Nonelastic behavior
In all that has been said so far, we have assumed that stress is proportional to strain; in general, that is not true. Figure 39–8 shows a typical stress-strain curve for a ductile material. For small strains, the stress is proportional to the strain. Eventually, however, after a certain point, the relationship between stress and strain begins to deviate from a straight line. For many materials—the ones we would call “brittle”—the object breaks for strains only a little above the point where the curve starts to bend over. In general, there are other complications in the stress-strain relationship. For example, if you strain an object, the stresses may be high at first, but decrease slowly with time. Also if you go to high stresses, but still not to the “breaking” point, when you lower the strain the stress will return along a different curve. There is a small hysteresis effect (like the one we saw between $B$ and $H$ in magnetic materials). The stress at which a material will break varies widely from one material to another. Some materials will break when the maximum tensile stress reaches a certain value. Other materials will fail when the maximum shear stress reaches a certain value. Chalk is an example of a material which is much weaker in tension than in shear. If you pull on the ends of a piece of blackboard chalk, the chalk will break perpendicular to the direction of the applied stress, as shown in Fig. 39–9(a). It breaks perpendicular to the applied force because it is only a bunch of particles packed together which are easily pulled apart. The material is, however, much harder to shear, because the particles get in each other’s way. Now you will remember that when we had a rod in torsion there was a shear all around it. Also, we showed that a shear was equivalent to a combination of a tension and compression at $45^\circ$. For these reasons, if you twist a piece of blackboard chalk, it will break along a complicated surface which starts out at $45^\circ$ to the axis. A photograph of a piece of chalk broken in this way is shown in Fig. 39–9(b). The chalk breaks where the material is in maximum tension. Other materials behave in strange and complicated ways. The more complicated the materials are, the more interesting their behavior. If we take a sheet of “Saran-Wrap” and crumple it up into a ball and throw it on the table, it slowly unfolds itself and returns toward its original flat form. At first sight, we might be tempted to think that it is inertia which prevents it from returning to its original form. However, a simple calculation shows that the inertia is several orders of magnitude too small to account for the effect. There appear to be two important competing effects: “something” inside the material “remembers” the shape it had initially and “tries” to get back there, but something else “prefers” the new shape and “resists” the return to the old shape. We will not attempt to describe the mechanism at play in the Saran plastic, but you can get an idea of how such an effect might come about from the following model. Suppose you imagine a material made of long, flexible, but strong, fibers mixed together with some hollow cells filled with a viscous liquid. Imagine also that there are narrow pathways from one cell to the next so the liquid can leak slowly from a cell to its neighbor. When we crumple a sheet of this stuff, we distort the long fibers, squeezing the liquid out of the cells in one place and forcing it into other cells which are being stretched. When we let go, the long fibers try to return to their original shape. But to do this, they have to force the liquid back to its original location—which will happen relatively slowly because of the viscosity. The forces we apply in crumpling the sheet are much larger than the forces exerted by the fibers. We can crumple the sheet quickly, but it will return more slowly. It is undoubtedly a combination of large stiff molecules and smaller, movable ones in the Saran-Wrap that is responsible for its behavior. This idea also fits with the fact that the material returns more quickly to its original shape when it’s warmed up than when it’s cold—the heat increases the mobility (decreases the viscosity) of the smaller molecules. Although we have been discussing how Hooke’s law breaks down, the remarkable thing is perhaps not that Hooke’s law breaks down for large strains but that it should be so generally true. We can get some idea of why this might be by looking at the strain energy in a material. To say that the stress is proportional to the strain is the same thing as saying that the strain energy varies as the square of the strain. Suppose we have a rod and we twist it through a small angle $\theta$. If Hooke’s law holds, the strain energy should be proportional to the square of $\theta$. Suppose we were to assume that the energy were some arbitrary function of the angle; we could write it as a Taylor expansion about zero angle  \begin{equation} \label{Eq:II:39:40} U(\theta)=U(0)+U'(0)\,\theta+\tfrac{1}{2}U''(0)\,\theta^2+ \tfrac{1}{6}U'''(0)\,\theta^3+\dotsb \end{equation} 
 \begin{align} U(\theta)=U(0)&+U'(0)\,\theta+\!\tfrac{1}{2}U''(0)\,\theta^2\notag\\[.5ex] \label{Eq:II:39:40} &+\tfrac{1}{6}U'''(0)\,\theta^3\!+\dotsb \end{align}  The torque $\tau$ is the derivative of $U$ with respect to angle; we would have \begin{equation} \label{Eq:II:39:41} \tau(\theta)=U'(0)+U''(0)\,\theta+\!\tfrac{1}{2}U'''(0)\,\theta^2\!+\dotsb \end{equation}  Now if we measure our angles from the equilibrium position, the first term is zero. So the first remaining term is proportional to $\theta$; and for small enough angles, it will dominate the term in $\theta^2$. [Actually, materials are sufficiently symmetric internally so that $\tau(\theta)=-\tau(-\theta)$; the term in $\theta^2$ will be zero, and the departures from linearity would come only from the $\theta^3$ term. There is, however, no reason why this should be true for compressions and tensions.] The thing we have not explained is why materials usually break soon after the higher-order terms become significant. 