四边简支正交各向异性波纹型夹心矩形夹层板的固有频率

给出了一种把具有波纹型夹心的正交各向异性夹层板的控制方程组化为仅包含一个位移函数的单一方程的简单方法,从而获得了四边简支条件下其自由振动固有频率的精确解,同时还对两种具有重要实际意义的特殊情况进行了讨论。     

Stability and free vibration analyses of double-bonded micro composite sandwich cylindrical shells conveying fluid flow

In this study, based on Reddy cylindrical double-shell theory, the free vibration and stability analyses of double-bonded micro composite sandwich cylindrical shells reinforced by carbon nanotubes conveying fluid flow under magneto-thermo-mechanical loadings using modified couple stress theory are investigated. It is assumed that the cylindrical shells with foam core rested in an orthotropic elastic medium and the face sheets are made of composites with temperature-dependent material properties. Also, the Lorentz functions are applied to simulation of magnetic field in the thickness direction of each face sheets. Then, the governing equations of motions are obtained using Hamilton's principle. Moreover, the generalized differential quadrature method is used to discretize the equations of motions and solve them. There are a good agreement between the obtained results from this method and the previous studies. Numerical results are presented to predict the effects of size-dependent length scale parameter, third order shear deformation theory, magnetic intensity, length-to-radius and thickness ratios, Knudsen number, orthotropic foundation, temperature changes and carbon nanotubes volume fraction on the natural frequencies and critical flow velocity of cylindrical shells. Also, it is demonstrated that the magnetic intensity, temperature changes and carbon nanotubes volume fraction have important effects on the behavior of micro composite sandwich cylindrical shells. So that, increasing the magnetic intensity, volume fraction and Winkler spring constant lead to increase the dimensionless natural frequency and stability of micro shells, while this parameter reduce by increasing the temperature changes. It is noted that sandwich structures conveying fluid flow are used as sensors and actuators in smart devices and aerospace industries. Moreover, carotid arteries play an important role to high blood rate control that they have a similar structure with flow conveying cylindrical shells. In fact, the present study can be provided a valuable background for more research and further experimental investigation.     

Transient dynamic response of sandwich shells with transversely compressible core

The present study provides an investigation of the effect of the transverse core compressibility on the dynamic buckling response of sandwich structures. The study utilizes a previous v. Kármán type higher-order model for shallow sandwich shells. An analytical solution is obtained by means of an extended Galerkin procedure in conjunction with an explicit fourth order Runge-Kutta algorithm to solve the transient problem. In an example analysis, it is observed that the transverse core compressibility can have strong effects even on the global response of sandwich structures. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Free vibration of sandwich plates with impact-induced damage

Free vibrations of impact-damaged sandwich plates with honeycomb and foam cores are studied. It is assumed that damages caused by impact events are to exist before the vibration start and to be constant during oscillations. The influence of the impact damage modes involving the core crushing (planar damage), face sheets fracture (indentation) and the core to face sheets interface degradation (debonding) on the natural frequencies and associated mode shapes of the sandwich plates was investigated using commercially available finite element code ABAQUS. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear statics and dynamics of antisymmetric composite laminated square plates supported on nonlinear elastic subgrade

Non-linear static and dynamic analysis is presented for composite laminated anti-symmetric square plates supported on non-linear elastic foundation subjected to uniformly distributed transverse and step loading, respectively. The formulation is based on first order shear deformation theory (FSDT) and Von-Karman non-linearity, subgrade interaction is modeled as shear deformable with cubic nonlinearity. The methodology of solution is based on the Chebyshev series technique. The coupled non-linear partial differential equations are linearized using quadratic extrapolation technique. Houbolt time marching scheme is employed for temporal discretisation. An incremental iterative approach is employed for the solution. The effects of foundation stiffness parameters and boundary conditions on the non-linear static and dynamic analysis on the central response are studied.     

周边面内压力作用下夹层圆板的非线性振动

基于von Krmn薄板理论,讨论了滑动固定基础上周边面内压力作用下夹层圆板的非线性振动问题,应用变分法导出了该问题的非线性特征方程和边界条件,给出了其精确静态解,并使用修正迭代法求解了该方程,导出了夹层圆板振幅和非线性振频的解析关系式．当周边面力使夹层圆板的最低固有频率为零时,就可获得临界载荷的值．     

多夹层板壳的非线性理论及应用（Ⅲ）：扁壳的弯曲和稳定

本文根据文[1]获得的多夹层扁壳非线性基本方程，求解了各种载荷及边界条件下矩形底面多夹层扁壳的非线性弯曲问题，多夹层板、扁柱亮在轴向压力作用下的稳定问题，以及一般形状的扁壳在边界作用力下的变形．     

On the continuation for variational inequalities depending on an eigenvalue parameter

In this paper we generalize recent theoretical results on the local continuation of parameter-dependent non-linear variational inequalities. The variational inequalities are rather general and describe, for example, the buckling of beams, plates or shells subject to obstacles. Under a technical hypothesis that is satisfied by the simply supported beam, we obtain the existence of a continuation of both the solution and the eigenvalue with respect to a local parameter. A numerical continuation method is presented that easily overcomes turning points. Numerical results are presented for the non-linear beam.     

Wave propagation in composite multilayered structures with delaminations

Using the Rayleigh–Ritz method, the natural frequencies of layered cylindrical shells with delaminations are derived. The results are compared with FE computations and with data available in the literature. It is demonstrated that the transverse shear effects can play a significant role in a correct and accurate evaluation of the natural frequencies of such shells. Numerical investigations into the problem of radial excitations of cylinders with delaminations have also been performed. From the results obtained, it follows that their optimal excitation frequency is related to the natural frequency of a local region containing the delamination.     

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.     

缓变厚度中厚板的自由振动

本文将中厚板的厚度函数按一小参数展开,并采用奇异摄动方法,把原来变系数的微分方程组化成一系列常系数微分方程组求解.文中给出了任意变厚度中厚板的自振频率计算显式表达式,由此式,我们不仅可以方便地计算出各种变厚度的自振频率值,而且也可以根据频率的要求来优化板的厚度.文中的算例表明,本文的方法具有较好的精度、方法简便、有效等其他优点,可以考虑作为分析各种变厚度板壳的振动及稳定特征问题的有效方法之一.     

Electro-thermo-mechanical vibration and stability analyses of double-bonded micro composite sandwich piezoelectric tubes conveying fluid flow

New sandwich panels and tubes have widely applications in nanotechnology such as transportation, naval, aerospace industries, micro and nanoelectromechanical systems and fluid storage. For example, carotid arteries play an important role to high blood rate control that they have a similar structure with flow conveying cylindrical shells. In the current study, stability and free vibration analyses of double-bonded micro composite sandwich piezoelectric tubes conveying fluid flow embedded in an orthotropic foundation under electro-thermo-mechanical loadings are presented. In fact, this work can be provided a valuable background for more research and further experimental investigation. It is assumed that the micro tubes are made of flexible material and smart piezoelectric composites reinforced by carbon nanotubes as core and face sheets, respectively. Energy method and Hamilton's principle are applied to derive the governing equations of motions based on Euler–Bernoulli beam model and using modified strain gradient theory. Moreover, generalized differential quadrature method is used to discretize and solve the governing equations of motions. Numerical results are investigated to predict the influences of length-to-radius, thickness of face sheets-to-thickness of core ratio, temperature changes, orthotropic elastic medium, Knudsen number, and carbon nanotubes volume fraction on the dimensionless natural frequencies and critical flow velocity of sandwich double-bonded piezoelectric micro composite tubes. The results of this article show that increasing the thickness ratio, volume fraction carbon nanotubes and orthotropic elastic constants lead to enhance the dimensionless natural frequency and stability of system, while decrease these parameters with increasing the temperature and length-to-radius ratio.     

弹性约束边界条件下矩形蜂窝夹芯板的自由振动分析

蜂窝夹芯板在飞行器、高速列车等领域有广泛的用途，对其开展振动分析具有明确的科学价值及工程意义.为区别于诸简支等传统约束边界，提出了弹性约束边界下蜂窝夹芯板结构的自由振动特性分析方法.具体来说，首先通过将蜂窝夹芯层等效为各向异性板，将夹芯板问题转变为三层板结构.进一步地，将板结构的位移场函数由改进的二维Fourier级数表示，并基于能量原理的Rayleigh Ritz法得到结构的固有频率和固有振型，理论预测结果与数值模拟分析吻合较好.提出的理论模型可用于系统讨论约束边界对蜂窝夹芯结构自由振动特性的影响，为此类结构的约束方案设计提供理论依据.     

The vibration and stability of non-homogeneous orthotropic conical shells with clamped edges subjected to uniform external pressures

In this paper an analytical procedure is given to study the free vibration and stability characteristics of homogeneous and non-homogeneous orthotropic truncated and complete conical shells with clamped edges under uniform external pressures. The non-homogeneous orthotropic material properties of conical shells vary continuously in the thickness direction. The governing equations according to the Donnell’s theory are solved by Galerkin’s method and critical hydrostatic and lateral pressures and fundamental natural frequencies have been found analytically. The appropriate formulas for homogeneous orthotropic and isotropic conical shells and for cylindrical shells made of homogeneous and non-homogeneous, orthotropic and isotropic materials are found as a special case. Several examples are presented to show the accuracy and efficiency of the formulation. The closed-form solutions are verified by accurate different solutions. Finally, the influences of the non-homogeneity, orthotropy and the variations of conical shells characteristics on the critical lateral and hydrostatic pressures and natural frequencies are investigated, when Young’s moduli and density vary together and separately. The results obtained for homogeneous cases are compared with their counterparts in the literature.     

Numerical analysis of free vibrations of laminated composite conical and cylindrical shells: Discrete singular convolution (DSC) approach

A numerical study on the free vibration analysis for laminated conical and cylindrical shell is presented. The analysis is carried out using Love's first approximation thin shell theory and solved using discrete singular convolution (DSC) method. Numerical results in free vibrations of laminated conical and cylindrical shells are presented graphically for different geometric and material parameters. Free vibrations of isotropic cylindrical shells and annular plates are treated as special cases. The effects of circumferential wave number, number of layers on frequencies characteristics are also discussed. The numerical results show that the present method is quite easy to implement, accurate and efficient for the problems considered.     

On the natural frequency of transversely isotropic magneto-electro-elastic plates in contact with fluid

In this paper, the vibration characteristics of transversely isotropic magneto-electro-elastic (MEE) rectangular plates in contact with fluid are investigated. The mathematical formulation on the determination of added virtual mass for water-contacting MEE rectangular plates with uniform thickness is performed. Based on the recently proposed differential equation governing the dynamical responses of the MEE rectangular plates, a fluid-structure interaction model is established and analyzed. First of all, the added virtual mass incremental (AVMI) factor of the system is calculated by using the proposed approach and the added virtual mass can then be obtained, furthermore, the natural frequencies of the MEE rectangular plates in contact with fluid with different boundary conditions are also investigated. It is noted that the natural frequencies based on the proposed method are very useful for those engineers or researchers who are engaged in the vibration analysis and design of the MEE plate in contact with fluid.     

Fundamental size dependent natural frequencies of non-uniform orthotropic nano scaled plates using nonlocal variational principle and finite element method

In the present study, a nonlocal continuum model based on the Eringen’s theory is developed for vibration analysis of orthotropic nano-plates with arbitrary variation in thickness. Variational principle and Ritz functions are employed to calculate the size dependent natural frequencies of non-uniform nano-plates on the basis of nonlocal classical plate theory (NCLPT). The Ritz functions eliminate the need for mesh generation and thus large degrees of freedom arising in discretization methods such as finite element (FE). Effect of thickness variation on natural frequencies is examined for different nonlocal parameters, mode numbers, geometries and boundary conditions. It is found that thickness variation accompanying small scale effect has a noticeable effect on natural frequencies of non-uniform plates at nano scale. Also a comparison with finite element solution is performed to show the ability of the Ritz functions in fast converging to the exact results. It is anticipated that presented results can be used as a helpful source in vibration design and frequency optimization of non-uniform small scaled plates.     

Buckling forms of homogeneous and sandwich plates in pure shear in tangential directions

The mathematical problem of the plane shear buckling form (BF) of sandwich plates and plates homogeneous across the thickness in pure shear is considered. The solution to this problem is compared with the solution to the problem of a flexural BF which is realized in these plates with the formation of oblique waves. It is established that, in the case of plates homogeneous across the thickness, the critical loads corresponding to the plane shear BF are maximum for isotropic ones. In real one-layer structural elements manufactured both of isotropic homogeneous and orthotropic composite materials, these critical loads cannot be reached since they exceed considerably the critical loads for the flexural BFs with oblique waves. The critical loads corresponding to the two BFs are comparable only for relatively thick plates. However, the plane shear BF can be realized in sandwich plates earlier than the flexural one even if the plates are thin. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 2, pp. 215–228, 2000.     

Problems of geometric non-linearity and stability in the mechanics of thin shells and rectilinear columns

An analysis of the current state of the geometrically non-linear theory of elasticity and of thin shells is presented in the case of small deformations but large displacements and rotations, the ratios of which are known as the ratios of the non-linear theory in the quadratic approximation. It is shown that they required specific revision and correction by virtue of the fact that, when they are used in the solution of problems, spurious bifurcation points appear. In view of this, consistent geometrically non-linear equations of the theory of thin shells of the Timoshenko type are constructed in the quadratic approximation which enable one to investigate in a correct formulation both flexural as well as previously unknown non-classical forms of loss of stability (FLS) of thin plates and shells, many of which are encountered in practice, primarily in structures made of composite materials with a low shear stiffness. In the case of rectilinear elastic whereas, which are subjected to the action of conservative external forces and are made of an orthotropic material, the three-dimensional equations of the theory of elasticity are reduced to one-dimensional equations by using the Timoshenko model. Two versions of the latter equations are derived. The first of these corresponds to the use of the consistent version of the three-dimensional, geometrically non-linear relations in an incomplete quadratic approximation and the Timoshenko model without taking account of the transverse stretching deformations, and the second corresponds to the use of the three- dimensional relations in the complete quadratic approximation and the Timoshenko model taking account of the transverse stretching deformations. A series of new non-classical problems of the stability of columns is formulated and their analytical solutions are found using the equations which have been derived with the aim of analyzing their richness of content. Among these are problems concerning the torsional, flexural and shear FLS of a column in the case of a longitudinal axial, bilateral transverse and trilateral compression, a flexural-torsional FLS in the case of pure bending and axial compression together with pure bending and, also, a flexural FLS of a column in the case of torsion and a flexural-torsional FLS under conditions of pure shear. Five FLS of a cylindrical shell under torsion are investigated using the linearized neutral equilibrium equations which have been constructed: 1) a torsional FLS where the solution corresponding to it has a zero variability of the functions in the peripheral direction, 2) a purely beam bending FLS that is possible in the case of long shells and is accompanied by the formation of a single half-wave along the length of the shell while preserving the initial circular form of the cross-section, 3) a classical bending FLS, which is accompanied by the formation of a small number of half-waves in the axial direction and a large number of half-waves in a peripheral direction which is true in the case of long shells, 4) a classical bending FLS which holds in the case of short and medium length shells (the third and fourth types of FLS have already been thoroughly studied in the case of isotropic cylindrical shells), 5) a non-classical FLS characterized by the formation of a large number of shallow depressions in the axial as well as in the peripheral directions; the critical value of the torsional moment corresponding to this FLS is practically independent of the relative thickness of the shell. It is established that the well-known equations of the geometrically non-linear theory of shells, which were formulated for the case of the mean flexure of a shell, do not enable one to reveal the first, second and fifth non-classical FLS.     

正交异性简支矩形底双曲扁壳大挠度问题的解析解

本文在文献[1,2]的基础上,进一步研究了富里叶级数在求解板壳大挠度问题中的应用。文中导出了简支边界条件下正交异性矩底双曲扁壳大挠度微分方程组的解析解。这个解可通用于板与壳、大挠度与小挠度、各向同性与正交异性在直角坐标下的多种情况。其数值结果与实验数据和其它解法结果相吻合。