For certain applications, such as steady state heat transfer and flow in porous media, it is possible to simulate a much larger (infinite) domain by imposing periodic boundary conditions on opposite boundary faces and dirichlet bc's on the remaining boundaries. For a 2D rectangular domain, the periodic condition can be interpreted as if the domain lies on the surface of a cylinder.
I'm curious if the same can be said for elasticity problems. I've noticed that standard linear elasticity problems are limited to finite domains and I've never seen an example where a periodic boundary condition is prescribed or implemented. I suspect there may be issues with the uniqueness of solutions to this problem due to rigid body motion (translation and/or rotation) induced by periodicity.
For simplicity, let's assume the linear isotropic planar elasticity case on a 2D rectangular domain. Let's say that I want to model a large (periodic) medium by using fixed displacement (dirichlet) conditions on two opposite boundaries and periodic displacement conditions on the remaining boundaries.
Is this problem well-posed? If not, are there strategies (e.g. additional constraints) can I use to make it well-posed, knowing that my ultimate goal is to simulate a much larger (infinite) medium with repetitive material properties?