TY - JOUR

T1 - Homogenization of Kirchhoff plates with oscillating edges and point supports

AU - Nazarov, S. A.

N1 - Publisher Copyright: © 2020 Russian Academy of Sciences (DoM) and London Mathematical Society. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/8

Y1 - 2020/8

N2 - We study deformations of a long (narrow after rescaling) Kirchhoff plate with periodic (rapidly oscillating) boundary. We deduce a limiting system of two ordinary differential equations of orders 4 and 2 which describe the deflection and torsion of a two-dimensional plate in the leading order. We also consider point supports (Sobolev conditions) whose configuration influences the result of homogenizing the biharmonic equation by decreasing the size of the limiting system of differential equations or completely eliminating it. The boundary-layer phenomenon near the end faces of the plate is studied for various ways of fastening as well as for angular junctions of two long plates, possibly by point clamps (Sobolev conjugation conditions). We discuss full asymptotic series for solutions of static problems and the spectral problems of plate oscillations.

AB - We study deformations of a long (narrow after rescaling) Kirchhoff plate with periodic (rapidly oscillating) boundary. We deduce a limiting system of two ordinary differential equations of orders 4 and 2 which describe the deflection and torsion of a two-dimensional plate in the leading order. We also consider point supports (Sobolev conditions) whose configuration influences the result of homogenizing the biharmonic equation by decreasing the size of the limiting system of differential equations or completely eliminating it. The boundary-layer phenomenon near the end faces of the plate is studied for various ways of fastening as well as for angular junctions of two long plates, possibly by point clamps (Sobolev conjugation conditions). We discuss full asymptotic series for solutions of static problems and the spectral problems of plate oscillations.

KW - asymptotic expansion

KW - biharmonic equation

KW - boundary layer

KW - narrow plate

KW - one-dimensional model

KW - point supports and rivets

KW - rapidly oscillating boundary

KW - Sobolev conditions at points

UR - http://www.scopus.com/inward/record.url?scp=85093972347&partnerID=8YFLogxK

UR - https://www.mendeley.com/catalogue/3efdb69a-eb25-3446-a7e7-24053239a658/

U2 - 10.1070/IM8854

DO - 10.1070/IM8854

M3 - Article

AN - SCOPUS:85093972347

VL - 84

SP - 722

EP - 779

JO - Izvestiya Mathematics

JF - Izvestiya Mathematics

SN - 1064-5632

IS - 4

ER -