Von Karman Equations Research Articles

Nonlinear statics of magneto-electro-elastic nanoplates considering flexomagnetoelectric effect based on nonlocal strain gradient theory

The nonlinear static behavior of hygrothermal magnetoelectroelastic(MEE) nanoplates considering the flexomagnetoelectric (FME) effect is researched by engaging the first-order shear deformation theory (FSDT). The constitutive equations of MEE nanoplates take into account the FME effect and hygrothermal effect. Leveraging the FSDT, von Karman's nonlinear equation and Hamilton's principle, the nonlinear control equation for hygrothermal MEE nanoplates can be derived by using the variational approach. The nonlocal strain gradient theory (NSGT) has application in the size effect of MEE nanoplates. The nonlocal nonlinear term in the control equation is handled by employing the Airy stress function, and then the non-linear mechanical model is solved by the Galerkin method. Thus, the nonlinear load-deflection curves (NLDC) of the MEE nanoplate are obtained via the introduction of material parameters, enabling an examination of the impact of various factors such as the FME effect, two small size parameters of NSGT and other parameters on the nonlinear bending behavior of MEE nanoplates. In conclusion, this study offers a theoretical foundation for taking into account the FME effect in the development of nanodevices, thereby contributing to advancements in this field.

Stability and nonlinear vibration of a fuel rod in axial flow with geometric nonlinearity and thermal expansion

The vibration of fuel rods in axial flow is a universally recognized issue within both engineering and academic communities due to its significant importance in ensuring structural safety. This paper aims to thoroughly investigate the stability and nonlinear vibration of a fuel rod subjected to axial flow in a newly designed high temperature gas cooled reactor. Considering the possible presence of thermal expansion and large deformation in practical scenarios, the thermal effect and geometric nonlinearity are modeled using the von Karman equation. By applying Hamilton's principle, we derive the comprehensive governing equation for this fluid-structure interaction system, which incorporates the quadratic nonlinear stiffness. To establish a connection between the fluid and structure aspects, we utilize the Galerkin method to solve the perturbation potential function, while employing mode expansion techniques associated with the structural analysis. Following convergence and validation analyses, we examine the stability of the structure under various conditions in detail, and also investigate the bifurcation behavior concerning the buckling amplitude and flow velocity. The findings from this research enhance the understanding of the underlying physics governing fuel rod behavior in axial flow under severe yet practical conditions, while providing valuable guidance for reactor design.

A general stability result for a viscoelastic von Karman equation

In this paper, we derive a general stability result for a viscoelastic von Karman equation without the condition that involves differential inequalities on the kernel. The kernel just stays below a certain known function as like where is a known function satisfying some specific properties. By considering such kernel functions, we expand earlier results for viscoelastic von Karman equations in the literature.

Well‐posedness and dynamics for the von Karman equation with memory and nonlinear time‐varying delay

This paper is concerned with mathematical analysis for the viscoelastic von Karman equation with memory and time‐varying delay. Based on the existence and uniqueness of the strong solution established by Galerkin method, the finite dimensional global attractor for the system without time‐varying delay has been shown by using the quasi‐stability theory. The difficulties for our desired well‐posedness are the limiting of fourth‐order derivative terms and the von Karman bracket term, which need some delicate estimates to achieve. The key point for the dynamics is to obtain the quasi‐stability for gradient system, which needs the higher regularity of trajectory especially the von Karman bracket.

On the von Karman's equation in the nonlinear theory of gas dynamics

On the von Karman's equation in the nonlinear theory of gas dynamics

Dynamic von Karman equations with viscous damping

In this paper we are interested to the dynamic von Karman equations coupled with viscous damping and without rotational forces, (α=0) [Chueshov I., Lasiecka I. (2010)], this problem describes the buckling and flexible phenomenon of small nonlinear vibration of vertical displacement to the elastic plates. Our fundamental goal is to establish the existence and the uniqueness to the weak solution for the so-called global energy, under assumption F0∈H3+ϵ(ω). Finally for illustrate our theoretical results we use the finite difference method.

Frequency response analysis of edge-cracked magneto-electro-elastic functionally graded plates using extended finite element method

This paper studies the frequency response of edge-cracked magneto-electro elastic functionally graded (ECMEE-FG) plates using the extended finite element method (XFEM). First-order shear deformation theory (FSDT), von Karman's nonlinear strain-displacement equations, and a modified power-laware used to develop the numerical model. The coupled equations are derived and analyzed using Hamilton's principle and extended finite element methods. The influence of B-rich bottom and F-rich bottom material gradation, crack orientation, crack length, and aspect ratio on the geometrically nonlinear frequency response was investigated after the current study was validated. Furthermore, crack propagation behavior in the ECMEE-FG plate was examined. The results could be helpful for the design of functionally graded magneto electro elastic structures and devices.

General decay for a viscoelastic von Karman equation with delay and variable exponent nonlinearities

In this paper, we consider a viscoelastic von Karman equation with damping, delay, and source effects of variable exponent type. Firstly, we show the global existence of solution applying the potential well method. Then, by making use of the perturbed energy method and properties of convex functions, we derive general decay results for the solution under more general conditions of a relaxation function. General decay results of solutions for viscoelastic von Karman equations with variable exponent nonlinearities have not been discussed before. Our results extend and complement many results for von Karman equations in the literature.

Nonlinear pyrocoupled dynamic response of functionally graded magnetoelectroelastic plates under blast loading in thermal environment

The nonlinear dynamic response of magnetoelectroelastic (MEE) functionally graded plates (MEEFG) subjected to blast loads and operating in a temperature environment is examined in this study. The goal of this research is to investigate the impact of pyrocoupling along with the blast loads on the coupled nonlinear behavior of smart MEEFG plates. The material properties of MEEFG plates are calculated using the well-known modified power law. The kinematics of the plate is supposed to follow Reddy's higher-order shear deformation theory (HSDT). Also, von-Karman's equations are used to bring nonlinearity into the model. The blast loads are calculated using an exponential function and are expected to be dispersed evenly across the plate's surface. The principle of virtual work is used to develop the governing equations using finite element (FE) methods, which is then solved using the direct iterative technique. The numerical examples are offered to show how gradient patterns, gradient index, and blast loading parameters affect the nonlinear dynamic response.

A GENERAL DECAY RESULT FOR A VON KARMAN EQUATION WITH MEMORY AND ACOUSTIC BOUNDARY CONDITIONS

<abstract> We study a viscoelastic von Karman equation of memory type with acoustic boundary conditions. Utilizing some properties of convex functions and the perturbed energy method, we build a general decay result when the kernel function $ k $ is a very general type. This work extends and complements some previous decay results of solutions for von von Karman equations. </abstract>

Linking the von Karman Equations to the Practical Design of Plates

AbstractThe Foppl-von Karman equations describe the highly nonlinear postbuckling behavior of elastic plates but are notorious for their unwieldiness. Owing to the lack of a sufficiently general so...

Blow-up for a viscoelastic von Karman equation with strong damping and variable exponent source terms

In this article, we deal with a strongly damped von Karman equation with variable exponent source and memory effects. We investigate blow-up results of solutions with three levels of initial energy such as non-positive initial energy, certain positive initial energy, and high initial energy. Furthermore, we estimate not only the upper bound but also the lower bound of the blow-up time.

General decay for a von Karman equation with memory and time-varying delay

This paper is concerned with a von Karman equation with memory and time-varying delay. For a more general type of relaxation function β, we prove the general decay results. This work generalizes and improves previous results for the von Karman equation.

Linking the von Karman equations to the design of steel plates

In this paper the von Karman system of partial differential equations describing the nonlinear postbuckling response of plates is simplified into a single equation, while taking caution to preserve the main mechanisms through which plates develop post-buckling reserve capacity. The resulting equation is solved for the case of a perfect plate using a single Fourier term, and for the case of an imperfect plate using two Fourier terms. Good agreement with finite element simulations, used as a benchmark, is obtained. The theory is further used to derive a closed form expression for the plate capacity as a function of the slenderness, which agrees very well with the well-known Winter equation.

Behavioural Study of Thin Plate Under Large Deflection Theory

For a thin plate, if the deformation is on the order of the thickness and stay elastic, linear theory might not turn out correct results because it does not predict the in plane movement of the member. Therefore, to account for the inconsistencies of geometric nonlinearity, large deflection theory is required [1]. This report pertains to the analytical study dispensed to check the behavior of thin plate under fixed and pinned edge conditions, and for diverse thicknesses, under the small and large deflection theories. The deformation is additionally studied, supported by Von-Karman equations. Non linear analysis has been performed on FE model using the ANSYS software. The consequences of geometric nonlinearities are mentioned. Outline on conclusion of the theoretical and experimental results obtained, are compared so as to review the similarity of the modeling and theory.

In-plane stiffness of imperfect thin rectangular plates subjected to biaxial loads in elastic post-buckling region

The elastic post-buckling behavior of thin plates covers a relatively vast region in which geometry nonlinearity (large deflection) and material linearity (Hooke's low) are realized. In this region, a thin rectangular plate has constant stiffnesses in both orthogonal directions. Few simplified analysis guidelines have been analytically represented for in-plane stiffnesses of an elastic post-buckled thin plate subjected to biaxial loads. In this study, Marguerre's equations (the generalized form of von Karman equations), which describe the elastic post-buckling behavior of imperfect thin plates, are solved. Galerkin's method is used to solve these equations in a semi-analytical procedure. Simply supported imperfect thin rectangular plates are considered, and the stresses and displacements functions are obtained in two orthogonal directions to determine corresponding in-plane stiffnesses of the plate. Also, the maximum applicable load is obtained so that the material's linear behavior is maintained. The semi-analytical procedure has accuracy enough to predict the in-plane stiffness of post-buckled plates and can be easily used for practical purposes.

Existence and decay results for a von Karman equation with variable exponent nonlinearities

In this article, we consider a von Karman equation with variable exponent nonlinearities in a bounded domain . We firstly discuss an existence result of solutions by utilizing Faedo‐Galerkin approximation technique. Then, we undertake an investigation of asymptotic stability to the solutions by making use of the multiplier method.

Hessian discretisation method for fourth-order semi-linear elliptic equations: applications to the von Kármán and Navier–Stokes models

This paper deals with the Hessian discretisation method (HDM) for fourth-order semi-linear elliptic equations with a trilinear non-linearity. The HDM provides a generic framework for the convergence analysis of several numerical methods, such as the conforming and nonconforming finite element methods (ncFEMs) and methods based on gradient recovery (GR) operators. The Adini ncFEM and GR method, a specific scheme that is based on cheap, local reconstructions of higher-order derivatives from piecewise linear functions, are analysed for the first time for fourth-order semi-linear elliptic equations with trilinear non-linearity. Four properties, namely, the coercivity, consistency, limit-conformity and compactness, enable the convergence analysis in HDM framework that does not require any regularity of the exact solution. Two important problems in applications, namely, the Navier–Stokes equations in stream function vorticity formulation and the von Karman equations of plate bending, are discussed. Results of numerical experiments are presented for the Morley and Adini ncFEMs, and the GR method.

Adaptive Morley FEM for the von Kármán Equations with Optimal Convergence Rates

The adaptive nonconforming Morley finite element method approximates a regular solution to the von Karman equations with optimal convergence rates for sufficiently fine triangulations and small bul...

Structural acoustic problem and dynamic nonlinear plate equations

ABSTRACT The purpose of this paper is to investigate a structural interaction model coupled with dynamic von Karman equations, without neither rotational inertia nor clamped boundary conditions. Our fundamental goal is to establish the existence and the uniqueness of the weak solution for the so-called global functional energy. The stability is also discussed. At the end, a numerical study based on the finite difference method is presented as well.