Boundary homogenization and reduction of dimension in a Kirchhoff-Love plate

Overview & Synopsis

We investigate the asymptotic behavior, as ε tends to 0+, of the transverse displacement of a Kirchhoff-Love plate composed of two domains Ωε+∪ Ωε-, contained in the (x1,x2)-coordinate plane and depending on ε in the following way. The first domain Ωε- is a thin strip with vanishing height hε (in direction x2), as ε tends to 0+. The second one Ωε+ is a comb with fine teeth, having small cross section εω and constant height, ε-periodically distributed (in direction x1) on the upper side of the thin strip (see Figure 1). The structure is assumed clamped on the top of the teeth, with a free boundary elsewhere, and subjected to a transverse load. As ε 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 x2 in the rescaled (with respect to hε) strip. We show that the displacement in the strip is equal to the displacement on the base of the teeth, if hε >> ε4. While, if the strip is thin enough (i.e. hε ≃ ε4), 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.