On the Free Boundary Conditions for a Dynamic Shell Model Based on Intrinsic Differential Geometry

The mathematical theory behind the modeling of shells is a crucial issue in many engineering problems. Here, the authors derive the free boundary conditions and associated strong form of a dynamic shallow Kirchhoff shell model based on the intrinsic geometry methods of Michael Delfour and Jean-Paul Zolésio. This model relies on the oriented distance function which describes the geometry. This is an extension of the work done in [J. Cagnol, I. Lasiecka, C. Lebiedzik and J.-P. Zolésio (2002). Uniform stability in structural acoustic models with flexible curved walls. J. Differential Equation, 186(1), 88–121.], where the model was derived for clamped boundary conditions only. In the current article, manipulations with the model result in a cleaner form where the displacement of the shell and shell boundary is written explicitly in terms of standard tangential operators.     

Justification of two dimensional model of shallow shells using gamma convergence

In this paper we derive the two dimensional model of elastic shallow shell using gamma convergence. We consider thin elastic shallow shells of “very small” thickness and we show that the sequence of functions minimizing the energy associated with the three-dimensional elastic shallow shells converges to the function which minimizes the energy associated with the two dimensional elastic shallow shell as the thickness of the shell goes to zero.     

Micro-vibrations of thin cylindrical shells with uniperiodic structure

Micro-vibrations of thin linear-elastic Kirchhoff-Love cylindrical shells, having a periodic structure along one direction tangent to the shell midsurface, are investigated using the new averaged non-asymptotic model of such shells proposed by Tomczyk (2008). This model describes the effect of microstructure size on the overall shell behaviour and makes it possible to analyze the shell's micro-dynamics independently of its macro-dynamics. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

正交异性劈锥面扁壳屋盖的计算

本文介绍工业建筑中常用的正交各向异性劈锥面扁壳屋盖的计算.由于该劈锥面扁壳的中曲面存在变曲率和扭率,故所得的基本方程组是变系数的,欲得到精确的解析解在数学上有很大的困难.本文用小参数法求出在均布载荷作用之下,四边简支的正交各向异性劈锥面扁壳屋盖的近似解答.     

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.     

Vibrations of thin cylindrical shells with a periodic structure

Free vibrations of thin linear–elastic Kirchhoff–Love cylindrical shells, having a periodic structure along one direction tangent to the shell midsurface, is considered. In order to take into account the effect of the periodicity cell size in this problem, a new averaged non–asymptotic model of such shells, proposed by Tomczyk (2006), is applied. The new additional higher–order free vibration frequencies dependent on the microstructure size will be derived and discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Calculation of the natural-vibration spectra of layered shells of moderate thickness

Conclusions Relations from a linear, kinematically nonuniform model of a layered shell were used to construct a system of motion equations for an M-layered shallow shell which considered all components of the stress-strain state and inertia of the shell. It was shown using sample calculations of the natural frequency spectrum of physically uniform and hybrid threelayer hells that this model makes it possible in a linear approximation to calculate the complete natural-frequency spectrum of layered shells. It can be used in engineering calculations of the dynamic characteristics of shells in which relatively thin and stiff bearing layers alternate in the packet with layers of a soft filler (structurally nonuniform hybrid shells).The use of simplified (classical) models, refined kinematically uniform models, and nonuniform models not accounting for compressive strains in the shell layers, etc. (see [1, 5]) is limited to the classes of physically uniform and quasiuniform shells and to cases of calculation of the dynamic characteristics determined by three fundamental frequencies of the shell when regarded as a three-dimensional body.Translated from Mekhanika Kompozitnykh Materialov, No. 2, pp. 298–304, March–April, 1985.     

Load and frequency interaction in dynamic buckling of soft-core sandwich plates and shells

The present study is concerned with an analysis of load-frequency interaction effects in the dynamic buckling response of soft-core sandwich plates and shells. Based on a higher-order geometrically non-linear sandwich shell theory in conjunction with an extended Galerkin procedure, a mathematical model together with analytical solution for simply supported sandwich plates and shells with with rectangular projection is obtained. By linearization of the non-linear solution with respect to the oscillating parts, the interaction of the natural frequencies with static preloads can be analyzed. The results reveal that the transverse compressibility of soft sandwich cores might have distinct effects on the natural frequencies. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

球壳的环向剪切屈曲

通过球壳微元初始屈曲的微分几何分析,推导出一组新的精确的屈曲分支方程,并且应用Galerkin变分法研究铰支球壳承受环向剪切力时的整体稳定性,构造了接近分支点变形状态的屈曲模式,首次求得了从扁球壳到半球壳大范围内的扭转屈曲临界特征值,临界荷载强度和临界应力.     

均布载荷作用下波纹扁壳的非线性振动

应用轴对称旋转扁壳的非线性大挠度动力学方程,研究了波纹扁壳在均布载荷作用下的非线性受迫振动问题．采用格林函数方法,将扁壳的非线性偏微分方程组化为非线性积分微分方程组．再使用展开法求出格林函数,即将格林函数展开为特征函数的级数形式,积分微分方程就成为具有退化核的形式,从而容易得到关于时间的非线性常微分方程组．针对单模态振形,得到了谐和激励作用下的幅频响应．作为算例,研究了正弦波纹扁球壳的非线性受迫振动现象．该文的解答可供波纹壳的设计参考．     

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·     

扁球壳在均布压力与均匀温度场联合作用下的屈曲

根据扁壳几何非线性理论,推导了均布压力与均匀温度场联合作用下的扁球壳的位移型几何非线性控制方程.考虑夹紧边界条件,采用打靶法得到了扁球壳轴对称弯曲与屈曲的数值结果.讨论了壳体几何参数对平衡路径、临界荷载的影响.给出了壳体临界几何参数.当几何参数大于临界几何参数时,上、下临界荷载都随几何参数增加而增加.给定几何参数时,考察了不同均匀温度场对壳体上、下临界荷载、临界几何参数以及平衡构型的影响.均匀升温会使上临界荷载显著增加,会使下临界荷载略有减小.均匀变温会使临界几何参数改变.     

扁薄球壳非对称大变形问题

本文用修正选代法研究了扁球壳非对称大变形问题，求得了在线性液体载荷作用下的扁球壳变形的二次近似解析解并绘出了摄动点的挠度与载荷的特征曲线族．应用本文方法还可对其他板壳的非轴对称大变形问题进行讨论．本文通过算例对平板及不同初挠度的扁球壳大挠度变形进行了讨论．     

Some features of the natural flexural vibrations of circular rings with initial imperfections

The effect of small initial deviations from an ideal circular shape on the frequencies and modes of the natural oscillations of a thin ring under conditions of plane deformation is investigated. The analysis is based on the equations of motion of the shallow shell theory when the length of the shell tends to infinity. A new approach is proposed to the construction of a finite-dimensional model of a ring in which it is assumed that initial imperfections lead to the interaction of flexural and radial vibrations. A number of assertions are formulated and proved regarding the specific features which accompany the vibrations of an imperfect ring that apparently also have to be taken into account when investigating the dynamic characteristics of thin circular cylindrical shells with initial imperfections.     

A new finite element model for the analysis of arbitrary stiffened shells

A new stiffened shallow shell finite element has been introduced for the static analysis of stiffened plates and shells. This approach has been presented to cater for stiffeners in which the positions and properties remain undisturbed in the formulation and the element can accomodate the stiffener anywhere within the shell element and in any direction, which introduces a considerable flexibility in the analysis. This is a distinct improvement over the existing models. Stiffened shells having various disposition of stiffeners as available in the literature, have been analysed by the proposed approach. Comparison obtained with the existing theoretical and/or experimental values have indicated good accuracy with relatively coarser mesh sizes and less CPU time.     

Numerical solution of multipoint boundary-value problems in improved shell theory

For some classes of stratified shells subject to transverse shear (shells of revolution with parameters varying along the generatrix, noncircular cylindrical shells with characteristics varying along the directrix, rectangular shallow shells), we propose an approach that reduces the solution of multipoint boundary-value problems to a number of two-point problems. As an example, we consider the stressed state of an open noncircular cylindrical shell supported in some section along the generatrix.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 62, pp. 35–38, 1987.     

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)     

双层网格开顶扁球壳的非线性稳定性分析

根据基于等效夹层壳思想的双层网格圆底扁球壳在极坐标下的平衡方程、相容方程,采用修正迭代法,对外边缘滑动固定、内边缘悬空和外边缘夹紧固定、内边缘悬空两种边界条件下,双层网格开顶圆底扁球壳的非线性稳定性进行了分析,得出了非线性载荷 位移关系及临界荷载的解析表达式,并讨论和分析了网壳几何参数对临界屈曲载荷的影响.