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

Abstract

We investigate the asymptotic behavior, as $\varepsilon$ tends to $0^+$, of the transverse displacement of a Kirchhoff–Love plate composed of two domains $\Omega_\varepsilon^+\cup\Omega^-_\varepsilon\subset{\mathbb R}^2$ depending on $\varepsilon$ in the following way. The set $\Omega_\varepsilon^+$ 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 with vanishing height $h_\varepsilon$, as $\varepsilon$ tends to $0^+$. 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 $0^+$, 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 base of the teeth if $h_\varepsilon\gg\varepsilon^4$. However, if the strip is thin enough (i.e., $h_\varepsilon\simeq\varepsilon^4$), we show that microscopic oscillations of the displacement in the strip, between the basis of the 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.