Normally, when you look at what a Neumann BC is, you get that it's a prescribed derivative of the solution in the direction of the outward surface normal at the boundary. However, in linear elasticity model, the Neumann boundary condition is prescribing traction force.
"Standard": $$\frac{\partial u}{\partial n} = a(x) \text{ for } x\in \Gamma \subset \partial\Omega$$
with $\Omega$ being the domain, $x$ the coordinate vector, $u$ our unknown field (displacement or whatever), $n$ the outer surface normal at given point and $a$ a given function of $x$.
Elasticity: $$\sigma n = t(x)\text{ for } x\in \Gamma \subset\partial\Omega $$
with $\sigma$ being the stress tensor, $n$ the normal as before, and $t$ some prescribed surface traction.
Now, I do realise that stress is related to strain and that is related to the derivative of $u$, so I guess those two versions are probably closely related. But I struggle to realise exactly how. Are they identical? Is one stronger than the other?
The only thing I could find on this was this post: https://scicomp.stackexchange.com/questions/29246/traction-stress-stress-displacement-gradient
which shows that yes, if you prescribe the BC in the traction "version", it matches naturally with the boundary integral you get after applying the Green's identity. However, for me the question still remains why cannot I define it as the derivative of the displacement field? What is the fundamental difference?