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

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.

scholarly journals THIN ELASTIC PLATES EXPERIENCING LARGE DEFLECTIONS ABOVE THE VON KARMAN LIMITS

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.

scholarly journals LOCALLY ORTHOTROPIC LAY-UP AS AN OPTIMAL SOLUTION FOR VAT POST-BUCKLED COMPOSITE PLATES EXPERIENCING LARGE DEFLECTIONS ABOVE VON KARMAN LIMITS

2021 ◽  
Author(s):  
Sergey Selyugin
Keyword(s):  
Optimal Solution ◽  
Composite Plates ◽  
Maximization Problem ◽  
Large Deflections ◽  
Von Karman ◽  
Particular Solution ◽  
Principal Directions ◽  
The Moment ◽  
Curvature Tensors ◽  
Von Kármán

The present paper deals with the optimization of post-buckled VAT (variable angle tow) composite plates with large deflections. The Kirchhoff assumptions are used. The plates have a symmetric lay-up. The large deflection geometrically nonlinear theory above the von Karman limits is employed. The structural potential energy is treated as a measure of structural stiffness. For the plate stiffness maximization problem, the first-order necessary conditions of the local optimality are derived. The mathematical analysis of the conditions is performed. The conditions contain two terms. One of them corresponds to the mid-plane strains; another one corresponds to the generalized plate curvatures. A locally orthotropic lay-up is identified as an optimal solution. The local ply material direction is clearly coupled with the principal directions of 2D-strains and generalized curvatures. A particular solution of the linear combination of the ply optimality conditions is indicated. For the solution two pairs of the structural tensors are co-axial: the force and the strain tensors, as well as the moment and the generalized curvature tensors.

On Von Kármán Equations and the Buckling of a Thin Circular Elastic Plate

2015 ◽  
Vol 15 (3) ◽  
Author(s):  
Joanna Janczewska ◽  
Anita Zgorzelska
Keyword(s):  
Elastic Plate ◽  
Compressive Load ◽  
Load Parameter ◽  
Elastic Plates ◽  
Plate Structures ◽  
Von Karman ◽  
Fredholm Map ◽  
Von Kármán Equations ◽  
Von Kármán ◽  
Nonlinear Deformations

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.

scholarly journals A virtual element method for the von Kármán equations

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.

A mixed variational principle for the Föppl–von Kármán equations

2020 ◽  
Vol 79 ◽  
pp. 381-391 ◽  
Author(s):  
Matteo Brunetti ◽  
Antonino Favata ◽  
Achille Paolone ◽  
Stefano Vidoli
Keyword(s):  
Variational Principle ◽  
Mixed Variational Principle ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Von Kármán

A low-order mixed variational principle for the generalized Marguerre–von Kármán equations

Meccanica ◽  
2019 ◽  
Vol 55 (4) ◽  
pp. 883-890 ◽  
Author(s):  
Matteo Brunetti ◽  
Antonino Favata ◽  
Achille Paolone ◽  
Stefano Vidoli
Keyword(s):  
Variational Principle ◽  
Mixed Variational Principle ◽  
Von Karman ◽  
Von Kármán Equations ◽  
Von Kármán ◽  
Low Order
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