Stress

Figure 2.3: Differential cube with components of stress on three of six faces.

Stress is the description of forces inside or on the surface of a body. The stress at a point in the body will change depending on which plane through the point one examines. Feynman uses the device of imagining that the body is cut with the plane - the body will deform, which means that there are stresses across the plane before it is cut. Stress is a vector function of both position and orientation throughout the body. If we examine the three planes that are perpendicular to coordinate axes at a point, then it is clear that nine quantities are needed: one vector in $\mathbb{R}^3$ for each plane. fig:StressCube shows the stresses on three of six sides of a small cube of material. If we assume the cube is infinitesimally small, then stresses on the other faces will simply be the negatives of those shown. We label the components of stress as $\sigma_{ij}$ , where $\sigma_{i}$ is the stress vector in the $i$ th direction. Note that, in general, $\sigma_{i}$ does not lie parallel to the $i$ th direction. A force-balancing argument can show that given the stresses in each of the coordinate directions, the stress across a plane with normal $n$ is simply $\sigma_{ji}n_j$ .

Figure 2.4: Torques on a unit cube. For illustration, stresses in the $z$ direction and stresses perpendicular to the faces are ignored, since they do not change the torque in question.

We can also show that the stress tensor is symmetric by observing the torques induced on the centre of the cube by the components of stress. fig:TorqueCube shows the situation schematically. The magnitude of the total torque induced on the centre is $\frac{1}{2} \sigma_{12} - \frac{1}{2}\sigma_{21} + \frac{1}{2} \sigma_{12} - \frac{1}{2} \sigma_{21} = \sigma_{12} - \sigma_{21}$ . There cannot be any torques on the cube, otherwise it would start spinning. Hence $\sigma_{12} = \sigma_{21}$ , and by extension the tensor $\sigma$ is symmetric.