Von Karman Equations Research Articles

General decay for a von Karman equation of memory type with acoustic boundary conditions

A Karman equation of memory type with acoustic boundary conditions is considered. This work is devoted to investigate the influence of kernel function g and prove general decay rates of solutions when g does not necessarily decay exponentially.

G030072 ひずみゲージと解析関数による大たわみ問題の応力場推定に関する研究

This study aims to estimate of stress fields of real notched plates that are subjected to large out-of-plane deflections. The general solutions of von Karman equations on this problem does not exist, therefore the authors try to estimate a stress fields of large deflection problem by superposing stress fields of two soluble linear problems, a bending problem and a plane problem. This paper provides the approximate procedure to estimate of stress fields in a circular-holed plate subjected to large out-of-plane deflections using strain gages and the equations on stress fields near a notch root in a plate subjected to out-of-plane bending and in-plane deformation. Unknown coefficients of the equations are determined by the stress values from rosette-type strain gages using the method of least squares. The rosette-type strain gages are located on circular arc whose radius is 1.5 times the notch root radius. The estimated maximum stress values at a notch root are compared with those by the finite element analyses. The estimated results are approximate to those by the finite element analyses in the cases when the ratio of the maximum deflection to the plate thickness is less than 0.8. In these cases, the membrane stress is large enough for problems of large deflection.

Large deflection analysis of laminated composite plates resting on nonlinear elastic foundations by the method of discrete singular convolution

This paper presents nonlinear static analysis of a rectangular laminated composite thick plate resting on nonlinear two-parameter elastic foundation with cubic nonlinearity. The plate formulation is based on first-order shear deformation theory (FSDT). The governing equation of motion for a rectangular laminated composite thick plate is derived by using the von Karman equation. The nonlinear static deflections of laminated plates on elastic foundation are investigated using the discrete singular convolution method. The effects of foundation and geometric parameters of plates on nonlinear deflections are investigated. The validity of the present method is demonstrated by comparing the present results with those available in the literature.

Large deflection analysis of circular and annular FGM plates under thermo-mechanical loadings with temperature-dependent properties

Large deflection analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads and under various boundary conditions is presented. Material properties are assumed to be temperature-dependent, and graded through the plate thickness according to a power-law distribution of the volume fraction of the constituents. The non-linear formulations are based on a first-order shear deformation plate theory (FSDT) and large deflection von Karman equations that includes the thermal effects. The numerical results are obtained using the dynamic relaxation (DR) method combined with the finite difference discretization technique. Numerical results for silicon nitride/stainless steel circular and annular plates are given in dimensionless graphical forms. The influences produced by temperature rise, material constant n, boundary conditions, thickness-to-radius ratio and temperature-dependency of material are studied in detail.

Asymptotics of solutions and modeling of the Von Karman equations in a singularly perturbed domain

In a plane domain with several small holes of diameter O(e), we consider the von Karman equations describing the flexure of a thin isotropic plate. We construct asymptotic expansions for solutions to the nonlinear and the corresponding linearized problems. The coefficients of expansions turn out to be holomorphic and rational functions of |ln e| −1 . The asymptotic results for the linear problem (the Kirchhoff plate) are interpreted within the framework of the theory of selfadjoint extensions of differential operators by using the tool of weighted spaces with separated asymptotics. We also present a model of a nonlinear singularly perturbed problem that provides high accuracy asymptotic formulas. This problem includes the generalized Sobolev conditions at the points to which the holes shrink. Bibliography: 34 titles. Illustrations: 1 figure.

Weak and strong solutions of a nonlinear subsonic flow–structure interaction: Semigroup approach

We consider a non-rotational, subsonic flow–structure interaction describing the flow of gas above a flexible plate. A perturbed wave equation describes the flow, and a second-order nonlinear plate equation describes the plate’s displacement. It is shown that the linearization of the model generates a strongly continuous semigroup with respect to the topology generated by “finite energy” considerations. An interesting feature of the problem is that linear perturbed flow–structure interaction is not monotone with respect to the standard norm describing the finite energy space. The main tool used in overcoming this difficulty is the construction of a suitable inner product on the finite energy space which allows the application of ω -maximal monotone operator theory. The obtained result allows us to employ suitable perturbation theory in order to discuss well-posedness of weak and strong solutions corresponding to several classes of nonlinear dynamics including the full flow–structure interaction with von Karman, Berger’s and other semilinear plate equations.

A hierarchical multiple plate models theory for laminated composites including delamination and geometrical nonlinear effects

Abstract Modeling and predicting the response and failure of laminated composite structures is a very challenging task. On one hand the designer is interested in the overall response of the structure, but on the other he is also interested in modeling phenomena such as damage which have a localized behavior. The ideal solution would be to carry out the analysis at a global level enhancing the model by adding localized information when needed. Plate models dimensionally reduce a finite element method (FEM) problem by imposing constraints on the through thickness variation of the displacement field. In this work, the displacement field is represented as the superposition of a number of plate fields, and a unifying strain field is derived. By appropriately defining boundaries to the ‘enhancing’ displacement fields, it is demonstrated that the superposition of displacement fields can be used to locally enrich the solution where more information is required. In this manner, an efficient global model can be used to determine gross displacements, and an enriching model can be used to determine stresses at lamina interfaces for the accurate prediction of localized phenomena. The proposed theoretical framework would allow an analyst to study both thin and thick walled laminated structures by introducing simple switch-over criteria between single and multiple layer theories. The model is implemented using an extended FEM (X-FEM) and superposition approach. Extra degrees of freedom are added to the model to represent the additional displacement fields, and the meshing process remains independent for each field. Both equivalent single layer theories and discrete layer theories are incorporated into the general unified framework. Geometric nonlinearities are also included in the form of the von Karman equations. Delaminations are integrated by performing an extrinsic enrichment of the assumed displacement fields. A cohesive law between laminae was also incorporated to calculate the tensile and shear tractions. The present paper outlines the various general theories to describe in-plane and out-plane displacements fields, then a unified theoretical framework is formulated and finally the finite element formulation which is based on a multilevel mesh superposition approach is reported.

Analysis of Nonlinear Free Vibration of Circular Plates With Cut-Outs Using R-Function Method

Geometrically nonlinear vibrations of circular plate with two cut-outs are simulated by the von Karman equations with respect to displacements. The combination of the Rayleigh–Ritz method and the R-function method, which allows satisfying all boundary conditions, is applied to obtain the vibration modes of the plate. The nonlinear vibrations are expanded using these vibrations modes. The dynamical system with three degrees of freedom is derived by Galerkin method. The influence of cut-outs size on linear and nonlinear vibrations of the plate is analyzed. For different parameters of cut-outs, different internal resonances occur in the plate. The nonlinear vibrations of the system for different internal resonances are analyzed by multiple scale method.

Dynamic contact problem for a bridge modeled by a viscoelastic full von Kármán system

The existence of solutions is proved for a full system of dynamic von Karman equations expressing vibrations of geometrically nonlinear viscoelastic plate, the viscosity of which has the character of a short memory. The system models the behaviour of a bridge. The in-plane acceleration terms are taken into account. The boundary contact conditions for plane displacements and possibly the contact with the rigid support are considered.

A Note on the Solution of the Von Kármán Equations Using Series and Chebyshev Spectral Methods

The classical von Karman equations governing the boundary layer flow induced by a rotating disk are solved using the spectral homotopy analysis method and a novel successive linearisation method. The methods combine nonperturbation techniques with the Chebyshev spectral collocation method, and this study seeks to show the accuracy and reliability of the two methods in finding solutions of nonlinear systems of equations. The rapid convergence of the methods is determined by comparing the current results with numerical results and previous results in the literature.

Global existence and stability for a von Karman equations with memory in noncylindrical domains

In this paper, we study the initial-boundary value problem for the von Karman equations inside domains with moving ends. Global existence, uniqueness of solutions, and the exponential decay to the energy are established provided the initial data are bounded.

Uniform Decay of Solutions for von Karman Equations of Dynamic Viscoelasticity with Memory

In this paper, we consider the dynamical von Karman equations for viscoelastic plates with nonlinear boundary dissipation. We show the existence of solutions using the Galerkin method and then prove the asymptotic behaviour of the corresponding solutions by choosing suitable Lyapunov functional.

An exact finite strip for the calculation of relative post-buckling stiffness of isotropic plates

This paper presents the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of isotropic flat plates. The so-called exact finite strip is assumed to be simply supported out-of-plane at the loaded ends. The strip is developed based on the concept that it is effectively a plate. The present method, which is designated by the name Full-analytical Finite Strip Method in this paper, provides an efficient and extremely accurate buckling solution. In the development process, the Von-Karman's equilibrium equation is solved exactly to obtain the buckling loads and the corresponding form of out-of-plane buckling deflection modes. The investigation of thin flat plate buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. It is noted that in the present method, only one of the calculated out-of-plane buckling deflection modes, corresponding to the lowest buckling load, i.e., the first mode is used for the initial post-buckling study. Thus, the postbuckling study is effectively a single-term analysis, which is attempted by utilizing the so-called semi-energy method. In this method, the Von-Karman's compatibility equation governing the behavior of isotropic flat plates is used together with a consideration of the total strain energy of the plate. Through the solution of the compatibility equation, the in-plane displacement functions which are themselves related to the Airy stress function are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. These in-plane and out-of-plane deflected functions are then substituted in the total strain energy expressions and the theorem of minimum total potential energy is applied to solve for the unknown coefficient. The developed method is subsequently applied to analyze the initial postbuckling behavior of some representative thin flat plates for which the results are also obtained through the application of a semi-analytical finite strip method. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is significantly promoted.

Parametrices and Exact Paralinearization of Semi-Linear Boundary Problems

The subject is parametrices for semi-linear problems, based on parametrices for linear boundary problems and on non-linearities that decompose into solution-dependent linear operators acting on the solutions. Non-linearities of product type are shown to admit this via exact paralinearization. The parametrices give regularity properties under weak conditions; improvements in subdomains result from pseudo-locality of type 1,1-operators. The framework encompasses a broad class of boundary problems in Hölder and L p -Sobolev spaces (and also Besov and Lizorkin–Triebel spaces). The Besov analyses of homogeneous distributions, tensor products and halfspace extensions have been revised. Examples include the von Karman equation.

An exact finite strip for the calculation of relative post-buckling stiffness of I-section struts

This paper presents the theoretical developments of an exact finite strip for the buckling and initial post-buckling analyses of I-section struts. The so-called exact strip is developed based on the concept that it is effectively a plate. The presented method, which is designated by the name Full-analytical Finite Strip Method, provides an efficient and extremely accurate buckling solution. In the development process, the Von-Karman's equilibrium equation is solved exactly to obtain the buckling loads and mode shapes for the I-section struts. The investigation of buckling behavior is then extended to an initial post-buckling study with the assumption that the deflected form immediately after the buckling is the same as that obtained for the buckling. The current post-buckling study is effectively a single-term analysis, which is attempted by utilizing the so-called semi-energy method. Through the solution of the Von-Karman's compatibility equation, the in-plane displacement functions which are themselves related to the Airy stress function are developed in terms of the unknown coefficient in the assumed out-of-plane deflection function. All the displacement functions are then substituted in the total strain energy expressions. The theorem of minimum total potential energy is subsequently applied to solve for the unknown coefficient. Finally, the developed method is subsequently applied to analyze the initial post-buckling behavior of some representative I-sections for which the results were also obtained through the application of a Semi-energy Finite Strip Method [Ovesy HR. The development and application of a semi-energy post-local buckling finite strip. PhD dissertation, Cranfield University, UK, 1998]. Through the comparison of the results and the appropriate discussion, the knowledge of the level of capability of the developed method is significantly promoted.

Nonlinear analysis of a thin circular functionally graded plate and large deflection effects on the forces and moments

The dynamic von Karman equations are used for nonlinear analysis of a thin circular plate made of a functionally graded material. The thickness of the plate is constant and the properties of the functionally graded material depend on temperature and vary throughout the thickness. It is assumed that the plate oscillates with large amplitudes. The forces and moments in the plate are determined in solving the equations for harmonic vibrations. Relevant results are obtained in the case of stead-state free vibrations. These results indicate that the volume fraction has a strong effect on the forces, moments, and material properties

Vibration amplitude and thermal effects on the nonlinear behavior of thin circular functionally graded plates

In this paper, the nonlinear free axisymmetric vibration of a thin circular functionally graded plate in thermal environment is formulated in terms of von-Karman's dynamic equations, and a semi-analytical approach is developed. The plate thickness is constant and the material properties of the functionally graded plate are assumed to vary continuously through the thickness, according to a power-law distribution of the volume fraction of the constituents. For harmonic vibrations, by using assumed-time-mode method and Kantorovich time averaging technique, governing equations are solved. The nonlinear frequencies and associated stresses are determined at large amplitudes of vibration. Effects of material compositions and thermal loads on the vibration characteristics and stresses are examined. The numerical results obtained here are compared with available published results, based on various approaches.

Elastic stability of inhomogeneous thin plates on an elastic foundation

We study the elastic stability of infinite inhomogeneous thin plates on an elastic foundation under in-plane compression. The elastic stiffness constants depend on the coordinate variable in the thickness direction of the plate. The elastic foundation is represented as a Winkler-type model characterized by linear and nonlinear spring constants. First we derive the Foppl–von Karman equations by taking variations of the elastic strain energy. Next we develop the linear stability analysis of the plate under uniform in-plane compression and explicitly derive the critical loads and wave numbers for particular three cases. The effects of the material inhomogeneity, material orthotropy and loading orthotropy on the critical states are examined independently. Finally, we perform a weakly nonlinear analysis of the plate at the onset of the buckling instability. With the multiple scales method, the amplitude equations for the unstable modes that provide insight into the mode type and its amplitude are derived and then the effect of the material inhomogeneity on buckling modes are evaluated qualitatively.

Description of the solution set of the von Kármán equations for a circular plate in a small neighbourhood of a simple bifurcation point

In this work we study the von Karman system for a thin circular elastic plate fixed to the elastic base and subjected to the compressing force along its boundary. The system is composed of two fourth-order nonlinear partial differential equations that give a valid mathematical description of the buckling of the plate. We intend to demonstrate the applicability of nonlinear functional analysis in the study of this problem. We describe the solution set of the von Karman equations in a small neighbourhood of a simple bifurcation point.

Buckling of a flush-mounted plate in simple shear flow

The buckling of an elastic plate with arbitrary shape flush-mounted on a rigid wall and deforming under the action of a uniform tangential load due to an overpassing simple shear flow is considered. Working under the auspices of the theory of elastic instability of plates governed by the linear von Karman equation, an eigenvalue problem is formulated for the buckled state resulting in a fourth-order partial differential equation with position-dependent coefficients parameterized by the Poisson ratio. The governing equation also describes the deformation of a plate clamped around the edges on a vertical wall and buckling under the action of its own weight. Solutions are computed analytically for a circular plate by applying a Fourier series expansion to derive an infinite system of coupled ordinary differential equations and then implementing orthogonal collocation, and numerically for elliptical and rectangular plates by using a finite-element method. The eigenvalues of the resulting generalized algebraic eigenvalue problem are bifurcation points in the solution space, physically representing critical thresholds of the uniform tangential load above which the plate buckles and wrinkles due to the partially compressive developing stresses. The associated eigenfunctions representing possible modes of deformation are illustrated, and the effect of the Poisson ratio and plate shape is discussed.