# Stress tensor between spatially separated layers

If I have an interface between two bodies (1 and 2). I would expect that the stress tensor is such that $$\boldsymbol \sigma_1\cdot \hat n=\boldsymbol\sigma_2\cdot \hat n$$ at the interface, where $$\hat n$$ is the vector normal to the interface and $$\sigma_{1,2}$$ are the Cauchy stress tensors from each side.

Now, suppose that the surfaces are not touching (separated by vaccuum) but that there is a force $$\mathbf F$$ between the two bodies (it could be that the two bodies are electrically charged).

How do I write the equations for the stress tensor in 2 ($$\boldsymbol\sigma_2\cdot \hat n_2$$) as a function of the stress in body 1 ($$\boldsymbol\sigma_1\cdot \hat n_1$$) accounting for the force ?

Can I match the stress tensor to something I can define in vaccuum? (like Maxwell's stress tensor)

Well, $$\sigma_1 \cdot n = \sigma_2 \cdot n$$ is basically Newton's actio est reactio for the surface force density when the bodies are touching, because the surface force density is $$f = \sigma \cdot n\,.$$ Take away the contact between the bodies and in the purely mechanical problem we are left with vanishing surface forces - because the vacuum is mechanically force-free. Now enter your additional body force. Generally, this might get hairy, but with the electromagnetic example we're in luck: You are right that we can use the Maxwell stress tensor, which is designed to work in a similar manner to the mechanical stress tensor. The above force equilibrium now becomes $$\left(\sigma_1 + T_1\right) \cdot n = T_v \cdot n\,,$$ $$\left(\sigma_2 + T_2\right) \cdot n = T_v \cdot n\,,$$ evaluated at the respective interface with $$T$$ denoting the Maxwell stress tensor evaluated in the medium (index 1 or 2) or the vacuum (index v) - right at the surface, of course.
So, do we have our relation between the stresses $$\sigma_1$$ and $$\sigma_2$$? Well, not quite: We further need information on the charge distributions, geometry of the problem, electric properties of the media to be able to compute the Maxwell stress tensor at both interfaces. Note that the purely local boundary condition of contact in the mechanical case has been replaced by a relation needing global information. :)