Hooke’s law
The subject of elasticity deals with the behavior of those substances which have the property of recovering their size and shape when the forces producing deformations are removed. We find this elastic property to some extent in all solid bodies. If we had the time to deal with the subject at length, we would want to look into many things: the behavior of materials, the general laws of elasticity, the general theory of elasticity, the atomic machinery that determine the elastic properties, and finally the limitations of elastic laws when the forces become so great that plastic flow and fracture occur. It would take more time than we have to cover all these subjects in detail, so we will have to leave out some things. For example, we will not discuss plasticity or the limitations of the elastic laws. (We touched on these subjects briefly when we were talking about dislocations in metals.) Also, we will not be able to discuss the internal mechanisms of elasticity—so our treatment will not have the completeness we have tried to achieve in the earlier chapters. Our aim is mainly to give you an acquaintance with some of the ways of dealing with such practical problems as the bending of beams. When you push on a piece of material, it “gives”—the material is deformed. If the force is small enough, the relative displacements of the various points in the material are proportional to the force—we say the behavior is elastic. We will discuss only the elastic behavior. First, we will write down the fundamental laws of elasticity, and then we will apply them to a number of different situations. Suppose we take a rectangular block of material of length $l$, width $w$, and height $h$, as shown in Fig. 38–1. If we pull on the ends with a force $F$, then the length increases by an amount $\Delta l$. We will suppose in all cases that the change in length is a small fraction of the original length. As a matter of fact, for materials like wood and steel, the material will break if the change in length is more than a few percent of the original length. For a large number of materials, experiments show that for sufficiently small extensions the force is proportional to the extension \begin{equation} \label{Eq:II:38:1} F\propto\Delta l. \end{equation}  This relation is known as Hooke’s law. The lengthening $\Delta l$ of the bar will also depend on its length. We can figure out how by the following argument. If we cement two identical blocks together, end to end, the same forces act on each block; each will stretch by $\Delta l$. Thus, the stretch of a block of length $2l$ would be twice as big as a block of the same cross section, but of length $l$. In order to get a number more characteristic of the material, and less of any particular shape, we choose to deal with the ratio $\Delta l/l$ of the extension to the original length. This ratio is proportional to the force but independent of $l$: \begin{equation} \label{Eq:II:38:2} F\propto\frac{\Delta l}{l}. \end{equation}  The force $F$ will also depend on the area of the block. Suppose that we put two blocks side by side. Then for a given stretch $\Delta l$ we would have the force $F$ on each block, or twice as much on the combination of the two blocks. The force, for a given amount of stretch, must be proportional to the cross-sectional area $A$ of the block. To obtain a law in which the coefficient of proportionality is independent of the dimensions of the body, we write Hooke’s law for a rectangular block in the form \begin{equation} \label{Eq:II:38:3} F=YA\,\frac{\Delta l}{l}. \end{equation}  The constant $Y$ is a property only of the nature of the material; it is known as Young’s modulus. (Usually you will see Young’s modulus called $E$. But we’ve used $E$ for electric fields, energy, and emf’s, so we prefer to use a different letter.) The force per unit area is called the stress, and the stretch per unit length—the fractional stretch—is called the strain. Equation (38.3) can therefore be rewritten in the following way: \begin{gather} \label{Eq:II:38:4} \frac{F}{A}=Y\times\frac{\Delta l}{l},\\[6pt] \text{Stress}=(\text{Young’s modulus})\times(\text{Strain}).\notag \end{gather}  There is another part to Hooke’s law: When you stretch a block of material in one direction it contracts at right angles to the stretch. The contraction in width is proportional to the width $w$ and also to $\Delta l/l$. The sideways contraction is in the same proportion for both width and height, and is usually written \begin{equation} \label{Eq:II:38:5} \frac{\Delta w}{w}=\frac{\Delta h}{h}=-\sigma\,\frac{\Delta l}{l}, \end{equation}  where the constant $\sigma$ is another property of the material called Poisson’s ratio. It is always positive in sign and is a number less than $1/2$. (It is “reasonable” that $\sigma$ should be generally positive, but it is not quite clear that it must be so.) The two constants $Y$ and $\sigma$ specify completely the elastic properties of a homogeneous isotropic (that is, noncrystalline) material. In crystalline materials the stretches and contractions can be different in different directions, so there can be many more elastic constants. We will restrict our discussion temporarily to homogeneous isotropic materials whose properties can be described by $Y$ and $\sigma$. As usual there are different ways of describing things—some people like to describe the elastic properties of materials by different constants. It always takes two, and they can be related to $\sigma$ and $Y$. The last general law we need is the principle of superposition. Since the two laws (38.4) and (38.5) are linear in the forces and in the displacements, superposition will work. If you have one set of forces and get some displacements, and then you add a new set of forces and get some additional displacements, the resulting displacements will be the sum of the ones you would get with the two sets of forces acting independently. Now we have all the general principles—the superposition principle and Eqs. (38.4) and (38.5)—and that’s all there is to elasticity. But that is like saying that once you have Newton’s laws that’s all there is to mechanics. Or, given Maxwell’s equations, that’s all there is to electricity. It is, of course, true that with these principles you have a great deal, because with your present mathematical ability you could go a long way. We will, however, work out a few special applications. 