Finite deflection analysis of elastic plate by the boundary element method

An integral equation formulation for finite deflection analysis of thin elastic plates is presented, based on general nonlinear differential equations which are equivalent to the von Kármán equations and by virtue of generalized Green identities. Boundary element discretization is applied and a relaxation iterative approach is employed to solve the nonlinear plate bending problems. A number of numerical examples are given; the results of computation are compared with the analytical solutions and good agreement is observed. It appears that the approach developed in this paper is effective.     

A unilateral contact model with buckling in von Kármán plates

The unilateral contact problem for the von Kármán plate including postbuckling is numerically studied in this paper. The mathematical model consists of a system of nonlinear inequalities and equations for the transversal displacements and the stress function on the middle plane of the plate, respectively. The boundary conditions correspond to simply supported or partially clamped plates. The lateral displacements are constrained by the presence of a rigid support. A variational principle with penalty is used to treat the mechanical model. Then the variational penalized problem is solved by a spectral method. For the obtained discrete model we develop an iterative scheme based on Newton's iterations, combined with numerical continuation coupled with an appropriate procedure for the choice of the penalty and regularization parameters. Numerical results demonstrate the effectiveness of the proposed method.     

Nonlinear dynamic characteristics of bi-graphene sheets/piezoelectric laminated films considering high order van der Walls force and scale effect

This paper presents an analytical method to investigate the nonlinear vibration characteristics of bi-graphene sheets/piezoelectric (BGP) laminated films subjected to electric loading based on a nonlocal continuum model, in which the two adjacent layers are coupled by van der Walls force. Utilizing von Kármán nonlinear geometric relation and nonlocal physical relation, the nonlinear dynamic equation of BGP laminated films under electric loading exerted on the piezoelectric layer is found, then the relation between the nonlinear resonant frequency and the nonlinear vibration amplitude for each layer of the BGP laminated films is obtained by using Galerkin method and harmonic-balance method. Results show that the nonlinear vibration amplitude for each layer of laminated films can be controlled by adjusting the electric potential exerted on piezoelectric layer, and the coupled effect of van der Walls force between graphene sheet and piezoelectric layer on the vibration amplitude of each layer depends on the order number of nonlinear resonant frequency and the mode shape.     

The necessary and sufficient condition for bifurcation in the von Kármán equations

The paper is devoted to the study of bifurcation in the von Kármán equations with two parameters that describe the behaviour of a thin round elastic plate lying on an elastic base under the action of a compressing force. The problem appears in the mechanics of elastic constructions. We prove the necessary and sufficient condition for bifurcation at points of the set of trivial solutions. Our proof is based on reducing the von Kármán equations to an operator equation in Banach spaces with a nonlinear Fredholm map of index 0 and applying the Crandall-Rabinowitz theorem on simple bifurcation points or a finite-dimensional reduction and degree theory. RID="h1" ID="h1"This research was supported by grant BW of UG no. 5100-5-0153-1 and by grant KBN no. 5 P03A 020 20.     

Nonlinear stability and vibration of pre/post-buckled multilayer FG-GPLRPC rectangular plates

The present study examines the nonlinear stability and free vibration features of multilayer functionally graded graphene platelet-reinforced polymer composite (FG-GPLRPC) rectangular plates under compressive in-plane mechanical loads in pre/post buckling regimes. The GPL weight fractions layer-wisely vary across the lateral direction. Furthermore, GPLs are uniformly dispersed in the polymer matrix of each layer. The effective Young's modulus of GPL-reinforced nanocomposite is assessed via the modified Halpin–Tsai technique, while the effective mass density and Poisson's ratio are attained by the rule of mixture. Taking the von Kármán-type nonlinearity into account for the large deflection of the FG-GPLRPC plate, as well as utilizing the variational differential quadrature (VDQ) method and Lagrange equation, the system of discretized coupled nonlinear equations of motions is directly achieved based upon a parabolic shear deformation plate theory; taking into account the impacts of geometric nonlinearity, in-plane loading, rotary inertia and transverse shear deformation. Afterwards, first, by neglecting the inertia terms, the pseudo-arc length approach is used in order to plot the equilibrium postbuckling path of FG-GPLRPC plates. Then, supposing a time-dependent disturbance about the postbuckling equilibrium status, the frequency responses of pre/post-buckled FG-GPLRC plate are obtained in terms of the compressive in-plane load. The influences of various vital design parameters are discussed through various parametric studies.     

The geometrically nonlinear dynamic responses of simply supported beams under moving loads

This paper presents a method for determining the nonlinear dynamic responses of structures under moving loads. The load is considered as a four degrees-of-freedom system with linear suspensions and tires flexibility, and the structure is modeled as an Euler–Bernoulli beam with simply supported at both ends. The nonlinear dynamic interaction of the load–structure system is discussed, and Kelvin−Voigt material model is employed for the beam. The nonlinear partial differential equations of the dynamic interaction are derived by using the von Kármán nonlinear theory and D'Alembert's principle. Based on the Galerkin method, the partial differential equations of the system are transformed into nonlinear ordinary equations, which can be solved by using the Newmark method and Newton−Raphson iteration method. To validate the approach proposed in this paper, the comparison are performed using a moving mass and a moving oscillator as the excitation sources, and the investigations demonstrate good reliability.     

The spectral method and numerical continuation algorithm for the von Kármán problem with postbuckling behaviour of solutions

In this paper a spectral method and a numerical continuation algorithm for solving eigenvalue problems for the rectangular von Kármán plate with different boundary conditions (simply supported, partially or totally clamped) and physical parameters are introduced. The solution of these problems has a postbuckling behaviour. The spectral method is based on a variational principle (Galerkin’s approach) with a choice of global basis functions which are combinations of trigonometric functions. Convergence results of this method are proved and the rate of convergence is estimated. The discretized nonlinear model is treated by Newton’s iterative scheme and numerical continuation. Branches of eigenfunctions found by the algorithm are traced. Numerical results of solving the problems for polygonal and ferroconcrete plates are presented. Communicated by A. Zhou.     

Numerical analysis of the generalized von Kármán equations

The ‘generalized von Kármán equations’ constitute a mathematical model for a nonlinearly elastic plate subjected to boundary conditions ‘of von Kármán type’ only on a portion of its lateral face, the remaining portion being free. We establish here the convergence of a conforming finite element approximation to these equations. The proof relies in particular on a compactness method due to J.-L. Lions and on Brouwer's fixed point theorem. To cite this article: P.G. Ciarlet et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

On the compatibility relation for the Föppl–von Kármán plate equations

By using a coordinate-free approach we propose a new derivation of the compatibility equation for the Föppl–von Kármán nonlinear plate theory.     

Asymptotic modelling of a Signorini problem of generalized Marguerre-von Kármán shallow shells

Chacha and Bensayah [Asymptotic modeling of a Coulomb frictional Signorini problem for the von Kármán plates, C. R. Mécanique 336 (2008), pp. 846–850] have studied the asymptotic modelling of Coulomb frictional unilateral contact problem between an elastic nonlinear von Kármán plate and a rigid obstacle. The main result obtained is that the leading term of the asymptotic expansion is characterized by a two-dimensional Signorini problem but without friction. In this article, we extend this study to the case of a shallow shell under generalized Marguerre-von Kármán conditions.     

Un théorème d'existence pour les équations de von Kármán généralisées

Using techniques from formal asymptotic analysis, the first two authors have recently identified “generalized von Kármán equations”, which constitute a two-dimensional model for a nonlinearly elastic plate where only a portion of the lateral face is subjected to boundary conditions of von Kármán's type, the remaining portion being free.In this Note, we establish an existence theorem for these equations. To this end, we notably adapt a compactness method due to J.-L. Lions.     

On the Cauchy problem for the full von Kármán system

We consider here the Cauchy problem for the full system of dynamic Von Kármán equations, which is a model for the vibrations of a nonlinear elastic plate. We prove global existence and uniqueness of finite energy solutions in the case of an infinite plate. We show then that our methods and results still hold for a rectangular plate which is simply supported or clamped at the boundary. Moreover we obtain continuous dependence on the initial data. Received December 25, 1995     

Duality Theory in Nonlinear Buckling Analysis for von Kármán Equations

The nonlinear eigenvalue problem in buckling analysis is studied for von Kármán plates. By using the general duality theory developed by Gao-Strang [1, 2] it is proved that the stability criterion for the bifurcated state depends on a reduced complementary gap function. The duality theory is established for nonlinear bifurcation problems. This theory shows that the nonlinear eigenvalue problem is eventually equivalent to a coupled quadratic dual optimization problem. A series of equivalent variational principles are constructed and a lower bound theorem for the first eigenvalue of the buckling factor is proved.     

Conservative numerical methods for the Full von Kármán plate equations

This article is concerned with the numerical solution of the full dynamical von Kármán plate equations for geometrically nonlinear (large‐amplitude) vibration in the simple case of a rectangular plate under periodic boundary conditions. This system is composed of three equations describing the time evolution of the transverse displacement field, as well as the two longitudinal displacements. Particular emphasis is put on developing a family of numerical schemes which, when losses are absent, are exactly energy conserving. The methodology thus extends previous work on the simple von Kármán system, for which longitudinal inertia effects are neglected, resulting in a set of two equations for the transverse displacement and an Airy stress function. Both the semidiscrete (in time) and fully discrete schemes are developed. From the numerical energy conservation property, it is possible to arrive at sufficient conditions for numerical stability, under strongly nonlinear conditions. Simulation results are presented, illustrating various features of plate vibration at high amplitudes, as well as the numerical energy conservation property, using both simple finite difference as well as Fourier spectral discretizations. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1948–1970, 2015     

Generalized Marguerre-von Kármán Equations for a Nonlinearly Elastic Shallow Shell

Using techniques from asymptotic analysis, we identify equations that generalize the classical Marguerre-von Kármán equations. These more general equations are justified by means of the leading term of a formal asymptotic expansion of the solution to the three-dimensional equations of nonlinear elasticity associated with a specific class of boundary conditions that characterizes "generalized Marguerre-von Kármán shallow shells".     

Effects of partial slip on the steady Von Kármán flow and heat transfer of a non-Newtonian fluid

The steady Von Kármán flow and heat transfer of a non-Newtonian fluid is extended to the case where the disk surface admits partial slip. The constitutive equation of the non-Newtonian fluid is modeled by that for a Reiner-Rivlin fluid. The momentum equations give rise to highly nonlinear boundary value problem. Numerical solutions for the governing nonlinear equations are obtained over the entire range of the physical parameters. The effects of slip and non-Newtonian fluid characteristics on the velocity and temperature fields have been discussed in detail and shown graphically.     

The time-dependent von K��rm��n plate equation as a limit of 3d nonlinear elasticity

The asymptotic behaviour of solutions of three-dimensional nonlinear elastodynamics in a thin plate is studied, as the thickness h of the plate tends to zero. Under appropriate scalings of the applied force and of the initial values in terms of h, it is shown that three-dimensional solutions of the nonlinear elastodynamic equation converge to solutions of the time-dependent von Kármán plate equation.     

正交异性矩形板后屈曲摄动分析

本文从正交异性板Kármán型大挠度方程出发,以挠度为摄动参数,采用直接摄动法,研究了正交异性矩形板在面内压缩作用下的后屈曲性态.本文讨论了两种面内边界条件,同时考虑了初始挠度的影响.本文给出了多种复合材料板的计算结果.所得结果与实验结果的比较表明二者是一致的.