-0-691-08342-1
CHAPTER 1
The Mechanical Properties of Materials and the Structure of Bone
1.1 What Is Bone for?
Any biological material has an enormous number of mechanical properties that it is possible for scientists to investigate and that may also be tested by natural selection. Not all are likely to be of importance. I shall here discuss a question that will recur, by implication, frequently in this book: what are bones for? It is not worthwhile discussing at length the philosophical question of whether bone can be said to be designed for anything at all. As a convinced Darwinist, I believe that all living organisms are nicely designed for the conditions that their ancestors of the last few thousand generations have lived in. Organisms have evolved organs that help them to survive and to pass on their genes to the next generation. It is one of the jobs of biologists to find out what these functions are, and how the organs perform them. For a contrary view of the perfection of organisms see Gould and Lewontin (1979).
Often the function seems obvious, and discovering it seems trivial. However, this apparent triviality may be misleading. It would usually be possible, with the materials available, to design an organ that would perform the apparent function better. A heart would pump blood more smoothly, and with more reserves of power, if it were larger. Therefore, the fact that hearts are the size they are, and not larger, must imply to a Darwinist that there is some disadvantage in having a larger heart that would outweigh the hydraulic advantage. It may be, for instance, that the reserve of power is, in selective terms, not worth the metabolic cost of keeping a larger heart healthy.
We shall particularly attack this problem in chapter 4 when we discuss minimum mass analysis. However, it will be seen in that chapter that we have to assume certain functions, and then see whether bones seem to be performing them efficiently. What are these functions? Baldly, they are to be stiff enough, and not to break under either static or dynamic loading. There are some other functions, which we shall touch on later, but these are usually less important. It is perhaps natural to think that the strength of a bone, that is, the load it can bear before breaking, is its most important mechanical attribute. But it must surely be true that bones function mainly by not deforming appreciably under load. It is customary and correct to think of bones, especially limb bones, as acting as levers. The muscles, by their contraction and relaxation, alter the distance between fixed points on different bones, and so the bones are constrained to move. If the bones were floppy, they would not be constrained in the same way, and the movements of the muscles would be futile (figure 1.1). On the other hand, if the bone is stiff, but breaks, it becomes useless, and so the strength is of great but secondary importance.
The stiffness of a bone and its strength depend on two factors: the stiffness and strength of the bone material itself and also the build of the whole bone. By "build" I mean the amount of the bone material and how it is distributed in space. The question of the build of bones will be discussed in chapter 4. In this chapter we shall be concerned with the basis of the whole book: the mechanical properties of materials in general and the structure of bone itself. In the second chapter we can then talk about the mechanical properties of bone material.
1.2 Mechanical Properties of Materials
Because many readers may not be too clear about the mechanical matters to be discussed, I shall run through various basics now. Later, slightly more recondite mechanical concepts will be introduced. Only the minimum necessary for the understanding of this book will be given. Readers wishing to go further should read Gordon (1976) or Wainwright et al. (1982), and a good book on strength of materials.
1.2.1 Stress, Strain, and Their Relationship
First: stress and strain. Consider a bar acted on by a force tending to stretch it (figure 1.2). If its original length is L, it will undergo some change in length, ΔL. It will also get thinner, and its breadth B will decrease by an amount ΔB. The proportional changes in length, ΔL/L and ΔB/B, are called normal strains. They are often given the Greek letter ε. Note first that strains refer to changes in length in particular directions, and second that the strains tell us nothing, directly, about the forces causing them.
There is another way in which a material can distort: in such a way as to cause changes in angles between imaginary lines in the material (figure 1.3). A cubical block of material, which is part of a larger block, is acted on by a shearing force S. (For reasons of equilibrium, if S acts on the top and bottom surfaces, it must also act, in the way shown, on the right and left surfaces. This does not mean that forces have to be applied externally; if the cube is in a mass of material, the material takes care of it itself.) The shearing force will tend to distort the material in the manner shown in figure 1.3. The material is said to be showing shear strain. Shear strain is quantified as the change in angle undergone by two lines originally at right angles. The angle is measured in radians. It is frequently denoted by γ. In bone we are usually dealing with rather small strains, e less than 0.005, γ less than 0.1. It may be helpful to show some loading situations in bones together with the kinds of strains that are produced (figure 1.4).
The basic idea of strain is fairly simple, that of stress less so. Stress is best thought of as the intensity of a force acting across a particular plane. Imagine an area in some plane in a body that is sufficiently small for the forces acting across it to be considered uniform, and to be represented by a vector F (figure 1.5). This vector can be resolved into three mutually perpendicular vectors, each in the direction of one of the three axes we have set up. The component normal to the plane, and parallel to the Z axis is Fz. The other two components are in the X–Y plane and are shear forces Fzx and Fzy. There is no necessary relation between these forces, their relative magnitudes depending on the angle F makes with the X, Y, and Z axes. If our little area has an area A, then we define the stresses acting on it thus: a normal stress Fz/A, and two shear stresses Fzx/A and FzyA.
It is most important to be clear that the stresses acting in a small region of a body depend upon the orientation of the plane we consider. To show this, look at a very simple loading system: a long bar loaded in tension by a force F. The bar has a square section of area A. The vector of the force acting on the cross section has no shear components, and so there are no shear stresses across the section. The only stress is the normal stress F/A (figure 1.6, middle). However, there is nothing in the situation that we have set up that necessarily makes the cross-sectional plane the important one. Consider instead a plane B lying right along the length of the bar (figure 1.6, right). There are no normal forces and so no normal stresses. Nor are there any shear stresses, because the forces do not act upward on one side of the plane and downward on the other, which is necessary if shear stresses are to occur. Therefore, there are no stresses acting across the B plane. What about planes at intermediate angles? Suppose the plane C is inclined at an angle U to A (figure 1.6, left). The force F can be resolved into a normal force F x cos U and a shear force F x sin U. The area of C is greater than that of A; it is A/cos U. The normal stress will be (F x cos U)/(A/cos U) = F x cos2U/A. The shear stress will be (F x sin U)/(A/cos U) = F x cos U x sin U/A. The normal stress is greatest when U = 0°, as seems obvious, and declines rapidly as U increases beyond about U= 30°. The shear stress is, less obviously, greatest when U = 45°, where it has a value half that of the maximum normal stress.
We have envisaged here a simple loading system. In fact, a small block of material in a body can be subjected to complex loading situations: tension on some faces, compression on others, and shear in various directions on all faces. Nevertheless, the stress across any plane in the blocks is always resolvable into just three, at right angles.
Now that we have some idea of stress and strain, we can consider the relationship between them. Suppose we load a small specimen of bone in tension until it breaks. We can do this by gripping it at the ends and pulling the grips apart at some constant rate, meanwhile measuring the load on the grips, which will be the same as the load on the specimen. There are devices for measuring the extension of a portion of the length of the test specimen, and the output can be displayed on an oscilloscope or similar instrument. The output will show a curve of load as a function of deformation. It will look something like figure 1.7. Starting from the origin, there is a part where the load varies linearly with the deformation. At the so-called yield point the curve flattens considerably, and now increasing deformation involves little extra load. Eventually the bone specimen will break.
A load-deformation curve is all very well for giving a rough idea of what is going on, but does not allow us to put any values on the variables that will tell us anything about the mechanical properties of the test piece. However, in a simple tensile test this is easily remedied. The force/(cross-sectional area) is the normal stress acting across the cross section, on which there are no shear stresses. Similarly, the increase in length of the part of the test piece being measured (the gage length), divided by the original length, is the strain. So the load-deformation curve can, with minimal arithmetic, be turned into a stress-strain curve. (Such simplicity does not hold for such loading systems as bending or torsion.)
The stress-strain curve gives a considerable amount of information. The linear part shows that there is a region where stress and strain are proportional to each other. For many biological materials this is not true. The stress-strain curve for cartilage, for instance, has no linear region; the curve is always curved, right from the origin. The less the deformation produced by a particular load, the stiffer the material. (Many people worry that stress appears on the Y axis, and strain on the X axis, because they feel that the stress "causes" the strain. In fact, it is the strain that makes things happen in bone, as in other materials. For instance, one can show that bone breaks when the strain gets too big, not when the stress becomes too big.) The modulus of elasticity, or Young's modulus, is defined as stress/strain in the linear region of the curve of an ordinary tensile or compressive test. It is expressed in pascals (Pa), which are newtons per square meter, and in bone it has a value of roughly 2 × 109 Pa, or 20 GPa. It is usually denoted by E. The steeper the initial part of the curve, the greater the modulus of elasticity. In fact, the modulus of elasticity is the tangent of the angle made by the curve to the abscissa. Although Young's modulus of elasticity is often referred to as "the" modulus, we shall see that a number of independent moduli are needed to define the elastic behavior of a material. There is no reason why Young's modulus should have preference except that it is the most straightforward to measure and the easiest to understand.
It is found, near enough, that if a specimen of bone has been loaded only into the linear part of the curve, and the strain is then reduced, the force falls to zero when the strain falls to zero. If a material returns to its original state when loads are removed, it is said to behave elastically. If the material is also like bone in that the first part of the stress-strain curve is straight, it is said to be linearly elastic. Beyond the place where the curve bends over, something happens to the bone because beyond this point it is found that the stress will fall to zero before the strain falls to zero (figure 2.9). This residual strain is called plastic strain, and the material is said to be behaving plastically. As we shall see later, in chapter 2, so-called plastic behavior can be caused by a great variety of phenomena at the microscopic level; plasticity in bone has a different cause from plasticity in aluminum, for instance.
The point, or more properly the region, where the deformation changes from being elastic to being at least partially plastic is the yield point. Although a bone reaching this point still has far to go before it breaks, it is damaged to some extent once it enters the plastic region, and healing will have to take place if it is to be as it was before. It is, therefore, not surprising that what information we have shows that the loads placed on bones in life normally load them only into the elastic region. In the plastic region the bone extends with little extra stress. Therefore, the difference between the fracture stress, which is usually considered to be the strength of the bone, and the yield stress is usually rather small. So, if a bone were subjected to an ever-increasing load, it would make little difference to the load at failure whether the bone broke at the end of the linear part of the curve rather than at the end of the long plastic region. (Although this is true for tension, if the bone is loaded in bending, the plastic region does help to increase the load necessary to cause fracture.)
A material that breaks without showing any plastic deformation is said to be brittle. Characterizing a material as brittle gives no information about whether it is weak or strong in the sense of what load it can bear. However, the presence or absence of a reasonable amount of plastic deformation is an extremely significant feature of the mechanical properties of a material. Materials that show a reasonable amount of plastic deformation are very often "tough." A tough material is not weakened by small scratches on its surface, as brittle materials are. Bone may behave more or less brittlely according, among other things, to the direction in which it is loaded (figure 2.16). We shall see more of this later. In bone, toughness is important in determining how it breaks in life. What usually happens is that the bone is subjected to violence. This can be direct, as when a small boy falls off a wall onto his head, or it can be indirect, as when a galloping horse puts its hoof down a rabbit hole. In the latter case the bone is not struck directly, like the boy's skull. Instead, the mass of the horse's body, moving on while the leg is stuck fast, produces a violent bending load on the leg. Sometimes, as in the spiral fracture of the tibia in a skiing fall, the bone is loaded by the ligaments that attach it to its neighboring bones, which may not themselves break. What happens in these cases is that the bone, and the surrounding tissues, are given a quantity of mechanical energy. This energy has to be dissipated in the bone and surrounding tissues without anything breaking. Now the energy absorbed by a material per unit volume is proportional to the area under the load-deformation curve up to the point considered. The reason for this is as follows: look at the load-deformation curve in figure 1.8. If we take a very narrow strip, its area is nearly F1(L2 – L1). In this part of the curve a force F1 has acted over a distance (L2 – L1), and therefore, the amount of work done is F1(L2 – L1). If the work done on the material is summed over the whole curve, the work done, or energy absorbed, per unit volume is the area under the stress-strain curve. So, the flat top of the curve of bone, although having little effect on the load at which the material breaks, can greatly increase the work that has to be done on it to break it.
1.2.2 Anisotropy
The information given by a stress-strain curve is considerable, but by no means exhausts what can be found out. Bone, like most real materials, is anisotropic. A material is anisotropic if its properties are different when measured in different directions. In this book we are really only interested in mechanical anisotropy, though anisotropy can occur in many physical properties. The stress-strain curve of an anisotropic material gives the value of Young's modulus in one direction, but it could be different in a different direction.
(Continues...)
Excerpted from The Mechanical Adaptations of Bones by JOHN D. CURREY. Copyright © 1984 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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Table of Contents
- FrontMatter, pg. i
- Contents, pg. v
- Preface, pg. ix
- 1.. The Mechanical Properties of Materials and the Structure of Bone, pg. 3
- 2. The Mechanical Properties of Bone, pg. 38
- 3. The Limits of the Mechanical Properties of Bone, pg. 88
- 4. The Shapes of Bones, pg. 98
- 5. Articulations, pg. 158
- 6. Bonesr Tendons, and Muscles, pg. 185
- 7. Safety Factors and Scaling Effects in Bones, pg. 223
- 8. Construction and Reconstruction, pg. 245
- References, pg. 271
- Index, pg. 287