Convergence of equilibria for incompressible elastic plates in the von Kármán regime

AbstractWe shall be concerned with the buckling of a thin circular elastic plate simply supported along a boundary, subjected to a radial compressive load uniformly distributed along its boundary. One of the main engineering concerns is to reduce deformations of plate structures. It is well known that von Kármán equations provide an established model that describes nonlinear deformations of elastic plates. Our approach to study plate deformations is based on bifurcation theory. We will find critical values of the compressive load parameter by reducing von Kármán equations to an operator equation in Hölder spaces with a nonlinear Fredholm map of index zero. We will prove a sufficient condition for bifurcation by the use of a gradient version of the Crandall-Rabinowitz theorem due to A.Yu. Borisovich and basic notions of representation theory. Moreover, applying the key function method by Yu.I. Sapronov we will investigate the shape of bifurcation branches.

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A virtual element method for the von Kármán equations

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2020085 ◽

2021 ◽

Vol 55 (2) ◽

pp. 533-560

Author(s):

Carlo Lovadina ◽

David Mora ◽

Iván Velásquez

Keyword(s):

Optimal Error Estimates ◽

Transverse Displacement ◽

Discrete Problem ◽

Elastic Plates ◽

Virtual Element Method ◽

Von Karman ◽

Von Kármán Equations ◽

Virtual Element ◽

Von Kármán ◽

Element Method

In this article we propose and analyze a Virtual Element Method (VEM) to approximate the isolated solutions of the von Kármán equations, which describe the deformation of very thin elastic plates. We consider a variational formulation in terms of two variables: the transverse displacement of the plate and the Airy stress function. The VEM scheme is conforming inH2for both variables and has the advantages of supporting general polygonal meshes and is simple in terms of coding aspects. We prove that the discrete problem is well posed forhsmall enough and optimal error estimates are obtained. Finally, numerical experiments are reported illustrating the behavior of the virtual scheme on different families of meshes.

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THIN ELASTIC PLATES EXPERIENCING LARGE DEFLECTIONS ABOVE THE VON KARMAN LIMITS

10.31224/osf.io/hr79j ◽

2021 ◽

Author(s):

Sergey Selyugin

Keyword(s):

Virtual Work ◽

Plate Thickness ◽

Geometrically Nonlinear ◽

Large Deflections ◽

Elastic Plates ◽

Equilibrium Equations ◽

Virtual Work Principle ◽

Von Karman ◽

Von Kármán ◽

Thin Elastic Plates

Thin elastic plates (homogeneous or composite) experiencing large deflections are considered. The deflections are much larger than the plate thickness. The geometrically nonlinear elasticity theory and the Kirchhoff assumptions are employed. Small elongations and shears are assumed. Following Novozhilov, the strain expressions are derived. Then, under a small in-plane rotation assumption and using the virtual work principle, the equilibrium equations and the boundary conditions are obtained. The equations/conditions become the known von Karman ones for the case of moderate deflections. The solutions of the obtained equations may be used as benchmarks for the nonlinear structural analysis (e.g., FEM) software in the case of large deflections.

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Convergence of Equilibria of Thin Elastic Plates - The Von Karman Case

Communications in Partial Differential Equations ◽

10.1080/03605300701629443 ◽

2008 ◽

Vol 33 (6) ◽

pp. 1018-1032 ◽

Cited By ~ 18

Author(s):

S. Muller ◽

M. R. Pakzad

Keyword(s):

Elastic Plates ◽

Von Karman ◽

Von Kármán ◽

Thin Elastic Plates

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A KINEMATIC VARIATIONAL PRINCIPLE FOR THIN METALLIC AND COMPOSITE PLATES EXPERIENCING LARGE DEFLECTIONS ABOVE THE VON KARMAN LIMITS

10.31224/osf.io/e2vgj ◽

2021 ◽

Author(s):

Sergey Selyugin

Keyword(s):

Variational Principle ◽

Plate Thickness ◽

Composite Plates ◽

Geometrically Nonlinear ◽

Large Deflections ◽

Elastic Plates ◽

Equilibrium Equations ◽

Von Karman ◽

Von Kármán ◽

Thin Elastic Plates

Thin elastic plates (metallic or composite) experiencing large deflections are considered. The plate deflections are much larger than the plate thickness. The geometrically nonlinear elasticity theory and the Kirchhoff assumptions are employed. The elongations, the shears and the in-plane rotations are assumed to be small. A kinematic variational principle leading to a boundary value problem for the plate is derived. It is shown that the principle gives proper equilibrium equations and boundary conditions. For moderate plate deflections the principle is transformed to the case of the von Karman plate.

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