Degenerate scales for thin elastic plates with Dirichlet boundary conditions

A degenerate scale occurs when a loss of uniqueness of the solution of the boundary integral equations happens for that scale of the problem. We consider here the biharmonic 2D problem with Dirichlet boundary conditions which models the bending behavior of a clamped isotropic Kirchhoff plate. We ext...

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Bibliographic Details
Published in:Acta mechanica Vol. 234; no. 4; pp. 1503 - 1532
Main Authors: Corfdir, Alain, Bonnet, Guy
Format: Journal Article
Language:English
Published: Vienna Springer Vienna 01.04.2023
Springer
Springer Nature B.V
Springer Verlag
Subjects:
Biharmonic equations
Boundary conditions
Boundary integral method
Classical and Continuum Physics
Complex variables
Control
Dirichlet problem
Dynamical Systems
Elastic plates
Engineering
Engineering Fluid Dynamics
Engineering Thermodynamics
Heat and Mass Transfer
Integral equations
Lame functions
Mathematics
Numerical analysis
Original Paper
Physics
Solid Mechanics
Symmetry
Theoretical and Applied Mechanics
Vibration
Elasticity
Degenerate scale
Boundary integral equation
Plate
ISSN:0001-5970, 1619-6937
Online Access:Get full text
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Summary:A degenerate scale occurs when a loss of uniqueness of the solution of the boundary integral equations happens for that scale of the problem. We consider here the biharmonic 2D problem with Dirichlet boundary conditions which models the bending behavior of a clamped isotropic Kirchhoff plate. We extend several results about degenerate scales previously found for the Laplace and Lamé equation to the biharmonic equation in 2 dimensions. The degenerate scales are obtained from a 4 × 4 discriminant matrix whose shape is provided for different kinds of domain symmetry. We show that degenerate scales can be obtained from a minimization problem. Then, we compare the degenerate scales of two boundaries, one included within the other. For smooth star-shaped curves, we show that there are only two degenerate scales, give sufficient conditions for not being at a degenerate scale and produce bounds to the degenerate scales. For symmetric smooth simply connected curves there are also only two degenerate scales. For these symmetric cases, the use of complex variables allows us to go further and to link the problem of biharmonic equation to the one of plane elasticity and to give information, including bounds and exact values of degenerate scales, for many cases, while until now only very few ones were known. Our results have some consequences for the biharmonic outer radius defined by Pólya and Szegö. The numerical computation of degenerate scales by using boundary elements confirms the theoretical results.
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ISSN:0001-5970
1619-6937
DOI:10.1007/s00707-022-03472-4