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It is my understanding that the entries in the stress tensor are forces per unit area. I assume this means the

*total*force per unit area. Considering the case of a cylinder with axis along the x axis...

Say you press the two flat ends towards each other, i.e, exert a force F on the left surface and -F on the right, such that the cylinder does not move. Is the T_xx entry in the stress tensor then zero for any point inside the cylinder, since the surface forces should cancel so that the point remains stationary?

In other words, if I have an object that experiences only surface forces, no body forces, should the stress tensor have all zeros along the diagonal?