Junction of a periodic family of elastic rods with a 3d plate. Part II. ha l-0 6, v er sio n - 2 De c

We consider a set of elastic rods periodically distributed over a 3d elastic plate (both of them with axis x3) and we investigate the limit behavior of this problem as the periodicity ε and the radius r of the rods tend to zero (see fig.1 below). We use a decomposition of the displacement field in the rods of the form u = U + u where the principal part U is a field which is piecewise constant with respect to the variables (x1, x2) (and then naturally extended on a fixed domain), while the perturbation u remains defined on the oscillating domain containing the rods. We derive estimates of U and u 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 ε and r, of the constant in Korn’s inequality in such an oscillating domain. To deal with the field u, 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