Asymptotic analysis for a dynamic piezoelectric shallow shell

The paper deals with the asymptotic formulation and justification of a mechanical model for a dynamic piezoelastic shallow shell in Cartesian coordinates. Starting from the three‐dimensional dynamic piezoelastic problem and by an asymptotic approach, the authors study the convergence of the displacement field and of the electric potential as the thickness of the shell goes to zero. In order to obtain a nontrivial limit problem by asymptotic analysis, we need different scalings on the mass density. The authors show that the transverse mechanical displacement field coupled with the in‐plane components solves an problem with new piezoelectric characteristics and also investigate the very popular case of cubic crystals and show that, for two‐dimensional shallow shells, the coupling piezoelectric effect disappears. Copyright © 2017 John Wiley & Sons, Ltd.     

2D modelling of thin skeletonal elastic shallow shells

The subject of this contribution is a certain thin skeletonal elastic shallow shell. The aim of analysis is to derive and apply a 2D-macroscopic model for shallow shells with the non-uniformly oscillating microstructure. The main feature of the proposed mathematical model is that the microstructure length parameter λ is similar compared to thickness δ of the shell (λ ≃ δ). The formulation of approximate mathematical model of these shells is based on a tolerance averaging approximation [5]. The general results of the contribution will be illustrated by the analysis of a specific problem. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Justification of a simplified model for shells in nonlinear elasticity

We introduce a simplified model for the minimization of the elastic energy in thin shells. The thickness of the shell remains a parameter in this new model.     

Vibrations of thin piezoelectric shallow shells: Two-dimensional approximation

In this paper we consider the eigenvalue problem for piezoelectric shallow shells and we show that, as the thickness of the shell goes to zero, the eigensolutions of the three-dimensional piezoelectric shells converge to the eigensolutions of a twodimensional eigenvalue problem.     

变厚度扁锥壳的非线性固有频率

借助于变厚度圆薄板非线性动力学变分方程和协调方程,给出了变厚度扁薄锥壳的非线性动力学变分方程和协调方程·　假设薄膜张力由两项组成,将协调方程化为两个独立的方程,选取变厚度扁锥壳中心最大振幅为摄动参数,采用摄动变分法,将变分方程和微分方程线性化·　对周边固定的圆底变厚度扁锥壳的非线性固有频率进行了求解;一次近似得到了变厚度扁锥壳的线性固有频率,三次近似得到了变厚度扁锥壳的非线性固有频率,且绘出了固有频率与静载荷、最大振幅、变厚度参数的特征曲线图·　为动力工程提供了有价值的参考·     

ASYMPTOTIC ANALYSIS OF LINEARLY ELASTIC SHALLOWSHELLS WITH VARIABLE THICKNESS

The author considers a linearly elastic shallow shell with variable thickness and shows that, as the thickness of the shell goes to zero, the solution of the three-dimensional equations converges to the solution of the two-dimensional shallow shell equations with variable thickness.     

ASYMPTOTIC ANALYSIS OF LINEARLY ELASTIC SHALLOWSHELLS WITH VARIABLE THICKNESS

g1. IntroductionIn this paper j we identify the two-dimensiona1 model of a shallow shell with variablethickness. More precisely we consider a family of linearly elastic shallow shells with variablethickness. We show that, if the aPplied forces are of specific order of magnitude, the covariaatcomponents of the scaled displacement field converge, as the thickness of the shell goesto zerQ, to a two dimensional problem that constitutes the model of a shallow shell withvariable thickness. The key …     

On the investigation of free vibrations of nonthin cylindrical shells of variable thickness by the spline-collocation method

For determining the dynamic characteristics of free vibrations of circular unclosed cylindrical shells of variable thickness in two coordinate directions, we have used the spline-collocation method together with the method of discrete orthogonalization. The problem has been solved within the framework of the refined Timoshenko–Mindlin theory. We have also investigated the influence of different laws of change in the shell thickness on the character of its natural vibrations. Our calculations have been carried out for different geometrical and elastic parameters of the shell under study and different boundary conditions.     

The possibility of a theoretical confirmation of the experimental values of the external critical pressure of thin-walled cylindrical shells

The problem of the buckling of elastic, isotropic, thin-walled cylindrical shells with small initial shape defects that are under the action of an external pressure is solved in a geometrically non-linear formulation. Equations that are identical to Marguerre's equations for a shallow cylindrical shell are used in formulating the problem. The solution is constructed by the Rayleigh–Ritz method with the points of the middle surface of the shell approximated by double functional sums over trigonometric and beam functions. The system of non-linear equations obtained is solved by arc-length methods. Cases of the clamped and supported shells when loading with a lateral and uniform hydrostatic pressure are considered. Its deflections from the limit points of the postbuckling branches of its loading trajectory are used as the initial imperfections. An inspection of the different forms of the initial imperfections when they have maximum values of up to 30% of the shell thickness made it possible to obtain practically the whole range of experimentally found critical pressures.     

Nonlinear flexural vibrations of composite shallow open shells

This paper addresses geometrically nonlinear flexural vibrations of open doubly curved shallow shells composed of three thick isotropic layers. The layers are perfectly bonded, and thickness and linear elastic properties of the outer layers are symmetrically arranged with respect to the middle surface. The outer layers and the central layer may exhibit extremely different elastic moduli with a common Poisson's ratio ν. The considered shell structures of polygonal planform are hard hinged supported with the edges fully restraint against displacements in any direction. The kinematic field equations are formulated by layerwise application of a first order shear deformation theory. A modification of Berger's theory is employed to model the nonlinear characteristics of the structural response. The continuity of the transverse shear stress across the interfaces is specified according to Hooke's law, and subsequently the equations of motion of this higher order problem can be derived in analogy to a homogeneous single-layer shear deformable shallow shell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

斜放四角锥扁网壳的非线性弯曲理论

双层网壳是大型空间结构的主要结构形式,斜放四角锥扁网壳就是其中一种.它主要依靠上、下表层承受载荷,网壳腹部则比较空而且柔.根据斜放四角锥扁网壳的几何和力学特点,在三个基本假定的基础上,把它连续化并等效成一夹层扁壳.先从能量和内力等效的角度来分析它的本构关系,然后运用虚功原理,推导出斜放四角锥扁网壳几何非线性弯曲理论的基本方程.     

双层网格圆底扁球壳的非线性稳定问题

从基于等效夹层壳思想的双层网格扁壳,非线性弯曲理论的变分方程出发,利用坐标变换方法和驻值余能原理,导出双层网格圆底扁球壳,在均布压力作用下的轴对称大挠度方程和边界条件．采用修正迭代法,求得了两类边界条件下双层网格圆底扁球壳的非线性载荷-位移关系式和临界屈曲载荷的解析表达式,并讨论了几何参数对临界屈曲载荷的影响．     

波纹壳的格林函数方法

应用轴对称旋转扁壳的基本方程,研究了在任意载荷作用下具有型面锥度的浅波纹壳的非线性弯曲问题·　采用格林函数方法,将扁壳的非线性微分方程组化为非线性积分方程组·　再使用展开法求出格林函数,即将格林函数展成特征函数的级数形式,积分方程就成为具有退化核的形式,从而容易得到非线性代数方程组·　应用牛顿法求解非线性代数方程组时,为了保证迭代的收敛性,选取位移作为控制参数,逐步增加位移,求得相应的载荷·　在算例中,研究了具有球面度的浅波纹壳的弹性特征·　结果表明,由于型面锥度的引入,特征曲线发生显著变化,随着荷载的增加,将出现类似扁球壳的总体失稳现象·　本文的解答符合实验结果·     

TWO-DIMENSIONAL APPROXIMATION OF EIGENVALUE PROBLEMS IN SHELL THEORY： FLEXURAL SHELLS

1.IntroductionInthispaper,westudythelimitingbehaviourofeigenvaluesandeigenfunctionsdescribingthevibrationsofathinlinearlyelasticshell,clampedalongitslateralsurface,underageometricassumptiononthemiddlesurfaceoftheshellthatthespaceofinextensionaldisplacements(of.(4.2))isnonzero.Inthestationarycasegunderadditionalassumptionsontheorderofmagnitudeofthebodyforces,thisleadstothetwo-dimensionalmodelofthe"fie-curalshell"asshownbyCiarlet,LodsandMiara[5].Examplesofclampedshellswhichobeytheabovegeometric…     

Exact Controllability and Asymptotic Analysis for Shallow Shells

The authors consider the exact controllability of the vibrations of a thin shallow shell, of thickness 2εwith controls imposed on the lateral surface and at the top and bottom of the shell. Apart from proving the existence of exact controls, it is shown that the solutions of the three dimensional exact controllability problems converge, as the thickness of the shell goes to zero, to the solution of an exact controllability problem in two dimensions.     

On a dynamical hierarchical model for prismatic shells in the theory of elastic mixtures

The present paper is devoted to the design of a hierarchy of two‐dimensional models for dynamical problems within the theory of multicomponent linearly elastic mixtures in the case of prismatic shells with thickness which may vanish on some part of its boundary. The hierarchical model is obtained by a semidiscretization of the three‐dimensional problem in the transverse direction. In suitable weighted Sobolev spaces we investigate the well‐posedness of the two‐dimensional problems, prove pointwise convergence of the sequence of approximate solutions restored from the solutions of the reduced problems to the exact solution of the original problem and estimate the rate of convergence. Copyright © 2005 John Wiley & Sons, Ltd.     

The limits of hamiltonian structures in three-dimensional elasticity,shells, and rods

Summary This paper uses Hamiltonian structures to study the problem of the limit of three-dimensional (3D) elastic models to shell and rod models. In the case of shells, we show that the Hamiltonian structure for a three-dimensional elastic body converges, in a sense made precise, to that for a shell model described by a one-director Cosserat surface as the thickness goes to zero. We study limiting procedures that give rise to unconstrained as well as constrained Cosserat director models. The case of a rod is also considered and similar convergence results are established, with the limiting model being a geometrically exact director rod model (in the framework developed by Antman, Simo, and coworkers). The resulting model may or may not have constraints, depending on the nature of the constitutive relations and their behavior under the limiting procedure. The closeness of Hamiltonian structures is measured by the closeness of Poisson brackets on certain classes of functions, as well as the Hamiltonians. This provides one way of justifying the dynamic one-director model for shells. Another way of stating the convergence result is that there is an almost-Poisson embedding from the phase space of the shell to the phase space of the 3D elastic body, which implies that, in the sense of Hamiltonian structures, the dynamics of the elastic body is close to that of the shell. The constitutive equations of the 3D model and their behavior as the thickness tends to zero dictates whether the limiting 2D model is a constrained or an unconstrained director model. We apply our theory in the specific case of a 3D Saint Venant-Kirchhoff material andderive the corresponding limiting shell and rod theories. The limiting shell model is an interesting Kirchhoff-like shell model in which the stored energy function is explicitly derived in terms of the shell curvature. For rods, one gets (with an additional inextensibility constraint) a one-director Kirchhoff elastic rod model, which reduces to the well-known Euler elastica if one adds an additional single constraint that the director lines up with the Frenet frame. This paper is dedicated to the memory of Juan C. Simo This paper was solicited by the editors to be part of a volume dedicated to the memory of Juan Simo.     

弹性地基上双曲率扁壳的优化设计问题

本文讨论位于弹性地基上的双曲率扁壳优化设计的一个方法。其实质是取扁壳的初始挠曲函数作为待求的控制函数或设计变量,以载荷的势能作为判定双曲率扁壳优化设计的质量准则,故势能泛函即为目标函数,而优化条件及等周条件均作为约束条件,从而得到本问题优化设计的必要条件。同时引入一共轭函数,最后将问题归结为求解共轭函数的微分方程及初始挠曲函数两个边值问题。     

Investigation of the Stress State of Pressure Vessels of Inhomogeneous Structure Made of Composite Materials

The problem on the stress-strain state of layered cylindrical shells with bottoms of intricate shape under the action of internal pressure is considered. The elastic system examined is formed by spiral-circular winding. Two variants of the shell bottom structure are investigated. In the first variant, one spiral layer is installed, which leads to great variations in the bottom thickness along the meridian. In the second one, the bottoms are formed according to the zone-winding scheme. The stress state of the shell constructions of the classes considered is determined by solving boundary-value problems for systems of ordinary differential equations. The solution results for cylindrical shells with elliptic bottoms for the two types of winding are given. It is shown that the zone winding leads to smaller deflections and stresses than the conventional ways of reinforcing shell bottoms.     

From the classical to the generalized von Kármán and Marguerre–von Kármán equations

In this work, we describe and analyze two models that were recently proposed for modeling generalized von Kármán plates and generalized Marguerre–von Kármán shallow shells.

First, we briefly review the “classical” von Kármán and Marguerre–von Kármán equations, their physical meaning, and their mathematical justification. We then consider the more general situation where only a portion of the lateral face of a nonlinearly elastic plate or shallow shell is subjected to boundary conditions of von Kármán type, while the remaining portion is free. Using techniques from formal asymptotic analysis, we obtain in each case a two-dimensional boundary value problem that is analogous to, but is more general than, the classical equations.

In particular, it is remarkable that the boundary conditions for the Airy function can still be determined on the entire boundary of the nonlinearly elastic plate or shallow shell solely from the data.

Following recent joint works, we then reduce these more general equations to a single “cubic” operator equation, which generalizes an equation introduced by Berger and Fife, and whose sole unknown is the vertical displacement of the shell. We next adapt an elegant compactness method due to Lions for establishing the existence of a solution to this operator equation.