Solving some problems in the theory of thin shells of revolution with complex boundary conditions

An approach is proposed to solving linear boundary-value problems for shells of revolution that are closed in the circumferential direction, with complex boundary conditions in which the coefficients of the solving functions depend on the circumferential coordinate. The approach relies on reduction of the boundary-value problem to a number of boundary-value problems for systems of ordinary differential equations and systems of algebraic equations. We solve a specific problem for the stressed state of a conical shell with one of its ends supported by an elastic foundation with a variable modulus.Institute of Mechanics, Ukrainian Academy of Sciences. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 68, pp. 85–93, 1989.     

Calculation of the Thermoelastoplastic Nonaxisymmetric Stress-Strain State of Layered Orthotropic Shells of Revolution

The methods for determining the nonaxisymmetric thermoelastoplastic stress-strain state of layered orthotropic shells of revolution are developed. It is assumed that the layered package deforms without mutual slippage or separation of layers. The problem is solved using the geometrically nonlinear theory of shells based on the Kirchhoff-Love hypotheses. In the isotropic layers, plastic deformations may appear, whereas the orthotropic layers deform in the elastic region. It is assumed that the mechanical properties of the materials are temperature-dependent. The thermoplasticity equations are presented in a form corresponding to the method of additional deformations. The order of the system of partial differential equations obtained is reduced with the help of trigonometric series in the circumferential coordinate. The resulting systems of ordinary differential equations are solved by the Godunov technique of discrete orthogonalization. The nonaxisymmetric thermoelastoplastic stress-strain states of layered shells of revolution are considered as examples.     

Nonaxisymmetric thermal stress state of laminated rotational bodies of orthotropic materials under nonisothermic loading

A procedure for numerical investigation of nonaxisymmetric temperature fields and the elastic stress-strain state of laminated rotational bodies of cylindrically and rectilinearly orthotropic materials under nonisothermal loading is proposed. The deformation of orthotropic materials is described by the equations of anisotropic elasticity theory. The equations of state are written in the form of Hookes law for homogeneous materials, with additional terms which take into account the thermal deformation, changes in the mechanical properties of materials in the circumferential direction, and their dependence on temperature. A semianalytic finite-element method in combination with the method of successive approximations is used. An algorithm for numerical solution of the corresponding nonlinear boundary problem is elaborated, which is realized as a package of applied FORTRAN programs. Some numerical results are presented.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 731–752, November–December, 2004.     

The vibrations of a thin elastic orthotropic circular cylindrical shell with free and hinged edges

The problem of the existence of natural oscillations of a thin elastic orthotropic circular closed cylindrical shell with free and hinge-mounted ends and of an open cylindrical shell with free and hinge-mounted edges, when the two boundary generatrices are hinge-mounted is investigated. Dispersion equations and asymptotic formulae for finding the natural frequencies of possible vibration modes are obtained using the system of equations corresponding to the classical theory of orthotropic cylindrical shells. A mechanism is proposed by means of which the vibrations can be separated into possible types. Approximate values of the dimensionless characteristic of the natural frequency and the attenuation characteristic of the corresponding vibration modes are obtained using the examples of closed and open orthotropic cylindrical shells of different lengths.     

Closed-form elasticity solution for smart curved sandwich panels with soft core

Due to the rapid development of intelligent engineering structure, the demand for high performance smart structures has been increased in recent years. In this paper, an exact elasticity solution for bending analysis of sandwich panel with generally orthotropic facings and core is developed. Two electro-orthotropic piezoelectric layers in the top and bottom surfaces of sandwich panel as sensor and actuator are considered. From practical point of view, an initially curved sectorial geometry is considered for smart sandwich panel. In this regard, the constitutive relations and governing equations are considered in polar coordinate and coupled Euler–Cauchy Equations are derived. The characteristic equations are determined and closed-form basis functions of displacements and stresses are achieved for various material and geometrical conditions.Furthermore, based on classical and first-order shell theories, the governing equations of smart curved sandwich panels are derived. The governing equations are solved analytically and compared with exact elasticity solution.Several parametric studies are performed on both material and geometrical properties such as angular span, facing and core thickness, and external electrical voltage.     

Stability of Orthotropic Corrugated Cylindrical Shells under Axial Compression

The problem of stability of orthotropic noncircular cylindrical shells whose cross section can be described by a function in the form of superposition of a constant and a multiperiodic cosinusoid is considered. The solution of this problem is based on a rigorous consideration of the shell geometry and on the representation of the resolving functions in terms of trigonometric series in the circumferential coordinate. The determination of the bifurcation load is reduced to finding the minimum eigenvalue of a sequence of infinite systems of homogeneous algebraic equations. The effect of corrugation on the critical load of thin glass- and boron-reinforced shells of arbitrary length is analyzed. The data on the efficiency of corrugated shells, in comparison with circular ones, in relation to the mechanical properties of materials are obtained. The accuracy of the calculation procedure is estimated in the case where the corrugated shell is simulated by an equivalent circular one.     

Simplified equations and solutions for the free vibration of an orthotropic oval cylindrical shell with variable thickness

Based on the framework of the Flügge's shell theory, the transfer matrix approach and the Romberg integration method, this paper presents the vibration behavior of an isotropic and orthotropic oval cylindrical shell with parabolically varying thickness along its circumference. The governing equations of motion of the shell, which have variable coefficients are formulated and solved. The analysis is formulated to overcome the mathematical difficulties related to mode coupling, which comes from variable curvature and thickness of shell. The vibration equations of the shell are reduced to eight first‐order differential equations in the circumferential coordinate and by using the transfer matrix of the shell, these equations can be written in a matrix differential equation. The proposed model is adopted to get the vibration frequencies and the corresponding mode shapes for the symmetrical and antisymmetrical modes of vibration. The sensitivity of the frequency parameters and the bending deformations to the shell geometry, ovality parameter, thickness ratio, and orthotropic parameters corresponding to different type modes of vibration is investigated. Copyright © 2011 John Wiley & Sons, Ltd.     

The three-dimensional problem of the axisymmetric deformation of an orthotropic spherical layer

A 3D problem of the deformation of an elastic orthotropic spherical layer that is subjected to normal pressure applied to its outer and inner surfaces is analyzed. Asymptotic first-order approximation solutions are obtained for a slightly orthotropic layer for which the elastic moduli in the meridional and circumferential directions have similar values. The solutions that are obtained are used for analyzing the scleral shell under intraocular pressure; however, they can also be used for solving the inverse problem of analyzing the stress–strain state of a human eye during intravitreal injections. The influence that the meridional and circumferential elastic moduli have on the magnitudes of changes in the relative layer thickness and in the length of the anteroposterior eye axis due to elevated intraocular pressure is studied.     

Determination of the Steady-State Response of Viscoelastically Point-Supported Rectangular Orthotropic Plates

In the present study, the steady-state response to a sinusoidally varying concentrated force acting at the center of a rectangular point-supported orthotropic elastic plate is analyzed. The equation of the plate is represented by finite-difference expressions. By using the ordinary finite-difference technique, the problem is reduced to the solution of a system of algebraic equations. The influence of mechanical properties of the plate material on the vibration modes and the steady-state response of the plate is investigated numerically. Also, the effect of location of the point supports is studied. The problems considered are solved within the framework of the Kirchhoff-Love hypothesis.     

New Approach to Solving a Two-Dimensional Problem for an Orthotropic Plate

A new approach to solving a two-dimensional problem for an orthotropic multiply connected plate is proposed in place of the standard approach which reduces the problem to integrating a fourth-order differential equation. The new method reduces the problem to integrating second-order differential equations that can be solved successively using a perturbation technique.     

考虑高阶横向剪切正交各向异性板非线性弯曲的微分求积分析

采用微分求积方法（DQ方法）讨论了计及高阶横向剪切的正交各向异性弹性板的非线性弯曲问题．导出了非线性控制方程的DQ形式,利用推广的DQWB技巧处理了高阶矩的边界条件．进一步推广并运用新的分析技术简化了非线性方程的计算．为说明该方法的可靠性和有效性,将考虑剪切变形及不计剪切变形的薄板的数值结果与三维弹性解析解及其它数值解进行了比较,同时研究了数值结果的收敛性,并考察了不同的节点分布对收敛速度的影响A·D2还考察了几何、材料参数及横向剪切效应对正交各向异性板非线性弯曲的影响．分析结果表明横向剪切效应对正交各向异性中厚板的影响是显著的．     

An Analysis of the Decay of Saint-Venant End Effects in a Transversely Isotropic Matrix with an Isotropic Covering

A calculation model is proposed for a numerical analysis of the decay of Saint-Venant end effects in a laminate material of irregular structure to which corresponds a transversely isotropic matrix with an isotropic covering. The elastic properties of the matrix correspond to those of a unidirectional glass-fiber-reinforced plastic. The problem is investigated within the framework of the concept of a representative element of the material. The decay of the end effect in the direction perpendicular to the isotropy plane of a transversely isotropic matrix for the case of symmetric deformation of the material is considered. The source of the end effect is simulated by a piecewise-constant periodic surface load. This load is local for the calculated region and changes within a part of the boundary comparable with the typical size of the structural heterogeneity of the material. The equations of the linear elasticity theory of orthotropic bodies, a model of piecewise-homogeneous media, and quantitative criteria of decay of the end effect are used. Starting from the base system of equations, a discrete problem is constructed and solved. The results of a computing experiment for the choice of a representative element and the results of determining the maximum extent of the end-effect zone for this element are presented.     

Numerical solution of the propagation problem of longitudinal elastic waves in an orthotropic hollow cylinder

Wave processes in an orthotropic cylinder are analyzed. Harmonic oscillations are considered for which the original problem is reduced to a system of ordinary differential equations. This system is solved by the numerically stable discrete-orthogonalization method. Discrete curves for specific problems are given.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 56, pp. 75–78, 1985.     

Boundary-Value Problems of the Theory of Thin and Nonthin Orthotropic Shells with Account of Nonlinearly Elastic Properties and Low Shear Rigidity of Composite Materials

The basic geometric and physical relations and resolving equations of the theory of thin and nonthin orthotropic composite shells with account of nonlinear properties and low shear rigidity of their materials are presented. They are derived based on two theories, namely the theory of anisotropic shells employing the Timoshenko or Kirchhoff-Love hypothesis and the nonlinear theory of elasticity and plasticity of anisotropic media in combination with the Lagrange variational principle. The procedure and algorithm for the numerical solution of nonlinear (linear) problems are based on the method of successive approximations, the difference-variational method, and the Lagrange multiplier method. Calculations of the stress-strain state for a spherical shell with a circular opening loaded with internal pressure are presented. The effect of transverse shear strains and physical nonlinearity of the material on the distribution of maximum deflections and circumferential stresses in the shell, obtained according to two variants of the shell theories, is studied. A comparison of the results of the problem solution in linear and nonlinear statements with and without account of the shell shear strains is given. The numerical data obtained for thin and nonthin (medium thick) composite shells are analyzed.     

Two boundary value problems for a strongly anisotropic inhomogeneous elastic ring

The second boundary value problem (displacements are given on the boundary) and the improper mixed problem for a cylindrically orthotropic ring are studied. It is assumed that the coefficients of elasticity are continuously differentiable functions of the coordinates and depend on a small parameter in a specific manner. The form of the dependence of the coefficients on the small parameter is selected in such a way that in the case of constant coefficients it describes bonding of the ring by two families of very rigid fibers located along the radius vectors and concentric circles, where the stiffness of the fiber families is of identical order. Consequently, the coefficients of elasticity are represented in the form of products of constants which will later be called provisionally the “stiffnesses”, and functions of the coordinates. It is assumed that the stiffnesses in the radial and circumferential directions are equal and exceed and shear stiffness considerably. The asymptotic form of the solution of the boundary value problems under consideration is constructed when the ratio between the shear stiffness and the stiffness in the radial direction is used as the small parameter. In the case of the second boundary value problem the limit boundary value problem is described by a hyperbolic system of equations and is not solvable uniquely, since one of the families of characteristics is parallel to the boundary. When constructing the asymptotic form the necessity arises to average the coefficients of elasticity with respect to the circumferential coordinate. In this respect, there is an analogy with the results obtained in /1/ where the boundary value problem was studied for a second-order elliptic equation.     

Buckling and vibration of cross-ply laminated circular cylindrical panels

Summary The free vibration problem of a thin cross-ply composite laminated circular cylindrical panel subjected to arbitrary combinations of axial and circumferential initial compressions is studied. The equations of motion are derived, in the framework of the Donnell-type theory, in terms of the panel middle surface displacements components. Closed form solutions are obtained, for simply-supported panels, and numerical results for both antisymmetric and unsymmetric laminated cross-ply panels are presented.

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Résumé On étudie le problème de libre vibration d'une coque cylindrique ouverte, composée de fines lamelles croisées, soumise aux combinaisons arbitraires des compressions initiales de la circonférence et de l'axe. Les équations de vibrations dérivent, en accord avec la théorie de Donnell, des composantes de déplacement de la surface moyenne de la coque ouverte. On obtient des solutions exactes pour les coques cylindriques ouvertes posées librement et on présente des résultats numériques pour les coques cylindriques ouvertes composées de fines lamelles croisées antisymétriques et non symétriques.

     

正交各向异性板的非对称大变形问题

从各向异性板的基本理论出发,推导出正交各向异性圆板的非对称大变形基本方程,利用Fourier级数把问题的偏微分方程转化为一组可积分求解的非线性常微分方程,并给出利用迭代法求解该问题的基本方法.     

Combined effect of magnetic field and heat absorption on unsteady free convection and heat transfer flow in a micropolar fluid past a semi-infinite moving plate with viscous dissipation using element free Galerkin method

The fully developed electrically conducting micropolar fluid flow and heat transfer along a semi-infinite vertical porous moving plate is studied including the effect of viscous heating and in the presence of a magnetic field applied transversely to the direction of the flow. The Darcy-Brinkman-Forchheimer model which includes the effects of boundary and inertia forces is employed. The differential equations governing the problem have been transformed by a similarity transformation into a system of non-dimensional differential equations which are solved numerically by element free Galerkin method. Profiles for velocity, microrotation and temperature are presented for a wide range of plate velocity, viscosity ratio, Darcy number, Forchhimer number, magnetic field parameter, heat absorption parameter and the micropolar parameter. The skin friction and Nusselt numbers at the plates are also shown graphically. The present problem has significant applications in chemical engineering, materials processing, solar porous wafer absorber systems and metallurgy.     

Solution of problems of free torsional vibrations of thick-walled orthotropic inhomogeneous cylinders

With the use of the 3D theory of elasticity, we investigate the problem of free torsional vibrations of an anisotropic hollow cylinder with different boundary conditions at its end faces. We have proposed a numerical-analytic approach for the solution of this problem. The original partial differential equations of the theory of elasticity with the use of spline approximation and collocation are reduced to an eigenvalue problem for a system of ordinary differential equations of high order in the radial coordinate. This system is solved by the stable numerical method of discrete orthogonalization together with the method of step-by-step search. We also present numerical results for the case of orthotropic and inhomogeneous material of the cylinder for some kinds of boundary conditions.     

Analysis of Guided Lamb Wave Propagation (GW) in Honeycomb Sandwich Panels

The paper aims to introduce the guided lamb wave propagation (GW) in a honeycomb sandwich panels to be used in the health monitoring applications. Honeycomb sandwich panels are well-known as lightweight structures with a good stiffness behavior and a wide range of applications in different industries. Due to the complex geometry and complicated boundary conditions in such a structure, the development of analytical solutions for describing the wave propagation and the interaction of waves with damages is hardly possible. Therefore dimensional finite element simulations have been used to model GW for different frequency ranges and different sandwich panels with different geometrical properties. The waves, which are highly dispersive, have been excited by thin piezoelectric patches attached to the surface of the structure. In the first step, the honeycomb panel has been simplified as an orthotropic layered continuum medium. The required material data have been calculated by applying a numerical homogenization method for the honeycomb core layer. The wave propagation has been compared in the homogenized model with the real geometry of a honeycomb sandwich panel. Such calculations of high frequency ultrasonic waves are costly, both in creating a proper finite element model as well as in the required calculation time. In this paper the influence of changes in the geometry of the sandwich panel on the wave propagation is presented. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)