# Stress tensor from forces

Consider a body that is acted on by a variable external force density $$\vec{f}(\vec{r})$$.

I want to know what the pressure and shear stress would be within the body as a result of these external forces.

However, the problem as stated seems to be under determined to actually solve for a stress or pressure tensor according to the equation:

$$f_i=-\nabla^j\sigma_{ij}$$

Furthermore, intuitively I would expect the internal pressures to somehow depend on the variation and therefore derivative of the external force, since if it was uniform the entire body would accelerate in unison (assuming uniform density) without any part having to pull another part.

Are there additional material constraints we must assume (isotropy?) to find the stress tensor, and is my intuition correct and somehow accounted for?

• You are saying that f is a body force density? Commented Apr 30 at 16:04
• @ChetMiller Yes, force per unit volume, but from an external source. I'm thinking of a charge distribution in an external electric field, or a mass under gravity. Though I now realize the equation I wrote is incorrect because it also accounts for the internal forces within the body. But the question stands for how to solve this in general. Commented Apr 30 at 16:16

The expression $$f_i=-\nabla^j\sigma_{ij}$$ is not determined if we don't know how exactly it is supported on the ground (or by a liquid). It is intuitive because in Newtonian model, gravity is a force, and only by the expression, it can even be in free fall, in which case the stresses would be zero (for $$g$$ uniform).