Why is stress a symmetric tensor?
The fact that the stress is a symmetric tensor follows from some simple considerations. The force on a small volume element will be the sum of all the stress forces over the surface area of that element. Suppose we have a volume element in the form of a long bar with a triangular cross section, where the triangle is a right triangle. We can neglect the forces on the ends of the bar, because they are small compared to the faces of the bar. Let be the (vector) area of one face of the bar, be the area of the other, and be the area of the “hypotenuse face” of the bar. It can be seen that Lets say is the force on area and likewise for the other faces. Since the stress is by definition the force per unit area, it is clear that The total force on the volume element will be: Let’s suppose that the volume element contains mass, at a constant density. The important point is that if we make the volume smaller, say by halving all lengths, the area will decrease by a factor of four, while the volum
