Approximation theory of three dimensional elastic plates and its boundary conditions without using Kirchhoff-Love assumptions

1995 ◽  
Vol 16 (3) ◽  
pp. 203-224 ◽  
Author(s):  
Chien Wei-zang
Keyword(s):  
Boundary Conditions ◽  
Approximation Theory ◽  
Three Dimensional ◽  
Elastic Plates

Flutter of Rectangular Plates in Three Dimensional Incompressible Flow With Various Boundary Conditions: Theory and Experiment

2012 ◽  
Author(s):  
Ivan Wang ◽  
Samuel C. Gibbs ◽  
Earl H. Dowell
Keyword(s):  
Boundary Conditions ◽  
Structural Model ◽  
Three Dimensional ◽  
Flexible Plate ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Test Bed ◽  
Trailing Edge ◽  
Various Boundary Conditions ◽  
Trailing Edge Flap

The aeroelastic stability of rectangular plates are well-documented in literature for certain sets of boundary conditions. Specifically, wing flutter, panel flutter, and divergence of a plate that is clamped on all sides are well-understood. However, the ongoing push for lighter structures and novel designs have led to a need to understand the aeroelastic behavior of elastic plates for other boundary conditions. One example is NASA’s continuous mold-line link project for reducing the noise generated by commercial transport aircraft during landing; in order to reduce the noise generated by vortex shedding from the trailing edge flap during landing, the project proposes to connect the gap between the trailing edge flap and the rest of the wing with a flexible plate. This paper summarizes the aeroelastic theory, numerical results, and experimental results of a study on the flutter and/or divergence mechanisms of a rectangular plate for different sets of structural boundary conditions. The theory combines a three-dimensional vortex lattice aerodynamic model with a plate structural model to create a high-fidelity frequency domain aeroelastic model. A modular experimental test bed is designed for this study in order to test the different boundary conditions. The test bed is also designed to test different plate thicknesses and sizes with only a small number of modifications. The well-understood boundary conditions are used as test cases to validate the analysis results, and then results of additional configurations that have not been extensively explored are presented. The results of this paper can be used to support the design efforts of projects involving plates or plate-membranes. In addition, the paper adds to the fundamental understanding of plate aeroelasticity and provides a wealth of experimental data for comparison and future validation.

Three Dimensional Vibration Analysis of Rectangular Plates With Undamaged and Damaged Boundaries by the Spectral Collocation Method

2011 ◽  
Author(s):  
Ma’en S. Sari ◽  
Eric A. Butcher
Keyword(s):  
Boundary Conditions ◽  
Collocation Method ◽  
Vibration Analysis ◽  
Numerical Technique ◽  
Three Dimensional ◽  
Free Vibration Analysis ◽  
Rectangular Plates ◽  
Elastic Plates ◽  
Simply Supported ◽  
Grid Points

This paper presents a new numerical technique for the free vibration analysis of isotropic three dimensional elastic plates with damaged boundaries. In the study, it is assumed that the plates have free lateral surfaces, and two opposite simply supported edges, while the other edges could be clamped, simply supported or free. For this purpose, the Chebyshev collocation method is applied to obtain the natural frequencies of three dimensional plates with damaged clamped boundary conditions, where the governing equations and boundary conditions are discretized by the presented method and put into matrix vector form. The damaged boundaries are represented by distributed translational springs. In the present study the boundary conditions are coupled with the governing equation to obtain the eigenvalue problem. Convergence studies are carried out to determine the sufficient number of grid points used. First, the results obtained for the undamaged plates are verified with previous results in the literature. Subsequently, the results obtained for the damaged three dimensional plates indicate the behavior of the natural vibration frequencies with respect to the severity of the damaged boundary. This analysis can lead to an efficient technique for damage detection of structures in which joint or boundary damage plays a significant role in the dynamic characteristics. The results obtained from the Chebychev collocation solutions are seen to be in excellent agreement with those presented in the literature.

Theories for Elastic Plates Via Orthogonal Polynomials

1981 ◽  
Vol 48 (4) ◽  
pp. 900-904 ◽  
Author(s):  
S. Krenk
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Three Dimensional ◽  
Transversely Isotropic ◽  
Displacement Function ◽  
Shear Stresses ◽  
Transverse Displacement ◽  
Normal Strain ◽  
Two Dimensional ◽  
Elastic Plates

A complementary energy functional is used to derive an infinite system of two-dimensional differential equations and appropriate boundary conditions for stresses and displacements in homogeneous anisotropic elastic plates. Stress boundary conditions are imposed on the faces a priori, and this introduces a weight function in the variations of the transverse normal and shear stresses. As a result the coupling between the two-dimensional differential equations is described in terms of a single difference operator. Special attention is given to a truncated system of equations for bending of transversely isotropic plates. This theory has three boundary conditions, like Reissner’s, but includes the effect of transverse normal strain, essentially through a reinterpretation of the transverse displacement function. Full agreement with general integrals to the homogeneous three-dimensional equations is established to within polynomial approximation.

Theory of laminated elastic plates I. isotropic laminae

1988 ◽  
Vol 324 (1582) ◽  
pp. 565-594 ◽  
Keyword(s):  
Boundary Conditions ◽  
Exact Solutions ◽  
Plane Stress ◽  
Three Dimensional ◽  
Laminated Plates ◽  
Surface Boundary ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Dimensional Solution ◽  
Stress Theory

In this paper (part I) we establish a theory for stretching and bending of laminated elastic plates in which the laminae are different isotropic linearly elastic materials. The theory gives exact solutions of the three-dimensional elasticity equations that satisfy all the interface traction and displacement continuity conditions, with no traction on the lateral surfaces; the only restriction is that edge boundary conditions can be satisfied only in an average manner, rather than point by point. The method, which is based on a generalization of Michell’s exact plane stress theory, yields exact solutions for each lamina. These solutions are generated in a very straightforward manner by solutions of the approximate two-dimensional classical equations of laminate theory and contain sufficient arbitrary constants to enable all the continuity and lateral surface boundary conditions to be satisfied. The values of the constants depend only on the lamina thicknesses and the elastic constants. Thus, for a given laminate and for any boundary-value problem , it is necessary only to solve the appropriate two-dimensional plane problem, and the corresponding exact three-dimensional laminate solution follows by straightforward substitutions. The two-dimensional solution may be derived by any of the available methods, including numerical methods. An important feature of the theory is that it determines the interfacial shearing tractions, as well as the in-plane stress components. The procedure is illustrated by applying the theory to three problems involving stretching and bending of laminated plates containing circular holes.

scholarly journals Charge algebra in Al(A)dSn spacetimes

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Adrien Fiorucci ◽  
Romain Ruzziconi
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Three Dimensional ◽  
Dirichlet Boundary Conditions ◽  
Boundary Structure ◽  
Asymptotic Symmetry ◽  
Renormalization Procedure ◽  
Field Dependent ◽  
Odd Dimensions

Abstract The gravitational charge algebra of generic asymptotically locally (A)dS spacetimes is derived in n dimensions. The analysis is performed in the Starobinsky/Fefferman-Graham gauge, without assuming any further boundary condition than the minimal falloffs for conformal compactification. In particular, the boundary structure is allowed to fluctuate and plays the role of source yielding some symplectic flux at the boundary. Using the holographic renormalization procedure, the divergences are removed from the symplectic structure, which leads to finite expressions. The charges associated with boundary diffeomorphisms are generically non-vanishing, non-integrable and not conserved, while those associated with boundary Weyl rescalings are non-vanishing only in odd dimensions due to the presence of Weyl anomalies in the dual theory. The charge algebra exhibits a field-dependent 2-cocycle in odd dimensions. When the general framework is restricted to three-dimensional asymptotically AdS spacetimes with Dirichlet boundary conditions, the 2-cocycle reduces to the Brown-Henneaux central extension. The analysis is also specified to leaky boundary conditions in asymptotically locally (A)dS spacetimes that lead to the Λ-BMS asymptotic symmetry group. In the flat limit, the latter contracts into the BMS group in n dimensions.

scholarly journals On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

2021 ◽  
Author(s):  
Olivier Ozenda ◽  
Epifanio G. Virga
Keyword(s):  
Free Energy ◽  
Dimension Reduction ◽  
Three Dimensional ◽  
Energy Functional ◽  
The Other ◽  
Variational Model ◽  
Two Dimensional ◽  
Elastic Plates ◽  
Kinematic Constraint ◽  
Free Energy Functional

AbstractThe Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard $\varGamma $ Γ -convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate’s mid surface (alongside the expected measures of bending). The incompatibility with standard $\varGamma $ Γ -convergence also appears to be removed in the cases where contact with that method and ours can be made.

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