Let's say I have a lattice, and I impose periodic boundary conditions. I want to construct a tight-binding model on a strained lattice, and I can determine the change in the hopping parameter based on the change in the NN distances. If I'm using homogeneous strain, this is fairly straightforward. My question is, how should I impose strain if I have periodic boundary conditions and inhomogeneous strain? For example, let's say my strain field is the following mapping.
$$ x\rightarrow x + 2 c xy$$ $$ y\rightarrow y + c(x^2-y^2)$$
where $c$ is a parameter controlling the strain. It's not immediately obvious to me how I should prevent discontinuities at the boundary, but I'd like to keep periodic boundary conditions so I don't pick up edge states when I do a numerical simulation. Any thoughts?