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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?

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The example you give is well-posed. Korn's inequality holds, if the subset on which the displacement is fixed contains an open (in the topology of the boundary) subset of the boundary, which is true in your case.

The simple test is: if you fix a rigid body at your Dirichlet boundary, can it still be moved. For instance, if you fix a point in two dimensions, your object can rotate around it. If you fix a point or a line in 3 dimensions, the same.

If in the end you want periodic boundary conditions in $x$ and $y$ directions, you will have to impose an additional constraint, for instance that the mean value of the displacement over the whole rectangle is zero. Possibly, you will have to eliminate rotations as well.

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