The boundary integral equation method applied to shallow membrane shells of positive Gaussian curvature

The boundary integral equation method (BIEM) is developed for the analysis of shallow membrane shells with positive Gaussian curvatures. Shells with constant thickness and constant curvatures are considered. In the infinite domain, fundamental solutions are obtained which correspond to generalized concentrated tangential forces in the x and y coordinate directions. The Betti-Maxwell reciprocal theorem and Green's second identity are used to obtain the boundary integral equations of the solution presented.This approach, which is applied for the first time in membrane shell theory, seems to be a powerful alternative to domain type methods. Shells with various boundary conditions, loadings and arbitrary plan forms can be considered. It is also possible to add the effects of thermal fields and openings in the shells.The potential of the method is demonstrated by means of a worked example.     

Thermoelastic vibration and buckling analysis of functionally graded piezoelectric cylindrical shells

This paper presents the report of an investigation into thermoelastic vibration and buckling characteristics of the functionally graded piezoelectric cylindrical, where the functionally graded piezoelectric cylindrical shell is made from a piezoelectric material having gradient change along the thickness, such as piezoelectricity and dielectric coefficient et al. Here, utilizing Hamilton’s principle and the Maxwell equation with a quadratic variation of the electric potential along the thickness direction of the cylindrical shells and the first-order shear deformation theory, and taking into account both the direct piezoelectric effect and the converse piezoelectric effect, the thermoelastic vibration and buckling characteristics of functionally graded piezoelectric cylindrical shells composed of BaTiO₃/PZT − 4, BaTiO₃/PZT − 5A and BaTiO₃/PVDF are, respectively, calculated. The effects of material composition (volume fraction exponent), thermal loading, external voltage applied and shell geometry parameters on the free vibration characteristics are described, and the axial critical load, critical temperature and critical control voltage are obtained.     

Vibration analysis of truncated conical shells subjected to flowing fluid

A method to predict the dynamic behaviour of anisotropic truncated conical shells conveying fluid is presented in this paper. It is a combination of finite element method and classical shell theory. The displacement functions are derived from exact solutions of Sanders’ shell equilibrium equations of conical shells. The velocity potential, Bernoulli’s equation and impermeability condition have been applied to the shell–fluid interface to obtain an explicit expression for fluid pressure which yields three forces (inertial, centrifugal, Coriolis) of the moving fluid. To the best of the authors’ knowledge, this paper reports the first comparison made between two works which deal with conical shells subjected to internal flowing fluid effects. The results obtained by this method for conical shells with various boundary condition and geometries, in vacuum, fully-filled and when subjected to flowing fluid were compared with those of other experimental and numerical investigations and good agreement was obtained.     

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.     

Nonlinear free vibration analysis of SSMFG cylindrical shells resting on nonlinear viscoelastic foundation in thermal environment

In this paper, a semi-analytical method for the free vibration behavior of spiral stiffened multilayer functionally graded (SSMFG) cylindrical shells under the thermal environment is investigated. The distribution of linear and uniform temperature along the direction of thickness is assumed. The structure is embedded within a generalized nonlinear viscoelastic foundation, which is composed of a two-parameter Winkler-Pasternak foundation augmented by a Kelvin-Voigt viscoelastic model with a nonlinear cubic stiffness. The cylindrical shell has three layers consist of ceramic, FGM, and metal in two cases. In the first model i.e. Ceramic-FGM-Metal (CFM), the exterior layer of the cylindrical shell is rich ceramic while the interior layer is rich metal and the functionally graded material is located between these layers and the material distribution is in reverse order in the second model i.e. Metal-FGM-Ceramic (MFC). The material constitutive of the stiffeners is continuously changed through the thickness. Using the Galerkin method based on the von Kármán equations and the smeared stiffeners technique, the problem of nonlinear vibration has been solved. In order to find the nonlinear vibration responses, the fourth order Runge–Kutta method is utilized. The results show that the different angles of stiffeners and nonlinear elastic foundation parameters have a strong effect on the vibration behaviors of the SSMFG cylindrical shells. Also, the results illustrate that the vibration amplitude and the natural frequency for CFM and MFC shells with the first longitudinal and third transversal modes (m = 1, n = 3) with the stiffeners angle θ = 30°, β = 60° and θ = β = 30° is less than and more than others, respectively.     

Existence and uniqueness theorems for cusped prismatic shells in the Nth hierarchical model

We study the well posedness of boundary value problems for elastic cusped prismatic shells in the Nth approximation of I. Vekua's hierarchical models under (all reasonable) boundary conditions at the cusped edge and given displacements at the non‐cusped edge and stresses at the upper and lower faces of the shell. Copyright © 2008 John Wiley & Sons, Ltd.     

被动约束层阻尼圆柱壳振动和阻尼分析的一种新矩阵方法

基于线弹性薄壳理论和线粘弹性理论,考虑粘弹性层的剪切耗能作用和各层间的相互作用力,导出了被动约束层阻尼层合圆柱壳在谐激励作用下的一阶常微分矩阵控制方程．然后,借助作者提出的齐次扩容精细积分技术建立了一种新的矩阵方法,并利用该方法研究了层合圆柱壳的振动特性和阻尼特性．该方法与已提出的以位移及其导数作为状态向量的传统传递矩阵法的根本区别在于,控制方程中的状态向量中包含了层合壳的全部位移和整合内力变量,因此,可以方便地适用于各种位移和内力边界条件以及部分环状覆盖约束层阻尼圆柱壳的动态分析．数值算例与解析解和有限元解的结果比较有力说明了该方法的正确性和有效性．     

A central limit theorem for a weighted power variation of a Gaussian process*

Let ρ be a real-valued function on [0, T], and let LSI(ρ) be a class of Gaussian processes over time interval [0, T], which need not have stationary increments but their incremental variance σ(s, t) is close to the values ρ(|t ? s|) as t → s uniformly in s ∈ (0, T]. For a Gaussian processesGfrom LSI(ρ), we consider a power variation V _n corresponding to a regular partition π _n of [0, T] and weighted by values of ρ(·). Under suitable hypotheses on G, we prove that a central limit theorem holds for V _n as the mesh of π _n approaches zero. The proof is based on a general central limit theorem for random variables that admit a Wiener chaos representation. The present result extends the central limit theorem for a power variation of a class of Gaussian processes with stationary increments and for bifractional and subfractional Gaussian processes.     

Korn inequalities for shells with zero Gaussian curvature

We consider shells with zero Gaussian curvature, namely shells with one principal curvature zero and the other one having a constant sign. Our particular interests are shells that are diffeomorphic to a circular cylindrical shell with zero principal longitudinal curvature and positive circumferential curvature, including, for example, cylindrical and conical shells with arbitrary convex cross sections. We prove that the best constant in the first Korn inequality scales like thickness to the power 3/2 for a wide range of boundary conditions at the thin edges of the shell. Our methodology is to prove, for each of the three mutually orthogonal two-dimensional cross-sections of the shell, a “first-and-a-half Korn inequality”—a hybrid between the classical first and second Korn inequalities. These three two-dimensional inequalities assemble into a three-dimensional one, which, in turn, implies the asymptotically sharp first Korn inequality for the shell. This work is a part of mathematically rigorous analysis of extreme sensitivity of the buckling load of axially compressed cylindrical shells to shape imperfections.     

On the stability and domain of attraction of asymptotically nonsmooth stationary solutions to a singularly perturbed parabolic equation

A stationary solution to the singularly perturbed parabolic equation ?u _t + ε² u _xx ? f(u, x) = 0 with Neumann boundary conditions is considered. The limit of the solution as ε → 0 is a nonsmooth solution to the reduced equation f(u, x) = 0 that is composed of two intersecting roots of this equation. It is proved that the stationary solution is asymptotically stable, and its global domain of attraction is found.     

Well-behaved Beurling primes and integers

In this paper, we study generalised prime systems for which both the prime and integer counting functions are asymptotically well-behaved, in the sense that they are approximately li(x) and ρx, respectively (where ρ is a positive constant), with error terms of order O(x^θ₁) and O(x^θ₂) for some θ₁,θ₂<1. We show that it is impossible to have both θ₁ and θ₂ less than .     

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.     

Comparative analysis of two algorithms for numerical solution of nonlinearstatic problems for multilayered anisotropic shells of revolution 2. Account of transverse compression

Two algorithms for numerical solution of static problems for multilayer anisotropic shells of revolution are discussed. The first algorithm is based on a differential approach using the method of discrete orthogonalization, and the second one—on the finite element method with linear local approximation in the meridional direction. It is assumed that the layers of the shell are made of linearly elastic, anisotropic materials. As the unknown functions, six displacements of the shell are chosen, which often simplifies the definition of static problems for multilayer shells. The calculation of a cross-ply cylindrical shell stretched in the axial direction is considered. It is shown that taking account of the transverse compression, anisotropy, and geometrical nonlinearity is important for the given class of problems.Tambov State Technical University, Tambov, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 4, pp. 435–446, May–June, 1999.     

Contact of shells with a sharp linear stamp of finite length

The action of a plane, absolutely rigid stamp on a transversely isotropic shell is investigated. The use of the equations of shells with finite shear stiffness enables the correct formulation of the problem of the action on a shell by a stamp of fixed length. The problem is reduced to an integral equation. Applying the Fourier transform, the kernel of the integral equation is represented in the form of an expansion with respect to Chebyshev polynomials. By the representation of the solution of the integral equation in the form of a product, of a series of Chebyshev polynomials and a function that takes into account the singularities of the solution at the boundary of the contact zone, the considered problem is reduced to the solving of an infinite system of linear algebraic equations, whose coefficients have been determined by the methods of numerical integration. As an example a problem for a cylindrical shell has been solved.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 59–63, 1989.     

Thermal stresses in anisotropic cylindrical elastic shells

We investigate the deformation of general anisotropic and inhomogeneous shells, under the action of a given temperature distribution. We assume that the temperature field is a polynomial in the axial coordinate, and we establish the displacements produced by the prescribed thermal field. The results are obtained in the framework of the linear theory for Cosserat thermoelastic shells. The solution is used to study the special case of orthotropic cylindrical shells. Copyright © 2009 John Wiley & Sons, Ltd.     

Comparative analysis of two algorithms for numerical solution of nonlinear static problems for multilayer anisotropic shells of revolution. 1. Account of transverse shear

The discussion focuses on two numerical algorithms for solving the nonlinear static problems of multilayer composite shells of revolution, namely the algorithm based on the discrete orthogonalization method and the algorithm based on the finite element method with a local linear approximation in the meridian direction. The material of each layer of the shell is assumed to be linearly elastic and anisotropic (nonorthotropic). A feature of this approach is that the displacements of the face surfaces of the shell are chosen as unknown functions, i.e., the functions which allows us to formulate the kinematic boundary conditions on these surfaces. As an example, a cross-ply cylindrical shell subjected to uniform axisymmetric tension is considered. It is shown that the algorithms elaborated correctly describe the local distribution of the stress tensor over the shell thickness without an expensive software based on the 3D anisotropic theory of elasticity.Tambov State Technical University, Tambov, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 3, pp. 347–358, May–June, 1999.     

Estimates of the solutions of difference-differential equations of neutral type

In this paper, we study scalar difference-differential equations of neutral type of general form $$\sum\limits_{j = 0}^m {\int_0^h {u^{(j)} (t - \theta )d\sigma _j (\theta ) = 0,t > h,} } $$ where the σ_j(θ) are functions of bounded variation. For the solutions of this equation, we obtain the following estimate: $$\left\| {u(t)} \right\|W_2^m (T,T + h) \leqslant CT^{q - 1} e^{\kappa T} \left\| {u(t)} \right\|W_2^m (0,h),$$ where C is a constant independent of u ₀(t) and the values of q and ? are determined by the properties of the characteristic determinant of this equation. Earlier, this estimate was proved for equations of less general form. For example, for piecewise constant functions σ _j(θ) or for the case in which the function σ _m(θ) has jumps at both points θ = 0 and θ = h. In the present paper, this estimate is obtained under the only condition that σ _m(θ) experiences a jump at the point θ = 0; this condition is necessary for the correct solvability of the initial-value problem.     

Axisymmetrical vibrations of shells of revolution under abrupt loading

A frequency method is proposed for solving the problem of the vibrations of shells of revolution taking into account the energy dissipation under arbitrary force loading and on collision with a rigid obstacle. The Laplace transform is taken of the equation of the vibrations of a shell of revolution with non-zero initial conditions. For the inhomogeneous differential equation obtained, a variational method is used to solve the boundary-value problem, which consists of finding the Laplace-transformed boundary transverse and longitudinal forces and bending moments as functions of the boundary displacements. The equations of equilibrium of nodes, i.e. the corresponding equations of the finite-element method, are then compared, using results obtained earlier [1–4]. Amplitude-phase-frequency characteristics (APFCs) for the shell cross-sections selected are plotted. An inverse Laplace transformation is carried out using the clear relationship between the extreme points of the APFCs and the coefficients of the corresponding terms of the series in an expansion vibration modes [3]. In view of the fact that the proposed approach is approximate, numerical testing is used.     

Buckling Instability of Sectional Noncircular Cylindrical Composite Shells under Axial Compression

The problem of buckling instability of cylindrical shells under axial compression is considered. The shells consist of cylindrical sections of smaller radius. The geometrical parameters of the shells are approximated by Fourier series on a discrete point set. A Timoshenko-type shell theory is used. The solution is obtained in the form of trigonometric series. It is shown that shells consisting of cylindrical sections have considerable advantages over circular ones. At a constant shell weight, the choice of suitable parameters of shell sections leads to a significant increase in the critical load. The composite shells considered possess higher efficiency indices in comparison with isotropic ones.     

Vibration monitoring of cylindrical shells using piezoelectric sensors

From linear vibration theory for beams and plates, one can express the response as a linear combination of its natural modes. For beams, these eigenfunctions can be shown to be mutually orthogonal for any boundary conditions. For plates, orthogonality of the modes is not guaranteed, but can be proven for various boundary conditions. Modal analysis for beams and plates allows the system response to be broken down into simpler vibration models, due to the orthogonality of the modes. Here the modal analysis approach is extended to the vibration of thin cylindrical shells. The longitudinal, radial, and circumferential displacements are coupled with each other, due to Poisson's ratio and the curvature of the shell. As will be shown, the mode shapes can be solved analytically with numerically determined coefficients. The immediate application of this work will be for modal sensing of cylindrical shell vibrations using thin piezoelectric films.