Approximation of Thin Three-Dimensional Plates with Smooth Lateral Surface by Polygonal Plates

S. A. Nazarov, G. A. Chechkin

Research outputpeer-review

Abstract

In a thin isotropic homogeneous three-dimensional plate of thickness h, we consider the limit passage of elastic fields as h → +0. It is assumed that the connected component ΓNof the boundary of the median cross section ω is a broken line with links of length ah, where a > 0 is a fixed parameter. We consider the Lamé equations with the Neumann condition (a free surface) on the plate bases, the Dirichlet condition (a rigidly clamped surface) on a smooth part, and some linear contact conditions on a ribbed part of the lateral surface. For the solution to the boundary value problem we obtain an asymptotic expansion with different boundary layers. In the two-dimensional model, new boundary conditions, different from the contact ones, arise on the limit smooth contour Γ0, thereby for the spatial elasticity problem, we confirm the Babuška paradox caused by linear boundary conditions on the plate edge, but not unilateral constraints of Signorini type. Bibliography: 42 titles. Illustrations: 5 figures.

Original languageEnglish
Pages (from-to)399-428
Number of pages30
JournalJournal of Mathematical Sciences (United States)
Volume210
Issue number4
DOIs
Publication statusPublished - 1 Oct 2015

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Lateral
Three-dimensional
Approximation
Broken line
Boundary conditions
Contact
Unilateral Constraint
Neumann Condition
Dirichlet conditions
Elasticity Problem
Bibliographies
Paradox
Connected Components
Free Surface
Boundary value problems
Asymptotic Expansion
Boundary Layer
Elasticity
Figure
Boundary layers

Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Applied Mathematics

Cite this

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abstract = "In a thin isotropic homogeneous three-dimensional plate of thickness h, we consider the limit passage of elastic fields as h → +0. It is assumed that the connected component ΓNof the boundary of the median cross section ω is a broken line with links of length ah, where a > 0 is a fixed parameter. We consider the Lam{\'e} equations with the Neumann condition (a free surface) on the plate bases, the Dirichlet condition (a rigidly clamped surface) on a smooth part, and some linear contact conditions on a ribbed part of the lateral surface. For the solution to the boundary value problem we obtain an asymptotic expansion with different boundary layers. In the two-dimensional model, new boundary conditions, different from the contact ones, arise on the limit smooth contour Γ0, thereby for the spatial elasticity problem, we confirm the Babuška paradox caused by linear boundary conditions on the plate edge, but not unilateral constraints of Signorini type. Bibliography: 42 titles. Illustrations: 5 figures.",
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N2 - In a thin isotropic homogeneous three-dimensional plate of thickness h, we consider the limit passage of elastic fields as h → +0. It is assumed that the connected component ΓNof the boundary of the median cross section ω is a broken line with links of length ah, where a > 0 is a fixed parameter. We consider the Lamé equations with the Neumann condition (a free surface) on the plate bases, the Dirichlet condition (a rigidly clamped surface) on a smooth part, and some linear contact conditions on a ribbed part of the lateral surface. For the solution to the boundary value problem we obtain an asymptotic expansion with different boundary layers. In the two-dimensional model, new boundary conditions, different from the contact ones, arise on the limit smooth contour Γ0, thereby for the spatial elasticity problem, we confirm the Babuška paradox caused by linear boundary conditions on the plate edge, but not unilateral constraints of Signorini type. Bibliography: 42 titles. Illustrations: 5 figures.

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