A model of a plane deformation state of a two-dimensional plate with small almost periodic clamped parts of the edge

S. A. Nazarov ; J. Taskinen

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 51, Tome 506 (2021), pp. 130-174

Résumé

We construct asymptotics, as the small positive parameters $h$ and $\varepsilon$ tend to zero, of the displacement and stress fields in a planar isotropic body whose boundary is rigidly fixed at $h$-periodially posed boundary parts of length $O(h \varepsilon)$. We propose an asymptotic model that involves the Winkler–Robin boundary conditions connecting the displacement vector and the vector of normal stresses at the boundary, and provides acceptable approximation for the solution of the original problem for a wide range of the parameters $h$ and $\varepsilon$. Error estimates are based on various weighted inequalities.