Junction of a periodic family of elastic rods with a 3d plate. Part I

We consider a set of elastic rods periodically distributed over a 3d elastic plate (both of them with axis x(3)) and we investigate the limit behavior of this problem as the periodicity 8 and the radius r of the rods tend to zero. We use a decomposition of the displacement field in the rods of the form u = U + (u) over bar where the principal part U is a field which is piecewise constant with respect to the variables (x(1), x(2)) (and then naturally extended on a fixed domain), while the perturbation (u) over bar remains defined on the domain containing the rods. We derive estimates of U and (u) over bar in term of the total elastic energy. This allows to obtain a priori estimates on u without solving the delicate question of the dependence, with respect to epsilon and r, of the constant in korn's inequality in a domain with such a rough boundary. To deal with the field (u) over bar, we use a version of an unfolding operator which permits both to rescale all the rods and to work on the same fixed domain as for U to carry out the homogenization process. The above decomposition also helps in passing to the limit and to identify the limit junction conditions between the rods and the 3d plate. (c) 2007 Elsevier Masson SAS. All rights reserved.