Question

I am struggling to understand how to derive a stress tensor. In the picture above, I was asked to compute the stress tensor and the stress acting on the plane Σ. The x-axis and y-axis are in the horizontal and the vertical direction, while the z-axis is through the monitor.

First, I drew a vertical line going through the center of the plane Σ
and considered the force balance of the right half of the rectangle given by the vertical line (as in the picture). Because the half of the rectangle does not move, the force acting on the left side must be f
in the negative x direction. So, I thought the first column of the stress tensor must be [−f,0,0]T, which was not correct. The answer was [f,0,0]. If I were to consider the force balance of the left half of the rectangle, I can get the answer by the same argument. So, the stress tensor is not unique? What is wrong with my argument?

Answer 1

The stress tensor is unique and describes the internal forces in a material. The force balance for a body in equilibrium dictates that the normal component of the stress vector is positive in the direction of the applied force and not negative, leading to the correct stress tensor components being [f,0,0].

Step-by-step explanation:

The stress tensor is a mathematical construct used to describe internal forces experienced by a material body. When deriving the stress tensor for a rectangle separating into two halves, it's crucial to consider the Newton's third law which states that for every action, there is an equal and opposite reaction. If the force on one side of the rectangle is f in the negative x-direction, then the force on the other side must be f in the positive x-direction because the two forces balance each other out. Thus, the stress vector normal to the cut will have components [f, 0, 0] and not [-f, 0, 0]. The stress tensor itself is unique and does not depend on which half of the material we consider; both halves exert equal and opposite forces on each other due to internal stresses.