Classical and Nonclassical Dynamic Problems of Sandwich Shells with a Transversely Soft Core

Based on the hypothesis of similarity of transverse displacements in thin-walled sandwich shells with a transversely soft core under dynamic and static loads, refined geometrically nonlinear dynamic equations of motion are constructed in the case of large variations in the parameters of the stress-strain state (SSS) in the tangential directions. For shells structurally symmetric across the thickness and loaded with initial static loads, linearized dynamic equations are derived, which, upon introducing the synphasic and antiphasic functions of displacements and forces, can be used to describe the synphasic and antiphasic buckling forms in the transverse and tangential directions. For nonshallow cylindrical and shallow spherical shells, the nonclassical problems on all possible vibration forms realized at zero indices of variability of the SSS parameters in the tangential directions are formulated and solved. For shallow shells of symmetric structure, the resolving equations are obtained by introducing, instead of tangential displacements and transverse tangential stresses in the core, the corresponding potential and vortex functions.     

Refined equations of elliptic boundary layer in shells of revolution under normal shock surface loading

This paper continues the studies on the construction of the first order approximation of equations for a boundary layer in the vicinity of a surface Rayleigh wave front in shells of revolution under normal shock surface loading. Since the first order asymptotic approximation is insufficient for determination of all components of the stress-deformed state, we obtain refined asymptotic equations for construction of solutions for all components of displacements and stresses with an asymptotic error of the order of the relative shell thickness.     

The method of I. N. Vekua for nonshallow cylindrical shells

The refined theory of shells is constructed by reducing three-dimensional problems of the theory of elasticity to two-dimensional problems. I. Vekua obtained the equations of shallow shells. This means that the interior geometry of the shell does not vary in thickness. This method for nonshallow shells in the case of geometrical and physical nonlinear theory was generalized by T. Meunargia. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 51, Differential Equations and Their Applications, 2008.     

The equations of the geometrically non-linear theory of elasticity and momentless shells for arbitrary displacements

To validate earlier results for the case of arbitrary deformations and displacements in orthogonal curvilinear coordinates, kinematic and static relations of the non-linear theory of elasticity are set up which, in the limit of small deformations, lead, unlike the known relations, to correct and consistent relations. The same relations are also constructed for momentless shells of general form for the case of arbitrary displacements and deformations on the basis of which the problem of the static instability of a cylindrical shell with closed ends, made of a linearly elastic material and under conditions of an internal pressure (the problem of the inflation of a cylinder), is considered. It is shown that, in the case of momentless shells, the components of the true sheat stresses are symmetrical, unlike the three-dimensional case. All the above-mentioned relations are constructed for the loading of deformable bodies both by conservative external forces of constant directions and, also, by two types of “following” forces.     

A finite element model for laminated piezoelectric shell structures

Thin piezoelectric laminates are used for a wide range of technical applications. A four-node piezoelectric shell element is presented to analyse such structures effectively. In case of bending dominated problems incompatible approximation functions of the electrical and mechanical fields cause incorrect results. In order to overcome this problem the finite element formulation is based on a mixed variational principle implying six independent fields: displacements, electric potential, strains, electric field, mechanical stresses and dielectric displacements. This allows for an interpolation of the strains and the electric field in thickness direction independent of the bilinear interpolation functions. A piecewise quadratic approach for the shear strains in thickness direction and the corresponding electric field is proposed for arbitrarily layered shells. Regarding coupling of electrical and mechanical fields this yields to an appropriate balance of the approximation functions. Numerical examples show more precise results in contrast to standard elements with lowest order interpolation functions. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

厚壳理论及其在圆柱壳中的应用

本文从Hellinger-Reissner广义变分原理出发,以位移和应力的假设为基础,建立了厚壳理论.文中把壳体的位移展开为其厚度方向的幂级数,对平行和垂直于中面的位移分别保留其级数的前四项和前三项.并假定壳体的法向挤压和横向剪切应力沿壳厚为三次曲线,使其满足上下壳面上的应力条件,利用变分原理推导出分析厚壳所需的物理方程,平衡方程和边界条件.文中对圆柱壳的情况作了实例计算,并作了光弹性实验,结果表明理论和实验符合良好.     

The non-linear deformation and stability of elliptical cylindrical shells under transverse bending

A finite-element formulation of the solution of problems of the stability of non-circular cylindrical shells taking into account their bending moments and the non-linearity of their precritical stress-strain state in proposed. Explicit expressions for the displacements of the elements of non-circular cylindrical shells as rigid bodies are derived by integration of the equations obtained by equating the components of the linear strains to zero. These expressions are used to construct shape functions of an effective quadrilateral finite element of the natural curvature. An effective algorithm is developed for investigating the non-linear deformation and stability of the shells. The stability of a cylindrical shell of elliptical cross-section under transverse bending is investigated. The influence of the ellipticity and non-linearity of the deformation on the shell's stability is determined. The results of the analysis are compared with experimental data.     

球面各向同性弹性体的平衡问题

本文将位移和体积力同时进行分解,把含体积力的球面各向同性三维弹性理论平衡问题,化为一个二阶微分方程和一个四阶微分方程.利用球面函数的性质和级数展开方法,得到了相应于这两个方程齐次方程的级数解,可用于解决整球体和整球壳的平衡问题.最后,给出了旋转球的特解.     

The contact problem for a geometrically non-linear Timoshenko-type shell

An algorithm is developed for the numerical solution of the contact problem of an elastic Timoshenko-type shell subjected to arbitrarily large displacements and rotations, using mixed finite-element approximations. It is essential that six displacements of the faces of the shell are chosen as the required functions. This enables one, first, to simplify the formulation of contact problems in the mechanics of thin-walled structures, since functions by means of which the conditions for the non-penetration of the bodies are formulated are chosen as the required functions and, second, to obtain relations for the components of the Green-Lagrange strain tensor in curvilinear, orthogonal coordinates which accurately represent arbitrarily large displacements of a shell as a rigid body.     

Correction of the formulas for calculating thick-walled cylinders and spherical shells

A new formula is given for calculating the stresses and radial displacements of thickwalled cylinders and spherical shells.     

A method of integral equations in nonlinear boundary-value problems for flat shells of the Timoshenko type with free edges

We prove the existence theorem for solutions of geometrically nonlinear boundary-value problems for elastic shallow isotropic homogeneous shells with free edges under shear model of S. P. Timoshenko. Research method consists in the reduction of the original system of equilibrium equations to a single nonlinear equation for the components of transverse shear deformations. The basis of this method are integral representations for the generalized displacements, containing an arbitrary holomorphic functions, which are determined by the boundary conditions involving the theory of one-dimensional singular integral equations.     

The theory of orthotropic layered cylindrical shells

The author examines orthotropic layered cylindrical shells for which the moduli of elasticity of the load-carrying layers substantially exceed the shear modulus between layers. This class of structure includes, in particular, shells made of orthotropic glass-reinforced plastic. In this case the classical theory based on the Kirchhoff-Love hypotheses requires refinement; the corresponding equations obtained as a result of approximating the distribution of shear stresses or displacements over the thickness of the shell by a certain known function are presented in [4, 7, 8]. In this paper equations that make it possible to construct the stress distribution over the shell thickness are obtained within the framework of the engineering theory on the basis of the hypothesis of the incompressibility of a normal element.Mekhanika Polimerov, Vol. 4, No. 1, pp. 136–144, 1968     

Optimal position of a strut on a cylindrical shell

A solution is given for the problem of the optimal design of a rigid structure consisting of a spherical shell and two cylindrical shells linked by a strut. The volume of the structure is minimized under constraints on the maximal equivalent stresses in each of the shells and on the geometric parameters of the strut. Among the regulating parameters are the thickness of the shells, the geometric parameters of the strut, and the length of the first of the cylindrical shells. The problem is solved by the method of geometric programming.Translated from Matematicheskie Metody i Fiziko-Mekhanicheski Polya, No. 29, pp. 76–80, 1989.     

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.     

在任意不定常温度场和任意法向动载荷联合作用下中心开孔圆底扁球壳的动力问题

本文得出了在任意不定常温度场和任意法向动载荷联合作用下中心开孔圆底扁球壳的动力问题的解析解.我们假设温度沿壳体厚度直线分布.在第一部分.我们研究了常用边界条件下的中心开孔圆底扁球壳的自由振动.作为例子,我们计算了一边缘夹紧的扁球壳的自然基频(m=0),所得结果与E.Reissner[1]的结果作了比较.频率方程的解法是钱伟长[2]提出来的.这将在附录3中介绍.在第二部分,我们研究了在任意谐温度场和任意谐法向动载荷联合作用下的中心开孔圆底扁球壳的强迫振动.在第三部分,我们研究了在任意不定常温度场和任意法向动载荷联合作用下的具有初始条件的上述壳体的强迫振动.在附录1和2中,我们讨论了如何用应力函数来表示位移边界条件和m=1情形的边界条件.     

Mathematical model of the pipeline with angular joint of elements

A mathematical model of the pipeline as a Koiter-Vlasov moment shell with kink lines of the surface at the junctions of the pipe segments was constructed and substantiated. The following tasks are solved: The geometric parameters of the mechanical system as a three-dimensional elastic body and as a shell are found; force factors of the shell are expressed in terms of displacements of the middle surface of the wall, taking into account the presence of a kink line; equations of pipe equilibrium are derived as Koiter-Vlasov shells with an edge along the line; forces on oblique sections are expressed as functions of shell movements; the conjugation conditions on the pipe joint line for displacements and the angle of rotation of the normal are imposed and justified; conjugation conditions for bending moments, shear forces, transverse and normal forces are imposed and justified. The presence of the solution singularity at points on the connection line of the pipe segments is theoretically established and illustrated by the numerical example.     

On a method of studying the stress state of orthotropic shells and plates weakened by a hole with cracks extending to its edge

On the basis of the Timoshenko kinematic hypothesis for shells we give a formulation of the problem of studying the stress-strain state of orthotropic plates and shells weakened by a combined stress concentrator (a hole with two symmetric cracks extending to its edge). We propose a method of solving such problems on the basis of the finite-element method. To simulate the singularity of the stresses and displacements in a neighborhood of the tip of a crack we apply special finite elements with degenerate faces and nodes displaced by 1/4 the length of an edge. The stress intensity factors are found in terms of the displacements of the nodes of such elements. We give the results of computation of the concentration coefficients and the stress intensity factors for spherical and cylindrical shells loaded by internal pressure and for a cylindrical shell and a plate under the action of a distending load with various concentrators: a circular hole, an isolated crack, and a combined concentrator. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 48–54.     

Revised analysis of the thermal stresses in a shallow spherical shell with a foreign inclusion

We carry out a revised analysis of the concentration of thermal stresses in a shallow spherical shell with a small foreign inclusion. We show that near the interface of the materials the stressed state has a sharply expressed three-dimensional character; thus the stresses differ significantly (both qualitatively and quantitatively) from the shells predicted by the standard theories. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Investigation of Locally Loaded Multilayer Shells by a Mixed Finite-Element Method. 1. Geometrically Linear Statement

The Hu-Washizu functional is constructed for analyzing prestressed multilayer anisotropic Timoshenko-type shells. As unknown functions, six displacements and eleven strains of the faces of the shells are chosen. Based on mixed finite-element approximations, a numerical algorithm is developed for solving linear static problems of prestressed multilayer composite shells. The results of solving the well-known test problem on a cylindrical shell subjected to two opposite point forces and the problem on local loading of a toroidal multilayer rubber-cord shell are presented.     

壳体的非线性应变分量

壳体的非线性应变分量是非线性壳体力学的基础,在壳体的稳定以及各种大变形问题中须要用到它们.由于壳体几何复杂,在现有文献中,还未见到较全面地表示此种非线性应变分量的一般公式,现在导出六个用正交曲线坐标表示的包括线性与非线性部分的壳体应变分量的公式,其中三个为拉伸应变分量,另三个为剪切应变分量.