Boundary Homogenization and Reduction of Dimension in a Kirchhoff-Love Plate

We investigate the asymptotic behavior, as $\varepsilon$ tends to zero, of the transverse displacement of a Kirchhoff-Love plate composed by the union of two domains contained in the plane: $A_\varepsilon$ and $B_\varepsilon$ and depending on $\varepsilon$ in the following way. The set $A_\varepsilon$ is a union of fine teeth, having small cross section of size $\varepsilon$ and constant height, $\varepsilon$ periodically distributed on the upper side of a horizontal thin strip $B_\varepsilon$ with vanishing height $h_\varepsilon$, as $\varepsilon$ tends to zero. The structure is clamped on the top of the teeth, with a free boundary elsewhere, and subjected to a transverse load. As $\varepsilon$ tends to zero, we obtain a “continuum” bending model of rods in the limit domain of the comb, while the limit displacement is independent of the vertical variable in the rescaled (with respect to $h_\varepsilon$) strip. We show that the displacement in the strip is equal to the displacement on the basis of the teeth if $h_\varepsilon>>\varepsilon^4$. However, if the strip is thin enough (i.e., $h_\varepsilon=\varepsilon^4$), we show that microscopic oscillations of the displacement in the strip, between the basis of then teeth, may produce a limit average field different from that on the base of the teeth; i.e., a discontinuity in the transmission condition may appear in the limit model.