Boundary conditions for the high order homogenized equation: laminated rods, plates and composites

International audienceThe high order homogenization technique generates the so called infinite order homogenized equation. Its coefficients were widely discussed in composite mechanics literature because they are closely related to the so called high order strain gradients theories. However, it was not clear what is the correct mathematical setting for this equation and what are the asymptotically exact boundary conditions. In the present Note we give a variational formulation for the high order homogenized equation by the projection of the initial problem on the "ansatz subspace". This formulation generates the appropriate boundary conditions for the high order homogenized equation. The error estimates for the solution of the original problem and the homogenized one are obtained