stress, strain, and elastic moduLi
The rigid body is a useful idealized model, but the stretching, squeezing, and twisting of real bodies when forces are applied are often too important to ignore. Figure 11.12 shows three examples. We want to study the relationship between the forces and deformations for each case.
You don’t have to look far to find a deformable body; it’s as plain as the nose on your face (Fig. 11.13). If you grasp the tip of your nose between your index finger and thumb, you’ll find that the harder you pull your nose outward or push it inward, the more it stretches or compresses. Likewise, the harder you squeeze your index finger and thumb together, the more the tip of your nose compresses. If you try to twist the tip of your nose, you’ll get a greater amount of twist if you apply stronger forces.
These observations illustrate a general rule. In each case you apply a stress to your nose; the amount of stress is a measure of the forces causing the deformation, on a “force per unit area” basis. And in each case the stress causes a deformation, or strain. More careful versions of the experiments with your nose suggest that for relatively small stresses, the resulting strain is proportional to the stress: The greater the deforming forces, the greater the resulting deformation. This proportionality is called Hooke’s law, and the ratio of stress to strain is called the elastic modulus:
The value of the elastic modulus depends on what the body is made of but not its shape or size. If a material returns to its original state after the stress is removed, it is called elastic; Hooke’s law is a special case of elastic behavior. If a material instead remains deformed after the stress is removed, it is called plastic. Here we’ll consider elastic behavior only; we’ll return to plastic behavior in Section 11.5. We used one form of Hooke’s law in Section 6.3: The elongation of an ideal spring is proportional to the stretching force. Remember that Hooke’s “law” is not really a general law; it is valid over only a limited range of stresses. In Section 11.5 we’ll see what happens beyond that limited range.